Estimating Passive Mechanical Properties in a Myocardial Infarction using MRI and Finite Element Simulations

Abstract

Myocardial infarction (MI) triggers a series of maladaptive events that lead to structural and functional changes in the left ventricle. It is crucial to better understand the progression of adverse remodeling, in order develop effective treatment. In addition, being able to assess changes in-vivo would be a powerful tool in the clinic. The goal of the current study is to quantify the in-vivo material properties of infarcted and remote myocardium 1 week after MI, as well as the orientation of collagen fibers in the infarct. This will be accomplished by using a combination of magnetic resonance imaging (MRI), catheterization, finite element modeling, and numerical optimization to analyze a porcine model (N=4) of posterolateral myocardial infarction. Specifically, properties will be determined by minimizing the difference between in-vivo strains and volume calculated from MRI and finite element model predicted strains and volume. The results indicate that the infarct region is stiffer than the remote region, and that the infarct collagen fibers become more circumferentially oriented 1 week post-MI. These findings are consistent with previous studies, which employed ex-vivo techniques. The proposed methodology will ultimately provide a means of predicting remote and infarct mechanical properties in-vivo at any time point post-MI.

1. Introduction

Nearly 8 million Americans have suffered a myocardial infarction (MI), with over 700,000 additional MI’s occurring each year (Go et al. 2014). Left ventricular (LV) remodeling is the maladaptive structural and functional sequelae of MI that leads to symptomatic heart failure. Infarct expansion (i.e. stretching) has been identified as the initiating and sustaining phenomenon that drives adverse LV remodeling after MI (Jackson et al. 2002). In spite of the importance of infarct material properties on infarct expansion, little data exists on the time-dependent changes in this critical parameter during the remodeling process. Such data is crucial to a full understanding of MI-induced LV remodeling as well as for the development of new and more effective therapies to treat and/or prevent the development of heart failure after MI.

There have been a number of experimental studies that have sought to characterize the structural and functional changes that occur at discrete time points after the induction of MI. Holmes et al. (Holmes et al. 1997) investigated the collagen content and strain patterns in isolated pig hearts 3 weeks after a lateral wall infarct was induced. In that study, markers were embedded transmurally through the LV wall, allowing for strain to be calculated during passive inflation. It was found that the circumferential component of strain decreased significantly, indicating increased stiffness along that direction. Histological examination found significant collagen deposition in the MI. In addition, the collagen fibers were highly aligned and oriented within 30 degrees of the circumferential axis. In a similar study, Fomovsky et al. (Fomovsky et al. 2012) evaluated the collagen structure in rats which had undergone either apical or anterior wall infarction. In both cases, collagen content increased due to infarction. However, collagen fibers in the anterior wall infarcts tended to align with the circumferential axis, while the apical infarcts had more randomly distributed fibers. These results indicate that the location of the infarct plays a crucial role in the structure of the collagen fibers.

Few studies have been conducted that experimentally quantify the stiffness of infarcted tissue by means of mechanical testing. Gupta et al. (Gupta et al. 1994) investigated the stiffness of anteroapical aneurysms in sheep over a 6 week period. Equibiaxial stretch testing was performed on infarcted and remote tissue within 4 hours of MI induction, as well as 1, 2, and 6 weeks post-MI. The results showed a significant increase in stiffness leading up to the 2 week mark, and then showed a significant decrease in stiffness at 6 weeks. Morita et al. (Morita et al. 2011) performed equibiaxial testing of infarcted tissue from control and treated sheep 8 weeks after anteroapical infarction. At that time point, the infarct stiffness was isotropic in the control case, but showed increased longitudinal stiffness in the treated case. Zhang et al. (Zhang et al. 2010) performed diffusion tensor magnetic resonance imaging (DTMRI) and equibiaxial testing on pig hearts 2 days after infarction. In this study, infarcts were generated in the lateral wall and apex of the LV, as well as in the lateral wall of the RV. The biaxial stress-stretch curves show anisotropic stiffness in the lateral wall infarcts and isotropic stiffness in the apical infarct. The results of the imaging study indicate that myofibers are not detectable within the infarcted region using DTMRI. Wu et al. (Wu et al. 2009) also performed DTMRI on porcine hearts with either lateral wall or apical infarcts, but at 4 and 13 weeks post-infarction. It was found that the fractal anisotropy and diffusivity in the infarct region were significantly altered relative to controls, and the fiber continuity was disrupted within the infarct. In addition, these parameters were found to correlate with infarct size. Due to current limitations with DTMRI in quantify collagen fiber orientation within the MI, histology is still required to assess these fiber angles post-infarction. However, DTMRI is making rapid advancements in the area of MI imaging.

