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A Computationally Efficient Formal Optimization of Regional

Myocardial Contractility in a Sheep with Left Ventricular

Aneurysm

Kay Sun1,4, Nielen Stander5, Choon-Sik Jhun1,4, Zhihong Zhang1,4, Takamaro Suzuki1,4,

Guan-Ying Wang3,4, Maythem Saeed3, Arthur W. Wallace2,4, Elaine E. Tseng1,4, Anthony J.

Baker3,4, David Saloner3,4, Daniel R. Einstein6, Mark B. Ratcliffe1,4, and Julius M.

Guccione1,4,*

1 Department of Surgery, University of California, San Francisco, CA

2 Department of Anesthesia, University of California, San Francisco, CA

3 Department of Radiology, University of California, San Francisco, CA

4 Department of Veterans Affairs Medical Center, San Francisco, CA

5 Livermore Software Technology Corporation, Livermore, CA

6 Biological Monitoring and Modeling, Pacific Northwest National Laboratory, Olympia, WA

Abstract

A non-invasive method for estimating regional myocardial contractility in vivo would be of great

value in the design and evaluation of new surgical and medical strategies to treat and/or prevent

infarction-induced heart failure. As a first step towards developing such a method, an explicit finite

element (FE) model-based formal optimization of regional myocardial contractility in a sheep with

left ventricular (LV) aneurysm was performed using tagged magnetic resonance (MR) images and

cardiac catheterization pressures. From the tagged MR images, 3-dimensional (3D) myocardial

strains, LV volumes and geometry for the animal-specific 3D FE model of the LV were calculated,

while the LV pressures provided physiological loading conditions. Active material parameters

(Tmax_B and Tmax_R) in the non-infarcted myocardium adjacent to the aneurysm (borderzone) and

in myocardium remote from the aneurysm were estimated by minimizing the errors between FE

model-predicted and measured systolic strains and LV volumes using the successive response surface

method for optimization. The significant depression in optimized Tmax_B relative to Tmax_R was

confirmed by direct ex vivo force measurements from skinned fiber preparations. The optimized

values of Tmax_B and Tmax_R were not overly sensitive to the passive material parameters specified.

The computation time of less than 5 hours associated with our proposed method for estimating

regional myocardial contractility in vivo makes it a potentially very useful clinical tool.

Keywords

tagged magnetic resonance imaging; finite element modeling; numerical optimization; cardiac

mechanics

*Corresponding Author: Julius M. Guccione, Ph.D., UCSF/VA Medical Center (112D), 4150 Clement Street, San Francisco, CA 94121,

GuccioneJ@surgery.ucsf.edu.

NIH Public Access

Author Manuscript

J Biomech Eng. Author manuscript; available in PMC 2009 December 15.

Published in final edited form as:

J Biomech Eng. 2009 November 1; 131(11): 111001. doi:10.1115/1.3148464.

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INTRODUCTION

A non-invasive method for estimating regional myocardial contractility is a “holy grail” of

cardiology. Such a method would be of great value in the design and evaluation of new surgical

and medical strategies to treat and/or prevent infarction-induced heart failure. Once the

constitutive equation for the myocardium is established the effect of therapeutic changes on

regional geometry (i.e., surgical remodeling) and/or material properties (i.e., medicine, gene

therapy, cell therapy) can be evaluated and the success or failure of a proposed therapy

predicted. With clinical experience, such a method could be used as a diagnostic modality to

risk stratify patients early after a myocardial infarction who are at risk for adverse remodeling

and the development of heart failure.

