# A comparative study of lumped heart models for personalized medicine through sensitivity and identifiability analysis

Marie Haghebaert, Pavlos Varsos, Roel Meiburg, Irene Vignon-Clementel

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Marie Haghebaert, Pavlos Varsos, Roel Meiburg, Irene Vignon-Clementel. A comparative study of lumped heart models for personalized medicine through sensitivity and identifiability analysis. 2025. [hal-04844897v3](https://hal.science/hal-04844897v3)

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# A comparative study of lumped heart models for personalized medicine through sensitivity and identifiability analysis

Marie Haghebaert<fnref n="1" />, Pavlos Varsos<fnref n="1" />, Roel Meiburg<fnref n="1" />, and Irene Vignon-Clementel<fnref n="1" />,<fnref n="2" />

**Keywords:** Lumped parameter modelling, Identifiability, Sensitivity analysis, Cardiac modelling, Time-Varying Elastance Model, Single Fibre Model, Pressure Volume loop

**Running Title:** Comparison of lumped heart models: sensitivity & identifiability

## Abstract

Numerical modelling of the cardiovascular system is becoming an increasingly accepted tool for clinical applications, with the ultimate goal of personalized medicine. Lumped parameter models are attractive due to their low computational cost but often do not directly incorporate physical properties, thus requiring calibration to (often sparse) clinical data. Furthermore, there exists a trade-off between physiological relevance and model complexity, making the choice of cardiac model non-trivial. For two established cardiac chamber models embedded in a hemodynamics whole circulation model, we perform sensitivity and identifiability analyses to examine the possibility of finding a unique subset of important parameters with varying levels of clinically measurable data, thereby examining their applicability in personalized medicine. To give a concrete clinical context, the case of treatment planning for a young pulmonary arterial hypertension patient is considered. The methodology is however relevant for other pathophysiologies. The results suggest that the single-fibre model, although a priori more complex than the time-varying elastance model, is more amenable for patient-specific modelling. This was found for representing the patient state from clinical data, defining parameter ranges for sensitivity analysis and in the results of the identifiability analysis. Sensitivity analysis also revealed the most influential parameters, which for the right (resp. left) heart chambers, mostly affect the hemodynamics of these chambers and the pulmonary (resp. systemic) circulation but also the ones of the left (resp. right) side. This highlights the importance of studying the whole circulation, especially in diseases traditionally thought to affect only one side.

<footnote n="1">¹Inria, Research Centre Saclay Ile-de-France, France;</footnote>

<footnote n="2">²Corresponding Author: Irene.Vignon-Clementel@inria.fr</footnote>
# Comparing Two Cardiac Modelling Approaches for Personalized Medicine

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<table><thead><tr><th>Time Varying Elastance<br>Heart Model</th><th>Single Fiber<br>Heart Model</th></tr></thead><tbody><tr><td><ul><li>Less complex</li><li>Lower physiological interpretability</li><li>Challenging to calibrate</li><li>Good for exploratory simulations</li></ul></td><td><ul><li>More complex</li><li>Greater physiological interpretability</li><li>Easier to calibrate</li><li>Good for personalized simulations</li></ul></td></tr></tbody></table>

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Comparison based on sensitivity and identifiability analysis

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Clinical example: Pulmonary Arterial Hypertension

The Journal of
**Physiology**

**Graphical abstract:** Comparative graphical summary of two cardiac chamber models applied in patient-specific cardiovascular simulations. This scheme contrasts the Time-Varying Elastance model and the Single Fibre model integrated within a whole-body circulatory framework. The comparison highlights differences in model complexity, physiological interpretability, and suitability for clinical application. The evaluation is based on sensitivity and identifiability analyses and is applied in the context of Pulmonary Arterial Hypertension as a clinical case study. The TVE model, being less complex, is suitable for exploratory simulations but more challenging to calibrate. In contrast, the SF model offers greater physiological fidelity and is more suited for personalized simulations.
## Key points

* Two established cardiac chamber computational models are compared in the context of pulmonary hypertension but the methodology holds for other pathophysiologies.

* Comprehensive patient-specific data, accurate parameter ranges, and thorough model validation are essential to enhance parameter identifiability, improve model personalization, and eventually lead to better prediction of haemodynamic changes after clinical intervention.

* Due to its inherent mathematical constraints and physiologically interpretable input parameters, the single-fibre model is easier to make patient-specific and more accurately captures the nonlinear dynamics of ventricular pressure-volume loops than the simpler time-varying elastance model, making it more suitable for personalized cardiac simulations.

* Accurate representation of both the left and right heart in cardiac modelling is important, as our results suggest that parameters of each side impact the hemodynamics of the other circulatory side, making it worthwhile to measure characteristics of both right and left chambers, even if the disease emerges from one side.

# 1. Introduction

Cardiac computational modelling of the heart is gaining acceptance in clinical practice. It can serve as a research tool to deepen the understanding of cardiac mechanics (Niederer, Lumens, and Trayanova 2019; Lopez-Perez, Sebastian, and Ferrero 2015; Hunter and Nash 2000; Regazzoni, Dedè, and Quarteroni 2020), as an educational resource (Heldt et al. 2010; Dassen et al. 2011; David Roy Warriner et al. 2017), and is evolving to the point where it can support clinical decision-making. Notably, the first cardiac hemodynamic computational model to receive CE marking as a medical device has now become commercially available in many countries (HeartFlow Inc, US).

Modelling cardiac function is extremely complex due to its multi-physics nature. A comprehensive cardiac model would incorporate the electrophysiological behaviour of ion-channels, sub-cellular activation mechanisms, tissue deformation, valve mechanics and fluid mechanics inside the heart cavities and in the myocardium. There are models that focus solely on one component (Trayanova 2011; Franzone, Pavarino, and Scacchi 2014; Papamanolis et al. 2021; Montino Pelagi et al. 2024) while others study the interactions of typically two components (e.g., electro-mechanics, fluid-solid interaction) (Feng, Gao, and Luo 2024; Hirschhorn et al. 2020; Astorino et al. 2009; Chapelle et al. 2010). One of the first three-dimensional (3D) models that successfully integrated these three different fields (electrophysiology, tissue and simplified valve mechanics, as well as fluid mechanics) was developed by Quarteroni et al. (2017). Over the past few years, various studies have been published on 3D multi-physics integrated cardiac models, towards a direction of personalization and clinical adaptation (Santiago et al. 2018; Peirlinck et al. 2021; Levine et al. 2022; Corrado, Gerbeau, and Moireau 2015; Zingaro et al. 2024). Currently, the primary drawback of these models is the significant computing power and time required to run them (although Machine Learning techniques could potentially alleviate this issue (Fresca et al. 2020;)
Regazzoni et al. 2022)), rendering them impractical for scenarios where rapid results are needed for clinical decision-making.

To address these limitations, reduced-order models, such as zero-dimensional (0D) models, have gained prominence. These models describe system dynamics with simplified equations that average out spatial variations, offering a computationally efficient alternative. For example, 0D cardiac chamber models relate pressures to volumes within the cardiac cavities, providing detailed insights into cardiac function with reasonable accuracy and reduced computational cost. When combined with lumped parameter models (LPMs), which represent the broader circulatory system, these models enable the simulation of hemodynamics throughout the entire circulatory system. This approach strikes a balance between accuracy, interpretability, and computational efficiency—a capability that remains challenging for detailed 3D models. LPMs employ the hydraulic-electric analogy, where flow rate (Q) is analogous to electrical current (I) and blood pressure (P) corresponds to voltage (V) (Formaggia, Quarteroni, and Veneziani 2010).

In the context of 0D heart modelling, its applications can be categorized mainly in two types. The first type pertains to derive relevant cardiac properties or unmeasurable/hard to measure properties (e.g. stroke work, contractility) (Meiburg et al. 2021), cluster patient traits (Bjørdalsbakke et al. 2023), or characterize certain pathologies (Pant et al. 2016). The second type consists in modelling the effect of different loads on the body (e.g., effect of pregnancy, tilt tests) (Corsini et al. 2017; Comunale, Susin, and Mynard 2022; Fois et al. 2022), effect of surgery (Golse et al. 2021; Pant et al. 2022; Esmaily Moghadam et al. 2012), and peri-operative monitoring (Le Gall et al. 2020).

The most often adopted heart chamber model is the time varying elastance (TVE) model, introduced by Suga, Sagawa, and Shoukas (1973). This model describes the instantaneous pressure-volume relationship of a cardiac chamber, which is seemingly independent of pre- and afterload. The pressure-volume (PV) loop, a key concept for cardiologists, is central to understanding cardiac function and is effectively represented by this model. It captures the dynamic interplay between electrophysiology, tissue mechanics, and cavity fluid dynamics, offering a comprehensive view of cardiac performance. The model is elegant and easy to implement, and has therefore already been employed in many different applications, such as investigating the factors influencing ventricular performance in heart failure (Faes and Kerkhof 2015; Liang and Liu 2005), predicting the risk of postoperative portal hypertension (Golse et al. 2021), and modelling the Norwood circulation (Migliavacca et al. 2001). However, its simplicity is also its major drawback, as its parameters have limited physiological interpretation.

To combat limitations of the TVE model, the single-fibre (SF) model was developed by T. Arts et al. (1991), under the assumption of a spherical or cylindrical chamber to relate chamber pressure and volume to myocardium fibre stress and strain, thereby incorporating physiological features and increasing interpretability of the model parameters. Originally developed for the left ventricle, it has since been employed in various applications, such as studying the mechanical mechanisms and treatment of right heart failure in pulmonary arterial hypertension (Lumens and Delhaas 2012), exploring the cardiovascular transition from fetal to neonatal life (Munneke, Lumens, and Delhaas 2022) within the CircAdapt framework (Theo Arts et al. 2005), and investigating valve regurgitation in single-ventricle patients (Pant et al. 2018).
Currently, research in computational modelling of the cardiovascular system is transitioning from 'generic' models (which describe an average individual within a population) to patient-specific models, which describe the hemodynamics of a specific patient. Generating a patient-specific model requires solving a so-called inverse problem, i.e., tuning model input parameters in such a way that the model best represents patient data. Solving these inverse problems is not trivial, as they depend on amount, type and quality of data available, the model formulation and of course the research question. While both the TVE and SF models have existed for decades and have been the basis for many applications in cardiovascular research, very little has been done in the way of formal assessment of finding a unique subset of important parameters with varying amounts of data that are clinically measurable. This is crucial for reliable patient-specific modelling and, consequently, for the clinical applicability of the models, particularly in terms of robustness and trustworthiness. To address this challenge, it is essential to analyse the sensitivity and identifiability of the model's parameters. These evaluations help determine how variations in parameters affect outcomes and assess the certainty of estimating non-measurable parameters with available data.

Sensitivity Analysis (SA) is a widely applied methodology that quantifies how variations or uncertainties in model inputs (parameters) affect model outputs. In this work, we apply SA to evaluate how TVE and SA parameters influence clinically relevant outputs. Various approaches can be employed to perform this analysis. One such method is the Sobol's method, a variance-based approach that decomposes the total model output variance to attribute portions of this variance to individual input parameters and their interactions (Sobol 2001; Saltelli et al. 2004). This method provides insights into each parameter's contribution and importance regarding model outputs of clinical interest. Such information enables the selection of a subset of key parameters—those most critical for accurate intervention predictions and, therefore, most important to personalize for individual patients.

Identifiability refers to the ability to uniquely estimate model parameters based on available data (Little, Heidenreich, and Li 2010; Raue et al. 2009; Vanlier et al. 2012). It assesses whether a given dataset contains enough information to reliably estimate these immeasurable parameters. In scenarios with variable data availability, identifiability analysis becomes a key tool for understanding the reliability of parameter estimation and the limits of the model's applicability.

This study presents, for the first time to our knowledge, a direct comparison between the TVE model and the SF model. This comparison is performed within a generic closed-loop 0D circulatory framework. To ground our investigation in a realistic clinical scenario, we focus on a pathophysiological case involving pediatric pulmonary arterial hypertension (PAH), where limited clinical data is the norm. Such constraints are common in pathologies and underscore the broader challenge of obtaining adequate parameter information for patient-specific models. Our comparison is conducted via an iterative pipeline that combines calibration, parameter-range refinement, sensitivity analysis, and identifiability assessment. First, we perform an initial calibration of each model to the available patient data. We then iteratively refined the parameter ranges so that sampling within these ranges continued to produce outputs encompassing the patient's measurements. With these final parameter ranges, we perform a sensitivity analysis to pinpoint those parameters with the greatest impact on clinically relevant outputs. Finally, we conduct an identifiability assessment to determine which of these sensitive parameters can be reliably estimated under varying levels of clinical data availability. This systematic approach guides the selection of the most appropriate model and parameter set for a given patient dataset. This pipeline thus provides the basis to establish a robust model-based medical data collection protocol. While our focus here is on
pediatric PAH, the same pipeline can be readily applied to other pathophysiologies. A schematic overview of the entire workflow is provided in Fig. 1, illustrating the sequence of steps from calibration to model selection.

<figure><img src="image_9.png" /><figcaption>**Figure 1:** Schematic overview of the methodology for comparing the SF and TVE cardiac models within a generic closed-loop LPM. The process begins with an inverse-problem calibration to establish physiologically plausible parameter ranges. Next, a sensitivity analysis (SA) identifies which parameters most strongly influence model outputs, and an identifiability assessment determines which parameters can be reliably estimated from the available clinical data. Finally, the results guide model selection—offering recommendations on whether SF or TVE is more suitable, given different levels of clinical data availability.</figcaption></figure>
## 2. Models and Methods

### 2.1. Heart chambers models: Pressure-Volume (PV) relationships

Heart chamber mathematical models represent the dynamic behaviour of the heart over one or several cardiac cycles, predicting how the heart responds to various conditions, such as changes in preload, afterload, or contractility. These models simulate the cyclic processes of contraction and relaxation, capturing the essential pressure-volume (PV) relationships that describe cardiac function.

The function of the heart is driven by a complex sequence of electrical and mechanical events, all of which are captured by pressure-volume models. The process begins with the electrical activation of cardiac cells, initiating a depolarization wave across the myocardium. This wave triggers contraction of myocardial fibres, reducing the volume of the heart chambers. At the cellular level, this contraction is driven by the sarcomere-the contractile unit of muscle tissue-where actin and myosin filaments generate the force required to reduce the volume of the heart chambers.

During systole, the myocardium contracts while the heart valves remain closed, causing the pressure within the chamber to rise. Once the outflow valve opens, blood is ejected and the chamber volume decreases. This rise in chamber pressure defines the systolic phase of the PV-loop, a graphical representation of the relationship between pressure and volume in a cardiac chamber over a single cardiac cycle (Fig. 2 right panel). Importantly, PV loops incorporate not only the heart's intrinsic contractile properties but also the circulatory system's influence. Preload (the initial stretching of cardiac fibres) and afterload (the resistance the heart must overcome) significantly affect the loop's shape and features.

Clinically, PV loops are a key tool for understanding cardiac function. They capture parameters such as end-systolic and end-diastolic volumes, stroke volume, ejection fraction, and indices of myocardial contractility, enabling the assessment of systolic and diastolic performance as well as valvular function. Moreover, they provide insights into pathological states—such as systolic or diastolic dysfunction—and the outcomes of various therapeutic interventions. It is precisely these PV loop principles that underpin the TVE model, discussed in detail later.

Following systole, the sarcomeres relax, and the myocardium enters diastole, allowing the chambers to refill with blood. During this phase, chamber pressure decreases as volume increases, completing the cycle in preparation for the next contraction. While the atrial cycle is more complex due to the conduit phase, where ventricular filling occurs and the atria serve as a conduit without active contraction, the same general principles of active and passive elastance still hold true.