Techniques have been developed that use a combination of magnetic resonance imaging (MRI), catheterization, and finite element modeling in order to estimate in-vivo myocardial material properties in a non-invasive manner. In all of these studies, the finite element models were based on anatomically correct geometry, which was contoured from MRI data. Moulton et al. (Moulton et al. 1996) used 2D tagged MRI, finite element models with an isotropic material law and linearized strain, and a Levenberg-Marquardt optimization method to estimate the passive material properties of the healthy canine heart. This work utilized a single short axis slice to determine in-plane strains and compare them to a 2D finite element model. Wang et al. (Wang et al. 2009) used 3D tagged MRI, nonlinear finite element modeling, and a sequential quadratic programming optimization method to estimate passive properties in a single canine heart. The material parameters were fitted to a transversely-isotropic hyperelastic constitutive law (Guccione et al. 1991). Fiber architecture from diffusion tensor imaging was incorporated into the finite element models, rather than using histological data. Recently, Xi et al. (Xi et al. 2013) used 3D tagged MRI, nonlinear finite element modeling, and the method of parameter sweeps to estimate passive properties in a healthy human ventricle and two diseased human ventricles. In that study, the Guccione transversely-isotropic hyperelastic constitutive law was also employed (Guccione et al. 1991), but the fiber angles were assigned to vary from −60 degrees at the epicardium to 60 degrees at the endocardium based on histological data. The results indicated that myocardial stiffness increased in the diseased cases. Sun et al. (Sun et al. 2009) used 3D tagged MRI, nonlinear finite element modeling, and a successive response surface methodology to estimate active material properties in the remote and border zone regions of sheep ventricles with anteroapical aneurysm 8 weeks post-MI. This same approach was employed by Wenk et al. (Wenk et al. 2011) to evaluate active properties in the remote region and border zone of sheep ventricles with posterobasal infarction. In both of these studies, the border zone contractility was found to be depressed, relative to the remote region.

Other work has been done to assess the passive material properties in isolated hearts. Okamoto et al. (Okamoto et al. 2000) used 3D tagged MRI and finite element modeling to estimate myocardial properties in isolated arrested canine hearts. In this work, an experimental suction apparatus was attached to the epicardial wall in order to generate deformations with shear strains. Augenstein et al. (Augenstein et al. 2006) used SPAMM (SPAtial Modulation of Magnetization) MRI, catheterization, and 3D finite element modeling to estimate the material properties in arrested pig hearts that underwent passive inflation. Unlike previous studies, fiber architecture from diffusion tensor imaging was incorporated into the finite element models, rather than using histological data. Nair et al. (Nair et al. 2007) used previously measured epicardial strain at 2 points on the surface of an isolated rabbit heart, along with a 3D finite element model, to estimate material parameters. The number of strain points used did not yield much information about myocardial deformation. However, the focus of this work was to present the use of a genetic algorithm optimization scheme for estimating material parameters.

Each of the aforementioned in-vivo and isolated heart studies focused on determining properties only in viable myocardium. To our knowledge, no such studies have determined the in-vivo passive material properties of a myocardial infarction. The goal of the current study is to quantify the in-vivo material properties of infarcted and remote myocardium in a porcine model 1 week after MI, as well as the orientation of collagen fibers in the infarcted zone, rather than assigning them a priori. This will be accomplished by using a combination of MRI, catheterization, and finite element modeling. Properties will be determined by using an optimization scheme to minimize the difference between in-vivo strains and volume calculated from MRI and finite element model predicted strains and volume. This investigation will ultimately provide a means of predicting remote and infarct mechanical properties at any time point post-MI.

2. Methods

2.1. Animal Model of MI

The animals used in this work received care in compliance with the protocols approved by the Institutional Animal Care and Use Committee at the University of Pennsylvania in accordance with the guidelines for humane care (National Institutes of Health Publication 85–23, revised 1996).

Myocardial infarction was induced in adult male Yorkshire swine (N=4) weighing approximately 40 kg. The pigs were anesthetized and the LV free wall exposed through a left thoracotomy. A posterior MI was induced by suture ligation of the left circumflex artery (LCX) and select obtuse marginal (OM) branches. An example of the ligature points for one of the cases is shown in Figure 1. In general, this ligation approach produced an infarct comprising 20–25% of the left ventricle. Ten MRI-compatible platinum wire markers were sutured to the epicardium around the infarct periphery immediately following infarction to enable infarct contouring from subsequent MRI acquisitions.

Figure 1.

Example of coronary artery ligation pattern for case 1. Note that the LV has been opened through the intraventricular septum and “unrolled”. The infarct involves the posterolateral wall of the LV; the anterior and septal walls are spared. Markers bound the infarct region. LAD=left anterior descending artery; D=diagonal branch artery; OM=obtuse marginal branch artery; PD=posterior descending artery: INF=infarct

2.2. MRI Acquisition and Image Analysis

Cardiac MRI was performed 1 week post-MI using a 3T Seimens Trio A Tim Magnetom scanner (Seimens; Malvern, PA) (Figure 2). For each MRI scan, a high fidelity pressure transducer (Millar Instruments; Houston, TX) was guided into the LV for cardiac gating. The measured pressure was later used to load the FE model (Figure 3). Images were collected using two techniques: 1) 3D SPAMM was performed in order to assess regional wall strain and 2) late-gadolinium enhanced (LGE) images were taken to further define the infarct area (Figure 2). The LV pressure was used to trigger the tagging sequence to occur at isovolumic relaxation, as opposed to the onset of systole, which is done using EKG triggering. As a result, the tags remain strong throughout diastole. The 3D SPAMM tagged sequence was performed with the following parameters: field of view = 260 mm × 260 mm, acquisition matrix = 256 × 128, pixel size = 1.015 mm × 1.015 mm, repetition time = 34.4ms, tag spacing = 6mm, bandwidth = 330Hz/pixel, slice thickness = 2mm, averages = 4.