It is impossible to determine myocardial material properties (in the form of three-dimensional

constitutive equations) from ventricular pressure-volume relations alone. Instead, our

laboratory recently used cardiac catheterization, magnetic resonance (MR) imaging with

myocardial tissue tagging [1], MR diffusion tensor imaging [2], and a finite element (FE)

method (developed specifically for cardiac mechanics [3]) to measure regional systolic

myocardial material properties in the beating hearts of four sheep with left ventricular (LV)

aneurysm [4] and six sheep with LV aneurysm repaired surgically [5]. With knowledge of these

myocardial material properties, we were able to quantify the effect of aneurysm plication on

regional myocardial stress distributions. The 3D stress distributions in the myocardium are

important to regional ventricular function because both regional coronary blood flow [6] and

myocardial oxygen consumption [7] are influenced by ventricular wall stress [8]. Changes in

ventricular wall stress are believed to be stimuli for hypertrophy and remodeling [9]. There

have been no successful methods developed to measure stress in the intact heart wall—

primarily because of its large deformations and the tissue injury caused by implanted

transducers [10].

Although our previous studies [4–5] represent significant advancements in FE modeling of

hearts with myocardial infarction, because of long computation times, they both employed a

manually directed pseudo-optimization. In other words, a formal nonlinear optimization of

material constants was not feasible. The objective of the present study is to describe a method

to formally optimize regional myocardial contractility in vivo that is at least an order of

magnitude faster than that used in our previous studies.

METHODS

The sheep used in this study was treated in compliance with the “Guide for the Care and Use

of Laboratory Animals” prepared by the Institute of Laboratory Animal Resources, National

Research Council, and published by the National Academy Press (revised 1996).

Experimental Measurements

Data collected from one male adult sheep [11] was used to demonstrate methodology and

accuracy of the FE optimization tool. Briefly, the sheep underwent anteroapical myocardial

infarct following the procedures described in Markovitz et al. [12]. At 14 weeks post-

myocardial infarction, a series of orthogonal short- and long-axis tagged MR images were

acquired as described in detail previously [11]. The tags were laid down at end of diastole by

synchronization to the R-wave of the electrocardiographic (ECG) signal and tagged images

were captured as the heart continues through systole. LV pressure was measured with a non-

ferromagnetic transducer-tipped pressure catheter (model SPC-320; Millar Instruments,

Houston, TX) inserted into the LV via sterile neck incisions as described previously [11]. The

end-diastolic and end-systolic LV pressures (PED, PSD) recorded were used to define the

endocardial boundary conditions of the FE model.

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A customized version of the MR image tagging post-processing software, FindTags

(Laboratory of Cardiac Energetics, National Institutes of Health, Bethesda, MD) was used to

contour the endocardial and epicardial LV surfaces and also to segment the systolic tags for

each image slice [13]. Systolic myocardial strains (6 Lagrangian Green’s strain tensor

components in cylindrical coordinates, circumferential, longitudinal and radial) at midwall and

around the circumference in each short-axis slice were calculated from tag-line deformation

using the four dimensional B-spline-based motion tracking technique [14] (Fig. 1).

FE Model

An FE model was created using early diastole as the initial unloaded reference state since the

LV pressure is lowest at this point and therefore stress is at a minimum. From the LV contours

at early diastole, aneurysm, remote and borderzone regions were determined based on the

ventricular wall thickness. Specifically, the borderzone region is defined as the steep transition

in wall thickness between remote and aneurysm regions [15]. Surface meshes were then created

from the LV contours to replicate the in vivo geometry (Rapidform, INUS Technology, Inc.,

Sunnyvale, CA). The spaces between the endocardium and epicardium surfaces were filled

with 8-noded brick elements with a single integration point for computational efficiency to

generate a volumetric mesh that is refined into three elements transmurally (Truegrid, XYZ

Scientific Applications, Inc., Livermore, CA). Each zone, remote, borderzone and infarct, were

assigned different material properties. The inner endocardial layer was lined with a layer of

shell elements that extends to LV base to form an enclosed volume for LV volume

measurements. The shell elements were modeled as an extremely soft linearly elastic material

(Young’s modulus of 1×10−10 kPa and Poisson’s ratio of 0.3) that offered no mechanical

response. The reliability of the model predictions was tested with a mesh convergence study

to find the minimum number of elements needed to produce accurate results within the fastest

computation time. The mesh convergence study determined that 2496 elements is required and

further mesh refinement only results in a 1% change in strain predictions (Fig. 2).

