####### A Proposal to the Seed Money Fund

**Cardiopulmonary Resuscitation Using Optimal Control**

E. Jung,^1^ S. M. Lenhart,^1,2^ V. A. Protopopescu,^1,2^ and C. F.
Babbs^3^

^1^Computer Science and Mathematics Division

^2^Center for Engineering Science Advanced Research

^3^Basic Medical Sciences, Purdue University

######### Requested Budgets and Duration

####### FY 2002 Budget: \$125,000

Total Budget: \$125,000

# Abstract

We propose to develop a new Cardiopulmonary Resuscitation (CPR) strategy
to significantly improve artificially maintained circulation of blood
with respect to current practice. The new strategy will be realized by
applying optimal control (OC) theory to a validated circulation model.
The OC approach will maximize the blood flow through vital organs by
suitably adjusting the compression rate and/or duration in the validated
model. The proposed research will provide a proof-of-principle for the
enhanced efficiency of the controlled CPR technique over the traditional
ad-hoc techniques used today. This result will be used and extended in
more refined circulation models to be addressed in the follow-up funding
phases.

**Background**

Each year, more than 250,000 people die in the United States alone from
cardiac arrest.^1^ Physicians and rescue personnel have long used CPR as
the method of choice for maintaining a reasonable fraction of normal
blood flow through vital organs during cardiac arrest. Despite
widespread use of CPR, the long-term survival for patients with cardiac
arrest remains dismal. Indeed, for patients who arrest in-hospital, the
rate of long-term survival is 10-15%.^28^ For out-of-hospital arrest,
where CPR is the only procedure immediately available, the odds are much
grimmer, namely around 3%.^7^ The rationale underlying this proposal is
that modified methods of resuscitation that generate greater blood flow
during CPR can improve long term survival from cardiac arrest - a
proposition for which there is already clinical evidence.^27^

Although there has been considerable interest in studying the mechanism
of blood flow during CPR, the practiced techniques have changed
relatively little since the 1960's. Most existing computer simulations
of CPR use an electrical lumped parameter model of the
circulation,^3,4,5,10^ governed by a system of ordinary differential
equations (ODEs). Various mathematical models^3^ describe the standard
CPR technique and various alternative CPR techniques such as: (i)
interposed abdominal compression (IAC), (ii) active
compression-decompression, and (iii) Lifestick CPR. Since all these
models use time independent parameters (for instance, a fixed
compression rate 80 bpm is chosen during the entire CPR cycle), they did
not and could not account for time dependent control strategies, even
though the condition of the patient deteriorates rapidly with successive
minutes of unsuccessful resuscitation.^2^ Hence in real arrests, the
*patient-dependent factors* such as vascular resistance and compliance
are likely to change with time. Moreover, drugs such as epinephrine may
be given, which can have dramatic effects upon the circulatory system,
lasting for several minutes. Based on our extensive previous work on OC
of various lumped and/or distributed systems, we are convinced that this
situation can be significantly improved, by bringing the suitable tools
to the table.

**Objective**

We propose to apply OC techniques on a validated circulation model to
develop new effective methods for improving hemodynamic efficacy of CPR
techniques. The *rescuer-dependent factors* such as compression rate or
compression duration, that significantly affect the magnitude of blood
flow during CPR, will be considered time dependent control functions.
These functions will be determined to maximize the blood flow through
vital organs. In turn, this will result in more effective and successful
CPR techniques.

The goals and expected results of this seed money project are: (i) to
determine whether OC approaches can be used to significantly increase
the blood flow in the circulation system; (ii) to use the results to
apply for an National Institutes of Health (NIH) grant; and (iii) to
integrate our research within related programs, (e.g., the Digital
Human, previously known as the Virtual Human initiative).

**Proposed Approach**

We shall use an existing validated multicompartment lumped parameter
model to calculate pressures and flows during CPR. The chosen model has
been developed and validated against real data by Charles F. Babbs,
M.D., Ph.D.,^3^ who is a collaborator on this project. The advantages of
this circulation model over similar models are: (i) it has lowest
dimensionality; and (ii) provides excellent comparison with real data.

