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J Biomech Eng. Author manuscript; available in PMC 2011 November 3.

Published in final edited form as:

PMCID: PMC3207355

NIHMSID: NIHMS331834

Lik Chuan Lee,^{1,}^{2,}^{5,}^{*} Jonathan F. Wenk,^{1,}^{2,}^{5} Doron Klepach,^{1,}^{2,}^{5} Zhihong Zhang,^{1,}^{5} David Saloner,^{3,}^{5} Arthur W. Wallace,^{4,}^{5} Liang Ge,^{1,}^{2,}^{5} Mark B. Ratcliffe,^{1,}^{2,}^{5} and Julius M. Guccione^{1,}^{2,}^{5}

The publisher's final edited version of this article is available at J Biomech Eng

See other articles in PMC that cite the published article.

Homogeneous contractility is usually assigned to the remote region, border zone (BZ) and the infarct in existing infarcted left ventricle (LV) mathematical models. Within the LV, the contractile function is therefore discontinuous. Here, we hypothesize that the BZ may in fact define a smooth linear transition in contractility between the remote region and the infarct. To test this hypothesis, we developed a mathematical model of a sheep LV having an anteroapical infarct with linearly–varying BZ contractility. Using an existing optimization method [1], we use that model to extract active material parameter T_{max} and BZ width d_{n} that “best” predict in–vivo systolic strain fields measured from tagged magnetic resonance images (MRI). We confirm our hypothesis by showing that our model, compared to one that has homogeneous contractility assigned in each region, reduces the mean square errors between the predicted and the measured strain fields. Because the peak fiber stress differs significantly (~ 15%) between these two models, our result suggests that future mathematical LV models, particularly those used to analyse myocardial infarction treatment, should account for a smooth linear transition in contractility within the BZ.

It has been known since the mid–1980s that systolic function (systolic shortening and wall thickening) is depressed in the nonischemic infarct border zone (BZ) [2, 3]. Using two-dimensional tagged magnetic resonance imaging (MRI), Moulton and co-workers [4] measured stretching of BZ fibers during isovolumic systole in an ovine model of left ventricular (LV) aneurysm. They hypothesized that this mechanical dysfunction was due to an abnormally high load on the BZ fibers and/or abnormally low contractility of the BZ fibers. Using a finite element (FE) model, Guccione *et al.* [5] suggested that BZ fiber contractility must be only 50% of that in the remote uninfarcted myocardium in order to simulate stretching of BZ fibers during isovolumic systole.

Walker and co-workers [6,7] used cardiac catheterization, three-dimensional MRI with tissue tagging [8], MR diffusion tensor imaging [9], and an FE method (developed specifically for cardiac mechanics [10]) to measure regional systolic myocardial material properties in the beating hearts of four sheep with LV aneurysm [6] and six sheep with LV aneurysm repaired surgically [7]. Although these previous studies [6, 7] represent significant advancements in FE modeling of hearts with myocardial infarction, they both employed a manually directed pseudo-optimization. Formal optimization methods based on mathematical models of infarcted LV have, on the other hand, been used by Sun *et al.* [1] and Wenk *et al.* [11] to characterize and quantify its regional contractility. However, these models have assumed that the contractility is homogeneous within the pre–defined BZ, remote and infarct regions. Consequently, contractility changes abruptly at the infarction–BZ and the BZ–remote boundaries.

In this work, we hypothesize that the BZ defines a smooth transition in contractility between the remote region and the infarct. To test this hypothesis, we developed a mathematical model of an infarcted sheep LV that has a continuous contractile function and examined if such a model can better predict the measured strain obtained from three–dimensional tagged MRI. Specifically, the active material parameter is defined to vary linearly within the BZ so as to ensure a smooth transition in the contractility from the infarction to the remote region. Using a non–invasive method based on optimization developed by Sun *et al.* [1], we determine 1) active material parameters in the remote and BZ regions and 2) the size of the BZ producing strain fields that closely match the ones previously measured from tagged MRI of the same sheep model. We demonstrate that a linear variation in contractility within the BZ, when compared to one having homogeneous BZ contractility, reduces the mean square errors between the measured and the predicted strain fields. In addition, we also show that the resultant myofiber stress is about 15% larger than that found when a homogeneous BZ contractility is used.

