201947

Jesús Carro Fernández

New Methodologies for theDevelopment andValidation ofElectrophysiologicalModelsDepartamento

Instituto de Investigación en Ingeniería [I3A]

Director/es

RODRIGUEZ MATAS, JOSE FELIX
PUEYO PAULES, ESTHER

Reconocimiento – NoComercial –SinObraDerivada (by-nc-nd): No sepermite un uso comercial de la obra
original ni la generación de obras
derivadas.

© Universidad de ZaragozaServicio de Publicaciones
ISSN 2254-7606

Tesis Doctoral

NEW METHODOLOGIES FOR THE DEVELOPMENT
AND VALIDATION OF ELECTROPHYSIOLOGICALMODELS

Autor

Jesús Carro Fernández
Director/es
RODRIGUEZ MATAS, JOSE FELIX
PUEYO PAULES, ESTHER

UNIVERSIDAD DE ZARAGOZA
Instituto de Investigación en Ingeniería [I3A]

2019
Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es

New Methodologies for theDevelopment and Validation ofElectrophysiological Models
Author: Jesús Carro Fernández
Supervisors:
Esther Pueyo Paules
José Félix Rodríguez Matas

PhD Thesis in
Biomedical Engineering

Zaragoza, September, 2018

New Methodologies forthe Development andValidation ofElectrophysiologicalModelsAuthor: Jesús Carro Fernández
Supervisors:
Esther Pueyo Paules
José Félix Rodríguez Matas

PhD Thesis in
Biomedical Engineering

Zaragoza, September, 2018

To Débora.
To Nacho, Samuel, and Lucía.

Acknowledgment
This thesis has been a long journey and, therefore, many people have put a bit toget here. My thanks to all of you.
To Esther and José Félix. It is very difficult to summarize everything received
from you in these years. How much we have done, how much remains to be done,
how much to know and to understand. You have generated me more questions
than answers. Thank you.
To Pablo Laguna. For your support and help, especially in the first phases ofthis thesis: the beginning, the search of founds, etc. Thank you.
To Juan, Pablo Vicente, David Sampedro, Carlos, Jesús Fernández and all themembers of BSICoS. Every time I met with you, it did not matter how much time
had passed since the last time, I was once again one more in the group. You spreadyour enthusiasm for research. Thank you.
To Pamela, Ana, and Arantxa. The least visible part of the group, but thosethat have made all the procedures easy. Thank you.
To Jorge, David, África, Jesús, Carlos, Paqui, Gabriel, Violeta, Jaime, Lorena,
and all those who have been encouraging and supporting me these years at the
USJ. Thank you.
To my parents-in-law. For all the help received during these years: to have time
to write, to travel, etc. For everything. Thank you.
To my parents, and my brothers and sisters; especially to my father. For all
the values that you have transmitted to me, for all your support, for helping me tobe who I am. Thank you.
To Nacho, Samuel and Lucia. I do not know who have taken me away more
sleep hours in these years: the thesis or you; but you have the ability to make meforget both things with a smile and a hug. I hope I did not steal too much of yourtime. Thank you.
To Débora. For your patience all these years. For getting me up in the hardest
moments, and putting my feet on the ground in the best ones. For being by myside on this road. Thank you.
Thank you very much to all.

vii

Agradecimientos
Esta tesis ha sido un viaje largo y, por ello, son muchas las personas que han puesto
un granito de arena para llegar hasta aquí. Mi agradecimiento a todos vosotros.
A Esther y José Félix. Es muy difícil resumir todo lo recibido de vosotros enestos años. Cuánto hemos hecho y cuánto queda por hacer, cuánto por conocer yentender. Me habéis generado más preguntas que respuestas. Gracias.
A Pablo Laguna. Por el apoyo y la ayuda recibida, especialmente en las primeras
fases de esta tesis: el arranque, la búsqueda de financiación, etc. Gracias.
A Juan, Pablo Vicente, David Sampedro, Carlos, Jesús Fernández y a todos los
miembros de BSICoS. Cada vez que me juntaba con vosotros, daba igual el tiempo
que hubiera pasado desde la última vez, volvía a ser uno más. Contagiáis vuestro
entusiasmo por investigar. Gracias.
A Pamela, Ana, y Arantxa. La parte menos visible del grupo, pero las que hanhecho que todos los trámites fueran fáciles. Gracias.
A Jorge, David, África, Jesús, Carlos, Paqui, Gabriel, Violeta, Jaime, Lorena,y todos aquellos que me habéis estado animando y apoyando estos años en la USJ.
Gracias.
A mis suegros. Por toda la ayuda recibida en estos años, para escribir, para loscongresos, para todo. Gracias .
A mis padres y a mis hermanos; especialmente a mi padre. Por todos losvalores transmitidos, por todo el apoyo dado, por haberme ayudado a ser quiensoy. Gracias.
A Nacho, Samuel y Lucía. No sé quién me habrá quitado más horas de sueño enestos años: la tesis o vosotros; pero tenéis la capacidad de hacerme olvidar ambas
cosas con una sonrisa y un abrazo. Espero no haberos robado yo demasiado tiempo
a vosotros. Gracias.
A Débora. Por tu paciencia todos estos años. Por levantarme en los momentosmás duros, y ponerme los pies en la tierra en los mejores. Por estar a mi lado eneste camino. Gracias.
Muchas gracias a todos.

ixAbstract
According to data from the World Health Organization (WHO), 17.7 million people were estimated to have died of cardiovascular diseases (CVDs) in 2015. That
represents 31% of all global deaths, making these diseases the leading cause ofdeath worldwide. The heart is a complex system that works due to the interactionof a large number of elements at different temporal and spatial scales. The mainfunction of the heart is to pump blood throughout the body, with this mechanical action being triggered by electrical impulses. Issues arising in the electrical ormechanical actions of the heart at any of the involved temporal and spatial scales
can lead to cardiac malfunctioning. Mathematical modeling and simulation of theheart’s electrical activity (so-called cardiac electrophysiology) and signal processing of bioelectrical signals provide an ideal framework to join the information fromclinical and experimental studies with the understanding of the mechanisms underlying them. Due to the high number of factors involved in the development andvalidation of cardiac computational electrophysiological models and the intricate
interrelationships between them, novel methodologies that help control the design,
update and validation of new models become of great advantage. These methodologies can target from the definition of ionic gating in the simulated cells to the
propagation of the electrical impulse in multi-scale models. This thesis aims to
improve the existing knowledge on heart’s electrophysiology by developing novel
techniques to develop and validate cardiac computational models, with evaluationof the effects of the modeled events by considering interactions between modelcomponents and by simulating a range of spatio-temporal scales.
In chapter 2, a new paradigm was introduced to develop a novel human ventricular cell model, the CRLP model, by departing from a previously publishedmodel and incorporating novel experimental measurements of potassium currents
and reformulating the L-type calcium current. The introduced paradigm was basedon the analysis of the model’s ability to replicate a set of well-established electrophysiological markers and on a sensitivity analysis of those markers to variations
in model parameters. A major advantage of the proposed paradigm was the possibility to identify model parameter values that do not directly depend on individual
current measurements or concentrations and which are commonly set in an ad hoc

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New Methodologies for the Development
and Validation of Electrophysiological Models

manner. The developed CRLP model was validated, and its improved capacity
to investigate arrhythmia-related properties, as compared to the cell model it was
based on, was corroborated.
In chapter 3, the CRLP model developed in chapter 2 was updated to introduce the formulation of intracellular potassium ([K + ]i ) dynamics. This is an
important characteristic for investigation of ventricular arrhythmias arising underconditions of hyperkalemia, one of the components of myocardial ischemia. Direct introduction of [K + ]i dynamics into the model generated an imbalance in thepotassium currents leading to a drift in [K + ]i . To correct for such an imbalance,
an optimization framework was proposed that allowed estimating the ionic current conductances of the CRLP model while guaranteeing physiologically plausible
values of selected electrophysiological properties, some of them highly relevant forinvestigation of ventricular arrhythmias.
As mentioned above, when proposing a new model, or when updating an existing model, consistency between simulated and experimental data should be verified
by considering all involved effects and scales. The closer the experimental conditions are reproduced in the computer simulations, the more robust the process ofmodel development and validation can be. In chapter 4, in silico simulation ofexperimental protocols was proposed to analyze: how interactions between modelcomponents affect the development and validation of mathematical ion channel
models; and how propagation affects action potential (AP)-based markers simulated in isolated cells and in tissue preparations, with identification of the ioniccontributors in each case. The CRLP model, developed in chapter 2 and updated inchapter 3, presented a rather atypical shape at the end of the depolarization phaseof the AP (phase 1). In silico simulations of experimental protocols described inchapter 4 and the optimization methodology introduced in chapter 3 were used to
improve the AP shape while validating the adjusted model at ionic, cell and tissue
scales.
In chapter 6 all the initial formulations and subsequent updates of the CRLP
model proposed in previous chapters were integrated and the ionic conductances of
the integrated model were readjusted to improve replication of experimental electrophysiological measures. All the methodologies introduced throughout the thesis
were used to build a novel human ventricular AP model. For model validation, arange of available experimental data at different scales and aimed at evaluating different electrophysiological properties was considered. Conditions underlying eachof the experimental studies were replicated as faithfully as possible. Results simulated with the final version of the CRLP model were in all cases compared with all
available experimental evidences and with the most recent human ventricular cell
models published in the literature.
Chapter 7 summarizes the main conclusions of the thesis and presents new linesof research that could be undertaken in future studies.

Universidad Zaragoza

Resumen y conclusiones
De acuerdo a los datos de la Organización Mundial de la Salud, se estima que 17,7
millones de personas murieron de enfermedades cardiacas en 2015, lo que supone el
31% de las muertes, haciendo de estas patologías la primera causa de muerte en elmundo. El corazón es un sistema complejo que trabaja gracias a la interacción deun gran número de elementos en diferentes escalas espaciales y temporales. La función principal del corazón es bombear sangre en todo el cuerpo, siendo esta acción
mecánica activada por la estimulación eléctrica. La aparición de problemas en el
funcionamiento eléctrico o mecánico del corazón en cualquiera de las escalas involucradas, temporal o espacial, puede dar lugar a un mal funcionamiento cardiaco. El
modelado matemático y la simulación de la actividad eléctrica del corazón (denominada electrofisiología cardiaca) y el procesado de señales bioeléctricas proporcionan
un marco ideal para unir la información clínica y los estudios experimentales conla comprensión de los mecanismos que subyacen a estos problemas. Debido al grannúmero de factores que se deben tener en cuenta a la hora de desarrollar y validar
un modelo computacional de electrofisiología cardiaca, asi como las complejas interacciones que existen entre ellos, hacen que nuevas metodologías que facilitan laconcepción, la actualización y la validación de nuevos modelos sean de gran valor.
Estas metodologías pueden enfocarse sea en la definición de las compuertas iónicas
de los modelos, como en la propagación del impulso eléctrico en modelos multiescala. Esta tesis pretende mejorar el conocimiento existente sobre electrofisiología
cardiaca proponiendo nuevas técnicas para desarrollar y validar modelos computacionales cardiacos, a través de la evaluación de los efectos de los eventos modelados
mediante la consideración de las interacciones entre los diferentes componentes delmodelo y la simulación de un rango de escalas espacio-temporales.
En el capítulo 2, se introdujo un nuevo paradigma para desarrollar un nuevo
modelo de potencial de acción de cardiomiocito humano, el modelo CRLP, partiendo de un modelo previamente publicado e incorporando nuevas mediciones
experimentales de corrientes de potasio y reformulando la corriente de calcio tipo
L. El paradigma introducido se basó en el análisis de la capacidad del modelo
para replicar un conjunto de marcadores electrofisiológicos bien establecidos y en
un análisis de sensibilidad de esos marcadores a las variaciones en los parámetros

xiii

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New Methodologies for the Development
and Validation of Electrophysiological Modelsdel modelo. Una de las ventajas del paradigma propuesto fue la posibilidad deidentificar parámetros del modelo que no dependen directamente de las mediciones
individuales de corrientes o concentraciones y que comúnmente se establecen ad
hoc. El modelo CRLP se validó y se midió su rendimiento en la capacidad parapredecir marcadores relacionados con la arritmia ventricular en comparación con
el modelo cellular en el que se había basado.
En el capítulo 3, se actualizó el modelo CRLP desarrollado en el capítulo 2 para
introducir la formulación de la dinámica de potasio intracelular ([K + ]i ). Esta es
una característica importante para la investigación de arritmias ventriculares que
surgen en condiciones de hiperpotasemia, uno de los componentes de la isquemia
de miocardio. La introducción directa de la dinámica de [K + ]i en el modelo generó un desequilibrio en las corrientes de potasio que condugeron a una deriva en[K + ]i . Para corregir tal desequilibrio, se propuso un algoritmo de optimización
que permitía estimar las conductancias de las corrientes iónicas del modelo CRLP
al tiempo que garantizaba valores fisiológicamente plausibles de una selección depropiedades electrofisiológicas, algunas de ellas muy relevantes en el estudio dearritmias ventriculares.
Como se mencionó anteriormente, al proponer un nuevo modelo o al actualizar
un modelo existente, la coherencia entre los datos simulados y experimentales debe
verificarse considerando todos los efectos y escalas involucradas. Cuanto mejor se
reproduzcan las condiciones experimentales en las simulaciones, más robusto seráel proceso de desarrollo y validación del modelo. En el capítulo 4, se propuso la
simulación de protocolos experimentales in silico para analizar cómo las interacciones entre los componentes del modelo afectan el desarrollo y la validación delos modelos matemáticos de canales iónicos; y cómo la propagación afecta los marcadores basados en el potencial de acción cuando son simulados en células aisladaso en preparaciones tisulares, identificando cómo contribuye cada corriente iónica
en cada caso.
El modelo CRLP, desarrollado en el capítulo 2 y actualizado en el capítulo 3,
presentaba una forma más bien atípica al final de la fase de despolarización delpotencial de acción (fase 1). La simulación in silico de los protocolos experimentales descritos en el capítulo 4 y la metodología de optimización presentada en el
capítulo 3 se utilizaron para mejorar la forma del potencial de acción al tiempo que
validaba el modelo ajustado a escalas iónicas, celulares y de tejido.
En el capítulo 6 se integraron todas las formulaciones iniciales y actualizaciones subsiguientes del modelo CRLP propuestas en los capítulos anteriores y se
reajustaron las conductancias iónicas del modelo para mejorar el comportamientodel modelo con respecto a medidas electrofisiológicas experimentales. Todas las
metodologías introducidas a lo largo de la tesis se utilizaron para obtener un nuevo
modelo de potencial de acción ventricular humano. Para la validación del modelo, se consideró un rango de datos experimentales disponibles a diferentes escalasy destinados a evaluar diferentes propiedades electrofisiológicas. Las condiciones
subyacentes a cada uno de los estudios experimentales se replicaron tan fielmentecomo fue posible. Los resultados simulados con la versión final del modelo CRLP se
compararon en todos los casos con todas las evidencias experimentales disponibles

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y con los modelos de células ventriculares humanas más recientes publicados en laliteratura.
El capítulo 7 resume las principales conclusiones de la tesis y presenta nuevas
líneas de investigación que podrían emprenderse en futuros estudios. En conclusión,diferentes técnicas para mejorar el desarrollo y la validación de modelos electrofisiológicos cardíacos han sido propuestos y analizados en esta tesis. Basándose enel aumento de potencia computacional, se han considerado nuevas estrategias para
reducir el número de hipótesis y/o supuestos al construir un modelo de potencial deacción de cardiomiocito ventricular. La aplicación de un algoritmo de optimización
junto con la simulación in silico de los protocolos experimentales han ayudado aencontrar un modelo que represente mejor los resultados experimentales de losmarcadores electrofisiológicos de riesgo arrítmico.

JesúsCarroIndex

Contents

Acknowledgment

vii

Agradecimientos

ix

Abstractxi

Resumen y Conclusiones

xiii

Index

xvii

Contents

xix

List of Figures

xxv

List of Tables

xxix

Dissertation

1

1 Introduction
3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Electrical and Mechanical Activity of the Heart . . . . . . . . . . . . 4
1.2.1 Excitability of cardiac cells . . . . . . . . . . . . . . . . . . . 5
1.2.2 Electromechanical coupling . . . . . . . . . . . . . . . . . . . 6
1.2.3 Propagation of the electrical signal throughout the heart . . . 6
1.2.4 The electrocardiogram . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5 Acute myocardial ischemia . . . . . . . . . . . . . . . . . . . 8
1.3 Modeling the Electrical Activity of the Heart . . . . . . . . . . . . . 10
1.3.1 First electrophysiological model of an excitable cell . . . . . . 10
1.3.2 Evolution of electrophysiological models for cardiac cells . . . 14

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New Methodologies for the Development
and Validation of Electrophysiological Models

xx

1.3.3

Human ventricular electrophysiological models: main ioniccurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Human ventricular electrophysiological models: recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.5 The Grandi-Pasqualini-Bers model . . . . . . . . . . . . . . .
1.3.6 Acute ischemia models . . . . . . . . . . . . . . . . . . . . . .
Objectives and Outline of the Thesis . . . . . . . . . . . . . . . . . .
1.4.1 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 Specific objectives . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . .

17
18
21
22
22
22
23

2 A New Paradigm for Electrophysiological Model Development
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Selected markers . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Stimulation Protocols . . . . . . . . . . . . . . . . . . . . . .
2.2.3 The CRLP Model . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Conducted Simulations . . . . . . . . . . . . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Results for Control Conditions . . . . . . . . . . . . . . . . .
2.3.2 Results for Hyperkalemic Conditions . . . . . . . . . . . . . .
2.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 New model characteristics . . . . . . . . . . . . . . . . . . . .
2.4.2 Simulation of hyperkalemic conditions . . . . . . . . . . . . .
2.4.3 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Model limitations . . . . . . . . . . . . . . . . . . . . . . . . .

27
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28
29
31
37
38
38
45
48
49
49
50
50

3 Improvement of Existing Electrophysiological Models
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Optimization problem . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Upgrade of the CRLP model . . . . . . . . . . . . . . . . . .
3.2.3 Frequency-dependence behavior . . . . . . . . . . . . . . . . .
3.2.4 Simulation of changes in potassium concentration . . . . . . .
3.2.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Non-linear optimization . . . . . . . . . . . . . . . . . . . . .
3.3.3 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Frequency response . . . . . . . . . . . . . . . . . . . . . . . .
3.3.5 Response to changes in extracellular potassium . . . . . . . .
3.3.6 Computational Cost . . . . . . . . . . . . . . . . . . . . . . .
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Response surface optimization . . . . . . . . . . . . . . . . .
3.4.2 Database generation . . . . . . . . . . . . . . . . . . . . . . .

53
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60
65
65
66
66
66
68
68
71
71
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73
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74

1.4

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Contents
3.4.3 Computational cost . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Definition of the optimization problem . . . . . . . . . . . . .
3.4.5 Model comparison . . . . . . . . . . . . . . . . . . . . . . . .
3.4.6 Response to potassium concentration changes . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75
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76
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4 Limitations in Electrophysiological Model Development and Validation
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 L-type calcium channel gating . . . . . . . . . . . . . . . . .
4.2.2 Characterization of L-type calcium voltage-dependent inactivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Steady-state AP markers . . . . . . . . . . . . . . . . . . . .
4.2.4 Ionic contributors to AP markers . . . . . . . . . . . . . . . .
4.2.5 Computational simulations . . . . . . . . . . . . . . . . . . .
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Characterization of L-type calcium voltage-dependent inactivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Ionic contributors to AP markers . . . . . . . . . . . . . . . .
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Effect of submodel variable interactions in the evaluation ofvoltage-dependent L-Type Calcium Current inactivation . . .
4.4.2 Effect of cell-to-cell interactions in the evaluation of ioniccontributors to AP markers . . . . . . . . . . . . . . . . . . .
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79
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82

3.5

83
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90
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94
97
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5 Reformulation of Ionic Current Expression
99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Modification of ICaL voltage-dependent inactivation time constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 Adjustment of ICaL voltage-dependent inactivation steady-state100
5.2.3 Numerical methods and implementation . . . . . . . . . . . . 103
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Solution to the minimization problem . . . . . . . . . . . . . 104
5.3.2 New definition of ICaL current . . . . . . . . . . . . . . . . . . 104
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.4.1 Development and validation of a new ionic current formulation107
5.4.2 Reproducing experimental data from different laboratories . . 107
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 A Human Ventricular Electrophysiological Model with Improved Description of Experimental Behavior
109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

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and Validation of Electrophysiological Models

xxii

6.3

6.4

6.5

6.2.1 Adjustment of ionic conductances . . . . . . . . . . . . . . . . 111
6.2.2 Model comparison . . . . . . . . . . . . . . . . . . . . . . . . 120
6.2.3 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 120
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.1 Model optimization . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3.2 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 129
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.1 Model performance . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4.2 Optimization algorithm . . . . . . . . . . . . . . . . . . . . . 133
6.4.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

7 Conclusions and Future Work
137
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.1 Identification of major issues in electrophysiological model
development and validation . . . . . . . . . . . . . . . . . . . 137
7.1.2 New techniques for improved model development and validation138
7.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Appendix

143

A Publications
145
A.1 Publications in Journal Articles . . . . . . . . . . . . . . . . . . . . . 145
A.2 Publications in Conference Proceedings . . . . . . . . . . . . . . . . 145
A.3 Other Conferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
B Description of the First Model147
B.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
B.1.1 Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . 147
B.1.2 Enviromental Parameters . . . . . . . . . . . . . . . . . . . . 147
B.1.3 Fractional Currents . . . . . . . . . . . . . . . . . . . . . . . . 148
B.1.4 Ion Concentrations . . . . . . . . . . . . . . . . . . . . . . . . 148
B.1.5 Sodium Transport . . . . . . . . . . . . . . . . . . . . . . . . 148
B.1.6 Potassium Currents . . . . . . . . . . . . . . . . . . . . . . . 148
B.1.7 Chlorine currents . . . . . . . . . . . . . . . . . . . . . . . . . 149
B.1.8 Calcium Transport . . . . . . . . . . . . . . . . . . . . . . . . 149
B.1.9 SR Calcium Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 149
B.1.10 Buffering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
B.2 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.2.1 Nerst Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 151
B.2.2 INa :Fast Sodium Current . . . . . . . . . . . . . . . . . . . . . 151
B.2.3 INa,bk : Background Sodium Current . . . . . . . . . . . . . . . 152
B.2.4 INaK : Na-K Pump Current . . . . . . . . . . . . . . . . . . . . 153
B.2.5 IKr : Rapidly Activating Potassium Current . . . . . . . . . . 153

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xxiiiContents

B.2.6 IKs : Slowly Activating Potassium Current . . . . . . . . . . . 153
B.2.7 IKp : Plateau Potassium Current . . . . . . . . . . . . . . . . . 154
B.2.8 Ito : Transient Outward Potassium Current . . . . . . . . . . 154
B.2.9 IK1 :Inward Rectifier Potassium Current . . . . . . . . . . . . 155
B.2.10 IClCa : Calcium-Activated Chlorine Current . . . . . . . . . . . 155
B.2.11 ICl,bk : Background Chlorine Current . . . . . . . . . . . . . . . 155
B.2.12 ICaL : L-type Calcium Current . . . . . . . . . . . . . . . . . . 155
B.2.13 Incx : Na-Ca Exchanger Current . . . . . . . . . . . . . . . . . 158
B.2.14 IpCa : Sarcolemmal Calcium Pump Current . . . . . . . . . . . 158
B.2.15 ICa,bk : Background Calcium Current . . . . . . . . . . . . . . 159
B.2.16 SR Fluxes: Calcium Release, SR Calcium Pump, SR Calcium
Leak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.17 Ion Homeostasis . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2.18 Membrane Potential . . . . . . . . . . . . . . . . . . . . . . . 161
C Description of the Final Model
163
C.1 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
C.1.1 Enviromental Parameters . . . . . . . . . . . . . . . . . . . . 163
C.1.2 Fractional Currents . . . . . . . . . . . . . . . . . . . . . . . . 163
C.1.3 Sodium Transport . . . . . . . . . . . . . . . . . . . . . . . . 163
C.1.4 Potassium Currents . . . . . . . . . . . . . . . . . . . . . . . 164
C.1.5 Chlorine currents . . . . . . . . . . . . . . . . . . . . . . . . . 164
C.1.6 Calcium Transport . . . . . . . . . . . . . . . . . . . . . . . . 164
C.1.7 SR Calcium Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 164
C.2 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
C.2.1 Ito : Transient Outward Potassium Current . . . . . . . . . . 164
C.2.2 ICaL : L-type Calcium Current . . . . . . . . . . . . . . . . . . 164
C.2.3 SR Fluxes: Calcium Release, SR Calcium Pump, SR Calcium
Leak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
C.2.4 Ion Homeostasis . . . . . . . . . . . . . . . . . . . . . . . . . 166
C.2.5 Membrane Potential . . . . . . . . . . . . . . . . . . . . . . . 166
D Acronyms

167

References

169

JesúsCarroList of Figures

1.1

The anatomy of the heart and associated vessels . . . . . . . . . . .

5

1.2

Isochronic surfaces of the ventricular activation . . . . . . . . . . . .

7

1.3

Different waveforms for each of the specialized cells found in the heart7

1.4

Electrode positions in the standard 12-lead ECG . . . . . . . . . . .

8

1.5The equivalent circuit of the Hodgkin-Huxley model. . . . . . . . . . 11

1.6

Voltage dependence of the particles in the potassium channels of the
HH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7

Voltage dependence of the particles in the sodium channels of the
HH model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1

Graphical description (using simulation results) of the markers ofarrhythmic risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2

Characteristics of ICaL and IK1 in simulations and experiments . . . . 35

2.3

APs computed with the TP06, GPB and CRLP models for isolated
cells and 1D tissue at CL = 1000 ms . . . . . . . . . . . . . . . . . . 39

2.4

markers of arrhythmic risk from TP06, GPB and CRLP models . . . 43

2.5

Effect of blocking potassium currents for the GPB and CRLP models 44

2.6

CV and APD90 restitution curves for the TP06, GPB and CRLP
models in a simulated 1D fiber. . . . . . . . . . . . . . . . . . . . . . 45

2.7

Simulated hyperkalemia in cell and tissue using the CRLP model . . 46

2.8

Behavior of the CRLP model under hyperkalemic conditions in a 1D
fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1

Flow chart of the proposed algorithm. . . . . . . . . . . . . . . . . . 58

3.2

Temporal evolution of [K + ]i in CRLP and GPB models after introducing [K + ]i dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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New Methodologies for the Development
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xxvi
3.3

3.4
3.5
3.6
3.7

3.8

4.1
4.2

4.3

4.4

4.5

4.6

Results of the sensitivity analysis applied to the original CRLP
model. The blue scale indicates relative sensitivity for each marker.
White color indicates maximum relative sensitivity and dark blue
color indicates that property and parameter are independent. Percentages in the white boxes indicate the absolute sensitivity of theproperty. A minus sign in a box indicates that marker and model parameter vary inversely. ‘Trian’ stands for Triangulation, ‘Sys’ stands
for systolic and ‘Dia’ stands for diastolic. . . . . . . . . . . . . . . .
Model Comparison Results . . . . . . . . . . . . . . . . . . . . . . .
Model Comparison Results . . . . . . . . . . . . . . . . . . . . . . .
Steady-state AP D90 values as a function of pacing CL. . . . . . . . .
Results for an unstimulated cell simulated with the modified CRLP
model following changes in [K + ]o . a) Temporal evolution of membrane potential; b) Time to achieve 90% of the steady-state membrane potential (t90% ). . . . . . . . . . . . . . . . . . . . . . . . . . .
[K + ]i evolution following [K + ]o changes simulated with the modified CRLP model. a) [K + ]i dependence on [K + ]o at different time
instants; b) time course of [K + ]i for different [K + ]o values. . . . . .
Paired-pulse voltage clamp protocol and measurement of the steadystate value fss of voltage-dependent inactivation. . . . . . . . . . . .
Comparison between model definition (continuous and discontinuouslines), in silico simulations (yellow crosses) and experimental results(red circles) for fss . a) TP06 model and data from [1]. b) GPB
model and data from [2]. c) ORd model and data from [3]. d)
CRLP model and data from [1]. . . . . . . . . . . . . . . . . . . . . .
Temporal evolution of the ICaL gates during simulation of the pairedpulse protocol (see text for specific values of tpre , tsep and tpulse usedin the simulations): a) TP06 model. b) GPB model. c) ORd model.
d) CRLP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contribution of ionic conductances to electrophysiological markers
simulated in tissue and cell with the TP06 model. Dark colors (blueor brown) indicate maximum correlation between changes in a conductance and changes in a marker; white color indicates no correlation. Percentages in boxes indicate the contribution of changes ina conductance to changes in a marker. Minus signs indicate that
conductances and markers vary inversely; plus signs indicate that
conductances and markers vary in the same direction. . . . . . . . .
Contribution of ionic conductances to AP markers simulated in celland tissue with the GPB model. The legend of the figure is the sameas in see Figure 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contribution of ionic conductances to AP markers simulated in celland tissue with the ORd model. The legend of the figure is the sameas in see Figure 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . .

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67
69
70
71

71

72

84

88

89

92

92

93

List of Figures
4.7

4.8

5.1

5.2

5.3

5.4

5.5

6.1

6.2
6.3

6.4

6.5
6.6

xxvii

Contribution of ionic conductances to AP markers simulated in celland tissue with the ORdNa model. The legend of the figure is thesame as in see Figure 4.4. . . . . . . . . . . . . . . . . . . . . . . . . 93
Contribution of ionic conductances to AP markers simulated in celland tissue with the CRLP model. The legend of the figure is thesame as in see Figure 4.4. . . . . . . . . . . . . . . . . . . . . . . . . 94
Time constants of ICaL voltage-dependent inactivation in the CRLP
model and following the readjustment proposed in this chapter.
Comparison with experimental data [2, 4–7] is provided as well. . . . 100
Effect of varying βf 2 in the AP shape. Red line represents control
conditions. Dark grey lines correspond to reductions in βf 2 , whilelight grey lines correspond to increases in βf 2 . . . . . . . . . . . . . . 101
Effect of varying βf in the AP shape after having modified βf 2 in
the CRLP model. Red line represents control conditions. Dark greylines correspond to reductions in βf , while light grey lines correspondto increases in βf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Comparison between the results of the in silico simulations for the
CRLP model and the adjusted model of this chapter and the experimental results of [1–3]. . . . . . . . . . . . . . . . . . . . . . . . . . 106
Results of tissue simulations computed with the readjusted CRLP
model proposed in this chapter. a) AP at a cycle length of 1000
ms. b) [Ca2+ ]i at a cycle length of 1000 ms. c) APD adaptation toabrupt cycle length changes. . . . . . . . . . . . . . . . . . . . . . . . 106
Action Potential of the different models under control conditions insingle epicardial cells: a) optimized CRLP model; b) initial CRLP
model; c) GPB model; d) ORd model; e) TP06 model. . . . . . . . . 123
AP D90 variation with CL under the same conditions of the CLVPET
dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
AP D90 variation with CL under the same conditions of the CLVPET
dataset following ionic current inhibitions: A) 15% IKr block; B) 70%
IKr block; C) 95% IKs block; D) Zoomed graphic for 15% IKs block;E) 75% IK1 block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Simulations of the experiments by Drouin et al. [8]: A) AP D90
variation with CL for epicardial cells. B) Zoomed version of the
figure shown in A. C) AP D90 variation with CL for endocardial
cells. D) Zoomed version of the figure shown in C. . . . . . . . . . . 127
Variation in [N a+ ]i under the same conditions of the experiments
by Piekse et al. [9] for endocardial cells. . . . . . . . . . . . . . . . . 128
Variation in [Ca2+ ]i under the same conditions of the experiments
by Schmidt et al. [10]: A) Diastolic [Ca2+ ]i normalized by the value
when the frequency is 1Hz; B) Systolic [Ca2+ ]i normalized by thevalue when the the frequency is 1Hz. . . . . . . . . . . . . . . . . . . 128

JesúsCarroNew Methodologies for the Development
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xxviii
6.7

Variation in [Ca2+ ]i characteristics under the same conditions of theexperiments by Coppini et al. [11]: A) Difference between systolicand diastolic [Ca2+ ]i levels as a function of CL; B) Systolic [Ca2+ ]ifor different CLs normalized by the value when CL is 1000 ms; C)Time from [Ca2+ ]i peak to 50% as a function of CL; D) Time from
[Ca2+ ]i peak to 90% as a function of CL. . . . . . . . . . . . . . . . 130

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List of Tables

1.1

Differences between human ventricular cell models . . . . . . . . . . 19

2.1
2.2

Results of the sensitivity analysis applied to the GPB model . . . . . 41
markers of arrhythmic risk for the TP06, GPB and CRLP humanmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Percentages of variation in APD90 caused by blocking different potassium currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3

3.1
3.2
3.3
4.1
4.2
4.3

5.1

5.2

6.1
6.2
6.3

Electrophysiological markers evaluated in the proposed response surface approximation-based optimization. . . . . . . . . . . . . . . . . 64
Steady-state values of [K + ]i for different [K + ]o values as calculatedwith the optimized CRLP model. . . . . . . . . . . . . . . . . . . . . 73
Computational cost of the different protocols used for the optimization. 73
Paired-pulse test parameters in experimental protocols. . . . . . . . 85
Comparison of electrophysiological markers simulated in single celland 1D tissue under control conditions. . . . . . . . . . . . . . . . . 90
Absolute (Ea) and relative (Er) differences between electrophysiological markers in single cell and 1D tissue simulations. . . . . . . . . 91
Results of the minimization problem and subsequent validation. Improvement represents the error for each combination of constraints
normalized by the error before the optimization. . . . . . . . . . . . 105
Evaluation of electrophysiological markers in tissue for the initialand adjusted CRLP models (third and fourth columns). Available
physiological ranges are provided in the second column [12]. . . . . . 105
Evaluation of AP D90 and Vrest in the TESZT dataset. . . . . . . . . 113
Evaluation of variations in AP D90 in the CLVPET dataset. . . . . . 114
Evaluation of AP D90 in the study by Drouin et al. [8]. . . . . . . . . 115

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New Methodologies for the Development
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xxx

Evaluation of [N a+ ]i and its variations in the study by Pieske et al.
[9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.5 Evaluation of [Ca2+ ]i and its variations in the study by Schmidt
et al. [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.6 Evaluation of [Ca2+ ]i characteristics in the study by Coppini et al.
[11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.7 Evaluation of time for APD rate adaptation in the study by Pueyo
et al. [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.8 Results of steady-state AP markers for a CL of 1000 ms computedwith the optimized CRLP model and other published human ventricular cell models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.9 Results of steady-state [N a+ ]i and [Ca2+ ]i for a CL of 1000 ms computed with the optimized CRLP model and other published humanventricular cell models. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.10 Results of APD adaptation to abrupt CL changes computed with alltested human ventricular models under the same conditions as in [13].129
6.11 Validation results. Steady-state AP markers for a CL of 1000 ms
calculated in tissue fibers simulated with the optimized and initial
CRLP models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4

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Dissertation

Chapter 1

Introduction
1.1 Motivation
According to data of the World Health Organization, an estimated 17.7 million
people died of cardiovascular diseases (CVDs) in 2015. That represents 31% of all
global deaths, making these diseases the leading cause of death worldwide. CVDs
are caused by different disorders of the heart and blood vessels, which include coronary heart disease (heart attacks), cerebrovascular disease (stroke), raised bloodpressure (hypertension), peripheral artery disease. It is also important to note that
CVDs are the leading noncommunicable disease (NCD). In 2011, 36 million people
died from an NCD, with half caused by CVDs [14]. For this reason, it is necessary
to improve the current techniques to detect, diagnose, and prevent CVDs.
In particular, within the group of CVDs, one of the most important causesof sudden death is cardiac arrhythmia, when the heart beats with an abnormal
rhythm: too quickly, too slowly, or erratically. Types of arrhythmia include: atrial
fibrillation, bradycardia, conduction disorders, premature contraction, tachycardia, and ventricular fibrillation. Typically, a cardiac arrhythmia is caused by anabnormal behavior of the heart’s natural pacemaker, a problem in the conduction
system to contract the heart, or because another part of the heart starts working
as the pacemaker. These pathological situations preclude the heart carrying out
its main function: to pump deoxygenated blood to the lungs and oxygenated bloodto the rest of the body.
The heart is a complex system that works through the interaction of different
elements at different scales, in which its main function is performed by a mechanical action that is electrically controlled. Mathematical modeling and simulationof the heart’s electrical activity (so-called cardiac electrophysiology) and signal
processing of bioelectrical signals provide an ideal framework for joining the information from clinical and experimental studies with the understanding of themechanisms underlying them. Tools capable of testing different hypotheses and3

4

New Methodologies for the Development
and Validation of Electrophysiological Models

predicting potential abnormalities in the heart’s behavior can be produced withinsuch a framework, and, in a relatively near future, they may be used in clinicalpractice as complementary instruments to help in the prevention of cardiac diseases and in the improvement of their diagnosis and therapy. In particular, for the
study of ventricular arrhythmias, the utilization of an in silico model of the heart
could improve understanding of the heart function, how different pathologies aredeveloped, and how they are expressed. Nevertheless, it remains essential to havean action potential (AP) model able to reproduce experimentally and clinicallyobserved arrhythmic behaviors at the cell and tissue levels. This would allow simulation of the electrical activity of the heart from cell to body surface. Then, a total
in silico reconstruction of the most well known bioelectrical signal, the electrocardiogram (ECG), could be used for investigating the mechanisms and manifestationsof ventricular arrhythmias at different scales.
Due to the number of factors to be considered during the development of amodel, the interrelationships between them, and the need to preserve the improvements obtained by previous models, it is necessary to develop novel methodologies
that help control the design, modification, and validation of new models. These
new methodologies should help control all elements of the model during its development, from the definition of the ionic gates to the combination in multiscalemodels, and to consider the implication of any definition or modification in thesimulation of the entire heart.
In the following sections, a general introduction is presented on how the heart
works (mainly from an electrical point of view), how the behavior of the heart isaffected by disease or by pathological conditions and how mathematical models
can be used to study the heart in physiological and pathological conditions. Subsequently, a description is provided on the electrical conduction system of the heartand on the range of involved spatio-temporal scales. Next, the evolution of cardiac
electrophysiological models and the main components of a model of human ventricular cells are analyzed. Finally, the Grandi-Pasqualini-Bers (GPB) model ofhuman ventricular cells [15], used as a starting point for this thesis, is studied. The
chapter end with the main objectives and a short outline of the different studies
described in this thesis.
1.2 Electrical and Mechanical Activity of the HeartThe heart is a volumetric pump that beats at 75 beats per minute and moves morethan five liters of blood. It consists of four chambers, the left and right ventricles,
and the left and right atria (see Figure 1.1). Deoxygenated blood enters the heart
through the right atrium, and, from there, flows through the tricuspid valve to theright ventricle and is pumped to the lungs. Oxygenated blood returns from thelungs through the left atrium, then, flows to the left ventricle through the mitralvalve. Once, the blood is in the left ventricle, it is pumped through the aortic
valve to the systemic circulation. Under normal conditions, all of these phases are
timed and synchronized, and this permits the blood to circulate and to transportoxygen and other nutrients throughout the body. This mechanism works because

Universidad Zaragoza

1. Introduction

5

Figure 1.1: The anatomy of the heart and associated vessels, and orientation ofcardiac muscle fibers. Figure from [16].
of the electromechanical coupling of the cardiac cells, and to a complex system that
propagates the electrical signals to obtain a coordinated contraction of the heart.
1.2.1 Excitability of cardiac cells
Cardiac muscle cells (or cardiomyocytes) are excitable cells, that is, when a stimulation current pulse depolarizes the resting membrane of a cardiomyocyte beyond
the threshold voltage, then an action potential (AP) is generated. During an AP,
the membrane potential (the potential at the inner surface relative to that at theouter surface of the membrane), rises rapidly at the beginning (depolarization).
This first depolarization phase is followed by a small decrease in the action potential (dome) to then enter the third phase (plateau) where the membrane potential
is held at a high voltage for few milliseconds prior to enter the repolarization (thefourth phase), in which the membrane potential recovers its resting value (the fifth
and last phase).
The membrane potential is caused by the difference of concentration between
the ions in the intracellular and extracellular compartments. The three main ionspresent in the cardiomyocytes are sodium (N a+ ), potassium (K + ) and calcium(Ca2+ ). When an external stimulus is applied, an initial upstroke is generated by
the inward flux of sodium ions through the corresponding sodium channels. Afterthe first phase, the sodium channels close and the transient outward potassium
current is activated and attempts to return the cell to its resting state. The outward
potassium current is compensated by and inward calcium ion flux and therefore
slow down the decrease in the potential during the plateau phase. Calcium channels
begin closing while slow potassium currents begin acting until the cell recovers its
initial resting state. In cardiac muscle, the activation of one cell can propagate to
the neighbor cells, and, as a result, the activation generates complex wavefronts
that propagate through the cardiac tissue.