Figure 2.

Short axis view of 3D SPAMM images from case 1 at early diastole with (a) endocardial and (b) epicardial contours, and short axis view at end-diastole with (c) endocardial contour. Short axis view of LGE image at (d) early diastole, with marker circled and boundaries defined.

Figure 3.

Pressure vs. time collected from LV catheterization of case 2. The circles indicate the early diastolic and end-diastolic time points.

The endocardium and epicardium of the LV were contoured from the 3D SPAMM images using ImageJ software (NIH; Bethesda, MD). The reference contours were generated at early diastolic filling (Figure 2), following isovolumic relaxation, in order to generate the animal-specific ventricular geometry. Additionally, the endocardium was contoured at end-diastole in order to calculate LV volume. Infarct boundaries were created by identifying the platinum epicardial markers in the 3D SPAMM images. The total infarct area was confirmed by comparing marker positions with the enhanced region in the LGE images. Diastolic strain was calculated from the 3D SPAMM images using a custom optical flow plug-in for ImageJ to derive 3D displacement flow fields (Xu et al. 2010). It should be noted that the LV strain, volume, and contour data were all generated from the 3D SPAMM images and matched with simultaneous pressure measurements. This ensures continuity between all data used to generate the model.

2.3. FE Model Generation

The reference state for the FE model was taken to be at early diastole, where LV pressure is at a minimum and is therefore closest to an in-vivo stress-free state. Each set of endocardial and epicardial reference contours were fit with 3D surfaces to represent the animal-specific geometry (Rapidform; INUS Technology, Inc., Sunnyvale, CA). The boundary between the infarct and remote region was defined using 3D spline curves that were created by visually tracing infarct marker positions from MR images and then projecting onto the endocardial and epicardial surfaces. The FE mesh was produced by filling the volume between endocardial and epicardial surfaces with hexahedral brick elements incorporating a trilinear interpolation scheme (TrueGrid; XYZ Scientific, Inc., Livermore, CA, USA). After performing a mesh convergence study, the wall of each FE model consisted of 3 elements transmurally, 36 elements circumferentially, and 12 elements longitudinally, with a patch of 192 elements at the apex. A transmural thickness of three elements deep was previously found to be sufficient for accurate volume calculations while maintaining computational efficiency (Sun et al. 2009). The infarct and remote regions were designated as two separate materials, allowing for different properties to be assigned to each (Figure 4). The inner endocardial surface was lined with linearly elastic shell elements to create an enclosed airbag for computing left ventricle cavity volume. The airbag had no influence on the mechanical response of the myocardium, as it was modeled as extremely soft with a Young’s modulus of 1 × 10⁻¹⁰ kPa and Poisson’s ratio of 0.3. Myofiber angles were assigned to each hexahedral element using a custom MATLAB script. The angles were appointed at the endocardial and epicardial surfaces and varied linearly through the transmural thickness. For the remote region, angles were fixed at −37 degrees at the epicardium and 83 degrees at the endocardium, with respect to the circumferential direction, based on previous studies with porcine ventricles (Hooks et al. 2007, Lee et al. 2012). The fiber angles in the infarct region were parameterized as part of the optimization, which will be discussed in a later section. Boundary conditions were implemented at the base of the LV to fully constrain displacement on the epicardial-basal edge, while allowing the remaining basal nodes to move in the circumferential-radial plane. The endocardial wall was loaded with a pressure boundary condition, which was based on the time course of the measured animal-specific diastolic pressure.

Figure 4.

Finite element models of each infarcted porcine LV for (a) case 1, (b) case 2, (c) case 3, and (d) case 4. The infarcted regions are shown as blue element and the remote as red.