Following systole, the sarcomeres relax, and the myocardium enters diastole, allowing the chambers to refill with blood. During this phase, chamber pressure decreases as volume increases, completing the cycle in preparation for the next contraction.
### 2.1.1. Time varying elastance (TVE) model

TVE model, introduced by Suga, Sagawa, and Shoukas (1973), is a foundational concept for understanding the dynamic relationship between cardiac chamber pressure, volume, namely the time varying elastance of the chamber. This model postulates that the elastance of a cardiac chamber varies over time in a manner that is consistent across different physiological conditions. Specifically, the elastance curves maintain a consistent shape and reach their peak towards the end of the ejection phase (see Fig. 2 upper left panel), regardless of variations in mechanical load, fibre contractile state, or heart rate. This implies that the general form of the elastance function remains unchanged across different physiological scenarios.

At any given time $t \ge 0$, the TVE model describes the relationship between the pressure within the cardiac chamber $P(t)$, the volume of the chamber $V(t)$, and the time varying elastance $E(t)$ as follows:

$$P(t) = E(t)(V(t) - V_0), \qquad (1)$$

where $V_0$ is the unstressed volume of the cardiac chamber, representing the theoretical volume at which the chamber would exert zero pressure and $E(t)$ is expressed as:

$$E(t) = E_b + E_a \times e(t), \qquad (2)$$

where $E_b$ is the baseline elastance, representing the minimal elastance during diastole, and $E_a$ is the amplitude of elastance change, representing the increase in elastance during systole. The activation function $e(t)$ governs the time-dependent change in elastance and varies depending on the specific cardiac chamber.

The values of $E_a$, $E_b$, and $V_0$ are linked to PV loops (see Fig. 2 right panel). Specifically, $E_b$ is often derived from the end-diastolic pressure-volume relationship (EDPVR), and $E_{max}$ is obtained from the end-systolic pressure volume relationship (ESPVR). The amplitude $E_a$ is then calculated as the difference between $E_{max}$ and $E_b$, i.e., $E_a = E_{max} - E_b$. The EDPVR and ESPVR represent the intrinsic mechanical properties of the heart during diastole and systole, respectively. These relationships can be approximated with different mathematical forms, such as linear or exponential functions (Burkhoff, Mirsky, and Suga 2005; Claessens et al. 2006). For simplicity, a linear approximation is often applied, assuming a straight-line relationship between pressure and volume at end-diastole and end-systole. However, exponential approximations may provide a better fit for certain PV loops, capturing the nonlinear characteristics of ventricular filling and emptying (Claessens et al. 2006). Although nonlinear elastance models may better capture pathological conditions, the linear TVE framework remains widely utilized for modeling pathophysiological states (Shi and Korakianitis 2006; David R. Warriner et al. 2014; Dash et al. 2022). In this study, we adopted the commonly implemented linear elastance formulation to ensure consistency with existing literature and facilitate a direct comparison between models.

Suga, Sagawa, and Shoukas (1973) observed that, at a constant heart rate, both $E_{max}$ and the time to reach this peak from the beginning of systole ($T_{E_{max}}$) are unaffected by changes in preload and afterload. This reflects the heart's capability to achieve consistent peak elastance and timing despite different loading conditions. Furthermore, the model assumes that $E_{max}$ and $T_{E_{max}}$ are primarily influenced by biochemical and neural controls, such as epinephrine levels and heart rate adjustments.
It is important to note that $V_0$ can sometimes take non-physiological values (e.g., negative volumes; see Fig. 2). This occurs because the ESPVR is often approximated with linear relationships. When these approximations are extrapolated, they may intersect the volume axis at negative values (Claessens et al. 2006; Shoucri 1991).

Although the activation function $e(t)$ for ventricles and atria has been described following many forms, in this study, the choice is motivated by consistency with the SF model of Pant et al. (2022). These activation functions are designed to replicate the physiological timing and duration of cardiac muscle activation in different chambers. For the ventricles, the activation function is defined as:

$$e_v(t_m) = \left[ \frac{1}{2} \left( 1 - \cos \left( 2\pi \frac{t_m}{t_{\text{max}}^V} \right) \right) \right]^{\beta^V} \cdot \mathbf{1}[0, t_{\text{max}}^V](t_m), \quad (3)$$

where $t_m = t \mod T$ represents the current time within the cardiac cycle, $T$ is the duration of a cardiac cycle, $t_{\text{max}}^V$ is the activation duration for the ventricle (see Fig. 2) and $\beta^V$ controls the shape of the activation curve, influencing the rate of rise and fall of elastance. The function $\mathbf{1}[0, t_{\text{max}}^V](t_m)$ is the indicator function, which equals 1 if $t_m \in [0, t_{\text{max}}^V]$ and 0 otherwise. For the atria, the activation function is defined as:

$$e_A(t_m) = \left[ \frac{1}{2} \left( 1 - \cos \left( 2\pi \frac{(t_m + t_{\text{max}}^A - t_1) \mod T}{t_{\text{max}}^A} \right) \right) \right]^{\beta^A} \cdot \mathbf{1}[0, t_1) \cup [T - t_{\text{max}}^A + t_1, T)(t_m), \quad (4)$$

where $t_{\text{max}}^A$ is the activation duration for the atria, $t_1$ represents the overlap time between atrial and ventricular activation, and $\beta^A$ controls the activation curve's shape for the atria.

Both Eq. (3) and Eq. (4), (illustrated in 3), employ chamber-specific parameters $t_{\text{max}}$ and $\beta$, ensuring that differences in activation timing and contractile dynamics across the left and right atria and ventricles are accurately captured.

For clarity, the atrioventricular (AV) delay corresponds to $t_{\text{AVdelay}} = T - t_{\text{max}}^A + t_1$ in our formulation, assuming the cardiac cycle begins with ventricular activation. This formulation allows the modeler to adapt the duration of atrial contraction without interfering with the physiological AV delay, preserving flexibility while maintaining physiological relevance.
<figure><img src="image_10.png" /><figcaption>**Figure 2:** Illustration of key aspects of cardiac function: the time varying elastance ($E_{LV}$) of the left ventricle, highlighting maximum ($E_{max}$) and baseline ($E_b$) values, as well as left atrial elastance ($E_{LA}$) activation. Key time intervals are indicated: $T_{Emax}$ is the time from the beginning of ventricular activation to peak ventricular activation; $T_{ELA}$ is the time of peak atrial activation; $T$ is the duration of one cardiac cycle; and $t_{max}^V$ and $t_{max}^A$ are the activation durations for the ventricles and atria, respectively. The states of the mitral and aortic valves are shown, as well as the pressure-volume relationship during a single cardiac cycle. The markers indicate specific events: green star (end-diastole with the mitral valve closed), blue circle (aortic valve opening), yellow square (end-systole with the aortic valve closed), and red triangle (mitral valve opening). The dashed lines represent the linear approximations of the End-Systolic Pressure-Volume Relationship (ESPVR) and End-Diastolic Pressure-Volume Relationship (EDPVR), expressed as $P = E_{max}(V - V_0)$ and $P = E_b(V - V_0)$, respectively.</figcaption></figure>

### 2.1.2. Single-Fibre (SF) model

The SF model was developed in the 1980s with a series of publications by Arts and co-authors (Theo Arts, Reneman, and Veenstra 1979; Theo Arts, Veenstra, and Reneman 1982) that laid the groundwork for a detailed description of the model's principles, assumptions, and equations in (T. Arts et al. 1991).

The SF model is constructed to relate macroscopic cardiac hemodynamics to microscopic myocardial fibre function. It simplifies the complex three-dimensional structure of cardiac chambers while capturing myocardial contraction. The model relates cavity blood pressure with cavity volume, fibre stress, and wall volume. T. Arts et al. (1991) employed several key assumptions to establish the model.

First, the SF model assumes rotational symmetry of the cavity wall, which simplifies the complex geometry of the ventricles into a more manageable form. This symmetry allows for a consistent application of the model across different sections of the cardiac wall. Additionally, the model treats the
cardiac wall material as a soft, incompressible medium embedding stress-bearing muscle fibres. The stress in this non-fibrous bulk material is described by hydrostatic pressure, implying that the bulk material behaves fluid-like between the fibres. This means that the fibres bear the majority of the mechanical load, while the surrounding matrix provides structural support. Due to the incompressibility assumption, the myocardial wall volume remains constant during the cardiac cycle. Another key assumption is that fibre stress and strain are homogeneously distributed. Thus, the stress and strain experienced by the muscle fibres are uniform across the wall. The model also assumes no external forces are loading the cardiac chamber, allowing the shape to vary to minimize potential energy. Moreover, the fibres are assumed to be parallel to the isobars of tissue pressure in the wall. The principle of energy conservation is applied in the model, assuming that the mechanical work generated by myocardial fibres is equal to the pumping work of the cardiac chamber. Furthermore, the stress in the collagen matrix in the plane of a shell is assumed to be completely determined by the passive stress in the fibres (Bovendeerd et al. 2006). Lastly, shear stress components are neglected due to the low shear stiffness in myocardial tissue (Bovendeerd et al. 2006). This assumption reduces the complexity of the stress analysis, as the shear components are considered insignificant in the overall stress distribution.

All these assumptions lead to the formulation of the SF model detailed here for one cardiac chamber. The relationship between cavity pressure $P(t)$, cavity volume $V(t)$, wall volume $V_w$, and stress in the fibre $\sigma_f$ is given as:

$$ \frac{P(t)}{\sigma_f} = \frac{1}{3} \ln \left( 1 + \frac{V_w}{V(t)} \right). \qquad (5) $$

According to (T. Arts et al. 1991), this relation appears to be nearly perfectly linear and can be approximated by this simpler model:

$$ \frac{\sigma_f}{P(t)} = 1 + 3 \frac{V(t)}{V_w}, \qquad (6) $$

which was employed in this analysis, without loss of generality. Fibre stress $\sigma_f$ is the sum of active and passive components, respectively $\sigma_a$ and $\sigma_p$:

$$ \sigma_f = \sigma_a + \sigma_p. \qquad (7) $$

The active component was modelled to depend on contractility $0 \le c \le 1$, sarcomere length $l_s$, time $t_m = t \mod T$, and sarcomere shortening velocity $v_s$ as:

$$ \sigma_a = T_{a0} f(l_s) e_c(t_m) h(v_s), \qquad (8) $$

where $T_{a0}$ is the product of $\sigma_{a,max}$ (scaling constant associated with the maximum active sarcomere stress) and contractility $c$; $f(l_s)$ represents the force-length relationship of a sarcomere with length $l_s$; $e_c(t_m) \forall C \in \{V, A\}$ represents the time-dependent variation of active stress of the cardiac chamber, for which we applied the same activation functions as Eq. (3) and Eq. (4) defined previously for ventricles and atria; and $h(v_s)$ represents the force-velocity relationship.

The length of a generic sarcomere is given by the sarcomere length at zero transmural pressure of the chamber $l_{s,0}$ multiplied by the sarcomere stretch $\lambda$:
$$l_s = l_{s,0}\lambda \quad \text{with} \quad \lambda = \left( \frac{1 + 3V(t)/V_w}{1 + 3V_0/V_w} \right)^{1/3}, \qquad (9)$$

where $V_0$ is the cavity volume at zero transmural pressure. Different forms of the force-length relationship can be chosen (Bovendeerd et al. 2006; Pant et al. 2016). Here, in this analysis we apply the one described in (Pant et al. 2016) where for a sarcomere length $l_s$, $f(l_s)$ is described as:

$$f(l_s) = \begin{cases} 0 & \text{if } l_s \le l_{a0}, \\ (l_s - l_{a0})/(l_{am} - l_{a0}) & \text{if } l_{a0} < l_s \le l_{am}, \\ 1 & \text{if } l_{am} \le l_s \le l_{ae}, \\ (l_{af} - l_s)/(l_{af} - l_{ae}) & \text{if } l_s > l_{af}, \end{cases} \qquad (10)$$

with the material constants: $l_{a0}$, $l_{am}$, $l_{ae}$, and $l_{af}$.

Sarcomere shortening velocity is defined as:

$$v_s = -\frac{dl_s}{dt} = -\frac{l_s}{V_w}\left(1 + \frac{3V(t)}{V_w}\right)^{-1}\frac{dV(t)}{dt}, \qquad (11)$$

and the active viscous stress is expressed as:

$$h(v_s) = \frac{1 - (v_s/v_0)}{1 + c_v(v_s/v_0)}. \qquad (12)$$

where $v_0$ is the initial sarcomere shortening velocity and $c_v$ is a parameter that governs the shape of the velocity-stress relationship. Both $f(l_s)$ and $h(v_s)$ functions, alongside the activation functions for the ventricle and atria, $e_v(t_m)$ and $e_A(t_m)$ (Eq. (3) and Eq. (4)), are illustrated in Fig. 3.

Finally, following (Bovendeerd et al. 2006), the passive stress component, $\sigma_p$, is modelled by:

$$\sigma_p = T_{p0} \left( \exp \left[ c_p (\lambda - 1) \right] - 1 \right) \cdot \mathbb{1}_{[1,\infty)}(\lambda), \qquad (13)$$

where $T_{p0}$ and $c_p$ are sarcomere material constants and $\lambda$ is given by (9). Other forms of passive stress component can be found in (Theo Arts et al. 2005; Pant et al. 2018).

<figure><img src="image_11.png" /><figcaption>**Figure 3:** Illustration of active component functions. Left: The activation functions for the ventricle $e_v(t_m)$ and atria $e_A(t_m)$ (Eqs. (3) and (4)), showing the time-dependent variation in activation for both</figcaption></figure>
chambers. Middle: The force-length relationship $f(l_s)$ (Eq. (10)), illustrating sarcomere stress as a function of length. Right: The active viscous stress $h(v_s)$ (Eq. (12)), representing stress behaviour based on sarcomere shortening velocity.

The SF model has evolved since its original development incorporating a deeper understanding of cardiac mechanics. For example, Bovendeerd et al. (2006) improved the model by integrating anisotropic material properties and detailed fibre architecture. This update allowed for a more accurate representation of the heterogeneous mechanical behaviour of myocardial tissue.

To highlight the differences and similarities between the TVE and SF models, a side-by-side comparison is presented in Appendix Table 16.

## 2.2. A generic lumped parameter model (LPM) of the entire blood circulation

### 2.2.1. Closed-loop LPM

To compare TVE and SF models we employed a generic whole-body LPM, illustrated in Fig. 4, which incorporates both the systemic and pulmonary circulations. The LPM model features a simplified circulatory system that integrates the four cardiac chambers, where volume (V) and pressure (P) relationships are defined by either the TVE model (Eq. (2)) or the SF model (Eq. (6)). Additionally, the model includes the four cardiac valves, with their pressure-dependent dynamic flow rates (Q), as detailed in the following paragraph.