JesúsCarro

6

New Methodologies for the Development
and Validation of Electrophysiological Models

1.2.2 Electromechanical coupling
When the membrane of the myocyte is depolarized, the channels that allow thepassage of calcium ions through them are open. Most of these ions enter throughthe L-type calcium channel to a subspace called the dyadic space, which is formedby the T-tubule (an extension of the cell membrane) and the close proximity ofthe sarcoplasmic reticulum (the internal store for calcium). When calcium enters,
the concentration in the dyadic space elevates and it triggers the opening of theryanodine receptors in the sarcoplasmic reticulum. This launches a process known
as calcium-induced calcium release, during which, the calcium input into the cellis amplified with the calcium present in the sarcoplasmic reticulum (SR). Free calcium present in the cytosol binds to troponin and calmodulin favoring myofilamentcontraction. The joint action of many cardiac cells contracting in an orderly way
in a small period of time causes the blood to be pumped by the heart.
1.2.3 Propagation of the electrical signal throughout the heart
As explained in the previous sections, the cardiac tissue is electrically excitable andits excitation propagates through the tissue. In addition, when a cardiac tissue is
excited, it contracts because of the electromechanical coupling described previously.
These two phenomena are fundamental for the correct functioning of the heart,
but its proper operation also depends on the regulation and synchronization of thedifferent cardiac chambers.
The pacemaker of the heart is the sinoatrial (SA) node which is located in theright atrium at the superior vena cava. It consists of specialized self-excitatory
muscle cells that are self-activated at the rate of about 70 beats per minute. Fromthe sinus node, activation propagates throughout the atria, but it cannot propagate
directly to the ventricle because a barrier of fibrous tissue is present. To do this, theactivation propagates first through the atrioventricular (AV) node, which is locatedat the boundary between the atria and ventricles and which delays the excitationof the ventricular tissue due to a very low conduction velocity (CV): 2-5 cm/s (e.g.
the CV of the atria is between 80 and 100 cm/s) [16]. Activation wavefront takes
an estimated 75 ms to cross the AV node. Because of this, the atria contract andpump the blood to the ventricles when they are still relaxed.
Once the activation traverses the AV node, it travels through a common bundle,
called the bundle of His which later separates into two bundle branches: the rightand left bundle branches. Finally, these branches ramify into the Purkinje fibersthat are connected to the ventricular walls and, through them, the activation is
spread in the ventricle from many different locations. The ventricles, which havethe blood coming from the atria, contract with the propagation of the electricalactivation and pump the blood to the lungs and the rest of the body.
Figure 1.2 illustrates the isochronic surface of the activation in the ventricles.
The image is from [16] with data from [17].
Different AP shapes can be found at the various types of cardiac cells in theheart. Figure 1.3 illustrates them with a delay similar to the one that is normallyfound in a healthy heart.

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Figure 1.2: Isochronic surfaces of the ventricular activation. Image from [16] withdata from [17].

Figure 1.3: Different waveforms for each of the specialized cells found in the heart
are shown. The latency shown approximates that normally found in the healthy
heart. Image from [16].

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Figure 1.4: Electrode positions in the standard 12-lead ECG. Image created by
Nicholas Patche, Boston Medical Center (CC BY-SA 4.0).
1.2.4 The electrocardiogram
The ECG is a tool that can be used to assess the electrical and muscular activitiesof the heart. It records the electric potential on the surface of the thorax that has
been generated by the electrical activity of cardiac cells. To measure it, electrodes
can be placed on the thorax that will record various electric signals depending on
their positions. Figure 1.4 illustrates, the standard 12-lead ECG is presented.
As depicted in Figure 1.3, the depolarization and repolarization of the cardiac
cells is represented in the following parts of the ECG:
• The atrial depolarization generates the P-wave.
• The ventricular depolarization corresponds to the QRS-complex that hides
the atrial repolarization.
• The ventricular repolarization corresponds to the T-wave.
The use of the ECG is widespread in clinical practice because it is a non-invasive
test that provides significant information about cardiac electrical events. One of
the limitations of the ECG is that it offers simultaneous information in different
spatio-temporal scales. The main limitation of this measurement is that, in manycases, some information could be masked by other events in the heart or by theeffect of the propagation. In-silico models of the electrical activity of the heart isa powerful tool to solve this limitation, and to help in the better understandingof the ECG, because this technique permits simultaneous observation of what ishappening in each of the involved scale.
1.2.5 Acute myocardial ischemia
In previous sections, heart’s behavior under physiological conditions and its primary function of pumping blood to the body have been described. However, many
pathologies and diseases can lead to cardiac malfunctioning. From the electrical

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9

point of view, when an alteration of the normal conditions generates an abnormal
propagation of the electrical waves from the SA node to the rest of the heart, due
to the electromechanical coupling the heart would not be able to contract properly, and consequently, would not be able to pump blood to the rest of the body.
The arrhythmias are among the most important causes of sudden death. They
can be generated by two main mechanisms: abnormal automaticity and abnormal
conduction.
Injured or damaged tissue can conduct slowly or act as an obstacle (no conducting tissue) causing an abnormal anatomical conduction. Conversely, abnormal
behavior of myocardial cells associated with channelopathies or the influence of certain drugs can generate an abnormal automaticity. This could cause heart beatsthat originate not in the SA node (ectopic beats) but ones that can degenerate into
different types of arrhythmias grouped into two large families: supra ventriculararrhythmias and ventricular arrhythmias. Supra ventricular arrhythmias are associated with arrhythmic behaviour of the upper part of the heart (the atria). On thecontrary, ventricular arrhythmias are associated with the lower part of the heart,and are of two types –ventricular tachycardia or ventricular fibrillation– [18, 19].
Many pathological conditions can result in a ventricular tachycardia or fibrillation,but one of the most important perpetrators is ventricular ischemia caused by theobstruction of one of the main coronary arteries. When the flow of blood is partially interrupted by this blockage, the flow of oxygen and glucose to the muscle
cells and the washout are also interrupted within the affected region [20]. Thissituation generates mainly three effects: a reduction in oxygen (hypoxia), an increasing of the extracellular potassium (hyperkalemia), and an acidification of thetissue (acidosis). Under these effects, the AP suffers significant alterations: theexcitability of the cell and the conduction velocity in the tissue are reduced, andthe effective refractive period (ERP) is increased [20, 21]. In addition, all of thesechanges appear with a high degree of heterogeneity which makes it an important
pro-arrhythmic substrate.
Although the pro-arrhythmic mechanisms under acute ischemia have been extensively investigated, they are not yet fully understood. Many studies have attempted to shed some light on this phenomenon, primarily using animal models
instead of human ventricles, as for example Janse et al. in porcine [22] and canine
subjects [21]. Experimental studies demonstrated that, this heterogeneity (mainlyin the border between the ischemic and the healthy zones) leads to the establishment of reentries around the ischemic zone [21, 23]. When these phenomena appear
under ischemic conditions, they could separate the rhythm of the heart from theone marked by the SA node and cause it to beat in a disorderly fashion.
Myocardial ischemia is clinically recognized when a ST-segment depression isdetected in the ECG [24, 25]. However, the mechanisms behind this depression
remain under discussion and other techniques are being explored to attempt toincrease the detection capacity. Due to the complexity of the problem and to thedifferent scales involved, computer simulations of three dimensional human hearts
have been used to study this pathology to better understand the mechanisms ofreentry formations and maintenance under myocardial ischemia.

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1.3 Modeling the Electrical Activity of the Heart
Cardiac electrophyisiological models can be used to simulate the electrical behaviorof one myocyte, a fiber of cardiac cells, a tissue, or the entire heart. This wide rangeof possibilities enables their use in many different situations and for many purposes.
The primary reasons to use these models can be summarized in the following ideas:• Reduce animal experimentation: The development and validation of medical
drugs require different phases. One is the experimental validation of the drug:
before testing the drug in humans, it can first be tested in animal models.
• Increase electrophyisiological cardiac knowledge: One of the limitations of thein vivo or in vitro experiments is that only a few variables can be measured at
the same time during the experiment. The in silico simulation of the electricalactivity of the cardiac cells permits simultaneous study of all parameters ofinterest, improving understanding of the mechanisms behind a normal or apathological behavior of the cell. Given that models are limited by theirimperfections or the lack of knowledge of some parts of the cell, their proper
use must include awareness of the model limitations, consideration during
the research process, and careful use.
• Assist the decision-making process: Once a patient has been diagnosed, it isimportant to consider his specific characteristics to determine the most effective treatment. The dosage of a medical drug and the area of a tachycardic
heart to be ablated are among the many example situations in which thesimulation of the patient’s heart could result in a more accurate treatment.
1.3.1 First electrophysiological model of an excitable cell
The first action-reaction electrophysiological model was proposed by Alan L. Hodgkin
and Andrew F. Huxley and their colleagues at Cambridge [26]. This model was
possible because of their measurements of the electrical activity of the squid giant
fiber with the voltage clamp technique. With the results of the measurements,
they proposed a model in which the cell was described as an electrical circuit inwhich ionic currents pass through the ionic channels that act as resistances, andthe membrane work as a capacitor. The electrical activity of the squid giant fiber
was described using four currents: one of sodium (INa ), another of potassium (IK ),a small leakage current (Il ), that represents the current carried by other ions, andthe capacitive current. Each of these currents uses its own path or channel in thecell, and for this reason, this model is called the parallel conductance model or thechord conductance model [27]. In the original formulation of the Hodgkin-Huxley(HH) model a current from the outside to the inside was represented by a positive
current, but in this thesis the current notation that represents a current from theinside to the outside as a positive one is used [16] (see Figure 1.5).
Two forces move the ions into or out of the cell: diffusion and electrostatic
attraction. For a type of ion, when these two forces are equals in magnitude but

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Figure 1.5: The equivalent circuit of the Hodgkin-Huxley model. The voltage
sources show the polarity of the positive value. The calculated Nernst voltagesof sodium, potassium, and chloride designate the value of corresponding voltage
sources. Image from [16].
opposite, this ion is in equilibrium. The equilibrium voltage of an ion is called theNernst potential and is defined as follows [28]:
(
)
[k]i
R·T
log
(1.1)
VK = −
zk · F[k]owhere VK is Nernst potential for the ion k, R is the gas constant, T is theabsolute temperature, zk is the valence of the ion k, F is the Faraday’s constant,[k]i is the intracellular concentration of the ion k, and [k]o is the extracellularconcentration of the ion k.
The diffusion force is modeled as a voltage source in the HH model using the
Nernst potential, and the ion permeability of the membrane for the different ionsis modeled with a conductance per unit area (based on Ohm’s law) as follows:
GK =Ik
Vm − VK(1.2)

where GK is the membrane conductance per unit area for the ion k, Ik is the electric
current carried by the ion k per unit area, VK is the Nernst voltage for the ion k,
and Vm is the membrane voltage.
Hodgkin and Huxley assumed that each type of ion passes only through aspecific channel for it; they also supposed that the opening and closing of thechannels are controlled by electrically charged particles. The probability that oneof this particles is in an open state depends on the membrane voltage. So, they

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defined the ionic conductance for each channel as the product between the maximal
conductance of the channel (a constant value) and the fraction of particles in openstate, but, as is explained in [16], they did not make any assumptions on thechemical or anatomical nature of these particles.
They modeled mathematically the fraction of them in an open state with ordinary differential equations as follows:dg= αg · (1 − g) − βg · g
dt

(1.3)

where αg is the transfer rate coefficient for particles from closed to open state, βg is
the transfer rate coefficient for particles from open to closed state, g is the fractionof particles in the open state and (1 − g) is the fraction of particles in the closed
state.
Using the previous equation the steady-state values of the fraction of them inan open state and the constant time they need to reach the steady-state can beobtained:
αg
gss =
(1.4)
αg + βgτg =1
αg + βg(1.5)

During a voltage-clamp experiment, the fraction of particles in an open state
will follow this expression:g(t) = gss + (gss − g0 ) · e

− τtg

(1.6)

where g0 is the fraction of particles in an open state at the beginning of the experiment.
According to the definition proposed by Hodgkin and Huxley, each channel canhave more than one type of particles and can need more than one particle in openstate to permit to the ions to pass through itself. For example, in the HH model,
the potassium channels are formed by four particles of type n, and the sodium
ones are formed by three of type m and one of type h. The potassium and sodium
conductances in the HH model are defined as follows:
GK = GK,max · n4

(1.7)

GNa = GNa,max · m3 · h(1.8)

where GK,max and GN a,max are the maximal conductances of the potassium andsodium channel respectively.
The particles of type n and m are particles of activation. Under the resting
potential they are not in the open states and when the voltage increases, thefraction of particles in the open state increases (see Figure 1.6 and Figure 1.7).
On the contrary, the h particles are deactivation ones. When the voltage tend to

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Figure 1.6: Voltage dependence of the particles in the potassium channels of the
HH model. (A)Variation of transfer rate coefficients αn and βn , (B) Variation ofn and n4 as functions of membrane voltage. Image from [16].

Figure 1.7: Voltage dependence of the particles in the sodium channels of the HH
model. (A) αm and βm , (B) αh and βh , (C) m and h, and (D) m3 · h as a functionof membrane voltage. Image from [16].

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minus infinity all the particles are in the open state but when the voltage increases,
the fraction of particles in the open state is reduced (see Figure 1.7).
The total transmembrane current density in the HH model is the sum of thecapacitive and ionic currents as follows:
Im = Cm ·dVm+ (Vm − VNa ) · GNa + (Vm − VK ) · GK + (Vm − VK ) · GL
dt

(1.9)

where Im is the membrane current per unit area; Cm is the membrane capacitance
per unit area; Vm is the membrane voltage; VNa , VK and VK are the Nernst potential
for sodium, potassium and leakage ions respectively; and GNa , GK and GL are thesodium, potassium, and leakage conductance per unit area respectively.
When the cell is in an open circuit, there cannot be axial current since there
is no potential gradient in the axial direction. Under this situation, the previous
equation can be expressed as follows:dVm
1=−· ((Vm − VNa ) · GNa + (Vm − VK ) · GK + (Vm − VK ) · GL )dt
Cm

(1.10)

1.3.2 Evolution of electrophysiological models for cardiac cells
While the Hodgkin and Huxley equations were empirically derived, their formalism
continues to be used, and most of the electrophysiological models developed after
Hodgkin and Huxley (nerve models, cardiac models, and neuronal models, forexample) follow the formalism they proposed.
Ten years after the HH model, Denis Noble proposed the first model of cardiac cell in which he adapted the HH model to represent the electrophysiologicalbehavior of a Purkinje cell [29]. The first cardiac electrophysiological models weredeveloped to explain the main differences between the heart cells and the nervous cells: the AP duration [30]. Soon, these models became more complex byincorporating the different ionic currents that were being discovered. Initially, thepotassium current was separated into two new currents: IK1 and IK2 [31–33]. Later,
Sanguinetti and Jurkiewicz discovered that the IK2 current was made up of two
components: IKr and IKs [34]. However, it was not until 1964 that voltage-clamp
experiments allowed the discovery of the calcium current [35]. Step by step, otherelements were incorporated to the models as the sodium-calcium exchanger (Incx )and the homeostasis of the intracellular concentrations [36].
Similarly, the models were adapted to different cell types: ventricle [37], atrium[38]; and different species: rabbit [38, 39], guinea pig [40, 41], dog [42, 43], andhuman [44–47], for example.
One of the possibilities of cellular action potential models is that they can beincorporated into cellular network models to simulate cardiac tissue [48–51] andstudy the behavior of the entire organ under pathological situations or before theeffect of drugs.

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1.3.3 Human ventricular electrophysiological models: main ionic currents

An electrophysiological cardiac model describes the membrane potential of a cardiac cell and is composed of different currents that pass through the cell membrane.
As described for the HH model, these currents are generated by the passage of ionsthrough different channels, exchangers, and pumps that connect the intracellular
and the extracellular media. Each current is formulated mathematically to represent the various effects that occur in the channels (for example, activation andinactivation of ion channels) and these models also include the balance between
the intracellular and extracellular ionic concentrations, and between the different
compartments of the miocyte. In the cases of the human ventricular models, themost important ions for the modeling of the AP are: sodium (N a+ ), potassium(K + ), and calcium (Ca2+ ). While there are many differences between the existing
human ventricular cell models, the main currents are present in all of them. In thefollowing subsections, the principal ones are described.
Sodium current
The fast sodium current (INa ) is the main sodium current in human ventricular
cells. There exists also the late sodium current (INaL ) but this current plays arelevant role only under pathological conditions (for example, LQT3 or HF [52]).
INa is mainly modeled following the formalism proposed by Hodgkin and Huxley,with three activation gates of type m and two inactivation gates (j and h).
Potassium currents
Four potassium currents are mostly present in the human ventricular cell models. Three correspond to voltage-gated ion channels (Kv ), whereas the fourth isan inward rectifying current. Kv channels are the most diverse super-family of
ion channels and have a significant impact on AP shape and duration. These differences are related to the cell type and function, but also to differences among
various species [52]. These differences make it important to use data from humancardiomyocytes for the characterization of these currents.
Transient outward potassium current (Ito ): This voltage gated current activates and inactivates rapidly during the depolarization phase of the action potential. It can be divided into two components (Ito,f and Ito,s ).Both components
activate rapidly following membrane depolarization. On the contrary, Ito,f inactivates rapidly whereas Ito,s inactivates slowly (compared to Ito,f ). While not all of
the models divide the current into two components, all define it with two voltagedependent gates (one for activation and another one for inactivation). The majordifference between models lies in how they model the differences between the ventricular cell types: by varying the conductance, by modifying the proportion of thefast and the slow component, or both together.

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Rapidly activation potassium current (IKr ): This current activates rapidly upon
depolarization [53]. The activation occurs faster than inactivation (τact > 100ms,
τinact < 20ms) [52]. In all of the recent human ventricular cell models, this current
is modeled with two voltage-dependent gates (one for activation and another forinactivation) and uses the Nernst potential following the HH formalism.
Slowly activation potassium current (IKs ): This slow component of the total
potassium current is much smaller than IKr and plays an less significant role in
the APD [54]. When this current is blocked, and in the absence of sympathetic
stimulation, the role played by IKs is quite small. However, when the ventricular
muscle repolarization reserve is attenuated, the role of this current in limiting AP
prolongation becomes significant [55]. This current is generally modeled using the
Nernst potential and with two voltage-dependent activation gates. In most partsof the models, the two voltage-dependent gates are the same.
Inward rectifier potassium current (IK1 ): This current is present in all ventricularand atrial myocytes and plays a major role in the action potential repolarization[56]. It is usually modeled with one voltage-dependent and time-independent gate,
and the formulation used for this gate also makes use of the Nernst potential
following the HH formalism.
Calcium current
The L-type calcium current (ICaL ) is the principal current for the entry of calcium inthe cardiac cells and it is typically modeled using the Goldman-Hodgkin-Katz equation. This current plays an important role during the plateau phase of the action
potential, in the action potential duration, and in the shape of the repolarization.
The L-type calcium channels are frequently modeled by a voltage-dependent
activation gate (d), by one or more voltage-dependent inactivation gates (f , f 2),
and by calcium-dependent inactivation gates.
Pumps and exchangers
The previous currents modeled the behavior of ionic channels, through which theions pass without needing energy expenditure. However, there are also currents inthe cell caused by the movement of ions against gradient that require consuming
ATP (pumps), or by using the force of ions that pass in the direction of the gradient(exchangers). Two of these types of currents are present in most human ventricular cell models: the sodium-potassium pump current and the sodium-calcium
exchanger current.
Sodium-potassium pump(INaK ): The sodium-potassium pump current is the principal mechanism for active ion transport across the membrane of excitable cells[57]. In addition, it plays an important role in regulating cardiac electrophysiology

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under physiological and pathological conditions. Two schemes have become predominant for modeling this current: the first one proposed by Di Francesco andNoble in their seminal work [36], and the second one corresponding to it [58]. The
DiFrancesco–Noble formulation is mostly present in the majority of the models forthe specialized conduction system (sinoatrial node and Purkinje fiber models), and,on the contrary, the Luo–Rudy formulation results predominate in mathematicalmodels of atrial and ventricular cardiac myocytes.
Sodium-calcium exchanger current (Incx ): The sodium-calcium exchanger plays
an important role in the regulation of intracellular calcium concentration ([Ca2+ ]i ).
This exchanger regulates muscle relaxation in cardiac myocytes [59]. This current isusually modeled following the formulation proposed by Mullins [60] and simplified
by DiFrancesco and Noble [36]. In this formulation, the sodium calcium exchanger
current is modeled as a time-independent current, with the value of the current
being dependent on the intracellular and the extracellular concentration of N a+
and Ca2+ ions, and on the membrane potential.
1.3.4 Human ventricular electrophysiological models: recent developments
In recent years a number of mathematical models of electrical and ionic homeostasis
in human ventricular myocytes have been proposed. One of the most extensively
used models is that proposed by ten Tusscher-Panfilov (TP06) [51], which is an
improved version of the ten Tusscher-Nobel-Nobel-Panfilov (TNNP04) model [46],in which the calcium dynamics, the slow delayed rectifier potassium current (IKs )and the L-type calcium current (ICaL ) were reformulated. Both the TNNP04 andthe TP06 models are based on experimental human data for most of the main ioniccurrents. One of the major advantages of the TP06 model over previous humanventricular models is that it accurately reproduces restitution of the AP duration(APD). Conversely, the TP06 model presents the major shortcomings of being
insensitive to changes in certain ionic current densities [61] and of inadequately
representing rate dependence of intracellular calcium levels [12].
Another relevant model of human ventricular cells was proposed by Iyer et al.
(IMW04) [45]. This model provides a detailed description of calcium homeostasis
and reproduces diverse aspects of the excitation-contraction coupling (ECC). However, important limitations have been reported for this model, including the factthat it is extremely sensitive to variations in the sodium potassium pump (INaK )and to the inward rectifier potassium (IK1 ) current densities [61]. It also produces
quite long APs and relatively flat restitution curves in tissue [62]. In addition, themost relevant ionic currents in the IMW04 model are described using Markovian
chains, with the consequent increase in complexity (more than 60 state variables),which imposes serious restrictions on the size of the time step required for stability and increases the overall computational time when using this model in tissue
simulations [62].
At the beginning of this thesis, a new model of human ventricular AP was proposed by Grandi et al. (GPB) [15]. The development of this model departs from

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the rabbit ECC model proposed by Shannon et al. [63], which includes the subsarcolemmal and junctional compartments in the formulation of the currents andprovides a detailed description of calcium handling. The GPB model includes new
definitions of ionic current densities and kinetics according to recent experimental data on human myocytes. This model is defined for two types of cells, namely
endocardial and epicardial. These two cell types differ only by the maximal conductivity of the fast and slow components of the transient outward potassium current(Ito ). The GPB model improves the AP response to frequency changes and offers a better performance against blockades of potassium currents (IK1 , IKs and therapid delayed rectifier potassium current IKr ) with respect to the TP06 model. The
major drawback of the GPB model is that it does not adequately reproduce S1S2
restitution properties nor APD rate adaptation dynamics. These drawbacks can be
attributed to the fact that the GPB model was developed from a rabbit ventricularmodel, and the characteristics of S1S2 restitution and APD rate adaptation arenotably different in the two species.
Parallel to the development of this thesis, O’Hara et al. proposed the O’HaraRudy-dynamic (ORd) model [64]. This new ventricular cell model is based on
the most extended data set of healthy human cells. Data from more than 100
healthy human hearts were used to examine the steady state rate dependence, andrestitution of the ventricular AP. The model was also studied under the effect ofdrugs that selectively block different channels (for example, mexiletine, ryanodine,
nisoldipine, and E-4031). As it will be discussed later, the complexity of thismodel is greater than that of previous models. The major drawback of the ORdmodel is the rate dependence of the intracellular calcium ([Ca2+ ]i ): While underphysiological conditions [Ca2+ ]i increases about a 30% with frequency, in the ORdmodel it increases about 300%.
In 2015, Himeno et al. proposed a new human ventricular cell model [65]. As
opposed to the previously described models, the Himeno et al. model included
a description of excitation-contraction. The model comparison performed in thepresent study was restricted to purely electrophysiological models.
Table 1.1 presents a summary with the main differences among the four models
described above, i.e. TP06, IMW04, GPB and ORd.
1.3.5 The Grandi-Pasqualini-Bers model
When this thesis was started, the most detailed and recent human ventricularcell model published in the literature was the GPB model. Since the objective ofthe thesis was to computational modeling and simulation for the investigation ofventricular arrhythmias under ischemic conditions, the GPB model was taken as astarting point. After performing an in-depth evaluation on the ability of this model
to represent experimental behavior in relation to a number of electrophysiologicalproperties, the necessity to update the GPB model for the study of arrhythmias was
identified. In this thesis different methodologies for the development and validationof an electrophysiological cardiac model are introduced and applied to update the
GPB model. In this section a detailed description of the GPB model is presented.

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TP06♯ State 20
variables
Computation Short
time
Cell types Endo, mid,
epi
Steady-state PhysiologicalAP properties APD values,
very low triangulation
Steady-state Low diastoliccalcium levels and systolicvalues
APD90 adap- Very fast
tation toabrupt heart
rate changes
Sodium and Very sensitive
calcium rate to increasing
dependence stimulation
frequency
APD sen- Lowly sensisitivity to tive to INaK ,variations INa,Ca , IKs , IK1 ,in ionic cur- ICaL , IKr andrent densities INa

IMW04
67

GPB
38

ORd
41

Very long

Long

Long

Epi

Endo, epi

Physiologicalbehavior

Physiologicalbehavior

Endo, mid,epi
Physiological Physiological Short APD
APD values
APD values
values, but
and triangula- and triangula- physiological
tion
tion
triangulation
Low diastolic Very low diLow diastolicand systolic
astolic andand systolicvaluessystolic values values
Very slow
Rabbit charPhysiologicalacteristics
behavior

Highly sensitive to INaK
and IK1

Very sensitive
to increasing
stimulation
frequency
Lowly sensiLowly sensitive to IKr and tive to IK1 ,highly sensiICaL and INa,Ca ,tive to IK1and highly
sensitive to IKrTable 1.1: Differences between human ventricular cell models. Data are obtained
from [12, 61, 62, 64] and from simulations performed in this thesis.

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The GPB model describes the AP of endocardial and epicardial human myocytes. For both cell types the total transmembrane ionic current, Iion , is given
by:Iion = INatot + ICltot + ICatot + IK tot ,where each of the individual currents denotes the current generated by the passage
of sodium, chlorine, calcium, and potassium ions, respectively, through the cell
membrane.
Sodium current. Following the approach of Shannon et al. [63], the sodium
current was divided into two components: i) a subsarcolemmal component, andii) a junctional component, which correspond to the cell compartments where thecurrent is generated. Both components are formed by the same subcurrents:
INatotjunc = INajunc + INa,bkjunc + 3 · Incxjunc + 3 · INaK junc + ICaNajunc
INatotsl = INasl + INa,bksl + 3 · Incxsl + 3 · INaK sl + ICaNasl
INatot = INatotjunc + INatotsl
where INa is the fast sodium current, INa,bk is the background sodium current, Incx
is the current of the sodium-calcium exchanger, INaK is the current of the sodiumpotassium pump, and ICaNa is the sodium current through the L-type calcium channels.
Chlorine current. The chlorine current, which is not present in other previous
model as in TP06 or TP04. In the GPB models this current is described by:
ICltot = IClCa + ICl,bk
where IClCa is the calcium-activated chlorine current, and ICl,bk is the backgroundchlorine current.
Calcium current. The calcium current, as in the case of the sodium current,
has two components, junctional and subsarcolemmal:
ICatotjunc = ICajunc + ICa,bkjunc + IpCajunc − 2 · Incxjunc
ICatotsl = ICasl + ICa,bksl + IpCasl − 2 · Incxsl
ICatot = ICatotjunc + ICatotsl
where ICa is the calcium current through the L-type calcium channels, ICa,bk is the
background calcium current, IpCa is the sarcolemmal calcium pump current, andIncx is the current of the sodium-calcium exchanger.
Potassium current. The potassium current is defined as follows:
IK tot = Ito + IKr + IKs + IK1 − 2 · INaK + ICaK + IKp
where Ito is the transient outward potassium current (decomposed into a fast
and a slow component), IKr is the rapidly activating potassium current, IKs is
the slowly activating potassium current (decomposed into junctional and subsarcolemmal components), IK1 is the inward rectifier potassium current, INaK is the

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current of the sodium-potassium pump (decomposed into junctional and subsarcolemmal components), ICaK is the potassium current through the L-type calcium
channels (decomposed into junctional and subsarcolemmal components), and IKp
is the plateau potassium current. In the GPB model, differentiation between endocardial and epicardial cells is set by assigning different contributions to the fastand slow Ito components.
1.3.6 Acute ischemia models
Over the past 20 years, cardiac ischemia has been studied using mathematicalmodeling and computer simulations. These techniques have become a useful toolin analyzing the electrophysiology of the myocardial arrhythmias [66]. In one of thefirst works by Ferrero et al. [67], the authors studied how the metabolic changes
produced by the ischemia in the cell affect the electrophysiological behavior of the
cardiac tissue. This study was limited by the utilization of two dimensional tissues
and this limitation was also present in further studies [68]. Following these results,
different works have extended this concept, accounting for heterogeneity caused byischemia [69], realistic fiber orientation and transmural heterogeneity [70, 71], andhow reduced repolarization increases arrhythmic risk in the heterogeneous substratecaused by acute myocardial ischemia [72].
At the cell level, AP under myocardial ischemia is modeled adding the threemain effects to the electrophysiological cardiac models: hyperkalemia, hypoxia andacidosis. The hyperkalemia is simulated by increasing the extracellular potassium
to levels around 10 mM as it has been experimentally reported [73]. The hypoxia issimulated by adding the ATP-sensitivity [K + ]i current (IK ATP ), a current discoveredby Noma [74]. When distribution of oxygen is reduced, the cell is not able togenerate ATP from ADP. Due to this situation, ATP concentration is reducedin the cell and IK ATP channels open. Finally, acidosis is usually modeled by areduction of the ionic conductances of the INa and the ICaL currents according to
the experimental results of [67, 75]. While all three of these effects are present
during a cardiac ischemia, the hyperkalemia plays the most important role.
For the simulation of the ischemia, different scales need to be considered whenthe myocardial it is studied because some of the electrophysiological consequencesof ventricular ischemia must be analyzed in tissue or in the entire organ. For example, how the effective refractory period increase or how the conduction velocity
decreases under ischemic conditions must be studied at least in a fiber of ventricular myocytes. Another example is the study of reentrant patterns and vulnerable
windows in acutely-ischemic ventricle which must be analyzed on the organ level.
In this case, an ischemic region is placed in a bi-ventricle model with fiber orientation, and, according to the experimental findings from Coronel et al. [73], theischemic region is composed of a normal zone (NZ), transitional border zones (BZ),
and the central zone (CZ) of ischemia. The BZ corresponds to a linear variation inelectrophysiological properties between the NZ and the CZ as shown in experiments[73]. For this simulations, an important parameter in the simulation of ventricularischemia is the washed zone (not affected by ischemia) in the endocardium as a re-

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22

sult of the interaction between the blood inside of the ventricle and the endocardial
tissue [23].
In this complex scenario, the AP model represents the corner stone. A good
AP model that is able of reproducing the cardiac behavior under arrhythmias is a
key element for the in silico cardiac modeling.
1.4 Objectives and Outline of the Thesis
1.4.1 Aim of the thesis
The main objective of this thesis is the proposal of novel methodologies to improve development and validation of cardiac computational models. In particular,
this thesis focused on building a human ventricular cell model able to represent
experimental evidences regarding a range of electrophysiological properties, which
included well-established markers of arrhythmic risk. The developed model could
then be used to investigate ventricular arrhythmias and, in particular, those arisingunder conditions of hyperkalemia, a component of myocardial ischemia.
As described at the beginning of this general introduction, mathematical modeling and simulation of cardiac electrical activity provides an ideal framework where
to join clinical data and experimental observations to gain understanding of cardiac electrophysiology and underlying mechanisms. The use of cardiac modelingand simulation in combination with other techniques, such as signal processing ofcardiac electrical signals, can be a powerful tool to aid in the prevention of cardiac
arrhythmias as well as in the improvement of their diagnosis and therapy.
To this end, it is very important to have cardiac computational models able
to reproduce available clinical and experimental evidences. Cardiac action potential models are a key element in the development of whole-heart in silico models.
The more faithfully the action potential model represents experimental behavior,
the more useful it will be for investigation of cardiac electrical activity and thegeneration of cardiac arrhythmias.
1.4.2 Specific objectives
The general objectives of this thesis are:
• To develop new methodologies for the development and validation of electrophysiological models: During the development of a cardiac model, many
simplifications are required to be able to isolate parts of the model and to adjust the parameters of this part to the experimentally measured values. The
increase in computational capacity permits consideration of more elementsin the development process and could help reduce these simplifications. In
particular, the following situations have been considered in this thesis:– Considering the entire model of an ionic channel when the parametersof the channel are adjusted: The usual process to model the behaviorof an ionic channel involves in the measurement of different elements ofthe model considering them independent of each other because of the

Universidad Zaragoza

1. Introduction

23

experimental protocol used for the measurement. While in many cases
this approximation could be valid, with the increasing complexity ofthe models, the interaction between the different elements of the model
also increases. For this reason, it is necessary to consider the possible
interactions of the different elements of the model in the experimental
measurements.
– Considering simultaneously all of the ionic conductances when the modelis adjusted: One of the most important parts in the development of a
cardiac model is the selection of the value for the ionic conductances.
This action is notably difficult because ionic conductances cannot bemeasured directly. To select each value, they are usually considered
independently to adjust some electrophysiological properties. One of
the limitations of using this adjustment technique of one-at-a-time isthat the interactions or the cross effects are neglected. For this reason,
one of the objectives of this thesis is to develop techniques that allow
selecting the cardiac ionic conductances by modifying them at the sametime and considering all of the interactions or the cross effects.
– Considering the differences between the simulations in isolated cell andin tissue to fit as closely as possible to the experiments: Due to thesensitivity of some ionic currents to the process of cell isolation, theexperiments in cardiac cells are usually performed using small piecesof tissue. Nevertheless, due to computational reasons, most of the adjustments in cardiac models are performed using simulations at the celllevel. One of the objectives of this thesis is to study how this simplification could affect the development of a cardiac model and how it could
be overcome.
• To apply all of these methodologies for the development of a human ventricular action potential model: In recent years, several models of the humanventricular action potential have been proposed, because of the increase inavailable data. However, while the available data increases the models arelimited in representing certain physiological situations. The objective of thisthesis is to incorporate the new technique to the modification of a human ventricular AP model to try to approximate its behavior, as much as possible,
to what has been observed experimentally.
1.4.3 Outline of the thesis
The thesis is organized as follows:• Chapter 2 describes an initial analysis of the GPB model and a first update ofit. The GPB model was analyzed for the purpose of investigating ventriculararrhythmias occurring in the context of hyperkalemia, a component of myocardial ischemia. After evaluating the model, the necessity to modify it tosolve issues related to different electrophysiological properties, like APD rate
adaptation to abrupt frequency changes among others, was identified. The

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update of the GPB model, denoted as the CRLP model, was first presented
at the CinC Conference, while a full analysis of the updated model, including
evaluation of its performance under hyperkalaemic conditions, was publishedin a scientific journal:– J. Carro, J. F. Rodríguez Matas, P. Laguna, E. Pueyo. Analysis andImprovement of a Human Ventricular Cell Model for Investigation of
Cardiac Arrhythmias. XXXVII International Conference on Computers
in Cardiology, Belfast (UK), pp. 817 - 820 (Sep, 2010).
– J. Carro, J. F. Rodríguez, P. Laguna, E. Pueyo. A Human Ventricular Cell Model for Investigation of Cardiac Arrhythmias under Hyperkalaemic Conditions. Phil Trans A, n. 639, pp. 4205 - 4232 (Nov,
2011).
• Chapter 3 presents an algorithm to adjust the ionic conductances of a cardiac computational model by considering electrophysiological markers as constraints. The need for such an algorithm came from the fact that, after evaluation of the CRLP model developed in chapter 2, additional characteristics
were determined to be necessary for investigation of cardiac arrhythmias.
Specifically, in this chapter intracellular potassium dynamics were introduced
into the CRLP model developed in chapter 2 to investigate arrhythmias arising in the context of hyperkalemic conditions. To avoid the imbalance inthe total potassium current, a new algorithm was proposed to adjust ioniccurrent conductances of the updated computational model in such a way that
representation of experimentally measured electrophysiological markers is either improved or at least maintained with respect to the model before theupdate. The initial results of this chapter were presented at a scientific conference. The final algorithm with illustration of its application to the studyof conditions involving large changes in potassium concentration has been
accepted to be published in an international journal:– J. Carro, J. F. Rodríguez Matas, and E. Pueyo, A Response Surface
Optimization approach to adjust ionic current conductances of cardiac
electrophysiological models, in Gordon Research Conference in Cardiac
Arrhythmia Mechanisms, Lucca, Italy, 2015.
– J. Carro, E. Pueyo, J. F. Rodríguez. A response surface optimization
approach to adjust ionic current conductances of cardiac electrophysiological models. Application to the study of potassium level changes.
PLOS One (Accepted).
• In Chapter 4, limitations in the development and validation of electrophysiological models caused by the differences between the experimental protocols
and the simulations are analyzed. In the previous chapters of this thesis,
for the development and validation of the new versions of the model available experimental measures were readily used to adjust model parameters or
to compare with simulation results. To evaluate whether this is a suitable

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25

approximation, two scenarios involving scales ranging from ion channels totissue were studied. This study allowed us to determine when it is possible touse them directly and when more complicated simulations are required. The
results of this chapter at the ionic level were presented at an international
conference, and the complete study evaluating both scenarios was publishedin a scientific journal:– J. Carro, J. F. Rodríguez-Matas, V. Monasterio, and E. Pueyo. Limitations in electrophysiological model development and validation causedby differences between simulations and experimental protocols. Progress
in Biophysics and Molecular Biology, vol 129, pp. 53 - 64, (Oct, 2017).
– J. Carro, J. F. Rodríguez Matas, E. Pueyo. In Silico Simulations ofExperimental Protocols for Cardiac Modeling. XXXVI Annual International Conference of the IEEE Engineering in Medicine and Biology
Society, Chicago (USA), pp. 5695 - 5698 (Aug, 2014).
• In Chapter 5, a modification of the calcium current is proposed to improve
the shape of the AP before the plateau phase. Based on the results obtainedin Chapter 4, rather than directly comparing experimental and model characterizations, in silico simulation of the experimental protocol was used toadjust the new version of the model. To do that, the algorithm proposed in
Chapter 3 was adapted to update model expressions at the ionic level. In
addition, a new validation process considering the entire cell was described.
The obtained results were presented at two international conferences:– J. Carro, J. F. Rodríguez Matas, and E. Pueyo, A Methodology to
Improve Human Ventricular Models for the Investigation of Cardiac
Arrhythmias, in 61st Annual Meeting of the Biophysical Society, New
Orleans, USA, 2017.
– J. Carro, J. F. Rodríguez Matas, and E. Pueyo, Multiscale Methods
for Definition of Ionic Variables in Electrophysiological Models, in European Medical and Biological Engineering Conference (EMBEC) Tampere, Finland, 2017.
• Chapter 6 presents the final update of the model. All the modifications to
the CRLP model proposed in chapter 2, as described in Chapters 3 and 5,
were considered and the optimization algorithm proposed in Chapter 3 wasused to adjust ionic current conductances and calcium handling characteristics. In this chapter the optimization function was redefined and a large
number of experimental datasets recorded from healthy human ventricular
cells or tissues were used. Based on the results obtained in Chapter 4, only
those electrophysiological markers whose values are comparable when evaluated in simulated single cells and tissues were considered, as some of theexperimental data had been recorded in tissue and simulations were run incells. Additional electrophysiological markers were used for validation purposes by comparing experimental data with tissue simulation results. The

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results of this chapter are part of a paper that is in preparation and will be
submitted to an international journal.
• Finally, Chapter 7 presents the most important conclusions of the thesis anddiscusses possible future research lines.