Babbs' model is a lumped parameter model for the circulatory system,
wherein the heart and blood vessels in various parts of the body are
represented as resistance-capacitive networks, similar to electric
circuits. Following the analogy with Ohm's law, pressures in the chest,
abdomen, and vascular compartments are interpreted as voltages, blood
flow as an electric current, and cardiac and venous valves as diodes -
electrical devices that permit current flow in only one direction. The
analog of the capacitance is the compliance C, defined as
![](media/image1.wmf) where ![](media/image2.wmf) is the incremental
change in pressure within a compartment as volume ![](media/image3.wmf)
is introduced. Figure 1 shows an example of the elements of the
electrical model. Three major sections, namely head, thorax, and
abdomen, are considered.

In the state system, the temporal variation of the pressure is
calculated for each compartment from a system of ODEs. These equations
are derived from the fundamental properties of the circulatory system,
namely the relation between pressure gradient and flow (Ohm's law) and
the definition of compliance. The state system based on Babbs's model^3^
may be represented as follows:

![](media/image4.jpeg){width="5.1097222222222225in"
height="4.117361111111111in"}

*[Extra-thoracic Components:]{.underline}*

![](media/image5.wmf)

*[Thoracic Components:]{.underline}*

![](media/image6.wmf)

The valves, indicated by green arrows in figure 1, lead to discontinuous
functions depending on the values of ![](media/image7.wmf)'s in our
system. In the circulation system above, the step function
θ(![](media/image8.wmf)) represents a diodic valve, that is,
θ(![](media/image9.wmf)) = 1 if ![](media/image10.wmf) and = 0
otherwise. The function θ(![](media/image11.wmf)) describes the action
of venous valves (Niemann's valves) at the thoracic inlet. The functions
θ(![](media/image12.wmf)) and θ(![](media/image13.wmf)) represent the
actions of the aortic and pulmonic valves, respectively.

The terms![](media/image14.wmf) and ![](media/image15.wmf) represent
driving intrathoracic and intraabdominal pressures applied to outer
surfaces of blood vessels in the abdomen and chest by virtue of external
compression of either the chest or the abdomen of the victim by the
rescuer. Although any arbitrary function or waveform can be used for
these external forces, for illustration we shall refer to periodic
(sinusoidal) functions. A full compression/relaxation cycle, of length
T, comprises a compression period, of length α \< T, and a relaxation
period, of length T - α. The compression rate is given
by![](media/image16.wmf). In the standard CPR, we take
![](media/image17.wmf) \> 0 during compression and
![](media/image18.wmf)(t) = 0 during relaxation; we note that
![](media/image19.wmf)(t) = 0 during the entire cycle. In IAC-CPR, the
chest compression is the same as in the standard CPR. The abdomen
compression starts with a relaxation period, followed by a compression
waveform ![](media/image20.wmf)(t) = -b sin (ωt /2α), which is positive
during abdominal compression and zero otherwise. The parameters a and b
are the maximum amplitude of the chest and abdomen compression,
respectively. The other parameters are summarized in Table 1.

  ------------------------- ------------------------- ------------------------ ------------------------
  Pressures, Compliances                              Resistances              

  Abdominal aorta           ![](media/image21.wmf),   Aorta                    ![](media/image23.wmf)
                            ![](media/image22.wmf)                             

  Inferior vena cava        ![](media/image24.wmf)    Subphrenic organs        ![](media/image25.wmf)

  Carotid artery            ![](media/image26.wmf)    Subphrenic vena cava     ![](media/image27.wmf)

  Jugular veins             ![](media/image28.wmf)    Carotid arteries         ![](media/image29.wmf)

  Thoracic aorta            ![](media/image30.wmf)    Head + arm resistance    ![](media/image31.wmf)

  Right heart &Superior     ![](media/image32.wmf)    Jugular veins            ![](media/image33.wmf)
  vena cava                                                                    

  Chest pump                ![](media/image34.wmf)    Pump input (tricuspid    ![](media/image35.wmf)
                                                      valve)                   

                                                      Pump output (aortic      ![](media/image36.wmf)
                                                      valve)                   

                                                      Coronary vessels         ![](media/image37.wmf)
  ------------------------- ------------------------- ------------------------ ------------------------