The systemic and pulmonary circulations are represented by a three-element *RCR* Windkessel model for the arterial system and a subsequent two-element *RC* Windkessel model for the venous system. In both circulations, the Windkessel model includes a proximal resistance ($R^p$) to represent the large vessels, a distal resistance ($R^d$) for the microvasculature, and total arterial compliance (C) to account for the elasticity of the large vessels. For the venous system, the two-element *RC* Windkessel model comprises venous resistance ($R^v$) and total venous compliance ($C^v$). Moreover, a compliance element is integrated for both large vessels after ventricles for the aorta ($C_{Aorta}$) and the pulmonary artery ($C_{PulArt}$). The relationships between pressure, volume, and flow rate for resistance and compliance, applying the hydraulic-electrical analogy, are detailed in Fig. 5.
<figure><img src="image_12.png" /><figcaption>**Figure 4:** Schematic diagram of the generic closed-loop cardiovascular LPM for comparing TVE and SF models.</figcaption></figure>

<table><caption>**Figure 5:** Pressure (P), volume (V) and flow rate (Q) relationships applying the hydraulic-electrical analogy.</caption><thead><tr><th>Component</th><th>Relationship</th></tr></thead><tbody><tr><td>P<sub>1</sub> &rarr; R &rarr; Q P<sub>2</sub></td><td>P<sub>1</sub> &minus; P<sub>2</sub> = RQ</td></tr><tr><td>Q<sub>1</sub> P Q<sub>2</sub> &rarr; C</td><td>Q<sub>1</sub> &minus; Q<sub>2</sub> = C(dP/dt)</td></tr><tr><td>Q<sub>1</sub> P Q<sub>2</sub> &rarr; V</td><td>Q<sub>1</sub> &minus; Q<sub>2</sub> = dV/dt</td></tr><tr><td>P<sub>1</sub> &rarr; Q &rarr; P<sub>2</sub></td><td>P<sub>1</sub> &minus; P<sub>2</sub> = BQ|Q| + M(dQ/dt)</td></tr></tbody></table>
### 2.2.2. Valve model

The valve model in the generic LPM predicts the valve dynamics based on the pressure difference across each valve during the cardiac cycle. This model captures the fluid-solid interaction between the valve leaflets and the blood flow. As the pressure difference changes, the valve leaflets respond by altering their position, thus forming a variable area through which blood can pass. By ignoring viscous losses, the relationship between the pressure drop $\Delta P$ across the valve and the volumetric flow-rate $Q$ through the valve is expressed, following (Mynard et al. 2012), by the Bernoulli equation as:

$$ \Delta P = BQ|Q| + M \frac{dQ}{dt} \quad \text{with} \quad B = \frac{\rho}{2A_{\text{eff}}^2} \quad \text{and} \quad M = \frac{\rho l_{\text{eff}}}{A_{\text{eff}}}, \qquad (14) $$

where $B$ and $M$ represent the non-linear resistance and inertance of the valve, respectively. These parameters depend on the fluid density $\rho$ and valve's geometrical characteristics namely $l_{\text{eff}}$ the effective length of the valve and $A_{\text{eff}}$ the effective area given by (Pant et al. 2022):

$$ A_{\text{eff}}(t) = (A_{\text{eff}}^{\text{max}} - A_{\text{eff}}^{\text{min}})\xi(t) + A_{\text{eff}}^{\text{min}} \quad \text{and} \quad l_{\text{eff}}(t) = \sqrt{\frac{A_{\text{eff}}(t)}{\pi}}. \qquad (15) $$

where $\xi(t) \in [0,1]$ is a variable representing the valve state, and $A_{\text{eff}}^{\text{min}}$ and $A_{\text{eff}}^{\text{max}}$ are the minimum and maximum effective areas, respectively. An illustration of the valve state for the mitral and aortic valves is provided in Fig. 2, shown in the lower left panel. The valve state $\xi(t)$ depends on the pressure difference across the valve and is described by:

$$ \frac{d\xi(t)}{dt} = \begin{cases} (1 - \xi(t))K_{vo}\Delta P & \text{if } \Delta P \ge 0, \\ \xi(t)K_{vc}\Delta P & \text{otherwise,} \end{cases} \qquad (16) $$

where $K_{vo}$ and $K_{vc}$ are parameters that control the opening and closing rates of the valves, respectively.

This model can be adapted to handle different types of valve regurgitation (Pant et al. 2016). Additionally, unlike traditional diodic valve models, which assume instantaneous closure when the pressure gradient becomes negative, this valve model incorporates the dynamic closure process of the valve leaflets, represented by the state variable $\xi$ (see Eq. (16)). By allowing for a small amount of regurgitation during valve closure, this approach better reflects physiological behaviour, as also observed in Doppler velocity tracings (Pant et al. 2016). This realistic representation of valve dynamics includes a slight re-filling observed at the end of ejection.

## 2.3. Methods for model analysis

### 2.3.1. Inverse problem framework

To calibrate the models to patient-specific data, we formulated an inverse problem aimed at tuning a subset of input parameters $\theta \subseteq X$ to minimize the discrepancy between the simulated outputs and the measured data. Here $X$ is the full set of input parameters, and the remainders of these parameters $X_{\text{Fixed}} = X \setminus \theta$, are held constant. Let $Z$ denote the set of measured clinical data $\{Z_1, \dots, Z_N\}$. The corresponding
model outputs, given the unknown $\theta$ and fixed inputs $\mathbf{X}_{\text{fixed}}$ are $Y_i(\theta, \mathbf{X}_{\text{Fixed}})$ for $i = 1, \dots, N$. The objective (or cost) function to be minimized is defined as:

$$ \varepsilon(\theta) = \sqrt{\sum_{i=1}^{N} \left( \frac{Z_i - Y_i(\theta, \mathbf{X}_{\text{Fixed}})}{\sigma_i} \right)^2} + \sum_{j=1}^{N_p} \left[ \exp \left( 10 \left( \frac{Y_j(\theta, \mathbf{X}_{\text{Fixed}})}{Y_j^{\text{up}}} - 1 \right) \right) - 1 \right] \cdot 1_{\{Y_j(\theta, \mathbf{X}_{\text{Fixed}}) \ge Y_j^{\text{up}}\}} \quad (17) $$

The first term is a weighted least-square difference between model outputs and measured data, where the weights are inversely proportional to the measurement noise $\sigma_i$, which are considered independent and normally distributed. Table 1 lists the specific $\sigma_i$ values for each clinical measurement Z. The second term is a penalty function that imposes a steep cost when any of the $N_p$ penalized outputs $\{Y_j\}_{j=1}^{N_p}$ exceed their respective physiological upper bound $Y_j^{\text{up}}$, thus ensuring that the estimated parameters yield physiologically plausible results. The exponential factor grows rapidly if $Y_j(\theta, \mathbf{X}_{\text{Fixed}}) \ge Y_j^{\text{up}}$, forcing the optimizer away from parameter combinations that yield non-physiological predictions.

<table><caption>**Table 1:** Estimated noise for a given measurement of interest, incorporated in Eq. (17). All measurements are assumed normally distributed and proportional to the respective measurement, with a lower bound given for the pressure measurement of 1.0 mmHg.</caption><thead><tr><td>Parameter Name</td><td>Uncertainty</td><td>Source</td></tr></thead><tbody><tr><td>Volumes</td><td>5%</td><td>(O'Dell 2019)</td></tr><tr><td>Pressures</td><td>max (1.25%, 1.0 mmHg)</td><td>(Edwards 2024)</td></tr><tr><td>Flows</td><td>5%</td><td>(Oscanoa et al. 2023)</td></tr></tbody></table>

To solve the inverse problem, we utilized the covariance matrix adaptation evolution strategy (CMA-ES) (Hansen, Müller, and Koumoutsakos 2003), a stochastic, derivative-free optimization method capable of handling non-linear and non-convex optimization problems. The convergence criterion for the CMA-ES algorithm was set as follows: a maximum of 600 iterations (maxiter = 60θ), a stagnation tolerance of 50 iterations (tolstagnation = 5θ), a timeout of 15 hours, and a target function value of $10^{-2}$ (ftarget = $10^{-2}$).

### 2.3.2. Patient-specific data and model parametrization

#### 2.3.2.1. A 13-year-old patient with suprasystemic idiopathic pulmonary arterial hypertension

To compare cardiac models in simulating complex cardiac conditions, this study is based on a 13-year-old male patient diagnosed with suprasystemic idiopathic pulmonary arterial hypertension (siPAH) at Necker University Hospital for Sick Children, Paris, France. The patient-specific data acquired in this study were acquired in (Boudjemline et al. 2017) and are publicly available in (Pant et al. 2022).

PAH is a rare pediatric disease primarily caused by vasoconstriction of the pulmonary vessels (Humbert et al. 2022). This narrowing of the arterioles' vascular walls leads to a drastic increase in pulmonary vascular resistance (PVR), which induces an increase in right ventricular (RV) workload and pressure, sometimes
exceeding left ventricular (LV) levels (suprasystemic pulmonary pressure). If left untreated, it can progressively lead to RV failure and sudden death.

Regarding the patient, despite receiving drug therapy, he was diagnosed with end-stage RV failure, and given his situation, the medical team decided to perform a percutaneous Potts shunt procedure. This intervention creates a connection, by implanting a grafted conduit, between the left pulmonary artery (LPA) and the descending aorta (DAo). The procedure aims to equalize the pressures on the two sides of the heart, thus decompressing the overloaded right ventricle (Blanc, Vouhé, and Bonnet 2004). In this work, the patient's pre-operative state, before the Potts shunt implantation, is the study case to compare computational models of heart chambers within a whole-circulation model. This is an essential first step in order to perform model prediction of the shunt implantation (Pant et al. 2022).

Pre-operative measurements included electrocardiogram, pressure tracings from heart catheterization, Doppler flow velocity tracings from echocardiography, and CT-angiography imaging. Cardiac output (CO) was measured based on the thermodilution technique. Atrial and ventricular systole durations were obtained from Doppler echocardiography and electrocardiography, which also provided the cardiac cycle time-period. Three-dimensional reconstruction of cardiac CT images allowed for the determination of ventricular end-diastolic volumes (EDV), atrial end-systolic volumes (ESV), and myocardial wall volumes ($V_w$) for each cardiac cavity. Stroke volumes (SV) for the left and right ventricles were calculated based on the measured pre-operative CO and cardiac cycle time-period, which, in combination with the CT-reconstructed EDVs, yielded the corresponding ejection fractions (EF). Tables 2 and 3 present the detailed pre-operative measurements of the patient. For clarity, the clinical outputs listed in Table 2 are referred to as $Z_{Data}$ while the clinical parameters in Table 3 are denoted by $\theta_{Data}$.

<table><caption>**Table 2: Pre-operative clinical outputs $Z_{Data}$.** Notation: left atrium (LA), right atrium (RA), pulmonary artery (PulArt), left ventricle (LV), right ventricle (RV). (Pant et al. 2022)</caption><thead><tr><th rowspan="2">Measurement</th><th rowspan="2">Notation</th><th rowspan="2">Unit</th><th colspan="2">Arterial</th><th colspan="2">Atrial</th><th colspan="2">Ventricular</th></tr><tr><th>Aorta</th><th>PulArt</th><th>LA</th><th>RA</th><th>LV</th><th>RV</th></tr></thead><tbody><tr><td>Systolic pressure</td><td>maxP</td><td>[mmHg]</td><td>94</td><td>112</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Diastolic pressure</td><td>minP</td><td>[mmHg]</td><td>53</td><td>67</td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Mean pressure</td><td>P&#773;</td><td>[mmHg]</td><td>69</td><td>85</td><td>4</td><td>6</td><td>—</td><td>—</td></tr><tr><td>End-diastolic volume</td><td>EDV</td><td>[mL]</td><td>—</td><td>—</td><td>—</td><td>—</td><td>66.9</td><td>91.8</td></tr><tr><td>End-systolic volume</td><td>ESV</td><td>[mL]</td><td>—</td><td>—</td><td>22.7</td><td>51.9</td><td>—</td><td>—</td></tr><tr><td>Ejection fraction</td><td>EF</td><td>—</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0.76</td><td>0.55</td></tr><tr><td>Cardiac output</td><td>CO</td><td>[L/min]</td><td>—</td><td>—</td><td>—</td><td>—</td><td>3.4</td><td>—</td></tr><tr><td>Stroke volume</td><td>SV</td><td>[mL]</td><td>—</td><td>—</td><td>—</td><td>—</td><td>51.0</td><td>—</td></tr></tbody></table>
<table><caption>**Table 3: Pre-operative clinical parameters $\theta_{Data}$.** Notation: left atrium (LA), right atrium (RA), left ventricle (LV), right ventricle (RV). (Pant et al. 2022)</caption><thead><tr><th></th><th>Notation</th><th>Unit</th><th>LA</th><th>RA</th><th>LV</th><th>RV</th></tr></thead><tbody><tr><td>Activation duration</td><td>t<sub>max</sub></td><td>[s]</td><td>0.16</td><td>0.14</td><td>0.40</td><td>0.38</td></tr><tr><td>Myocardial volume</td><td>V<sub>w</sub></td><td>[mL]</td><td>6.4</td><td>10.3</td><td>51.7</td><td>69.0</td></tr><tr><td>Cardiac cycle time-period</td><td>T</td><td>[s]</td><td></td><td></td><td>0.9</td><td></td></tr></tbody></table>

### 2.3.2.2. Parametrization strategy specific to pathophysiological cases

We selected a patient presenting a specific cardiac pathophysiology to explore the applicability of both the SF and TVE approaches under realistic clinical conditions. A core challenge in such pathophysiological scenarios is the limited availability of parameter values and corresponding variation ranges in the literature. In our case, some parameters (e.g., certain sarcomere lengths) lacked both animal and human data on variability; for these, we adopted the fixed values reported in Pant et al. (2022). For most parameters, in order to define their ranges of study, we rely on measured parameters $\theta_{Data}$ (Table 3), data-derived parameters (circulatory parameters from pulmonary and systemic vascular resistances) or estimated parameters through inverse problem.

To estimate parameter ranges, our requirement was that the parameter space be broad enough so that the model outputs can reproduce the patient's clinical measurements $Z_{Data}$ (Table 2). To establish suitable parameter ranges, we proceeded as follows:

1. **Initial broad search through an inverse problem.** For parameters to optimize (detailed in the next paragraph), we employed the CMA-ES algorithm to align the simulated model outputs $Y_{SF}$ or $Y_{TVE}$ with $Z_{Data}$. Numerical resolution framework is addressed in the Appendix B.

2. **Iterative refinement of parameter variation ranges.** Guided by the inverse problem results, we iteratively adjusted the parameter ranges so that sampling within these ranges produced output distributions encompassing the patient measurements.

3. **Final ranges.** After confirming that the simulated outputs $Y_{SF}$ and $Y_{TVE}$ remained within physiologically realistic (and patient-specific) ranges, we finalized the parameter ranges for the subsequent SA and identifiability study. These final ranges are provided in the Appendix Tables 13 and 14.

*Note:* Model parameterization in this context is an intricate phase of the overall modelling process, commonly encountered by other modellers in the field.

In our simulations, certain clinically relevant outputs are not direct state variables but are derived from key events within the cardiac cycle. For instance, ventricular EDV are determined at the onset of mechanical activation, specifically when the activation function Eq. (3) begins its kick (see Fig. 2). For the atria, end-
diastolic (ED) and end-systolic (ES) times are defined by the opening and closing of the semilunar valves; ED corresponds to valve opening, and ES corresponds to valve closure. By tracking these specific instants in each cardiac cycle, we extract the relevant volumes and pressures. We computed SV as the difference between its maximum and minimum volumes during one cycle, and determine the CO by averaging the aortic flow.

### 2.3.2.3. Parameter specification

We categorized the parameters into four groups: (1) global circulation parameters (e.g., vascular resistances and compliances) denoted as $\theta_{Circ}$, (2) cardiac valve parameters (common to both cardiac-chamber models) denoted as $\theta_{Valve}$, (3) parameters specific to the SF model denoted as $\theta_{SF}$, and (4) parameters specific to the TVE model denoted as $\theta_{TVE}$.