Universidad Zaragoza

Chapter 2

A New Paradigm forElectrophysiologicalModel Development
2.1 Introduction
Despite the fact that the human ventricular AP models currently available in theliterature are able to reproduce certain electrophysiological properties observed experimentally, they fail in representing other relevant aspects, as pointed out in
Chapter1. To obtain a model suitable for the study of human ventricular electrophysiology, with a focus on the investigation of ventricular arrhythmias, in thischapter a human ventricular AP model was developed departing from the GPBmodel. The methodology proposed for model development was based on the computation of well-established electrophysiological markers and the results of a sensitivity analysis of the modulation of those markers by variations in the characteristics
of major ionic currents. Using that methodology key limitations of the GPB model
were unraveled and inadequacies in the definitions of specific ionic current characteristics were identified. Based on those findings, gating kinetics of the ICaL current
were reformulated by introducing both fast and slow voltage-dependent inactivation
gates [51], maximal INaK was reduced to best match experimental APD90 rate adaptation data, and the IK1 current was redefined by accounting for recent experimental
data published in the literature [76]. The new developed model, herein denoted by
CRLP, was tested and compared with the original GPB model and with the widelyused TP06 model. Model comparison was based on twelve cellular electrophysiological markers computed using five different stimulation protocols, as in [12].
Our results show that the introduced modifications brought most markers into thephysiological range, and considerably improved others with respect to the GPB and

27

New Methodologies for the Development
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28

TP06 models. In addition, the CRLP model was tested by simulating hyperkalemic
conditions in tissue. Under those conditions, potassium starts to pour out of ischemic myocardial cells causing depolarization of the myocardium at rest, slowing
conduction and decreasing excitability. Recovery of excitability is known to outlast repolarization, a phenomenon termed post-repolarization refractoriness. Our
results show that under hyperkalemia APD90 progressively decreased with the levelof extracellular potassium, [K + ]o , whereas the effective refractory period (ERP)
increased after a threshold in the level of [K + ]o was surpassed. Conduction velocity (CV) decreased with [K + ]o and conduction was blocked above [K + ]o = 10.4
mM. Additionally, APD90 alternans appeared for [K + ]o > 9.8 mM. These results
suggest that the model is suitable to represent areas of inhomogeneities of resting
potential and dispersion in recovery of excitability corresponding to different hyperkalemic levels within the tissue. Those areas of inhomogeneities could provide
a setting for unidirectional block that could become the substrate for arrhythmias
in ischemic hearts.
2.2 Methods
Electrophysiological markers were used in this study to guide the modifications
applied onto the GPB model to generate the proposed CRLP model. Specifically,markers related to S1S2 restitution and APD90 rate adaptation dynamics wereidentified as not being adequately reproduced by the GPB model and appropriate
modifications were performed. Those and other markers were also used to evaluate
the performance of the CRLP model and to compare it with that of the TP06 andGPB models.
In the following subsections, the cellular electrophysiological markers used inthis study are presented, and the protocols used to compute them are described.
2.2.1 Selected markers• APD90 : APD was traditionally considered as the main preclinical marker ofdrug cardiotoxicity. APD prolongation was linked to long QT syndrome andincreased risk of developing Torsades de Pointes [77, 78]. APD90 was used inthis study to denote AP duration at 90% repolarization (Figure 2.1.a)).
• Triangulation: This marker quantifies the shape of the final part of the AP
and was calculated as the difference between APD at 50% and 90% repolarization (Figure 2.1.a)). Low triangulation values indicate square APs, whilehigh values indicate triangular APs. Triangulation was proposed as a marker
of pro-arrhythmia [77], with long APD90 values without triangular APs considered as anti-arrhythmic, and triangular APs considered as pro-arrhythmic
in general.
• Systolic and diastolic intracellular calcium ([Ca2+ ]i ) levels: Calcium transient
properties evaluated at different frequencies was reported as risk markers

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development

29[79, 80]. In this study, diastolic (level at rest) and systolic (level at peak)calcium transient levels were evaluated at 0.5 Hz and 1 Hz (Figure 2.1.b)).
• Time constants of APD90 adaptation to abrupt changes in CL: Adaptationof ventricular repolarization duration to abrupt changes in cycle length (CL)
was proposed as an arrhythmic risk marker [81]. In this study, the dynamics
of APD90 adaptation to abrupt changes in CL were fitted by two exponentials
with associated time constants τf ast and τslow (Figure 2.1.c)), following themethodology proposed in [13].
• Maximum slope of APD90 restitution curves: The maximum slope of the
APD90 restitution (APDR) curve was suggested as an arrhythmic risk marker[82, 83]. The APDR curve describes the relationship between APD90 and thediastolic interval (DI) (Figures 2.1.d)-e)). In this study APDR was evaluatedusing the S1S2 and the dynamic protocols (see Section 2.2.2).
• Rate dependence of N a+ and Ca2+ concentrations: The importance of intracellular sodium, [N a+ ]i , and intracellular calcium, [Ca2+ ]i , dynamics inarrhythmogenesis was extensively reported [84, 85]. In this study, systolic[N a+ ]i and [Ca2+ ]i levels were evaluated for increasing stimulation frequencies (Figure 2.1.f)).
2.2.2 Stimulation Protocols
Five stimulation protocols were applied to evaluate the electrophysiological markers
presented in section 2.2.
• Steady-state protocol: A train of 3,000 stimulation pulses was delivered andsteady-state was guaranteed to be reached. Simulations were carried out
using two different CL values (1,000 and 2,000 ms). The evaluated markers
were: APD90 at 1 Hz, Triangulation at 1 Hz, Systolic [Ca2+ ]i at 0.5 and 1
Hz, and Diastolic [Ca2+ ]i at 0.5 and 1 Hz (Figures 2.1.a) and 2.1.b)).
• Abrupt changes in CL protocol: The cell was stimulated with a CL of 1,000
ms for 8 minutes, then with a CL of 600 ms for additional 8 minutes andfinally with a CL of 1,000 ms for 8 minutes (Figure 2.1.c)). The APD90
dynamics after the first abrupt CL change were fitted by two exponentials
with time constants τf ast and τslow . APD90 adaptation dynamics after thesecond abrupt CL change provided very similar results to those obtained forthe first CL change, in accordance with [12, 13], and were not reported.
• S1S2 protocol: A train of 10 basic stimulations (S1) was applied to the cellor tissue at a CL of 1,000 ms followed by an extra stimulus (S2) delivered at
coupling intervals (CI) ranging from 1,000 to 250 ms in variable steps (100 ms
for DI>300 ms, 5 ms for 150<DI<300 ms, and 1 ms for 5<DI<150 ms). The
S1S2 restitution curve was obtained by plotting the APD90 corresponding to
the extra stimulus versus the previous DI for each of the tested CI values(Figure 2.1.d)).

JesúsCarroNew Methodologies for the Development
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30

b)

APD90
0

100

200 300 400
Time (ms)

0

500

d)

0.2
0

200
400
Ti me (ms)

AP D 9 0 (ms)

280
260
240
0

200

400
600
D I (ms)

800

τslow

280

0

5

10
15
Time (min)20f)
D y nami c

300

τfast

300

260

600

e)
S1S2

AP D 9 0 (ms)

APD90 (ms)

−50

0.4

APD Rate Adaptation

300
250
200
0

200

400
600
D I (ms)

800

Sy stol i c [ C a 2 +] i (%)

APD50

sy s [ C a 2 +] i

Triangulation

0

0.6

di a [ C a 2 +] i

[ C a 2 +] i (µ M)

Vm (mV)50c)

Ste ady -State [ C a 2 +] i P rop e rti e s

Steady-State AP ShapeRate D e p e nde nc e
140

max i mum

a)

120
100
0.5

1 1.5 2 2.5
Fre q ue nc y (Hz )3

Figure 2.1: Graphical description (using simulation results) of the markers ofarrhythmic risk used in this study: a) Steady-state AP properties (APD90 , APtriangulation); b) Steady-state [Ca2+ ]i properties (systolic ‘sys’ and diastolic ‘dia’
values); c) APD90 daptation to abrupt changes in CL (from 1,000 ms to 600 ms
and again to 1,000 ms), characterized by time constants τf ast and τslow ; d) APDR
curve computed using the S1S2 protocol; e) APDR curve computed using the
dynamic protocol; f) Normalized rate dependence of systolic concentrations, withnormalization performed with respect to the lowest tested frequency, i.e. 0.25 Hz.
For a better understanding of the protocols used to compute the markers shown inpanels a)-f), see the description provided in section 2.2.2.

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development

31• Dynamic protocol: In this protocol 100 stimulations were applied at a given
CL, with the CL ranging from 1,000 to 50 ms. The APDR curve was obtainedby plotting the steady-state APD90 (corresponding to the 100th stimulus ofeach CL) versus the corresponding steady-state DI (Figure 2.1.e)).
• Rate dependence protocol: The systolic values of both [N a+ ]i and [Ca2+ ]iconcentrations were measured at different frequencies (0.25, 0.5, 1, 1.5, 2, 2.5
and 3 Hz) and normalized to the level of the minimum frequency. For eachfrequency, the cell was stimulated during 10 minutes to reach steady-stateand [N a+ ]i and [Ca2+ ]i were measured at the end of the 10-minute period.
The maximum normalized concentration value was computed and used as amarker (Figure 2.1.f)).
2.2.3 The CRLP Model
Taking the GPB model as a starting point, the CRLP model was built by reformulating two ionic currents (ICaL and IK1 ) and redefining a number of model
parameters, including GNa (maximum INa conductance) and GNaK (maximal INaK
value). Those changes were motivated by the possibility of improving the performance exhibited by the GPB model on two particular aspects: i) the shapeand maximum slope of the S1S2 APD90 restitution curve, and ii) the response of
the APD90 to abrupt changes in heart rate (see below). The failure of the GPBmodel in reproducing experimental data on those two aspects could be attributed
to an inheritance from the Shannon rabbit ventricular model from which the GPBmodel was developed. In this regard, the development of the CRLP model fromthe GPB model was made by considering human data only [61]. The following
paragraphs describe the introduced modifications, as well as the data and criteria
used in the development process. A full description of the CRLP model can befound in Appendix B.
To keep as much consistency as possible with the original GPB model, the
CRLP model was developed for the same type of myocardial cells as the GPBmodel, namely endocardial and epicardial. Differences between the two cell types
were based on the definition of the Ito fast and slow components, in the same way
as previously described for the GPB model.
L-Type Calcium Current (ICaL ). To obtain an APD90 adaptation response toabrupt changes in CL in better concordance with experimental data, the L-type
calcium current was reformulated. The voltage-dependent inactivation gate f of the
GPB model was replaced with the product of a fast, f , and a slow, f2 , inactivation
gates, as in [51]. The new equations for the ICaL current in the CRLP model were:()
ICajunc = FjuncCaL · I¯Caj · d · f · f2 · 1 − fCaBj
ICasl = FslCaL · I¯Casl · d · f · f2 · (1 − fCaBsl )()
ICaNajunc = FjuncCaL · I¯Naj · d · f · f2 · 1 − fCaBj
ICaNasl = FslCaL · I¯Nasl · d · f · f2 · (1 − fCaBsl )JesúsCarroNew Methodologies for the Development
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(
)
(
)
ICaK = I¯K · d · f · f2 · FjuncCaL · 1 − fCaBj + FslCaL · (1 − fCaBsl )ICa = ICajunc + ICasl
ICaNa = ICaNajunc + ICaNasl
ICaL = ICa + ICaK + ICaNa
As indicated in [61], the replacement of one current component by anotherone is not a simple task. Splitting the f gate in the GPB model into two newgates required the readjustment of the time constants τf2 associated with the slow
voltage-dependent inactivation gate f2 and the time constant τd associated withthe activation gate d in order to obtain a meaningful ICaL current-voltage curve(splitting the f gate without readjusting the time constants would have led to a
current-voltage curve having two relative minima).
Readjustment of τf2 was based on available experimental data [2, 4–7]. Figure 2.2.a) depicts τf and τf2 along with experimental data [2, 4–7]. Since theexperimental data does not cover the entire voltage range, the readjustment was
performed by best fitting the tail of the τf2 curve while obtaining a current-voltage
curve within the range of experimental data. The other voltage-dependent inactivation time constant, τf , was kept as in [51]. The equations describing f and f2
were:
fss =

1
Vm +2071+e

αf = 1102.5 · e−(
βf =
γf =

Vm +27
15

)

2

200
1+e
180

13−Vm
10

Vm +30 + 20
1 + e 10
τf = αf + βf + γfdf
fss − f=dt
τf

f2ss =

0.67
1+e

Vm +357

αf2 = 300 · e
βf2 =
γf2 =

Universidad Zaragoza+ 0.33−(Vm +25)2
170

31
25−Vm

1 + e 1016
1+e

Vm +3010

2. A New Paradigm for Electrophysiological Model Development33

τf2 = αf2 + βf2 + γf2
df2
f2 − f2= ssdtτf2
where Vm denotes transmembrane voltage.
Readjustment of τd was based on the description provided in [51]:
dss =

αd =

1
1+e

−(Vm +5)
6

1.4
1+e

βd =
γd =

−35−Vm13+ 0.25

1.4
1+e

Vm +5
5

1
1+e

50−Vm
20

τd = αd · βd + γddss − ddd=
dt
τd
The definitions of the calcium-dependent inactivation gates, fCaBj and fCaBsl ,were taken as in the GPB model:()
dfCaBj= 1.7 · Caj · 1 − fCaBj − 11.9 · 10−3 · fCaBj
dt
dfCaBsl= 1.7 · Casl · (1 − fCaBsl ) − 11.9 · 10−3 · fCaBsl
dt
Additionally, based on the results of the sensitivity of APD90 to variations in
the maximal ICaL conductivity and the actual APD90 value in the GPB model(see below, Tables 2.1 and 2.2), the relative permeabilities of the L-type calcium
channels were set to:
pCa = 1.9887 · 10−4 cm/s
pK = 5.4675 · 10−8 cm/s
pNa = 3.0375 · 10−9 cm/s()
pCa · Vm · F rdy · F oRT · Caj · e2·Vm ·F oRT − Cao¯ICaj =
e2·Vm ·F oRT − 1()
pCa · Vm · F rdy · F oRT · Casl · e2·Vm ·F oRT − Cao¯ICasl =
e2·Vm ·F oRT − 1()
pNa · Vm · F rdy · F oRT · N aj · eVm ·F oRT − N ao¯
INaj =
eVm ·F oRT − 1
JesúsCarroNew Methodologies for the Development
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34()
pNa · Vm · F rdy · F oRT · N asl · eVm ·F oRT − N ao¯
INasl =
eVm ·F oRT − 1()
pK · Vm · F rdy · F oRT · Ki · eVm ·F oRT − Ko¯
IK =eVm ·F oRT − 1
where F rdy is Faraday’s constant, F oRT is the ratio between Faraday’s constantand the product of the gas constant and the absolute temperature, I¯km is the
maximal current for ion k in the compartment m, and pk is the relative permeabilityof ion k.
The current-voltage curve obtained with the new definition of the ICaL current
was compared with experimental data [1–3] and with the original definition of
the GPB model. Figure 2.2.b) shows the different current-voltage curves, where
it can be observed that the new definition of ICaL was in good agreement withavailable experimental data [1–3], with the additional benefit of contributing tophysiological model response in terms of S1S2 APD90 restitution and APD90 rate
adaptation dynamics.
Inward Rectifier K Current (IK1 ). The IK1 current was redefined using recent
experimental data from human ventricular cells published in the literature [76].
The new definition for IK1 was as follows:
aK1 =

bK1 =

4.094
1.0 + e0.1217·(Vm −EK −49.934)

15.720 · e0.0674·(Vm −EK −3.257) + e0.0618·(Vm −EK −594.31)
1.0 + e−0.1629·(Vm −EK +14.207)
aK1
K1ss =
aK1 + bK1√
[K + ]o
IK1 = 0.5715 ·· K1ss · (Vm − EK )5.4

where EK is the [K + ] reversal potential. Figure 2.2.c) shows the modified IK1current compared with the experimental data from [76] and with IK1 from the
GPB model.
Na/K Pump Current (INaK ). In order to improve the APD90 adaptation response to abrupt changes in CL, maximal INaK was reduced by 45% with respect
to its value in the GPB model. That reduction was set based on the results of thesensitivity analysis for the τslow marker that is described above.
[K + ]i and GNa . A more physiological value for [K + ]i of 138 mM was used in ourmodel. Considering this value for [K + ]i made the model more excitable as compared to the original GPB model because of the increased availability of sodium
channels. Therefore, in order to obtain physiological values for the maximal upstroke velocity, dV /dt, in tissue, the maximum conductance of the sodium current,
GNa , was reduced to 18.864 mS/µF. This modification led to a maximal upstroke
velocity of 228 mV/ms in the simulated one-dimensional (1D) tissue (see below),which was in good agreement with the experimental results reported in [8].

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Developmentb)τ f CRLP
τ f 2 CRLP
τ f GP B
I nac t.
Sl ow I nac t.
Fast I nac t.

1000
500
0
−100

−50
0
V (mV)50

I C a L (pA/pF)

τ i (ms)

a)

35

0
GPB
CRLP
[Pelzmann1998][Li1999][Magyar2000]−5−10−40 −200
20
V (mV)

40

60

I K 1 (p A/p F )

c)

0
−10
−20






( 1
((((

0
XXX
aa
XX
aa
−1
aa
aa
GPB
−100
aa
CRLP
a
[Fink2008]

−80

−60

−40

−140 −120 −100 −80 −60 −40
V (mV)

Figure 2.2: Characteristics of ICaL and IK1 in simulations and experiments: a)
Simulated ICaL voltage-dependent inactivation time constants τi as a function ofvoltage, where i ∈ {f, f2 }. Experimental data, represented by markers, were takenfrom [2, 4–7]; b) ICaL current density versus voltage, with experimental data takenfrom [1–3]; c) IK1 current density versus voltage with experimental data taken from[76]. To compare simulated and experimental data under the same conditions,[K + ]i = 138 mM and [K + ]o = 4 mM were used for the curves shown in panel c),with [K + ]i denoting the intracellular potassium concentration.

JesúsCarroNew Methodologies for the Development
and Validation of Electrophysiological Models

36

Numerical methods and implementation
The cell membrane was modeled as an electrical circuit [26]. Mathematically, thetransmembrane voltage Vm was described by the following ordinary differential
equation:Cm

dVm= − (Iion − Istim ) ,
dt

(2.1)

where t is time, Iion is the sum of all transmembrane ionic currents, Istim is theexternally applied stimulation current and Cm is the cell capacitance per unit
surface area.
Propagation of the electrical activity in a 1D fiber was described by the following
parabolic reaction-diffusion equation [86]:
(
)
∂Vm
∂
∂V
Cm
= − (Iion − Istim ) +
σ
,
(2.2)
∂t
∂x
∂x
where σ is the diffusion coefficient of the tissue and x denotes the coordinate along
the fiber.
Physical units in the model were as follows: time t was given in milliseconds,
transmembrane potential Vm in millivolts, current densities in picoamperes perpicofarad and ionic concentrations in millimoles per liter.
The GPB and CRLP models were implemented in Fortran based on the originalcodes provided by Eleonora Grandi. The TP06 model was also implemented in
Fortran from the original C codes available at Kirsten ten Tusscher’s web page:www-binf.bio.uu.nl/khwjtuss.
Cells and tissue were stimulated with square transmembrane current pulses
twice the diastolic threshold in amplitude and 1-ms duration. For single cell simulations, forward Euler with a time step ∆t = 0.002 ms was used to integrate thesystem of ordinary differential equations governing the cellular electrical behavior.
For the 1D simulations, a homogeneous 4-cm long fiber composed of epicardial
cells was considered. A semi-implicit operator-splitting scheme [87] was used tosolve equation 2.2 with a space discretization of ∆x = 0.1 mm and a time stepof ∆t = 0.002 ms. A value of σ = 0.0013 cm2 /ms was used, corresponding toa maximum CV of 74 cm/s, which is in agreement with experimentally reported
CV values [88]. The cell capacitance was set to Cm = 1 µF/cm2 . Five positions
within the fiber were selected for further analysis in each simulation. Those positions were located at 1.5, 1.75, 2, 2.25 and 2.5 cm from the stimulation end of thecable. Reported APD90 and CV values were computed as mean values over thosepositions. In all cases, the Rush and Larsen integration scheme [89] was used tointegrate the Hodgkin-Huxley type equations for the gating variables of the various
time-dependent currents (m, h, and j for IN a ; xtof , xtos , ytof and ytos for Ito ; xkr
for IKr ; xks for IKs and d, f and f2 for ICaL ).
To test the sensitivity of the numerical solution to the time and space discretization indicated above, simulations were performed using a time increment of
0.5 µs and a spatial discretization of 0.05 mm. Differences of less than 0.2% for

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development37

the steady-state markers computed in isolated cells were obtained when comparedwith the default time and space steps used in this study. In tissue, CV and APD90
varied by less than 1.5% and 0.02%, respectively. Also, other stimulus durations(from 1 to 3 ms) and amplitudes (from 1.5 to 2 times the diastolic threshold) were
tested and were confirmed to provide very similar results (differences in terms of
APD90 , maximal upstroke velocity as well as CV in tissue were in all cases below
0.2%, 1.6% and 0.8%, respectively).
2.2.4 Conducted Simulations
Sensitivity Analysis
A sensitivity analysis was performed to the original GPB model following themethodology described in [12]. The aim was to elucidate how different ionic current
properties modulate the investigated electrophysiological markers. In the sensitivity analysis, each parameter was varied at a time by ±30%. Also, variations of±15% were considered to assess markers’ response to different degrees of variation.
These percentages are in agreement with variability levels reported in experiments
performed in human ventricular tissue [2, 76, 90–94].
The parameters considered in the sensitivity analysis were:• GCaL : Maximal conductance of the L-Type Calcium Current (ICaL ).
• τf : Time constant of the fast voltage-dependent inactivation gate (f ) of the
ICaL current.
• τf 2 : Time constant of the slow voltage-dependent inactivation gate (f2 ) of
the ICaL current.
• GKr : Maximal conductance of the rapidly activating potassium current (IKr ).
• GKs : Maximal conductance of the slowly activating potassium current (IKs ).
• τxs : Time constant of the activation gate (xKs ) of the IKs current.
• GK1 : Maximal conductance of the inward rectifier potassium current (IK1 ).
• GNaK : Maximal value of the sodium-potassium pump current (INaK ).
• Gncx : Maximal value of the sodium-calcium exchanger current (Incx ).
• Gto : Maximal conductance of the transient outward potassium current (Ito ).
For each marker c and parameter p, the percentage of change (Dc,p,a ) andsensitivity (Sc,p,a ) were calculated as follows [12]:Dc,p,a =Cp,a − Ccontrol· 100
CcontrolSc,p,a =Dc,p,a − Dc,p,−a· 100
2aJesúsCarroNew Methodologies for the Development
and Validation of Electrophysiological Models

38

where Cp,a is the value of marker c when the parameter p is varied by the percentage a, and Ccontrol is the value of marker c under control conditions. As shown
above, sensitivity (Sc,p,a ) was calculated as the ratio of the difference between thepercentage of change (Dc,p,a − Dc,p,−a ) and the interval of change (2a). In thisstudy we used this definition particularized for a = 30%:
Sc,p,30 =Dc,p,+30 − Dc,p,−300.6Larger percentages of variation (±60% and ±90%) in model parameters wereadditionally investigated to confirm the validity of the sensitivity analysis results
under more severe ionic alterations.
Potassium Current Blocks
The following cases of total and partial potassium current blocks were simulated:• IKs : To simulate the effect of 1 µM of HMR-1556, IKs was completely blocked.
• IKr : To simulate the effect of 50 nM of dofetilide, IKr was completely blocked.
• IK1 : To simulate the effect of 10 µM of BaCl2 , IK1 was reduced by 50%.
Restitution in Tissue
CV and APD90 restitution were additionally investigated in tissue by applying the
S1S2 protocol described in 2.2.2, with the only difference that the extra stimulus(S2) was in this case decreased in steps of 5 ms.
Hyperkalemic Conditions
Hyperkalemic conditions were simulated by increasing [K + ]o from 4 mM to 15
mM. For each [K + ]o value, the diastolic threshold used to compute the stimulus
amplitude was determined as follows: the model was first stabilized until the product of the gates h · j reached 99% of the steady-state value hss · jss . Stimulation
current pulses of variable amplitude were applied at one end of the cable and thediastolic threshold was defined as the minimum amplitude required for an AP topropagate along the fiber. With the model initialized at 1 Hz steady-state, ERP
was determined by stimulating the tissue with a train of 10 basic stimulations (S1)delivered at a CL of 1,000 ms, followed by an extra stimulus (S2) delivered at
different times taken in 1 ms steps. Steady-state CV and APD90 were measured
after the last S1 stimulus.
2.3 Results
2.3.1 Results for Control Conditions

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development

39

Isolated Cells
a)

b)
TP 06

0
−50
−100

0

200
400
Ti me (ms)

0
−50
−100

600

0

200
400
Ti me (ms)

CRLP

50
V (mV)50
V (mV)V (mV)

50

c)
GP B

0
−50
−100

600

0

200
400
Ti me (ms)600

1D Fiber
d)

e)
TP 06

0
−50
−100

0

200
400
Ti me (ms)

600

0
−50
−100

0

200
400
Ti me (ms)

CRLP

50
V (mV)50
V (mV)V (mV)

50

f)
GP B

600

0
−50
−100

0

200
400
Ti me (ms)600

Figure 2.3: APs computed with the TP06, GPB and CRLP models for isolated
cells and 1D tissue at CL = 1000 ms. a) TP06 cellular AP; b) GPB cellular AP;c) CRLP cellular AP; d) TP06 tissue AP; e) GPB tissue AP; f) CRLP tissue AP.

APs in Cell and Tissue
Figure 2.3 presents APs computed from the TP06, GPB and CRLP models for an
isolated epicardial cell and for an epicardial cell coupled in tissue. In all cases APs
correspond to steady-state pacing at CL = 1000 ms. As can be appreciated fromthe figure, the AP duration was of the same order in the three models. However,the AP shape was notably more triangular for the GPB and CRLP models ascompared to the TP06 model, which presented a flatter plateau phase. Also, phase
1 of the AP substantially differed from one model to another, particulary whencomparing TP06 with the other two models. Those differences were to some extent
attenuated when comparing APs measured in tissue. It is also clear from thefigure that maximal upstroke velocity was considerably reduced for tissue APs as
compared to cell APs. While for the GPB and CRLP models maximum voltage
occurred at the upstroke peak in both cell and tissue, for the TP06 model the
maximum voltage at the plateau was even larger than at the upstroke peak intissue AP, as previously noted in [62] for the TNNP04 model from which the TP06
model was developed.

JesúsCarro

40

New Methodologies for the Development
and Validation of Electrophysiological Models

Sensitivity Analysis
The results of the sensitivity analysis applied to the epicardial version of the GPBmodel are presented in Table 2.1. This table shows the dependence of a large number of electrophysiological markers on the ICaL current. Not only markers related
to systolic and diastolic [Ca2+ ]i levels, but also APD90 and S1S2 restitution slope
were mostly sensitive to variations in ICaL . Also, Table 2.1 shows the importance of
the IK1 current for AP triangulation. INaK was determinant for [N a+ ]i rate dependence as well as for the time constant of the slow phase of APD90 rate adaptation.
Note that APD90 rate adaptation dynamics computed from the GPB model did
not present a characteristic biphasic behavior as reported in experimental studies[95], but adaptation was completed in only one slow phase. For that reason themarker τf ast described in subsection 2.2 is not shown in Table 2.1. Finally, Incx
was relevant for [Ca2+ ]i rate dependence. Table 2.1 together with the analysis
presented in the following subsection were the road map to carry out the various
model improvements proposed in this work, which were described in subsection 2.2.
To confirm that the results of the sensitivity analysis presented in Table 2.1 were
not dependent on the limit (30%) considered for parameter variations, simulations
were performed considering larger percentages of variations of up to 90%. Despite
the fact that in some cases a highly nonlinear modulation of specific electrophysiological markers by certain ionic mechanisms was observed (e.g. triangulation
modulation by GK1 , systolic [Ca2+ ]i modulation by GCaL or τslow modulation by
GNaK ), results corroborated the mechanisms identified as determinant for the investigated markers in the sensitivity analysis.
Electrophysiological markers
Table 2.2 shows the values of the computed markers for the epicardial versionsof the three models analyzed in this study (TP06, GPB and CRLP). Simulation
results were compared against a variety of experimental data taken from [2, 8–
10, 79, 95–98].
Results indicated that the GPB model performed better than the TP06 modelwith respect to the following markers: AP triangulation, steady-state systolic anddiastolic [Ca2+ ]i levels at 0.5 Hz and rate dependence of [N a+ ]i and [Ca2+ ]i . Thosemarkers are illustrated for the three analyzed models in Figure 2.4.a), 2.4.b),2.4.f) and 2.4.g). However, according to other analyzed markers, the GPB model
rendered worse performance than the TP06 model, as was the case for the S1S2
restitution curve and the APD90 adaptation to abrupt changes in CL. Regarding
the APD90 adaptation, two phases of APD90 rate adaptation were experimentally
reported in [95]: a first phase with a short time constant and a second phasewith a longer time constant. As described above, the APD90 rate response in
the GPB model did not present the first fast phase (Figure 2.4.c)). For the S1S2
restitution curves (Figure 2.4.d)), the TP06 model was in better agreement withthe experimental data of [95, 98] than the GPB model.
For the CRLP model, seven out of the twelve analyzed markers were within
physiological range and other three markers were closer to the physiological range

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development

41

1
AP D 9 0

60. 88%0.9

Tri angul ati on
Sy s[ C a2+-62. 78%
0.8] i 1 Hz205. 40%

Sy s[ C a 2 +] i 0. 5 Hz

239. 61%

0.7

D i a[ C a 2 +] i 1 Hz

28. 09%0.6

D i a[ C a 2 +] i 0. 5 Hz

29. 41%

0.5

Sl op e max S1S2-62. 50%

0.4

Sl op e max D y n

134. 46%

τ s low

0.3-81. 45%

Max sy s[ C a 2 +] i

0.2
67. 71%

Max sy s[ N a + ] i

0.1-10. 52%
0
G C aL

τf

GKr

GKs

τxs

GK1

G N aKG nc xGto

Table 2.1: Results of the sensitivity analysis applied to the GPB model. The blue
scale indicates relative sensitivity for each marker. White color indicates maximum
relative sensitivity and dark blue color indicates that property and parameter are
independent. Percentages in the white boxes indicate the absolute sensitivity ofthe property. A minus sign in a box indicates that marker and model parametervary inversely. ‘Sys’ stands for systolic and ‘Dia’ stands for diastolic.
as compared to the GPB and TP06 models. As for systolic and diastolic [Ca2+ ]ilevels at 1 Hz, TP06 outperforms CRLP. A significant change in the shape of the
S1S2 restitution curve was observed for the CRLP model as compared to the GPBmodel (Figure 2.4.d)). The behavior shown by the CRLP model was in agreementwith experimental data [95].
Potassium Current Blocks
Table 2.3 summarizes the effect of potassium current blocks on epicardial APs computed using the three tested models. Results are given as percentage of variationof the APD90 value with respect to control conditions. In those cases where morethan one current was blocked, the variation refers to the APD90 value obtained
when only the first current was blocked. Experimental values are taken from [54].
It is clear from Table 2.3 that the GPB model performs better than the TP06
model regarding current blocks. The CRLP model performs similar to the GPBmodel. Figure 2.5 illustrates the effect of current blocks on the AP shape for the
GPB and CRLP models.
Restitution in Fiber
Figure 2.6 shows CV and APD90 restitution curves computed by applying the
S1S2 protocol to the simulated 1D fiber. Regarding CV restitution, the threemodels present similar behaviors, with only notable differences (>5 cm/s) observed
between the TP06 model and the other two models (GPB and CRLP) in the final

JesúsCarroNew Methodologies for the Development
and Validation of Electrophysiological Models

42markerAP D90 (ms)Triangulation (ms)Sys [Ca2+ ]i 1 Hz (µM)
Sys [Ca2+ ]i 0.5 Hz (µM)
Dia [Ca2+ ]i 1 HZ (µM)
Dia [Ca2+ ]i 0.5 Hz (µM)
SmaxS1S2 (—)
SmaxDYN (—)τf ast (s)τslow (s)Max. sys. [Ca2+ ]i (%)
Max. sys. [N a+ ]i (%)

TP06
301.2
28.4
0.886
0.199
0.104
0.068
1.3
1.0
13.3
124.8
1157
217

GPB
285.0
51.5
0.383
0.345
0.089
0.085
0.2
1.1
—
56.3
178
132

CRLP
305.6
78
0.602
0.523
0.097
0.091
1.0
0.9
10.2
105.6
147
134

Physiol.
271-366
44-112
1.59-2.01
0.71-1.68
0.20-0.33
0.14-0.32
0.79-4.25——
70-110130-170
145

Table 2.2: markers of arrhythmic risk for the TP06, GPB and CRLP human models. Black color indicates within physiological range. Dark gray color indicates outof physiological range but better than previous models. Light gray color indicatesout of physiological range. Experimental data are from [2, 8–10, 79, 95–98].

Current
IKs
0%
IKr
0%
IKr
0%
IKs
0%
IK1 50%IKr
0%
IK1 50%

Ref.
Control
Control

TP06
74.9
15.9

GPB
0.7
18.6

CRLP
0.6
14.1

Exper.
< 2.8
44±4

IKr block

—

1.4

0.8

9

Control

3.6

11.5

14.6

4.8±1.5

IKr block

8.9

15.6

21.1

33

Table 2.3: Percentages of variation in APD90 caused by blocking different potassium currents. Black color indicates within physiological range. Dark gray color
indicates out of physiological range but better than previous models. Light graycolor indicates out of physiological range.

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development

a)

b)

c)
Ste ady State [ C a 2 +] i Shap e

200
400Time (ms)

0.5

0

600

0

500
Ti me (ms)

d)

300
280
260

1000

0

5

10
15
Time (min)20e)
D y nami c

300

AP D 9 0 (ms)

AP D 9 0 (ms)

S1S2

250
200
0

200

400
600
D I (ms)

GP B
CRLP
TP 06

300
250
200

800

0

200

400
600
D I (ms)

800g)[Ca2+ ]i Rate Dependence
1200
1000
800
600
400
200
0.5

1 1.5 2 2.5
Frequency (Hz)3

Systolic [Na+ ]i (%)f)
Systolic [Ca2+ ]i (%)

APD Rate Adaptation

1
APD90 (ms)

40
20
0
−20
−40
−60
−80

[ C a 2 +] i (µ M)

Vm (mV)

Steady-State AP Shape

043[Na+ ]i Rate Dependence
200
150
100
0.5

1 1.5 2 2.5
Frequency (Hz)3

Figure 2.4: markers of arrhythmic risk from TP06, GPB and CRLP models: a)
Steady-state AP for CL = 1000 ms; b) Steady-state Ca transient for CL = 1000
ms. c) APD90 rate adaptation to abrupt changes in CL (from 1000 ms to 600 ms);d) S1S2 restitution curve; e) Dynamic restitution curve; f) Normalized [Ca2+ ]irate dependence; g) Normalized [N a+ ]i rate dependence. For panels f) and g)
normalization is performed with respect to the lowest tested frequency, i.e. 0.25
Hz.

JesúsCarroNew Methodologies for the Development
and Validation of Electrophysiological Models

44

GPB Modelb)Control
I K r Bl ock
I K s Bl ock
I K r & I K s Bl ock

40
20
0
−20
−40
−60
−80
0

200
400
Ti me (ms)V m (mV)V m (mV)a)Control
I K r Bl ock
I K 1 Bl ock
I K r & I K 1 Bl ock

40
20
0
−20
−40
−60
−80

600

0

200
400
Ti me (ms)600

CRLP Modeld)Control
I K r Bl ock
I K s Bl ock
I K r & I K s Bl ock

40
20
0
−20
−40
−60
−80
0

200
400
Ti me (ms)600

V m (mV)V m (mV)c)Control
I K r Bl ock
I K 1 Bl ock
I K r & I K 1 Bl ock

40
20
0
−20
−40
−60
−80
0

200
400
Ti me (ms)600

Figure 2.5: Effect of blocking potassium currents for the GPB and CRLP models:a) Effect of completely blocking IKr and IKs currents in the GPB model; b) Effect ofcompletely blocking IKr and partially blocking IK1 in the GPB model; c) Effect ofcompletely blocking IKr and IKs currents in the CRLP model; d) Effect of completely
blocking IKr and partially blocking IK1 in the CRLP model.

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Development

a)

45

b)

AP D 9 0 (ms)CV (c m/s)

80
70
60
50

0

200

400
600
D I (ms)

800

300
TP06
CRLP
GPB

250
200
0

200

400
600
D I (ms)

800

Figure 2.6: CV and APD90 restitution curves for the TP06, GPB and CRLP
models in a simulated 1D fiber.
part of the curve, i.e. for DI values above 200 ms. Regarding APD90 restitution,
both the TP06 and CRLP model present steep restitution curves, with high slope(close to 1) only for a narrow DI interval in the case of the CRLP model and for amuch wider DI range in the case of the TP06 model (with slope values above 1).
The GPB model shows a relatively flat APD90 restitution curve. Those results arein good correspondence with the observations from restitution curves measured inisolated cells (Figure 2.4.d)).
2.3.2 Results for Hyperkalemic Conditions
AP properties
Figures 2.7.a) and 2.7.c) show a comparison of the AP shape computed in single
cell and tissue for different levels of [K + ]o using the CRLP model. Figures 2.7.b)and 2.7.d) show the effect of varying [K + ]o on the resting potential in single celland tissue. As expected, hyperkalemia shortens the APD90 and leads to elevatedresting potential, in both cells and tissue.
From Figure 2.7.c) it is also clear that for high [K + ]o levels, the AP shapevaries between even and odd beats, thus leading to the occurrence of AP alternans.
Further analysis on the alternating AP behavior and the [K + ]o threshold for its
occurrence can be found below.
Conduction Velocity
Figure 2.8.a) shows CV as a function of [K + ]o computed from the simulated 1D
fiber using the CRLP model. As expected, hyperkalemia leads to a decrease in
CV. However, “supernormal conduction” is not observed for [K + ]o levels below
physiological values (5.4 mM) as has been observed in other studies [99, 100]. Our
results also show that conduction block occurs for [K + ]o > 10.4.

JesúsCarroNew Methodologies for the Development
and Validation of Electrophysiological Models

46

Isolated Cellsb)[ K +] o = 4[ K +] o = 6[ K +] o = 8[ K + ] o = 10

50
V (mV)

Re sti ng Pote nti al (mV)

a)

0
−50
−100

0

200
400
Ti me (ms)

600

−60

−80

−100

4

6
8
[ K + ] o (mM)

10

68
[ K + ] o (mM)

10

1D Fiberc)[K ]o = 4
[ K +] o = 6[ K +] o = 8[ K + ] o = 10

V (mv )

50
0
−50
−100

0

500

1000
Ti me (ms)

1500

2000

Re sti ng Pote nti al (mV)

d)
+

−60

−80

−100

4

Figure 2.7: Simulated hyperkalemia in cell and tissue using the CRLP model. a)
AP shape for varying [K + ]o in isolated cells; b) Resting potential for varying [K + ]oin isolated cells; c) AP shape for varying [K + ]o in 1D fiber; d) Resting potential
for varying [K + ]o in 1D fiber.

Universidad Zaragoza2. A New Paradigm for Electrophysiological Model Developmentb)

E RP, AP D 90 (ms)a)

47

CV (c m/s)

80
60
40
20

4

6
8
[ K + ] o (mM)

1000
E RP
AP D 9 0
500

0

10

4

6
8
[ K + ] o (mM)

10c)

Vm (mV)

50
0
−50
−100
6000

6500

7000

7500

8000

8500

8000

8500

Time (ms)d)

I C aLIN a[ C a 2 +] i

1

0

−1
6000

6500

7000

7500
Ti me (ms)Figure 2.8: Behavior of the CRLP model under hyperkalemic conditions in a 1D
fiber: a) CV versus [K + ]o ; b) dependence of ERP and APD90 with [K + ]o ; c)example of alternans for [K + ]o = 10.0 mM, each color representing a different
position in the fiber; d) value of different currents normalized to the maximal valueof each one in the same example as the previous results.