Cardiac myofiber angles of −37°, 23° and 83° were assigned at the epicardium, midwall and

endocardium, respectively, in the remote and borderzone regions [16]. At the aneurysm region,

fiber angles were set to 0° in order to use experimentally determined aneurysm material

parameters with respect to this direction [17]. In other words, the constitutive equation for the

aneurysm is in terms of strain components referred to cardiac (i.e., circumferential and

longitudinal) coordinates instead of fiber coordinates. Nodes at the LV base were restricted to

displace horizontally and circumferential displacements were constrained at the basal

epicardial nodes. The inner endocardium wall was loaded to the measured in vivo end-diastolic

and end-systolic LV pressures. Surface meshes and subsequent volumetric meshes were also

created from the LV contours at end-diastole and end-systole to provide the end-diastolic and

end-systolic LV volumes.

Constitutive Model

Nearly incompressible, transversely isotropic, hyperelastic constitutive laws for passive [18]

and active myocardium [19] were modeled in a user-defined material subroutine in the explicit

FE solver, LS-DYNA (Livermore Software Technology Corporation, Livermore, CA). The

passive myocardium mechanics is described by the strain energy function, W, that is

transversely isotropic with respect to the local fiber direction,

(1)

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where C, bf, bt and bfs are diastolic myocardial material parameters. E11 is strain in fiber

direction, E22 is cross-fiber in-plane strain, E33 is radial strain transverse to the fiber direction,

and the rest are shear strains.

Systolic contraction was modeled as the sum of the passive stress derived from the strain energy

function and an active fiber directional component, T0, which is a function of time, t, peak

intracellular calcium concentration, Ca0, sarcomere length, l, and maximum isometric tension

achieved at the longest sarcomere length, Tmax [19],

(2)

where S is the second Piola-Kirchoff stress tensor, p is the hydrostatic pressure introduced as

the Lagrange multiplier needed to ensure incompressibility and was calculated from the bulk

modulus of water, J is the Jacobian of the deformation gradient tensor, C is the right Cauchy-

Green deformation tensor and the Dev is the deviatoric projection operator,

(3)

W̃ is the deviatoric contribution of the strain energy function, W (Eq. 1). The assumption of

near incompressibility of the myocardium requires the decoupling of the strain energy function

into dilational and deviatoric components,

(4)

where U is the volumetric contribution.

The active fiber directional stress component is defined by a time-varying elastance model,

which at end-systole, is reduced to [20],

(5)

with m and b as constants, and the length-dependent calcium sensitivity, ECa50, is given by,

(6)

where B is a constant, (Ca0)max is the maximum peak intracellular calcium concentration, l0

is the sarcomere length at which no active tension develops and lR is the stress-free sarcomere

length. The material constants for active contraction were found to be [21]: Ca0 = 4.35 μmol/

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L, (Ca0)max = 4.35 μmol/L, B = 4.75 μm−1, l0 = 1.58 μm, m = 1.0489 secμm−1, b = −1.429 sec,

and lR was set at 1.85 μm, the sarcomere length in the unloaded configuration. Based on the

biaxial stretching experiments [22] and FE analyses [4,23], cross-fiber, in-plane stress

equivalent to 40% of that along the myocardial fiber direction was added.

Since tagged MR images were acquired during systole, only systolic myocardial strains could

be determined. This meant only systolic material parameters, Tmax in remote (Tmax_R) and

borderzone region (Tmax_B), could be optimized. Tmax in the aneurysm region was set to zero

as the coronary ligations were permanently put in place to create dyskinetic infarcts. Diastolic

materials parameters, bf, bt and bfs, were set to the average optimized values for sheep obtained

previously from Walker et al. [4]: bf = 49.25, bt = 19.25, bfs = 17.44. C in the aneurysm region

(CI) was defined as 10 times stiffer than that in the remote (CR) [4]. CR was determined by

calibrating it such that the predicted end-diastolic LV volume matched the measured value.