2.4. Constitutive model

The material response of the myocardium was assumed to be nearly incompressible, transversely isotropic, and hyperelastic. The diastolic mechanics are described by the strain energy function for passive myocardium developed by Guccione et al. (Guccione et al. 1991), in which a Fung-type exponential relation is used

$$W=\frac{1}{2}C(e^Q-1),$$ (1)

with transverse-isotropy given by

$$Q=b_f{E}_{11}^2+b_t(E_{22}^2+E_{33}^2+E_{23}^2+E_{32}^2)+b_{fs}(E_{12}^2+E_{21}^2+E_{13}^2+E_{31}^2).$$ (2)

The constants C, b_f, b_t, b_fs, are diastolic myocardial material parameters, E₁₁ is the Green-Lagrange strain in the fiber direction, E₂₂ is the cross-fiber in-plane strain, E₃₃ is the radial strain transverse to the fiber direction, and the rest are shear strains. The assumption of near incompressibility of the myocardium requires the decoupling of the strain energy function into dilatational and deviatoric components,

$$W=U(J)+\stackrel{\sim }{W}(\stackrel{\sim }{\mathbf{C}})$$ (3)

where J is the Jacobian of the deformation gradient tensor, C̃ is the deviatoric right Cauchy-Green deformation tensor, U is the dilatational (volumetric) contribution and W̃ is the deviatoric contribution of the strain energy function. Near incompressibility is enforced using a penalty method on the dilatational part. The stress is derived by taking the derivative of the strain energy function with respect to the deformation yielding,

$$\mathbf{S}=pJ{\mathbf{C}}^{-1}+2J^{-2/3}\text{Dev}\left(\frac{\partial \stackrel{\sim }{W}}{\partial \stackrel{\sim }{\mathbf{C}}}\right)$$ (4)

where S is the second Piola-Kirchoff stress tensor, p is the hydrostatic pressure introduced to ensure incompressibility (Wenk et al. 2010), and Dev is the deviatoric projection operator,

$$\text{Dev}(•)=(•)-\frac{1}{3}\left([•]:\mathbf{C}\right){\mathbf{C}}^{-1}$$ (5)

This strain energy function was used to describe both the remote and infarct regions, and was implemented through a material subroutine in the nonlinear FE solver LS-DYNA (Livermore Software Technology Corporation, Livermore, CA, USA). The FE simulations were conducted using an explicit (central difference) time integration scheme with adaptive time stepping for stability.

2.5. Optimization

A genetic algorithm (GA) technique was chosen for the optimization, due to the large number of parameters to be optimized, and was performed using the software LS-OPT (Livermore Software Technology Corporation, Livermore, CA). Genetic algorithms perform well as global optimizers, whereas gradient-based methods can get stuck in local optima when exploring a large parameter space. In a study by Nair et al. (Nair et al. 2007) GAs were found to be a robust method for optimizing cardiac material parameters in 3D models.

A total of 10 parameters were optimized in the current study. These included the diastolic material parameters from the strain energy function (C, b_f, b_t, b_fs) for both the remote and infarct regions, as well as the epicardial and endocardial fiber angles in the infarct. For the infarct, the fiber angles were assumed to vary linearly, as in the remote region. The ranges that were defined for each parameter are given in Table 1. The bounds for the remote parameters were based on previous reported values for normal myocardium (Augenstein et al. 2006), whereas the range for the infarct parameters was based on numerous test runs of the optimization scheme. These values were found to provide sufficient space for the optimization to converge without the bounds driving the solution.

Table 1.

Parameter ranges used during the optimization.

		Min	Max
Remote	C (kPa)	0.1	10
	bf	1	100
	bt	1	50
	bfs	1	50
Infarct	C (kPa)	0.1	25
	bf	1	250
	bt	1	60
	bfs	1	60
	epi-angle (deg)	−50	0
	endo-angle (deg)	30	90

The objective function that was minimized during the optimization was taken to be the mean squared error (MSE) between experimentally measured and FE predicted strains at an array of points in the midwall of the LV. Specifically, the centroids of midwall elements were compared to the nearest LV points from the MRI data where strain was measured. This was possible because the LV contours used to generate the FE model and experimental strain points were from the same 3D SPAMM data. A total of seven rings of elements starting from the base, excluding the first ring, were used for the strain comparison. This led to a total of 252 points where strain was compared. Additionally the normalized difference in LV cavity volume was added to the objective function and carried an equal weight as that of the strain points. The MSE was defined as

$$\mathrm{M}\text{SE}=\sum _{n=1}^N\sum _{\begin{array}{l}i=1,2,3;\\ j=1,2,3;\end{array}}{w}_n{\left(E_{ij,n}-{\overline{E}}_{ij,n}\right)}^2+\sum _{m=1}^M{w}_m{\left(\frac{V_m-{\overline{V}}_m}{{\overline{V}}_m}\right)}^2$$ (6)

where n is the strain point, N is the total number of strain points, E_ij,n is the computed FE end-diastolic strain at each point, and V_m is the computed FE end-diastolic LV volume. The overbar represents animal-specific in-vivo measurements. The weights in the MSE (w_n and w_m) were assigned to be one for all of the optimizations performed in this study. An additional constraint was placed on the volume to ensure that it stayed bounded within ±5% of the experimentally measured volume. This constraint was enforced using the efficient constraint handling method (Deb 2000).