JesúsCarro

48

New Methodologies for the Development
and Validation of Electrophysiological Models

Action Potential Duration and Effective Refractory Period
Figure 2.8.b) shows ERP and APD90 of the last basic (S1) beat for different [K + ]ovalues computed in the simulated 1D fiber using the CRLP model. Results showthat hyperkalemia causes postrepolarization refractoriness. The difference between
ERP and APD90 remains almost constant for [K + ]o ≤ 6 mM, with this difference
increasing rapidly for [K + ]o > 6 mM. For [K + ]o > 9.8 mM, alternans appear
in the tissue, causing the ERP and APD90 curves to branch out (Figure 2.8.b)).
This characteristic of the ERP and APD90 curves has not been reported in otherhuman ventricular simulation studies of the literature, but is in agreement withexperimental results in other species and with clinical data [101–103]. We tested
the same hyperkalemic conditions in simulated TP06 and GPB 1D fibers and we
could not observe branching out of the ERP or APD90 curves after increasing [K + ]oup to 15 mM.
Figure 2.8.c) depicts AP propagation in the simulated CRLP 1D fiber for[K + ]o = 10 mM. Alternans were observed for all computed positions within thefiber. Figure 2.8.d) shows ICaL , IN a and [Ca2+ ]i , each normalized to its maximum value, for the same time period shown in Figure 2.8.c). Results indicate that
conduction relies on IN a even for the highly depressed APs. For the long APs,
the AP upstroke is divided into two components, the first one characterized by asteep upstroke and supported by IN a , and the second one supported by ICaL , inagreement with the results reported in [67].
2.4 Discussion and conclusions
In this chapter a new human ventricular cell model, the CRLP model, was developed based on the recently proposed GPB model. The CRLP model combines
recent experimental measurements of potassium currents and reformulates the Ltype calcium current by introducing fast and slow voltage-dependent inactivation
gates, similar to the formulation proposed in the TP06 model. That new formulation led to ICaL dynamics that were in good agreement with experimental data
and, more notably, allowed the CRLP model to accurately reproduce S1S2 restitution and APD90 rate adaptation characteristics. All the modifications were madewith the aim of retaining the advantages that the GPB model has over previous
models, such as the TP06 model, while improving the performance on other electrophysiological aspects. The novelty of this study was that model development was
not based only on the incorporation of new data from experiments on individual
ionic currents or concentrations. More importantly, the development was driven
by a new paradigm consisting in analyzing the ability of the model to replicate a
set of well-established electrophysiological markers and on a sensitivity analysis of
those markers to variations in model parameters. The proposed model development
framework allows assessing model performance under a variety of conditions, either
physiological or pathological. This provides useful information for the adjustment
of different model parameters that do not directly depend on individual current
measurements or concentrations, and which would be otherwise set in an ad hoc

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manner.
2.4.1 New model characteristics
The introduction of fast and slow ICaL inactivation gates in the CRLP model led
to a more physiological APD90 rate adaptation response to abrupt changes in CL
as compared to the GPB model. However, by doing so, a larger ICaL current was
obtained for the CRLP model, which led to the reformulation of IK1 and readjustments of other parameters to keep APD90 within physiological values. The
resulting APD90 value for the CRLP model was closer to the APD90 value of the
TP06 model, without significantly altering the AP triangulation of the GPB model,
which was well within the reported physiological range. The other modifications
introduced into the CRLP model rendered most of the investigated markers to bewithin the physiological range or otherwise closer to the physiological range as compared to the GPB and TP06 models. In this regard, only for systolic and diastolic[Ca2+ ]i levels at 1 Hz the TP06 model outperformed CRLP. In addition, the CRLP
model showed improved performance with respect to the GPB model regarding the
S1S2 restitution, not only by bringing the maximum S1S2 restitution slope within
the physiological range, but also by providing a shape for the restitution curve ingood agreement with experimental measurements [95].
Even though the model modifications introduced in this work significantly improved the performance of the CRLP model over the GPB model in reproducing
experimental data related to a number of electrophysiological markers, they haveminimally affected the behavior of the model against potassium current blocks. In
this regard, the CRLP model still provides results in very good agreement withavailable experimental data regarding AP response to IKr , IKs and IK1 blocks [54].
2.4.2 Simulation of hyperkalemic conditions
Simulations under hyperkalemic conditions showed that APD90 progressively decreased with the level of [K + ]o , whereas ERP increased after a threshold in [K + ]owas reached ([K + ]o ≈ 6 mM). CV decreased with hyperkalemia and conduction
was blocked above [K + ]o = 10.4 mM. These results suggest that the longer ERP
values and the occurrence of conduction block above [K + ]o = 10.4 mM in thecentral zone of acutely ischemic tissue could cause areas of conduction block that
could set a substrate for reentrant arrhythmias. Additionally, alternans were observed in ERP and APD90 for [K + ]o > 9.8 mM. These results are in agreementwith experimental studies where alternans were observed in the cells of the heart’s
ischemic zone [101, 102]. To the authors’ knowledge, this is the first time thatthose observations are reported using a human ventricular AP model. That typeof behavior was not observed in either the TP06 or the GPB models. For the TP06
model, conduction block occurred for [K + ]o ≈ 15 mM without the occurrence ofalternans [104]. For the GPB model, conduction block occurred for [K + ]o ≈ 9 mM
and alternans were not observed either.

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2.4.3 Data sources
As described in [61], one of the problems on reutilization of model components inelectrophysiology is the inclusion of components adjusted with data from different
species. For that reason, in this work all model developments were made by using
model components formulated based on human data only [51] or by fitting equations to additional human data available from the literature [76]. We did not try
modifying other ionic currents or concentrations for which human data was notavailable. Regarding the types of human data used in the study, they all corresponded to adult humans, except for the data taken from the study by Pelzmann
et al. [1], in which ICaL was measured in children. Those data were used because thestudy clearly showed that there were not significant differences in the ICaL current
characteristics measured in children as compared to adults.
2.4.4 Model limitations
The developed CRLP model has a number of limitations, some of them inheritedfrom the GPB model on which it was based, and others independent of it. Inherited from the GPB model is the absence of intracellular potassium dynamics([K + ]i is kept constant in the GPB and CRLP models). [K + ]i dynamics play animportant role in modeling hyperkalemic conditions, and is, therefore, a pivotal
element for the accurate computational investigation of ventricular arrhythmias
under such conditions. In addition, phase 1 of the AP shows an atypical behavior for the CRLP model. This may be caused by the Ito current. However, noattempt to modify this current has been made mainly because new human data
was not available for that purpose. This current happens to be the only one whose
definition differentiates endocardial and epicardial cells in the GPB model and in
the CRLP model as well. In this regard, both models lack a description for theelectrophysiology of midmyocardial cells. Another limitation of the CRLP modelhas to do with calcium handling. The results of the analysis conducted in thisstudy point out the importance that the ICaL current as well as [Ca2+ ]i dynamics
have over different electrophysiological markers. Many studies have attempted toprovide accurate definitions of [Ca2+ ]i dynamics in human ventricular cells, but
this is still an area of intensive research. The present study suggests the need forcontinuing with the development of more reliable calcium dynamics models thatallow improving the performance of whole cell AP models.
On top of the above described limitations, a comment is in order regarding
the comparison of human ventricular AP models. It is clear that each model has
been developed to reproduce certain electrophysiological behaviors observed fromexperimental recordings. Also, considering the huge variability in the experimental data, model comparison becomes a challenging task for which a model should
never be qualified as being an improvement over another one just because it is able
to reproduce specific experimental observations that the other model is not able
to reproduce. In this study we used the term improvement exclusively to refer tothe fact that, considering the set of investigated electrophysiological markers, the
CRLP model provides a response that is more in line with the averaged experimen-

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tal data used for evaluation purposes. The following chapters attempt to resolve
some of the limitations enumerated above. For this purpose, novel methodologies
have been developed and will be presented accordingly.

JesúsCarroChapter 3

Improvement of ExistingElectrophysiologicalModels3.1 Introduction
Departing from the CRLP model developed in Chapter 2, this chapter aims at
improving its performance by including a relevant characteristic for investigationof ventricular arrhythmias arising under conditions of hyperkalemia, one of thecomponents of myocardial ischemia. Such a characteristic refers to formulation of
intracellular potassium dynamics, which were lacking both in the CRLP model aswell as in the GPB model from which it was built. When potassium dynamics
are incorporated into a model, an imbalance in the potassium currents may beoriginated, which may in turn lead to a drift in the intracellular potassium concentration. A way to overcome this is by readjusting model parameters to maintainpotassium homeostasis.
As mentioned in Chapter 1, an AP model is described by a system of coupled differential equations that involve the transmembrane current flow as wellas dynamic changes in intracellular ionic concentrations. For each ionic current
a mathematical formulation is employed to represent characteristics like gatingor permeability of the corresponding ion channels. Those formulations rely on anumber of model parameters whose values are commonly estimated based on experimental data obtained from specific protocols for each ionic current. In this context
parameter estimation presents various shortcomings intrinsic to the voltage-clamp
experimental protocols [105]: i) insufficient separation of ionic current activationand inactivation processes; ii) use of non-selective pharmacological ion channel
blockers; and iii) measurement of non-physiological solutions.

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Despite these limitations, most parameter values of individual ionic currents
have been identified in the past using independent sets of experiments. However,parameters describing ionic current conductances have not always been successfully characterized from experimental datasets, in large part due to the sensitivityof some ion channels to the cell isolation procedure performed before application of
the voltage-clamp protocols [106]. Also, experimental studies may have purposely
selected cells with large ionic currents, which interfere with model development
based on such data [47]. For those reasons, a common procedure to identify individual ionic current conductances is based on model parameter fitting aimed at
reproducing plausible values of electrophysiological AP properties like APD, triangulation or upstroke velocity [15, 46]. Other models include also constraints
based on experimental data measured after blocking certain ionic currents [107],
imposed to obtain a correct input resistance [47] or based on additional properties
like APD rate adaptation and rate dependence of ionic concentrations as proposed
in Chapter 2.
Cardiac electrophysiological computational models are commonly used to analyze the effect of drugs or to study the mechanisms underlying pathological conditions. Under those circumstances, the correct characterization of ionic current
conductances is crucial. Overestimation or underestimation of current conductance values could lead to completely erroneous predictions. For this reason, while
for many years the adjustment of current conductances in cardiac computationalmodels has been frequently performed manually [108], a number of techniques havebeen developed in recent years to help in the process of parameter identification.
Gradient descent methods [109, 110] and genetic algorithms [111–114] have beenused for this purpose in several studies. Gradient descent methods rely on computing local gradients of the error function to determine a search direction towards
a minimum. On the other hand, genetic algorithms use ideas from evolutionary
biology and test different combinations (populations) of parameters to optimize
the error function. In each iteration they create a new population based on thebest results from previous iterations.
Gradient descent methods and genetic algorithms have a small degree of parallelization and, thus, other techniques with higher parallelization capability could
better exploit the possibilities of current computing infrastructures. In this chapter, a novel methodology is proposed for identification of parameters describing
ionic current conductances of cardiac electrophysiological models by using an optimization algorithm with trust region management [115]. The algorithm minimizes
an objective function subject to non-linear constraints associated with a numberof electrophysiological properties. The objective function and constraints are approximated by polynomial response surfaces that allow the optimization algorithm
searching for a solution to the problem without performing a direct evaluation of
the model. The use of a response surface helps to reduce the number of iterationsof the algorithm and, thus, reduce the time cost of optimizing the model. To guarantee the ability of the model to represent physiological behavior, the region where
the solution is searched for is limited based on experimentally reported values ofelectrophysiological properties like APD, AP triangulation, diastolic and systolic

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intracellular calcium concentrations and time constants of APD rate adaptation
to abrupt changes in cycle length (CL). The proposed methodology is applied toadjust the ionic conductances of the human ventricular CRLP model proposed in
Chapter 2 after incorporation of intracellular potassium, [K + ]i , dynamics. The
modified CRLP model is compared against the most commonly used human ventricular AP models, namely the TP06 [51], GPB [15], ORd [64] models as well aswith the CRLP model.
The relevance of accounting for [K + ]i dynamics is demonstrated by simulating a
number of transitions in the extracellular potassium concentration, as in hemodialysis studies [116], and comparing the behavior of the original and modified CRLP
models.
3.2 Materials and Methods
3.2.1 Optimization problem
General formulation
A minimization problem with constraints can be written as:
min f (x)(3.1)

h(x) = 0,g(x) ≥ 0,(3.2)

xL ≤ x ≤ xU ,(3.4)

subject to(3.3)

where x is the vector of decision variables, xL and xU their respective lower andupper bounds, f(x) is the objective function, and h(x) and g(x) are vector functionsof equality and inequality constraints, respectively.
Particularization to ionic conductances estimation
The general minimization problem described in previous section was particularized
in this study for estimation of ionic current conductances. Since different ioniccurrent conductances Gi may differ by orders of magnitude, the following scaling
was adopted for each of the elements of vector x:
xi = 20 · log10

Gi
,
G0i

where Gi is the unscaled ionic current conductance and G0i is its initial value inthe model.
The objective function is defined for each particular problem. In this study,
it was represented by the integral of the total potassium current during a cardiac
cycle at 1 Hz steady-state pacing, as fully described in 3.2.2. No equality constraint function was defined in this case. The inequality constrain function was

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defined by imposing a number of electrophysiological properties to remain within
experimentally observed ranges:mLL ≤ m(x) ≤ mU U

(3.5)

where mLL and mU U are the lower and upper bounds delimiting physiologicalranges, respectively, and m(x) is the vector of electrophysiological properties calculated for x.
Eq. 3.5 was reformulated to be consistent with the general formulation of theinequality constraints described in eq. 3.3:
gj (x) =mj (x) − mj,CR− 1 ≤ 0,mj,RRwhere mj,CR = (mj,U L + mj,LL )/2 is the central point of the experimental range forthe electrophysiological property j and mj,RR = (mj,U L − mj,LL )/2 is the distance
between mj,CR and each of the limits of the experimental range.
Optimization using response surfaces
The optimization algorithm used in this study seeks for a solution of the minimization problem, Eqs. 3.1-3.4, by solving a sequence of approximate constrained
optimization problems within a region of trust. For the iteration k of the algorithm,
the approximate problem reads as:
min f˜(x)(3.6)

h̃(x) = 0,

(3.7)g̃(x) ≥ 0,

(3.8)

∥x∥ ≤ ∆ ,(3.9)

subject tokwhere f˜(x), h̃(x) and g̃(x) are response surface approximations (RSA) of f (x),h(x), and g(x) around xk , respectively, and ∆k is the corresponding trust region
radius. The trust region radius defines a region of trust in which the RSA isassumed to be reliable. The validity of the RSA can be evaluated through thetrust region ratio test. For the objective function f (x), the trust region ratio for
iteration k was defined as:ρk0 =f (xk ) − f (xk,∗ ),f˜(xk ) − f˜(xk,∗ )(3.10)

where xk,∗ is a solution of the minimization problem given by Eqs. 3.6-3.9. Thisdefinition of the trust region ratio is based on the consideration that ρk0 → 1 as
the trust region radius ∆k → 0. If ρk0 ≤ 0, the approximation is deemed as
unacceptable and, consequently, the trust region radius is reduced. If ρk0 ≈ 1 and∥xk,∗ − xk ∥ = ∆k , the RSA is evaluated as acceptable and ∆k is increased. If
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0 < ρk0 < 1, ∆k is kept unchanged, but the RSA is rebuilt around xk,∗ . Note thatρk0 >> 1 is also indicative of poor RSA and ∆k is reduced, but the optimizer isconsidered to be moving in the right direction. The update of the trust region
radius offers significant flexibility depending on the implementation. A particular
updating rule will be presented in the following section.
The algorithm proposed in this work is based on the one described in [115] but
it simplifies some aspects on how to manage and use the response surfaces andit reduces the number of times the database needs to be generated. Instead of
considering the augmented Lagrangian of the problem, the proposal of this study
is based on generating the response surface of the model and looking for the minimum in this surface. Subsequently, this solution is validated by using the trust
region test in both the minimization function and the constraints of the model.
This notably reduces the computational cost, as the database is generated once
per iteration rather than every time there is a change in the parameters of theaugmented Lagrangian.
Database generation and response surface approximation
One of the key elements of the minimization algorithm is building the RSA of
the objective and constraint functions that will be used to solve the approximate
minimization problem. To construct a response surface for a function of n variables,a large number of sample points are required. Also, to construct a reliable RSA,
the sample points must be adequately distributed. Sampling techniques based on
design of experiments are commonly used for this purpose [117]. In this study,
full factorial designs, which require 3n sampling points, were considered. A full
factorial design consists of a three-level factorial: the corner points of a hypercube,plus the center of the hypercube and of each facet, plus the midpoints of the lines
connecting the corners of the hypercube. In the next sections, the set of results ofthe sampling technique in an iteration of the algorithm for both the minimizationand constraint functions will be referred to as database. Second order polynomial
response surfaces were used in all cases.
Minimization algorithm
The minimization algorithm implemented in this work is shown in Fig. 3.1. The
database was computed only once per iteration of the algorithm or whenever thesolution was rejected and the trust region radius was reduced. The trust region
ratio was used to validate the RSA for the objective function and constraints independently.
The basic steps of the algorithm shown in Fig. 3.1 are briefly described in thefollowing. The index k indicates the iteration of the algorithm, i indicates thevariable index and j represents the constraint index.

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Figure 3.1: Flow chart of the proposed algorithm.

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Step 0: Initialization
Define values for the tolerances δ1 (solution tolerance), δ2 (gradient tolerance) andδ3 (improvement tolerance), the minimum acceptable value ρL for the trust region
ratio to consider a trust region as acceptable and an upper limit ∆max for thetrust region radius. Define an initial value ∆0 for the size of the trust region anda starting point x0 for the variables of the problem. Set the iteration counter at
k = 0.
Step 1: Database generation
Compute the database as described in section 3.2.1.
Step 2: Response surfaces
Build the RSA of the objective function and constraints of the problem around thecurrent iterate xk .
Step 3: Find the minimum of the approximate problem
Find the minimum xk,∗ of the constrained minimization problem defined by Eqs. 3.43.9 in the trust region limited by ∆k .
Step 4: Trust region test
Evaluate f and each of the properties mi in xk,∗ and calculate the trust region
ratio for the objective and constraint functions as in Eq. 3.10.
ρk0=

f (xk,∗ ) − f (xk )fe(xk,∗ ) − fe(xk )ρkj=mj (xk,∗ ) − mj (xk ), 1 ≤ j ≤ M.
me j (xk,∗ ) − me j (xk )The solution xk,∗ was accepted when ρk0 > 0 and one of the following two criteria
was met: i) the constraints in g(x) were feasible or; ii) |ρkj − 1| ≤ ρL , ∀j. For thiswork, ρL = 0.25 was used. If the solution was accepted, the flag f lgTk R = true wasset and xk+1 := xk,∗ , otherwise f lgTk R = f alse and xk+1 := xk .
Step 5: Trust region radius update
The trust region radius was updated as follows: 0.25∆k if f lgTk R = f alse,
k+1∆k
if f lgTk R = true and ∥xk,∗ − xk ∥ < ∆k ,
∆
=

k
2∆
if f lgTk R = true and ∥xk,∗ − xk ∥ = ∆k .
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Step 6: Algorithm termination

The algorithm ended if one of the following conditions was satisfied:
1. ∥xk+1 − xk ∥ < δ1 and f lgTk R = true: The distance between the last two
accepted iterates was too small to be significant.
2. ∥∇ϕ(xk+1 )| < δ2 and f lgTk R = true: The gradient of the Lagrangian function
was too small. The Lagrangian function was defined as:
ϕ(x) = f (x) +M∑

λkj · mj (x)j=1

where λkj are the Lagrange multipliers obtained in the constrained minimization problem defined by Eqs. 3.4-3.9 in the trust region (limited by ∆k ) in
Step 3.
3. ∆k+1 < δ3 : The trust region radius became too small.
If none of these criteria was satisfied, then set k := k + 1 and go to step 1.
3.2.2 Upgrade of the CRLP model[K + ]i dynamics in the CRLP model
One of the differences between the CRLP model and the ORd and TP06 models
lies in the representation of [K + ]i dynamics. As mentioned above, in the CRLP
model, as well as in the GPB model, [K + ]i was assumed to be constant. Under
steady-state conditions, [K + ]i variations during an AP can be neglected. Nevertheless, when there is a change in the investigated conditions or if the model issomewhat altered, [K + ]i may reach a new steady-state value. This is the case,for instance, when the stimulation frequency is varied, as [K + ]i becomes reduced
in response to increased stimulation frequency [64, 118]. Another scenario where[K + ]i dynamics are relevant is in the simulation of acute ischemia. Dutta et al.
showed the limitations of using the GPB model with constant [K + ]i [119]. In thepresent chapter, the methodology described in section 3.2.1 was used to incorporate [K + ]i dynamics into the CRLP model while retaining its capacity to reproduce
experimentally observed values of arrhythmic risk-related markers.
[K + ]i dynamics were introduced through the following equation:d +Cmem[K ]i = −IK tot ·dt
Vmyo · Frdy

(3.11)

where IK tot is the total potassium current, Cmem is the membrane specific capacitance, Vmyo is the bulk cytosol volume and Frdy is Faraday’s constant.
IK tot was calculated as the sum of all potassium currents and the stimulus current:
IK tot = Ito + IKr + IKs + IK1 − 2 · INaK + ICaK + IKp + Istim(3.12)
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a)

61

b)
CRLP model with [K+]i dynamics

CRLP model with [K+]i dynamics−10∫<CL> IK,tot dt (ms*pA/pF)[K+]i (mM)

150

145

140

0

2000

4000
6000
time (s)

8000

−15
−20
−25

10000

c)

0

2000

4000
6000
time (s)

8000

10000

d)
GPB model with [K+]i dynamics∫<CL> IK,tot dt (ms*pA/pF)

GPB model with [K+]i dynamics[K+]i (mM)

135
130
125
120
0

2000

4000
6000
time (s)

8000

10000

−20
−30
−40
−50
−60
−70
0

2000

4000
6000
time (s)

8000

10000

Figure 3.2: Temporal evolution of [K + ]i in CRLP and GPB epicardiacl cell
models after introducing [K + ]i dynamics: a) and c) show [K + ]i evolution whenthe models (CRLP and GPB, respectively) are stimulated with a CL of 1000 ms.
b) and d) show the evolution of the integral of IK tot during one cycle for the same
simulation as in a) and c).
where Ito is the transient outward potassium current, IKr is the rapidly activating
potassium current, IKs is the slowly activating potassium current, IK1 is the inward
rectifier potassium current, INaK is the sodium-potassium pump current, ICaK is thecurrent generate by potassium ions passing through the L-type calcium channels,
IKp is the plateau potassium current, and Istim is the stimulus current.
If Eq. 3.11 is directly added to the CRLP and GPB models, [K + ]i drifts(Fig. 3.2, panels a) and c), respectively). The continuous increase in [K + ]i is due
to a larger potassium influx in each cardiac beat, which can in turn be explainedby an imbalance in the potassium current. Although it is not straightforward to
establish a physiological range for [K + ]i because there are no experimental measurements of this concentration, all human ventricular cell models published in
the literature, despite having different [K + ]i values under steady-state conditions,they all lie within the range 120-145 mM. This range is in agreement with thecorresponding experimentally reported values for the resting membrane potential,
which is close to the Nernst potential for potassium [64].

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In Fig. 3.2, panels b) and d), the integral of the total potassium current during
one cardiac cycle is shown, which gives an idea of how far the models are from equilibrium. The above noted issue associated with the introduction of [K + ]i dynamics
into the CRLP model can be sorted out by readjusting the current conductances tomaintain [K + ]i homeostasis. Since, at the same time, the performance of the modelin terms of representation of electrophysiological behavior wants to be retained, aconstrained optimization problem was formulated in this study. This problem wastackled by applying the response surface-based optimization methods presented in
3.2.1.
For further model assessment, optimization was performed as described in thenext paragraph while considering different initial [K + ]i values in the range 120-145
mM.
Optimization-driven model parameter update
A new set of current conductances for the modified CRLP model was sought suchthat potassium homeostasis was preserved while a number of electrophysiologicalproperties were guaranteed to remain within experimental limits. This was statedas a constrained optimization problem of the form:
(∫
)2
min
IK tot (x)(3.13)<CL>subject tomLL ≤ m(x) ≤ mU U ,(3.14)x ≤x≤x ,(3.15)

L

U

where x is the vector of scaled current conductances(see 3.2.1), with lower an∫upper bounds xL and xU , respectively, <CL> represents the integral during a
cardiac cycle and m is a vector function of selected electrophysiological properties
with lower and upper physiological bounds defined by vectors mLL and mU U .
Initialization was defined based on the epicardial CRLP model (where [K + ]i takes
a constant value of 138 mM). Other initializations were additionally tested, as
described in 3.2.2.
Constraint functions
Seven ventricular electrophysiological properties of epicardial cells were considered
for definition of inequality constraints in 3.2.2:• APD90 : APD is considered the main preclinical marker of drug cardiotoxicity. APD prolongation has been linked to long QT syndrome and increased
risk of developing Torsades de Pointes [77, 78]. In this study, APD90 wasused to denote AP duration at 90% repolarization.
• Triangulation: This marker quantifies the shape of the final part of the
AP and is defined as the difference between APD at 90% and 50% of repolarization. Low triangulation values indicate square APs, while high values

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indicate triangular APs. Triangulation has been proposed as a marker of proarrhythmia [77], with long APD90 values without triangular APs considered
as anti-arrhythmic and with triangular APs considered as pro-arrhythmic ingeneral.
• Systolic and diastolic intracellular calcium ([Ca2+ ]i ) levels: Calcium
transient properties evaluated at different pacing frequencies have been reported as arrhythmic risk markers [79, 80]. In this study, diastolic (restinglevel) and systolic (peak value) calcium transient levels were evaluated at 0.5
Hz and 1 Hz steady-state pacing.
• Time constants of APD90 adaptation to abrupt changes in CL: Adaptation of ventricular repolarization duration to abrupt changes in CL has been
proposed as an arrhythmic risk marker [81]. In this study, the dynamics of
APD90 adaptation to abrupt changes in CL were fitted by two exponentials
with associated time constants τf ast and τslow , following the methodology
proposed in Pueyo et al. [13]. Only τslow has been considered since there is
no available quantitative experimental data on τf ast values.
Two stimulation protocols were used to calculate the above described markers:• Steady-state protocol: A train of 3,000 stimulation pulses was deliveredat a given CL. Simulations were carried out using two different CL values:
1,000 and 2,000 ms. The evaluated markers were: APD90 and triangulation
at 1,000 ms; and diastolic and systolic [Ca2+ ]i at 1,000 and 2,000 ms.
• Abrupt changes in CL protocol: The cell was stimulated with a CL
of 1,000 ms for 8 minutes and then with a CL of 600 ms for additional 8
minutes. APD90 dynamics after the abrupt CL change were best fitted by
two exponentials, the second of which was characterized by τslow .
Physiological bounds for constraint functions
Lower and upper bounds mLL and mU U were defined according to physiologicalexperimental values reported in the literature for the analyzed properties. The
selected range for APD90 was 280-310 ms. Even though some studies have reported
wider ranges, up to 271-366 ms [2, 8, 96], a restricted interval was used to keep
the model within the same range of existing human ventricular cell models [15, 46,
51, 64, 120]. The range for triangulation was 44-112 ms [2, 96]. For the diastolicand systolic [Ca2+ ]i levels at 1 Hz and 0.5 Hz stimulation, the CRLP model, as
occurs with other recent human ventricular AP models (GPB, ORd, TP06, TP04)[12, 15, 46, 51, 64, 120, 121], is out of the physiological range reported in theliterature [10, 97]. For that reason, the physiological range was extended to ensurea feasible solution of the optimization problem. Nevertheless, the optimal solution
was evaluated to confirm that either it let the model lie within physiological range
or at least improved the results of the CRLP model. The physiological range forτslow was set to 70-110 s [95]. Table 3.1 summarizes the evaluated markers andJesúsCarroNew Methodologies for the Development
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APD90 (ms)Triangulation (ms)Sys. [Ca2+ ]i 1Hz (µM)
Sys. [Ca2+ ]i 0.5Hz (µM)
Dia. [Ca2+ ]i 1Hz (µM)
Dia. [Ca2+ ]i 0.5Hz (µM)∫τslow (s)I· dt (ms·pA/pF)
CL K tot

Physiol.
Range
271-366
44-112
1.59-2.01
0.71-1.68
0.20-0.33
0.14-0.32
70-110

Selected
Range
280-31044-1120.602-2.01
0.523-1.68
0.097-0.33
0.091-0.32
70-110

CRLP
Model
305.6
78
0.602
0.523
0.097
0.091
105.6
-28.52

This
work
280.9
84.6
0.701
0.664
0.100
0.096
106.0
-3.9·10−9

Table 3.1: Electrophysiological markers evaluated in the proposed response surface approximation-based optimization.
their physiological ranges in the second column, whereas the third column indicatesthe lower and upper bounds used for the optimization.
Sensitivity Analysis
Prior to solving the minimization problem, a sensitivity analysis was performed on
the CRLP model following the methodology described by Romero et al.[12]. The
aim was to identify the conductances that most notably influenced the objectivefunction (the integral of the total potassium current) and the constraints (electrophysiological properties) in order to define the set of decision variables for the
optimization.
For the sensitivity analysis, the conductances were varied by ±30% one at atime. Fourteen conductances were considered in the sensitivity analysis, namely:
Gto , the maximal conductance of the transient outward potassium current; GKs ,the maximal conductance of the slowly activating potassium current; GpCa , themaximal conductance of the sarcolemmal calcium pump; GKr , the maximal conductance of the rapidly activating potassium current; GKp , the maximal conductanceof the plateau potassium current; GNaK , the maximal activity of the Na-K pump;Gncx , the maximal activity of the Na-Ca exchanger; GCaL , the maximal conductance of the L-Type calcium current; GK1 , the maximal conductance of the inward
rectifier potassium current; GN a , the maximal conductance of the fast sodium current; GClCa , the maximal conductance of the calcium-activated chloride current;
GN a,Bk , the maximal conductance of the background sodium current; GCl,bk , themaximal conductance of the background chloride current; and GCa,bk , the maximal
conductance of the background calcium current.
For each property m and conductance p, the percentage of change (Dm;p;a ) andsensitivity (Sm;p;a ) were calculated as follows [12]:Dm;p;a =Universidad Zaragoza

Mp;a − Mcontrol· 100,
Mcontrol3. Improvement of Existing Electrophysiological Models

65

Dm;p;a − Dm;p;−a· 100,
2·a
where Mp;a is the value of property m when the conductance p is varied by thepercentage a and Mcontrol is the value of property m under control conditions. As
shown above, sensitivity (Sm;p;a ) is calculated as the ratio of the difference between
the percentage of change (Dm;p;a − Dm;p;−a ) and the interval of change (2 · a). In
particular, for this study a = 30%:
Sm;p;a =Sm;p;0.3 =Dm;p;0.3 − Dm;p;−0.3
Mp;0.3 − Mp;−0.3 105· 100 =·
2 · 0.3
Mcontrol6

Sensitivity was calculated for each of the electrophysiological markers describedin section 3.2.2 and for the integral of the total potassium current (IK tot ) during
one cardiac cycle at 1 Hz steady-state pacing.
Model Comparison
Considering the difficulty of directly comparing the absolute value of each conductance for different computational models, due to the dependence on the wholecurrent formulation, ionic current traces calculated during a cardiac cycle at 1 Hz
steady-state pacing were compared for the CRLP, TP06, GPB and ORd models.
The comparisons aimed at verifying that the modified CRLP model had current
densities of similar magnitude and ionic currents of similar shapes as calculatedwith other published models.
3.2.3 Frequency-dependence behavior
Pacing at different frequencies was simulated to assess whether the modified CRLP
model retained the frequency-dependence behavior of the original CRLP model. To
this end, AP D90 was calculated at steady-state for CLs ranging from 250 to 5000
ms (in each case computed after pacing for 3,000 cycles).
3.2.4 Simulation of changes in potassium concentration
Hypokalemic and hyperkalemic conditions were simulated with the original andthe modified CRLP models. Two types of simulations were performed to compare
the models with the experimental results.
First, the effect of changing [K + ]o was assessed when the cell was not stimulated to compare with the experimental results described in [122]. Subsquently,
and following the experiments performed in [116], the extracellular potassium concentration, [K + ]o , was increased from 2 mM to 7 mM in 0.5 mM steps using
a homogeneous 4-cm long fiber composed of epicardial cells. The physiological[K + ]o value of 5.4 mM replaced the value of 5.5 mM in the simulations [116]. The
conductivity σ, defining the velocity of the stimulus propagation, was set to rendera Conduction Velocity (CV) close to 65 cm/s under control conditions [88]. The
cell capacitance was set to Cm = 1 µF/cm2 . For each [K + ]o value, the model was
first stabilized (without stimulation) until the product of the gates h·j reached 99%

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New Methodologies for the Development
and Validation of Electrophysiological Modelsof the steady-state value hss · jss , which was subsequently followed by 1 Hz pacing
simulation. Stimulus current pulses twice the diastolic threshold were applied at
one end of the cable. The diastolic threshold was defined as the minimum amplitude required for propagating five pulses along the fiber. The model was studied
by stimulating the tissue with a train of 100 basic stimulations (S1) delivered at a
CL of 1,000 ms. [K + ]i was measured at the end of each stimulus at five different
positions along the cable situated at distances from the origin of 1.5, 1.75, 2, 2.25
and 2.5 cm. The average value over the five locations was calculated.
3.2.5 Implementation
The CRLP model was implemented in Matlab based on the original CellML file[120]. The code used to compute the simulations and to calculate the different
markers is freely available at https://github.com/ChusCarro/MatCardiacMLab/releases/tag/0.1.0.
The minimization algorithm was implemented in Matlab. For 1D tissue simulations, a semi-implicit operator-splitting scheme implemented in Fortran was used
to solve the propagation [87] with a space discretization of ∆x = 0.1 mm and atime discretization of ∆t = 0.002 ms, as in [120].
All computations were performed by using the high performance computing
cluster (ICTS) “NANBIOSIS”. Details about the cluster can be found at: http://www.nanbiosis.es/u27-e1-hermes-cluster
3.3 Results
3.3.1 Sensitivity Analysis
Results of the sensitivity analysis applied to the epicardial version of the CRLP
model are shown in Figure 3.3. The value of the integral of the total potassium
current, IK tot , in one cardiac cycle was highly dependent on the conductance ofthe background chloride current, GCl,bk , and the L-type calcium current, GCaL (S30
values of -412.62% and 396.83%, respectively). Other currents were also found toplay an important role in regulating potassium homeostasis: GNaK , 199.46%; GK1 ,-152.06%; Gto , -110.57%; GKr , -81.16%; and Gncx 62.72%. The currents with thelargest effect on the APD were found to be the same as for the objective function,although the sensitivities were notably reduced: GCl,bk , -48.56%; and GCaL , 33.41%.
Triangulation was mainly modulated by: GK1 , -57.7664%; and GKr , -20.1113%.
Diastolic and systolic [Ca2+ ]i values were mostly influenced by GCaL (27.65% and
171.71%, respectively). These markers were also influenced by changes in thefollowing conductances: GNaK (-16.08%, -78.62%), Gncx (-15.74%, -74.48%), GCl,bk(-14.53%, -80.66%) and GCa,bk (11.51%, 47.46%). Regarding τslow , GNaK was the
conductance with the largest effect on this marker (-128.37%), with the remaining
conductances not playing a significant role.
Based on these results, five conductances were selected as decision variables forthe minimization problem described in 3.2.2, namely: GCl,bk , GCaL , Gncx , GK1 , and
Gto . Of note, GNaK was discarded because any modification on this conductance

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67

Figure 3.3: Results of the sensitivity analysis applied to the original CRLP model.
The blue scale indicates relative sensitivity for each marker. White color indicatesmaximum relative sensitivity and dark blue color indicates that property and parameter are independent. Percentages in the white boxes indicate the absolutesensitivity of the property. A minus sign in a box indicates that marker and model
parameter vary inversely. ‘Trian’ stands for Triangulation, ‘Sys’ stands for systolicand ‘Dia’ stands for diastolic.

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significantly influenced τslow , with this effect not being counterbalanced by changes
in any other conductance (see Figure 3.3).
3.3.2 Non-linear optimization
The optimization algorithm found a minimum after four iterations. Variations in
the updated current conductances with respect to their initial values were: GCl,bk ,+19.34%; GCaL , +3.93%; Gncx , -33.25%; GK1 , -14.08%; and Gto , +22.35%.
With this new set of conductance values, and after introducing [K + ]i dynamics
in the model, [K + ]i stabilized at 138 mM. The value of the electrophysiologicalmarkers are described in table 3.1. All markers were within physiological ranges orcloser than with the original CRLP model (last column in table 3.1).
When the model was optimized while considering different initial [K + ]i values
in the range 120-145 mM, results were practically coincident with those reported
above for [K + ]i initialized at 138 mM. At steady-state, [K + ]i reached a value of
138 mM as well, differences in the values of the markers where below 1% and the
maximum difference between optimal conductances was lower than 2.5%. In thiscase, the algorithm required six iterations rather than four to reach the optimal
solution.
3.3.3 Model Comparison
To evaluate the impact of the new values for the estimated ionic conductances in
the modified CRLP model, the AP and the associated ionic currents (ICaL , IK1 ,Incx , ICl,bk and Ito ) were compared with the equivalent properties in the original
CRLP model and in other human ventricular cell models. The results are shownin Fig. 3.4. Other currents that were not considered for the optimization but that
could be altered as well, like IKr , IKs and IKp , are shown in Fig. 3.5.
The main differences in the AP between the original and modified CRLP models
were a decrease in the membrane potential during the plateau phase as well as
shortening of the APD. There were small differences in the peak value of somecurrents, but in all cases the modified CRLP model was within the range of theother models or within experimentally reported bounds. ICaL peak was larger in themodified model with respect to the GPB model and the original CRLP model, but
smaller than in the TP06 model. IK1 peak was slightly smaller in the modified modelwith respect to the original CRLP model, which had the smallest peak value among
all ventricular models, but in any case within the range reported experimentally(0.4-1.8 pA/pF[76, 96]).
An important finding was that, while Gncx and GCl,bk in the modified CRLP
model were significantly different from those in the original CRLP model (-33.25%
and +19.34% respectively), the effects in terms of ionic current traces led to both
Incx and ICl,bk not being notably different from those in the original CRLP model.
The IKp current remained as in the original model, while IKs was different but took
really small values in both cases. IKr was diminished in amplitude by 32.6% and
its integral during a cycle was reduced by 30% with respect to the CRLP model.
The extent of APD prolongation following blockade of this current was 10% for the

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3. Improvement of Existing Electrophysiological Modelsa)b)Action Potential

Intracellular Potassium Concentration
150[K+]i (mM)

50
V (mV)

69

0
−50
−100

0

100

200 300
time (ms)

400

140
130
120

500

c)

0

100

200 300
time (ms)

400

500

d)
L−Type Calcium current
0ICa,L (pA/pF)

Incx (pA/pF)

Sodium−Calcium Exchanger current
0.5
0
−0.5
−1

0

100

200 300
time (ms)

400

−5

−10

500

e)

0

100

200 300
time (ms)

400

500

f)
Inward Rectifier Potassium current

Transient Outward current
20
Ito (pA/pF)

IK1 (pA/pF)

2
1
0
−1

0

100

200 300
time (ms)400

V (mV)

ICl,Bk (pA/pF)

1
0

200 300
time (ms)

400

500

400

500

This work
CRLP
GPB
TP06
ORd

0
−50
−100

200 300
time (ms)10050

Background Chloride current

100

0Action Potential

2

0

0
−10

500

g)

−1

10

0

100

200 300
time (ms)

400

500

Figure 3.4: Model comparison. Blue lines represent the results of this workwith inclusion of [K + ]i dynamics, green lines represent the CRLP model, red lines
represent the GPB model, cyan lines represent the TP06 model and purple lines
represent the ORd model. All the figures show results at steady-state (after pacingfor 3,000 cycles at a CL of 1000 ms). a) Action potential; b) Intracellular potassium
concentration; c) Sodium-calcium exchanger Current; d) L-type calcium current; e)Inward potassium current; f) Transient outward potassium current; and g) Background chloride Current.