The optimization method used to determine myocardial material parameters from experimental strain measurements has been discussed in Sun *et al* [1]. Here, we briefly describe that methodology along with key differences in the FE model i.e. the treatment of myocardial contractility in the BZ.

A male adult sheep model at 14 weeks postmyocardial infarction described by Zhang *et al.* [12] is used to test the hypothesis stated in §1. *In–vivo* strains are obtained using a series of orthogonal short and long axes tagged MRI. These tags are laid down at end diastole (ED) and the images are acquired as the heart continues through end systole (ES). Thereafter, the images are post–processed using FINDTAGS (Laboratory of Cardiac Energetics, National Institutes of Health, Bethesda, MD) and the systolic strain fields are calculated at the midwall and around the circumference in each short–axis slice. Contours of the endocardial and epicardial LV surfaces are also obtained and used to create the FE model.

Figure 1a shows the FE model of the sheep LV at 14 weeks postmyocardial infarction created using early diastole as the initial unloaded reference state. In that model, the aneurysm region of myocardial infarction is defined based on ventricular wall thickness. A BZ with depressed contractility is defined to be a region within a distance *d _{n}* measured from the infarct, whereas the section in the LV at a distance

Sheep LV with anteroapical aneurysm (a) finite element mesh (b) contractility *T*_{max} in BZ. Dotted line: homogeneous *T*_{max} in BZ. Solid line: linearly–varying *T*_{max} with distance *d* in BZ.

Homogenous passive hyperelastic constitutive law for a nearly incompressible and transversely isotropic material is assigned to the infarct, BZ and remote regions using the strain energy function

$$W=\frac{C}{2}\left({e}^{{b}_{f}{{E}_{ff}}^{2}+{b}_{t}({{E}_{ss}}^{2}+{{E}_{nn}}^{2}+{{E}_{ns}}^{2}+{{E}_{sn}}^{2})+{b}_{fs}({{E}_{fs}}^{2}+{{E}_{sf}}^{2}+{{E}_{fn}}^{2}+{{E}_{nf}}^{2})}-1\right).$$

(1)

In Eq. (1), *C*, *b _{f}*,

Contraction during systole is modelled by adding an active stress component *T*_{0} in the fiber direction **f** to the passive stress derived from Eq. (1). The resultant 2nd Piola stress tensor during systole is thus given by

$$\mathbf{S}=\kappa (J-1){J\mathbf{C}}^{-1}+2{J}^{-2/3}\text{Dev}\left(\frac{\partial \stackrel{\sim}{W}}{\partial \stackrel{\sim}{\mathbf{C}}}\right)+{T}_{0}(t,{Ca}_{0},l,{T}_{max})\mathbf{f}\otimes \mathbf{f}.$$

(2)

Here, κ is the bulk modulus, is the isochoric contribution to the strain energy function *W* given in Eq. (1), **C** is the right Cauchy stretch tensor, is the deviatoric part of **C**, *J* is the Jacobian of the deformation gradient, Dev(·) is the deviatoric projector [1] and *T*_{0} is the active stress developed during systole. The active stress *T*_{0} is a function of time *t*, peak intracellular calcium concentration *Ca*_{0}, sarcomere length *l* and maximum isometric tension achieved at the longest sarcomere length *T*_{max} [13]. In the infarct, *T*_{max} is set to zero (*T*_{max_}* _{I}* = 0) because the infarcted aneursym region does not contract i.e. the infarct is dyskinetic. Other than

In contrast to the FE model described in Sun *et al.* [1], our model differs in the description of contractility in the BZ. Within the BZ, the contractility *T*_{max_}* _{BZ}* increases linearly with distance

$${T}_{max\_BZ}=\left(1-\frac{d}{{d}_{n}}\right){T}_{max\_I}+\frac{d}{{d}_{n}}{T}_{max\_R}.$$

(3)

The resultant variation of *T*_{max} in the BZ is illustrated by the solid line in Fig. 1b, and is shown in contrast to the dotted line illustrating a homogeneous *T*_{max_}* _{BZ}* found in Sun