**Circulatory System Parameters** $\theta_{Circ}$. Following Pant et al. (2022) we derived key systemic and pulmonary parameters (e.g., pulmonary vascular resistance PVR, systemic vascular resistance SVR) from the measured cardiac output and mean pressures. The relationships are:

$$PVR = \frac{\overline{P_{PulArt}} - \overline{P_{LA}}}{CO} \quad \text{and} \quad SVR = \frac{\overline{P_{Aorta}} - \overline{P_{RA}}}{CO}, \qquad (18)$$

where $\overline{P_{PulArt}}$ and $\overline{P_{LA}}$ are mean pulmonary artery and left atrial pressures (and similarly for the aorta and right atrium). For each vascular bed, we partitioned resistances $R^p$, $R^d$, and $R^v$ in proportions of 8.2%, 81.8%, and 10% of the total resistance (pulmonary or systemic) (LaDisa et al. 2011; Pant et al. 2022).

Compliances $C_{i \in \{sys,pul\}}$ in the three-element Windkessel models are related to the time constant $\tau_{i \in \{sys,pul\}}$ of the exponential pressure decay (Spilker and Taylor 2010; Mackenzie Ross et al. 2013) such that $C_i = \tau_i/R_i^d$ $i \in \{sys,pul\}$ where:

$$\tau_{sys} = \frac{SVR \times SV}{PP_{sys}} \quad \text{and} \quad \tau_{pul} = \frac{PVR \times SV}{PP_{pul}},$$

where SV is the stroke volume, $PP_{sys} = \max P_{Aorta} - \min P_{Aorta}$ and $PP_{pul} = \max P_{PulArt} - \min P_{PulArt}$ are pulse pressure (PP).

Venous compliances ($C_i^v$ for $i \in \{sys,pul\}$) were obtained from $C_i^v = \tau_i^v/R_i^v$ for $i \in \{sys,pul\}$, where $\tau_{sys}^v = 2$ s and $\tau_{pul}^v = 1$ s (Pant et al. 2022).

Although these methods provided reasonable initial estimates for the systemic parameters, they did not adequately capture the patient's elevated pulmonary pressures and altered flow dynamics. To address this gap, we included certain circulation parameters in our inverse-problem optimization alongside the chamber-level parameters (SF or TVE). Specifically, the systemic resistances and compliance ($C_{sys}^d, R_{sys}^p, R_{sys}^d$) with the aortic and pulmonary arterial compliances ($C_{Aorta}, C_{PulArt}$), which were manually tuned in Pant et al. (2022). Table 4 summarizes the initial values from SVR estimates and the final optimized values for these five parameters denoted as $\theta_{Circ}$.
<table><caption>**Table 4:** Circulatory parameters obtained from inverse problem optimization and indication of the first estimated value from SVR.</caption><thead><tr><th>Set</th><th>Parameter</th><th>Unit</th><th>TVE</th><th>SF</th><th>Initial estimates from SVR</th></tr></thead><tbody><tr><td rowspan="4">θ<sub>Circ, Opt</sub></td><td>C<sub>Aorta</sub></td><td>[cm<sup>4</sup>·s<sup>2</sup>/g]</td><td>3.56 × 10<sup>-4</sup></td><td>2.27 × 10<sup>-4</sup></td><td>-</td></tr><tr><td>C<sub>PulArt</sub></td><td>[cm<sup>4</sup>·s<sup>2</sup>/g]</td><td>2.75 × 10<sup>-4</sup></td><td>1.78 × 10<sup>-4</sup></td><td>-</td></tr><tr><td>C<sub>sys</sub><sup>d</sup></td><td>[cm<sup>4</sup>·s<sup>2</sup>/g]</td><td>1.11 × 10<sup>-3</sup></td><td>1.14 × 10<sup>-3</sup></td><td>1.14 × 10<sup>-3</sup></td></tr><tr><td>R<sub>sys</sub><sup>p</sup></td><td>[g/cm<sup>4</sup>·s]</td><td>1.30 × 10<sup>2</sup></td><td>1.29 × 10<sup>2</sup></td><td>1.22 × 10<sup>2</sup></td></tr><tr><td></td><td>R<sub>sys</sub><sup>d</sup></td><td>[g/cm<sup>4</sup>·s]</td><td>1.28 × 10<sup>3</sup></td><td>1.21 × 10<sup>3</sup></td><td>1.21 × 10<sup>3</sup></td></tr></tbody></table>

**Valve model parameters** $\theta_{\text{Valve}}$. The valve parameters for both the SF and TVE models are adapted from Pant et al. (2022). They represent healthy adolescent valves without stenosis or regurgitation. Table 5 presents the opening ($K_{vo}$) and closing ($K_{vc}$) rates, as well as the maximum and minimum effective areas ($A_{\text{eff}}^{\text{max}}$ and $A_{\text{eff}}^{\text{min}}$) for the four valves: mitral valve (MiV), aortic valve (AoV), tricuspid valve (TrV), and pulmonary valve (PuV). These parameters, collectively referred to a $\theta_{\text{Valve}}$ also include the blood density $\rho$.

<table><caption>**Table 5:** Patient-specific pre-operative parameters for the valve model: mitral valve (MiV), aortic valve (AoV), tricuspid valve (TrV), and pulmonary valve (PuV).</caption><thead><tr><th>Set</th><th>Parameters</th><th>Unit</th><th>MiV</th><th>AoV</th><th>TrV</th><th>PuV</th></tr></thead><tbody><tr><td rowspan="4">θ<sub>Valve</sub></td><td>K<sub>vo</sub></td><td>[cm s/g]</td><td>0.03</td><td>0.012</td><td>0.03</td><td>0.02</td></tr><tr><td>K<sub>vc</sub></td><td>[cm s/g]</td><td>0.04</td><td>0.012</td><td>0.04</td><td>0.02</td></tr><tr><td>A<sub>eff</sub><sup>max</sup></td><td>[cm<sup>2</sup>]</td><td>2.1</td><td>2.2</td><td>2.1</td><td>2.3</td></tr><tr><td>A<sub>eff</sub><sup>min</sup></td><td>[cm<sup>2</sup>]</td><td>1 × 10<sup>-3</sup></td><td>1 × 10<sup>-3</sup></td><td>1 × 10<sup>-3</sup></td><td>1 × 10<sup>-3</sup></td></tr><tr><td></td><td>ρ</td><td>[g/cm<sup>3</sup>]</td><td></td><td></td><td>1.06</td><td></td></tr></tbody></table>

**SF Model parameters** $\theta_{\text{SF}}$. To fit the patient-specific data for the SF model, we applied the optimization framework described in Section 2.3.1 to estimate parameters that were manually tuned in (Pant et al. 2022), along with the subset of circulatory parameters $\theta_{\text{Circ,Opt}}$. Table 6 summarizes the final values of all SF model parameters, distinguishing those that were optimized ($\theta_{\text{SF,Opt}}$) from those that remained fixed ($\theta_{\text{SF,Fixed}}$) to the values described in Pant et al. (2022). Since the SF model outputs inherently remained within physiological ranges, we did not include any additional penalty term in the objective function Eq. (17).
<table><caption>**Table 6:** Summary of SF model parameters. The top section lists the optimized parameters ($\theta_{SF,Opt}$); the bottom section lists the fixed parameters ($\theta_{SF,Fixed}$).</caption><thead><tr><th>Set</th><th>Parameters</th><th>Value</th><th>Parameters</th><th>Value</th><th>Parameters</th><th>Value</th><th>Parameters</th><th>Value</th></tr></thead><tbody><tr><td rowspan="2">θ<sub>SF, Opt</sub></td><td>V<sub>0</sub><sup>LA</sup> [ml]</td><td>25.0</td><td>V<sub>0</sub><sup>RA</sup> [ml]</td><td>37.8</td><td>V<sub>0</sub><sup>LV</sup> [ml]</td><td>31.1</td><td>V<sub>0</sub><sup>RV</sup> [ml]</td><td>43.6</td></tr><tr><td>T<sub>a0</sub> [kPa]</td><td>75.0</td><td>T<sub>p0</sub> [kPa]</td><td>0.9</td><td>c<sub>p</sub> [-]</td><td>12.8</td><td>-</td><td>-</td></tr><tr><td rowspan="3">θ<sub>SF, Fixed</sub></td><td>v<sub>0</sub> [µm/s]</td><td>10.0</td><td>c<sub>v</sub> [-]</td><td>0.0</td><td>l<sub>s,0</sub> [µm]</td><td>2.0</td><td>l<sub>a0</sub> [µm]</td><td>1.5</td></tr><tr><td>l<sub>am</sub> [µm]</td><td>2.0</td><td>l<sub>ae</sub> [µm]</td><td>2.4</td><td>l<sub>af</sub> [µm]</td><td>2.7</td><td>t<sub>1</sub> [s]</td><td>0.02</td></tr><tr><td>β<sup>LA</sup> [-]</td><td>1.0</td><td>β<sup>RA</sup> [-]</td><td>1.0</td><td>β<sup>LV</sup> [-]</td><td>1.05</td><td>β<sup>RV</sup> [-]</td><td>0.7</td></tr></tbody></table>

**TVE Model parameters** $\theta_{TVE}$. Our goal for the TVE model was to achieve hemodynamic predictions comparable to those of the SF model. Unlike the SF model—which could be calibrated using only the clinical data $Z_{Data}$—the TVE model required additional atrial constraints to ensure physiological solutions. To address this need, we supplemented the limited clinical atrial data with end-diastolic pressures and volumes derived from the calibrated SF model, denoted by $Z_{Y_{SF}}$. This approach enabled the TVE model to capture realistic atrial dynamics while still matching the patient's measured hemodynamics, a necessity later confirmed through identifiability analyses.

We then solved an inverse problem to estimate both the TVE-specific parameters ($\theta_{TVE, Opt}$) and the circulatory parameters ($\theta_{Circ, Opt}$) under the objective function Eq. (17) with the penalty term to prevent non-physiological outputs (See Table 7). Table 8 summarizes the resulting parameter values, distinguishing between those that were optimized ($\theta_{TVE, Opt}$) and those kept fixed ($\theta_{TVE, Fixed}$).

<table><caption>**Table 7:** Imposed penalty to ensure physiological outputs when performing the inverse problem for the TVE model tuning (Sarikouch et al. 2011; Kawel-Boehm et al. 2020).</caption><thead><tr><th>Upper bound</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>Y<sub>PLA</sub><sup>up</sup></td><td>20</td><td>[mmHg]</td></tr><tr><td>Y<sub>VRA</sub><sup>up</sup></td><td>90</td><td>[mmHg]</td></tr><tr><td>Y<sub>PRA</sub><sup>up</sup></td><td>25</td><td>[mmHg]</td></tr><tr><td>Y<sub>VLA</sub><sup>up</sup></td><td>60</td><td>[mmHg]</td></tr></tbody></table>

<table><caption>**Table 8:** Summary of TVE model parameters. The top section lists the optimized parameters ($\theta_{TVE,Opt}$); the bottom section lists the fixed parameters ($\theta_{TVE,Fixed}$). Notation: left atria (LA), right atria (RA), left ventricle (LV), and right ventricle (RV).</caption><thead><tr><th>Set</th><th>Parameters</th><th>Unit</th><th>LA</th><th>RA</th><th>LV</th><th>RV</th></tr></thead><tbody><tr><td rowspan="3">θ<sub>TVE, Opt</sub></td><td>E<sub>a</sub></td><td>[mmHg/mL]</td><td>2.9 × 10<sup>-1</sup></td><td>5.5 × 10<sup>-1</sup></td><td>1.9</td><td>3.3</td></tr><tr><td>E<sub>b</sub></td><td>[mmHg/mL]</td><td>1.1 × 10<sup>-1</sup></td><td>1.4 × 10<sup>-1</sup></td><td>9.8 × 10<sup>-2</sup></td><td>1.2 × 10<sup>-1</sup></td></tr><tr><td>V<sub>0</sub></td><td>[mL]</td><td>-9.4</td><td>31.5</td><td>-29.0</td><td>8.8</td></tr></tbody></table>
<table><tr><td rowspan="2">θ<sub>TVE, Fixed</sub></td><td>β</td><td>[-]</td><td>1.0</td><td>1.0</td><td>1.05</td><td>0.7</td></tr><tr><td>t<sub>1</sub></td><td>[s]</td><td>0.02</td><td>0.02</td><td>-</td><td>-</td></tr></table>

**Summary of Inverse Problem Framework.** Table 9 provides an overview of the parameter sets and target data considered in the inverse problem for both the SF and TVE models. It specifies which parameters were optimized ($\theta$) and which remained fixed ($X_{Fixed}$) in each scenario, along with the corresponding data sets ($Z$).

<table><caption>**Table 9:** Summary of optimized ($\theta$), fixed parameter ($X_{Fixed}$) and target data ($Z$) sets for SF and TVE model scenario when performing the inverse problem (See Eq. (17)). Parameters sets are defined across Tables 4-8.</caption><thead><tr><th rowspan="2">Set</th><th colspan="2">SF</th><th colspan="2">TVE</th></tr><tr><th><b>X</b><sub>Fixed</sub></th><th><b>θ</b></th><th><b>X</b><sub>Fixed</sub></th><th><b>θ</b></th></tr></thead><tbody><tr><td><b>θ</b><sub>Circ</sub></td><td><b>θ</b><sub>Circ</sub>\<b>θ</b><sub>Circ,Opt</sub></td><td><b>θ</b><sub>Circ,Opt</sub></td><td><b>θ</b><sub>Circ</sub>\<b>θ</b><sub>Circ,Opt</sub></td><td><b>θ</b><sub>Circ,Opt</sub></td></tr><tr><td><b>θ</b><sub>Valve</sub></td><td><b>θ</b><sub>Valve</sub></td><td>—</td><td><b>θ</b><sub>Valve</sub></td><td>—</td></tr><tr><td><b>θ</b><sub>SF</sub></td><td><b>θ</b><sub>SF,Fixed</sub></td><td><b>θ</b><sub>SF,Opt</sub></td><td>—</td><td>—</td></tr><tr><td><b>θ</b><sub>TVE</sub></td><td>—</td><td>—</td><td><b>θ</b><sub>TVE,Fixed</sub></td><td><b>θ</b><sub>TVE,Opt</sub></td></tr><tr><td><b>θ</b><sub>Data</sub></td><td><b>θ</b><sub>Data</sub></td><td>—</td><td><b>θ</b><sub>Data</sub>\<b>V</b><sub>W</sub></td><td>—</td></tr><tr><td><b>Z</b></td><td><b>Z</b><sub>Data</sub></td><td></td><td><b>Z</b><sub>Data</sub> ∪ <b>Z</b><sub>Y<sub>SF</sub></sub></td><td></td></tr></tbody></table>

### 2.3.3. Sensitivity analysis

Sensitivity analysis (SA) is a general methodology, widely applied across various scientific fields, that examines how variations or uncertainties in the inputs $X$ of a model influence selected outputs $\bar{Y}$, which are either a subset of the model variables $Y$ or derived from them. In the settings of cardiovascular modelling, this concept extends beyond mathematics to explore the factors that affect the progression of a disease or the outcome of a surgery (Sala et al. 2023). In the present study, we aim to quantify the impact of the different input parameters such as ventricular wall and unstressed volumes on clinically relevant outputs, including ventricular stroke volumes and pressure levels (diastolic, systolic, mean). The SA is an integral step in the proposed methodological pipeline. It does not only identify influential parameters but also establishes a foundation for the subsequent identifiability analysis, which evaluates the certainty of estimating these parameters with the available data.