A flowchart of the optimization process can be found in Figure 5. Briefly, the process begins with the generation of an initial population of 100 random designs. A “design” is a unique combination of the 10 parameters within each respective range, which are assigned to the FE model. After some initial testing, it was found that the population size of 100 FE simulations over 30 generations (iterations) allowed the genetic algorithm to produce consistently optimized results. After the initialization, a custom MATLAB script takes the fiber angles and modifies the fiber architecture of the FE model for each of the 100 designs. LS-DYNA then runs the FE simulations for each model, after which LS-OPT begins the optimization process (Stander et al. 2012). The designs are analyzed using the objective function defined previously. The 2 designs with the lowest MSE are saved for the next generation in a process called elitism. A tournament selection was chosen as the reproduction operator; in accordance, two individual designs are randomly selected and the one with the lowest MSE is chosen to participate in mating. The designs of the next generation, the children, are created through an exploration operator referred to as crossover. This operator randomly selects a number of parents to mate to produce a number of children. The children carry qualities of each of their parents in the sense that their parameter values are the same as that of a parent or somewhere between the ranges of the parents. Using the standard (2,2) strategy for crossover, two children are created by and replace two parents. Therefore the sole victor decided in the tournament selection mates with each of the other 99 designs in the generation. Mutation is also involved in the creation of the child population and incites random changes in parameters based on a designated mutation probability. Once the child population of 100 is created, the 2 with the highest MSE are replaced by the 2 elites saved from the parent population. Hence the new generation is fully created and the process continues with the creation of the next set of models and simulations.

Figure 5.

Flowchart of the optimization procedure showing the loop for a single generation.

Once the optimization was fully converged, the 95% confidence intervals were calculated for the last iteration of each case. The intervals were based on the parameters used to generate the 100 simulations that were run during the last iteration and represent the range over which the optimization is 95% confident that the parameter values exist.

2.6. Simulated Equi-biaxial Extension

In order to facilitate a clearer interpretation of the material parameters, equi-biaxial extension tests were simulated using the parameters determined from the optimization. The constitutive law, given in Eqn. (1) and (2), was formulated for the case of planar equi-biaxial extension, as shown in Guccione et al. (Guccione et al. 1991). The resulting stress-strain equations were implemented into an in-house MATLAB script, where stress was calculated at the same set of strain points for each case. In the formulation, the fiber and cross-fiber directions were assumed to align with the loading directions. Additionally, the simulated sample was assumed to be a thin, incompressible segment of tissue in which there were no changes in fiber orientation through the thickness.

3. Results

Each of the 4 cases converged to an optimal set of material parameters before reaching 30 generations. Since there are 10 parameters in each case, it is not possible to visualize the entire parameter space. Therefore, Figure 6 shows of a portion of the parameter space generated in the optimization run for case 4. Specifically, Figure 6a shows the resulting MSE values for every combination of C,_infarct and b_f,infarct within the bounds of those parameters. This type of plot could be generated for any combination of parameters. Each point represents a unique design generated by the optimization. It can be seen in Figure 6a that the GA systematically searched the full parameter space, which is why the points are scattered throughout. Note that the designs from early generations are evenly distributed throughout the parameter space, but that later generations begin to cluster around the optimal solution. Figure 6b shows a close-up of the convergence of parameter b_f,infarct. Note that a cluster forms over the final value of b_f,infarct and that later generations have lower MSE values compared to earlier generations. Figure 7 shows how the best design found at each generation changes over the course of the optimization for case 2. The values of the non-angle parameters were normalized by their respective parameter ranges given in Table 2. Note that after generation 24 the parameter values do not change, indicating that the GA has converged to the optimal solution. All 4 of the cases converged with the same trends presented in the representative examples. Additionally, each optimization was run multiple times, leading to different sets of initial random designs within the parameter space. However, the final converged results were nearly identical each time.

Figure 6.

Optimization results for case 4 depicting two infarct parameters. (a) 3D plot of b_f versus C with the MSE on the vertical axis. Each point represents a unique design generated by the optimization. (b) 2D slice of plot (a) with b_f versus MSE. The converging cluster of points coincides with the red boxed area in (a).

Figure 7.

Optimization results by parameter for case 2. The non-angle values in Table 1 were normalized by their respective ranges allowed by the optimization. Results for the remote region can be seen in (a), whereas the infarct results are in (b) with the angles in (c).

Table 2.

Optimization results for each case, along with the mean and standard deviation for each of the parameters.

		Case
		1	2	3	4	Mean ± Std
Remote	C (kPa)	1.99	4.89	0.917	0.367	2.04 ± 2.02
	bf	9.93	8.08	12.49	99.31	32.45 ± 44.61
	bt	2.32	2.78	6.00	38.87	12.49 ± 17.66
	bfs	46.53	7.77	19.81	4.90	19.75 ± 18.98
Infarct	C (kPa)	0.109	1.01	1.20	1.22	0.88 ± 0.53
	bf	221.8	182.3	112.3	179.8	174.07 ± 45.47
	bt	55.53	12.02	11.07	31.72	27.58 ± 20.92
	bfs	48.16	11.79	24.62	25.31	27.47 ± 15.13
	epi-angle (deg)	−49.21	−22.31	−0.45	−0.829	−18.20 ± 23.06
	endo-angle (deg)	43.28	41.31	44.66	41.32	42.65 ± 1.64
	MSE	9.30	9.80	11.62	4.28	8.75 ± 3.14

