Multiscale Fiber Remodeling in the Infarcted Left Ventricle using a Stress-Based Reorientation Law

Abstract

The organization of myofibers and extra cellular matrix within the myocardium plays a significant role in defining cardiac function. When pathological events occur, such as myocardial infarction (MI), this organization can become disrupted, leading to degraded pumping performance. The current study proposes a multiscale finite element (FE) framework to determine realistic fiber distributions in the left ventricle (LV). This is achieved by implementing a stress-based fiber reorientation law, which seeks to align the fibers with local traction vectors, such that contractile force and load bearing capabilities are maximized. By utilizing the total stress (passive and active), both myofibers and collagen fibers are reoriented. Simulations are conducted to predict the baseline fiber configuration in a normal LV as well as the adverse fiber reorientation that occurs due to different size MIs. The baseline model successfully captures the transmural variation of helical fiber angles within the LV wall, as well as the transverse fiber angle variation from base to apex. In the models of MI, the patterns of fiber reorientation in the infarct, border zone, and remote regions closely align with previous experimental findings, with a significant increase in fibers oriented in a left-handed helical configuration and increased dispersion in the infarct region. Furthermore, the severity of fiber reorientation and impairment of pumping performance both showed a correlation with the size of the infarct. The proposed multiscale modeling framework allows for the effective prediction of adverse remodeling and offers the potential for assessing the effectiveness of therapeutic interventions in the future.

Graphical Abstract

1. Introduction

The architecture of myofibers and collagen within the myocardium plays a significant role in the mechanical behavior of the heart [1–3]. Epicardial myofibers in the left ventricle (LV) are arranged in a left-handed helical configuration, which gradually transitions into a right-handed helical arrangement in the endocardium [2, 4–6]. The myofibers also exhibit a transverse angle relative to the LV cavity. At the mid-wall of the LV, the myofiber transverse angles display negative values (inward toward the LV cavity) near the apex and positive values (outward from the LV cavity) near the base [4]. In addition, myofibers are surrounded by a network of collagen fibers, which are composed of different types. Endomysial collagen struts interconnect myofibers and help transmit force during systole, whereas perimysial collagen runs parallel to myofibers and serves as a reinforcing element (protecting myofibers from overstretch) during diastole [7, 8].

The effect of myofiber architecture on cardiac muscle mechanics directly influences global pumping performance. Early geometric studies have demonstrated that the helical structure of the myofibers leads to more efficient ejection of blood during systolic shortening, as compared to purely circumferential or longitudinal distributions [1, 9]. Another study evaluated the Starling relationship and apex twist angle (a measure of torsion) by utilizing cardiac finite element (FE) models to assess variations in myofiber distributions. It was found that helical angles closer to healthy physiological values were associated with superior stroke volume generation and twist performance [10]. The transmural variation of myofibers across the cardiac wall also determines the distribution of mechanical quantities such as active muscle fiber stress, strain, and shear deformation [11]. Therefore, cardiac function is highly sensitive to the architecture of myofibers [12–17], and any alteration in their organization can disrupt optimal pumping performance.

The arrangement of myofibers and collagen can remodel due to adverse conditions such as myocardial infarction (MI). Experimental studies have demonstrated the disruption of cardiac myofiber architecture in all regions of the heart following an infarction, including the infarct itself, the adjacent border zone (BZ), and the remote regions [18–28]. In addition, the heart undergoes replacement fibrosis, which is the result of the healing process after cardiomyocyte necrosis. This preserves structural integrity after the removal of the dead cells and leads to the formation of a dense scar of collagen [29]. A notable observation is the increased proportion of fibers oriented in a left-handed helical configuration inside the infarcted segments of the LV [22–24]. Moreover, as measured by diffusion tensor magnetic resonance imaging (DTMRI), the infarcted myocardium exhibits decreased diffusion anisotropy, which is associated with microscopic fiber dispersion (i.e., increased variation in fiber angles within a region) [19, 22–25, 28, 30–36]. The degradation of myocardial tissue post infarction extends beyond mere fiber reorientation. Numerous studies have demonstrated alterations in material properties and volumetric growth within the infarcted regions of the myocardium [37–39].

Finite element modeling has become a widely used approach for investigating cardiac structure and function. This includes studies of healthy myocardium, as well as modeling the effects of MI and possible therapeutic strategies for treating the disease [14, 37–45]. Many of these cardiac models rely on rule-based assignment of myofiber angles that were derived from existing experimental findings [12, 17, 42, 43, 46–48], while others have used warping and projection methods to assign geometry-specific myofiber angles from DTMRI [49–51]. Given the impact of cardiac myofiber organization on pumping performance and the presence of pathological conditions that disrupt myofibers, it is crucial to adopt a flexible approach when assigning the initial myofiber distribution in cardiac models. Additionally, since myofiber and collagen architecture can change over time, due to pathological conditions such as MI, it is necessary to incorporate remodeling algorithms into the FE models to capture these effects.

Kroon et. al (2009) introduced a cardiac FE model capable of adapting myofiber orientation according to a remodeling law that minimizes shear strain. This approach is based on the assumption that shear strain in the myocardium breaks some of the extracellular matrix (ECM), and the formation of new ECM leads to permanent reorientation of the myofibers [52]. According to their findings, the chosen myofiber adaptation law improved the pumping performance of a normal LV model. Similarly, Washio et al. (2015) used an adaptive model to predict myofiber orientation in a healthy heart. The fiber reorientation was based on instantaneous branching of myofibers, which was determined by two independent optimization parameters: (1) local muscle workload and (2) contractile load impulse [53]. Since both optimization parameters were a function of “active” cardiac contraction, the prediction of infarction behavior may not be captured in such models. This is due to the lack of contraction in regions of pure scar tissue. In a later study involving physiological and pathological cases, Washio et al. (2020) demonstrated that the impulse-based mechanism is more suitable for myofiber reorientation [44]. However, in that study, it was assumed that the infarct region had some level of contractility, which may not occur in vivo. Recently, another study employed an agent-based model to describe fibroblast driven collagen remodeling that was regulated by chemical and mechanical cues in response to MI. The cell level model was upscaled into a 3D FE model of the LV that was restricted to simulating only the diastolic phase [54]. Additionally, other studies have used a stress-based law to investigate collagen remodeling in arterial tissue [55, 56].

The current study presents a stress-based fiber remodeling algorithm that was inspired by the previously discussed adaptation laws. Our approach extends the previous methods by incorporating active and passive stress as adaptation drivers, allowing the simulation to capture the remodeling that occurs throughout the entire cardiac cycle in both the myofibers and extracellular matrix. In this approach, fibers are inclined to reorient towards local traction vectors, which maximizes force generation during myofiber contraction (resulting in optimal pumping capacity) [10] and maximizes the load bearing capacity of the collagen fibers (by redirecting them in the direction of axial stress) [55]. This enables the model to predict the normal fiber architecture in a healthy LV and facilitates the model’s ability to remodel fibers under adverse conditions such as MI. The aim of the current study is to investigate fiber remodeling during the initial phases of infarction, preceding any long-term modifications in the LV geometry. We hypothesize that the proposed model can effectively predict fiber remodeling, and that MI exhibits notable changes in fiber orientation as the size of the infarct increases.

2. Materials and methods

2.1. Overview of the MyoFE framework

The current study presents a comprehensive modeling approach to investigate cardiac behavior using a modular multiscale FE framework, called MyoFE (Figure 1). This framework integrates various characteristics of cardiovascular function, including calcium handling, myofiber active contraction, and circulatory hemodynamics, in a closed loop model of the LV and systemic circulation. It should be noted that all model parameters are provided in Table S1 of the Supplementary Material.

