A Novel Method for Quantifying In-Vivo Regional Left Ventricular Myocardial Contractility in the Border Zone of a Myocardial Infarction

Abstract

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 ante-roapical infarct with linearly–varying BZ contractility. Using an existing optimization method (Sun et al., 2009, “A Computationally Efficient Formal Optimization of Regional Myocardial Contractility in a Sheep With Left Ventricular Aneurysm,” J. Biomech. Eng., 131(11), pp. 111001), 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 analyze myocardial infarction treatment, should account for a smooth linear transition in contractility within the BZ.

1. Introduction

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) [1,2]. Using two-dimensional tagged magnetic resonance imaging (MRI), Moulton and coworkers [3] 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. [4] 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 [5,6] used cardiac catheterization, three-dimensional MRI with tissue tagging [7], MR diffusion tensor imaging [8], and an FE method (developed specifically for cardiac mechanics [9]) to measure regional systolic myocardial material properties in the beating hearts of four sheep with LV aneurysm [5] and six sheep with LV aneurysm repaired surgically

[6]. Although these previous studies [5,6] 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. [10] 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. [10], 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.

2. Materials and Methods

The optimization method used to determine myocardial material parameters from experimental strain measurements has been discussed in Sun et al. [10]. Here, we briefly describe that methodology along with key differences in the FE model, i.e., the treatment of myocardial contractility in the BZ.

2.1. Experimental Measurements.

A male adult sheep model at 14 weeks postmyocardial infarction described by Zhang et al. [12] is used to test the hypothesis stated in Sec. 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.

2.2. Finite Element Model.

Figure 1(a) 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 d ≥ d_n measured from the infarct is defined to be the remote region with normal contractile function. Fiber angles −37°, 23° and 83° are assigned to the epicardium, midwall, and endocardium, respectively.

Fig. 1.

Sheep LV with anteroapical aneurysm. (a) Finite element mesh and (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

$$W=\frac{C}{2}\left(e^{b_f{E}_{\mathit{f}\mathit{f}}^2+b_t\left(E_{\mathit{s}\mathit{s}}^2+E_{\mathit{n}\mathit{n}}^2+E_{\mathit{n}\mathit{s}}^2+E_{\mathit{s}\mathit{n}}^2\right)+b_{\mathit{f}\mathit{s}}\left(E_{\mathit{f}\mathit{s}}^2+E_{\mathit{s}\mathit{f}}^2+E_{\mathit{f}\mathit{n}}^2+E_{\mathit{n}\mathit{f}}^2\right)}-1\right)$$ (1)

In Eq. (1), C, b_f, b_t and b_fs are the diastolic myocardial material parameters and E_ij, with subscripts $\{i,j\}\in \{f,s,n\}$, are the components of the Green–Lagrange strain tensor E taken in the fiber f, cross–fiber s, and transverse–fiber n directions. Following Sun et al. [10], the value of the exponents are b_f = 49.25, b_t = 19.25, and b_fs = 17.44, and the material parameter C in the infarct (C_I) is defined to be ten times stiffer than that in the remote (C_R) and in the BZ (C_BZ), i.e. C_I = 10C_R = 10C_BZ.

Contraction during systole is modeled by adding an active stress component T₀ 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

$$\mathrm{S}=\kappa \left(J-1\right)J{C}^{-1}+2J^{-2/3}\text{Dev}\left(\frac{\partial \tilde{W}}{\partial \tilde{C}}\right)+T_0\left(t,C{a}_{0,}l,T_{\max }\right)\mathbf{f}\otimes \mathbf{f}$$ (2)

Here, κ is the bulk modulus, $\tilde{W}$ is the isochoric contribution to the strain energy function W given in Eq. (1), C is the right Cauchy stretch tensor, $\tilde{\mathrm{C}}$ is the deviatoric part of C, J is the Jacobian of the deformation gradient, Dev (.) is the deviatoric projector [10], and T₀ is the active stress developed during systole. The active stress T₀ is a function of time t, peak intracellular calcium concentration Ca₀, 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 T_max in the BZ (T_{max_BZ}) and T_max at the remote (T_{max_R}), all the active material constants required in T₀ have the same value as those found in Sun et al. [10].

In contrast to the FE model described in Sun et al. [10], our model differs in the description of contractility in the BZ. Within the BZ, the contractility T_max _BZ increases linearly with distance d from the infarct so that T_max is continuous in the entire LV, i.e.

$$T_{\max \_BZ}=\left(1-\frac{d}{d_n}\right)T_{\max \_I}+\frac{d}{d_n}{T}_{\max \_R}$$ (3)

The resultant variation of T_max in the BZ is illustrated by the solid line in Fig. 1(b), and is shown in contrast to the dotted line illustrating a homogeneous T_{max_BZ} found in Sun et al. [10]. The finite element model is implemented and solved in the FE software LSDYNA using an explicit formulation.