Initial ranges for Tmax_R were between 0.1 and 1000.0 kPa, and between 0.1 and 500.0 kPa for

Tmax_B.

Material Parameter Optimization

The commercial FE optimization software, LS-OPT (Livermore Software Technology

Corporation, Livermore, CA), uses a systematic search methodology to automatically explore

the parameter space and find an optimum design [24]. In this application, the optimum design

is the ideal myocardial material parameters that will produce 3D myocardial strains that come

closest to matching those measured in vivo. LS-OPT is based on the successive response surface

method (SRSM). It works by first selecting experimental design points which consist of systolic

myocardial material parameters, Tmax_R and Tmax_B as components, within their specified

ranges (Fig. 3a). The selection process is based on the Design of Experiments approach using

the D-Optimality criterion [25]. This method uses a subset of all the possible design points as

a basis to solve

where X is the coefficient matrix of the normal equations:

The vector â represents the coefficients of the basis functions (in this case linear monomials)

and vector y represents the computed values (e.g. strain or volume values) at all the

experimental design points.

The subset is usually selected from an ℓn -factorial design where ℓ is chosen a priori as the

number of grid points in any particular dimension and n is the number of variables. A genetic

algorithm is used to solve the resulting discrete maximization problem. The proper selection

of experimental points results in the most accurate response surface and therefore a faster

convergence rate and greater accuracy of the optimization solution [24].

Using the selected experimental points, FE simulations were performed and the strains and LV

volumes were calculated (Fig. 3b) for all points. A linear response surface was then fitted to

each strain and volume by means of a least square fitting method. The approximate MSE is

defined as the difference between the response surface predicted and experimental results (end-

diastolic and end-systolic volumes and strains).

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(7)

where n is the in vivo strain point, N is the total number of in vivo strain points, Ei j,S are the

predicted strains at each strain point, VED and VSD are the predicted end-diastolic and end-

systolic LV volumes, respectively. The overbar represents experimental in vivo measurements.

Strain in the radial direction, E33, is excluded as it cannot be measured with sufficiency

accuracy with tagged MR images [26,27]. The goal of the optimization is to minimize the MSE.

The leap-frog dynamic trajectory method was applied towards the minimization of the MSE

constructed from these response surfaces to determine the optimum parameter set [28–30] (Fig.

3c). This optimum then becomes the initial experimental point for the next iteration. The range

for each parameter was adjusted by either contraction or translation or both depending on how

close the current optimum is to the previous one and the degree of oscillation of the optimization

solution across the iterations [24]. For the next sub-region of parameter space, the D-Optimality

criterion was applied again to find a new set of experimental points for the Response Surface

Method (RSM) to be applied once more [31]. By repeating these steps for each successive

iteration, subregions of the parameter space were reduced and shifted until a final optimum

design was found (Fig. 4). LS-OPT is also able to compute confidence intervals of the optimized

material parameters in order to assess their reliability [24]. Fig. 5 summarizes the methodology.

It is important to note here that a two-step process was required to compute Green’s strain

referred to the end-diastolic state. Using index notation, the Green’s strain to be compared with

the experimentally measured values is defined as:

(8)

where F is the deformation gradient tensor, δ is the kronecker delta, Y refers to the end-diastolic

(ED) state and x to the end-systolic (ES) state. Each run starts from a given undeformed state

X, so that the deformation gradient referred to ED can be computed as:

(9)

where FmK and FiK are the deformation gradient components at the ED and ES phases

respectively, both referred to the undeformed configuration X.

The executable ‘mri2lso’ (which reads a text file containing the measured 3D strain data in

simple tabular format and converts it to an LS-OPT command file) is available at

ftp://ftp.lstc.com/user/ls-opt as part of the LS-OPT distribution.