The goal of the optimization is to obtain values of *T*_{max_}* _{R}* and

$$\text{OBJ}=\sum _{n=1}^{N}\sum _{\begin{array}{c}i=1,2,3;\\ j=1,2,3;\\ i\ne 3\&j\ne 3\end{array}}{({E}_{ij,n}-{\overline{E}}_{ij,n})}^{2}+{\left(\frac{{V}_{\text{ED}}-{\overline{V}}_{\text{ED}}}{{V}_{\text{ED}}}\right)}^{2}+{\left(\frac{{V}_{\text{ES}}-{\overline{V}}_{\text{ES}}}{{V}_{\text{ES}}}\right)}^{2},$$

(4)

where *n* is the *in–vivo* strain point and *N* is the total number of strain points measured in the LV. Radial strain is excluded in computing OBJ because it cannot be measured accurately with tagged MRI [14, 15]. Successive response surface method implemented using LS-OPT (Livermore Software Technology Corporation, Livermore, CA) is used to perform the optimization [1].

An ED pressure of 10.85 mmHg and an ES pressure of 99.55 mmHg is applied to the endocardium in the FE model. Displacements at the base are constrained in the longtitudinal direction, with the epicardial–basal edge being constrained in all directions. Because *V*_{ED} depends only on diastolic material parameters, *C _{R}*(=

To determine the optimal size of the BZ width *d _{n}*, the optimization is carried out with

Figure 2 shows the convergence of OBJ with *d _{n}* as a parameter. In Fig. 2a, the value of OBJ converges quickly; within ~ 6 iterations for all cases. Figure 2b shows that, except for the case when

Optimization results: (a) convergence with BZ width *d*_{n} as a parameter (b) converged value of OBJ vs. *d*_{n}.

The resultant optimal distribution of *T*_{max} in the border zone (i.e for *d _{n}* = 3cm) is shown in Fig. 3. In the figure,

Figure 4 shows a comparison of myofiber stress distribution at ES between (a) the optimal linearly–varying *T*_{max_}* _{BZ}* FE model and (b) the optimal homogeneous

Comparison of stress in myofiber direction at ES. (a) Linearly–varying *T*_{max_}_{BZ} with *d*_{n} = 3cm and *T*_{max_}_{R} = 186.3kPa. (b) Homogeneous *T*_{max_}_{BZ} with *T*_{max_}_{R} = 190.1kPa and *T*_{max_}_{BZ} = 60.3kPa. Unit of fringe levels in kPa.

By contrast, the peak myofiber stress found in the linearly–varying *T*_{max_}* _{BZ}* model (92.5kPa) is about 15% higher than that found in the homogeneous

Our results confirm the hypothesis that the transition in contractility between the remote region and the infarct is smooth and linear. In particular, we show that the measured strain of a sheep LV having an anteroapical infarction can be better predicted using a mathematical model with continuous contractile function such that *T*_{max} varies linearly with distance within the BZ. Our result thus provides the first evidence that the contractility varies smoothly throughout the LV via a linear transition within the BZ.

Besides being able to better quantify the contractility in the LV, we are also able to use our mathematical model to predict the size of the BZ *d _{n}*, which may be an important predictor of post myocardial infarction mortality [16]. For the sheep LV at 14 weeks postmyocardial infarction, we predict

Compared to the mathematical model having a homogeneous *T*_{max_}* _{BZ}* [1], our model predicts the peak fiber stress to be about 15% higher. We believe that the stress difference is related to our model’s ability to better fit the measured strain. Because regional coronary blood flow [17] and myocardial oxygen consumption are influenced by ventricular wall stress, whereas changes in wall stress are believed to be a stimuli for hypertrophy and LV remodeling [18, 19, 20], our result suggests that it is important to model

In conclusion, we have shown that *in–vivo* strain data obtained from tagged MRI of an infarcted sheep LV can be fitted more accurately using a mathematical model with a linearly–varying *T*_{max} in the BZ. To corroborate our results *ex–vivo*, we will, in the near future, measure skinned fiber force in tissues taken from different locations of an explanted infarcted heart. In addition, we will also use the linearly–varying *T*_{max} BZ mathematical model to analyze biomaterial injection therapies and to predict long term remodeling processes of the myocardium.

This study was supported by the following National Institutes of Health research grants: R01 HL077921 and R01 HL086400 (JMG) and R01 HL063348 and R01 HL084431 (MBR). This support is gratefully acknowledged.

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