The optimized parameters for each of the 4 cases are given in Table 2. On average, the exponential parameters (b_f, b_t, b_fs) were found to be greater for the infarct region than the remote region. However, the average value of the C parameter was greater for the remote region, compared to the infarct region. The collagen fiber orientation at both the endocardium and epicardium became more circumferential in the infarct region, compared to the myofiber orientation in the remote region. The 95% confidence intervals for each parameter are given in Table 3. It should be noted that the intervals are narrow relative to the parameter values reported in Table 2. This implies that after the initial broad search, the optimization converged to a small region of the parameter space where the final converged values were found. Additionally, the MSE values ranged from 4.28 to 11.62 and are comparable to MSE values achieved in previous studies that employed a similar approach (Sun et al. 2009, Wenk et al. 2011). In fact, the current approach used 1.5 to 2 times more strain components in the MSE calculation than the previous studies. This implies that the agreement, between the experimentally measured and FE predicted quantities, is improved in the current work. In each of the 4 cases, the FE predicted LV end-diastolic volume was within ± 5% of the measured value when loaded to the specified measured pressure.

Table 3.

The 95% confidence intervals are given for each parameter, which were calculated during the last iteration of each optimization.

		Case
		1	2	3	4
Remote	C (kPa)	± 0.14	± 0.016	± 0.015	± 0.008
	bf	± 1.94	± 0.14	± 0.24	± 4.97
	bt	± 0.54	± 0.055	± 0.083	± 0.46
	bfs	± 0.59	± 0.21	± 0.84	± 0.18
Infarct	C (kPa)	± 0.032	± 0.064	± 0.027	± 0.043
	bf	± 1.85	± 1.29	± 1.38	± 1.81
	bt	± 0.55	± 0.64	± 0.20	± 0.25
	bfs	± 0.53	± 0.72	± 0.17	± 0.82
	epi-angle (deg)	± 1.1	± 1.23	± 0.14	± 0.088
	endo-angle (deg)	± 0.29	± 1.50	± 0.29	± 1.32

The MSE values reported in Table 2 represent the sum of all local MSE values that were calculated using strain from contributing midwall elements and the LV volume at end-diastole. While the global MSE value is useful for understanding the overall robustness of the parameter fitting, visualizing the spatial heterogeneity in the element level MSE values within the LV allows for interpreting where the FE predicted and MRI measured strains coincide. Figure 8a shows a short axis slice near mid-ventricle of the FE model in case 1. Figure 8b shows the corresponding MSE values for the midwall elements within that slice. These results are consistent with all of the cases in the current study. It can be seen that the MSE values are lowest in the free wall and infarct regions, and are largest in the regions were the RV wall connects to the LV wall (RV insertion points). The MSE is also elevated in the septum, relative to the free wall and infarct.

Figure 8.

(a) Short axis slice near mid-ventricle of the FE model for case 1. Note that several regions are labeled around the circumference to indicate where the LV interacts with the RV. (b) Distribution of MSE values within each midwall element around the circumference of the LV. The labels correspond to the regions in (a) and the circumferential numbering is clockwise from 1.

The reference state, deformed state, and maximum principal strains for case 3 are shown in Figure 9. It should be noted that these results are representative of the trends seen in the other cases analyzed in the current study. The dilation in the short axis slice of the FE model (Figure 9a) is comparable to that seen in-vivo from the contoured MR images (Figures 2a and 2c). In order to quantify the deformation more directly, the maximum principal strain distributions are given in Figures 9c and 9d. It can be seen that the deformation in the infarct region is reduced relative to the remote region, as indicated by the decrease in strain. This is intuitive, since the endocardium is subjected to the same pressure load, but the stiffness of the infarct region is greater than the remote region. As an additional check on the accuracy of the model, the pressure-volume relationship was evaluated from early diastole to end-diastole (Figure 10). This was done by comparing the cavity volume of the LV measured from MRI to that calculated from the FE model. These values are directly compared, since the model was loaded with the experimentally measured pressure. It can be seen that the FE model is within ± 5% of the measured value at all time points during diastolic filling. This was true for all cases.

Figure 9.

Reference and deformed state of the FE model for case 3, which is representative of all cases. (a) Short axis slice at mid-ventricle and (b) long axis slice through the remote and infarct region. The wire frame represents the reference state (yellow = remote, green = infarct) and the shaded regions represent the deformed state (red = remote, blue = infarct).

Figure 10.

Pressure-volume relationship for case 3. Square markers represent the experimentally measured volume at the measured pressure, the horizontal bars represent ±5% of the experimentally measured volume, and the triangular markers represent the volume calculated from the FE model at the same pressure.