The first step when conducting a SA corresponds to identifying the independent model input parameters. The ranges and distributions of these inputs need to be identified, either from literature, or common sense (Hose et al. 2019). In this work, the variability of the input parameters, when possible, takes into account the characteristics of a young population, ranging from infants to young adults, in conjunction with PAH. Given the high PVR from hypertension, initial parameter ranges from the literature not specific for PAH had to be revised to achieve pressures and cardiac output distributions that were overall physiological (See Section 2.3.2). The model independent inputs and ranges are presented in Appendix Tables 13 and 14. The distributions of the inputs for this study were assumed uniform. SA was performed exclusively on chamber-related inputs because we aimed to compare the two heart chamber models, excluding valvular
and circulatory parameters from our study. Parameters for which no variability data are available, either from animal or human studies (e.g., different sarcomere lengths), were assumed to be fixed. It is also worth noting that the activation duration of the right and left ventricles, as well as the atria, were assumed to be equal.

Once all model inputs and their distributions are defined, the DAE model L is run several times depending on the parameter sampling method of choice. The sampling algorithm is crucial as it is associated with the sensitivity measures or indices that will evaluate the inputs' impact on the outputs. In general, there is a variety of SA methods, spanning from the one-at-a-time (OAT) method to derivative-based and variance-based methods (Saltelli et al. 2004). When performing SA on a large number of parameters (e.g., when the whole-body circulation is considered), an initial screening, employing the Morris method (Morris 1991) for example, could be valuable. Although these screening methods are quite fast in their predictions, they only provide a qualitative ranking, since the interactions among the inputs and their non-linear nature are not well taken into account. In our case, where we only focus on chamber-related inputs, the Sobol Indices method is explored (Sobol 2001; Saltelli et al. 2010). The latter is a variance decomposition-based method, which quantifies the contribution of a specific input or a combination of inputs to the overall variance of an output.

Therefore, let the input parameters X be random independent variables following a probability distribution, employed to compute the random output vector $\bar{X}$. The first and the total order Sobol indices are then provided by:

$$S_{ij} = \frac{\mathrm{var}[\mathbb{E}(Y_i|X_j)]}{\mathrm{var}[Y_i]} \quad \forall i \in [1, \dots, N_{\mathrm{outputs}}], \qquad (20)$$

$$S_{ij}^{\mathrm{tot}} = 1 - \frac{\mathrm{var}[\mathbb{E}(Y_i|\mathbf{X}_{(-j)})]}{\mathrm{var}[Y_i]} = 1 - S_{-i}(j) \quad \forall i \in [1, \dots, N_{\mathrm{outputs}}], \qquad (21)$$

where var corresponds to the variance, E to the expected value, and $\mathbf{X}_{(-j)} = \mathbf{X} \setminus X_j$. Equation (20) can be interpreted as: var[Y_i] corresponds to the overall variability of Y_i, including nonlinear effects, while var[E(Y_i|X_j)], which is the variance of conditional expectation E(Y_i|X_j), represents the main or first-order effect of X_j on Y_i. For example, if Y_i is very sensitive to X_j, then E(Y_i|X_j) is going to vary significantly, in the sense that it will cover most of the variance of Y_i, resulting in a value of the first order Sobol index close to 1.

The total order Sobol index in Eq. (21), on the other hand, describes the influence of the specific input X_j and its interaction with the other inputs $\mathbf{X}_{(-j)}$ with respect to the output Y_i. If the value of the total order Sobol index $S_{ij}^{\mathrm{tot}}$ is close to 0, then the output Y_i is considered insensitive to the input X_j.

The sampling is performed with Saltelli's extension of the Sobol' sampling approach (Saltelli et al. 2004) via the open-source Python library SALib (Herman and Usher 2017), which is an algorithm aimed exploring a space with the minimal amount of points. The computational cost (e.g., the number of model evaluations) for this approach is $N_{\mathrm{sim}} = (k+2)N_{\mathrm{samples}}$, k signifies the model's input variables, and $N_{\mathrm{samples}}$ denotes the pre-determined sample size.
However, it is important to note that the sampling, by exploring the whole parameter space, does not guarantee physiological outputs value of the model. Therefore, a filtering approach is applied to discard non-physiological outputs along with their corresponding input parameters. This filtering process can significantly reduce the number of simulations in the generated sample, to the extent that traditional Monte-Carlo approaches are no longer valid. To address this issue, as also proposed in (Sala et al. 2023), an alternative method, Polynomial Chaos Expansion (PCE), is employed. PCE leverages the properties of orthogonal polynomials to create a surrogate model. It is particularly useful in the context of sensitivity analysis, as it approximates the variance decomposition of model output with respect to model input, thereby generating Sobol indices (Eck et al. 2016).

The applied physiological filters consisted of two main criteria: first, samples with ventricular volumes smaller than 5 mL were removed (to ensure the ejection fraction remains within physiological limits), and secondly samples with negative atrial volumes were removed. Negative volumes, although physiologically unrealistic, can occur when simulating the TVE model because it is not constrained by physiological limits. In contrast, this phenomenon is not observed in the SF model. Due to these filters, a significant portion of simulations was excluded from our generated dataset.

After applying these filters, at least 30,000 simulations remained for both models to conduct the sensitivity analysis (SA). Convergence was monitored via the $Q^2$-metric (Quicken et al. 2016), which is given by

$$Q^2 = 1 - \frac{N_s - 1}{N_s} \frac{\sum_{k=1}^{N_s} (Y(X_k) - f_{PCE\setminus k}(X_k))^2}{\sum_{k=1}^{N_s} (Y(X_k) - \bar{Y})^2} \quad (22)$$

Here, $N_s$ equals the number of samples, $Y(X_k)$ the output of the 0D-model with a given input parameter set, $\bar{Y}$ the mean value of output Y and $f_{PCE\setminus k}(X_k)$ the output of the PCE-metamodel with the same input parameter set. Note that the testing data-set was not included in the generation of the PCE-metamodel, as denoted by the $\setminus k$. Convergence was achieved in both models, as evidenced by a $Q^2$ value exceeding 0.95 for all outputs of interest with the aforementioned number of simulations. Values above 0.95 suggest strong predictive accuracy and reliable model performance. Notably, the SF model required more simulations to reach convergence compared to the TVE, probably due to its non-linear behaviour.

Once all sensitivity indices are calculated, we then select only the set of parameters which have a significant influence on the outputs of interest. This subset of parameters are the most important parameters to tune to ensure that the model represents the patient in his/her current state and the most important in any form of prediction. Remaining parameters are often deemed unimportant and kept at literature values.

### 2.3.4. Identifiability

Identifiability of the sensitive parameter set derived from SA results can be assessed conducting Profile Likelihood analysis (PLA) (Vanlier et al. 2012). It is a step-wise process, which involves solving the inverse problem as defined in Section 2.3.1 multiple times. An example can be found in Fig. 6. To determine the likelihood profile of any parameter, first the values of all the sensitive parameters corresponding to the lowest error function must be found, often referred to as the optimal set of input parameters (blue $\times$). Then the identifiability of the parameter of interest is assessed by stepwise variation
of that parameter in place and re-performing the optimization strategy, leaving the remaining sensitive
parameters as degrees of freedom for the algorithm (orange squares/green triangles markers). This
stepwise variation continues until either the boundary of the parameter value or the maximum error
threshold (dashed line) is reached, generating a curve relating the parameter value and the lowest
corresponding error. These resulting value-error curves can then be examined, and serve to determine
possible follow-up steps. Three example value-error curves are shown in Fig. 6 displaying different
identifiability behaviours. The left curve shows the ideal behaviour, a convex curve where any deviation
from the initially found value leads to an increased error value, and is thereby considered identifiable. The
right curve shows no increases in error function regardless of parameter value, indicating either no outputs
contributing to the error-function are sensitive to this input parameter, or this input parameter is correlated
to other input parameters, meaning the change in this input parameter can be compensated by changing
another parameter. Therefore, this parameter can be considered unidentifiable. Finally, the centre curve
shows identifiable behaviour on the left-hand side, but unidentifiable on the right-hand side, and is
considered practically unidentifiable. To counteract identifiability issues in there are several options, such
as fixing unidentifiable parameters to reasonable values or determine pseudo-parameters in case of strong
interactions. Alternatively, one could review which additional measurements could possibly improve
parameter identifiability and assess their feasibility.

<figure><img src="image_13.png" /><figcaption>**Figure 6:** Profile Likelihood analysis examples. First, the full optimization problem is run, resulting in a set of optimal parameters θ. This is the starting point for the PLA (blue X). For each parameter θj, the parameter value increased incrementally and the optimization problem is re-performed (orange square). The problem should be run until the stopping criterium is reached (dashed line), and subsequently performed using decreasing values of θj (green triangle). **Left** - In ideal case, a convex curve is found, meaning the parameter is fully identifiable. The values of θj where ε exceeds the thresholds can be considered the upper and lower ranges the estimated parameter. **Centre** - If the profile is only convex one hand, there exists only one bound, while the other constraint is the defined upper limit of θi. These profiles mean the parameter is practically unidentifiable. **Right** - If the profile remains completely below the threshold, the parameter is considered unidentifiable.</figcaption></figure>
# 3. Results

## 3.1. Models' outputs from the PAH patient-specific simulations

Using the parameters listed in Tables 4-8 (see Section 2.3.2), we carried out simulations for both the SF and TVE models. Table 10 summarizes the key outputs from these final simulations, their partial error contributions as defined by Eq. (17), and the corresponding clinical measurements of the PAH patient.

Overall, the SF model closely reproduces the measured aortic and pulmonary artery pressures (minimum, maximum, and mean), as well as $\bar{P}_{LA}$, $\bar{P}_{RA}$, and cardiac output (CO). Its predictions for end-diastolic volume (EDV), end-systolic volume (ESV), and stroke volume (SV) also match the clinical data.

By comparison, the TVE model shows larger partial error contributions for certain outputs. While aortic pressures (both max and min), $\bar{P}_{RA}$, left ventricular EDV, and ventricular SV still remain at acceptable partial error levels, other variables deviate more noticeably. For instance, the mean pulmonary artery pressure ($\bar{P}_{PulArt}$), $\bar{P}_{LA}$ and CO have a higher partial error. Although the TVE model demonstrates reasonable agreement with the augmented data set $Z_{YSF}$ for certain atrial variables, notably end-diastolic ventricular pressure, some discrepancies persist.

Fig. 7 compares the pressure-volume loops for both models. The SF model, with a more detailed representation of myocardial mechanics, reproduces the nonlinear diastolic filling phase accurately, while the TVE model's linear approximation underrepresents the steep rise in ventricular pressure during atrial contraction. In Fig. 7 a slight decrease in volume is observed, where end-ejection volume appears lower than the volume at which isovolumic relaxation occurs. This physiological phenomenon arises from the choice of valve model (Eq. (14)) employed in this study allowing for a small regurgitant volume to re-enter the ventricle during this phase.

Fig. 8 displays the Wiggers diagrams for the left and right hearts, respectively. Despite differences in the dynamic profiles of these signals, both models achieve the fitting targets for diastolic and systolic volumes, as well as maximum and mean pressures in the aorta and right atria.
<figure><img src="image_14.png" /><figcaption>**Figure 7:** Pressure-volume loops for the four cardiac chambers (left atrium, left ventricle, right atrium, right ventricle) generated by the SF and TVE models.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 8:** Wiggers diagram of the left and right heart, illustrating on the left panel the left atrial pressure, left ventricular pressure, aortic pressure, and left ventricular volume over time, and on the right panel the right atrial pressure, right ventricular pressure, pulmonary artery pressure, and right ventricular volume over time, as simulated by the SF and TVE models.</figcaption></figure>
<table><caption>**Table 10:** Comparison of measured/derived data versus the two model outputs (SF, TVE). The columns Err(SF) and Err(TVE) give the partial error contribution $(Z_i - Y_i)/\sigma_i$ from Eq. (17) for each output $i$, using Table 1 to define $\sigma_i$. Pressures are in [mmHg], volumes in [mL], and CO in [L/min].</caption><thead><tr><th>Variable</th><th>Z<sub>Data</sub></th><th>Y<sub>SF</sub></th><th>Err(SF)</th><th>Y<sub>TVE</sub></th><th>Err(TVE)</th></tr></thead><tbody><tr><td>maxP<sub>Aorta</sub></td><td>94</td><td>94.6</td><td>-0.51</td><td>94.3</td><td>-0.26</td></tr><tr><td>minP<sub>Aorta</sub></td><td>53</td><td>53.6</td><td>-0.60</td><td>54.1</td><td>-1.10</td></tr><tr><td>P&#773;<sub>Aorta</sub></td><td>69</td><td>67.8</td><td>1.20</td><td>67.4</td><td>1.60</td></tr><tr><td>maxP<sub>PulArt</sub></td><td>112</td><td>112.8</td><td>-0.57</td><td>113.3</td><td>-0.93</td></tr><tr><td>minP<sub>PulArt</sub></td><td>67</td><td>67.5</td><td>-0.50</td><td>66.5</td><td>0.50</td></tr><tr><td>P&#773;<sub>PulArt</sub></td><td>85</td><td>84.0</td><td>0.94</td><td>81.7</td><td>3.11</td></tr><tr><td>P&#773;<sub>LA</sub></td><td>4</td><td>4.0</td><td>0.00</td><td>6.7</td><td>-2.70</td></tr><tr><td>P&#773;<sub>RA</sub></td><td>6</td><td>6.0</td><td>0.00</td><td>6.2</td><td>-0.20</td></tr><tr><td>EDV<sub>LV</sub></td><td>66.9</td><td>66.9</td><td>0.00</td><td>65.1</td><td>0.54</td></tr><tr><td>EDV<sub>RV</sub></td><td>91.8</td><td>93.5</td><td>-0.37</td><td>86.4</td><td>1.18</td></tr><tr><td>ESV<sub>LA</sub></td><td>22.7</td><td>22.8</td><td>-0.09</td><td>23.7</td><td>-0.88</td></tr><tr><td>ESV<sub>RA</sub></td><td>51.9</td><td>52.0</td><td>-0.04</td><td>48.3</td><td>1.39</td></tr><tr><td>SV<sub>LV</sub></td><td>51.0</td><td>51.4</td><td>-0.16</td><td>49.5</td><td>0.59</td></tr><tr><td>SV<sub>RV</sub></td><td>51.0</td><td>51.3</td><td>-0.12</td><td>48.5</td><td>0.98</td></tr><tr><td>CO<sub>LV</sub></td><td>3.4</td><td>3.4</td><td>0.00</td><td>3.1</td><td>1.76</td></tr><tr><td></td><td>-</td><td>Z<sub>Y<sub>SF</sub></sub></td><td>-</td><td>Y<sub>TVE</sub></td><td>Err(TVE)</td></tr><tr><td>EDP<sub>LV</sub></td><td>-</td><td>16.2</td><td>-</td><td>9.1</td><td>7.10</td></tr><tr><td>EDP<sub>RV</sub></td><td>-</td><td>16.1</td><td>-</td><td>9.0</td><td>7.10</td></tr><tr><td>EDP<sub>LA</sub></td><td>-</td><td>4.2</td><td>-</td><td>6.2</td><td>-2.00</td></tr><tr><td>EDP<sub>RA</sub></td><td>-</td><td>6.7</td><td>-</td><td>6.0</td><td>0.70</td></tr><tr><td>EDV<sub>LA</sub></td><td>-</td><td>46.2</td><td>-</td><td>47.1</td><td>-0.39</td></tr><tr><td>EDV<sub>RA</sub></td><td>-</td><td>76.7</td><td>-</td><td>72.7</td><td>1.04</td></tr><tr><td>maxP<sub>LA</sub></td><td>-</td><td>16.6</td><td>-</td><td>18.4</td><td>-1.80</td></tr><tr><td>maxP<sub>RA</sub></td><td>-</td><td>19.4</td><td>-</td><td>20.1</td><td>-0.70</td></tr></tbody></table>
## 3.2. Sensitivity analysis

In order to better understand which parameters have more impact on the model behaviour, results from the Sobol sensitivity analysis are presented in Fig. 9. The model outputs were selected for their clinical relevance, as they assess key cardiac functions such as pressures, volumes, and ejection fractions. While measurable, they are rarely available in full within a single dataset. Additionally, given the focus on PV loops, these outputs offer critical insights into how model parameters influence key aspects like ESP, EDP, and SV. To complement the Sobol sensitivity analysis, we investigated the impact of parameter variations on the LV PV-loops, focusing on the most influential parameters identified in the analysis. The parameters were varied within the ranges derived from the sensitivity analysis, as detailed in Appendix Tables 13 and 14. The resulting PV loops are presented in Fig. 10.