Figure 1:

MyoFE framework schematic. A: The fiber reorientation algorithm reorients fibers using the myocardial stress from the LV model in the central framework. The updated fiber vectors are then passed to the central framework for the subsequent time step. The unit vector ${\mathit{f}}_0$ represents the initial direction of a fiber. ${\mathit{f}}_0$ reorients toward the local traction vector $\mathit{S}{\mathit{f}}_0$, where $\mathit{S}$ is the total stress tensor, encompassing both passive and active stresses. B: The central framework consists of the lumped parameter circulatory model and modules on the left. Detailed information for each module is provided in the text.

The active contraction mechanism is implemented using MyoSim, a molecular-level contraction model [57, 58]. At each time step, the MyoSim module calculates the active stress generated by sarcomeres using the mechanical configuration of the myofibers as well as the calcium concentration ([Ca2+]), generating myocardial contraction. In the current study, the calcium transient is defined a priori and is based on experimental measurements from previous studies [58, 59]. As the calcium concentration increases, contraction will initiate during the isovolumic phase, then shortening will occur during ejection, and finally relaxation will occur after the calcium concentration diminishes. It should be noted that this module can also be defined with a dynamic model, such as [60], if desired. The mechanical configuration of the myofibers, including changes in sarcomere length and total stress, are obtained through the 3D FE model of the LV (more details are in the FE formulation section).

The Windkessel circulation model [52] is used to impose the hemodynamic boundary condition associated with the LV cavity volume. Briefly, the total stress state in the LV wall of the 3D FE model (which includes the combined effects of passive and active stress) is balanced by the pressure in the LV cavity. By coupling the FE model and circulatory model, via a Lagrange multiplier, the cavity pressure can be determined. The LV pressure is then incorporated into the systemic circulation model to compute the LV cavity volume for the subsequent time step, thus enabling the model to capture the hemodynamic coupling between the LV and vasculature.

In the present study, a fiber reorientation module is implemented within the MyoFE framework. At each integration point in the FE mesh, the fiber orientation is computed from a stress-based reorientation law (more details in the Stress-Based Reorientation Law section). This formulation utilizes the myocardial stress state provided by the central framework to determine the updated fiber orientation (See Figure 1). The updated fiber configuration is then used in the central framework for the subsequent time step.

2.2. Circulation model

To establish hemodynamic boundary conditions within the LV cavity, the FE model is integrated with a Windkessel model of the closed systemic circulation. This circulation model is represented by a set of resistance (R) and capacitance (C) parameters, which correspond to various compartments within the circulatory system [52, 61]. In the current study, we employed a 3-compartment model of the circulation, which includes LV, arterial, and venous compartments. In each compartment, the rate of change in blood volume is defined as the difference between the amount of blood per unit time flowing into and out of each compartment as follows:

$$\begin{gathered} \frac{d{V}_{\mathit{\text{arteries}}}}{dt}=Q_{LVto\mathit{\text{arteries}}}-Q_{\mathit{\text{arteries}}to\mathit{\text{veins}}} \\ \frac{d{V}_{\mathit{\text{veins}}}}{dt}=Q_{\mathit{\text{arteries}}to\mathit{\text{veins}}}-Q_{\mathit{\text{veins}}toLV} \\ \frac{d{V}_{LV}}{dt}=Q_{\mathit{\text{veins}}toLV}-Q_{LVto\mathit{\text{arteries}}} \end{gathered}$$ (1)

Utilizing Ohm’s law, the blood flow from one compartment to another is equal to the pressure gradient between those neighboring compartments divided by the resistance between the compartments:

$$\begin{gathered} Q_{LVto\mathit{\text{arteries}}}=\{\begin{array}{cc}\frac{P_{LV}-P_{\mathit{\text{arteries}}}}{R_{\mathit{\text{arteries}}}}& \mathit{\text{when}}{P}_{LV}\ge P_{\mathit{\text{arteries}}}\\ 0& \mathit{\text{otherwise}}\end{array} \\ {Q}_{\mathit{\text{arteries}}to\mathit{\text{veins}}}=\frac{P_{\mathit{\text{arteries}}}-P_{\mathit{\text{veins}}}}{R_{\mathit{\text{peripheral}}}} \\ {Q}_{\mathit{\text{veins}}toLV}=\{\begin{array}{cc}\frac{P_{\mathit{\text{veins}}}-P_{LV}}{R_{\mathit{\text{veins}}}}& \mathit{\text{when}}{P}_{\mathit{\text{veins}}}\ge P_{LV}\\ 0& \mathit{\text{otherwise}}\end{array} \end{gathered}$$ (2)

Except for the LV, each compartment’s blood pressure is equal to its stressed blood volume – defined as the difference between its instantaneous and slack volumes– divided by its compliance:

$$\begin{gathered} P_{\mathit{\text{arteries}}}=\frac{V_{\mathit{\text{arteries}}}\left(t\right)-V_{\mathit{\text{arteries}},\mathit{\text{slack}}}}{C_{\mathit{\text{arteries}}}} \\ {P}_{\mathit{\text{veins}}}=\frac{V_{\mathit{\text{veins}}}\left(t\right)-V_{\mathit{\text{veins}},\mathit{\text{slack}}}}{C_{\mathit{\text{veins}}}} \end{gathered}$$ (3)

where ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right),{\mathrm{V}}_{\mathrm{i},\text{slack}}$ and ${\mathrm{C}}_{\mathrm{i}}$ are the instantaneous blood volume, the slack volume, and the compliance of compartment i, respectively. For the LV, cavity pressure is calculated through the FE model by utilizing LV cavity volume as a boundary condition, which is enforced with a Lagrange multiplier.

2.3. Finite element formulation

The FE model implemented in this study solves for LV mechanics using an implicit backward Euler scheme facilitated by the open-source FE library FEniCS [62]. Cardiac tissue is assumed to be incompressible. Quadratic interpolation functions are used to describe the displacement field, while linear interpolation functions are used to describe the hydrostatic pressure field. Furthermore, a second-degree quadrature scheme is applied within the tetrahedral elements used in the mesh. The FE formulation of the LV mechanics problem is based on the minimization of a Lagrangian functional, as described below:

$$\begin{gathered} ℒ\left(\mathit{u},p,P_{LV},c_x,c_y,c_z\right)={\int }_{{\Omega }_0}W\left(\mathit{u}\right)dV-{\int }_{{\Omega }_0}p\left(J-1\right)dV- \\ {P}_{LV}\left(V_{LV}\left(\mathit{u}\right)-V_{LV}\right)-c_x\cdot {\int }_{{\Omega }_0}{u}_{x}dV- \\ {c}_y\cdot {\int }_{{\Omega }_0}{u}_{y}dV-c_z\cdot {\int }_{{\Omega }_0}\mathit{z}\times \mathit{u}dV \end{gathered}$$ (4)

where $\mathit{u}$ is the displacement field of the myocardium, $W$ is the total strain energy, and $p$ is a Lagrange multiplier used to govern the incompressibility of the tissue by enforcing the Jacobian of the deformation gradient tensor to be unity (J=1). The LV cavity volume $V_{LV}\left(\mathit{u}\right)$ is constrained via the Lagrange multiplier $P_{LV}$ according to the hemodynamic boundary condition from the circulatory model $V_{LV}$. Lagrange multipliers $c_x$ and $c_y$ constrain rigid body translation in the $\mathbf{x}$ and $\mathbf{y}$ directions, respectively, and $c_z$ constrains rigid body rotation.