The results of the simulated equi-biaxial extension tests for all of the cases are shown in Figure 11a and 11b. It can be seen that there is variability between each of the cases, with case 4 showing the highest stiffness in both the remote and infarct region. In order to give a general comparison of stiffness in the remote and infarct regions, the stress-strain curves for all of the cases were averaged in each region. It can be seen in Figure 12 that the infarct region is stiffer than the remote region. Additionally, there appears to be more anisotropy in the infarct region compared to the remote region.

Figure 11.

Simulated equi-biaxial extension tests of optimized parameters. The values in Table 2 were input into the constitutive equation, and these plots were then generated using an in-house MATLAB script for biaxial testing. The remote results can be seen in (a) and the results from the infarct region in (b).

Figure 12.

Average stress-strain curves from simulated equi-biaxial extension testing. For each strain point in Figure 10, the corresponding stresses of each case were averaged for the remote and infarcted regions, respectively.

4. Discussion

The goal of the current study was to quantify the in-vivo passive material properties of infarcted and remote myocardium 1 week after MI using a combination of MRI, catheterization, FE modeling, and numerical optimization. To our knowledge, no previous studies have predicted the in-vivo passive material properties, or the collagen fiber orientation, in a myocardial infarction. The results of the current study indicate that the infarct region is stiffer than the remote region, and that the infarct collagen fibers become more circumferentially oriented 1 week post-MI. These findings are consistent with previous studies, which employed ex-vivo techniques.

The infarct collagen fiber architecture predicted in the current study indicates that both the epicardial and endocardial angles become more circumferentially oriented at 1 week after a posterolateral MI, compared to myofiber angles in the remote region. The endocardial angle shifted from 83 degrees to approximately 43 degrees, while the epicardial angle shifted from −37 degrees to approximately −18 degrees, a difference of 40 degrees and 19 degrees, respectively. This result agrees with findings reported by Holmes et al. (Holmes et al. 1997) that showed a change in endocardial collagen angles of roughly 32 degrees towards the circumferential direction in pigs 3 weeks after a posterolateral MI, compared to sham controls. In addition, Rouillard and Holmes (Rouillard et al. 2012) reported a change of nearly 40 degrees towards the circumferential direction in both the endocardial and epicardial collagen fiber angles in rats 3 weeks post-MI. While there is little data in the literature regarding changes in collagen fiber orientation due to MI, the general trend is that fibers become more circumferential over time as the MI remodels (Holmes et al. 2005) when the infarct is located away from the LV apex.

The stiffness in the infarct region was found to be greater than the remote myocardium. This can be inferred when comparing the exponential parameter values for the infarct region to those for the remote (Table 2). However, it is easier to visualize the change in stiffness and anisotropy by using simulated equibiaxial stress-strain curves, as shown in Figure 12. Though few studies have performed experimental biaxial testing on infarcted tissue, Gupta et al. (Gupta et al. 1994) showed that the stiffness of apical aneurysms was greater than the remote myocardium in sheep 1 week post infarction. In addition, Zhang et al. (Zhang et al. 2010) performed biaxial testing on infarcted tissue from the lateral wall of pig ventricles 2 days post-MI. Those results showed increased extensibility in the cross-fiber direction, which was attributed to fiber disruption and necrosis that occurs within the first few days post-MI. The current results show much greater stiffness in both the fiber and cross-fiber directions at 1 week post-MI, which are consistent with the timeline of infarct remodeling outlined by Holmes et al. (Holmes et al. 2005).

The material parameters determined in the current study, in both the remote and infarct regions, showed variability from case to case (Table 2). Specifically, case 4 in the current study was much stiffer than the other cases. It was found that the change in volume from early diastole to end-diastole was lower for this animal compared to the other cases, but the end-diastolic pressure for this animal was comparable to the other cases. These experimental results imply that the ventricle would be stiffer than the others, which was confirmed by the material parameters that were estimated from the optimization. It should be noted, this type of variability was seen in several previous studies of normal myocardium, where the values of C, b_f, b_t, and b_fs varied by as much as a factor of 5 within the same group of animals (Okamoto et al. 2000, Walker et al. 2005, Augenstein et al. 2006). Given the results of the current study, as well as previous studies, there appears to be natural variability in ventricular stiffness within an animal group.

In general, the material parameters determined for the remote region in the current study were stiffer than those found in previous studies of normal myocardium (Guccione et al. 1991, Augenstein et al. 2005, Wang et al. 2009, Kichula et al. 2013). Since remodeling also takes place in the remote myocardium, this could alter the stiffness as the MI progresses. To our knowledge, no other studies have sought to quantify the material parameters in a posterolateral infarct by fitting a constitutive model with experimental data. Wenk el al. (Wenk et al. 2010) modeled a sheep LV with a posterobasal infarct that was 10 times stiffer than the remote region. This value was based on data reported from apical aneurysms and was imposed by scaling the C parameter, which stiffened all directions by the same amount. However, the current results show that anisotropy changes due to infarction (Figure 12). The current study also sought to estimate fiber orientations in the infarct region using the optimization scheme. Kroon et al. (Kroon et al. 2009) used optimization to estimate fiber orientations in a FE model of the LV by minimizing fiber-cross fiber shear strain. However, that work was implemented for normal myocardium.