RESULTS

The pressure catheter recorded PED and PSD at 12.89 mmHg and 101.59 mmHg, respectively.

These recorded pressures were offset by the minimum recorded LV pressure of 2.04 mmHg

to get PED of 10.85 mmHg and PSD of 99.55 mmHg, which were applied to the inner

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endocardium layer in order to create an initial stress-free model. CR was calibrated to be 0.95

kPa and CI defined as 10 times stiffer at 9.5 kPa such that VED was accurately predicted at

123.6 mL with the measured value at 123.4 mL.

The minimum MSE of 4.17, consisting of 960 strain and 2 LV volume data points, was reached

in 10 iterations. The optimized Tmax_R and Tmax_B for this sheep are 190.1 kPa and 60.3 kPa,

respectively, with 90% confidence intervals at 14.9% and 16.9%, respectively. The precision

of the optimized parameters brought on by the narrowing of the parameter space over the 10

iterations (Fig. 6). VSD was also accurately predicted at 110.8 mL, only 4.9% higher than the

measured value of 105.6 mL. The predicted systolic strains using the optimized material

parameters were in generally decent agreement with the in vivo measured strains. The insertion

points of the right ventricle (RV) to the LV showed the largest difference between the measured

and predicted strains since the RV was not included in the model. Fig. 7 illustrates the

similarities in the circumferential strains that were measured in vivo from tagged MR images

and predicted from this FE model. The root mean square (RMS) error for the circumferential

strain component between the 137 pairs of measured and predicted strains in the remote zone

was 0.048 and in the borderzone, the RMS error was 0.070 with 55 pairs of strain points.

The optimization process was performed once using the upper bounds of the parameters as the

initial guess and repeated using the lower bounds as the initial guess. Both optimizations with

different starting points produced the same converged results (Table 1). Each forward

simulation of the FE model takes about 10 minutes on a single 2.4 GHz processor. For the

optimization of 2 material parameters, 5 experimental points were selected using a D-Optimal

method for each iteration. The entire optimization process involving 10 iterations or 50

experimental points plus 10 leap-frog optimizations required about 4.5 hours on four 2.4 GHz

processors.

DISCUSSION

A very efficient or fast method was developed in order to formally optimize regional

myocardial contractility from tagged MR images and cardiac catheterization pressures. Our

approach was demonstrated for data from a single sheep, 14 weeks after anteroapical

myocardial infarction. The proposed method involves performing FE simulations using the

customized commercial FE solver (LS-DYNA) that was programmed with the passive and

active myocardial material laws. The forward FE solutions are fed into the optimization

software (LS-OPT), which was customized to determine the systolic myocardial material

parameters using the SRSM approach by targeting the in vivo systolic strains and LV volumes.

The in vivo systolic strains and LV volumes were determined from tagged MR images, which

also provided the LV endocardial and epicardial contours that were used to generate the FE

model. Finite element model loading conditions were obtained from cardiac catheterization

measurements of LV pressures.

Validation of the Method

In order to really validate our method for estimating regional myocardial contractility in

vivo, Tmax_R and Tmax_B need to be estimated independently using another method (preferably

one in which forces can be directly measured). We have had great difficulty in making

comparisons of Tmax_R and Tmax_B obtained in the same hearts with our biaxial stretcher

because LV regions that are thick enough for tagged MRI are too thick for biaxial testing and

LV regions that are thin enough for biaxial testing are too thin for tagged MRI. This forced us

to seek another method which involves measurement of active stress developed in

“skinned” (de-membraned using Triton-X) ovine LV myofibers over a range of calcium

concentrations in the muscle bath. Direct measurements of maximal force (normalized by

cross-sectional area) developed by skinned myofibers dissected from remote regions of three