Furthermore, the functional relationship between the volume of the LV cavity and the displacement field is given by the following equation:

$$V_{LV}\left(\mathit{u}\right)={\int }_{{\mathrm{\Omega }}_{k,\mathit{\text{endo}}}}dV=-\frac{1}{3}{\int }_{{\mathrm{\Gamma }}_{k,\mathit{\text{endo}}}}\overline{\mathit{x}}\cdot \mathit{n}da$$ (5)

where the volume enclosed by the endocardial surface ${\mathrm{\Gamma }}_{k,\mathit{\text{endo}}}$ and the basal surface at $\mathrm{z}=0$ is denoted by ${\mathrm{\Omega }}_{k,\mathit{\text{endo}}}$, and $\mathit{n}$ is the direction of the outward normal vector. The position vector $\overline{\mathit{x}}$ is relative to the origin of the global coordinate system.

By taking the first variation of the Lagrangian functional, the weak formulation of the mechanics problem can be written as:

$$\begin{gathered} \delta ℒ\left(\mathit{u},p,P_{LV},c_x,c_y,c_z\right)={\int }_{{\Omega }_0}\mathit{F}\mathit{S}:\mathit{\nabla }\mathit{\delta }\mathit{u}dV-{\int }_{{\Omega }_0}(pJ{\mathit{F}}^{-\mathit{T}}:\mathit{\nabla }\mathit{\delta }\mathit{u}-\delta p\left(J-1\right)dV- \\ {P}_{LV}{\int }_{{\Omega }_0}J{\mathit{F}}^{-\mathit{T}}:\mathit{\nabla }\mathit{\delta }\mathit{u}dV-\delta P_{LV}\left(V_{LV}\left(\mathit{u}\right)-V_{LV}\right)-\delta c_x\cdot {\int }_{{\Omega }_0}{u}_{x}dV-c_x\cdot {\int }_{{\Omega }_0}\delta u_{x}dV- \\ \delta c_y\cdot {\int }_{{\Omega }_0}{u}_{y}dV-c_y\cdot {\int }_{{\Omega }_0}\delta u_{y}dV-\delta c_z\cdot {\int }_{{\Omega }_0}\mathit{z}\times \mathit{u}dV-c_z\cdot {\int }_{{\Omega }_0}\mathit{z}\times \mathit{\delta }\mathit{u}dV=0 \end{gathered}$$ (6)

where $\mathit{F}$ denotes the deformation gradient tensor, $\mathit{S}$ is the 2^nd Piola Kirchhoff stress tensor, $\delta u\in H^1\left({\Omega }_0\right)$, $\delta p\in L^2\left({\Omega }_0\right),\delta P_{Lv}\in R,\delta c_x\in R,\delta c_y\in R$, and $\delta c_z\in R$ are test functions corresponding to $u,p,P_{LV},c_x,c_y$, and $c_z$, respectively. Additionally, we considered a Dirichlet boundary condition ${\left.\mathit{u}\cdot \mathit{n}\right|}_{\mathit{\text{base}}}=0$, which constrains the deformation of the base to be in-plane. We note that the Lagrange multiplier $P_{LV}$ is used in Equation 2, which couples the FE and circulatory models.

2.4. Mechanics of cardiac muscle

In the present study, the mechanical behavior of the LV is modeled by incorporating both active and passive characteristics of the myocardium. As such, the 2nd Piola Kirchhoff stress tensor is additively decomposed into active $\left({\mathit{S}}_{\mathit{a}}\right)$ and passive $\left({\mathit{S}}_{\mathit{p}}\right)$ components, as follows:

$$\mathit{S}={\mathit{S}}_{\mathit{a}}+{\mathit{S}}_{\mathit{p}}$$ (7)

To account for the force-dependent behavior of myofibers within the myocardium, it is necessary to decompose the passive response of the myocardium into two separate components: the myofiber component and the remaining bulk material. Additionally, a passive component is included to account for the incompressibility of the myocardium. By taking the derivative of the strain energy function with respect to the Green-Lagrange strain tensor, $\mathit{E}$, the passive stress tensor for each component is calculated, as below:

$$\begin{gathered} {\mathit{S}}_{\mathit{p}}={\mathit{S}}_{\mathit{\text{vol}}}+{\mathit{S}}_{\mathit{\text{bulk}}}+{\mathit{S}}_{\mathit{\text{myofiber}}} \\ =\frac{\partial {\Psi }_{\mathit{\text{vol}}}}{\partial \mathit{E}}+\frac{\partial {\Psi }_{\mathit{\text{bulk}}}}{\partial \mathit{E}}+\frac{\partial {\Psi }_{\mathit{\text{myofiber}}}}{\partial \mathit{E}} \end{gathered}$$ (8)

where the volumetric strain energy function is ${\mathrm{\Psi }}_{\mathit{\text{vol}}}=-p(J-1)$, with a Lagrange multiplier $p$ to impose incompressibility exactly. The strain energy function of the bulk tissue, which represents the distribution of collagen, elastin, and microvasculature, is modeled as a transversely isotropic material based on the Guccione constitutive law [63]:

$$\begin{gathered} {\Psi }_{\mathit{\text{bulk}}}=\frac{c}{2}\left(e^Q-1\right) \\ \mathit{\text{with}}Q=b_{ff}{E}_{ff}^2+b_{xx}\left(E_{ss}^2+E_{nn}^2+E_{sn}^2+E_{ns}^2\right)+b_{fs}\left(E_{fs}^2+E_{sf}^2+E_{fn}^2+E_{nf}^2\right) \end{gathered}$$ (9)

where $C,b_{ff},b_{xx}$, and $b_{fs}$ are passive material parameters for the bulk tissue. $\mathit{E}$ represents the Green-Lagrange strain tensor, consisting of $f,s$, and $n$ components, denoting the fiber, sheet, and sheer-normal directions, respectively. Lastly, the myofiber strain energy function ${\Psi }_{\mathit{\text{myofiber}}}$ is given by the following equation:

$${\mathrm{\Psi }}_{\mathit{\text{myofiber}}}=\left\{\begin{array}{ll}{C}_1\left(e^{C_2(\alpha -1{)}^2}\right)& \alpha >1\\ 0& \alpha \le 1\end{array}\right.$$ (10)

In this equation, $\alpha =\sqrt{{\mathit{f}}_0\cdot \mathit{C}{\mathit{f}}_0}$ is the stretch along the primary fiber direction, where $\mathit{C}={\mathit{F}}^{\top }\mathit{F}$ is the right Cauchy-Green deformation tensor, ${\mathbf{f}}_0$ is the referential fiber direction, and $C_1,C_2$ are material constants.

The active stress component $\left({\mathit{S}}_{\mathit{a}}\right)$ presented in this study is determined using the MyoSim framework [57]. MyoSim models the mechanical properties of dynamically coupled myofilaments within a half sarcomere. A schematic of the cross-bridge scheme is illustrated in Figure 2 [12]. It should be noted that the MyoSim parameters are based on previous validation studies with experimental data (MRI, pressure catheterization) collected in rats [47]. To explain the MyoSim modeling approach, we first define the transition fluxes of the binding sites on the actin filaments, called ${\mathrm{J}}_{\text{on}}$ and ${\mathrm{J}}_{\text{off}}$. Subsequently, we describe the formulation of the transition fluxes for the myosin filaments, which capture different states of the myosin heads and are represented by ${\mathrm{J}}_1,{\mathrm{J}}_2,{\mathrm{J}}_3$, and ${\mathrm{J}}_4$ [47].