Due to the nature of the constitutive equation, there are coupled interactions between the material coefficients. In particular, the parameter C affects stiffness in all directions, since it scales the stress in an isotropic manner. This issue has been addressed in previous studies, where MRI, finite element methods, and optimization were used to estimate passive myocardial properties. In work by Wang et al. (Wang et al. 2009) the optimization routine was designed to first determine C and then sequentially determine the remaining exponential parameters. In work by Xi et al. (Xi et al. 2013) the value of C was fixed to be 1 kPa, based on average values reported in other studies, which allowed the remaining parameters to be optimized independent of C. In the current work, the C parameters for both the remote and infarct regions were included as part of the design variables to be determined by the genetic algorithm. It was found that both C parameters converged to their final value after 10 iterations and remained steady as the other parameters continued to be optimized (Figure 7). This trend occurred in all 4 cases that were run for this study, which implies that the genetic algorithm consistently determined the C parameter before the others. However, additional preliminary tests were conducted where the value of C was fixed in both the remote and infarct regions to be a value of 1 kPa. The resulting MSE values either increased by 3% (indicating a worse fit) or they decreased by 3% (indicating a better fit). These results are intriguing, but require additional investigation that will be pursued in future work. For example, the use of other constitutive laws (Holzapfel and Ogden 2009) will be examined in order to determine which produces the lowest MSE when compared to the experimental measurements. Another trend that was consistent between all 4 cases in the current work was that the remaining parameters converged to their final value by the 24^th iteration (Figure 7). Thus, the final optimal set of parameters was fully determined well before the optimization terminated, leading to a steady MSE value for the final 7 iterations.

There are similarities and differences between the approach in the current study and previous work. MRI and catheterization were used to quantify wall motion and ventricular pressure, respectively, in both the current and previous studies (Sun et al. 2009, Wang et al. 2009, Wenk et al. 2011, Xi et al. 2013). In every case, finite element models were generated from the MRI-based geometry and loaded with the measured pressure, in order to simulate ventricular deformation. However, in work by Wang et al. (Wang et al. 2009) and Xi et al. (Xi et al. 2013) wall displacement was used in the optimization, whereas in the current study, and in work by Sun et al. (Sun et al. 2009) and Wenk et al. (Wenk et al. 2011), strain was used to quantify wall deformation in the optimization. Since displacement is the primary value measured by MRI and strain quantifies only deformation without rigid motion, each approach has distinct advantages.

There are several limitations associated with the approach presented in this work. In the current study, and the study by Xi et al. (Xi et al. 2013), average myofiber angles were assigned to the myocardium, which does not fully represent the spatially varying fiber architecture. In the future, fiber angles will be based on DTMRI data, similar to the study by Wang et al. (Wang et al. 2009), in order to incorporate more accurate spatial variation. Another limitation that affects the current study, as well as several previous studies, is the absence of a pericardium and a right ventricle (RV) in the FE model. The RV not only affects the pressure loading on the septum, but also the deformation that is induced by the interaction between the LV wall and RV insertion points, as seen in Figure 8 of the current study. These effects could alter the resulting parameters and will be incorporated in future studies. Additionally, the reference configuration of the FE model was based on early diastole, where pressure is at a minimum, but the ventricle is still partially loaded. In the future, techniques for determining the unloaded geometry will be incorporated, similar to Krishnamurthy et al. (Krishnamurthy et al. 2013). It has been shown by Xi et al. (Xi et al. 2013) that there may be residual active tension in the myocardium at early diastole, which was not taken into account in the current work. While this is not a concern in the infarct region, since there are no contracting myocytes, this could affect the remote region leading to different mechanical properties. Alterations to the border zone properties were not addressed. In terms of the transition in material properties between the remote and infarct, it is difficult to see a gradient in fibrosis in the border zone when viewing the LGE images. The MI boundary appears to be very stark. Therefore, it was assumed the border zone gradient was not yet defined. Further experimental work is needed to define the border zone at 1 week post-MI. Finally, the strain field that was calculated from the 3D SPAMM images contained a small amount noise. This was minimized as much as possible by smoothing the strain at each point of interest with the neighboring values. Despite these limitations, the results of the current work are consistent with those of previous computational and experimental studies.

5. Conclusions

The technique presented in the current study was able to determine the in-vivo passive material properties and collagen fiber orientation in a porcine model of myocardial infarction. Additionally, the passive material properties in the remote region were also determined. The results showed that the stiffness in the infarct was higher than the remote region, which is consistent with previous ex-vivo studies. In future work, this technique will be used to assess changes in infarct stiffness and fiber architecture over a serial time course. Very few studies have investigated these types of serial changes, which are critical for understanding the mechanisms that drive remodeling. The ultimate goal is to use this technique in the design of therapies for treating MI.