2.3. Material Parameter Optimization.

The goal of the optimization is to obtain values of T_{max_R} and C_R of the FE model that “best” fit the strain measured experimentally using tagged MRI at ES (taken with reference to ED), as well as the LV cavity volumes at both ED and ES. To accomplish that, the objective function OBJ is defined to be the sum of mean squared error between measured strains and computed FE strains, ED volume V_ED and ES volume V_ES. Denoting measured quantities by ( $\overline{.}$), the objective function is given by

$$\text{OBJ}={\sum _{n=1}^N\sum _{\begin{array}{c}i=1,2,3;\\ j=1,2,3;\\ i\ne 3\hspace{0.17em}j\ne 3;\end{array}}\left(E_{ij,n}-{\overline{E}}_{ij,n}\right)}^2+{\left(\frac{V_{\mathit{E}\mathit{D}}-{\overline{V}}_{\mathit{E}\mathit{D}}}{{\overline{V}}_{\mathit{E}\mathit{D}}}\right)}^2+{\left(\frac{V_{\mathit{E}\mathit{S}}-{\overline{V}}_{\mathit{E}\mathit{S}}}{{\overline{V}}_{\mathit{E}\mathit{S}}}\right)}^2$$ (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 [10].

3. Results

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 longitudinal direction, with the epicardial–basal edge being constrained in all directions. Because V_ED depends only on diastolic material parameters, C_R (=C_BZ = 0.1C_I) is calibrated to 0.95 kPa so that the predicted value of V_ED equals the measured value V_ED = 123.4 ml.

To determine the optimal size of the BZ width d_n, the optimization is carried out with different values of d_n ranging from 1cm to 7 cm by an incremental change of 1 cm.

Figure 2 shows the convergence of OBJ with d_n as a parameter. In Fig. 2(a), the value of OBJ converges quickly; within ∼ 6 iterations for all cases. Figure 2(b) shows that, except for the case when d_n = 1 cm, adopting a linear variation of T_{max_BZ} with distance in the FE model produces a lower value of OBJ upon convergence when compared to that obtained from the FE model having a homogeneous T_{max_BZ} [10]. In that model, the BZ is defined as the steep transition in wall thickness between the infarct and the remote and has a width d_n ∼ 2 cm. Our result therefore implies that the measured in–vivo systolic strain can be better predicted by modeling T_max as a linearly–varying parameter within the BZ. The figure shows that the optimal BZ width for the sheep LV model at 14 weeks postmyocardial infarction is ∼ 3 cm. Though the change in OBJ is small, we show later that the impact on myofiber stress is substantial.

Fig. 2.

Optimization results: (a) convergence with BZ width d_n as a parameter and (b) converged value of OBJ versus d_n

The resultant optimal distribution of T_max in the border zone (i.e for d_n = 3 cm) is shown in Fig. 3. In the figure, T_max varies linearly from T_{max_I} = 0 kPa to T_{max_R} = 186.3 kPa within the BZ. In comparison, the optimal values found from assuming a homogenous distribution of T_max within the BZ are T_{max_R} = 190.1 kPa and T_{max_BZ} = 60.3 kPa [10].

Fig. 3.

Contractility in the border zone. d_n = 3 cm. T_{max- R} = 186.3 kPa

Figure 4 shows a comparison of myofiber stress distribution at ES between Fig. 4(a) the optimal linearly–varying T_{max_BZ} FE model and Fig. 4(b) the optimal homogeneous T_{max_BZ} FE model from Sun et al. [10]. Between them, the myofiber stress distribution S_ff is almost identical. Because the stress–strain relation is monotonic and the two models are optimized to have the same strain distribution measured from tagged MRI, we expect the stress distribution of the two models to be similar, i.e., regions having high myofiber stress should correspond to regions where the myofiber undergoes large strain.

Fig. 4.

Comparison of stress in myofiber direction at ES. (a) Linearly–varying T_{max- BZ} with d_n = 3 cm and T_{max- R} = 186.3 kPa. (b) Homogeneous T_{max- BZ} with T_{max- R} = 190.1 kPa and T_max- BZ = 60.3 kPa. Unit of fringe levels in kPa.

By contrast, the peak myofiber stress found in the linearly–-varying T_{max_BZ} model (92.5 kPa) is about 15% higher than that found in the homogeneous T_{max_BZ} model (80.1 kPa). Thus, despite the small difference in OBJ between these two cases, the magnitude of difference in the myofiber stress is substantial.

4. Discussion

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 d_n∼3 cm. This method, which utilizes only tagged MRI, is not only non–invasive but also helps to eliminate the subjectivity of defining BZ based on the transition of wall thickness [10].

Compared to the mathematical model having a homogeneous T_{max_BZ} [10], 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 T_max as a continuous material parameter varying linearly in the BZ. Moreover, our result suggests that existing FE models, which are used to analyze infarcted LV treatment, e.g., myosplint [21] and biopolymeric injection [22], can be further improved using our proposed linearly–varying T_max 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.