Figure 2:

Cross-bridge scheme. Sites on the thin filament switch between states that are available (${\mathrm{N}}_{\mathrm{o}\mathrm{n}}$) and unavailable (${\mathrm{N}}_{\text{off}}$) for cross-bridges to bind to. Myosin heads transition between a super-relaxed detached state $\left({\mathrm{M}}_{\mathrm{S}\mathrm{R}\mathrm{X}}\right)$, a disordered-relaxed detached state (${\mathrm{M}}_{\mathrm{D}\mathrm{R}\mathrm{X}}$), and a single attached force-generating state (${\mathrm{M}}_{\mathrm{F}\mathrm{G}}$). J terms indicate fluxes between different states. Reproduced with permission from Springer Nature from Sharifi et al. [12].

The binding sites on actin undergo transformation between an inactive state $\left({\mathrm{N}}_{\text{off}}\right)$, where no myosin heads can attach, and an active state $\left({\mathrm{N}}_{\text{on}}\right)$. The activated binding sites are further categorized into two configurations: ${\mathrm{N}}_{\text{unbound}}$, representing sites that are not bound to myosin heads, and ${\mathrm{N}}_{\text{bound}}$, representing sites that are attached to myosin heads and are unable to switch back to the ${\mathrm{N}}_{\text{off}}$ state. The flux governing the transition of binding sites from ${\mathrm{N}}_{\text{off}}$ to ${\mathrm{N}}_{\mathrm{o}\mathrm{n}}$ is defined by the following equation:

$$J_{on}=k_{\mathit{\text{on}}}\left[{Ca}^{2+}\right]\left(N_{\mathit{\text{overlap}}}-N_{\mathit{\text{on}}}\right)\left(1+k_{\mathit{\text{coop}}}\left(\frac{N_{\mathit{\text{on}}}}{N_{\mathit{\text{overlap}}}}\right)\right)$$ (11)

where $N_{\mathit{\text{on}}}$ represents the proportion of binding sites in the active state, $k_{\mathit{\text{on}}}$ denotes a rate constant, $N_{\mathit{\text{overlap}}}$ represents the proportion of binding sites in the vicinity of myosin heads, $k_{\mathit{\text{coop}}}$ is a constant factor that controls the cooperativity of the thin filament, and $\left[{\mathrm{C}\mathrm{a}}^{2+}\right]$ is the intracellular calcium signal that is based on experimental measurements from previous studies [58, 59] and is assumed to occur simultaneously across the entire LV.

The unbound activated sites transform back into the inactive state through the ${\mathrm{J}}_{\text{off}}$ flux, which is governed by a rate constant of ${\mathrm{k}}_{\text{off}}$:

$$J_{\mathit{\text{off}}}=k_{\mathit{\text{off}}}\left(N_{\mathit{\text{on}}}-N_{\mathit{\text{bound}}}\right)\left(1+k_{\mathit{\text{coop}}}\left(\frac{N_{\mathit{\text{overlap}}}-N_{\mathit{\text{on}}}}{N_{\mathit{\text{overlap}}}}\right)\right)$$ (12)

On the myosin filament, myosin heads can transition between three states: ${\mathrm{M}}_{\mathrm{S}\mathrm{R}\mathrm{X}}$ (detached super-relaxed), ${\mathrm{M}}_{\mathrm{D}\mathrm{R}\mathrm{X}}$ (detached disordered-relaxed), and ${\mathrm{M}}_{\mathrm{F}\mathrm{G}}$ (attached force generating). The fluxes governing the transitions between ${\mathrm{M}}_{\mathrm{S}\mathrm{R}\mathrm{X}}$ and ${\mathrm{M}}_{\mathrm{D}\mathrm{R}\mathrm{X}}$ are represented by ${\mathrm{J}}_1$ and ${\mathrm{J}}_2$, respectively:

$$\begin{gathered} J_1=k_1\left(1+k_{\mathit{\text{force}}}{F}_{\mathit{\text{total}}}\right)M_{\mathit{\text{SRX}}} \\ {J}_2=k_2{M}_{\mathit{\text{DRX}}} \end{gathered}$$ (13)

In this equation, $k_1$ and $k_2$ are the rate constants, $k_{\mathit{\text{force}}}$ is a parameter related to force dependent recruitment, $F_{\text{total}}$ is the total stress (passive myofiber plus active) in the ${\mathbf{f}}_0$ direction, and ${\mathrm{M}}_{\text{SRX}}$ and ${\mathrm{M}}_{\text{DRX}}$ are the proportions of myosin heads in the SRX and DRX states, respectively.

The attachment of myosin heads to binding sites is facilitated by the transition flux ${\mathrm{J}}_3$, while the detachment of myosin heads is governed by the transition flux ${\mathrm{J}}_4$:

$$\begin{gathered} J_3\left(x\right)=k_3{e}^{\frac{-k_{cb}{x}^2}{2k_{B}T}}\left(N_{on}-N_{\mathit{\text{bound}}}\right)M_{\mathit{\text{DRX}}} \\ {J}_4\left(x\right)=\left(k_{4,0}+k_{4,1}{x}^4\right)M_{FG}\left(x\right) \end{gathered}$$ (14)

where $k_3$ and $k_{\mathrm{4,0}}$ are rate constants, $k_{cb}$ is the stiffness of the cross-bridge link, $k_B$ is the Boltzmann constant, $T$ is temperature in Kelvin, $k_{\mathrm{4,1}}$ is the strain dependent cross-bridge detachment rate parameter, $M_{FG}$ is the proportion of myosin heads in the attached state, and $x$ is the cross-bridge length.

By utilizing the aforementioned transition fluxes, a system of ordinary differential equations (ODEs) is established that determines the temporal evolution of myosin head and binding site proportions at each configuration. This system of ODEs is partitioned with a spatial resolution of 1 nm, spanning a range of $-10\le x\le 10\mathrm{n}\mathrm{m}$ (i.e., n=21). Consequently, a total of 25 ODEs are solved at each integration point within the LV mesh, which can be expressed as follows:

$$\begin{gathered} \frac{d{N}_{\mathit{\text{off}}}}{dt}=-J_{on}+J_{\mathit{\text{off}}} \\ \frac{d{N}_{on}}{dt}=J_{on}-J_{\mathit{\text{off}}} \\ \frac{d{M}_{\mathit{\text{SRX}}}}{dt}=-J_1+J_2 \\ \frac{d{M}_{\mathit{\text{DRX}}}}{dt}=\left(J_1+{\sum }_{i=1}^n{J}_{4,x_i}\right)-\left(J_2+{\sum }_{i=1}^n{J}_{3,x_i}\right) \\ \frac{d{M}_{FG,i}}{dt}=J_{3,x_i}-J_{4,x_i}\mathit{\text{where}}i=1\dots n \end{gathered}$$ (15)

The active stress within each myofiber can be determined by utilizing the proportion of myosin heads in the force-generating state, as expressed by the following equation:

$$F_{\mathit{\text{active}}}=N_o{k}_{cb}{\sum }_{i=1}^n{M}_{FG,i}\left(x_i+x_{ps}\right)$$ (16)

In the above equation, $N_0$ represents the density of myosin heads [58], and $x_{ps}$ denotes the power stroke of an attached cross-bridge.

