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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
J Biomech Eng. Author manuscript; available in PMC Nov 3, 2011.
Published in final edited form as:
J Biomech Eng. Sep 2011; 133(9): 094506.
doi:  10.1115/1.4004995
PMCID: PMC3207355
A Novel Method for Quantifying In-Vivo Regional Left Ventricular Myocardial Contractility in the Border zone of a Myocardial Infarction
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. Guccione1,2,5
1Department of Surgery, University of California, San Francisco, CA
2Department of Bioengineering, University of California, San Francisco, CA
3Department of Radiology, University of California, San Francisco, CA
4Department of Anesthesia, University of California, San Francisco, CA
5Department of Veterans Affair Medical Center, San Francisco, CA
* Corresponding Author: Lik Chuan, Lee, Ph.D, Division of Surgical Services (112D), Department of Veterans Affair Medical Center, 4150 Clement Street, San Francisco, CA, 94121, likchuan/at/
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 Tmax and BZ width dn 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.
Keywords: finite element modeling, numerical optimization, cardiac mechanics, tagged magnetic resonance imaging
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.
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 §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 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 dn measured from the infarct, whereas the section in the LV at a distance ddn 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.
Figure 1
Figure 1
Sheep LV with anteroapical aneurysm (a) finite element mesh (b) contractility Tmax in BZ. Dotted line: homogeneous Tmax in BZ. Solid line: linearly–varying Tmax 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
equation M1
In Eq. (1), C, bf, bt and bf s are the diastolic myocardial material parameters and Ei j, with subscripts {i, j} [set membership] {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. [1], the values of the exponents are bf = 49.25, bt = 19.25 and bf s = 17.44, and the material parameter C in the infarct (CI) is defined to be ten times stiffer than that in the remote (CR) and in the BZ (CBZ) i.e. CI = 10CR = 10CBZ.
Contraction during systole is modelled 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
equation M2
Here, κ is the bulk modulus, W is the isochoric contribution to the strain energy function W given in Eq. (1), C is the right Cauchy stretch tensor, C is the deviatoric part of C, J is the Jacobian of the deformation gradient, Dev(·) is the deviatoric projector [1] 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 Tmax [13]. In the infarct, Tmax is set to zero (Tmax_I = 0) because the infarcted aneursym region does not contract i.e. the infarct is dyskinetic. Other than Tmax in the BZ (Tmax_BZ) and Tmax at the remote (Tmax_R), all the active material constants required in T0 have the same values as those found in Sun et al. [1].
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 Tmax_BZ increases linearly with distance d from the infarct so that Tmax is continuous in the entire LV, i.e.
equation M3
The resultant variation of Tmax in the BZ is illustrated by the solid line in Fig. 1b, and is shown in contrast to the dotted line illustrating a homogeneous Tmax_BZ found in Sun et al. [1]. The finite element model is implemented and solved in the FE software LS-DYNA using an explicit formulation.
2.3 Material Parameter Optimization
The goal of the optimization is to obtain values of Tmax_R and CR 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 and computed FE strains, ED volume VED and ES volume VES. Denoting measured quantities by ([bardot]), the objective function is given by
equation M4
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 VED depends only on diastolic material parameters, CR(= CBZ =0.1CI) is calibrated to 0.95 kPa so that the predicted value of VED equals the measured value VED = 123.4ml.
To determine the optimal size of the BZ width dn, the optimization is carried out with dn ranging from 1cm to 7cm by an incremental change of 1cm.
Figure 2 shows the convergence of OBJ with dn 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 dn = 1cm, adopting a linear variation of Tmax_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 Tmax_BZ [1]. In that model, the BZ is defined as the steep transition in wall thickness between the infarct and the remote and has a width dn ~ 2cm. Our result therefore implies that the measured in–vivo systolic strain can be better predicted by modeling Tmax 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 ~ 3cm. Though the change in OBJ is small, we show later that the impact on myofiber stress is substantial.
Figure 2
Figure 2
Optimization results: (a) convergence with BZ width dn as a parameter (b) converged value of OBJ vs. dn.
The resultant optimal distribution of Tmax in the border zone (i.e for dn = 3cm) is shown in Fig. 3. In the figure, Tmax varies linearly from Tmax_I = 0kPa to Tmax_R = 186.3kPa within the BZ. In comparision, the optimal values found from assuming a homogenous distribution of Tmax within the BZ are Tmax_R = 190.1kPa and Tmax_BZ = 60.3kPa [1].
Figure 3
Figure 3
Contractility in the border zone. dn = 3cm. Tmax_R = 186.3kPa.
Figure 4 shows a comparison of myofiber stress distribution at ES between (a) the optimal linearly–varying Tmax_BZ FE model and (b) the optimal homogeneous Tmax_BZ FE model from Sun et al. [1]. Between them, the myofiber stress distribution Sf f 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 stresses should correspond to regions where the myofiber undergoes large strain.
Figure 4
Figure 4
Comparison of stress in myofiber direction at ES. (a) Linearly–varying Tmax_BZ with dn = 3cm and Tmax_R = 186.3kPa. (b) Homogeneous Tmax_BZ with Tmax_R = 190.1kPa and Tmax_BZ = 60.3kPa. Unit of fringe levels in kPa.
By contrast, the peak myofiber stress found in the linearly–varying Tmax_BZ model (92.5kPa) is about 15% higher than that found in the homogeneous Tmax_BZ model (80.1kPa). Thus, despite the small difference in OBJ between these two cases, the magnitude of difference in the myofiber stress is substantial.
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 Tmax 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 dn, which may be an important predictor of post myocardial infarction mortality [16]. For the sheep LV at 14 weeks postmyocardial infarction, we predict dn ~ 3cm. 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 [1].
Compared to the mathematical model having a homogeneous Tmax_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 Tmax as a continuous material parameter varying linearly in the BZ. Moreover, our result suggests that existing FE models which are used to analyse infarcted LV treatment e.g. myosplint [21] and biopolymeric injection [22] can be further improved using our proposed linearly–varying Tmax 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 Tmax 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 Tmax 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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