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sheep hearts with LV aneurysm (38.1 ± 2.3 kPa) were significantly greater than that developed

by skinned myofibers dissected from borderzone regions (22.2 ± 3.0 kPa; p < 0.03). Part of

the reason why both of the optimized Tmax values reported in the Results (Tmax_R = 190.1 kPa;

Tmax_B = 60.3 kPa) are so much greater than the skinned fiber values is the latter are fit to a

much simpler (modified Hill) equation than Eq. 5 in the Methods. Specifically, the modified

Hill equation does not include the factor of 0.5 in front of Tmax, nor the factor in the parentheses,

the value of which depends on sarcomere length. This sarcomere length-dependence can be

removed from Eq. 5 by setting the parameter m = 0 and b ≥ 2000. Rerunning the optimization

with the modified Hill equation in place of Eq. 5 results in Tmax_R = 100.0 kPa and Tmax_B =

36.3 kPa. It is not difficult to explain the higher in vivo contractility than ex vivo contractility

in terms of tissue damage caused by the dissection process. Nevertheless, the significant

depression in optimized Tmax_B relative to Tmax_R was confirmed by direct ex vivo force

measurements from skinned fiber preparations.

Sensitivity Analysis

Effect of passive and active parameters specified on Tmax—To estimate how

sensitive the model formal optimization is to the specified passive material parameter values,

simulations were rerun after increasing or decreasing by 10% (one-at-a-time) the values

specified in the Methods for bf, bt, and bfs. The greatest sensitivity of the optimized Tmax_R or

Tmax_B to these three passive material parameters was only a 5.6% increase in Tmax_R in

response to a 10% decrease in bt and a 2.1% decrease in Tmax_B in response to a 10% decrease

in bfs. The model formal optimization was much more sensitive to some of the specified active

material parameter values, especially l0 and lR. Specifically, a 5% increase in l0 resulted in a

32% increase in Tmax_R and 5% decrease in lR resulted in a 195% increase in Tmax_R. However,

we are quite confident that the values specified in the Methods for l0 and lR are very well

defined by experimental data in the literature. The active material parameter Ca0 also “controls”

model myocardial contractility (like Tmax) but instead of simply scaling active stress

development, it affects the shape of the active stress versus sarcomere relationship (i.e., linear

at Ca0 = 2 μmol/L, “concave up” at Ca0 < 2 μmol/L; concave down at Ca0 > 2 μmol/L; see

Fig. 8 of Guccione and McCulloch 1993). Not surprisingly, 10% and 20% decreases in Ca0

resulted in increases of 7.7% and 18.9% in Tmax_R and increases of 4.6% and 10.5% in

Tmax_B, respectively. Unfortunately, to the best of our knowledge, there is no way to reliably

measure peak intracellular calcium concentration in the intact beating LV myocardium.

Effect of errors in strain measurement on Tmax—To estimate how sensitive the model

formal optimization is to errors in strain measurement, simulations were rerun after adding or

subtracting fixed offsets of 0.015 and 0.03 in circumferential strain (Ecc) measurements. The

greatest sensitivity of the optimized Tmax_R or Tmax_B to uncertainty in circumferential strain

measurement was a 112% increase in Tmax_R in response to a fixed decrease of 0.03 in Ecc and

a 33% increase in Tmax_B in response to fixed increase of 0.03 in Ecc. Not surprisingly, these

regional “indices” of myocardial contractility are very sensitive to the degree of systolic LV

circumferential shortening.

Relative Contributions of Terms in Objective Function (Eq. 7)

We performed a detailed study of the effect of the relative contributions of strains and volume

on Tmax_R and Tmax_B by varying the weights over a range of two orders of magnitude (LS-

OPT could not handle weights below 1%). Under all (weighting) conditions Tmax_R = 190 kPa

and Tmax_B = 60 kPa. The only “measurable” effect was in the 3rd significant figure of

Tmax_B (60.3 kPa for equal weighting versus 60.4 kPa for weighting of 98% for strain and 1%

for each of the two volumes). Moreover, the confidence intervals were equally “tight” for these

two (extreme weighting) cases.