JesúsCarroNew Methodologies for the Development
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Rapid Delayed Rectifier Potassium current

1

0.2

IKr (pA/pF)

Rapid Delayed Rectifier Potassium current

0.5

I

Kr

(pA/pF)

a)

0

0

100

200 300
time (ms)

400

0.1
0

500

0

100

200 300
time (ms)

400

500

b)
Slow Delayed Rectifier Potassium current
0.01
IKs (pA/pF)(pA/pF)

Slow Delayed Rectifier Potassium current
1
0.5

I

Ks

0
−0.5

0

100

200 300
time (ms)

400

0

−0.01

500

0

100

200 300
time (ms)

400

500

c)
Plateau Potassium current

Plateau Potassium current
0.1
IKp (pA/pF)

IKp (pA/pF)

2

1

0

100

V (mV)

0200 300
time
50(ms)

0400Action
500 Potential

0100

200 300
time (ms)

400

500

This work
CRLP
GPB
TP06
ORd

0
−50
−100

0.05

0

100

200 300
time (ms)

400

500

Figure 3.5: Model comparison. The colors in this figure are the same as thoseused in Fig.3.4. Results are shown at steady-state (after pacing for 3,000 cycles at
a CL of 1000 ms). The graphics on the right are zoomed versions of the graphics
on the left. a) Rapid delayed rectifier potassium current; b) Slow delayed rectifierpotassium current; and c) Plateau potassium current.

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3. Improvement of Existing Electrophysiological Models

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150

100
APD90 (%)

APD90 (%)

150

50

This work
CRLP

100

50
2000 3000 4000 5000
CL (ms) 01000 2000 3000 4000 5000
CL (ms)Figure 3.6: Steady-state AP D90 values as a function of pacing CL.
0

1000

a)

b)
K = 2.5 mM

-75

-75

o

200

K = 3.0 mMoV (mV)-85o

K = 4.0 mMo-85

K = 4.5 mMoKo = 5.0 mM

-90

-90

Ko = 5.4 mM
Ko = 6.0 mM

-95

-95
0

100

0

200

100

time (ms)

300

200

time (ms)t 90% (ms)

-80

-80

V (mV)

K = 3.5 mM

150
100
50
0

Ko = 6.5mM
300
Ko = 7.0 mM

3

4

5

6

7

[K +]o (mM)

Figure 3.7: Results for an unstimulated cell simulated with the modified CRLP
model following changes in [K + ]o . a) Temporal evolution of membrane potential;b) Time to achieve 90% of the steady-state membrane potential (t90% ).
modified CRLP model, as compared to 14% for the original CRLP model, being
both values within experimentally defined limits [54].
3.3.4 Frequency response
Fig. 3.6 shows steady-state normalized AP D90 values calculated with the original
and modified CRLP models. Normalization was performed with respect to the
AP D90 value for CL=1000 ms. Both models presented very similar behaviors,with the only relevant difference being observed for CL=250 ms. For this CL, themodified CRLP model showed alternans, while the original CRLP model did not.
3.3.5 Response to changes in extracellular potassium
Fig. 3.7a shows the temporal evolution of membrane potential following changes in[K + ]o for an unstimulated cell simulated with the modified CRLP model. During
the first 100 ms, membrane potential varied following the corresponding change
in [K + ]o , in line with the results reported in [122]. Figure 3.7b shows the time

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72

b)
138.6

138.6

138.4

138.4

138.2

138.2

138

138

137.8
CRLP
This work (t = 20 s)
This work (t = 40 s)
This work (t = 60 s)
This work (t = 80 s)
This work (t = 100 s)

137.6
137.4
137.2
137
3

4

5

[K +]o (mM)

67

[K +]i (mM)[K +]i (mM)

a)

137.8
137.6
K = 2.5 mM
K = 3.5 mM
K = 4.5 mM
K = 5.4 mM
K = 6.5 mM
K = 7.0 mM

137.4
137.2
137
0

20

40

60

80

100

Time (s)Figure 3.8: [K + ]i evolution following [K + ]o changes simulated with the modified
CRLP model. a) [K + ]i dependence on [K + ]o at different time instants; b) timecourse of [K + ]i for different [K + ]o values.
needed to reach 90% of the steady-state membrane potential (t90% ). The value fort90% in the simulation for the same change in [K + ]o than in [122] (from 4.4 mM to
6.6 mM) was 16.2 ms, which is somewhat faster than the experimentally reported
results in [122] for rabbit (80 ms) and rat (147 ms) ventricular myocytes.
Fig. 3.8a shows a snapshot of [K + ]i values calculated with the modified CRLP
model at different times following changes in [K + ]o as described in 3.2.4. Fig. 3.8bshows the temporal evolution of [K + ]i in response to changes in [K + ]o . Under control conditions, [K + ]i remained constant in time. As expected, following simulated
hyperkalemia ([K + ]o > 5.4 mM ), [K + ]i increased as [K + ]o increased, whereas
for simulated hypokalemia ([K + ]o < 5.4 mM ), [K + ]i decreased in response to[K + ]o decreases. For [K + ]o ≤ 2.0 mM , tissue became inexcitable. In table 3.2,
the steady-state value of [K + ]i in the optimized CRLP model is shown for eachsimulated [K + ]o value.
3.3.6 Computational Cost
Each iteration of the optimization algorithm involved computations associated witha number of stimulation protocols. In Table 3.3, a summary of the computational
cost is shown for each of the protocols. In each iteration, the protocols were run
twice: one for the generation of the database (243 simulations in parallel) andanother one for the evaluation of the solution (6 simulations in parallel). The
variance in the computation times is due to the fact that computations were run indifferent nodes of a high performance computing cluster, namely the one describedin section 3.2.5. The four iterations of the optimization algorithm were computedin 23 hours and 16 minutes.

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3. Improvement of Existing Electrophysiological Models[K + ]o (mM)
2.5
3.0
3.5
4.0
4.5
5.0
5.4
6.0
6.5
7.0

73

[K + ]i (mM)
73.5
84.8
95.8
107.2
118.3
129.3
138.0
151.4
161.6
172.5

Table 3.2: Steady-state values of [K + ]i for different [K + ]o values as calculatedwith the optimized CRLP model.
ProtocolSteady-state CL = 1000 ms
Steady-state CL = 2000 ms
APD Rate Adaptation

Mean
01:31:45
02:35:38
00:28:36

STD
00:41:31
01:03:47
00:14:17

Max
02:29:31
03:41:11
00:53:57

Table 3.3: Computational cost of the different protocols used for the optimization.
3.4 Discussion
In this chapter a mathematical framework to adjust ionic current conductances ofcardiac AP models by using response surface approximation-based optimization is
proposed. The framework has been applied to update the CRLP human ventricular AP model, described in Chapter 1, by incorporating [K + ]i dynamics. Suchincorporation involves readjustment of ionic current conductances to avoid drifting
in intracellular potassium. While adjusting for current conductances, a number ofelectrophysiological properties are required to remain within available experimental
ranges by properly defining constraints to the optimization problem that is being
solved. The proposed framework can be easily extended to solve other problems
frequently arising when developing cardiac computational models, for instance related to the update of existing models to reproduce the response to pharmacologicalion channel inhibitions on top of baseline conditions. In the following subsections,the main aspects of the proposed methodology and the results obtained in thiswork are discussed.
3.4.1 Response surface optimization
The use of response surfaces to minimize an objective function subject to a numberof constraints, as proposed in this chapter, allows reaching a solution to the proposed problem in a small number of iterations, thus reducing the computational

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New Methodologies for the Development
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cost with respect to other previously published approaches. In [115], Rodriguez
et al. proposed an algorithm that uses augmented Lagrangian techniques andrelies on successively building RSA of the augmented Lagrangian to solve a minimization problem like that defined by Eqs. 3.1-3.4. In this chapter, the trust region
ratio was used to independently validate the RSA for the objective function and theconstraints instead of considering the augmented Lagrangian of the problem. Our
proposal led to a reduction in the number of times the database was generated and,thus, simplified the implementation. The model that is optimized in the present
work is much more complex than those used in [115], but the number of times the
database was generated was smaller than in all the cases described in [115].
Also, differently to other optimization algorithms based on e.g. Genetic Algorithms [112, 114], the response surface approach proposed in this chapter guarantees the convergence to a local minimum of the problem when the response surfaces
are built such that they match the objective function up to the first order aroundthe point where the approximation is built. In addition, our response surface approach allows identifying a potential minimum anywhere within the trust region,
not necessarily contained in the database.
Finally, in this chapter the proposed optimization algorithm was used to solve a
problem where a new characteristic was added to a human ventricular cell model,
the CRLP model, while a number of electrophysiological markers were guaranteed
to remain within physiological range or closer to it than with the original CRLP
model. In other studies, like [123], the aim of the analysis is similar, but, as opposedto the present study, the objective function is defined to include multiple electrophysiological properties, for some of which a physiological range may be available
while for others it may not, rather than considering the electrophsyiological properties as elements to define the constraints imposed in the optimization. In any case,the same response surface approximation-based optimization algorithm employedin the present study could be applied to the problem defined in [123].
3.4.2 Database generation
The size of the database depends on the selected sampling technique. In thischapter a three-level full factorial design was used for sampling, which involved
3N sample points, with N the number of variables to be adjusted (five in thisstudy). However, the proposed optimization framework is independent of the sampling strategy or the size of the database as long as the database provides enoughinformation to fit the response surface. For instance, increasing the factorial number from 3 to 5 (the database size increased from 35 = 243 to 55 = 3125 sample
points) only reduced the total number of iterations from 4 to 3, without changing
the identified optimum. However, the computational cost increased tenfold.
The advantage of using factorial designs is that they provide unbiased information for parameter identification with polynomial RSAs since they provide optimal
space filling [117]. However, other sampling strategies are also possible. Monte
Carlo simulations represent a very flexible sampling strategy where the optimal
number of sampling points can be defined in terms of the type of RSA to be used.

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In this regard, Monte Carlo simulations were used in [124] to perform sensitivity
analysis in electrophysiological models by means of linear multivariate regression.
In addition to generating the required data for model fitting, Monte Carlo simulations were also used to perform population-based simulation studies, as in Wamsley
et al. [125].
In this chapter, the full factorial technique was implemented in order to guarantee that the algorithm was not limited by the sampling technique since thegenerated database is sufficient to build a second order response surface. However,as mentioned above, other techniques could be tested to reduce the computational
cost when a larger number of current conductances are involved or when large
parallel computing capacity is not available.
3.4.3 Computational cost
The major benefit of the methodology proposed in this chapter in terms of computational cost is the high degree of parallelization. It should be noted that the
global computation time depends mainly on two factors: the employed cell modeland the stimulation protocols that are performed. As compared to the CRLP or
GPB models, which take long computation times, the computational time for amodel like TP06 could be reduced to just a few hours. For the computations of
this study a high performance computing cluster was used. The number of simulations involved per iteration makes it possible to use other types of infrastructures
such as GPUs.
Nevertheless, the computation times associated with the presented methodology
based on response surface approximations are short if compared with those for thesame optimization solved by using a genetic algorithm, which could require around
100 generations and would cost more than 350 hours, even if all individuals in the
population were simulated in parallel. In [112], 500 generations were needed to fit
the model, which would still involve longer computation times.
3.4.4 Definition of the optimization problem
In this chapter, the integral of the total potassium current (IK tot ) during one cardiac
cycle was selected as the objective function for minimization. This selection wasmade to maintain [K + ]i homeostasis after incorporation of [K + ]i dynamics into
the CRLP human ventricular model. Optimal current conductances were identified, with additional constraints imposed to guarantee that fundamental electrophysiological properties took values within experimental limits. For some of those
properties the physiological range reported in experimental studies of the literature
is extremely large. To avoid a solution for the modified CRLP model that, althoughin physiological range, was very far from the average experimental behavior, therange of allowed APD values was somewhat restricted, as described in 3.2.2.

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3.4.5 Model comparison
The adjusted CRLP model developed in this study was compared with previous
human ventricular AP models published in the literature, namely the CRLP, GPB,
TP06 and ORd models. This comparison was made in terms of AP traces, [K + ]iconcentrations and ICaL , IK1 , Incx , ICl,bk and Ito currents. In practically all cases,
the results obtained for the adjusted CRLP model were along the lines of thoseobtained with the other models. The only current that was found to be smaller forthe modified CRLP model as compared to the other models, including the original
CRLP model, was IKr . This reduced current led to shorter APD prolongation
following IKr blockade, but still within experimentally reported limits. Specifically,in [54] an average APD prolongation of 56.3% with a standard deviation of 34.6%
was reported. The difference between the mean experimental prolongation andthat obtained for the modified CRLP model was within twice the experimental
standard deviation.
One of the differences between the GPB / CRLP models and the other humanventricular models is the inclusion of the ICl,bk current. This current was expected tohave a minor role in modulating electrophysiological properties, but, as reported in[121], APD showed the highest sensitivity (-86%) to changes in the maximal ionic
conductance of the ICl,bk current. In the CRLP model, this sensitivity was reduced(-46%) but it was still large. As there is similarity between the sensitivities of
the APD and the integral of the total potassium current to variations in ionic
conductances (see Figure 3.3), the ICl,bk current ended up having a very important
indirect role in modulating potassium balance.
As shown in the presented results, major differences between the modified andthe original CRLP models include APD shortening and decrease in the membranepotential during the plateau phase. Regarding the latter, experimental results [126]show that the plateau voltage takes values around 18 mV, while in the modified
CRLP model the plateau voltage is 8 mV. Differences in Incx and ICl,bk , even if notlarge, could still underlie those major deviations of the modified CRLP model withrespect to the original one. Future studies could include the membrane potential
during the plateau phase as an electrophysiological marker to be included in the
optimization algorithm for model refinement. This could definitely have an impact
on balancing the currents involved in this phase of the AP.
Our results demonstrate that with the proposed methodology it was possible
to successfully update the CRLP model by incorporating [K + ]i dynamics while
keeping the model’s ability to reproduce risk-related markers for investigation ofcardiac arrhythmias. As described in Table 3.1, the evaluated risk markers remain
within experimental limits or are closer to their physiological values than with theoriginal CRLP model.
3.4.6 Response to potassium concentration changes
While for some investigations the assumption of constant [K + ]i may be valid,
there are many situations where [K + ]i varies, for instance following changes in thefrequency of stimulation or in [K + ]o values. As shown in 3.2.2, the assumption of

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constant [K + ]i may also hide an imbalance between potassium currents and [K + ]i ,as the latter is not allowed to change.
Simulation results obtained when varying the level of [K + ]o allowed confirmingthat both the incorporation of [K + ]i dynamics and the use of the optimization algorithm to readjust the ionic conductances improved the CRLP model for simulationof abnormal conditions (see section 3.3.5). Although there are no experimental
measurements of [K + ]i , the model can be assessed by using measurements thatindirectly validate its behavior. On the one hand, the model is able to reproduce
behaviors similar to those of the experimental results presented in [122]. On theother hand, the inclusion of [K + ]i dynamics make it possible to evaluate variations
in [K + ]i in response to [K + ]o changes like those reported in hemodialysis scenarios[116]. In this latter case, simulated [K + ]i variations were in the expected direction
as to compensate for the effects of changing [K + ]o .
When [K + ]o increases, there is an increase in the Nernst potential for potassium and, thus, a reduction in the driving force of potassium, which diminishes
the amount of potassium leaving the cell. Additionally, an increase in [K + ]o alters the current through the sodium-potassium pump. Both effects contribute toincrease [K + ]i , which leads to a reduction in the Nernst potential for potassium,
thus returning the cell to an equilibrium state.
3.5 Conclusions
The optimization framework proposed in this chapter allows estimating ionic current conductances of AP models at an affordable computational cost while guaranteeing physiologically plausible values of selected electrophysiological properties.
This framework has been applied to introduce [K + ]i dynamics into the CRLP
human ventricular AP model developed in Chapter 1. Solving the formulated
optimization problem avoids the imbalance generated in the CRLP model after
introducing [K + ]i dynamics, without deteriorating the performance of the modelin terms of representation of electrophysiological behavior and arrhythmic riskrelated properties. The updated CRLP model built in this chapter is thus suitablefor investigation of cardiac arrhythmias, particularly under conditions involving
large changes in [K + ]i .

JesúsCarroChapter 4

Limitations inElectrophysiologicalModel Development
and Validation
4.1 Introduction
Chapter 2 and chapter 3 deal with methodologies for the development and updateof electrophysiological models. Importantly, methodologies are also required forvalidation of new or updated model formulations based on available experimental
data. Any methodology proposed for such purpose should consider all variables
and spatio-temporal scales involved in the corresponding modeling and simulation.
This becomes a challenging task, particularly for complex biophysically detailed
models. From the earliest mathematical model of an electrical cell’s action potential (AP) developed by Hodgkin and Huxley in the fifties, the complexity of current
AP models has grown considerably. The advent of new experimental techniques
has made large sets of experimental data readily available, which has motivatedthe development of more complex models to accurately describe cellular electrical activity. Whereas growing in model complexity is a natural consequence of
the increased knowledge [127], the more complex the model, the more difficult theidentification of model parameters tends to be. An AP model involves the sum ofdifferent transmembrane ionic currents and the balance between intra- and extracellular ionic concentrations. Each ionic current follows a mathematical formulationin which several effects are modeled, e.g., ion channel activation and inactivation
gating or current conductance. For each effect, a number of model parameters are

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identified based on data from experimental protocols specific for each particularionic current.
The experimental protocols used to obtain most of the parameters of each ioniccurrent are performed in isolated cells. But, due to the sensitivity of some ionicchannels to the cell isolation procedure applied prior to voltage-clamp experiments[106], the conductances of the ionic currents in cardiac models are often not estimated from direct measurements of the current density. Instead, individual channel conductances are adjusted so that measures from model-generated APs closely
match experimental AP measurements in tissue such as AP duration (APD) orothers. In the Courtemanche-Ramirez-Nattel (CRN) model [47], the ionic conductances GNa , GK1 , Gto , GKr and GKs were fitted to obtain a correct input resistance,
AP morphology, AP amplitude (APA) and upstroke velocity (dV /dtmax ). In a lateversion of the Luo-Rudy (LR) model [107], GKs was fitted to get the right APD
prolongation when the IKs current was blocked. Taking those models as an example, in the tenTusscher-Noble-Noble-Panfilov (TNNP04) model [46], GKs was set
to obtain physiologically plausible APD values for each cell type (epicardial, midmyocardial and endocardial). In the Grandi-Pasqualini-Bers (GPB) model [15],GNa was set so as to reproduce experimental measurements of APA and maximum value of the transmembrane potential (Vmax ). In the O’Hara-Rudy dynamic(ORd) model [64], the potassium current conductances were fitted to reproduce theexperimental effect on the APD when they were blocked. In the Carro-RodríguezLaguna-Pueyo (CRLP) model [120], using the sensitivity analysis proposed in [12],GK1 , GNaK , GCaL , and GNa were fitted to obtain not only APD values within physiological ranges, but also other markers of arrhythmic risk, including time constantsof APD rate adaptation or rate dependence of ionic concentrations.
On the contrary, the parameters that model current kinetics (gating parameters) are usually identified from single-cell experiments. The calibration process
is usually performed using a nonlinear least square fitting of voltage clamp databy assuming that each parameter’s effect is independent from the rest (e.g., thesteady-state of an inactivation gate is calibrated against experimental results whileconsidering that the time constant of the gate does not affect such results, whichmight not be correct). However, when the complexity of the model increases, theinteraction between effects becomes increasingly important. Therefore, assuming
independence of the effects when identifying model parameters may be misleading.
While other techniques have been proposed in recent years to improve the fitting of
the gating parameters [109, 128–130], none of the models analyzed in the present
thesis have used such techniques.
Once model parameters have been identified, the resulting AP models are validated against experimental measurements commonly obtained also from tissue
preparations. Characteristics such as resting membrane potential (Vmin ) and upstroke velocity (dV /dtmax ) are usually compared between model-generated andexperimental APs. In the CRN model, the role of different ionic conductances,the morphology of the AP and the behavior of the model for different pacing cycle
lengths (CLs) were compared with experimental observations. In the updated version of the LR model [107], the theoretical APD restitution curve was compared

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with experimental restitution curves obtained from optical recordings of cardiac
APs. In the TNNP04 model, simulated APD restitution curves (at 90% repolarization, AP D90 ) were evaluated in single cells to validate the model against
experimental results measured in tissue preparations. Also in this model, propagation in a homogeneous one-dimensional (1D) tissue was simulated to validate
the model in terms of Conduction Velocity (CV). In a subsequent version of themodel, the ten Tusscher-Panfilov (TP06) model [51], simulated APD restitution
curves (at 90% and 50% repolarization) in single cells were compared with experimental results. The GPB model was validated by comparing the predicted AP D90
prolongation caused by blockade of different potassium currents with experimental
results. The CRLP model, as the GPB model, was validated by comparing AP D90
prolongation caused by potassium current blockades with experimental results, butalso by comparing a number of computed markers not used in the fitting process.
For the aforementioned reasons, issues may appear in the calibration and/or
validation of electrophysiological models caused by two related aspects: submodel
variable relationships and cell-to-cell interactions during AP propagation. Regarding the first of such aspects, model parameters related to ion channel gating kinetics are commonly obtained by considering each gate of the channel independently.
Regarding the second one, ionic conductances are adjusted or validated against
experimental data obtained from tissue preparations while considering single cellcomputer simulations. In both circumstances, differences caused by not considering the corresponding interactions introduce non-negligible cross-effects between
model parameters that are not considered in the fitting process.
Other studies have analyzed issues related to the two aspects mentioned above.
In [131] differences between models of the same species to represent electrophysiological properties and how these differences propagate to tissue simulations wereanalyzed. In [132] how the uncertainty in the definition of gating variables propagates in multi-scale models was studied and the need to use observations acrossmultiple scales was highlighted. In [133] problems caused by the use of observations across multiple scales and how to characterize the relationships between
model parameters and the effect they have in the multi-scale model outputs was
explored.
In this chapter, experimental protocols are simulated in silico to analyze theconsequences of the corresponding interactions in two scenarios involving scales
ranging from ion channels to tissue: 1) characterization of L-type calcium voltagedependent inactivation; 2) identification of ionic current conductances with thelargest contribution to steady-state AP markers. In the first case, differences between the mathematical model, simulation results and experimental measurements
are analyzed to evaluate how interactions between model components affect thedevelopment and validation of mathematical ion channel models. In the second
case, differences between AP-based markers simulated in isolated cells and in homogeneous 1D tissue are analyzed to evaluate how propagation affects their values
as well as to assess the contribution of each ionic current to each marker.

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4.2 Materials and Methods

4.2.1 L-type calcium channel gating
In AP models, ionic currents are controlled by activation and inactivation gates.
Gates are modeled by functions varying between 0 and 1. Below a threshold potential, the activation gates are closed (the function value is 0) and the inactivation
gates are open (value 1). When transmembrane potential increases, the activation gates open (their values increase towards 1) and the inactivation gates close(their values decrease towards 0). In particular for the ICaL current, all human ventricular cell models analyzed in this thesis have one voltage-dependent activationgate d. This activation gate d multiplies the expression for the inactivation gating, whose formulation largely differs between models. In this chapter four of themost recently developed human ventricular cell models were considered: the GPBmodel, the TP06 model, the ORd model and the CRLP model. Voltage-dependent
inactivation of the L-type calcium current was studied in all of them.
The TP06 model is one of the most extensively used human ventricular AP
models. It represents an improved version of the model published in 2004 [46]in which the calcium dynamics, the slow delayed rectifier potassium current (IKs )and ICaL were reformulated. The TP06 model is based on experimental data fromhuman cardiomyocytes for most ionic currents and is defined for three types ofcells: epicardial, midmyocardial and endocardial. In the TP06 model, voltagedependent ICaL inactivation is modeled as the product of a fast, f2 , and a slow, f ,voltage-dependent inactivation gates as well as a calcium-dependent inactivation
gate, fCa :InactTP06
Ca,L = f · f2 · fCa
These gates have different time constants (τf2 , τf and τfCa ) and steady-state values(f2,∞ , f∞ and fCa,∞ ). This formulation is based on experiments that indicate thepresence of both fast and slow voltage-dependent ICaL recovery processes [1, 134,
135].
The GPB model is based on the rabbit AP model proposed in [63], which
includes subsarcolemmal and junctional compartments in the formulation of thecurrents and provides a detailed description of calcium handling. The GPB model
includes new definitions of ionic current densities and kinetics and is defined for
endocardial and epicardial cells. In the GPB model, voltage-dependent ICaL inactivation is modeled by a single voltage-dependent gate f and two calcium-dependent
gates, one for the subsarcolemmal compartment, fCa,sl , and another one for thejunctional compartment, fCa,j . The formulation of ICaL inactivation in the GPBmodel is as follows:
InactGPB
Ca,L = f · (Aj · fCa,j + Asl · fCa,sl )where Aj and Asl are the ratios of calcium-dependent inactivation channels in thejunctional and the subsarcolemmal compartment, respectively (Aj + Asl = 1).
The CRLP model is a modification of the GPB model. As in the original model,
the modified CRLP model describes the AP of endocardial and epicardial human

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ventricular myocytes. As described in previous chapters, the CRLP model includes
reformulation of ICaL , readjustment of the parameters describing IK1 and redefinitionof a number of other model parameters, including GNa (maximum INa conductance)
and GNaK (maximal INaK value). In the CRLP model, the formulation of voltagedependent ICaL inactivation is similar to that of the TP06 model. However, theassociated voltage-dependent inactivation time constants were adjusted to better
match experimental evidences on APD adaptation to CL changes. The calciumdependent inactivation gates were maintained as in the original GPB model. The
formulation of ICaL inactivation in the CRLP model is as follows:
InactCRLP
Ca,L = f · f2 · (Aj · fCa,j + Asl · fCa,sl )The ORd model is the most recent human ventricular AP model out of thefour analyzed models and it has the most complex ICaL definition. ICaL inactivation
is modeled as a weighted average involving voltage-dependent gates. The gates
ff and fs represent the behavior of the voltage-dependent gates when there isno calcium. The gates fCa,f , fCa,s and jCa represent the behavior of the voltagedependent gates when calcium is present. The calcium-dependent gate n modulates
the relative weight of both families of gates. The n gate is modeled via a Markov
chain. The gates ff , fs , fCa,f , fCa,s and jCa have different time constants, τf,f ,τf,s , τf,Ca,f , τf,Ca,s and τj , but the same steady-state value fss . As discussed byThomas O’Hara in the online version of the article presenting the model [64], there
are some issues related to the INa current formulation in the model. The authors
propose to replace the INa current formulation of the original ORd model with theformulation proposed in the TP06 model. In this work we analyze the original
ORd model and the ORd model with the INa current of the TP06 model (ORdNa ).
The formulation of ICaL inactivation in the ORd and ORdNa models is as follows:
InactORd
Ca,L = (Af,f · ff + Af,s · fs ) · (1 − n)+
jCa · (Af,Ca,f · fCa,f + Af,Ca,s · fCa,s ) · n
where Af,f and Af,Ca,f are the ratios of the fast voltage-dependent inactivation
gates ff and fCa,f , respectively, and Af,s and Af,Ca,s are the ratios of the slow
voltage-dependent inactivation gates fs and fCa,s , respectively.
4.2.2 Characterization of L-type calcium voltage-dependent inactivation
Voltage-dependent ICaL inactivation is experimentally characterized using the pairedpulse protocol, which is illustrated in Figure 4.1. This protocol consists of clamping
the membrane potential from a holding potential Vhold to different pre-pulse potentials Vpre during a specified time interval tpre . After this time, the potential isclamped to a pulse potential Vpulse during an interval tpulse . Following this, themembrane potential is clamped back to Vhold . Variations of this protocol introduce
a separation between the pre-pulse and the pulse potential, during which the potential is set to Vhold for a short period of time, tsep , before clamping the membrane

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Figure 4.1: Paired-pulse voltage clamp protocol and measurement of the steadystate value fss of voltage-dependent inactivation.

potential to Vpulse . Steady-state voltage-dependent ICaL inactivation is obtained
during the tpulse interval. In this part of the test, peak ICaL current is measured
for each potential used as pre-pulse and results are normalized by the peak valueof the current measured for the minimum pre-pulse potential. The resulting curveis used in the models as the steady-state value of voltage-dependent inactivation(fss ).
Human ventricular cell models use different experimental datasets to define
their voltage-dependent ICaL inactivation functions. With the aim of comparing
the simulation results of steady-state ICaL inactivation with experimental results,
the same experimental dataset that was used to adjust each model in the originalarticles was considered. For the TP06 model no indication was provided on theexperimental dataset used to fit steady-state voltage-dependent ICaL inactivation,
whereas in the previous version of the model [46] data from [1] was used. For the
TP06 model the authors redefined the ICaL expression by adding a second voltagedependent inactivation gate. Only the I/V curve for ICaL was compared withexperimental results from [3]. The GPB model used experimental results from [2]and the ORd model from [3].
Each set of experimental results was based on a different configuration of thepaired-pulse protocol to characterize steady-state voltage-dependent ICaL inactivation (Table 4.1). Also, different calcium concentrations in the extracellular solution
and in the patch pipette solution, varying from 1.8 to 5.4 mM and from 0 to 2.0
mM, respectively, were used for each experimental dataset.

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Vhold (mV )Vpulse (mV )Vpre,ini (mV )Vpre,f in (mV )∆Vpre (mV )tpre (ms)tsep (ms)tpulse (ms)

[1]
-45
10
-40
40
5
400
10
400

[2]
-80
10
-100
60
10
400
5
300

85
[3]
-80
5
-55
15
5
500
0
390

Table 4.1: Paired-pulse test parameters in experimental protocols.
In this chapter, in silico simulations were performed in which the experimental
paired-pulse tests were implemented for ICaL characterization. Intracellular andextracellular concentrations were set at the values used in the experimental protocols against which simulated data were compared. In particular, free intracellular
calcium ([Ca2+ ]i ) was set as the value of the pipette solutions of the experiments,
with the time derivative being set to zero (i.e. d[Ca2+ ]i /dt = 0). All the calcium buffers and the diffusion and transportation of calcium inside the cell weremaintained as in the original models. This allowed calcium concentrations in all
remaining intracellular compartments to follow the dynamics described in the corresponding model equations. The same values for Vhold , Vpre , and Vpulse as in thecorresponding experiments (see Table 4.1) were used. All currents except for ICaL
were blocked.
4.2.3 Steady-state AP markers
To study the effect of considering data measured at tissue level for characterizationof single cell models, AP markers calculated at single cell and tissue levels withthe four ventricular models were compared. The following markers were analyzed:i) Action potential duration at different percentages of repolarization (90%, 75%,
50%, 25%); ii) Triangulation; and iii) Minimum and Maximum Transmembrane
Potential and Transmembrane Potential Velocity. The maximum value of the AP
is used in many models to adjust the cardiac conductances [15, 47] whereas theminimum potential is used for model validation [64]. These values are, however,very sensitive to cellular electrophysiological changes like, for instance, those associated with hyperkalemia, where the resting membrane potential can be increased
from -85 mV to up to -60 mV and the maximum potential falls, in combination
with a potential splitting of the AP upstroke into more than one component [20].
4.2.4 Ionic contributors to AP markers
To evaluate the role played by each ionic current in contributing to each electrophysiological marker, the results of the simulations carried out by varying individual

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ionic conductances were adjusted by a first order response surface model:
Mj ≈ C j +N∑∆i · mj,ii=1

where Mj is the value of the electrophysiological marker j under the analyzed
condition, Cj is the value of the marker j under control conditions, mi,j is theweight of the current i in contributing to the marker j and ∆i is the variation of
the ionic conductance:
Gi − Gi,0∆i =Gi,0where Gi,0 and Gi are the values of the ionic conductance under control and under
the analyzed condition, respectively.
For each individual marker, if the results for the different evaluated conditions
k = 1, . . . , K are concatenated, the above system can be expressed in matrix form
as:

 
 



Mj,1
∆1,1 · · · ∆N,1mj,11 Mj,2   ∆1,2 · · · ∆N,2   mj,2 
 1 

 
 



≈
·
+
C
·






..
..
.
.
j  . ..
  ..
  ..  .. 
.
.
.
Mj,K

∆1,K

···

∆N,K

mj,K

1

↓
M j ≈ ∆ · mj + Cj · J K,1
where Mj,k is the value of the marker j and ∆i,k is the variation of the ionic conductance i at condition k. Using this notation, the weights of the ionic conductances
can be calculated as follows:
(
)−1
T
T (
)
mj = ∆ · ∆· ∆ · M j − Cj · J K,1
4.2.5 Computational simulations
The models were stimulated with square current pulses with an amplitude twice thediastolic threshold and a duration of 1 ms. Depending on the simulation scale, thediastolic threshold was defined as the minimum amplitude required to (a) generate
five APs (single cell simulations) or (b) propagate five APs (1D tissue simulations).
To simulate steady-state conditions, models were stabilized with a train of 100
stimulations at a CL of 1000 ms. Electrophysiological markers were calculated fromthe last simulated beat. Every model was first simulated under control conditions
and subsequentyly after varying each ionic conductance by ±15 and ±30%.
For 1D tissue simulations, a homogeneous 3-cm fiber composed of epicardial cells
was used. The value of the conductivity, σ, was set to obtain a Conduction Velocity(CV) close to 65 cm/s [88]. The cell capacitance was set to Cm = 1 µF/cm2 . The

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AP markers were computed as the mean value of the markers measured at fivedifferent positions within the fiber located at: 1, 1.25, 1.5, 1.75 and 2 cm.
For single cell simulations, a forward Euler scheme was used to solve the models
with a time step of ∆t = 0.002 ms for GPB, ORd and CRLP, and ∆t = 0.02 ms
for TP06. For 1D tissue simulations, a semi-implicit operator-splitting scheme wasused to solve the propagation [87] with a space discretization of ∆x = 0.1 mm andthe same ∆t used in single cell simulations.
The results from single cell and in 1D tissue simulations were compared using
the absolute (Ea ) and relative (Er ) differences:
Ea = Mjcell − Mjtissue
Er (%) =Mjcell − Mjtissue· 100
Mjtissue

where Mjcell and Mjtissue are the values of the electrophysiological marker j in celland tissue for the condition under analysis.
4.3 Results
4.3.1 Characterization of L-type calcium voltage-dependent inactivation
Results of the paired-pulse protocol for steady-state voltage-dependent ICaL inactivation are shown in Figure 4.2. Each model was compared with the set of experimental data that was used to characterize voltage-dependent ICaL inactivation in
the original article were the model was presented.
The TP06 model was able to adequately reproduce the experiments from [1](see Figure 4.2.a)). In this case, the modeled steady-state voltage-dependent ICaLinactivation is the product of the steady-states values (f2,ss , fss ) of the two inactivation gates (f2 , f ). The results from the in silico simulations, which accountfor additional interactive effects, were closer to the experimental data than thoseobtained by evaluation of the corresponding ICaL mathematical equations. Figure 4.3.a) shows that, for the TP06 model, the slow inactivation gate f did notreach steady-state for high Vpre potential values (Vpre = -5 mV and Vpre = 40 mV).
Figure 4.3.a) also shows that the calcium-dependent inactivation gate has a lesssignificant effect than the other gates, being this effect slightly more pronounced atlower Vpre potentials. Finally, one of the most significant differences between themodel outcome (multiplication of the steady-state values of the gates, black dashedline in Figure 4.2.a) and the result of the in silico simulations (yellow crosses) wasthat the curves were shifted.
Simulation results obtained with the GPB model fit quite well the model definition (see Figure 4.2.b)). Differences between model outcome and in silico simulation increased as fss decreased. As shown in Figure 4.3.b), the separation pulse
produced different effects depending on the pre-pulse potential. For Vpre = -100
mV, the effect was minimal, but in the other two cases shown in the figure (Vpre =-20 mV and Vpre = 60 mV), the value of f increased during the separation pulse.

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Figure 4.2: Comparison between model definition (continuous and discontinuouslines), in silico simulations (yellow crosses) and experimental results (red circles)
for fss . a) TP06 model and data from [1]. b) GPB model and data from [2]. c)
ORd model and data from [3]. d) CRLP model and data from [1].
Calcium-dependent inactivation gates had minimal effects, as these gates remained
open during most of the protocol duration. Finally, differences between experiments [2] and in silico simulations increased for positive potentials (Figure 4.2.b).
Simulation results with the ORd model showed a significant discrepancy withexperimental data for potentials between -20 mV and -10 mV (see Figure 4.2.c)).
The largest difference was observed for a pre-pulse potential of -15 mV. The simulated probability at this potential was 18% larger than the one obtained in theexperiments. The evolution of the voltage-dependent ICaL gate variables during the
pre-pulse potential is shown in Figure 4.3.c). For potentials below -15 mV, the ngate was nearly closed and the effects of fs and ff gates dominated. On the contrary, for positive potentials, the effects of the gates fCa,f , fCa,s and jCa prevailed.
This figure also shows that for potentials above -15 mV, the fast inactivation gates,
but not the slow ones, reached a steady-state value during the pre-pulse. For highpre-pulse potentials, the gate fCa was not able to reach steady-state (close to 0),
but the gate j was. However, the multiplication of both gates almost achieved asteady-state value.
The results obtained with the CRLP model were very similar to those found
for the TP06 model. This was expected, since both models have a very similar
formulation for ICaL inactivation. However, the difference between the results of
the CRLP model and the experiments from [1] was smaller than for the TP06
model (see Figure 4.2.d)).