Regarding the SF model, it is evident that the biomechanical properties of the myocardium, $C_p$ and, to a lesser extent, $T_{p0}$, have the greatest impact on the selected outputs, particularly influencing left heart volumes and pressures. As shown in the left ventricular PV loops (Fig. 10), an increase in $C_p$ and $T_{p0}$ leads to higher end-diastolic pressure (EDP) and reduced ventricular filling. Conversely, decreasing these parameters results in more compliant filling, allowing the ventricle to accommodate a greater volume at lower pressures. Additionally, $T_{a0}$, which accounts for the contractility and maximum active sarcomere strength, also shows to have a strong impact on many of the outputs, especially on pulmonary artery and aortic pressures. This aligns with the PV-loop analysis, where an increase in $T_{a0}$ enhances myocardial contractility, leading to higher maximum pressures and larger stroke volumes (Fig. 10). Notably, in the current model, $T_{a0}$ represents the contractile function of all chambers. Given the disease discussed in this study, where only the right side is impaired, one would expect the right heart parameters to have a greater influence. The right ventricular wall volume $V_W^{LV}$ stands out as another input parameter with considerable influence on various outputs, mainly on the right side, yet with a noticeable effect on the left too, more likely due to its hypertrophic state. Concerning the unstressed volumes $V_0$, it is evident that they only affect the volumes and pressures of their respective chambers. Looking at their impact on left ventricular PV-loops, these volume-related parameters affect the size, shape, and position of the loops. Modifying $V_0^{LV}$ results in a horizontal shift of the PV loop, with larger $V_0^{LV}$ values shifting the loop towards higher volumes while maintaining a similar shape (Fig. 10). This indicates changes in the ventricular unstressed volume, which affects the relationship between chamber volume and sarcomere length, in turn affecting the volume at which the ventricle begins to generate pressure. Variations in $V_W^{LV}$ alter both the size and shape of the PV loop. Increases in $V_W^{LV}$ lead to larger stroke volumes and changes in end-systolic and end-diastolic volumes, due to the increased contractile force from the additional myocardial mass. Similar to increased contractility $T_{a0}$, the systolic part of the PV-loop increases, as higher pressures are generated. Lastly, as demonstrated by the sensitivity analysis, variations in the ventricular and atrial activation times $t_{max}$ (which we set equal for the right and left sides) have minimal impact on the simulation outcomes. This finding supports our decision to assume equal $t_{max}$ values for corresponding atrial and ventricular chambers. Similarly, the shape factor $\beta$ of the left and right ventricular activation curves also exhibit minimal influence on the outputs.

In the TVE model, the right ventricular elastances $E_a^{RV}$ and $E_b^{RV}$, which are associated with systolic and diastolic ventricular function, have, compared to the other model input, the most significant influence on the selected outputs, including on the left side. This is also comparable to the impact of $T_{a0}$ and $C_p$ in the
SF model. Interestingly, the unstressed volumes $V_0$ in this model have a more pronounced effect on the outputs (compared to the SF model), particularly on their respective chambers. This can be attributed to the broader parameter ranges defined (Appendix Table 13), which even include negative (virtual) values. For the elastances of the LV and the atria, no significant impacts are observed other than for their respective chambers' volumes and pressures. Yet, the impact of the passive elastance of the LV on the LA pressures is interesting. Lastly, similar to the SF model, the ventricular and atrial activation times, as well as the ventricular shape factors of the activation curves, have a negligible impact on the model outputs.

Impact of LV specific parameters $E_a^{LV}$, $E_b^{LV}$ and $V_0^{LV}$ variations on PV-loops are illustrated in Fig. 10 panel B). Changes in $E_a^{LV}$ lead to significant changes in contractile function: increasing $E_a^{LV}$ results in a higher maximal pressure, an increased stroke volume, and a leftward shift of the PV loop, reflecting reduced volumes. The passive elastance parameter $E_b^{LV}$ primarily influences diastolic filling properties, where increased values correspond to higher end-diastolic pressures. The unstressed volume parameter $V_0^{LV}$ has a distinct effect, predominantly shifting the PV loops to the right. As $V_0^{LV}$ increases, maximal pressures decrease while stroke volume remains largely unaffected. However, at extreme values, this parameter can lead to non-physiological states where chamber volumes become negative, reinforcing the importance of parameter ranges in ensuring physiologically relevant simulations.

An attempt to estimate the TVE parameters $E_a^{LV}$, $E_b^{LV}$, and $V_0^{LV}$ with pressure and volume time series resulting from SF model simulation and the general form of the elastance function (Eq. (1) and Eq. (3)), is provided in the Appendix. This analysis highlights the challenges of calibrating the TVE model.
Time Varying Elastance Model

<figure><img src="image_16.png" /></figure>

Single Fibre Model

<figure><img src="image_17.png" /><figcaption>**Figure 9:** Sensitivity maps for the two models highlighting the influence of key parameters on cardiac function outputs.</figcaption></figure>
<figure><img src="image_18.png" /></figure>

The Journal of
Physiology

**Figure 10:** Pressure-volume loops illustrating the impact of parameter variations on left ventricular dynamics. **A)** SF Model: **Upper panel:** Variations in the active stress parameter $T_{a0}$. **Middle panels:** Variations in the passive stress parameters $T_{p0}$ (left) and $C_p$ (right). **Lower panels:** Variations in the volume-related parameters $V_0^{LV}$ (left) and $V_W^{LV}$ (right). **B)** TVE Model: **Upper panel:** Variations in the amplitude of elastance $E_a^{LV}$. **Middle panel:** Variations in the baseline elastance $E_b^{LV}$. **Lower panel:** Variations in the unstressed volume $V_0^{LV}$. Each plot includes variations corresponding to the minimum, mean, and maximum values from the sensitivity analysis range given in Appendix Tables 13 and 14, as well as the PV loop obtained based on patient-specific optimized parameters (**opt**).
### 3.3. Identifiability

<figure><img src="image_19.png" /><figcaption>**Figure 11:** Likelihood profiles of the cardiac parameters for the TVE model. Note that, for visualization purposes, the error function $\epsilon$ is scaled by the number of measurements included for better comparison. The PLA starts with the augmented data-set, iteratively fixing input parameters until all remaining parameters are identifiable. After, all augmented data are removed, and the process repeats. Finally, less feasible data (atrial volumes and pressures) are removed to mimic clinical routine.</figcaption></figure>

Fig. 11 shows the likelihood profiles for all TVE model parameters, obtained through several iterations of fixing unidentifiable parameters and gradually removing measurements in order of lowest to highest clinical feasibility. For direct comparison across scenarios with varying numbers of measurements, the error metric $\epsilon$ has been scaled by the number of data points in each case.

For the first step, the full augmented data-set is considered (Table 10). First, measurable parameters $t_{max}^{A/V}$ are fixed to their measured values, and non-sensitive parameters $\beta^V$ are fixed to reasonable values (blue). In this case, none of the input parameters appear fully identifiable within their defined input ranges, although all unloaded volumes $V_0$ show a distinct boundary on one side. This is to be expected, since it is
impossible to fit the full TVE on a single beat without time-series data (see Fig. 2). Furthermore, ventricular elastances have a more distinct convex curve than atrial elastances, meaning ventricular elastances have a higher influence on the quality of fit with respect to the measurements, which is to be expected. Next, $V_0$ was fixed for each chamber, since $V_0$ is often considered to be 0 or low in most applications of the TVE model (orange). This led to an identifiable set of parameters for the ventricles, although some parameters tend towards the boundaries of the input ranges. Atrial parameters show little improvement, likely due to their relatively low contribution to the error function. Subsequently, we removed data augmentations ($EDP_V$, $EDP_A$, $EDV_A$, $\max P_A$) (green). This led to the loss of identifiability of $E_a$, $E_b$ for both atria, with only minor impact on ventricular elastances. Based on the sensitivity indices, active elastances $E_a^A$ were fixed for both atria (red). This only led to minor improvements of the atrial parameters, indicating that atrial measurements had a relatively small contribution to the total error function, or that changes in atrial elastances showed little improvement with respect to the measured values. Regardless, these results show that atrial elastances are hard to uniquely identify, even relevant clinical measurements such as pressures and volumes are included in the dataset. Next, baseline atrial elastances $E_b^A$ were also fixed. Removing atrial data (brown, purple) showed to have little effect on the active ventricular elastances $E_a^V$, but decreased identifiability of baseline ventricular elastances $E_b$, indicating measurements of atrial function are necessary for characterization of baseline ventricular elastances in the TVE model.

A concise overview of each step in this iterative procedure, including which parameters and data were fixed or removed, is provided in Table 11.

<table><caption>**Table 11:** Summary of TVE parameter identifiability under successive data/parameter modifications. Rows (1)–(7) reflect the iterative steps described in the text: (1) Augmented dataset; (2) Fixed $V_0$; (3) Remove augmented data; (4) Fixed $E_a^A$; (5) Fixed $E_b^A$; (6) Remove atrial ESV; (7) Remove mean atrial pressure. Symbols: + = fully identifiable, × = non-identifiable, - = fixed, LB = lower bound only, UB = upper bound only.</caption><thead><tr><th>Sc.</th><th>Data</th><th>E<sub>a</sub><sup>LA</sup></th><th>E<sub>b</sub><sup>LA</sup></th><th>V<sub>0</sub><sup>LA</sup></th><th>E<sub>a</sub><sup>RA</sup></th><th>E<sub>b</sub><sup>RA</sup></th><th>V<sub>0</sub><sup>RA</sup></th><th>E<sub>a</sub><sup>LV</sup></th><th>E<sub>b</sub><sup>LV</sup></th><th>V<sub>0</sub><sup>LV</sup></th><th>E<sub>a</sub><sup>RV</sup></th><th>E<sub>b</sub><sup>RV</sup></th><th>V<sub>0</sub><sup>RV</sup></th></tr></thead><tbody><tr><td>1</td><td>Z<sub>Data</sub> ∪ Z<sub>Y<sub>SF</sub></sub></td><td>×</td><td>×</td><td>UB</td><td>×</td><td>LB</td><td>LB</td><td>LB</td><td>UB</td><td>UB</td><td>LB</td><td>+</td><td>UB</td></tr><tr><td>2</td><td>Z<sub>Data</sub> ∪ Z<sub>Y<sub>SF</sub></sub></td><td>LB</td><td>LB</td><td>-</td><td>×</td><td>LB</td><td>-</td><td>+</td><td>+</td><td>-</td><td>+</td><td>+</td><td>-</td></tr><tr><td>3</td><td>Z<sub>Data</sub></td><td>×</td><td>×</td><td>-</td><td>×</td><td>×</td><td>-</td><td>+</td><td>+</td><td>-</td><td>+</td><td>+</td><td>-</td></tr><tr><td>4</td><td>Z<sub>Data</sub></td><td>-</td><td>LB</td><td>-</td><td>-</td><td>×</td><td>-</td><td>+</td><td>+</td><td>-</td><td>+</td><td>+</td><td>-</td></tr><tr><td>5</td><td>Z<sub>Data</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>-</td><td>+</td><td>+</td><td>-</td></tr><tr><td>6</td><td>Z<sub>Data</sub> \ ESV<sub>A</sub></td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>-</td><td>+</td><td>+</td><td>-</td></tr><tr><td>7</td><td>Z<sub>Data</sub> \ {ESV<sub>A</sub>, P&#773;<sub>A</sub>}</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>-</td><td>+</td><td>+</td><td>-</td></tr></tbody></table>
<figure><img src="image_20.png" /><figcaption>**Figure 12:** Likelihood profiles of the cardiac parameters for the SF model. Note that, for visualization purposes, the error function $\epsilon$ is scaled by the number of measurements included for better comparison. The PLA starts with the augmented data-set, iteratively fixing input parameters until all remaining parameters are identifiable. After, all augmented data are removed, and the process repeats. Finally, less feasible data (atrial volumes and pressures) are removed to mimic clinical routine.</figcaption></figure>

Fig. 12 shows the likelihood profiles of the SF model parameters. In the first step (blue), the full augmented data-set is considered, while non-sensitive parameters $t_{max}^{A/V}$ and $\beta^V$ are fixed. All curves are mildly convex, indicating poor identifiability, likely due to the interaction between $C_p$ and $T_{p0}$, which both describe the passive stress-strain behaviour. As such, $T_{p0}$ was fixed, due to its low sensitivity and identifiability, which significantly improved the identifiability of the remaining parameters (orange). Subsequent removal of the augmented data (green) did not dramatically alter the identifiability, due to the shared parameters $C_p$ and $T_{a0}$, which hold for all 4 chambers. These results show that, whereas in the TVE atrial behaviour was poorly identifiable, this does not hold true for the SF atria. However, subsequent removal of atrial ESV (red) and mean atrial pressure (purple) significantly reduced the identifiability of
unloaded volumes of the atria $V_0^A$, meaning these data are required for successfully identifying atrial behaviour. Removal of atrial data had a relatively low effect on the identifiability of the unloaded volumes of the ventricles $V_0^V$, contractility $T_{a0}$ and non-linear stiffness $C_p$, indicating that the ventricular parameters in the SF model are more robust to the lack of pre-load data.

A concise overview of each step in this iterative procedure, including which parameters and data were fixed or removed, is provided in Table 12.

<table><caption>**Table 12:** Summary of SF model parameter identifiability under successive data/parameter modifications. Rows (1)–(5) reflect the iterative steps described in the text: (1) Augmented dataset; (2) Fixed $T_{p0}$; (3) Remove augmented data; (4) Remove atrial ESV; (5) Remove mean atrial pressure. Symbols: + = fully identifiable, × = non-identifiable, = fixed, LB = lower bound only, UB = upper bound only.</caption><thead><tr><th>Sc.</th><th>Data</th><th>T<sub>a0</sub></th><th>T<sub>p0</sub></th><th>c<sub>p</sub></th><th>V<sub>0</sub><sup>LA</sup></th><th>V<sub>0</sub><sup>RA</sup></th><th>V<sub>0</sub><sup>LV</sup></th><th>V<sub>0</sub><sup>RV</sup></th></tr></thead><tbody><tr><td>1</td><td>Z<sub>Data</sub> ∪ Z<sub>Y<sub>SF</sub></sub></td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>2</td><td>Z<sub>Data</sub> ∪ Z<sub>Y<sub>SF</sub></sub></td><td>+</td><td>-</td><td>UB</td><td>+</td><td>+</td><td>LB</td><td>LB</td></tr><tr><td>3</td><td>Z<sub>Data</sub></td><td>+</td><td>-</td><td>UB</td><td>+</td><td>+</td><td>LB</td><td>LB</td></tr><tr><td>4</td><td>Z<sub>Data</sub> \ ESV<sub>A</sub></td><td>+</td><td>-</td><td>UB</td><td>+</td><td>×</td><td>LB</td><td>LB</td></tr><tr><td>5</td><td>Z<sub>Data</sub> \ {ESV<sub>A</sub>, P&#773;<sub>A</sub>}</td><td>+</td><td>-</td><td>UB</td><td>×</td><td>×</td><td>LB</td><td>LB</td></tr></tbody></table>

## 4. Discussion

In this study, we compared two cardiac modelling approaches, the TVE model and the SF model, within a full circulatory system simulation. Our goal was to assess the advantages and limitations of each model in capturing the physiological behaviour of the heart. We investigated how variations in key SF parameters affect the ventricular PV loops and attempted to relate these findings to the parameters of the TVE model.