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Comparison to Previous Work

Previous attempts at estimating myocardial material parameters from an intact heart using

strains have been oversimplified by assigning non-physiological material properties [32] and

loading conditions [33]. Some studies with more realistic FE models however only used 2

strain measurements to fit for the optimum material parameters [34-34], while others that used

500 to 700 strain measurements for optimization required a significant amount of computing

time and power [4,36]. There have also been labor intensive, trial-and-error approaches towards

optimization [4,34] as well as automated optimization algorithms. The elegant study by

Augenstein et al. [37] is an example of a nonlinear automated optimization procedure, which

was independently validated using phantoms.

The optimization algorithms used by previous studies [32–33,36,38] were numerical gradient-

based, which may lead to local minima rather than the global one due to local sensitivities of

the nonlinear problem and require a good initial guess to the solution. This study employed

the gradient-free optimization approach called the successive response surface method

(SRSM), which is better able to reach the global minimum [39,40]. It is based on the iterative

construction and minimization of response surfaces that have been fitted onto the residuals

between the measured and FE predicted strains. The approximations avoid local minima and

the shrinkage of the parameter space with each iterative response surface ensures the detailed

variations of the nonlinear problem near the global minimum are captured. The optimized

myocardial contractility in the remote region was found to be 190.1 kPa while in the

borderzone, the contractility was 3.15 times less at 60.3 kPa. The marked difference in

contractility in the two regions is greater than the 50% reduction reported by Guccione et al.

[34]. The reason for the discrepancy can be attributed to the greater accuracy and precision of

the present FE model and optimization method. 960 in vivo systolic strain measurements (12

circumferentially, 16 longitudinally, 5 cylindrical strain components) and 2 LV volumes

formed the objectives of optimization in this study while the other study only used 2 strain

measurements (only circumferential strains in just the anterior and posterior borderzone

regions). The more data points available for optimization, the more precise the optimized

results will be. This relationship is reflected by the relatively small 90% confidence intervals

for the optimized Tmax_R and Tmax_B, 14.9% and 16.9%, respectively. Guccione et al. [34] did

not report any analysis on the confidence of the optimized results.

Advantages of SRSM

The SRSM optimization algorithm is a gradient-free approach that is suitable for highly

nonlinear problems with multiple local minima. SRSM is based on the sequential optimization

of the linear response surfaces constructed from the FE simulation results. The size of the

successive sub-regions, within which the response surfaces are formed, is adapted based on

contraction and panning parameters designed to prevent oscillations and premature

convergence [41]. As SRSM starts with a large sub-region of the parameter space that is

contracted automatically as the optimum is approached, local sensitivities do not cause large

departures from the previous design. This inherent adjustment limit of the parameter space in

the algorithm eliminates the step-size dilemma in gradient-based optimization methods [41].

In previous myocardial material optimization studies that used numerical gradients [32–33,

36,38], finite differences were taken at 5 to 10% interval for each material parameter. If the

interval was set too large, accuracy is lost. Conversely, if the interval was set too small, the

gradient may be spurious and miss the true optimum.

Compared to another gradient-free optimization approach, like the genetic algorithm used by

Nair et al. [35], SRSM is more efficient. Even though that previous study only used 2 strain

measurements for optimization, 5400 FE simulations and 150 “chromosomal generations” or

iterations were required for convergence and the total computation time took 25 to 40 days.

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Although this study optimized 2 fewer material parameters, a much larger number of data

points, 960 strain measurements and 2 LV volumes, were used for optimization and the total

computation time was only 4.5 hrs. The genetic method takes longer to compute probably

because it adds randomness into each material parameters selection process for FE analysis in

order to widen the parameter space to find the global minimum but doing so also slowed the

rate of convergence.