Finally, the active stress tensor is defined by projecting $F_{\text{active}}$ along the myofiber direction:

$${\mathit{S}}_{\mathit{a}}=F_{\mathit{\text{active}}}{\mathit{f}}_{\mathbf{0}}\otimes {\mathit{f}}_{\mathbf{0}}$$ (17)

2.5. Stress based reorientation law

Here, we show the motivation for the choice of reorientation law. We begin by defining the relationship between the 1^st Piola Kirchhoff stress tensor, $\mathit{P}$, and the 2^nd Piola Kirchhoff stress tensor, $\mathit{S}$, which is given by $\mathit{S}={\mathit{F}}^{-1}\mathit{P}$. We also note the relationship between the 1^st Piola Kirchhoff stress tensor and its traction vector associated with the arbitrary unit normal vector $\mathit{N}$, which is given by ${\mathit{p}}_{\left(\mathit{N}\right)}=\mathit{P}\mathit{N}$. For convenience with respect to the finite element implementation, we define a referential traction vector with respect to the 2^nd Piola Kirchhoff stress tensor, ${\mathit{s}}_{\left(\mathit{N}\right)}$. This is accomplished by performing a pullback of ${\mathit{p}}_{\left(\mathit{N}\right)}$ to the reference configuration, which is given by:

$${\mathit{s}}_{\left(\mathit{N}\right)}={\mathit{F}}^{-1}{\mathit{p}}_{\left(\mathit{N}\right)}=\mathit{S}\mathit{N}$$ (18)

It should be noted that in general, ${\mathit{s}}_{\left(\mathit{N}\right)}\nparallel \mathit{N}$. Thus, this traction vector can be separated into normal and shear components. By projecting the traction vector onto the normal vector $\mathit{N}$, this yields the component of the traction vector that acts in the direction $\mathit{N}$, which is given as:

$${\mathit{s}}_{\left(\mathit{N}\right)}^{\Vert }=(\mathit{N}\otimes \mathit{N}){\mathit{s}}_{\left(\mathit{N}\right)}$$ (19)

The shear component is attained by subtracting the normal component from the total traction vector, which leads to the following projection:

$${\mathit{s}}_{\left(\mathit{N}\right)}^{\perp }={\mathit{s}}_{\left(\mathit{N}\right)}-(\mathit{N}\otimes \mathit{N}){\mathit{s}}_{\left(\mathit{N}\right)}=(\mathit{I}-\mathit{N}\otimes \mathit{N}){\mathit{s}}_{\left(\mathit{N}\right)}$$ (20)

The principal stresses and principal directions associated with the 2^nd Piola Kirchhoff stress tensor can be found by solving the referential eigenproblem, which leads to $\mathit{S}\widehat{\mathit{N}}=\hat{S}\widehat{\mathit{N}}$, where $\widehat{\mathit{N}}$ is the principal direction (eigenvector) and $\hat{S}$ is the associated principal stress (eigenvalue). If the referential traction vector aligns with the principal direction, then equation (20) reduces to:

$$\begin{gathered} {\mathit{s}}_{\left(\widehat{\mathit{N}}\right)}^{\perp }=\left(\mathit{I}-\widehat{\mathit{N}}\otimes \widehat{\mathit{N}}\right){\mathit{s}}_{\left(\widehat{\mathit{N}}\right)} \\ =\left(\mathit{I}-\widehat{\mathit{N}}\otimes \widehat{\mathit{N}}\right)\mathit{S}\widehat{\mathit{N}} \\ =\left(\mathit{I}-\widehat{\mathit{N}}\otimes \widehat{\mathit{N}}\right)\hat{S}\widehat{\mathit{N}} \\ =\hat{S}\widehat{\mathit{N}}-\widehat{S}\widehat{\mathit{N}}=0 \end{gathered}$$ (21)

This verifies that if the traction vector and principal direction align, then the traction leads to a state of pure normal (axial) stress, i.e., the shear component vanishes. Thus, if the fiber direction aligns with the traction, then the fiber will experience a state of pure axial stress.

In Figure 3, ${\mathit{f}}_0$ represents the unit vector along the fiber direction in the reference configuration, which is shared by the myofibers and perimysial collagen fibers (as seen in the constitutive equations). The traction vector $\left(\mathit{S}{\mathit{f}}_0\right)$ is associated with the cross-sectional face of the fibers at a point in the LV. Our stress-based reorientation law incrementally reorients the fibers at each integration point in the LV mesh to be aligned with the traction vector as below:

$$\frac{d{\mathit{f}}_{\mathit{0}}}{dt}=\frac{1}{\kappa }\left(\frac{\mathit{S}{\mathit{f}}_{\mathbf{0}}}{‖\mathit{S}{\mathit{f}}_{\mathbf{0}}‖}-{\mathit{f}}_{\mathbf{0}}\right)$$ (22)

where $\kappa$ is a time constant that is employed to enforce the separation of time scales. More specifically, changes in fiber orientation occur over an extended period of time (e.g., days to weeks), whereas the current study achieves comparable fiber reorientation within several heartbeats. For example, a rat heart will beat approximately 6,000,000 times over a 2-week period. Since it is not practical to simulate this many beats with a FE code, the scaling of time, i.e., separation of time scales, acts like a map between model time and real time. This is a numerical necessity to speed up the calculation of remodeling, which has been employed in prior studies of the heart [52, 64, 65]. As long as the relationship between how much reorientation is produced over a given number of representative beats and the actual number of days needed to produce that amount of reorientation is known, then time can be scaled appropriately. It should be noted that $\kappa$ only affects the rate at which the fibers reorient, not the final steady state configuration (Figure S1 in Supplementary Material). Through a gradual reorientation process, the difference between the fiber direction and the direction of traction experienced by the fibers is minimized. To account for the reorientation that occurs in both the myofibers and collagen, it is necessary to employ the total stress tensor, which encompasses both active and passive stress components. This is especially important in the MI region where the scar tissue is devoid of contracting myofibers. Thus, our stress-based law can capture alterations in fiber orientation in the remote, border zone, and infarct regions.

Figure 3:

Stress based fiber reorientation. The unit vector ${\mathit{f}}_0$ represents the initial direction of the myofiber and collagen. ${\mathit{f}}_0$ reorients toward the local traction vector $\mathit{S}{\mathit{f}}_{\mathbf{0}}$, which is associated with the cross-sectional face of the fiber. $\mathit{S}$ is the total stress tensor, encompassing both passive and active stresses.

2.6. Models of healthy and infarcted LVs

An ellipsoidal geometry with ~1000 quadratic tetrahedral elements is used to model the LV (see Supplementary Figure S2 for mesh sensitivity analysis). The slack (unloaded) chamber volume of the LV is 0.1 ml, which approximates the LV of a rat, and its dimensions are illustrated in Figure 4A. The initial helical angle was defined using a Laplace-Dirichlet rule-based algorithm, changing linearly from 60° at the endocardium to −60° at the epicardium transmurally across the wall [66]. The initial transverse angle is set to zero. The helical angle and the transverse angle are defined in terms of local longitudinal, circumferential, and radial directions as shown in Figure 4B. The adaptation time constant $\kappa$ is set to 200 ms (Supplementary Material). The cardiac cycle time is 200 ms (mimicking 300 bpm), with a time step of 0.1 ms. Therefore, during each simulation time step, a $5\times {10}^{-4}(\mathrm{d}\mathrm{t}/\kappa )$ fraction of the difference between the traction vector and fiber vector is used to update the fiber direction.