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Figure 4.3: Temporal evolution of the ICaL gates during simulation of the pairedpulse protocol (see text for specific values of tpre , tsep and tpulse used in the simulations): a) TP06 model. b) GPB model. c) ORd model. d) CRLP model

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AP D90 (ms)
AP D75 (ms)AP D50 (ms)AP D25 (ms)T rian. (ms)Vmax (mV)
Vmin (mV)dV
dt max (V/s)
dV
dt min (V/s)

TP06
GPB
CRLP
ORd
ORdNa
Cell Fiber Cell Fiber Cell Fiber Cell Fiber Cell Fiber
300.5 300.2 286.1 283.5 306.2 306.7 224.3 224.2 223.2 224.0
292.1 292.4 272.6 271.8 281.1 284.5 208.1 208.6 206.2 208.5
272.4 277.0 235.0 243.8 228.1 239.6 177.8 179.1 174.4 179.2
212.3 242.1 127.3 180.4 111.8 179.1 137.0 144.1 126.5 144.3
28.1 23.2 51.1 39.7 78.1 67.1 46.5 45.1 48.8 44.8
38.5 24.6 38.7 23.4 38.1 21.9 36.2 31.1 41.9 30.9
-85.4 -85.4 -81.4 -81.4 -84.1 -84.2 -87.8 -87.8 -87.8 -87.9
292.6 193.7 322.8 234.5 335.4 227.5 234.9 78.1 364.2 237.9-9.5 -3.3 -6.2 -1.4 -5.9 -0.8 -1.8 -1.1 -2.9 -1.2

Table 4.2: Comparison of electrophysiological markers simulated in single cell and1D tissue under control conditions.
4.3.2 Ionic contributors to AP markers
The results of the analyzed AP markers under control conditions are shown in
Table 4.2 for the four analyzed models. Table 4.3 shows absolute and relative
differences between cell and tissue values for each electrophysiological marker. APD
was very similar in cell and tissue simulations at 90% and 75% repolarization. The
differences between cell and tissue increased at lower percentages of repolarization:
the relative differences for APD at 50% repolarization were between -0.7% and-3.6% and considerably higher at 25% repolarization, between 4.9% and 37.4%.
The differences in Triangulation were mainly due to differences in APD50 . Dueto the smaller values of Triangulation, the relative differences for this AP marker
were higher (between 3.0% and 28.8%). The GPB model and the CRLP model
were the ones that showed larger differences between cell and tissue results. On
the contrary, the ORd model was the one showing less differences between cell andtissue in the AP-based electrophysiological markers.
The value of the resting potential (Vmin ) was the same in cell and tissue. On
the contrary, peak AP value (Vmax ) showed large differences between cell and tissue
simulations (from 16.3% to 73.5%). Similarly, large differences were quantified fordV /dtmax and dV /dtmin . The ORd model showed the smallest differences in thosevoltage-related electrophysiological markers except for dV /dtmax . The other threemodels showed similar values, in particular for the absolute differences.
For the TP06 model, the ICaL current played different roles in cell and tissue
simulations (see Figure 4.4). While in tissue it was the most important current
to determine Vmax , in cell its effect on Vmax was negligible. A similar effect wasobserved for the T riangulation, where ICaL was a highly relevant current in tissue
simulations, whereas in cell simulations it had almost no effect on T riangulation.
On the contrary, the impact of INa current in cell was notably diminished in tissue
simulations. For Vmax , the weight of the INa current in cell was 51.6%, while intissue it was nearly 5%. This also happened with the Ito current, although to alesser extent. Gto conductance affected dV /dtmin in cell simulations, whereas in

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Ea
Er (%)
TP06 GPB CRLP ORd ORdNa TP06 GPB CRLP ORd ORdNa
AP D90 (ms)
0.3 2.6 -0.5 0.1-0.8 0.1 0.9 -0.2 0.0-0.3
AP D75 (ms) -0.3 0.8 -3.4 -0.5
-2.3 -0.1 0.3 -1.2 -0.2
-1.1
AP D50 (ms) -4.6 -8.8 -11.5 -1.3
-4.8 -1.7 -3.6 -4.8 -0.7
-2.7
AP D25 (ms) -29.8 -53.9 -67.3 -7.1 -17.8 -12.3 -29.4 -37.6 -4.9 -12.3
T rian. (ms)
4.9 11.4 11.0 1.4
4.0 20.7 28.8 16.4 3.0
9.0
Vmax (mV)
13.9 15.3 16.2 5.1
11.0 56.4 65.7 73.5 16.6
35.5
Vmin (mV)
0.0 0.0
0.1 0.0
0.1 0.0 0.0
0.0 0.0
0.0
dV
(V/s)
98.9
88.3
107.9
156.8
126.3
51.1
37.7
47.5
200.7
53.1
dt maxdV-6.2 -4.8 -5.1 -0.7
-1.7 187.8 343.2 616.3 52.5 143.1
dt min (V/s)
Table 4.3: Absolute (Ea) and relative (Er) differences between electrophysiologicalmarkers in single cell and 1D tissue simulations.
tissue simulations this marker was mainly controlled by ICaL .
The more influential currents in the GPB model differed from the TP06 model(Figure 4.5). In this model, the ICl,bk current, which is not present in the TP06
model, was the most relevant contributor to APD in both cell and tissue simulations. Other effects were similar in both models. As an example, the INa current
contribution was reduced in the GPB model when the electrophysiological markers were calculated in tissue as compared to cell. This reduction was found forall markers except for dV /dtmax . Similarly to the TP06 model, one current hada larger contribution to Vmax in tissue than in cell, but in the GPB model thatcurrent was Ito . The same behavior was found for T riangulation and IK1 .
The APD of the ORd model was mainly controlled by IKr in cell, whereas
in tissue the INa current also played an important role. The original ORd model
showed large differences in the role played by the INa current in tissue compared withthe one in cell (Figure 4.6). In cell simulations, when INa conductance was increased,
the APD at different percentages of repolarization was slightly decreased. On thecontrary, in tissue, when INa conductance was increased, the APD was considerably
prolonged. These differences in the ORd model disappeared when the INa current
formulation was replaced with the one described in the TP06 model. For Vmax ,when INa was replaced with the one defined in the TP06 model (Figure 4.7), thedifferent behavior between cell and tissue simulations was interchanged.
The APD of the CRLP model highly depended on the conductance of the ICl,bk
current (see Figure 4.8). This behavior is inherited from the GPB model but its
weight in the CRLP model was found to be slightly smaller than in the GPB model.
Likewise, the role of the ICaL current was very similar to the one in the GPB model.
The only difference was the role in dV /dtmin : in the CRLP model it was larger thanin the GPB model due to the particular AP shape at the beginning of the plateau
phase for the CRLP model, which is related to the definition of ICaL formulation.
Also, the contribution of IK1 to Triangulation was 16% larger in tissue than in cell
simulations.

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Figure 4.4: Contribution of ionic conductances to electrophysiological markers
simulated in tissue and cell with the TP06 model. Dark colors (blue or brown)
indicate maximum correlation between changes in a conductance and changes ina marker; white color indicates no correlation. Percentages in boxes indicate thecontribution of changes in a conductance to changes in a marker. Minus signs
indicate that conductances and markers vary inversely; plus signs indicate that
conductances and markers vary in the same direction.

Figure 4.5: Contribution of ionic conductances to AP markers simulated in celland tissue with the GPB model. The legend of the figure is the same as in see
Figure 4.4.

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Figure 4.6: Contribution of ionic conductances to AP markers simulated in celland tissue with the ORd model. The legend of the figure is the same as in see
Figure 4.4.

Figure 4.7: Contribution of ionic conductances to AP markers simulated in celland tissue with the ORdNa model. The legend of the figure is the same as in see
Figure 4.4.

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Figure 4.8: Contribution of ionic conductances to AP markers simulated in celland tissue with the CRLP model. The legend of the figure is the same as in see
Figure 4.4.
4.4 Discussion
This chapter has been focused on analyzing issues arising during model development and validation, namely: i) the effect of submodel variable interactions in thecharacterization of ion channel gating, and ii) the effect of cell-to-cell interactions in
the evaluation of ionic contributors to AP markers. A methodology for validating
computational model formulations has been introduced. The proposed methodology consists in performing in silico simulations using the same protocol as in theexperiments used to characterize a given current or AP marker. The methodology
accounts for all variables and scales involved in modeling and simulation as well as
their interactions. It has been applied to evaluate voltage-dependent ICaL inactivation and contributors to electrophysiological AP markers in four human ventricularmodels presenting different ionic formulations.
4.4.1 Effect of submodel variable interactions in the evaluation of voltage-dependent L-Type Calcium Current inactivation
In the case of voltage-dependent ICaL inactivation, our results show large differences
between ICaL inactivation as calculated from the model equation and ICaL inactivation from the in silico simulations. Such differences were due to the interaction
between voltage-dependent ICaL inactivation gating and other effects such as ICaL
activation gating, protocol definition or duration of the voltage pulses used to calculate inactivation properties. This suggests that, when proposing any new model
formulation, consistency between such formulation and the corresponding experimental data that is aimed at being reproduced needs to be first verified considering
all involved factors.
In the TP06 model, the product of the steady-state values of the two voltagedependent ICaL inactivation gates was very different from the experimental behav-

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ior. However, the results of the in silico simulations obtained with this model werein good agreement with the experiments. There are two effects that explain theseresults: i) normalization by ICaL current peak at the minimum pre-pulse potential,
as performed in the experimental protocol [1], is done for a transmembrane potential value where the product of the two voltage-dependent inactivation gates is lessthan 1; ii) voltage-dependent inactivation gates do not reach a steady-state value
at the end of the pre-pulse interval for some potentials. One of the most significant
differences between the model outcome (multiplication of the steady-state values of
the gates) and the result of the in silico simulations is that the curves representing
steady-state voltage-dependent inactivation (Figure 4.2.a)) were shifted. This shift
made the in silico simulations approximate the experimental behavior due to theperformed current peak normalization at a potential where inactivation probability
is less than one.
The GPB model and the associated in silico simulations provided similar results to the experiments used to fit the steady-state value of voltage-dependent
inactivation gating. Differences were mainly caused by the protocol used to measured fss , as explained in the following: the separation pulse had a different effectdepending on the pre-pulse potential. For pre-pulse potentials similar to Vhold thevalue of fss obtained at Vpulse is minimally affected since the steady-state valuesof f for Vhold and Vpre are very similar. However, for larger values of Vpre , andtherefore larger differences between Vhold and Vpre , the steady-state value of fcorresponding to Vpre is very different to the steady-state value corresponding to
Vhold , which corresponds to one. This causes that during the separation pulse thevalue of f increases (see Figure 4.2.b)) and the difference in fss between the model
definition and the simulation increases with Vpre . In addition, the interaction with
voltage-dependent activation, represented by gate d, always reduced the measured
value, but the normalization, using the result of the minimum pre-pulse potential,
nearly corrected this behavior. Finally, differences between experiments [2] and in
silico simulations get larger where differences between the model definition for fss
and experimental data are larger (see Figure 4.2.b) for positive potentials). These
results suggest that a simple ICaL inactivation formulation like that in the GPBmodel, with only one voltage-dependent inactivation gate, suffices to reproduce
the experimental behavior of steady-state voltage-dependent ICaL inactivation [2].
In the ORd model, the definition of voltage-dependent ICaL inactivation as aweighted sum of fast and slow inactivation gates is consistent with how experimentalists calculate inactivation time constants [1, 64]. Time constants are calculated
by fitting a biexponential function to the inactivation phase of the experimental
current traces. However, important discrepancies between in silico ORd simulations and the experiments used to develop the model [3] were found for pre-pulse
potentials between -20 mV and -10 mV. There are two possible causes behind suchdiscrepancies: i) the time constant of the slow voltage-dependent inactivation gate;ii) the definition of voltage-dependent ICaL inactivation as the sum of two gates.
The first cause has to do with the fact that ORd model time constants are identified based on a simple-pulse test protocol of 75 ms duration, whereas the slow
time constant of voltage-dependent ICaL inactivation in the ORd model is of the

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order of 10 seconds. The slowness of the inactivation gate prevents reaching thesteady-state value by the end of the pre-pulse, which overestimates the value of fss
at Vpulse . This is reinforced by the fact that voltage-dependent ICaL inactivation isformulated as a sum of two gates. If inactivation is expressed as a product, as in thecase of the TP06 model, when one of the gates reaches zero, the product reaches
zero. However, for the ORd model the situation is more complex. Due to theinteraction between the inactivation gates and the n and jCa gates in this model,
as shown in section 4.3.1, the result for potentials above -10 mV is equivalent to
the product of two gates since the jCa gate is almost zero, and therefore the result
from the simulation is similar to the model formulation of fss and experiments.
On the contrary, for potentials between -20 mV and -10 mV, where by the end of
the pre-pulse the jCa gate is far from zero and the slow inactivation gates, in particular fCa,s , are far from reaching their steady-state value, the fss value obtained
from the simulation is considerably larger than the fss value defined in the model
formulation and the experiments.
For the CRLP model, similarly to the TP06 model, the largest differences between model equations, in silico simulations and experiments are due to the time
constants of the activation and inactivation gates. Nevertheless, the results of
the in silico simulations with this model reproduced the experimental observations
slightly better than the TP06 model. In the CRLP model, the voltage-dependent
ICaL inactivation gate f2 is faster than in the TP06 model (as defined by the time
constant τf 2 ). Due to this gate being faster, the effect of the separation pulse waslarger than in the TP06 model, as during this interval the f2 gate opened morethan in the TP06 model (see Figure 4.3.d)). As a consequence, the CRLP modelyielded a larger fss value in the in silico measurements, which was closer to theexperimental values. This effect was not very pronounced but it could be better
observed for voltages where f2 is larger (around -20 mV).
This chapter has focused on the difficulties of fitting model parameters associated with the steady-state value of ICaL inactivation using data obtained fromvoltage-clamp experiments. Such difficulties arise from submodel variable interactions in the ICaL formulation. This situation is a common problem to many gating
variables in cardiac electrophysiological models. In this regard, most experimental
protocols cannot measure the behavior of one gate at a time, but instead measure
the behavior of all gates of the same nature (activation, inactivation, etc.) together(e.g. for ICaL inactivation the experimental protocol measures fast and slow inactivation together). Therefore model development needs to consider the coexistenceof all gates during parameter identification to accurately reproduce experimental
results. This issue is also present in the estimation of the time constants associated with the different gates. In this regard, the gating interactions associatedwith an experiment implies that the identified time constants associated with theexperimental data cannot, in general, be associated with an individual gate of themodel.
Some studies have explored the effect of gating interactions on voltage-clamp
data [129, 130]. The voltage-clamp protocol is not designed to measure this directly, but the corresponding effects are included in the experimental results. Nev-

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ertheless, most of these techniques have been developed for currents with only one
inactivation and one activation gate. Approaches to extend those methods to better
calibrate complex gate combinations remain to be investigated.
4.4.2 Effect of cell-to-cell interactions in the evaluation of ionic contributors
to AP markers
While the ionic currents identified as having the largest contribution to each analyzed AP-based marker varied from one model to another, the differences between
single cell and tissue simulations were comparable for all the analyzed markers in
the four considered human ventricular models.
As expected, measurement of depolarization-related AP markers in single cell
simulations was significantly affected by the externally applied stimulation current.
In particular, dV /dtmax in single cell simulations was quite different from thatobtained in tissue for all considered models. Although some models use dV /dtmax
or Vmax measured in single cell simulations to adjust INa [15, 47, 64], our results
suggest that INa should be adjusted with the results obtained from, at least, a 1D
tissue simulation as in [120], particularly considering that many experiments are
performed in tissue.
For repolarization-related AP markers, the largest differences between cell andtissue were observed for Triangulation and APD25 . The results for APD90 and
APD75 were very consistent between cell and tissue, while small discrepancies were
found for APD50 . Although differences for APD90 and APD50 were small, theiradded contribution together with the small value of Triangulation caused a relatively large accumulated difference for this marker. For this reason, Triangulation
should be used with caution as a marker in single cell simulations when used tocalibrate an AP model.
Importantly, the differences between markers computed from single cell and 1D
tissue simulations were not only the reflect of a constant bias, as shown in Table 4.2
and in Table 4.3. For repolarization markers, those differences reflected the different
contribution of ionic currents to the markers. Computational studies using single
cell simulations to analyze the sensitivity of different ionic current conductanceson the repolarization behavior of the model should only consider markers witha validated correspondence between cell and tissue simulations or, otherwise, use
markers computed from tissue rather than from single cell simulations.
4.5 Conclusions
When proposing a new model, or when evaluating an existing model, consistency
between simulated and experimental data should be verified considering all involved effects and scales. The closer the experimental conditions are reproduced in
the computer simulations, the more robust the process of model development andvalidation will be.
As discussed in this paper, proper characterization and validation of a given
model should be performed with the in silico simulation of the experimental proto-

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col. In this way, the effects of the interaction between model variables are accounted
for and modeling inconsistencies are avoided.
While the information provided by electrophysiological markers is very valuable for model development and validation, markers should be computed using
simulations that resemble as closely as possible the conditions associated with theexperimental measurements, or, at least, the consistency of the marker at different
scales should be previously validated.
As a final remark, complex models represent a real challenge for parameter
identification and validation. This does not mean that models should be necessarily
simple, but that complex models require additional testing to fully verify adequate
performance.

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Chapter 5

Reformulation of Ionic
Current Expression
5.1 Introduction
The human ventricular AP model proposed in Chapter 2 and subsequently updated in chapter 3, despite reproducing experimental observations in relation toa range of electrophysiological characteristics (see also Chapter 4), renders an
AP lacking the peak-and-dome morphology characteristic of ventricular epicardial
cells. Specifically, the end of phase 1 is followed by an atypical low-magnitude
bump that precedes the AP plateau phase. In this chapter a readjustment of ICaL
voltage-dependent inactivation gates is proposed to provide a more physiologicalAP shape while guaranteeing an improved fitting of the model to experimental data
in relation to all the electrophysiological properties the model had been validated
against in previous chapters.
The proposed readjustment lies on the methodologies described in chapters 3
and 4. As a first step, the time constants of ICaL voltage-dependent inactivation
gates were modified to render a more realistic AP shape. Due to the implications that a partial modification of an ionic model may have on the performanceof the model, as highlighted in chapter 4, the steady-state value of ICaL voltagedependent inactivation gates was adjusted to better reproduce experimental current
data available from three different studies [1–3] by using the algorithm proposed in
the chapter 3 and the in silico simulation of these experimental protocols describedin chapter 4. Finally, the model was evaluated at cell and tissue levels to confirm
that its performance was maintained or improved with respect to the model ofchapter 2 for all evaluated electrophysiological properties.

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100

CRLP =f
CRLP =f 2
New =f
New =f 2
Exp. inact.
Exp. rec. fastExp. rec. slow

=f ; =f2 (ms)

1500

1000

500

0
-80

-60

-40

-20

0

20

40

V (mV )
Figure 5.1: Time constants of ICaL voltage-dependent inactivation in the CRLP
model and following the readjustment proposed in this chapter. Comparison withexperimental data [2, 4–7] is provided as well.
5.2 Methods
5.2.1 Modification of ICaL voltage-dependent inactivation time constants
To improve the AP shape of the CRLP model, the time constants of the slow
(τf ) and fast (τf 2 ) ICaL voltage-dependent inactivation gates were modified. Afterevaluating all model parameters defining the currents with a major contribution
to the AP part immediately following phase one, namely ICaL and Ito , those twotime constants were identified as the ones with the largest influence on the AP
shape in that specific part. In particular, exploration of the parameter space led
to redefinition of the β components of both time constants τf and τf 2 , as follows:
βf =

200
1+e

20.6−V
10

(5.1)
(5.2)

βf 2 = 0

where V is membrane potential.
The effect of varying βf and βf 2 is shown in Figure 5.2 and Figure 5.3. In
Figure 5.1, the new ICaL inactivation time constants are compared with the originalones and with the experimental data from [2, 4–7].
5.2.2 Adjustment of ICaL voltage-dependent inactivation steady-state
To best fit ICaL voltage-dependent inactivation steady-state after readjustment of
the time constants described in previous section, the expression of the steady-statevalue for inactivation gates f and f2 was parameterized as follows:gss =1 − gss,∞
V −Vg,1/2

1+e
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(5.3)

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5. Reformulation of Ionic Current Expression

C32

V (mV)

50
0
−50
0

50

100

150

time (ms)Figure 5.2: Effect of varying βf 2 in the AP shape. Red line represents control
conditions. Dark grey lines correspond to reductions in βf 2 , while light grey lines
correspond to increases in βf 2 .

C19

C19
50
V (mV)V (mV)

50
0
−50
0

100

200 300
time (ms)

400

500

0
−50
0

50

100
150
time (ms)200

Figure 5.3: Effect of varying βf in the AP shape after having modified βf 2 in the
CRLP model. Red line represents control conditions. Dark grey lines correspondto reductions in βf , while light grey lines correspond to increases in βf .

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where g represents f or f2 (depending on the gate under consideration), V ismembrane potential, gss,∞ is the steady-state value of gate g when V tends toinfinity, qg is a parameter controlling the maximum slope of gss and Vg,1/2 is halfinactivation voltage for gate g.
Optimization problem
A minimization problem was defined in which the value of the parameters fss,∞ ,f2,ss,∞ , qf , qf 2 , Vf,1/2 , Vf 2,1/2 are searched for to best fit experimental steadystate ICaL voltage dependent inactivation data. The response surface optimization
approach described in chapter 3 was used to solve the minimization problem. The
objective function of the minimization problem was defined in terms of the meansquare error between a set of experimental data [1–3] and the corresponding resultsof the in silico simulations (see chapter 4) as follows:i
1 ∑2(fss,expi (Vj ) − fss,simi (Vj ))ni j=1nmsei =(5.4)

where ni is the number of prepulse voltages used in experiment i, fss,exp (Vj ) is theexperimental result for voltage Vj in experiment i and fss,sim (Vj ) is the in silico
result for voltage Vj in the simulation of the experiment i.
Since several experimental measurements of ICaL inactivation were available,
the objective function was defined as the addition of the mean square errors forthe different experiments weighted by the number of cells used in each experiment to reduce the mean square error of the simulations in comparison with theexperimental results:
∑

Nexp

min

fss,∞ ,f2,ss,∞ ,qf ,qf 2 ,Vf,1/2 ,Vf 2,1/2

Ni · msei

(5.5)

i=1

where Nexp is the number of experimental studies used in the minimization problem
and Ni is the number of cells used in experiment i.
Constraints
For the above minimization problem, the following constraints were imposed on
the aforementioned gating parameters:• C1: Both gates must have the same slope (qf = qf 2 ).
• C2: Vf,1/2 must be smaller or equal to Vf 2,1/2 .
• C3: When V tends to infinity, the values of fss and f2,ss must be the sameas in the original model (0 and 0.33, respectively).
• C4: f2,ss must tend to 0 when V tends to infinity.
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Each constraint was approximated using a second order response surface builtwith a full factorial experimental design (see chapter 3). The second order response
surface was enough to find a solution within a reasonable time.
An additional step was included to validate that the solution to the minimization problem led to physiological results at current, cell and tissue levels. At theionic level, the in silico simulation of ICaL voltage-dependent inactivation steadystate was required to decrease monotonically and have only one inflection point at
negative voltages. At cell and tissue levels, a set of electrophysiological markers
were computed, including: systolic and diastolic intracellular calcium concentration([Ca2+ ]i ), AP duration (APD) at 90% of repolarization (APD90 ), AP triangulation
at 1Hz and 0.5 Hz and APD adaptation to abrupt cycle length changes (from acycle length of 1000 ms to a cycle length of 600 ms). A complete description ofthese markers and the protocols used to compute them can be found in section 2.2.
Each marker was computed for varying ICaL conductance ranging from -30% to+30% of the original value in the CRLP model. A solution of the minimization
problem was accepted at the cell level when, for some value of GCaL , all markers
were within physiological range or, in the case of markers that were originally outof physiological range, when they were at an equal or closer distance to the physiological range than for the CRLP model. For the definition of physiological range,
the same values described in chapter 2 (see Table 2.2) were used, as proposed in[12].
Solution for ICaL voltage-dependent inactivation
After selecting the minimum set of requirements as constraints, the minimization
problem was solved once more using a fourth order response surface for the objective function and constraints to increase the sampling of the searching space. Basedon the conclusions of chapter 4, the solution was additionally evaluated in tissue.
A homogeneous 3-cm long fiber composed of epicardial cells was considered. Cell
capacitance was set to Cm = 1 µF/cm2 . Values for evaluated electrophsyiologicalmarkers were computed as the mean of the values measured at five different positions within the cable located at: 1, 1.25, 1.5, 1.75 and 2 cm. Numerical simulations
were solved using ELVIRA software [87].
5.2.3 Numerical methods and implementation
Cells and tissues were stimulated with square transmembrane current pulses twice
the diastolic threshold in amplitude and 1-ms duration. The diastolic threshold infiber simulations was calculated as the minimum value that generated ten pulsesthat propagated through the fiber with a CL of 1000ms as described in [136]. For
cell simulations, the diastolic threshold was calculated as the minimum value thatgenerated ten pulses.
When comparing with experimental data, extracellular concentrations in themodel were defined to have the same values as those described for the bath solutionof each experimental study.

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Computations at cell and ionic levels were performed using MatCardiacMLab
Simulator (http://github.com/ChusCarro/MatCardiacMLab). This MatLab-based
software uses the ode15s solver with a maximum step of 0.1 ms to solve the
ODEs of the model. The original CellML file of the CRLP model was considered. Fiber simulations were implemented in FORTRAN and simulated using thesoftware ELVIRA. The value of the conductivity, σ, was calculated to obtain a CV
value close to 65 cm/s [88]. The fiber was stimulated at one end. A semi-implicit
operator-splitting scheme [87] was used to solve the model with a space discretization of ∆x = 0.1 mm and a time step of ∆t = 0.002 ms. Cell capacitance was set
to Cm = 1 µF /cm2 .
5.3 Results
5.3.1 Solution to the minimization problem
Table 5.1 summarizes the results of the optimization algorithm for each combinationof constraints. In each of the tested cases, which are identified by letters A to L,
one, two, three or four constraints in the set {C1, C2, C3, C4 } were imposed. As
can be observed from Table 5.1, all the solutions that allowed qf and qf 2 to bedifferent (C1 not satisfied) rendered non-valid results at the ionic level. Only one
solution where Vf,1/2 was larger than Vf 2,1/2 (C2 not satisfied) rendered a valid
solution at the ionic level. In that case, the distance between Vf,1/2 and Vf 2,1/2
was smaller than in simulations G, H, and I, which were associated with non-valid
solutions (either at ionic or cell levels). When constraint C2 was active, Vf,1/2 wasalways equal to Vf 2,1/2 .
Simulation results accepted as valid after evaluation at the ionic level (G, J,
K, L) were subsequently evaluated at cell and tissue levels by computation of aset of electrophysiological markers. The three solutions that increased the value off2,ss,∞ (H, J, K) greatly reduced the value of AP D90 and, of note, varying GCaL
within the percentages indicated in section 5.2.2 did not lead to AP D90 values
within physiological range. Only the solution obtained after having imposed thefour constraints rendered a valid solution at both cell and tissue scales.
5.3.2 New definition of ICaL current
The solution obtained from the minimization problem with all constraints led to
the following definition of ICaL voltage-dependent inactivation steady-state:
fss =f2,ss =

1
1+e

V +24.63
5.87

0.67
1+e

V +24.63
5.87+ 0.33

(5.6)

(5.7)

The mean square error between data simulated with the modified CRLP modeland experimental data, calculated according to the expression in equation 5.5, wasUniversidad Zaragoza

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C1

C2
No

C3
No
Yes

No
Yes

No
Yes

No

No
Yes

Yes
Yes

No
Yes

C4

Id

Improvement

No
Yes
Yes
No
Yes
Yes
No
Yes
Yes
No
Yes
Yes

A
B
C
D
E
F
G
H
I
J
K
L

62.74%
54.93%
41.59%
53.44%
54.27%
35.28%
43.06%
20.60%
19.79%
38.57%
18.66%
17.14%

Validation
Ionic Cell
No
—
No
—
No
—
No
—
No
—
No
—
No
—
Yes
No
No
—
Yes
No
Yes
No
Yes
Yes

Table 5.1: Results of the minimization problem and subsequent validation. Improvement represents the error for each combination of constraints normalized by
the error before the optimization.
20% lower than the corresponding error for the initial CRLP model. Fig. 5.4 shows
how the proposed solution satisfactorily reproduced available experimental data.
In tissue, the obtained solution rendered values of APD, triangulation and APD
adaptation to abrupt cycle length changes within physiological range, althoughsystolic and diastolic [Ca2+ ]i values were somewhat further away from physiologicalrange than for the initial CRLP model (see Table 5.2).
Marker
APD90 (ms)Triangulation (ms)Sys. [Ca2+ ]i @ 1Hz (mM)
Sys. [Ca2+ ]i @ 0.5Hz (mM)
Dia. [Ca2+ ]i @ 1Hz (mM)
Dia. [Ca2+ ]i @ 0.5Hz (mM)τslow (s)PhysiologicalRange
271-366
44-112
1.59-2.01
0.71-1.68
0.20-0.33
0.14-0.32
70-110

CRLP
305.7
67.1
0.605
0.538
0.097
0.092
101.9

This
Work
279.2
67.8
0.335
0.305
0.090
0.087
99.5

Table 5.2: Evaluation of electrophysiological markers in tissue for the initial andadjusted CRLP models (third and fourth columns). Available physiological ranges
are provided in the second column [12].
In Figure 5.5, simulated electrophysiological markers in tissue are shown. Following the readjustment proposed in this chapter, the AP presents a physiologicalshape at the end of phase 1.

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a)

b)

c)

Pelzmann et al. (1998)

0.6

0.6

0.4

0.4

0.2

0.2

0

−40

−20

0
V (mV)

20

Experiment
Original
New

0.8
0.6
0.4
0.2

0
−100

40

1

Experiment
Original
New

0.8
fss

fss

0.8

Magyar et al. (2000)

1

Experiment
Original
New

fss

1

Li et al. (1999)

−50

0

0

50

−50

−40

−30

V (mV)−20−10
V (mV)

010

Figure 5.4: Comparison between the results of the in silico simulations for the
CRLP model and the adjusted model of this chapter and the experimental resultsof [1–3].
a)

b)

c)
Intracellular Calcium

0

200

400
time (ms)600

APD Rate Adaptation
280

0.1
0

270

90

0.2

(ms)

0.3

APD

[Ca2+]i (µM)

V (mV)

Action Potential
20
0
−20
−40
−60
−80

0

200

400
time (ms)

600

260
250

0

5

10
time (min)15

Figure 5.5: Results of tissue simulations computed with the readjusted CRLP
model proposed in this chapter. a) AP at a cycle length of 1000 ms. b) [Ca2+ ]i ata cycle length of 1000 ms. c) APD adaptation to abrupt cycle length changes.
5.4 Discussion
In cardiac electrophysiological modeling and simulation, there are ionic currents
whose definition is particularly challenging. Many ionic parameters can not bedirectly measured due to the interaction with other model components or due totechnical limitations. As an example, some gating dynamics are so fast that its activity overlaps with the capacitive current, thus preventing its direct measurement.
In other cases, experimental protocols have been designed to specifically isolate
the effects of one parameter of interest in a given model, but, when the complexityof the model increases, it is not possible to isolate it completely. For those reasons: i) when an ionic current is aimed at being formulated, its definition, or atleast its validation, should be performed considering the whole model of the ioniccurrent; ii) when an element of an ionic current is modified, the other elements
should be readjusted to guarantee that model ionic behavior is in agreement withexperimental evidences.
This chapter focused on adjusting the formulation of the time constants of ICaL
voltage-dependent inactivation to improve the AP shape of the CRLP model at theend of phase 1. As shown in chapter 4, due to the interactions between different
elements of the ICaL current, simulation of in silico protocols should be run toadjust parameters defining ICaL voltage-dependent inactivation steady-state, thus
accounting for the effects on the whole ionic current.

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5. Reformulation of Ionic Current Expression
5.4.1 Development and validation of a new ionic current formulation

A minimization problem was mathematically defined to adjust the formulation of
the ICaL current in the CRLP model. Different sets of experimental data providing
steady-state measurements of ICaL voltage-dependent inactivation were considered
to fit the ionic gating parameters. If no constraints were imposed on the minimization problem, the obtained result was associated with a very small error, but
rendered unphysiological results due to the fact the two inactivation gates (fastand slow) did not act in a coordinated way as in the experiments. For that reason,
constraints were imposed when fitting the parameters describing voltage-dependent
inactivation steady-state. These constraints were selected based on the formulation of ICaL gating proposed in [51]. Our results showed that to all constraints werenecessary to reproduce experimental results on ICaL inactivation and at the sametime guarantee that electrophysiological markers (APD90 , AP triangulation, APD
adaptation to abrupt cycle length changes and systolic and diastolic [Ca2+ ]i levels)
were within physiological ranges.
This chapter considered evaluations at ionic, cell and tissue scales that were
performed as a model validation step once the solution to the above described
constrained minimization problem had been obtained. Future studies could develop
the methods to integrate at least part of the validation rules as constraints in theformulation of the minimization problem.
5.4.2 Reproducing experimental data from different laboratories
One of the difficulties in using different experimental datasets to adjust the parameters of an ionic current is that each dataset is commonly measured under
specific experimental conditions. For this reason, the results of different experimental studies can not be directly compared and, consequently, they cannot be
used concomitantly when for parameter estimation without considering such differences. On the other hand, the benefit of having experimental datasets recorded
under different conditions provides a wider range of data that can be taken into
account for model adjustment. Thus, if used correctly, the fact of having highly
heterogeneous experimental sets of data can be seen as an advantage rather thana limitation.
Specifically, in this chapter the use of the described experimental datasets madeit possible to improve the overall behavior of ICaL voltage-dependent inactivation
steady-state without modifying the theoretical expression of the involved gates but
rather by modifying the values of the associated parameters. Replication of theexperimental data in [1] and [3] with the optimized ICaL model led to differences
between in silico and experimental results were remarkably smaller than for the
initial CRLP model. Only when comparing with the experimental data in [2],the observed differences were small. In overall terms, the optimization problem
proposed and solved in this chapter led to relevant improvements in almost all
performed evaluations.

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5.5 Conclusions
In this chapter the expression of the ICaL current in the CRLP model has beenadjusted to correct for the unphysiological shape of the AP following phase 1.
Although the CRLP model, developed in chapter 2 and updated in chapter 3,
had been tested against various sets of experimental data, considering different
scales and interactions between model components as reported in chapter 4, it
presented a rather uncommon bump following AP transient repolarization in phase
1. In silico simulations of experimental protocols described in chapter 4 were ran
and the corresponding results were included as part of an optimization problem
with a number of carefully defined constraints, which was solved to adjust model
variables in such a way that experimental outcomes were more closely represented.
The methodology described in chapter 3 was used for optimization solving. To
guarantee that model overfitting was avoided, validation was performed at ionic,
cell and tissue scales.

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Chapter 6

A Human VentricularElectrophysiologicalModel with Improved
Description ofExperimental Behavior
6.1 Introduction
In chapter 3 and chapter 5, updates to the model described in chapter 2 were addressed. The update in chapter 3 allowed the inclusion of intracellular potassium
dynamics, while the update in chapter 5 involved refining the expression of theL-type calcium current, ICaL , to render an AP shape closer to experimental shapes,
particularly regarding AP phase 1. When developing the CRLP model in chapter 2,
departing from the GPB model, a paradigm based on reproducing a set of experimentally measured electrophysiological markers was used to guide the development
process. Within that paradigm, model variables or parameters were updated one
at a time along the iterations of the proposed methodology. When [K + ]i dynamics
were included in the CRLP model (chapter 3), an optimization algorithm was used
to update several ionic conductances concomitantly, while guaranteeing that the setof electrophysiological markers remained within physiological ranges or, otherwise,
their values were closer to the physiological range than with the original CRLP
model. When updating the expression of the ICaL current (chapter 5), the methodologies described in chapter 4 were used and parameters related to the steady-state

109

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New Methodologies for the Development
and Validation of Electrophysiological Models

value of voltage-dependent inactivation gates were modified while considering theeffect on global gating behavior and the improvement in in silico simulations ofexperimental results.
As discussed in chapter 4, when comparing simulations of electrophysiologicalmodels against experimental results, in silico simulations should replicate the corresponding experimental protocols as closely as possible. For that purpose, theconditions in which the experiments were performed should be carefully taken into
account. In addition, the spatial scales involved in the experiments should beconsidered when comparing, for instance, simulated and experimental electrophysiological properties. Regarding the latter, experiments performed to characterize
the AP usually employ small pieces of tissue to avoid the problem of sensitivity ofsome ion channels to the cell isolation procedure [106]. However, for computational
reasons, simulations are commonly carried out using single cell models. As shownin chapter 4, this may represent a limitation, particularly for a number of markers
whose values highly differ when evaluated in single cells or tissues. Importantly,
not only do results differ from cell to tissue but the underlying ionic contributions
can be different as well. This makes identification of ionic current conductanceseven more challenging.
Another important point to consider when comparing in silico results with experimental measurements is the definition of experimentally-based physiologicalranges. In chapter 2, physiological ranges were defined following the criteria usedin [12]. Specifically, the upper and lower limits of the physiological range of a given
electrophysiological property were determined by the maximum and the minimum
value of that property found in the literature. This definition has two main limitations. The first limitation refers to the fact that in many experimental studies only
an average value of a set of experiments is provided. Consequently, if the maximum
or the minimum value is selected based on such average values, the physiologicalrange will surely be smaller than the experimental range. The second limitation isin regards of the fact that, in each experimental study, electrophysiological properties are measured by using different bath solutions. This means that part of thedifferences found between experimental studies could be due to differences in theextracellular ionic concentrations.
The main objective of this chapter is to integrate all the initial formulations
and subsequent updates of the CRLP model proposed in previous chapters andto readjust the ionic conductances of the integrated model to improve replicationof experimental electrophysiological measures. All the methodologies introduced
throughout the thesis are used to build this human ventricular AP model. In particular, to compare the simulation and experimental results, each experiment was
initially considered individually, making sure that the conditions underlying theexperiment were replicated as faithfully as possible. In line with the conclusions
reported in chapter 4, a physiological range was established for each individual experiment by using the mean value and the standard deviation reported in the corresponding study (see section 6.2 for details). Additionally, based on the mentioned
differences found between cell and tissue in many electrophysiological properties,
only those that remain essentially the same in one and another scale were consid-

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Description of Experimental Behavior

111

ered for model development, while other properties were included in the validation
process, in which tissue simulation results were contrasted with experimental tissue
data.
The final version of the CRLP model built in this chapter is compared withthat of the most recent human ventricular cell models: ten Tusscher-Panfilov(TP06) [51], Grandi-Pasqualini-Bers (GPB) model [15], O’Hara-Rudy-dynamic(ORd) model [64], and the original Carro-Rodríguez-Laguna-Pueyo (CRLP) model[120]. Results are in all cases compared against experimental results and conditions
where one or another model best represents experimental outcomes are highlighted.
6.2 Methods
6.2.1 Adjustment of ionic conductances
The CRLP model with the updates of chapter 3 and chapter 5 was considered as astarting point. The ionic conductances of that model were redefined following theprocedure introduced in chapter 3 with the following modifications:• All the ionic conductances and the calcium diffusion coefficients between theintracellular spaces were considered as variables to be adjusted.
• A large number of markers obtained from different experimental datasets
were considered. In the simulation of each dataset, the stimulation protocol
was reproduced and the specific experimental conditions were considered by
setting the values of the extracellular ionic concentrations as those in theexperiments.
• A new optimization function was proposed to make as many markers as
possible lie within established physiological ranges.
Details on each of the above aspects are provided in the following paragraphs.
Minimization function
With the aim of developing a model that closely reproduces the average behavior of human ventricular epicardial cells, a minimization function was introduced
that accounts for the sum of errors associated with the different evaluated electrophysiological markers. For each marker, mi , the error was defined in terms of theexperimental mean and standard error of the mean (SEM) as follows:
errmi (x) =µi − mi (x)
SEMi

(6.1)

where µi is the experimental mean value of marker mi ; mi (x) is the simulated
value of marker mi obtained for model parameter values in vector x; and SEMi isthe experimental standard error of the mean of marker mi .

JesúsCarroNew Methodologies for the Development
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112

The objective function was defined as the sum of the square of the errors associated with all evaluated electrophysiological markers as follows:
f (x) =

M
∑
i=1

2

(errmi (x)) =

)2
M (
∑
µi − mi (x)
i=1

SEMi

(6.2)

where M is the total number of electrophysiological markers evaluated in the optimization problem.
Of note, SEM represents the confidence in the experimental average value ofa marker and depends on the number of experiments and on the sample standard
deviation. If the sample standard deviation is high but there are a larger number ofexperiments, the mean experimental value could be trusted more than another one
for which the sample standard deviation was lower but was obtained from a fewer
number of experiments. When the difference between the simulated and mean
experimental values of the marker is within the SEM, the error for that marker isbetween -1 and 1. Consequently, The defined objective function reduces the error
for markers within that range and increases it for markers out of that range.
Experimental data sets
The following experimental datasets were included in the optimization problem
defined to fit ionic current conductances:• TESZT: AP traces from healthy human ventricular endocardial tissues atsteady-state were acquired at the University of Szeged. After reaching steadystate following pacing at a CL of 1000 ms, 10 pulses were recorded. Following
this, different drugs were added to inhibit potassium currents: 1 µM of HMR1556 and 100 nM of L735,821 corresponding to approximately 95% IKs block[137, 138]; 50 nM of dofetilide corresponding to 15% IKr block [139]; 1 µM of
dofetilide and 1 µM of E-4031 corresponding to 70% IKr block [34, 139]; 100µM of BaCl2 corresponding to 75% IK1 block [140]. After reaching steadystate following each of the above drug actions, 10 pulses were recorded at apacing CL of 1000 ms.
For each AP recording, two markers were measured: AP D90 and resting
membrane potential (Vrest ). According to the results presented in chapter 4,
these two markers show similar values and underlying mechanisms when evaluated at cell or tissue levels. In Table 6.1, evaluation of these two markers
in the TESZT dataset, both at baseline and following each of the describedpharmacological potassium inhibitions, are summarized. For potassium inhibition experiments, results are expressed as percentage of variation withrespect to control conditions.
The bath solution in the experiments of the TESZT dataset is describedin [64]. Values of ion concentrations in the bath solution were as follows:[Ca2+ ]o = 1.8 mM , [K + ]o = 4.0 mM , [N a+ ]o = 135.0 mM , [Cl− ]o =124.6 mM .

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6. A Human Ventricular Electrophysiological Model with Improved
Description of Experimental Behavior
ExperimentControl
IK1 block 75%
IKr block 15%
IKr block 70%
IKs block 95%

MarkerAP D90 (ms)
Vrest
Variation in AP D90 (%)
Variation of Vrest (%)
Variation in AP D90 (%)
Variation of Vrest (%)
Variation in AP D90 (%)
Variation of Vrest (%)
Variation in AP D90 (%)
Variation of Vrest (%)

Mean
293.85-86.75+19.25-4.56+51.17+0.67+51.17
+0.67
+1.53
-0.19

SEM
10.41
0.92
3.09
1.95
8.57
1.11
8.57
1.11
1.42
1.59

113
n
23
6
11
11
6

Table 6.1: Evaluation of AP D90 and Vrest in the TESZT dataset.