Our findings indicate that the SF model, despite considering a larger initial set of parameters, was easier to personalize and fit to clinical data than the TVE model. The SF model, due to its physiological derivation and therefore the interpretability of its parameters, allows the user to better differentiate between different physiological effects that drive similar behaviour. Furthermore, it distinguishes between tissue characteristics (stress-strain curves) and geometrical properties (myocardial mass, unloaded volume), which the TVE cannot. This physiological representation ultimately allows the user to make more substantiated assumptions on which parameters should or should not be included in the personalization of the model based on known physiology or other literature. It also allows for additional sources of data to be readily applicable, such as the measurement of myocardial mass from imaging data. In contrast, the TVE model, despite its simpler mathematical form and ease of use, proved more challenging to optimize. Its lack of physiological interpretability meant that no assumptions could be made to fix any parameters before analysis, and the linear assumptions inherent in the TVE model limited its ability to fit the nonlinear PV loop shapes produced by the SF model.

While this speaks in favour of SF model, the non-linear diastolic pressure-volume behaviour is well known in clinical literature. Hence, an extension to the non-linear passive pressure-volume behaviour of
cardiac chambers would improve TVE model accuracy, such as the exponential function implemented by (Artrip, Oz, and Burkhoff 2001; Burkhoff, Mirsky, and Suga 2005). While this evidently introduces similar issues as the assessment of $E_a$ in the TVE model, since it requires multiple PV-loops at different loads or time-series data for accurate parameter identification, these data are clinically measurable.

One limitation in this study, is the absence of the characteristic figure-eight-shaped PV loops for the atria. While our analysis focused on a patient with PAH—a condition where ventricular dynamics are of primary clinical interest—this limitation did not compromise our objectives. However, in scenarios where atrial function plays a central role, this atrial representation may be insufficient. Model adaptations that reproduce the figure-eight atrial PV-loop patterns have been proposed by (Pant et al. 2018; Regazzoni, et al. 2022).

One of the significant challenges we encountered was the lack of comprehensive parameter ranges in the medical literature for these models, mainly for the TVE model. The choice of some fixed parameters and the determination of appropriate ranges for others were not straightforward, and initially lead to parameter output ranges that were mostly outside of physiological behaviour. These issues could be attributed to the pulmonary hypertension cohort, which exhibits significantly different cardiac behaviour than can be achieved with the initially reviewed parameter ranges. This highlights the need for more patient-specific studies with sufficiently extensive data. Such studies would enhance the identifiability of model parameters and improve the personalization of cardiac simulations. After the initial challenges regarding parameter ranges were overcome, the sensitivity analysis revealed that right ventricular parameters not only influence the right side and pulmonary artery, but also affect the left heart, emphasizing the importance of a full circulatory model. Moreover, these observations were made despite not including any explicit left-right interaction between the ventricles or the atria via their respective septum. This cross-impact, particularly in the context of right-side disease, highlights how even localized impairments can have system-wide effects and future studies should ideally always characterize right and left heart, regardless of disease aetiology. The sensitivity analysis also showed the same parameters consistently are the most sensitive for multiple outputs. While this is interesting not only from a mathematical and physiological point of view, it also implies that the user can leverage this information to determine a pre-selection of which data to acquire, as to best personalize model parameters he/she is interested in. For instance, if the user would have been interested in the shape parameter $\beta$, it is evident that it can not be determined from the provided possible dataset, as no component is sensitive to $\beta$.

However, for consistent and reliable model predictions, finding a unique set of input parameters that describes the patient data is necessary. As evidenced from the identifiability analyses, the set of sensitive parameters does not equal the set of identifiable parameters, as there often exist multiple combinations of sensitive input parameters that lead to similar model outputs. It should be noted that in our identifiability analyses, the final set of clinical data is still quite extensive compared to standard clinical practice. Even though all measurements are clinically measurable, often not all data are available for all chambers for a single patient. Furthermore, the assumed measurement noise in the optimization is mostly based on the Signal to Noise Ratio of the measuring modality. If beat-to-beat variability is taken into account (Meiburg et al. 2021), this would lead to less convex likelihood profiles, and therefore wider confidence intervals for each parameter of interest. We assumed myocardial properties such as stiffness and contractility to be constant across all cardiac chambers. Instead, hemodynamic function of the individual chambers is only characterized by their myocardial mass and unloaded volume. From a physiological viewpoint, this
assumption means that one cannot distinguish between two pieces of tissue excised from the heart in two
different locations. This has the benefit of significantly improving the identifiability of chamber specific
properties (e.g., V₀). This approach led to a small set of identifiable and measurable parameters for the SF
model (V₀, V_W, T_a0, C_p) with only clinically feasible data (V_W, EDV_V, ESV_V, max P_Aorta, min P_Aorta,
P̅_Aorta, max P_PulArt, min P_PulArt, P̅_PulArt). In contrast, all parameters of the TVE model were
considered unique, since they do not distinguish between tissue and geometrical parameters. The TVE
model was not able to utilize geometrical information, and thus required more measurements
(EDV_V, ESV_V, max P_Aorta, min P_Aorta, P̅_Aorta, max P_PulArt, min P_PulArt, P̅_PulArt, P_A, V_A) to find a unique
set of E_a^V, E_b^V. We would like to reiterate that reducing the input parameter set does not negatively
influence the predictive capabilities of the models in question. In fact, finding these reduced sets of
identifiable inputs parameters will lead to a more robust and consistent optimization of this model patient
data, i.e., running the optimization multiple times will yield the same set of optimized parameters and will
therefore predict the same outcome, making patient specific model-based predictions more trustworthy. In
this work non-identifiable parameters were set to a fixed, physiological value. Alternative strategies exist,
such as pseudo-parameterization (e.g., linking multiple input parameters to a set of scaling parameters)
(Golse et al. 2021). However, this choice should be based on the clinical question at hand, and could be
informed by applying a combination of physiological knowledge, sensitivity and identifiability analyses.

5. Conclusion

This study provides, to our knowledge for the first time, a direct comparison between the TVE and SF
cardiac models in the context of PAH. Using an iterative pipeline that integrates calibration, parameter-
range refinement, sensitivity analysis, and identifiability assessment, we establish a robust framework for
model-based medical data collection protocol.

Our findings suggest that while the TVE model benefits from simplicity, its limited physiological
interpretation necessitates additional clinical data to adequately personalize parameters. In contrast, the SF
model's richer physiological representation better aligns with clinical data, making it more suitable for
personalized cardiac simulations.

Overall, this work highlights the importance of selecting an appropriate cardiac model based on the
specific disease context and available data, as well as the need for comprehensive data collection,
sensitivity and identifiably analysis, and model validation. Addressing these considerations is pivotal for
advancing personalized medicine.

6. Data Availability

Data necessary to reproduce the results are reported in the main text and in Appendix.
# 7. Authors contributions

All authors reviewed and approved the final manuscript. Conceptualization of the study was led by RM and IVC. Data curation was conducted by MH and PV, who organized and maintained the dataset. Formal analysis was performed collaboratively by MH, PV, and RM, ensuring robust data interpretation. Funding acquisition was managed by IVC. Investigation involved contributions from MH, PV, and RM. MH and PV took the lead in drafting the manuscript, with input and editing from RM and IVC. RM and IVC provided revisions. All authors agree to be accountable for the work, ensuring its accuracy and integrity.

# 8. Conflict of interest

The authors declare no conflict of interest.

# 9. Funding

We acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Program (Grant Agreement No. 864313 – MoDeLLiver). This project has also received funding from the European Union within the Horizon Europe program under the grant agreement No. 101136299 and from BPIFrance through the MEDITWIN project sustained by France 2030.

# 10. Ethics

This article did not require ethical approval from a human subject or animal welfare committee.
# APPENDIX

## A. Models' inputs and ranges

### A.1. Single Fibre Model

<table><caption>**Table 13:** Parameters of the Single Fibre Model.</caption><thead><tr><td>No.</td><td>Variable</td><td>Full Name</td><td>Range</td><td>Reference</td></tr></thead><tbody><tr><td>1</td><td>V<sub>WLA</sub></td><td>Left Atrial Wall Volume</td><td>[6, 14] mL</td><td>(Flink et al. 2020)</td></tr><tr><td>2</td><td>V<sub>WLV</sub></td><td>Left Ventricular Wall Volume</td><td>[38, 52] mL</td><td>(Kuehne et al. 2004; Buechel et al. 2009)</td></tr><tr><td>3</td><td>V<sub>WRA</sub></td><td>Right Atrial Wall Volume</td><td>[7, 15] mL</td><td>(Querejeta Roca et al. 2015)</td></tr><tr><td>4</td><td>V<sub>WRV</sub></td><td>Right Ventricular Wall Volume</td><td>[52, 80] mL</td><td>(Kuehne et al. 2004; Haarman et al. 2023)</td></tr><tr><td>5</td><td>V<sub>LAo</sub></td><td>Left Atrial Unstressed Volume</td><td>[10, 30] mL</td><td>(Pant et al. 2022)</td></tr><tr><td>6</td><td>V<sub>LVo</sub></td><td>Left Ventricular Unstressed Volume</td><td>[18.5, 33.2] mL</td><td>(Pant et al. 2022)</td></tr><tr><td>7</td><td>V<sub>RAo</sub></td><td>Right Atrial Unstressed Volume</td><td>[20.25, 60] mL</td><td>(Pant et al. 2022)</td></tr><tr><td>8</td><td>V<sub>RVo</sub></td><td>Right Ventricular Unstressed Volume</td><td>[25, 50.4] mL</td><td>(Pant et al. 2022)</td></tr><tr><td>9</td><td>T<sub>a</sub></td><td>Atrial Activation Duration Time</td><td>[0.1, 0.16] s</td><td>(Merckx et al. 2005; Müller et al. 2021)</td></tr><tr><td>10</td><td>T<sub>v</sub></td><td>Ventricular Activation Duration Time</td><td>[0.35, 0.44] s</td><td>(Boudoulas 1990; Hsieh et al. 1986)</td></tr><tr><td>11</td><td>β<sub>LV</sub></td><td>Left Ventricular Activation Function Shape</td><td>[0.8925, 1.2075]</td><td>(Pant et al. 2022)</td></tr><tr><td>12</td><td>β<sub>RV</sub></td><td>Right Ventricular Activation Function Shape</td><td>[0.55, 0.805]</td><td>(Pant et al. 2022)</td></tr><tr><td>13</td><td>T<sub>ao</sub></td><td>Maximum Active Sarcomere Stress</td><td>[50, 85] kPa</td><td>(Pant et al. 2022, 2016; Bovendeerd et al. 2006)</td></tr><tr><td>14</td><td>T<sub>po</sub></td><td>Sarcomere Material Constant</td><td>[0.6, 2.5] kPa</td><td>(Pant et al. 2022; Palit et al. 2018)</td></tr><tr><td>15</td><td>C<sub>p</sub></td><td>Material Constant</td><td>[8, 19]</td><td>(Pant et al. 2022; Palit et al. 2018)</td></tr></tbody></table>
## A.2 Time Varying Elastance Model

<table><caption>**Table 14:** Parameters of the Time-Varying Elastance Model.</caption><thead><tr><td>No.</td><td>Variable</td><td>Full Name</td><td>Range</td><td>Reference</td></tr></thead><tbody><tr><td>1</td><td>E<sub>a,LV</sub></td><td>Left Ventricular Active Elastance</td><td>[1, 1.88] mmHg·mL<sup>-1</sup></td><td>(Burkhoff, Mirsky, and Suga 2005; Blaudszun and Morel 2011; TS 2002)</td></tr><tr><td>2</td><td>E<sub>b,LV</sub></td><td>Left Ventricular Passive Elastance</td><td>[0.045, 0.15] mmHg·mL<sup>-1</sup></td><td>(Burkhoff, Mirsky, and Suga 2005; Blaudszun and Morel 2011; TS 2002)</td></tr><tr><td>3</td><td>V<sub>0,LV</sub></td><td>Left Ventricular Unloaded Volume</td><td>[-50, 50] mL</td><td>(Dell'Italia and Walsh 1988; Pagoulatou et al. 2021)</td></tr><tr><td>4</td><td>E<sub>a,RV</sub></td><td>Right Ventricular Active Elastance</td><td>[0.66, 3.30] mmHg·mL<sup>-1</sup></td><td>(Dell'Italia and Walsh 1988; Kuehne et al. 2004)</td></tr><tr><td>5</td><td>E<sub>b,RV</sub></td><td>Right Ventricular Passive Elastance</td><td>[0.01, 0.22] mmHg·mL<sup>-1</sup></td><td>(Kuehne et al. 2004)</td></tr><tr><td>6</td><td>V<sub>0,RV</sub></td><td>Right Ventricular Unloaded Volume</td><td>[-50, 50] mL</td><td>(Dell'Italia and Walsh 1988; Pagoulatou et al. 2021)</td></tr><tr><td>7</td><td>E<sub>a,LA</sub></td><td>Left Atrial Active Elastance</td><td>[0.015, 0.375] mmHg·mL<sup>-1</sup></td><td>(Hoit et al. 1994; Stefanadis et al. 1998)</td></tr><tr><td>8</td><td>E<sub>b,LA</sub></td><td>Left Atrial Passive Elastance</td><td>[0.03, 0.11] mmHg·mL<sup>-1</sup></td><td>(Hoit et al. 1994; Stefanadis et al. 1998)</td></tr><tr><td>9</td><td>V<sub>0,LA</sub></td><td>Left Atrial Unloaded Volume</td><td>[-50, 50] mL</td><td>(Hoit et al. 1994; Stefanadis et al. 1998)</td></tr><tr><td>10</td><td>E<sub>a,RA</sub></td><td>Right Atrial Active Elastance</td><td>[0.55, 1.125] mmHg·mL<sup>-1</sup></td><td>(Hoit et al. 1994; Stefanadis et al. 1998)</td></tr><tr><td>11</td><td>E<sub>b,RA</sub></td><td>Right Atrial Passive Elastance</td><td>[0.03, 0.22] mmHg·mL<sup>-1</sup></td><td>(Hoit et al. 1994; Stefanadis et al. 1998)</td></tr><tr><td>12</td><td>V<sub>0,RA</sub></td><td>Right Atrial Unloaded Volume</td><td>[-50, 50] mL</td><td>(Hoit et al. 1994; Stefanadis et al. 1998)</td></tr><tr><td>13</td><td>T<sub>v</sub></td><td>Ventricular Activation Duration Time</td><td>[0.35, 0.44] s</td><td>(Boudoulas 1990; Hsieh et al. 1986)</td></tr><tr><td>14</td><td>T<sub>a</sub></td><td>Atrial Activation Duration Time</td><td>[0.1, 0.16] s</td><td>(Roithinger et al. 2001; Müller et al. 2021)</td></tr><tr><td>15</td><td>β<sub>RV</sub></td><td>Right Ventricular Activation Function Shape</td><td>[0.55, 0.805]</td><td>(Pant et al. 2022)</td></tr><tr><td>16</td><td>β<sub>LV</sub></td><td>Left Ventricular Activation Function Shape</td><td>[0.8925, 1.2075]</td><td>(Pant et al. 2022)</td></tr></tbody></table>
## B. Numerical resolution of the mathematical models

From a mathematical standpoint, a LPM, denoted by $L$, can be described by a system of differential-algebraic equations (DAEs) (Formaggia, Quarteroni, and Veneziani 2010) as follows for $t \ge 0$:

$$L(y, \bar{y}, \mathbf{X}, t) = \begin{cases} \frac{dy}{dt} = b(y, \bar{y}, \mathbf{X}, t) \\ G(y, \bar{y}, \mathbf{X}, t) = 0. \end{cases} \qquad (23)$$

In this formulation, $y$ represents the vector of state variables, and $\bar{y}$ includes the remaining network variables, so the complete set of variables for the model $L$ is denoted by $\mathbf{Y} = \{y, \bar{y}\}$. The vector $\mathbf{X}$ represents the input parameters. The function $b(y, \bar{y}, t)$ defines the right-hand side of the differential equations, representing the dynamics of the state variables over time. The term $G(y, \bar{y}, t) = 0$ corresponds to the algebraic equations that capture instantaneous constraints within the system.