Figure 4:

LV models. A: Dimensions of the ellipsoidal model of the rat LV. B: The myofiber direction ${\overrightarrow{e}}_f$ is defined by the helical angle ${\alpha }_h$ and the transverse angle ${\alpha }_t$, respectively, in terms of local longitudinal, circumferential, and radial directions ${\overrightarrow{e}}_l,{\overrightarrow{e}}_c$, and ${\overrightarrow{e}}_r$. C: A total of four LV models were simulated in this study, including a baseline model and three infarct models with circular infarctions varying in size. The baseline LV is shown on the left, followed by LV models with a small infarct (infarct diameter of 3 mm), a medium infarct (infarct diameter of 4 mm), and finally a large infarct (infarct diameter of 5 mm). A border zone is included in all infarct models in which the cross-bridge density linearly varies from 50% of normal myocardium in the inner diameter to 100% in the outer diameter in a 0.5 mm translation band around the infarct. Passive material properties of the border zone are the same as those in the normal myocardium.

In this study, we investigate the reorientation of fibers in healthy and infarcted myocardium (Figure 4C). Previous studies have shown that myocardial passive and active behavior undergoes significant changes following an infarction. To accurately capture these alterations, we incorporate relevant material properties into the passive and contractile constitutive laws for different regions, namely the remote, border zone, and infarct (Table 1). For the baseline (healthy) model, material constants were chosen such that the model produces a realistic pressure volume loop observed in the rat’s heart.

Table 1:

Regional material properties of cardiac muscle in the baseline and infarcted LV models

	Passive properties						Active properties
Constants	$\mathbf{C}\left(\mathbf{P}\mathbf{a}\right)$	${\mathbf{C}}_1\left(\mathbf{P}\mathbf{a}\right)$	${\mathbf{C}}_2$	${\mathbf{b}}_{\mathbf{f}}$	${\mathbf{b}}_{\mathbf{t}}$	${\mathbf{b}}_{\mathbf{f}\mathbf{s}}$	Cross-bridge density (m−2)
Remote (baseline)	180	250	15	8.0	3.58	1.63	6.96e+16
BZ	180	250	15	8.0	3.58	1.63	3.48e+16 to 6.96e+16
Infarct	2700	250	15	10	8	8	0

For the infarcted LV models, we implement regional passive and active properties based on previous experimental findings. To match the modeling assumptions for an MI, prior to when significant volumetric growth occurs, we choose the time point of two weeks post-MI for estimating material properties. The passive material properties within the infarct region are defined such that the stiffness is 8 times greater than normal myocardium [39]. Furthermore, we consider that the cross-bridge density in the infarct region is zero, representing the scar tissue that is left behind after myocyte necrosis, which is based on studies with a transmural infarct. To account for the border zone, which is the poorly contracting nonischemic region adjacent to the infarct, we implemented passive material properties that were the same as those in the remote region. However, according to a previous experimental study that examined border zone myocardium two weeks post-MI, there exists a linear variation in myocardial contractile stress as a function of distance from the infarct. In addition, it was hypothesized that the reduction in stress generation was due to a reduction in cross-bridges that form between myosin and actin [67]. Based on this study, we incorporated a border zone where the cross-bridge density linearly decreased starting at a distance 0.5 mm from the infarct (corresponding to 100% of the remote value) to 50% at the direct edge of the infarct region (Figure 4C) [67].

The simulation protocol for the healthy baseline case was developed with the following steps: (1) the initial fiber configuration was assigned to the LV using a rule-based approach, as described above, then (2) the system reached hemodynamic steady state by simulating the first two consecutive cardiac cycles without permitting fiber reorientation, finally (3) fiber reorientation was simulated over the following 20 cardiac cycles to obtain the case-specific fiber configuration. It was then confirmed that fiber remodeling reached steady state during those 20 cycles. To generate the fiber architecture for the infarct models, we modified the initialization process (step 1 above), while keeping the remaining steps the same. For the infarct cases, the final healthy baseline fiber architecture was used as the initial configuration for all infarct models. This allowed us to better quantify the changes in helical and transverse angles induced by different size infarcts compared to the baseline.

3. Results:

3.1. Estimating fiber architecture in the healthy baseline model

In this study, we first examined the ability of the reorientation law to establish the fiber architecture in a healthy rat LV (baseline model). A rule-based approach was used to define the initial helical and transverse angles as shown in Figures 5A and 5B on the left. The fibers were reoriented until reaching steady state to obtain the final angles illustrated in Figures 5A and 5B on the right. A slight reorientation of the fibers was observed in the helical angle when comparing the distributions in Figure 5A. On average, the fibers became more circumferential near the endo and more longitudinal near the epi. However, as seen in Figure 6E (bottom), in both the endocardium and epicardium, the apical fibers are more longitudinal than those in the base and mid-ventricle regions. The transverse angles undergo a more notable reorientation as seen in Figure 5B, showing variations in reorientation when moving from apex to base. Figure 6E (top) also illustrates the transmural variation in the transverse angle, which exhibits positive values in the mid-wall of the basal region and negative values in the mid-wall of the apical region.

Figure 5:

Myocardial fiber reorientation in baseline LV model. A: Initial helical angle (left) ranging from −60 degrees at the epicardium to 60 degrees at the endocardium vs final helical angle (right) after fiber reorientation. B: Initial transverse angle (left) of 0 degrees vs transverse angle (right) after fiber reorientation.

Figure 6:

Comparison of Pressure-Volume (PV) loops and fiber configuration in baseline LV models with a rule-based fiber configuration and evolving fiber configuration. A: PV loops are plotted for the 1st, 5th, 10th, and 20th cycles of fiber reorientation as well as rule-based fiber configuration. B-E: The corresponding fiber configurations after the 1st, 5th, 10th, and 20th FR cycles, including transmural TA and HA. FR: fiber reorientation, TA: transverse angle, HA: helical angle.

To assess the impact of fiber reorientation on pumping performance, Figure 6A compares the pressure-volume (PV) loops of an LV model with a rule-based fiber configuration against an LV model with evolving fiber configuration after 1, 5, 10, and 20 cycles of fiber reorientation. A notable shift of the PV loop towards the left is evident during fiber reorientation. A 21% increase in ejection fraction was observed between the model with a rule-based fiber configuration and the final configuration (cycle 20) of the model with fiber reorientation. This confirms that the fiber architecture plays an important role in optimizing cardiac pumping performance. Additionally, Figure 6B–6E illustrates a direct comparison of fiber configuration changes for the same cycles of fiber reorientation, both in helical and transverse angles. As noted above, there is a progressive transmural variation that occurs in both the transverse and helical angles. Note how the PV loops shift as the fiber angles evolve between cycles.

3.2. Fiber reorientation due to myocardial infarction

Figure 7 demonstrates the magnitude of the 3D angle between the initial and reoriented fibers, which was computed by taking the dot product of the two direction vectors. This measurement captures the influence of both helical and transverse remodeling. We observed that the severity of fiber reorientation in the infarcted region of the LV increases with the size of the infarction, whereas fiber remodeling in remote regions is not significantly influenced by the size of the infarction.

Figure 7:

Total angle of fiber reorientation (magnitude of the 3D angle between initial and reoriented fibers as shown in the diagram in the bottom left corner) in LV models with small infarct (A), medium infarct (B) and large infarct (C).

In our investigation of the infarction models, we made several noteworthy observations regarding the distribution of fiber orientation. As shown in Figure 8, in all the infarct models there is a discernible decrease in the proportion of right-handed helical fibers in the endocardium when moving from the remote region to the adjacent border zone and infarct regions. The aforementioned observation exhibits a proportional relationship with the size of the infarct, i.e., the fibers become less right-handed as the infarct increases in size. Furthermore, we observed an increase in the proportion of left-handed fibers on the epicardium of the infarct regions, especially in the large infarct case. The fiber orientation in the remote region of all models did not show a strong dependence on the size of the infarct.