Limitations

There are a few limitations in this study. First, since tagged MR images were acquired during

the systolic phase, only systolic strains were calculated and therefore just the systolic

myocardial material parameters can be optimized. Due to the lack of diastolic strains, the

diastolic myocardial material parameters, bf, bt, bfs, were taken from Walker et al. [4], which

had optimized those parameters for four sheep. The other diastolic material parameters, CR

and CI, had to be calibrated to the measured end-diastolic LV volume. For future studies, 2

sets of tagged MR images acquired during diastole and systole will be required. Strain analysis

will be performed on both image sets to produce diastolic and systolic strains, which will then

be used separately to optimize for diastolic and systolic myocardial material parameters,

respectively.

The second limitation of this study is the assumption of an initial stress-free state, which is

present in all previous FE simulations of the heart based on in vivo images. Not all the blood

in the heart is ejected out during systole and it is this residual blood at early diastole that results

in a small amount of stress on the heart. Our previous FE model study suggests that residual

stress produced by surgical remodeling has little effect on ventricular function and regional

mechanics [42].

The next limitation is with the use of transversely isotropic material properties to describe the

myocardium, which is consistent with biaxial tests that observed stiffness in the muscle fiber

direction is greater than in cross-fiber direction. However, histological sections of myocardium

show myofibers arranged into branching laminae, implying stiffness is lower normal to the

laminar plane than within it [43,44]. Direct measurements of material orthotropy have yet been

determined due to experimental limitations of triaxial testing of excised myocardium.

Orthotropic material properties will be modeled in future studies and optimized diastolic

material parameters will be obtained to determine the degree of anisotropy.

The last limitation is the absence of the RV in the FE model. Only the LV was modeled since

it experiences the largest stress and strain as it has to eject blood from the heart with enough

pressure to reach the rest of the body. Although the RV just needs to contract with sufficient

force for the blood to reach the nearby lungs, the deformation of the RV seems to have a

significant effect on the LV wall strains at the RV insertion points. This effect is evident by

the largest deviations between the measured and predicted strains right at the RV insertion

points. Future studies will need to include the RV in the model and strain analysis.

CONCLUSIONS

In summary, we have presented a method to formally optimize regional myocardial

contractility in vivo that is at least an order of magnitude faster than that used in our previous

studies [4–5]. The significant depression in optimized Tmax_B relative to Tmax_R was confirmed

by direct ex vivo force measurements from skinned fiber preparations. The optimized values

of Tmax_B and Tmax_R were not overly sensitive to the passive material parameters specified.

The computation time of less than 5 hours associated with our proposed method for estimating

regional myocardial contractility in vivo makes it a potentially very useful clinical tool.

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Acknowledgments

The authors acknowledge financial support from NIH grants R01-HL-77921, R01-HL-63348 and R01-HL-84431.

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Fig. 1.

3D cardiac strain analysis from in vivo tagged MR images. Endocardial and epicardial contours

as well as segmented tag-lines were traced from (a) short and (b) long-axis MR images to create

(c) a 3D geometry. (d) Each short axis slice was divided into 12 sectors and a 4D B-spline-

based motion tracking technique was applied to the tag-line (dotted lines) deformations in order

to calculate the Lagrangian Green’s strains in cylindrical coordinates. For each sector of each

short axis slice, longitudinal, radial, (e) circumferential and shear strains throughout systole

were determined.

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Fig. 2.

Creation of the FE model of the LV using geometry from in vivo tagged MR images.

Endocardial and epicardial contours extracted from (Fig. 1a) short and (Fig. 1b) long-axis MR

images were used to generate (a) a surface mesh with three distinct LV regions (remote,

borderzone and aneurysm). The boundaries between these three LV regions are based on wall

thickness. The surface meshes provide projection surfaces for (b) the volumetric mesh, which

is refined into three elements transmurally. A layer of shell elements line the endocardial

surface and cap off the top of the LV to form a closed volume for LV volume measurements.

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