To solve the DAEs described by Eq. (23), the initial conditions are specified as $y(t_0) = y_0$. Numerical methods for solving DAEs typically integrate strategies to handle both the differential and algebraic components effectively.

In this work, the DAEs are solved with the Diffeqpy library in Python, which leverages the DifferentialEquations.jl suite from the Julia programming language for its core numerical routines (Rackauckas and Nie 2017). The solver employed is based on the backward differentiation formulas method to ensure stability, especially for stiff systems. The relative tolerance (reltol) and absolute tolerance (abstol) were set to $1 \times 10^{-5}$ for both in the SF model, and to $1 \times 10^{-5}$ and $1 \times 10^{-6}$, respectively, in the TVE model, to ensure the accuracy and stability of the numerical solutions.

Simulations were performed iteratively until cycle convergence was achieved. The convergence criterion was based on the relative Euclidean norm ($\ell^2$ norm) of the difference between the maximum values of the model variables in consecutive cycles. Mathematically, the convergence error $\varepsilon_{cycle}$ was calculated as:

$$\varepsilon_{cycle} = \left\| \frac{\mathbf{Y}_{\max}^{(n)} - \mathbf{Y}_{\max}^{(n-1)}}{\mathbf{Y}_{\max}^{(n-1)}} \right\|_2, \qquad (24)$$

where $\mathbf{Y}_{\max}^{(n)} = \max_{t \in T^{(n)}} \mathbf{Y}^{(n)}(t)$ denotes the vector containing the maximum values of the variables over time during the $n$-th cardiac cycle; $T^{(n)}$ is the time interval corresponding to the $n$-th cardiac cycle; and $\| \cdot \|_2$ represents the Euclidean norm. The simulations were considered to have converged when $\varepsilon_{cycle}$ dropped below the convergence threshold of $1 \times 10^{-2}$. This iterative approach ensures that the system reaches a periodic steady state.

Within this simulation framework, the SF model required approximately three times more computational time than the TVE model to converge to a stationary periodic state. This increased computational demand arises due to the non-linearities of the SF model and the adaptive time-stepping scheme employed by the solver to handle its DAEs. Additionally, constructing the surrogate model for the SF model using PCE required more simulations than for the TVE model, reflecting the SF model's greater complexity and non-linear behaviour.
To initialize the simulations, the system was set to start at the ventricle end-diastolic moment with initial conditions estimated based on the study by (Pant et al. 2022). Some state variables were initialized to ensure consistency with the model based on the flow-pressure relationships in Fig. 5. A nonlinear solver was employed to determine $P_{LA,0}$, $P_{RA,0}$, $Q_{pul,0}^v$, and $Q_{sys,0}^v$. Specifically, the fsolve function from the spipy.optimize package was utilized for these calculations. Moreover $V_{ra,0}$ and $P_{sys,0}^v$ where obtained from an inverse problem perform to fit the patient data for both models using CMAes algorithm (Hansen, Akimoto, and Baudis 2019). Table 15 summarizes initial conditions values.

<table><caption>**Table 15:** Initial Conditions for the Simulation.</caption><thead><tr><td>Variable</td><td>Symbol</td><td>Initial Value</td><td>Source / Rationale</td></tr></thead><tbody><tr><td>Left Atrium Volume</td><td>V<sub>la,0</sub></td><td>21.1 ml</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Left Ventricle Volume</td><td>V<sub>lv,0</sub></td><td>68.0 ml</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Right Atrium Volume (SF model)</td><td>V<sub>ra,0</sub> (SF)</td><td>87.33 ml</td><td>Optimization</td></tr><tr><td>Right Atrium Volume (TVE model)</td><td>V<sub>ra,0</sub> (TVE)</td><td>35.51 ml</td><td>Optimization</td></tr><tr><td>Right Ventricle Volume</td><td>V<sub>rv,0</sub></td><td>90.0 ml</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Aortic Valve Flow</td><td>Q<sub>AoV,0</sub></td><td>0 ml/s</td><td>End-Diastole in ventricles</td></tr><tr><td>Mitral Valve Flow</td><td>Q<sub>MiV,0</sub></td><td>0 ml/s</td><td>End-Diastole in ventricles</td></tr><tr><td>Pulmonary Valve Flow</td><td>Q<sub>PuV,0</sub></td><td>0 ml/s</td><td>End-Diastole in ventricles</td></tr><tr><td>Tricuspid Valve Flow</td><td>Q<sub>TrV,0</sub></td><td>0 ml/s</td><td>End-Diastole in ventricles</td></tr><tr><td>Aortic Valve Opening</td><td>ξ<sub>AoV,0</sub></td><td>0</td><td>End-Diastole in ventricles</td></tr><tr><td>Mitral Valve Opening</td><td>ξ<sub>MiV,0</sub></td><td>0</td><td>End-Diastole in ventricles</td></tr><tr><td>Pulmonary Valve Opening</td><td>ξ<sub>PuV,0</sub></td><td>0</td><td>End-Diastole in ventricles</td></tr><tr><td>Tricuspid Valve Opening</td><td>ξ<sub>TrV,0</sub></td><td>0</td><td>End-Diastole in ventricles</td></tr><tr><td>Aortic Pressure</td><td>P<sub>Aorta,0</sub></td><td>52 mmHg</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Systemic Arterial Pressure</td><td>P<sub>sys,0</sub><sup>a</sup></td><td>32 mmHg</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Systemic Venous Pressure (SF model)</td><td>P<sub>sys,0</sub><sup>v</sup> (SF)</td><td>14.08 mmHg</td><td>Optimization</td></tr><tr><td>Systemic Venous Pressure (TVE model)</td><td>P<sub>sys,0</sub><sup>v</sup> (TVE)</td><td>16.49 mmHg</td><td>Optimization</td></tr><tr><td>Pulmonary Artery Pressure</td><td>P<sub>PulArt,0</sub></td><td>68 mmHg</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Pulmonary Arterial Pressure</td><td>P<sub>pul,0</sub><sup>a</sup></td><td>48 mmHg</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Pulmonary Venous Pressure</td><td>P<sub>pul,0</sub><sup>v</sup></td><td>13.8 mmHg</td><td>Based on (Pant et al. 2022)</td></tr><tr><td>Left Atrium Pressure (initial guess)</td><td>P<sub>LA,g</sub></td><td>6 mmHg</td><td>Initial guess for fsolve</td></tr><tr><td>Right Atrium Pressure (initial guess)</td><td>P<sub>RA,g</sub></td><td>4 mmHg</td><td>Initial guess for fsolve</td></tr><tr><td>Pulmonary Venous Flow (initial guess)</td><td>Q<sub>pul,g</sub><sup>v</sup></td><td>64 ml/s</td><td>Initial guess for fsolve</td></tr><tr><td>Systemic Venous Flow (initial guess)</td><td>Q<sub>sys,g</sub><sup>v</sup></td><td>29 ml/s</td><td>Initial guess for fsolve</td></tr></tbody></table>
### C. Relation between SF model parameters, PV-loops and elastances

We attempted to estimate the time-varying elastance (TVE) model parameters $E_a^{LV}$, $E_b^{LV}$, and $V_0^{LV}$ to fit pressure $P_{SF}^{LV}(t)$ and volume $V_{SF}^{LV}(t)$ resulting from SF model simulations with $t \in [0,T]$. The optimisation was performed using the TVE model equations (Eq. (1) and Eq. (3)) with `scipy.optimization` and the `least_squares` function. However, the optimization was not successful in uniquely recovering the TVE model parameters. The primary limitation lies in the TVE model's assumption of linear elastance behavior, which is unable to fully capture the non-linear shapes of the PV loops generated by the SF model. These differences highlight the inherent discrepancies between the two models: the SF model's non-linear stress-strain relationships and the TVE model's linear framework. Figure 13 compares the PV loops obtained from the SF model simulations with those predicted by the fitted TVE model, where pressure is deduced through optimization and volume is taken directly from the SF simulation. Although the fits demonstrate trends—for instance, $E_a$ increases with $T_{a0}$ and $V_W$, while $E_b$ increases with $T_{p0}$ and $C_p$—the overall inability of the TVE model to replicate the SF model's PV loop shapes underscores the limitations of the approach.

<figure><img src="image_21.png" /><figcaption>**Figure 13:** PV-loops comparing SF model simulation and the fitted TVE model on pressure and volumes time series obtained with SF model simulation. PV-loops are obtained for different values of the active stress parameter $T_{a0}$.</figcaption></figure>

### D. Side-by-side comparison of TVE and SF models

**Table 16:** Comparison of TVE and SF models.
<table><thead><tr><th>Time-varying Elastance Model</th><th>Single-Fibre Model</th></tr></thead><tbody><tr><td><strong>Assumptions:</strong><br>1. <strong>Consistency of elastance curve shape:</strong> The elastance function E(t) maintains a consistent shape over time, reaching its peak near the end of the ejection phase, regardless of mechanical load, contractile state, or heart rate.<br>2. <strong>Independence of peak elastance and timing:</strong> At a constant heart rate, the peak elastance E<sub>max</sub> and time to peak T<sub>Emax</sub> are unaffected by changes in preload and afterload. Suga, Sagawa, and Shoukas (1973) observed that E<sub>max</sub> and T<sub>Emax</sub> remain constant despite variations in mechanical loading.<br>3. <strong>Modulation by biochemical and neural factors:</strong> The parameters E<sub>max</sub> and T<sub>Emax</sub> are primarily influenced by biochemical and neural controls, such as epinephrine levels and heart rate changes, rather than mechanical loading.</td><td><strong>Assumptions:</strong><br>1. <strong>Rotational symmetry and free shape variation:</strong> The cardiac chamber has rotational symmetry and is free from external forces, allowing its shape to vary to minimize potential energy.<br>2. <strong>Wall incompressibility and uniform fibre stress:</strong> The myocardial wall is incompressible, so its volume V<sub>w</sub> remains constant, and fibre stress σ<sub>f</sub> and strain are uniformly distributed across the wall.<br>3. <strong>Fibres in fluid-like matrix aligned with pressure isobars:</strong> Stress-bearing muscle fibres are embedded in a fluid-like non-fibrous matrix (hydrostatic pressure) and are aligned parallel to pressure isobars within the tissue.<br>4. <strong>Energy conservation and matrix stress determined by passive fibre stress:</strong> Mechanical work by myocardial fibres equals the pumping work of the chamber, and stress in the collagen matrix is determined solely by the passive stress of the fibres (Bovendeerd et al. 2006).<br>5. <strong>Negligible shear stress components:</strong> Shear stresses are neglected due to the low shear stiffness of myocardial tissue (Bovendeerd et al. 2006).</td></tr><tr><td><strong>Equations:</strong><br>Pressure - Volume relationship:<br>P(t) = E(t)(V(t) - V<sub>0</sub>) t ≥ 0<br>where V<sub>0</sub> is the unstressed volume of the cardiac chamber.<br>Chamber specific elastance for one cardiac cycle lasting T sec:<br>E(t<sub>m</sub>) = E<sub>b</sub> + E<sub>a</sub> × e(t<sub>m</sub>), where c ∈ V, A and t<sub>m</sub> = t mod T<br>where E<sub>b</sub> is the baseline and E<sub>a</sub> is the amplitude of elastance.</td><td><strong>Equations:</strong><br>Pressure-Volume relationship:<br>σ<sub>f</sub>/P(t) = 1 + 3V(t)/V<sub>w</sub> t ≥ 0<br>where V<sub>w</sub> is the wall volume.<br>Total fibre stress: σ<sub>f</sub> = σ<sub>a</sub> + σ<sub>p</sub></td></tr></tbody></table>
**Active contraction:**

* Ventricle:

$$e_v(t_m) = \left[ \frac{1}{2} \left( 1 - \cos \left( 2\pi \frac{t_m}{t_{\max}^V} \right) \right) \right]^{\beta^V} \cdot \mathbf{1}[0, t_{\max}^V](t_m)$$

where $t_{\max}$ is the duration of activation, $\beta^V$ controls the shape of the activation curve and the function $\mathbf{1}[0, t_{\max}^V](t_m)$ is the indicator function, which equals 1 if $t_m \in [0, t_{\max})$ and 0 otherwise.

* Atria:

$$e_A(t_m) = \left[ \frac{1}{2} \left( 1 - \cos \left( 2\pi \frac{(t_m + t_{\max}^A - t_1) \mod T}{t_{\max}^A} \right) \right) \right]^{\beta^A} \cdot \mathbf{1}[0, t_1) \cup [T - t_{\max}^A + t_1, T)(t_m),$$

with $t_1$ the overlap time between ventricular and atrial activation.

**Active fibre stress:** $\sigma_a = T_{a0}f(l_s)e_C(t_m)h(v_s)$,

with $T_{a0}$ the contractility parameter.

* Sarcomere length: $l_s = l_{s,0}\lambda$ with $\lambda = \left( \frac{1+3V/V_w}{1+3V_0/V_w} \right)^{1/3}$,

where $l_{s,0}$ is the sarcomere length at zero transmural pressure of the chamber and $V_0$ is the cavity volume at zero transmural pressure.

* Sarcomere shortening velocity $v_s = -\frac{l_s}{V_w} \left( 1 + \frac{3V}{V_w} \right)^{-1} \frac{dV}{dt}$

* Force-sarcomere length relationship:

$$f(l_s) = \begin{cases} 0 & \text{if } l_s \le l_{a0}, \\ \frac{(l_s - l_{a0})}{(l_{am} - l_{a0})} & \text{if } l_{a0} < l_s \le l_{am}, \\ \frac{1}{(l_{af} - l_s)} & \text{if } l_{am} \le l_s \le l_{ae}, \\ \frac{l_{af} - l_s}{(l_{af} - l_{ae})} & \text{if } l_s > l_{ae}, \end{cases}$$

where $l_{a0}$, $l_{am}$, $l_{af}$, and $l_{ae}$ are sarcomere material constants.

* viscous stress:

$$h(v_s) = \frac{1 - (v_s/v_0)}{1 + c_v(v_s/v_0)},$$

where $v_0$ is the initial sarcomere shortening velocity and $c_v$ governs the shape of the velocity-stress relationship.

* Time-dependent active contraction $e_C(t_m) \in V, A$ see TVE column

**Passive fibre stress:**

$$\sigma_p = T_{p0} \left( \exp \left[ c_p (\lambda - 1) \right] - 1 \right) \cdot \mathbf{1}_{[1, \infty)}(\lambda),$$

where $T_{p0}$ and $c_p$ are sarcomere material constants.
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