Figure 8:

Regional fiber distributions in the various infarcted LV models. Location of visualized fibers within the LV muscle are shown in the left with I (infarct), B (border zone) and R (remote) on the models with small, medium, and large infarcts from top to bottom. To better visualize the endocardium, the front half of the LV has been hidden from view in the images on the left.

As shown in Table 2, fibers in the infarct region exhibit substantial helical reorientation toward the circumferential direction near the endocardium, rotating by ~22 degrees in the small infarct and ~28 degrees in the large infarct. Additionally, the fiber orientation near the epicardium of the infarct region became more longitudinal, with the large infarct showing the most remodeling with a rotation of ~15 degrees. Overall, the transmural distribution of fibers in the infarct region became more left-handed as the infarct increased in size. The border zone region also exhibited a slight left-handed shift in the endocardial fibers, with rotation ranging from 6 to 8 degrees. The remote region underwent the least remodeling with rotation ranging from 0 to 2 degrees.

Table 2:

Helical and transverse fiber angles from the baseline and MI models after reorientation. The values are reported in the infarct and border zone (BZ) regions for the three MI cases. In each region, the fiber angle represents the mean value for that area. The fiber angles for the baseline case were calculated over a circular region that is the same size as the large infarct. The transverse angles were calculated at a 15% thickness of the myocardium near the endocardial and epicardial surfaces, as well as the mid-wall.

	Helical angle (degrees)			Transverse angle (degrees)
		Endo	Epi	Endo	Mid	Epi
Baseline		49.1	−71	−5.2	−6.4	−3.8
Small MI	Infarct	26.9	−67.8	−3.7	−1.7	−5.4
	BZ	40.9	−73.4	−1.7	−5.3	−0.4
Medium MI	Infarct	25.4	−74	−1.5	−1.6	−7.1
	BZ	40.6	−70.8	−0.7	−3.6	0.9
Large MI	Infarct	21.2	−85.5	−0.9	0.3	−8.5
	BZ	42.8	−69.7	0.3	7	2.2

As shown in Table 2, the transverse angles in the sub-endocardium reoriented towards the endocardial surface (became more positive) in both the infarct and BZ regions, as the infarct increased in size. However, the transverse angles in the sub-epicardium showed the opposite trend, reorienting away from the epicardial surface in the infarct (became more negative) and rotated towards the epicardial surface in the BZ region. In both regions the amount of reorientation ranged from 1 to 6 degrees. The remote region exhibited minimal remodeling with fiber rotations of ~1 degree.

3.3. Fiber dispersion after infarction

As a quantitative measure of fiber dispersion, the angular deviation (AD) of the helical and transverse angles were analyzed in each region of the LV models. AD is defined as the regional standard deviation of the helical and transverse angles. An epicardial layer of 15% thickness was used to determine the AD [19, 28]. In almost all regions, the infarcted LV models demonstrated elevated levels of fiber dispersion compared to the baseline model, as seen in Figure 9. The AD of the helical angle is higher in the infarct region compared to the BZ region. The AD of the transverse angle, however, is higher in the BZ than in the infarct region. The extent of helical and transverse fiber dispersion in the infarct region is proportional to the size of the infarct.

Figure 9:

Angular deviation of helical and transverse angle in different regions of infarcted and baseline LV models.

3.4. Myofiber stress distribution after infarction

Using the active stress distribution at end-systole, we illustrate the effects of depressed contractility in the infarcted LV (Figure 10). Accordingly, the infarcted myocardium expands outward due to the loss of contractility, making it locally thinner (since there is no radial thickening due to contraction). This, in turn, contributes to the increase in LV cavity volume at end-systole (Figure 11).

Figure 10:

A comparison of myofiber active stress distributions in the baseline and infarcted left ventricles at the end of systole.

Figure 11:

Pressure-volume (PV) loops of baseline and infarcted LV models with and without fiber reorientation (FR).

3.5. Cardiac pumping performance after infarction

The infarction affects the PV loop in a manner proportional to the infarct size. As seen in Figure 11, the stroke volume (SV) for fiber reorientation (FR) models reduced by as much as 25%, from a value of 0.16 mL at baseline to 0.12 mL for the large infarction. Moreover, a substantial reduction in end-diastolic (ED) volume was associated with the size of the infarct, particularly since the infarcted tissue is stiffer than the healthy remote tissue leading to impaired filling. It is also evident that the infarction affects the pressure in a way that is proportional to the size. The absence of cross-bridges in the infarct region reduced the active stress in the LV thereby lowering end-systolic pressure by ~15.5%, from 99 mmHg at baseline/FR to 83.5 mmHg for the large infarct. In addition, the end-systolic volume increased, since reduced contractility led to less blood being ejected during systole, in addition to the effect of infarct bulging. To evaluate the effect of fiber remodeling on cardiac performance, PV loops were plotted for LV models with and without fiber reorientation. The PV loops shrink in both cases, but when fiber reorientation is activated, the loops (regardless of infarct size) shift to the left.

4. Discussion

In the present study, we developed a stress-based fiber reorientation algorithm and integrated it into a multiscale finite element model of the LV pumping blood through the systemic circulation. By using the total stress (passive and active), the reorientation algorithm can capture the remodeling of both myofibers and collagen fibers. The results show that the model can predict realistic helical and transverse fiber distributions in the healthy LV and capture the pathological fiber remodeling patterns that occur following an MI. In addition, our results indicated that the size of the MI region substantially influences remodeling severity in the infarct region as compared to remote regions.

In the baseline LV model, the myofiber angles showed transmural variation consistent with previous experimental findings in rats, with a global average of 50 degrees at the endocardium and −70 degrees at the epicardium [18, 19]. Additionally, we observed that (1) the helical angle in the apical region tends to be more longitudinal (endo: 62 degrees, epi: −76 degrees) compared to the fiber orientation in the mid-ventricle and basal regions, and (2) the transverse angle varies within a ~20 degree range. Both of these findings align well with previous investigations utilizing DTMRI on rat myocardium [19] and other simulation-based studies [68]. Furthermore, we observed a notable variation in the transverse angle along the LV (mid-wall), transitioning from positive values near the basal region to negative values towards the apex (Figure 5B and Figure 6E). This longitudinal variation compares well with previous DTMRI findings [4] and other simulation studies [52, 69], further validating the accuracy and robustness of our modeling approach. In terms of the PV loops, in the current study the baseline EDV, ESV, and peak pressure after fiber remodeling were 0.24 ml, 0.073 ml, and 103 mmHg, respectively. From previous experimental measurements in n=5 anesthetized rats [46, 47], the values were 0.25 ml, 0.09 ml, and 105 mmHg, respectively, which is in good agreement with the current study. Furthermore, as shown in Figure 6, fiber reorientation significantly affects pumping performance by shifting the PV loops to the left and enhancing the ejection fraction. Similar optimization in cardiac performance has been noted in prior modeling studies of fiber reorientation [52, 68]. The leftward shift in ESV after reorientation is primarily driven by more efficient contraction (due to more force acting along the fiber direction), while the shift in EDV is related to the duration of diastolic filling.