###### Table 1. Model Parameters

The main goal of CPR is to restart and maintain a reasonably high blood
flow in the vital organs until the heart resumes its normal functions or
definite therapy, such as electrical ventricular defibrillation, is
applied. In fact, one would like to *maximize* the blood flow, by
adjusting the compression parameters. This formulation is very congenial
to the formulation of OC problems. The OC of ODEs was essentially
developed by Pontryagin^25^ and has since been successfully extended and
applied to many physical, engineering, ecological, and biological
situations. Time dependent control strategies have been studied for HIV
immunology models^9,15^ and a two-strain tuberculosis epidemic model.
^13^

We shall choose the (time dependent) controls among the crucial
rescuer-dependent factors, namely the shape of the compression function
and the compression duration. The objective functional, J, to be
maximized, is an integral over time of the total blood flow, expressed
as a function of the state P, the chosen controls, and - possibly - of
time:

![](media/image38.wmf).

The necessary conditions that the optimal solution must satisfy are
derived from Pontryagin\'s Maximum Principle.^25^ Upon applying this
principle, the control problem is converted into the problem of
maximizing a Hamiltonian function, H, pointwise, with respect of the
chosen control functions. The Hamiltonian is canonically constructed as:

![](media/image39.wmf)

where f is the integrand in ![](media/image40.wmf)J,
![](media/image41.wmf) i = 1,...., n, are the right hand sides in the
state system (1) - (2), ![](media/image42.wmf), i = 1,...., n, are the
solutions of the adjoint system that is computed by
![](media/image43.wmf)with ![](media/image44.wmf)at the final time,
![](media/image45.wmf), and n is the dimensionality of the state system
(in our case n = 7). The optimal solution is obtained by solving the
optimality system, which is a 2n-dimensional ODE system consisting of
the n state equations and the n adjoint equations, coupled together by
the (explicit) characterization of the OC, as obtained from the
optimality condition ![](media/image46.wmf). Due to the initial
condition in the state equation and the final condition in the adjoint
equation, the optimality system has opposite time orientations, which
usually present computational challenges.

**Research Tasks & Deliverables**

[*Task 1. Control Synthesis* -- *New Aspects due to
Discontinuities*]{.underline}: Some elements of the state system have
discontinuities (depending on the states) caused by the motions of the
cardiac and venous valves. Since the standard theory covers only smooth
functions, the OC theory will be extended to this situation. The OC will
be determined for various strategies and several choices of features to
be controlled.

*[Task 2. Computational Modeling:]{.underline}* We will implement
numerically the optimal strategies for standard or one alternative CPR
technique. Various important rescuer-dependent factors such as frequency
and compression duration of the external driving intrathoracic or
intraabdomenal forces, will be chosen as the time dependent control
functions. Due to the opposite time orientations of the equations
involved, an alternate iterative method will be used for solving the
resulting optimality systems.

*[Task 3. Validation and Analysis of Results:]{.underline}* Babbs's CPR
model^3^ has been calibrated to an actual physical electrical circuit
and validated in dog studies^20,26^ and human clinical trials.^23^ The
proposed research provides the optimal CPR strategy using the time
dependent control functions based on this validated model. After
numerically solving the optimality systems, pressure and flow curves in
the different compartments will be compared for the blood circulation
during CPR with and without controls, to ascertain the improvement
obtained by using OC.

**Risks vs. Benefits**

Our proposal starts from a simple idea: The very low success rate of
existing CPR techniques makes us believe that there is ample room for
improvement. Clinical studies of alternative CPR methods show that,
under certain conditions, at least twice standard survival rates can be
obtained with nonstandard methods that have been shown in the laboratory
to improve blood flow during cardiac arrest.^24,27^ These methods
however, are ad-hoc and certainly not optimal. Systematic comparisons of
various CPR methods in the SAME experimental model are quite rare.^3^
Hence it is reasonable that the development of a sophisticated
mathematical test bed can be of value in suggesting better resuscitation
methods for the twenty first century. On the other hand, it may turn out
that the OC applied to the circulation model discussed above would not
significantly improve the blood flow.

In our opinion, this risk is strongly outweighed by the benefits of a
positive outcome. First, our own experience shows that OC problems
solved numerically with iterative methods and applied to a host of
physical, engineering, biological, and economic situations does indeed
result in *substantial* improvement over ad-hoc procedures. Second, for
the present application, *even a minor* (10-15 %) increase in blood flow
may represent the difference between life and death for many of the
250,000 people who presently die prematurely of cardiac arrest. We
expect that the controlled CPR model developed in this seed money
project will rapidly become an essential tool for understanding complex
circulatory physiology during CPR and developing effective CPR
techniques, and will have a significant impact on computational sciences
and biomedical engineering sciences.