• CLVPET: AP traces from healthy human ventricular endocardial tissues
under variable CL stimulation were acquired at the University of Szeged.
After reaching steady-state following pacing at a CL of 1000 ms, stimulati at
CLs varying from 300 ms to 5000 ms were applied. For each CL, 25 pulses
were delivered and the AP corresponding to the last pulse was recorded. As
in the TESZT dataset, drugs were subsequently added to inhibit potassium
currents. The same drugs, and at the same concentrations, were supplied tothe tissues in this case.
For AP recordings under control conditions, the variation in the APD withrespect to steady-state APD at CL = 1000 ms was quantified. For AP recordings under potassium inhibitions, the variation in the APD with respect to the
APD at the same CL under control conditions was quantified. In Table 6.2,
evaluation of variations in AP D90 are summarized.
The bath solution in the experiments of the CLVPET dataset is describedin [64]. Values of ion concentrations in the bath solution were as follows:[Ca2+ ]o = 1.8 mM , [K + ]o = 4.0 mM , [N a+ ]o = 135.0 mM , [Cl− ]o =124.6 mM .
• Drouin et al. (1995) [8]: APD90 values from epicardial and endocardial
cells of human ventricular tissues were measured. APD90 was quantified at
different CLs spanning from 1000 ms to 10000 ms. In Table 6.3, evaluationof AP D90 is summarized.
The bath solution used in this experiments is described in [8]. Values ofion concentrations in the bath solution were as follows: [Ca2+ ]o = 2.7 mM ,[K + ]o = 4.0 mM , [N a+ ]o = 150.8 mM , [Cl− ]o = 141.4 mM .
• Pieske et al. (2002) [9]: Intracellular sodium ([N a+ ]i ) concentrations were
measured at the following CLs: 333, 500, 1000 and 4000 ms. In this study,
the absolute value of [N a+ ]i at CL = 1000 ms and the variations in [N a+ ]i

JesúsCarroNew Methodologies for the Development
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114

ExperimentMarkerControl∆AP D90 (%)

IK1 Blocked∆AP D90 (%)

IKr Blocked∆AP D90 (%)

IKs Blocked∆AP D90 (%)

CL
300 ms
400 ms
500 ms
700 ms1000 ms
1500 ms
2000 ms
3000 ms
5000 ms
400 ms
500 ms
700 ms1000 ms
1500 ms
2000 ms
3000 ms
5000 ms
500 ms
700 ms1000 ms
1500 ms
2000 ms
3000 ms
5000 ms
300 ms
400 ms
500 ms
700 ms1000 ms
1500 ms
2000 ms
3000 ms
5000 ms

Mean-24.98-20.35-14.57
-8.61
-3.12
+4.28
+8.75
+13.86+16.32-9.68-0.37+10.93+20.55+29.32+32.56+37.49+41.93+19.51+41.52+53.68+68.72+82.65+100.00+127.33-23.80-17.94-14.22
-8.22
-2.62
+2.82
+6.59
+9.13
+11.96

SEM
1.38
1.48
1.05
1.00
0.60
1.02
1.16
2.22
1.55
5.35
4.74
3.75
3.66
4.33
4.96
6.09
7.17
5.45
8.43
9.05
10.41
11.10
12.80
15.73
2.68
2.52
2.37
1.81
1.59
1.52
2.02
2.71
3.79

n

21

6

9

5

Table 6.2: Evaluation of variations in AP D90 in the CLVPET dataset.

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6. A Human Ventricular Electrophysiological Model with Improved
Description of Experimental Behavior
ExperimentMarkerENDO

AP D90 (ms)EPI

AP D90 (ms)CL1000 ms
2000 ms
3000 ms
4000 ms
5000 ms
6000 ms
10000 ms
1000 ms
2000 ms
3000 ms
4000 ms
5000 ms
6000 ms
10000 ms

Mean
329.5
345.3
362.2
369.5
378.1
393.0
406.5
350.6
382.1
392.0
406.5
418.8
424.4
463.1

SEM
15.6
17.2
20.5
21.2
20.8
22.8
25.5
14.6
19.5
19.9
18.9
17.2
17.2
14.6

115
n

14

14

Table 6.3: Evaluation of AP D90 in the study by Drouin et al. [8].
Experiment
CL = 1000 ms
CL = 4000 ms
CL = 500 ms
CL = 333 ms

Marker[N a+ ]i (mM)
Variation in [N a+ ]i (%)
Variation in [N a+ ]i (%)
Variation in [N a+ ]i (%)

Mean
18.58
-15.34+10.66+21.74

SEM
3.91
17.5
22.8
25.3

n
7
7
7
7

Table 6.4: Evaluation of [N a+ ]i and its variations in the study by Pieske et al.
[9].
for other CLs expressed as a percentage with respect to CL = 1000 ms were
quantified. In Table 6.4, evaluation of these markers is summarized.
Values of ion concentrations in the bath solution were as follows: [Ca2+ ]o =1.0 mM , [K + ]o = 5.4 mM , [N a+ ]o = 152.0 mM , [Cl− ]o = 157.8 mM .
• Schmidt et al. (1998) [10]: Intracellular calcium ([Ca2+ ]i ) concentrations
were measured at the following CLs: 333, 400, 500, 575, 666, 750, 1000, 1400,
2000, 3000 and 8000 ms. In this study, the absolute systolic and diastolicvalues of [Ca2+ ]i for CL = 1000 ms and the variations in those values forother CLs expressed as a percentage with respect to CL = 1000 ms were
quantified. In Table 6.5, evaluation of these markers is summarized.
The concentration of the main ions in the bath solution was: [Ca2+ ]o =2.5 mM , [K + ]o = 5.9 mM , [N a+ ]o = 126.2 mM , [Cl− ]o = 133.3 mM .
• Coppini et al. (2013) [11]: Intracellular calcium ([Ca2+ ]i ) concentrations
were measured at the following CLs: 1000, 2000, and 5000 ms. The following
markers were evaluated:

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116

Experiment
CL = 1000 ms
CL = 333 ms
CL = 400 ms
CL = 500 ms
CL = 575 ms
CL = 666 ms
CL = 750 ms
CL = 1400 ms
CL = 2000 ms
CL = 3000 ms
CL = 8000 ms

MarkerDiastolic [Ca2+ ]i (µM)
Systolic [Ca2+ ]i (µM)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. of Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)
Var. in Diastolic [Ca2+ ]i (%)
Var. of Systolic [Ca2+ ]i (%)

Mean
1.798
0.256
+48.26-2.22+34.45+8.89+25.83+14.11+18.96+10.92
+6.87
+8.41
+6.32
+5.61
-4.60
-5.71
+2.85
-16.63+1.09-21.95+3.43-27.56SEM
0.08870.0295
17.83
5.22
13.23
5.51
13.23
5.61
12.06
5.71
9.75
5.22
9.21
5.028
10.93
4.74
11.51
3.97
9.21
3.48
12.06
3.29

n
6
6
6
6
6
6
6
6
6
6
6

Table 6.5: Evaluation of [Ca2+ ]i and its variations in the study by Schmidt et al.
[10].

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6. A Human Ventricular Electrophysiological Model with Improved
Description of Experimental Behavior
Experiment
CL = 1000 ms
CL = 2000 ms
CL = 5000 ms

MarkerDiastolic [Ca2+ ]i (µM)∆[Ca2+ ]i (µM)
tCai ,50% (ms)tCai ,90% (ms)Var. of Diastolic [Ca2+ ]i (%)
Var. of Diastolic [Ca2+ ]i (%)∆[Ca2+ ]i (µM)
tCai ,50% (ms)tCai ,90% (ms)

Mean
0.139
0.3528170.27356.21
-3.66
-7.82
0.343
239.33517.57

117
SEM
0.00994
0.0176
26.37
46.15
6.70
5.83
0.0163
35.16
52.73

n
6
6
6

Table 6.6: Evaluation of [Ca2+ ]i characteristics in the study by Coppini et al. [11].
Markert90%,1000→600 (ms)t90%,600→1000 (ms)

Mean
119.5
243.0

SEM
21.57
59.40

n
2

Table 6.7: Evaluation of time for APD rate adaptation in the study by Pueyo
et al. [13].
– Diastolic [Ca2+ ]i : Absolute value for CL = 1000 ms and percentage ofvariation at CL = 2000 and 5000 ms with respect to CL = 1000 ms.
– Difference between systolic and diastolic [Ca2+ ]i levels, denoted by ∆[Ca2+ ]i :Absolute values for CL 1000 and 5000 ms.
– Time constants: Time from the peak to 50% and 90% recovery of the[Ca2+ ]i transient, denoted by tCai ,50% and tCai ,90% , respectively: Absolute values for CL = 1000 and 5000 ms.
In Table 6.6, evaluation of these markers is summarized.
Values of ion concentrations in the bath solution were as follows: [Ca2+ ]o =1.8 mM , [K + ]o = 4.0 mM , [N a+ ]o = 132.0 mM , [Cl− ]o = 142.0 mM .
• Pueyo et al. (2010) [13]: APD rate adaptation to abrupt CL changes whenvarying CL from 1000 to 600 ms and from 600 to 1000 ms were measured.
In this study, following a period of steady-state pacing at a CL of 1000 ms,
stimuli were delivered at a CL of 600 ms for 10 minutes and, subsequently,
at a CL of 1000 ms for additional 10 minutes. In each transition, the time
to reach 90% of the final APD value was measured, denoted by t90%,1000→600
and t90%,600→1000 , respectively. In Table 6.7, evaluation of this marker issummarized.
Values of ion concentrations in the bath solution were as follows: [Ca2+ ]o =1.0 mM , [K + ]o = 4.0 mM , [N a+ ]o = 144.0 mM , [Cl− ]o = 128.0 mM .

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Simulation of experimental protocols
The experimental protocols were simulated in isolated cells using a biphasic stimulation current as described in [118]. While biphasic stimulation does not remove
differences between cell and tissue simulation results as discussed in chapter 4, it
provides a better approximation for the variations in [N a+ ]i and [K + ]i with thefrequency of stimulation, owing the fact that the ionic load during each cycle is 0.
The amplitude of the biphasic stimulation current was set to twice the diastolic
threshold of the model. The duration of the positive square pulse of the biphasic
stimulation was set to 1 ms.
The model was first stabilized until steady-state conditions were reached. In
particular, the following stimulation protocols were applied when comparing withthe different experimental datasets:• For the TESZT and CLVPET datasets, the model was first stimulated during
50 minutes at a CL of 1000 ms. The TESZT protocol involved evaluation
during the last cycle. The CLVPET protocol involved replication at different
CLs after this pre-stimulation.
• For the experiments of Drouin et al., Pieske et al., Schmidt et al. and Coppini et al., the model was stimulated during 50 minutes at different CLs.
Electrophyisiological markers were evaluated during the last cycle.
• For the experiments published by Pueyo et al., the model was first stimulated
during 50 minutes at a CL of 1000 ms, CL was subsequently reduced to 600
ms and stimulation applied for 10 minutes and, finally, CL was increased to1000 ms and stimulation applied for other 10 minutes.
Optimization algorithm
The optimization algorithm was as described in chapter 3. Design variables included all the ionic conductances of the model as well as parameters related todiffusion of intracellular calcium, namely:• GNa : conductance of the fast sodium current (INa ).
• GNa,bk : conductance of the background sodium current (INa,bk ).
• GKr : conductance of the rapidly activating potassium current (IKr ).
• GKs : conductance of the slowly activating potassium current (IKs ).
• GKp : conductance of the plateau potassium current (IKp ).
• Gto : conductance of the transient outward potassium current (Ito ).
• GK1 : conductance of the inward rectifier potassium current (IK1 ).
• GClCa : conductance of the calcium-activated chloride current (IClCa ).
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6. A Human Ventricular Electrophysiological Model with Improved
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119• GCl,bk : conductance of the background chloride current (ICl,bk ).
• GCaL : conductance of the L-type calcium current (ICaL ).
• GCa,bk : conductance of the background calcium current (ICa,bk ).
• I¯NaK : maximum current carried through the sodium-potassium pump.
• I¯ncx : maximum current carried through the sodium-calcium exchanger.
• I¯PMCA : maximum current carried through the sarcolemmal calcium pump.
• JCajunc,sl : diffusion of calcium from the junctional to the subsarcolemmal compartment.
• JCasl,myo : diffusion of calcium from the subsarcolemmal compartment to the
myoplasm.
• JSRleak : diffusion of calcium from the sarcoplasmatic reticulum to the myoplasm.
• JSRCarel : calcium release from the sarcoplasmatic reticulum to the junctional
compartment.
• Jser Ca : calcium release from the sarcoplasmatic reticulum to the myoplasm.
As in chapter 3, all these variables were scaled to deal with their different ordersof magnitude:( )yixi = 20 · log10
,
(6.3)
yi0
where yi0 and yi are the initial and current values of the ith variable and xi is its
scaled value.
The initial radius for the trust region was selected as 2.5 for all the variables.
The minimum tolerance for the scaled variables, δ1 , was set to 0.01. The minimum
tolerance for the optimization function, δ2 , was set to 0.01. The minimum gradient,δ3 , was taken as 0.01. To create the database of markers in each iteration, 500
randomly-selected combinations of the design variables in the trust region were
simulated in each iteration of the optimization algorithm. For each marker, afirst-order response surface was calculated. To guarantee positivity of the response
surface of the objective function, a response surface of first order was created forthe error and later, it was squared.
Valid ranges for electrophysiological markers were defined based on experimental observations and simulated values of the electrophysiological markers undercontrol conditions. Specifically, the upper and lower limits were defined as follows:{}
mj,U L = max µmj ,exp + 2 · σmj ,exp ; mj,control{}
mj,LL = min µmj ,exp − 2 · σmj ,exp ; mj,control

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where mj,U L is the upper limit for marker mj , mj,LL is the lower limit for marker
mj , µmj ,exp is the mean of the experimental observations for marker mj , σmj ,exp
is the standard deviation of the experimental observations for marker mj andmj,control is the simulated value of the marker under control conditions.
All markers were considered as constraints for the optimization algorithm. As
in chapter 3, these constraints were defined to be consistent with the general formulation of the inequality constraints:
gj (x) =mj (x) − mj,CR− 1 ≤ 0,mj,RRwhere mj,CR = (mj,U L + mj,LL )/2 is the central point of the valid range for theelectrophysiological marker j and mj,RR = (mj,U L − mj,LL )/2 is the distance between
mj,CR and each of the limits of the valid range.
Implementation
The model was implemented in CellML using the original CellML model published in [120]. For the simulations, the model was exported to C++ and included in the application of the DENIS project (http://denis.usj.es). The tool
to export the model can be found here: https://bitbucket.org/usj_bsicos/
denis-cellml2c/src/v0.1/. The original code of the DENIS application can befound here: https://bitbucket.org/usj_bsicos/denis-myocyte/src/beta_v1.
0/. The equations of the model were solved using Fordward-Euler integrator witha time step of 0.002 ms. Post-processing of simulation results and implementationof the optimization algorithm were developed in MatLab® 2017a (The MathworksInc.).
6.2.2 Model comparison
The behavior of the optimized model was compared against the original CRLP
model and other commonly employed human ventricular cell models (TP06, GPB,
ORd). Electrophysiological markers were computed in single cells for each of thosemodels.
6.2.3 Model validation
The optimized model was validated against experimental results by using a set ofelectrophysiological markers different from the ones used for model development.
In chapter 4, markers like triangulation and upstroke or maximal potential wereshown to be highly different when evaluated in cells or tissues. These markers, nothaving been used for model optimization, were used for validation. Evaluation was
performed by using a 3 cm-long homogeneous fiber model of endocardial ventricular
cells. The conditions of the experiments against which simulation results werecompared were reproduced by adjusting the external ionic concentrations. The
value of σ was calculated to obtain a CV close to 65 cm/s [88] under control

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121

conditions. The cell capacitance was set to Cm = 1 µF /cm2 . Each marker was
evaluated at five positions within the fiber. Those positions were located at 1.00,
1.25, 1.50, 1.75 and 2.00 cm from the stimulation end of the cable. The mean valueof each marker over the five positions was calculated.
The same evaluation described for the optimized CRLP model was performed
for the initial CRLP model to assess the effect of the model optimization introducedin this chapter.
Experimental dataset
The TESZT dataset described above provided experimental results on the following
AP markers:• Triangulation: Difference between the AP D90 and the AP D50 .
• Upstroke: Maximum of the temporal derivative of the membrane potential.
• Maximum potential: Maximum of the membrane potential.
On top of evaluating these markers under control conditions, the percentages ofvariation in each marker following ionic current inhibitions (IK1 , IKr or IKs blockades)with respect to control conditions were measured. The bath solution in theseexperiments is described in [64]: [Ca2+ ]o = 1.8 mM , [K + ]o = 4.0 mM , [N a+ ]o =135.0 mM , [Cl− ]o = 124.6 mM .
For the steady-state simulations in fiber, we simulate 5 minutes of activity. We
reduced the simulation time for the fiber simulations in order to be able to obtain
the results in a reasonable time.
Implementation
Simulations were performed in a homogeneous 3-cm long fiber composed of endocardial cells. A semi-implicit operator-splitting scheme [87] was used to solve themodel with a space discretization of ∆x = 0.1 mm and a time step of ∆t = 0.002
ms. The value of σ was selected to get a maximum CV of 65 cm/s, which is inagreement with experimentally reported CV values [88]. The cell capacitance was
set to Cm =1µF/cm2 . Fiber simulations were implemented in FORTRAN.
6.3 Results
6.3.1 Model optimization
Following model optimization, the variation in the ionic conductances of the optimized model with respect to the initial CRLP model varied between -23% and30%. The original model had an error of 1783.4 (according to the formula definedin section 6.2). After 13 iterations of the optimization algorithm, the error was
1360.5, thus representing a reduction in the error of 23.7%.
While in the initial CRLP model 77.2% of the markers were in physiologicalrange, this value increased to 81.2% after the optimization. Figure 6.1 shows theJesúsCarroNew Methodologies for the Development
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Blockage Marker
Control
75% IK1
15% IKr
70% IKr
95% IKs

Physiol.
This
CRLP ORd GPB TP06
min mean max work
AP D90 (ms) 194.0 293.8 393.6 328.6 334.2 279.6 283.0 285.4
Vmin (mV)-95.6 -86.8 -77.9 -87.4 -90.7 -96.2 -81.6 -93.8∆AP D90 (%) 4.1 19.3 34.4 13.3 34.1 10.1 26.1 6.3∆Vmin (%)
-14.1 -4.6 5.0 -32.4 -25.0 1.0 -4.6 -3.9
∆AP D90 (%) 9.3 39.1 68.8 2.7
2.9
9.6 2.4 2.4∆Vmin (%)
-5.1 -1.2 2.8 0.2
0.0
0.0 0.0 0.0∆AP D90 (%) -5.5 58.1 121.7 15.2 15.9 71.6 13.1 13.0
∆Vmin (%)
-6.4 1.7 9.8 0.8-0.1
0.0 0.0 0.0∆AP D90 (%) -5.4 1.5 8.5 0.5
0.6
6.3 0.7 55.9
∆Vmin (%)
-8.0 -0.2 7.6 0.0
0.0
0.0 0.0 0.2

Table 6.8: Results of steady-state AP markers for a CL of 1000 ms computed withthe optimized CRLP model and other published human ventricular cell models.
AP of a single epicardial cell simulated under control conditions with the optimized
CRLP model and with other published human ventricular cell models. As can benoted, the peak and dome features improved in chapter 5 for the CRLP model were
not modified following the model optimization introduced in this chapter.
The values of the optimized ionic conductances and the final description of themodel can be found in Appendix C.
Steady-state AP markers
Table 6.8 summarizes the results of steady-state AP markers for a CL of 1000 ms
under control conditions and following ionic current inhibitions. These results werecompared with experimental measurements from the University of Szeged and theresults of other recent human ventricular cell models, as described in 6.2. The
first version of the CRLP model, proposed in chapter 2, had only two steady-state
AP markers out of physiological range following inhibition of IK1 or IKr currents.
These two markers were still out of physiological range for the new CRLP model.
For IK1 inhibition, the variation in the resting membrane potential was slightly
larger for the optimized CRLP model than for the initial model version. The lackof improvement for the new CRLP model can be explained by the modifications
introduced into the model prior to the optimization, namely the introduction of[K + ]i dynamics and the modification of the ICaL current, whereas subsequent optimization tried to compensate the effect of such variations on the resting membranepotential. Similar conclusions to those described above for IK1 block apply to thecase of IKr block.
APD dependence on CL
Figure 6.2 shows the variation in AP D90 for different CLs under control conditions
as computed with the optimized CRLP model, the original CRLP model and other

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6. A Human Ventricular Electrophysiological Model with Improved
Description of Experimental Behavior

b)
50

50

0

0

V (mV)V (mV)

a)

-50

-100

-50

-100
0

100

200

300

400

500

0

100

200

t (ms)

300

400

500

400

500

t (ms)

c)

d)
50

50

0

0

V (mV)V (mV)

123

-50

-100

-50

-100
0

100

200

300

400

500

0

100

200

t (ms)300

t (ms)e)

V (mV)

50

0

-50

-100
0

100

200

300

400

500

t (ms)

Figure 6.1: Action Potential of the different models under control conditions insingle epicardial cells: a) optimized CRLP model; b) initial CRLP model; c) GPB
model; d) ORd model; e) TP06 model.

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140

APD 90/APD 90,CL=1000,SS (%)

130
120
110
100
90

CRLP
TP06
GPB
ORd
New model
CLVPET (Physiological Range)
CLVPET (mean SE)

80
70
60
0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

CL (ms)Figure 6.2: AP D90 variation with CL under the same conditions of the CLVPET
dataset.

published human ventricular cell models. Experimental results evaluated for the
CLVPET dataset are presented for comparison. The AP D90 value in cycle 25of the CLVPET protocol expressed as a percentage of the AP D90 value understeady-state conditions for a CL of 1000 ms is presented. While the initial CRLP
model and the GPB model showed APD increases for decreasing CLs below 500 ms,
the optimized CRLP model already corrected this type of behavior, showing APD
decrases for decreasing CLs no matter the value of the CL. In fact, for long CLs,
the optimized CRLP model shows results very close to the experimental outcomes.
Figure 6.3 shows results of APD variations with CL changes, in this case following ionic current inhibitions. APD values are normalized with respect to thosecorresponding to the same CL under control conditions. As can be observed, there
is very good agreement between the new version of the CRLP model and mostexperimental results. Out of all the evaluated current inhibitions, the optimized
CRLP model renders results that are within experimental limits in all cases except
for the one corresponding to IKr inhibition.
Simulation results for AP D90 variations with CL changes were also comparedwith the experimental results published in [8]. Figure 6.4 illustrates the results for
endocardial and epicardial cells. The optimized CRLP model was within physiological ranges for all CLs. Endocardial cells showed longer APs than epicardial
cells for the new CRLP model as well as for all other published human ventricular

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Description of Experimental Behavior

85% of IKr

B

200
180
160
140
120
100
0

1000

2000

3000

4000

5000

APD 90/APD 90,Control (%)

APD 90/APD 90,Control (%)

A

25% of IKr
300
250
200
150
100
0

1000

CL (ms)5% of IKs

D

250
200
150
100
50
0
0

1000

2000

3000

4000

5000

CL (ms)APD 90/APD 90,Control (%)

E

2000

3000

4000

5000

4000

5000

CL (ms)APD 90/APD 90,Control (%)

APD 90/APD 90,Control (%)

C

125

5% of IKs
115
110
105
100
95
0

1000

2000

3000

CL (ms)25% of IK1
160

160

140

140

120

120

100

100

80
0

1000

2000

3000

4000

80
5000

CRLP
TP06
GPB
ORd
New model
CLVPET (Physiological Range)
CLVPET (mean SE)

CL (ms)Figure 6.3: AP D90 variation with CL under the same conditions of the CLVPET
dataset following ionic current inhibitions: A) 15% IKr block; B) 70% IKr block; C)
95% IKs block; D) Zoomed graphic for 15% IKs block; E) 75% IK1 block.

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Physiol.
This
CRLP ORd GPB TP06
min mean max work
[N a+ ]i (mM) [9]0.00 18.58 39.27 9.04 10.06 6.92 8.10 8.06
Sys. [Ca2+ ]i (µM) [10] 1.36 1.80 2.23 0.98 1.45 0.63 0.77 1.77
Dia. [Ca2+ ]i (µM) [10] 0.11 0.26 0.40 0.11 0.12 0.09 0.10 0.13
Dia. [Ca2+ ]i (µM) [11] 0.07 0.14 0.21 0.10 0.10 0.09 0.09 0.10
MarkerTable 6.9: Results of steady-state [N a+ ]i and [Ca2+ ]i for a CL of 1000 ms computed with the optimized CRLP model and other published human ventricular cell
models.
cell models, including the initial CRLP model.
Intracellular sodium concentrations
In Table 6.9, steady-state [N a+ ]i for a CL of 1000 ms is presented as calculated
for different human ventricular cell models, including the optimized CRLP model
introduced in this chapter. Figure 6.9 shows the variation in [N a+ ]i for different
CLs, compared with the physiological ranges measured in the experiments of [9].
All the tested AP models are within experimental ranges.
Intracellular calcium concentrations
In Table 6.9, steady-state diastolic and systolic [Ca2+ ]i values for a CL of 1000 ms
are presented as calculated for different human ventricular cell models, including
the optimized CRLP model introduced in this chapter, while reproducing the conditions of the experiments described in [10, 11]. Whereas the diastolic [Ca2+ ]i levelin the optimized model was within the physiological ranges of both experimental
studies, the systolic [Ca2+ ]i level was somewhat below experimentally reported
limits. Even if the optimization algorithm moved systolic [Ca2+ ]i towards thephysiological range, previous modifications introduced into the model had separated the simulated systolic level to such an extent that it was not possible for the
optimization algorithm to completely move it into the expected range. Nevertheless, other commonly used human ventricular AP models present systolic [Ca2+ ]ilevels similar to those of the optimized CRLP model.
Figure 6.6 shows the variation in diastolic and systolic [Ca2+ ]i levels for different
CLs under the conditions of the experiments of [10]. Results are expressed aspercentages with respect to the case when the pacing frequency was 1 Hz. For thecomparison with the experimental data of [10], the optimized CRLP model is the
only one with all simulated values lying within the experimentally reported limits.
Figure 6.7 shows variations in [Ca2+ ]i characteristics for different CLs under theconditions of the experiments of Coppini et al. [11]. The difference between systolicand diastolic [Ca2+ ]i levels was within experimental limits when computed for the
optimized CRLP model (Figure 6.7.A), while all the other human ventricular cell
models rendered results out of those limits. Variations in systolic [Ca2+ ]i , as well

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6. A Human Ventricular Electrophysiological Model with Improved
Description of Experimental Behavior

A

B
500

2000

APD 90 (ms)

APD 90 (ms)

2500

1500
1000
500
0

400

300

200
0

2000 4000 6000 8000 10000

0

2000

4000

CL (ms)

6000

8000 10000

CL (ms)

C

D
2500

500

2000

APD 90 (ms)

APD 90 (ms)

127

1500
1000
500
0

400

300

200
0

2000 4000 6000 8000 10000

0

2000

4000

CL (ms)

6000

8000 10000

CL (ms)
CRLP
TP06
GPB
ORd
New model
Drouin et al. (Physiological Range)
Drouin et al. (mean SE)

Figure 6.4: Simulations of the experiments by Drouin et al. [8]: A) AP D90
variation with CL for epicardial cells. B) Zoomed version of the figure shown
in A. C) AP D90 variation with CL for endocardial cells. D) Zoomed version of the
figure shown in C.

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150[Na +]i /[Na +]i (f=1Hz) (%)[Na +]i /[Na +]i (f=1Hz) (%)

128

100

50
0.5

1

1.5

2

2.5

150
CRLP
TP06
GPB
ORd
New model
Pieske (Physiological Range)
Pieske (mean SE)

100

50
3

f (Hz)

Figure 6.5: Variation in [N a+ ]i under the same conditions of the experiments byPiekse et al. [9] for endocardial cells.

300

200

100

0
0.5

1

1.5

2

2.5

3

Dia. [Ca 2+ ]i /[Ca 2+ ]i,1Hz (%)

Dia. [Ca 2+ ]i /[Ca 2+ ]i,1Hz (%)

A
300
CRLP
TP06
GPB
ORd
New model
Schmidt (Physiological range)
Schmidt (mean SE)

200

100

0

f (Hz)

400
300
200
100
0
0.5

1

1.5

f (Hz)

2

2.5

3

Sys. [Ca 2+ ]i /[Ca 2+ ]i,1Hz (%)

Sys. [Ca 2+ ]i /[Ca 2+ ]i,1Hz (%)

B
150
100
50
0
0.5

1

1.5

2

2.5

3

f (Hz)

Figure 6.6: Variation in [Ca2+ ]i under the same conditions of the experiments by
Schmidt et al. [10]: A) Diastolic [Ca2+ ]i normalized by the value when the frequency is 1Hz; B) Systolic [Ca2+ ]i normalized by the value when the the frequency
is 1Hz.

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Physiol.
This
CRLP ORd GPB TP06
min mean max work
t90%,1000→600 (s) 58.5 119.5 180.5 260.5 314.5 254.2 157.9 299.9
t90%,600→1000 (s) 75.0 243.0 411.0 265.3 346.4 352.2 184.4 350.1
MarkerTable 6.10: Results of APD adaptation to abrupt CL changes computed with alltested human ventricular models under the same conditions as in [13].
as time from [Ca2+ ]i peak to 50% and 90% recovery, for different CLs was in very
good agreement with experimental behavior (Figure 6.7.B, C and D).
APD adaptation to abrupt changes in the CL
Table 6.10 presents the results corresponding to APD adaptation to abrupt CLchanges as calculated for all tested human ventricular cell models and the experiments of [13]. Only the GPB model rendered values of time for 90% adaptation
lying within physiological ranges. The optimized CRLP model, despite being closer
to the experimental limits than the initial CRLP model, only provided physiological results for CL increases, but not decreases, as happened with all other model
except for the GPB model.
6.3.2 Model validation
Table 6.11 summarizes validation results for the optimized and the initial CRLP
models. Under control conditions, both models rendered triangulation values out ofphysiological range, but upstroke velocity and Vmax values within range. The effectsof ionic current inhibitions over triangulation were practically within physiologicalrange for both the new and the initial CRLP models. Only IK1 inhibition renderedresults out of physiological range, particularly for the upstroke velocity and Vmax .
Of note, when IK1 was blocked in the optimized model, the ERP increased over1000 ms and only out of two consecutive stimulations was able to propagate anAP.
6.4 Discussion
6.4.1 Model performance
The optimized CRLP model has been shown to produce a remarkably improved
behavior with respect to the initial CRLP model when comparing simulated datawith experimental evidences for a set of evaluated electrophysiological properties.
A detailed discussion on this is presented in the following paragraphs, together
with a comparison with other human ventricular cell models already published in
the literature.

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B[Ca 2+ ]i ( M)

0.8
0.6
0.4
0.2
0
1000

2000

3000

4000

5000

Sys [Ca 2+ ]i /[Ca 2+ ]i,1000ms (%)

A

150

100

50

0
1000

2000

CL (ms)

3000

4000

5000

4000

5000

CL (ms)C

D
1000

t 90% (ms)

800

200
100
0
1000

600
400
200

Sys [Ca 2+ ]i /[Ca 2+ ]i,1000ms (%)

t 50% (ms)

300

2000

3000
CL150
(ms)
100

50

0

4000

5000

0
1000

2000

3000

CL (ms)
CRLP
TP06
GPB
ORd
New model
Coppini (Physiological range)
Coppini (mean SE)

Figure 6.7: Variation in [Ca2+ ]i characteristics under the same conditions of theexperiments by Coppini et al. [11]: A) Difference between systolic and diastolic[Ca2+ ]i levels as a function of CL; B) Systolic [Ca2+ ]i for different CLs normalized
by the value when CL is 1000 ms; C) Time from [Ca2+ ]i peak to 50% as a functionof CL; D) Time from [Ca2+ ]i peak to 90% as a function of CL.

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6. A Human Ventricular Electrophysiological Model with Improved
Description of Experimental Behavior
BlockageControl

75% IK1

15% IKr

70% IKr

95% IKs

MarkerT riangulation (ms)
U pstroke (mV/ms)
Vmax (mV)∆T riangulation (%)∆U pstroke (%)∆Vmax (%)
∆T riangulation (%)∆U pstroke (%)∆Vmax (%)
∆T riangulation (%)∆U pstroke (%)∆Vmax (%)
∆T riangulation (%)∆U pstroke (%)∆Vmax (%)

min
44.6
132.6
15.7
21.3
-21.6
-7.3
-3.5
-37.0
-44.8
3.1
-23.8
-25.7
-9.6
-13.3
-31.1

Physiol.
mean max
65.2
85.8
283.5 434.4
28.0
40.3
58.2
95.2
-5.6
10.4
6.4
20.0
66.6 136.8
-1.7
33.6
-3.1
38.7
138.7 274.4
2.6
28.9
-5.1
15.5
2.4
14.4
-5.2
2.8
12.1
55.2

This
work
137.9
275.9
25.1
20.2
-98.6
-69.3
4.5
0.1
0.0
24.7
-1.5
0.2
0.2
-0.1
0.0

131

CRLP
111.1
278.6
27.4
27.7
0.8
-0.1
0.8
-0.7
0.4
8.3
1.9
1.0
2.3
-0.6
-7.1

Table 6.11: Validation results. Steady-state AP markers for a CL of 1000 ms
calculated in tissue fibers simulated with the optimized and initial CRLP models.

Response to potassium current inhibitions
One of the limitations of the initial CRLP model was related to APD prolongation
following potassium current inhibitions. Specifically, when the initial CRLP model
was paced at low frequencies, IK1 inhibition prolonged the APD to a larger extent
than experimentally observed. The optimization algorithm reduced the amount of
APD prolongation induced by such inhibition and rendered percentages of APDchange well within physiological ranges. On the other hand, the optimized CRLP
model did not completely reproduce experimental evidences related to the effects
of IK1 inhibition on resting membrane potential. This is due to the fact thatthe modifications introduced into the model before the optimization procedure(i.e. incorporation of [K + ]i dynamics and ICaL current refinement) led to resting
potentials further away from experimental data. While the optimization somewhat
improved that behavior, this was not enough to provide resting membrane potential
values lying within experimental limits. As a consequence, the ERP was extremely
large and not all stimulations were able to propagate an AP in a simulated tissue
fiber.
Following IKr inhibition, the optimized CRLP model led to APD prolongation
values of lower magnitude than experimentally reported. To reach an APD prolongation of the same extent as in the experiments, the optimization algorithm
employed to develop the new CRLP model was required to change some model
parameters to a remarkably large extent as compared with the trust region of thealgorithm. In the end, this effect was neglected by the algorithm. This can be

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considered as a limitation of how the algorithm was applied by using initially thesame trust region for all model conductances. More details on this limitation are
discussed below (section 6.4.2).
APD dependence on CL
One limitation of the initial CRLP model, inherited from the GPB model, was the
APD response to increasingly shorter CLs. Specifically, for increasingly shorter CLs
below 500 ms, the APD increased rather than decreasing. For the optimized CRLP
model, the APD decreased monotonically for increasingly shorter CLs, even if theslope of such a relationship was slightly lower than experimentally observed. APD
dependence on CL was additionally investigated following ionic current inhbitions.
The behavior of the optimized model was generally consistent with experimental
data (see Figure 6.3). Nevertheless, there is still some room for improvement,
particularly in regards of the relationship between APD and CL following IK1 inhibition.
Also, differences in the APD between endocardial and epicardial cells for different CLs were analyzed. The initial CRLP model proposed in this thesis was able
to adequately replicate the experimental APD results reported by Drouin et al. [8]for epicardial cells, but it presented a significant error for endocardial cells. For the
optimized model, epicardial cells presented APD values shorter than the original
CRLP model while endocardial cells had APD values closer to the mean value ofthe experiments. While for all tested human ventricular cell models, APD values
for endocardial cells were longer than for epicardial cells, Drouin et al. showed nostatistically differences in the APD values for both celly types, even being slightly
larger for epicardial cells. The optimized CRLP model rendered APD differences
between both cell types lower than the initial CRLP model and therefore closer toexperimental observations.
Intracellular sodium and calcium concentrations[N a+ ]i values present a large variation when measured in different experiments [9].
This results in a very wide physiological range for [N a+ ]i , with any concentration
below 39.3 mM being considered as physiologically plausible. In the optimization
algorithm applied to develop the new CRLP model, when a given electrophysiological marker presents large experimental variance, its weight in the objectivefunction is reduced. It is thus not surprising that [N a+ ]i values in the optimized
CRLP model are further away from mean experimentally reported value than theinitial CRLP model, but in any case of very similar magnitude than the values
reported for other human ventricular cell models of the literature.
Regarding [Ca2+ ]i values, calculated under the same conditions as in [10], theoptimized CRLP model presented systolic values lower than the first version of themodel and slightly out of physiological range. This can be attributed to the ICaL
current refinement proposed in chapter 5, which the optimization algorithm was
not completely able to compensate for. Nevertheless, similar results were obtainedwith other common models, like the ORd and GPB models. On the other hand, theUniversidad Zaragoza

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optimized CRLP model was the only model able to replicate experimental behavior regarding changes in [Ca2+ ]i characteristics with the frequency of stimulation.
Specifically, when compared with the experimental data by Coppini et al. [11],the new CRLP model showed physiological values for systolic and diastolic [Ca2+ ]imarkers as well as for time from [Ca2+ ]i peak to 90% recovery, with only timefrom [Ca2+ ]i peak to 90% recovery presenting a less marked changed with pacing
frequency than the mean experimental value.
APD adaptation to abrupt CL changes
The adaptation of ventricular repolarization duration to abrupt changes in CL
has been proposed as an arrhythmic risk marker [81]. The adaptation time wasconsidered in the development of the new CRLP model. The main limitation ofincluding this marker in the optimization algorithm is that there are not systematicstudies analyzing it. In [95], ten different combinations of CLs were studied but
none of these combinations was used more than once in the experiments. Onlyin [13], two experiments conducted under the same experimental conditions werereported that served to define a physiological range for the adaptation time in
the process of model development. Large differences were found between results
simulated with the initial CRLP model and the experimental range. Although the
optimized CRLP model rendered results still out of the experimental limits, theoptimization algorithm allowed reducing the difference between the model and themean experimental value by more than 27%. The difference between simulated
and maximal experimental value of the adapation time was reduced by as much as
40.3% with respect to the initial CRLP model.
6.4.2 Optimization algorithm
The following paragraphs contain a discussion on a number of aspects of the optimization algorithm used in the development of the CRLP model.
Initialization
In this work, we used an initial trust region with the same radius for each ionic conductance aimed to be updated by the optimization algorithm. After optimization,some currents were shown to contribute to AP changes to a lower extent than expected based on experimental evidences. In particular, the effect of IKr block on the
APD was remarkably lower than in available experimental data, and, while suchan effect was within physiological range, the dependence with pacing frequency
shown in the experiments was not reproduced by the model. Also, the increase in
IKs conductance required for this conductance to have notable effects on the AP
was far away from the range the algorithm considered. This could be explained by:i) all the radius are reduced when one electrophysiological marker is out of range;ii) the effect of variations in IKs conductance over the tested electrophysiologicalmarkers was very small, so the algorithm did not consider updating this variable.