Consistent with previous findings, we observed a correlation between the severity of fiber reorientation and the size of the infarction (Figure 7) [30, 31], as well as a reduction in pumping performance (Figure 11). As shown in Figure 11, the LV models of MI exhibit a considerable shrinkage of the PV loop, which has also been reported in previous models of infarction [70]. Similarly, Leong et al. [71] reported a comparable decrease in end-diastolic volume (EDV) and increase in end-systolic volume (ESV) in infarct models. It can be seen that the PV loops shrink with and without the fiber reorientation being activated. However, it should be noted that when fiber reorientation is active, the PV loops (no matter the size of the infarct) shift to the left. This leads to an increase in pumping performance via increased EF and peak systolic pressure. Thus, fiber remodeling post-MI may help to compensate for the decrease in contractile function due to the infarcted scar tissue. Furthermore, our results indicate a substantial reduction in right-handed fibers in the endocardium of the infarct, reducing from ~49 degrees at baseline to ~27 degrees for the small infarct and ~21 degrees for the large infarct. Additionally, there was an increase in left-handed fibers within the epicardium of the infarcted myocardium, with little reorientation for the small infarct but an increase in magnitude from ~−71 degrees at baseline to ~−86 degrees for the large infarct (as shown in Figure 8 and Table 2). This overall left-handed shift of the infarct helix angles is in strong agreement with previous experimental studies that utilized DTMRI and histology, as well as the left-handed shift of fibers in the border zone region [19, 22–24, 28, 30, 32, 34]. In a direct comparison with the work by Chen et al. [19], the fiber angles changed by approximately 15 degrees in the sub-endocardial region, whereas in the current study the angle changed by approximately 22 degrees for the small infarct case. In the sub-epicardial region, the measured change was approximately 7 degrees, while the angle changed by approximately 3 degrees for the small infarct case in the current study. In terms of the angular deviation (AD), the value measured by Chen et al. [19] was approximately 22 degrees, while the current study predicted 17 degrees for the small infarct case. Our model also predicted increased fiber dispersion (as shown in Figure 9), which agrees with findings from previous animal and human studies [19, 22–25, 28, 30–36].

Several studies have investigated cardiac fiber remodeling in healthy and pathological conditions using computational modeling approaches. In one of the pioneering studies, Kroon and colleagues applied a deformation-driven law to reorient myofibers within a healthy LV, which was able to produce a realistic distribution of helical and transverse angles [52]. Their approach was motivated by minimizing the shear strain experienced by a myofiber during the cardiac cycle. This was done locally at each integration point in the model. In contrast, the current study is based on minimizing the shear stress experienced by myofibers and collagen, i.e., aligning the fiber direction with the traction vector at each local integration point. Incorporating a stress-based fiber reorientation law establishes a framework for future cardiac modeling studies that could utilize a unified stress-driven law that accounts for both volumetric growth [65] and fiber remodeling. In a follow-up study to [52], Pluijmert et al. [68] explored the long-term effects of initial and boundary conditions on the final fiber configuration of the LV. It was found that while the influence of initial conditions faded with adaptation, the impact of boundary conditions remained significant, highlighting the importance of the interaction between the myocardium and valvular annulus for predicting the LV fiber configuration. The shift in the endocardial and epicardial helix fiber angles seen in the current study agrees with the FULL+LIN case shown in the study by Pluijmert et al. [68], in addition to the distribution of transverse angles from base to apex. In another study, Washio et al. utilized impulse and workload as reorientation drivers, which were functions of active stress only, to establish the myofiber distribution in a healthy heart [53]. In a follow-up study, Washio et al. considered myofiber reorientation during isovolumic contraction, which was driven by active tension and the principal direction of a branching network [44]. This approach was applied to simulations of normal function and a model of myocardial infarction. However, this work did not address passive remodeling or the effects of altered material properties, particularly within the infarct region. By utilizing a reorientation law that only accounts for the effects of active stress, the model would be unable to capture the remodeling of ECM in the absence of contractile myofibers, leading to possible misrepresentation of the scar tissue configuration. Lastly, the study by Zhuan et al. implemented an agent-based model, embedded within an FE model of the LV, to capture the deposition and reorientation of collagen in the infarct region after an MI [64]. This work showed the circumferential shift of collagen in the infarct, which matched experimental measurements. However, it did not account for the reorientation that takes place in the myofibers during MI (such as the border zone) and focused only on passive deformation during the diastolic phase, which does not consider the full cardiac cycle.

To address the limitations of previous approaches, the current study integrated several important characteristics into the multiscale FE model. For instance, we simulated the entire cardiac cycle using a mechanistic model of myocardial contraction and utilized the total stress (passive and active) to facilitate fiber alignment with local traction vectors at each integration point in the mesh, which captures the remodeling in both myofibers and collagen fibers. In the MyoSim contraction law, the flux of myosin heads from the super-relaxed state to the disordered-relaxed state is dependent on the total myofiber stress (Equation 13). This flux plays a major role in determining the number of myosin heads that are available for binding and thus affects the magnitude of myofiber active stress. By incorporating a stress-based fiber reorientation law, this results in a unified stress-based modulation of contraction magnitude and direction. We incorporated the infarct region into the LV geometry by utilizing material properties derived from experimental findings at two weeks post-MI [39, 67]. These experimental studies showed increased stiffness in the infarct region as well as impaired contractility in the border zone that transitions linearly from the remote region to the infarct. Specifically, Baker et al. [67] noted that the reduction in myofilament force generation in the border zone myocardium was linked to a reduction in the number of myosin cross-bridges that bind to actin. This dysfunction is captured by the MyoSim model, which incorporates the specific mechanism of cross-bridge formation. By including all these features into the MyoFE framework, an effective approach for predicting fiber remodeling has been proposed, providing insight into post-MI cardiac behavior and subsequent pumping performance.

4.1. Limitations

This study has some limitations. First, the predictive capabilities of the MyoFE framework for fiber reorientation and resulting cardiac performance were investigated using an idealized LV geometry. A more realistic geometry will be used in future studies. Second, it should be noted that the current investigation did not consider volumetric growth, which typically occurs concurrently with myocardial remodeling in cases of infarction. This limitation will be addressed in future works by incorporating a growth model into the modular framework. Another limitation of the current model is that we did not tune the time constant $\kappa$ in the reorientation law. This would require fitting to experimental data to find the correct rates of change that mimic the true time course of remodeling, i.e., find a time constant such that one cardiac cycle represents a certain number of days, weeks, etc. However, this will also be the focus of future studies. Lastly, in our contraction model the intracellular calcium signal is assumed to occur simultaneously across the entire LV, without considering the inherent minor spatiotemporal delay.

5. Conclusion

In conclusion, the current study demonstrates the effectiveness of a stress-based law, within the multiscale cardiac modeling framework, to predict fiber reorientation in the LV. The adapted fiber configuration in the baseline LV model exhibited realistic helical and transverse angles, and could potentially be used as a replacement for rule-based approaches for establishing fiber distributions in the model. Moreover, in response to the adverse passive and active stress alterations in the myocardium caused by infarction, we observed substantial fiber remodeling in the LV, with a predominantly left-handed shift of the helix angles. Notably, the severity of fiber remodeling was found to be closely associated with the size of the infarction. These findings hold potential to enhance the comprehension and prediction of myocardial behavior in pathological conditions, providing valuable insights for advancing cardiac research and potentially aiding in the development of therapeutic strategies to mitigate adverse remodeling effects post-MI.

Statement of Significance.

The organization of muscle and collagen fibers within the heart plays a significant role in defining cardiac function. This organization can become disrupted after a heart attack, leading to degraded pumping performance. In the current study, we implemented a stress-based fiber reorientation law into a computer model of the heart, which seeks to realign the fibers such that contractile force and load bearing capabilities are maximized. The primary goal was to evaluate the effects of different sized heart attacks. We observed substantial fiber remodeling in the heart, which matched experimental observations. The proposed computational framework allows for the effective prediction of adverse remodeling and offers the potential for assessing the effectiveness of therapeutic interventions in the future.