**Follow-On Funding**

Most biomedical problems today are best addressed using a
multi-disciplinary approach that extends beyond traditional physiology
and clinical sciences. Multi-disciplinary research based on principles
and techniques drawn from mathematics, physics, computer sciences, and
engineering is providing deeper understanding and original results that
improve human health and quality of life. Recognizing the importance of
research in public health, future opportunities in biomedical area are
imminent in the Bioengineering Consortium (BECON) at the NIH. The
proposed research is well aligned to BECON's mission. Follow-up funding
on the simulations of blood flow during CPR has been discussed with
Richard E. Swaja, who is a program manager of BECON at the NIH. The
follow-up proposal will be submitted to either the National Institute of
General Medical Sciences (NIGMS) or the National Institute of Nursing
Research (NINR). We have been in contact with James Cassatt, who is the
director of the division of Cell Biology and Biophysics at NIGMS and
Hilary Sigmon, who is a program manager of the Cardiopulmonary Health
and Critical Care Division at NINR. The follow-up proposal will extend
this proof of principle to a more refined circulation model, based on
PDEs & ODEs and capable of accounting for additional specific features
of the patient.

We realize of course, that while actually performing clinical CPR today
there is no direct way to measure blood flow in a routine manner. In
fact, carrying out such direct measurements is a major technical
objective of the [Post-Resuscitation and Initial Utility in Life Saving
Efforts](http://www.nhlbi.nih.gov/meetings/pulse/index.htm) initiative
and of non-governmental institutions (e.g., Institute for Critical Care
Medicine in Palm Springs, California) that will be developed
independently and in parallel with our study. However, indirect measures
of blood flow, such as carbon dioxide excretion ("end tidal
![](media/image47.wmf)"), oxygen blood content by clip-on ear sensors,
or pressure measurements carried out in the hospital, under monitored
circumstances, can be used as approximate measures of the blood flow.
Despite the present lack of routine clinical measures of blood flow in
cardiac arrest and CPR (except a few ongoing research protocols), we
think that it is reasonable to explore, develop, and validate control
strategies against existing data. Data from direct measurements on CPR
patients themselves would be used as soon as they become available and
the two technologies would be merged on a routine basis.

**Eligibility**

There are no funded programs at ORNL related to the proposed research,
but the proposed work is aligned with the lab's core competencies and
future directions in biomedical sciences and computational biology.

**Personnel**

The team has the expertise required to successfully complete this
proposal. The PI, Jung, has developed various analytic and computational
pumping models.^12,13^ Lenhart and Protopopescu have a strong record in
OC to many types of PDEs & ODEs models.^14-19^ Babbs is a M.D., Ph.D. at
the Purdue University and an expert in the areas of the applied
cardiovascular physiology and biomedical engineering including
cardiopulmonary resuscitation and mathematical modeling of CPR
hemodynamics.^3,4,5,26,28^ He has had a Research Career Development
Award from the National Heart Lung and Blood Institute as well as
several grants related to CPR and biomedical engineering.

**Budget**

+---------------------------+---------------------------+--------------+
| **Cost element**          | **PERSONNEL and Time**    | **Budget     |
|                           |                           | (\$K)**      |
+---------------------------+---------------------------+--------------+
| Task 1: Control Synthesis | Lenhart - 0.1 FTE         | 30           |
|                           |                           |              |
|                           | Protopopescu - 0.05 FTE   |              |
+---------------------------+---------------------------+--------------+
| Task 2: Computational     | Jung -- 0.45 FTE          | 70           |
| Modeling                  |                           |              |
+---------------------------+---------------------------+--------------+
| Task 3: Validation        | Babbs (consultant)        | 10           |
|                           |                           |              |
|                           | Jung, Lenhart, and        | 10           |
|                           | Protopopescu              |              |
+---------------------------+---------------------------+--------------+
| Material and Supplies     |                           | \-           |
+---------------------------+---------------------------+--------------+
| Travel                    |                           | 5            |
+---------------------------+---------------------------+--------------+
| Total Budget Request      |                           | 125          |
+---------------------------+---------------------------+--------------+

The project will start upon award and will be completed in one year from
the start date.

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