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New Methodologies for the Development
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A different strategy could be used to initialize the trust region. One possibility
could be to initialize the trust region with a different radius for each variable tobe updated by the optimization algorithm. The radius for each variable could, forinstance, be made inversely proportional to the greatest sensitivity of the evaluatedelectrophysiological markers and directly proportional to the experimental varianceof the marker with the largest sensitivity. Based on this or other ideas, futurestudies should address how to specifically define radius for each variable.
Objective function
In the optimization algorithm used in chapter 3 for incorporation of [K + ]i dynamics into the original CRLP model, the objective function was just defined by the
integral of the total potassium current during a cardiac cycle. In this chapter,the objective function is defined as a sum of mulitple error functions reflecting
differences between simulated and experimental data in relation to a number ofelectrophysiological properties. The weight assigned to each of the errors corresponding to different properties is challenging and has a large impact in the results
obtained from the optimization procedure. In this chapter all errors were weighted
by the same amount. As a consequence, if a specific AP phase is represented
by several evaluated electrophysiological properties, the algorithm could be biased
to fit that phase. Future studies could address how to weight the different error
functions composing the objective function in such a way that the model renders
improved results according to the criteria of interest.
The use of electrophysiological markers as part of an objective function to be
optimized for cardiac AP model development has been explored in [123]. In thatstudy the authors defined the objective function to be zero when the distance
between the marker and the mean value is lower than the experimental standard
error of the mean and to increase linearly out of this region. Limitations of thisapproach include the fact that several combinations of parameters could result inthe same value of the objective function or that improvements in one marker thatis close to physiological range could lead to a major deterioration in the value ofother markers completely out of physiological range, as the error increases linearly.
For this reason, in this chapter a minimization function was defined that assigns anerror value different from zero when the marker is within physiological range andpenalizes much more the fact of being out of range by using a quadratic function.
6.4.3 Validation
The main goal of the validation performed in this chapter was to evaluate thatthe model did not show abnormal behavior for electrophysiological properties notconsidered during the optimization phase. Future studies could extend validation
to incorporate other properties or to standardize a benchmark for the evaluationof human ventricular models. This could help to improve our current understanding on the benefits and limitations of any proposed model and to improve model
selection according to the specific problem to be investigated.

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135

6.5 Conclusions
In this chapter, an optimization algorithm was proposed to update the CRLP model
developed in previous chapters. The optimized CRLP model improved replicationof electrophysiological properties measured experimentally. Simulated results of
those properties were either within physiologically plausible ranges or at least closer
to them than before the optimization. A comparison with other published humanventricular cell models was established.

JesúsCarroChapter 7

Conclusions and FutureWork
7.1 Conclusions
In this thesis different techniques to improve the development and validation ofcardiac electrophysiological models have been proposed and analyzed. Based on theincrease in computational power, new strategies to reduce the number of hypothesis
and/or assumptions when building an electrophysiological model were considered.
The major conclusions of the thesis are summarized in the following.
7.1.1 Identification of major issues in electrophysiological model development
and validation
The development and validation of a computational electrophysiological model is acomplex process associated with a number of difficulties. Two main issues are: i) the
limited amount of experimental data the model can be based; ii) the way in which
information is extracted from experiments and used to build the models. Regarding
the former, a reduced number of experimental samples and the possibly limited
information available from those experimental samples may hamper the evaluationof cell and tissue electrical activities for some range of parameters or under specificconditions. In the case of human samples, this limitation is even more accentuated,as there are really few available sets of data, and even more so from healthy humanhearts. Importantly, the experimental protocols applied over those data may notalways provide the information necessary for model development or validation.
Regarding the second limitation, the difficulty in obtaining valuable insights fromexperimental studies may be due to the interaction between different elements
underlying cell or tissue electrical responses. For instance, it may be challenging toinvestigate specific characteristics of a particular ionic current, as in some cases it

137

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New Methodologies for the Development
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may not be possible to completely isolate the effects of other currents. Or it may
happen that interactions between activation and/or inactivation gating processes
for a given ionic current hinders the evaluation of specific gating characteristics
necessary to theoretically formulate ionic current expressions in the models.
The above described issues make it complex to develop and validate a computational electrophysiological model that can be used to reproduce experimental
evidences at different scales. In this thesis, a number of techniques were proposed
to minimize the impact of the above described issues and, when possible, to integrate them in the process of development and validation of the model. Of note,throughout the thesis the importance of considering the interactions between modelcomponents and spatio-temporal scales has been pointed out when evaluating simulated data. This involves the possibility of considering processes covering from ion
channel characteristics and diffusion of ions in the cell up to the propagation of theelectric wave in excitable cardiac tissue. Since sometimes it may not be possible toconsider all model components at the same time due to computational restrictions,in this thesis the focus has been on showing when specific elements should necessarily be considered, when it is possible to simplify the model and when it may be
convenient to consider only part of the available information. In particular, thisthesis has highlighted the benefits of using in silico simulations of experimental
protocols to evaluate all those possibilities, as long as in silico simulations replicate
as closely as possible the conditions of the experiments against which the simulated
data is compared. This is further discussed in the next section.
7.1.2 New techniques for improved model development and validation
Throughout this thesis, the uncertainty associated with experimental data used
to build computational models has been analyzed and, when possible, integrated
into the process of model development. Such an uncertainty is in relation to thelarge variance commonly observed over different experiments, the relatively low
number of conducted experiments or the limited amount of cells/tissues used forcharacterization of the electrical properties of interest.
In some cases there is an additional intrinsic difficulty in obtaining experimental
measures of specific model parameters or in extracting such measures from therecorded data. This would, for instance, apply to the evaluation of some gating
characteristics of ionic currents. In this thesis, in silico simulation of experimental
protocols is proposed to get indirect estimates of such characteristics. By using
in silico simulation of experimental protocols, together with the application ofother techniques, a model providing an improved representation of experimental
observations was built. Even if this technique turns out to be very useful for
identification of model ionic gating parameters, it becomes necessary to validate
the model with the new ionic expression as a whole so as to assess possible effects
at higher scales.
Another important challenge in computational model development and validation regards the identification of ionic current conductances. When investigating
cardiac diseases or assessing the effects of drugs, accurate evaluation of the contri-

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7. Conclusions and Future Workbution of each ionic current to the AP is very relevant. If a model with a highly
unbalanced representation of ionic currents is used, the simulation of the effects ofa drug or a disease that block one or several currents may provide totally meaningless results. As discussed throughout this thesis, it is possible to accurately
estimate ionic current conductances by using techniques such as those proposed inthis study. In particular, formulation of an optimization problem that integrates anumber of relevant electrophysiological properties together with available experimental data can to a great extent serve to the purpose of identifying physiologicallyplausible conductance values that render the model suitable for investigation of cardiac diseases or drug effects. Since optimization solving is highly computationallydemanding, approaches based on response surface approximations were proposedin this thesis, which allowed model optimization in a reasonable computational
time. Also, such developed approaches are amenable to parallel computing, whichmeans the time required to solve the optimization problem can be reduced by using computational architectures specifically designed to parallel programs, such as
GPUs or high-throughput computer clusters.
Based on the above, one of the main conclusions of this thesis is the needfor considering as many model elements as possible in both the development andvalidation processes. In some cases, the experimental data measured to study the
behavior of a given ionic current are affected by other ionic currents, intracellular
calcium handling or others factors, which may preclude investigation of certain cellcharacteristics without accounting for those other influences. A good practice canthus consist in developing/adjusting the model, or a part of it, by considering as
many involved elements as possible, while trying to replicate as faithfully as possible
the conditions underlying the experiments that are aimed to be reproduced.
In some cases the conditions or protocols of those experiments highly differ from
one to another study, yet more in the case of data recorded at different laboratories.
Although this might be seen as a limitation for model development and validation,
the techniques proposed in this thesis have shown the capacity to integrate all
those data no matter how different the experimental protocols underlying themare. An important aspect in the description of an experimental protocol is thevalue of the ion concentrations in the bath solution. Sometimes this aspect has
been overlooked, but it can be relevant when accounting for data recorded undervarious conditions and a single in silico AP model is expected to reproduce different
available evidences.
7.2 Future work
Some future research lines identified during the development of this thesis are:
• Extend the proposed methodologies to populations of models representing
electrophysiological variability: In this thesis, an average computational AP
model was considered. This average model is representative of the mean experimentally observed behavior in human ventricular cells, but, by definition,
it does not aim to represent the well known cell-to-cell variability present inbiological data, in general, and electrophysiological data, in particiular. All

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140

the approaches proposed in this thesis for the development and validation of
a computational electrophysiological model could be extended to consider apopulation of models aimed at covering the variability reported for different
cells in experiemntal studies.
• Application of the proposed methodologies to other human ventricular electrophysiological models: Published human ventricular AP models present widely
varied formulations of ionic currents, calcium handling and others. Whereas
those models faithfully reproduce experimental observations regarding someelectrophysiological properties, they fail to represent other properties, as highlighted in this thesis. The optimization methodologies presented in this thesis,
and in particular those described in chapter 6, could be applied onto thoseother human ventricular AP models to improve their overall performance.
Also, if the definition of a specific model component wants to be adjusted forone of those models, the optimization methodology could be as well applied
to guarantee accurate representation of the corresponding experimental data
following model adjustment.
• Definition of a benchmark for validation of electrophysiological models: Basedon the developments presented in this thesis, a benchmark could be established to evaluate human ventricular AP models and identify those satisfying
the criteria of interest. This benchmark could incorporate tests from the ionicto the tissue level, involving different stimulation protocols and considering
different measurement conditions. This could be applied not only to average
AP models, like the TP06, GPB, ORd or CRLP models, but also to models
being part of populations generated based on those other AP models. For
each research question to be investigated, the use of a set of selected models
defined based on such a developed benchmark could be of major help.
• Improved optimization solving by adapting the trust region radius to each estimated variable: As indicated in chapter 6, the radius of the trust region would
benefit from a definition adapted to the individual characteristics of each variable. Without such an adapted definition, some model variables end up being
barely modified during the optimization solving due to the way the impact on
the considered electrophysiological markers is accounted for. Future strategies could be investigated that allowed including variable normalization in
the optimization to improve overall efficiency of the optimization algorithm.
• Further update of model components: In this thesis a response-surface-basedalgorithm was used for optimization of ionic current conductances. Also, improved definition of ICaL voltage-dependent inactivation gates was addressed.
The methodologies presented in this thesis could be extended to update othermodel components.
• Development of new stimulation pulse: In chapter 4, a comparison of a number of electrophysiological properties measured in simulated cells and tissues
was provided. As described in that chapter, some properties presented highly

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7. Conclusions and Future Work141

different values depending on the evaluated spatial scales. Since in some studies the computational cost of conducting tissue simulations may be a limiting
factor, definition of stimulation protocols that rendered simulated results
similar to those measured in tissue while in fact simulating single cells would
be of major advantage. Those stimulation protocols could, for instance, bebased on the stimulation that one cell receives from its neighbors in the tissue(electrotonic coupling). In addition, the influence of different ionic currentson the propagation of the electric wave should could be accounted for in thedefinition of those stimulation protocols.

JesúsCarroAppendix

Appendix A

Publications derived
from this Thesis
A.1 Publications in Journal Articles• J. Carro, J. F. Rodríguez, P. Laguna, E. Pueyo. A Human Ventricular
Cell Model for Investigation of Cardiac Arrhythmias under Hyperkalaemic
Conditions. Phil Trans A, n. 639, pp. 4205 - 4232 (Nov, 2011).
doi:10.1098/rsta.2011.0127• J. Carro, J. F. Rodríguez-Matas, V. Monasterio, and E. Pueyo. Limitations
in electrophysiological model development and validation caused by differences
between simulations and experimental protocols. Progress in Biophysics andMolecular Biology, v. 129, pp. 53-64, 2017.
doi: 10.1016/j.pbiomolbio.2016.11.006• J. Carro, J. F. Rodríguez, E. Pueyo. A response surface optimization approach to adjust ionic current conductances of cardiac electrophysiologicalmodels. Application to the study of potassium level changes. PLOS One(Acepted).
doi: 10.1371/journal.pone.0204411• J. Carro, J. F. Rodríguez, E. Pueyo. A Human Ventricular Electrophysiological Model with Improved Description of Experimental Behavior. (Inpreparation).
A.2 Publications in Conference Proceedings• J. Carro, J. F. Rodríguez Matas, P. Laguna, E. Pueyo. Analysis and Improvement of a Human Ventricular Cell Model for Investigation of Cardiac

145

New Methodologies for the Development
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146

Arrhythmias. XXXVII International Conference on Computers in Cardiology, Belfast (UK), pp. 817 - 820 (Sep, 2010).
• J. F. Rodríguez Matas, J. Carro, E. Pueyo, K. Burrage, B. Rodríguez.
Impact of Multiple Ionic Changes in Arrhythmic Risk Biomarkers in Human
Ventricular Electrophysiology. Biophysical Society 56th Annual Meeting, San
Diego (USA), v. 102, p. 543a (Ene, 2012).
doi:10.1016/j.bpj.2011.11.2965• J. Carro, J. F. Rodríguez Matas, E. Pueyo. In Silico Simulations of Experimental Protocols for Cardiac Modeling. XXXVI Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Chicago(USA), pp. 5695 - 5698 (Aug, 2014).
• J. Carro, J. F. Rodríguez Matas, and E. Pueyo, A Methodology to Improve
Human Ventricular Models for the Investigation of Cardiac Arrhythmias, in
61st Annual Meeting of the Biophysical Society, New Orleans, USA, 2017.
A.3 Other Conferences• J. Carro, J. F. Rodríguez Matas, and E. Pueyo, A Response Surface Optimization approach to adjust ionic current conductances of cardiac electrophysiological models, in Gordon Research Conference in Cardiac Arrhythmia
Mechanisms, Lucca, Italy, 2015.
• J. Carro, J. F. Rodríguez Matas, and E. Pueyo, Multiscale Methods for Definition of Ionic Variables in Electrophysiological Models, in European Medical
and Biological Engineering Conference (EMBEC) Tampere, Finland, 2017.

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Appendix B

Description of the FirstModel
B.1 Model Parameters
B.1.1

Physical Constants

Name
R
F rdy
T
F oRT
Cmem

Value
8314
96485
310
F rdy/(R · T )1.381 · 10−10

B.1.2

Units
J/(mol · K)
C/mol
K
mV −1
F

Enviromental Parameters

Name
Lengthcell
Radiuscell
Vcell
Vmyo
Vsr
Vsl
Vjunc
JCajunc,slJCasl,myo
JNajunc,sl
JNasl,myoValue
100 · 10−5
10.25 · 10−5
π · Radius2cell · Lengthcell
0.65 · Vcell
0.035 · Vcell
0.02 · Vcell
5.39 · 10−4 · Vcell
8.2413 · 10−13
3.7243 · 10−12
1.8313 · 10−14
1.6386 · 10−12

Units
dm
dm
L
L
L
L
L
L/ms
L/ms
L/ms
L/ms

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New Methodologies for the Development
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148
B.1.3 Fractional Currents
Name
Fjunc
Fsl
FjuncCaL
FslCaLValue
0.11
1 − Fjunc
0.9
1 − FjuncCaL

Units
[−]
[−]
[−]
[−]

B.1.4 Ion Concentrations
Name
Ki
Ko
Cli
Clo
Cao
M gi
N ao

Value
138
5.4
15
150
1.8
1
140

Units
mM
mM
mM
mM
mM
mM
mM

B.1.5 Sodium Transport
Name
GNa
GNa,bkI¯NaK
KmKoKmNaipValue
18.86
0.597 · 10−3
0.99
1.5
11

Units
mS/µF
mS/µF
pA/pF
mM
mM

B.1.6 Potassium Currents
Name
pNaK
GK1
GKp
GKr
GKsjunc
GKssl
Gtof ,EPI
Gtos ,EPI
Gtof ,ENDO
Gtos ,ENDO

Value
1.833 · 10−2
5.7153 · 10−1
2 · 10−3
3.5 · 10−2
3.5 · 10−3
3.5 · 10−3
1.144 · 10−1
1.56 · 10−2
1.404 · 10−3
3.7596 · 10−2

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Units
[−]
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF

B. Description of the First Model
B.1.7
Name
GClCa
GCl,bkKdClCaB.1.8

Chlorine currents
Value
0.054813
9 · 10−3
100 · 10−3

Units
mS/µF
mS/µF
mM

Calcium Transport

Name
pCa
pK
pNa
I¯ncx
KmCai
KmCao
KmNai
KmNao
ksat
nu
Kdact
I¯PMCA
KmpCa
GCa,bk

B.1.9

149

Value
1.9887 · 10−4
5.4675 · 10−8
3.0375 · 10−9
4.5
3.59 · 10−3
1.3
12.29
87.5
0.32
0.27
0.15 · 10−3
0.0673
0.5 · 10−3
5.513 · 10−4

Units
L/(F · ms)
L/(F · ms)
L/(F · ms)
pA/pF
mM
mM
mM
mM
[−]
[−]
mM
pA/pF
mM
mS/µF

SR Calcium Fluxes

Name
VmaxSRCaP
ec50SR
hillSRCaP
kiCa
kim
koCa
kom
ks
Kmf
Kmr
M axSR
M inSRValue
5.3114 · 10−3
0.45
1.787
0.5
0.005
10
0.06
25
0.246 · 10−3
1.7
15
1

Units
mM /msmM[−]
mM −1 · mS −1
mS −1−2
mM · ms−1
mS −1
mS −1
mM
mM
[−]
[−]

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150
B.1.10 Buffering
Name
BmaxCaM
BmaxSR
BmaxT nChigh
BmaxT nClow
Bmaxmyosin
BmaxN aj
BmaxN asl
BmaxSLlow
sl
BmaxSLlowj

BmaxSLhighsl
BmaxSLhighj

Bmaxcsqn
kof fcsqn
kof fCaM
kof fSR
kof fT nChCa
kof fT nCh

Mg

Value
24 · 10−3
17.1 · 10−3
140 · 10−3
70 · 10−3
140 · 10−3
7.561
1.65
37.4 · 10−3 · Vmyo /Vsl
4.6 · 10−4 · Vmyo /Vjunc

Units
mM
mM
mM
mM
mM
mM
mM
mM
mM

13.4 · 10−3 · Vmyo /Vsl
1.65 · 10−4 · Vmyo /Vjunc

mM
mM

140 · 10−3 · Vmyo /Vsr
65
238 · 10−3
60 · 10−3
0.032 · 10−3
3.33 · 10−3

mM
mS −1
mS −1
mS −1
mS −1
mS −1

kof fT nCl
kof fmyoCa
kof fmyoM g
kof fslh
kof fsll
kof fN akoncsqn
konCaM
konSR
konT nCh
Ca
konT nCh

19.6 · 10−3
0.46 · 10−3
0.057 · 10−3
30 · 10−3
1.3
1 · 10−3
100
34
100
2.37
3 · 10−3

mS −1
mS −1
mS −1
mS −1
mS −1
mS −1
−1
mM · mS −1
mM −1 · mS −1
mM −1 · mS −1
mM −1 · mS −1
mM −1 · mS −1

konT nCl
konmyoCa
konmyoM gkonslhkonsllkonN a32.7
13.8
0.0157100
100
0.1 · 10−3

mM −1 · mS −1
mM −1 · mS −1
mM −1 · mS −1
mM −1 · mS −1
mM −1 · mS −1
mM −1 · mS −1

Mg

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B. Description of the First Model
B.2 Model Equations

B.2.1

Nerst Potentials

1
N ao
· ln
F oRT
N aj
N ao
1
· ln
ENasl =
F oRT
N asl
1
Ko + pNaK · N ao
EKs =
· ln
F oRT
Ki + pNaK · N ai
1
Ko
EK =
· ln
F oRT
Ki
1
Cao
ECajunc =· ln
2 · F oRT
Caj
Cao
1
· ln
ECasl =
2 · F oRT
Casl
1
Cli
ECl =
· ln
F oRT
Clo

ENajunc =B.2.2

INa :Fast Sodium Current

1

mss = (
1+e−(56.86+V )
9.03

)2
2

V −4.8232

τm =0.1292 · e−( 15.54 ) + 0.06487 · e−( 51.12 )dm mss − m=dtτm
V +45.79

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152
{

0
−(V +80)
0.057 · e 6.8

ah ={if V ≥ −40
otherwise
if V ≥ −40

5.9231bh =1+e−(V +10.66)
11.1

2.7 · e+ 3.1 · 10 · e
1τh =ah + bh1
hss = (
)2
V +71.55
1 + e 7.43
0.079·V

5

0.3485·V

otherwise

dh hss − h
=
dt
τh

{
aj =
{
bj =

0
(−2.5428·104 ·e0.2444·V −6.948·10−6 ·e−0.04391·V )·(V +37.78)
1+e0.311·(V +79.23)
0.057·V0.6·e1+e−0.1·(V +32)
0.02424·e−0.01052·V
1+e−0.1378·(V +40.14)

if V ≥ −40
otherwise1
τj =aj + bj1

jss = (
1+e

V +71.55
7.43)2

dj jss − j
=
dt
τj

(
)
INajunc =Fjunc · GNa · m3 · h · j · V − ENajunc
INasl =Fsl · GNa · m3 · h · j · (V − ENasl )INa =INajunc + INasl

B.2.3 INa,bk : Background Sodium Current()
INa,bkjunc =Fjunc · GNa,bk · V − ENajunc
INa,bksl =Fsl · GNa,bk · (V − ENasl )INa,bk =INa,bkjunc + INa,bksl

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if V ≥ −40
otherwise

153

B. Description of the First Model
B.2.4

INaK : Na-K Pump Current
N aoσ=
fNaK =

e 67.3 − 1
71
1 + 0.1245 · e−0.1·V ·F oRT + 0.0365 · σ · e−V ·F oRT

INaK junc = (Fjunc · I¯NaK · fNaK · Ko(
)4 )KmNaip1+· (Ko + KmKo )N aj

INaK sl = (Fsl · I¯NaK · fNaK · Ko
)4 )Km
1 + N aNaip· (Ko + KmKo )sl(

INaK =INaK junc + INaK sl
B.2.5

IKr : Rapidly Activating Potassium Current
xrss =

1
1+e−(V +10)5τxr = (
1+e

230
3300
) (
)+
V +40
V +11
1 + e 20
· 1+e 9−22−V
9dxKr xrss − xKr
=
dt
τxr
1
rKr =
V +74
1 + e 24√
Ko
IKr =GKr ·· xKr · rKr · (V − EK )
5.4

B.2.6

IKs : Slowly Activating Potassium Current
xsss =
τxs =1−(V +3.8)
14.25

1+e
990.1

−(V +2.436)

1 + e 14.12
dxKs xsss − xKs=dtτxs
IKsjunc = Fjunc · GKsjunc · x2Ks · (V − EKs )IKssl =Fsl · GKssl · x2Ks · (V − EKs )IKs =IKsjunc + IKssl

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154

B.2.7 IKp : Plateau Potassium Current

kpKp =

1V
1 + e7.488− 5.98

IKp =GKp · kpKp · (V − EK )B.2.8 Ito : Transient Outward Potassium Current

xtoss =
τxtos =11+e
9−(V −19)
13

V +3 + 0.5
1 + e 15
dxtos xtoss − xtos
=
dt
τxtos
1
ytoss =
V +19.5
1+e 5800
τytos =
V +60 + 30
1 + e 10dytos ytoss − ytos
=
dt
τytos

2

V +45
τxtof =8.5 · e−( 50 ) + 0.5
dxtof xtoss − xtof
=
dt
τxtof
−(V +40)2

τytof =85 · e 220
dytof ytoss − ytof
=
dt
τytof

+7

Itos =Gtos · xtos · ytos · (V − EK )Itof =Gtof · xtof · ytof · (V − EK )Ito =Itos + Itof

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B. Description of the First Model
B.2.9

IK1 :Inward Rectifier Potassium Current
aK1 =4.0938
1 + e0.12165·(V −EK −49.9344)

15.7197 · e0.06739·(V −EK −3.2571) + e0.06175·(V −EK −594.31)
1 + e−0.16285·(V −EK +14.2067)
aK1
K1ss =
aK1 + bK1√
Ko
· K1ss · (V − EK )IK1 =GK1 ·
5.4
bK1 =

B.2.10

IClCa : Calcium-Activated Chlorine Current
IClCajunc =Fjunc · GClCa· (V − ECl )ClCa
1 + KdCaj

IClCasl =Fsl · GClCa· (V − ECl )ClCa
1 + KdCasl

IClCa =IClCajunc + IClCasl
B.2.11

ICl,bk : Background Chlorine Current
ICl,bk =GCl,bk · (V − ECl )B.2.12

ICaL : L-type Calcium Current

Voltage-Dependent Activation Gate
dss =
αd =
βd =
γd =

1
−(V +5)

1+e 6
1.4

−35−V

+ 0.25

1 + e 13
1.4
V +5

1+e 5
1

50−V

1 + e 20
τd =αd · βd + γd
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Voltage-Dependent Inactivation Gates

fss =

1
1+e

V +20
7
2

αf =1102.5 · e−( 15 )
200
βf =
13−V
1 + e 10180γf =V +30 + 20
1 + e 10
τf =αf + βf + γf
df fss − f
=
dt
τf
V +27

f2ss =

0.67
1+e

V +35
7

+ 0.33−(V +25)2

αf2 =300 · e 170
31
βf2 =
25−V
1 + e 10
16
γf2 =
V +30
1 + e 10
τf2 =αf2 + βf2 + γf2
df2 f2ss − f2
=
dt
τf2

Calcium-Dependent Inactivation Gates()
dfCaBj=1.7 · Caj · 1 − fCaBj − 11.9 · 10−3 · fCaBj
dt
dfCaBsl=1.7 · Casl · (1 − fCaBsl ) − 11.9 · 10−3 · fCaBsl
dt

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157

B. Description of the First Model
Currents()
pCa · V · F rdy · F oRT · Caj · e2·V ·F oRT − CaoI¯Caj =
e2·V ·F oRT − 1()
2·V ·F oRTp−
Ca
Ca · V · F rdy · F oRT · Casl · eoI¯Casl =e2·V ·F oRT − 1()
V ·F oRTp−Na
Na · V · F rdy · F oRT · N aj · eoI¯Naj =
eV ·F oRT − 1()
pNa · V · F rdy · F oRT · N asl · eV ·F oRT − N ao¯
INasl =
eV ·F oRT − 1()
pK · V · F rdy · F oRT · Ki · eV ·F oRT − Ko¯
IK =eV ·F oRT − 1()
ICajunc =FjuncCaL · I¯Caj · d · f · f2 · 1 − fCaBj
ICasl =FslCaL · I¯Casl · d · f · f2 · (1 − fCaBsl )()
ICaNajunc =FjuncCaL · I¯Naj · d · f · f2 · 1 − fCaBj
ICaNasl =FslCaL · I¯Nasl · d · f · f2 · (1 − fCaBsl )()()
ICaK =I¯K · d · f · f2 · FjuncCaL · 1 − fCaBj + FslCaL · (1 − fCaBsl )ICa =ICajunc + ICasl
ICaNa =ICaNajunc + ICaNasl
ICaL =ICa + ICaK + ICaNa

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B.2.13

Incx : Na-Ca Exchanger Current(

Kajunc =1+(

Kasl =
1+

1
Kdact
Caj

1
Kdact
Casl

)2
)2

s1junc =enu·V ·F oRT · N a3j · Caos1sl =enu·V ·F oRT · N a3sl · Caos2junc =e(nu−1)·V ·F oRT · N a3o · Caj
s2sl =e(nu−1)·V ·F oRT · N a3o · Casl
(
(
)3 )
(
)
N
a
Caj
j
3
3
s3junc =KmCai · N ao · 1 ++ KmNao · Caj · 1 +
KmNai
KmCai
+ KmCao · N a3j + N a3j · Cao + N ao3 · Caj
(
(
)3 )
)
(
Casl
N asl
3
+ Km3Nao · Casl · 1 +s3sl =KmCai · N ao · 1 +
KmNai
KmCai
+ KmCao · N a3sl + N a3sl · Cao + N a3o · Casl

Fjunc · I¯ncx · Kajunc · (s1junc − s2junc )()
s3junc · 1 + ksat · e(nu−1)·V ·F oRT
Fsl · I¯ncx · Kasl · (s1sl − s2sl )()
Incxsl =s3sl · 1 + ksat · e(nu−1)·V ·F oRTIncxjunc =Incx =Incxjunc + Incxsl

B.2.14

IpCa : Sarcolemmal Calcium Pump Current

IpCajunc =IpCasl =Fjunc · I¯PMCA · Ca1.6
j
1.6
Km1.6
pCa + Caj
Fsl · I¯PMCA · Ca1.6
sl

1.6
Km1.6
pCa + Casl

IpCa =IpCajunc + IpCasl

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159

B. Description of the First Model
B.2.15ICa,bk : Background Calcium Current()
ICa,bkjunc =Fjunc · GCa,bk · V − ECajunc
ICa,bksl =Fsl · GCa,bk · (V − ECasl )ICa,bk =ICa,bkjunc + ICa,bksl

B.2.16 SR Fluxes: Calcium Release, SR Calcium Pump, SR Calcium LeakkCaSR =M axSR −koSRCa =M axSR − M inSR
(
)2.5
SR
1 + ec50
CaSR

koCa
kCaSR

kiSRCa =kiCa · kCaSR
RI =1 − RyRr − RyRo − RyRi
(
)
dRyRr
=kim · RI − kiSRCa · Caj · RyRr − koSRCa · Ca2j · RyRr − kom · RyRodt
dRyRo=koSRCa · Ca2j · RyRr − kom · RyRo − (kiSRCa · Caj · RyRo − kim · RyRi )
dt
)
(
dRyRi
=kiSRCa · Caj · RyRo − kim · RyRi − kom · RyRi − koSRCa · Ca2j · RI
dt
JSRCarel =ks · RyRo · (CaSR − Caj )(()hillSRCaP ))hillSRCaP (CaSR
CaiVmaxSRCaP ·
−
Kmf
Kmr
Jser Ca =)hillSRCaP()hillSRCaP (
CaSR
Cai
+
1 + KmKmrf
JSRleak =5.348 · 10−6 · (CaSR − Caj )B.2.17 Ion Homeostasis
Sodium Buffers()
dN aBj=konN a · N aj · BmaxN aj − N aBj − kof fN a · N aBj
dt()
dN aBsl=konN a · N asl · BmaxN asl − N aBsl − kof fN a · N aBsl
dt

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160
Cytosolic Calcium Buffers()
dT nCl=konT nCl · Cai · BmaxT nClow − T nCl − kof fT nCl · T nCl
dt()
dT nChc=konT nCh · Cai · BmaxT nChigh − T nChc − T nChm
Cadt
− kof fT nCh · T nChc
Ca()
dT nChm=konT nCh · M gi · BmaxT nChigh − T nChc − T nChm
Mg
dt
− kof fT nCh · T nChm
Mg

dCaM
=konCaM · Cai · (BmaxCaM − CaM ) − kof fCaM · CaM
dt()
dM yoc=konmyoCa · Cai · Bmaxmyosin − M yoc − M yom
dt− kof fmyoCa · M yoc()
dM yom=konmyoM g · M gi · Bmaxmyosin − M yoc − M yom
dt− kof fmyoM g · M yom
dSRB=konSR · Cai · (BmaxSR − SRB) − kof fSR · SRB
dt
dT nChc
dT nChm
dCaM
dT nCl+++JCaBcytosol =
dt
dt
dt
dt
dM yomdSRB
dM yoc
+
+
+
dt
dt
dt

Junctional and SL Calcium Buffers
(
)
dSLLj
=konsll · Caj · BmaxSLlow − SLLj − kof fsll · SLLjjdt()
dSLLsl=konsll · Casl · BmaxSLlow − SLLsl − kof fsll · SLLsl
sl
dt
(
)
dSLHj
=konslh · Caj · BmaxSLhigh − SLHj − kof fslh · SLHjjdt()
dSLHsl=konslh · Casl · BmaxSLhigh − SLHsl − kof fslh · SLHsl
sl
dt
dSLHj
dSLLj
+
JCaBjunction =dt
dt
dSLLsldSLHslJCaBsk =+dt
dt

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161

B. Description of the First Model
Sodium Concentrations
JNajunc,sldN aj −INatotjunc · CmemdN aBj=+· (N asl − N aj ) −dt
Vjunc · F rdy
Vjunc
dt
−I
·
C
JNajunc,sldN asl
Natot
mem
sl
=
+
· (N aj − N asl )dt
Vsl · F rdy
Vsl
JNasl,myodN aBsl+· (N ai − N asl ) −Vsl
dt
dN ai JNasl,myo=· (N asl − N ai )dt
Vmyo
Calcium Concentrations()
dCsqnb=koncsqn · CaSR · Bmaxcsqn − Csqnb − kof fcsqn · Csqnb
dt
JCajunc,sldCaj −ICatotjunc · Cmem=+
· (Casl − Caj ) − JCaBjunction
dt
Vjunc · 2 · F rdy
Vjunc
JSRCarel · Vsr
JSRleak · Vmyo
+
+
Vjunc
Vjunc
·
C
−I
J
dCasl
mem
Catot
Cajunc,slsl=+· (Caj − Casl )dt
Vsl · 2 · F rdy
Vsl
JCasl,myo+· (Cai − Casl ) − JCaBsk
Vsl
JCasl,myodCai −Jser Ca · Vsr=− JCaBcytosol +· (Casl − Cai )
dt
Vmyo
Vmyo
(
)
dCaSR
JSRleak · VmyodCsqnb=JserCa −+ JSRCarel −
dt
Vsr
dt

B.2.18 Membrane Potential
ICatotjunc =ICajunc + ICa,bkjunc + IpCajunc − 2 · Incxjunc
ICatotsl =ICasl + ICa,bksl + IpCasl − 2 · Incxsl
ICatot =ICatotjunc + ICatotsl

IK tot =Ito + IKr + IKs + IK1 − 2 · INaK + ICaK + IKp

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INatotjunc = INajunc + INa,bkjunc + 3 · Incxjunc + 3 · INaK junc + ICaNajunc
INatotsl =INasl + INa,bksl + 3 · Incxsl + 3 · INaK sl + ICaNasl
INatot =INatotjunc + INatotsl

ICltot =IClCa + ICl,bk

Itot =INatot + ICltot + ICatot + IK tot
dV= − (Itot − Istim )dt

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Appendix C

Description of the Final
Model
This appendix includes only the parameters or equations that were modified withrespect to the version described in Appendix B.
C.1 Model Parameters
C.1.1

Enviromental Parameters

Name
JCajunc,slJCasl,myo
C.1.2

Value
7.861586 · 10−13
3.734497 · 10−12

Units
L/ms
L/ms

Fractional Currents

Name
Fjunc
Fsl
FjuncCaL
FslCaL
Fto,s
Fto,f

Value
0.11
1 − Fjunc
0.9
1 − FjuncCaL
0.12
1 − Fto,f

C.1.3

Sodium Transport

Name
GNa
GNa,bkI¯NaK

Value
18.90240.5181448 · 10−3
1.140666

Units
[−]
[−]
[−]
[−]
[−]
[−]

Units
mS/µF
mS/µF
pA/pF

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New Methodologies for the Development
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164
C.1.4 Potassium Currents
Name
GK1
GKp
GKr
GKs
GtoEPI
GtoENDOValue
4.253697 · 10−1
2.173516 · 10−3
3.507879 · 10−20.0034
1.558487 · 10−1
4.674482 · 10−2

Units
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF
mS/µF

C.1.5 Chlorine currents
Name
GClCa
GCl,bkValue
4.757296 · 10−2
9.417983 · 10−3

Units
mS/µF
mS/µF

C.1.6 Calcium Transport
Name
GCaL
pCa
pK
pNa
I¯ncx
I¯PMCA
GCa,bkValue
4.7628 · 10−4
11.3746417 · 10−4
7.63689 · 10−6
2.747111
5.84106 · 10−2
4.820643 · 10−4

Units
mS/µF
L · Ω/sL · Ω/sL · Ω/s
pA/pF
pA/pF
mS/µF

C.1.7 SR Calcium Fluxes
Name
VmaxSRCaPks
kSR,leakValue
4.609837 · 10−3
24.75075.483405 · 10−6Units
mM /ms
mS −1
mS −1

C.2 Model Equations
C.2.1 Ito : Transient Outward Potassium Current
Itos =Gto · Fto,s · xtos · ytos · (V − EK )Itof =Gto · Fto,f · xtof · ytof · (V − EK )Ito =Itos + Itof
C.2.2 ICaL : L-type Calcium Current
Voltage-Dependent Activation Gate

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C. Description of the Final Model
Voltage-Dependent Inactivation Gates

fss =

1
1+e

V +24.63
5.87
2

αf =1102.5 · e−( 15 )200
βf =20.6−V
1 + e 10180γf =V +30 + 20
1 + e 10
τf =αf + βf + γf
df fss − f
=
dt
τf
V +27

f2ss =

0.67
1+e

V +24.63
5.87+ 0.33−(V +25)2

αf2 =300 · e 170
16
γf2 =
V +30
1 + e 10
τf2 =αf2 + γf2
df2 f2ss − f2=dtτf2
Currents()
Caj · e2·V ·F oRT − Cao
V · F rdy · F oRT¯·ICaj =GCaL · pCa ·2e2·V ·F oRT − 1()
Casl · e2·V ·F oRT − Cao
V · F rdy · F oRT¯ICasl =GCaL · pCa ··2e2·V ·F oRT − 1()
N aj · eV ·F oRT − N ao¯
INaj =GCaL · pNa · V · F rdy · F oRT ·eV ·F oRT − 1()
N asl · eV ·F oRT − N ao
I¯Nasl =GCaL · pNa · V · F rdy · F oRT ·eV ·F oRT − 1()
V ·F oRT
K
·
e
−
K
i
o
I¯K =GCaL · pK · V · F rdy · F oRT ·eV ·F oRT − 1

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(
)
ICajunc =FjuncCaL · I¯Caj · d · f · f2 · 1 − fCaBj
ICasl =FslCaL · I¯Casl · d · f · f2 · (1 − fCaBsl ))(ICaNajunc =FjuncCaL · I¯Naj · d · f · f2 · 1 − fCaBj
ICaNasl =FslCaL · I¯Nasl · d · f · f2 · (1 − fCaBsl ))(()
ICaK =I¯K · d · f · f2 · FjuncCaL · 1 − fCaBj + FslCaL · (1 − fCaBsl )ICa =ICajunc + ICasl
ICaNa =ICaNajunc + ICaNasl
ICaL =ICa + ICaK + ICaNa
C.2.3 SR Fluxes: Calcium Release, SR Calcium Pump, SR Calcium Leak
JSRleak =kSRleak · 10−6 · (CaSR − Caj )C.2.4 Ion Homeostasis
Potassium Concentrations
Cmem
dKi= − IK tot ·dt
Vmyo · F rdy

C.2.5 Membrane Potential
IK tot =Ito + IKr + IKs + IK1 − 2 · INaK + ICaK + IKp − Istim
Itot =INatot + ICltot + ICatot + IK tot
dV= − Itot
dt

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Appendix D

Acronyms• 1D: one-dimensional.
• AP: action potential.
• APA: action potential amplitude.
• APD: action potential duration.
• APDR: action potential duration restitution.
• AV : atrioventricular.
• BZ: border zone.
• CI : coupling interval.
• CL: cycle length.
• CRLP: Carro-Rodríguez-Laguna-Pueyo human ventricular cell model [120].
• CRN : Courtemanche-Ramirez-Nattel model [47].
• CV : conduction velocity.
• CVDs: cardiovascular diseases.
• CZ: central zone.
• DI : diastolic interval.
• ECC: excitation-contraction coupling.
• ECG: electrocardiogram.

167

New Methodologies for the Development
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168• ERP: effective refractive period.

• GPB: Grandi-Pascualini-Bers human ventricular model [15].
• HH : Hodgkin-Huxley model [26].
• IMW04: Iyer et al. human ventricular cell model [45].
• LR: Luo-Rudy model [107].
• NCD: noncommunicable disease.
• NZ: normal zone.
• ORd:O’Hara-Rudy-dynamic human ventricular cell model [64].
• RSA: response surface approximations.
• S1: basic stimulus.
• S2: extra stimulus.
• SA: sinoatrial.
• SEM : standard error of the mean.
• SR: sarcoplasmic reticulum.
• TNNP04: tenTusscher-Noble-Noble-Panfilov human ventricular cell model[46].
• TP06: tenTusscher-Panfilov human ventricular cell model [51].

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JesúsCarroThe development and validation of a computational electrophysiologicalmodel is a complex process associated with a number of difficulties. Twomain issues are: i) the limited amount of experimental data the model canbe based; ii) the way in which information is extracted from experiments
and used to build the models. Based on the increase in computational
power, new strategies to reduce the number of hypothesis and/or assumptions when building an electrophysiological model were considered. In
particular, the application of an optimization algorithm together with thesimulation of in silico simulation of experimental protocols, helped to finda model that better represents the experimental results.
This thesis highlighted the benefits of using in silico simulations of experimental protocols to evaluate all those possibilities, as long as in silico
simulations replicate as closely as possible the conditions of the experiments against which the simulated data is compared.