## Abstract

Mathematical models provide a suitable platform to test hypotheses on the relation between local mechanical stimuli and responses to cardiac structure and geometry. In the present model study, we tested hypothesized mechanical stimuli and responses in cardiac adaptation to mechanical load on their ability to estimate a realistic myocardial structure of the normal and situs inversus totalis (SIT) left ventricle (LV). In a cylindrical model of the LV, *1*) mass was adapted in response to myofiber strain at the beginning of ejection and to global contractility (average systolic pressure), *2*) cavity volume was adapted in response to fiber strain during ejection, and *3*) myofiber orientations were adapted in response to myofiber strain during ejection and local misalignment between neighboring tissue parts. The model was able to generate a realistic normal LV geometry and structure. In addition, the model was also able to simulate the instigating situation in the rare SIT LV with opposite torsion and transmural courses in myofiber direction between the apex and base [Delhaas et al. (6)]. These results substantiate the importance of mechanical load in the formation and maintenance of cardiac structure and geometry. Furthermore, in the model, adapted myocardial architecture was found to be insensitive to fiber misalignment in the transmural direction, i.e., myofiber strain during ejection was sufficient to generate a realistic transmural variation in myofiber orientation. In addition, the model estimates that, despite differences in structure, global pump work and the mass of the normal and SIT LV are similar.

- cardiac mechanics
- myocardium
- cardiac development

mathematical models provide a suitable platform to test hypotheses on the relation between local mechanical stimuli and responses to cardiac structure and geometry, such as changes in mass (18, 22), shape (23), and internal myofiber arrangement (2, 10, 17, 26). Model-based prediction of a realistic myocardial structure and geometry could substantiate the role of hypothesized stimuli and responses in real physiology.

Through mechanical feedback in the environment of the cell, Arts et al. (2) were able to generate a realistic normal left ventricle (LV) structure and geometry, i.e., myofiber orientations changed gradually across the wall from a subendocardial right-handed helix to a subepicardial left-handed one. In the normal LV, there is little change in the transmural variation in myofiber orientation from the apex to base. As a consequence, torsion in the normal LV is about equal at all levels from the apex to base (6). As shown by Delhaas et al. (6), in the case of situs inversus totalis (SIT), the LV, on average, exhibits a dualistic torsion pattern, with the torsion at the apex being as in the normal LV but opposite in sign at the base. The myocardial structure is dualistic as well, being the same as in the normal heart at the apex but mirrored at the base (see Fig. 3*a*, obtained from Ref. 15).

The aberrant nature of the myocardial structure in SIT provides an additional possibility to further test hypotheses regarding the mechanisms through which the cardiac wall adapts to mechanical load. In the present study, we investigated whether the hypotheses on adaptation used by Arts et al. (2) are also able to generate the myocardial torsion and structure of the SIT LV (6). Special focus was on the transition zone between the apex and base.

To be able to incorporate the apex-to-base changes in myofiber orientation as present in the SIT LV, the one-dimensional model by Arts et al. (2) was extended to include structural variations in the apex-to-base direction. Model-predicted LV structure and torsion were compared with experimental data reported by Delhaas et al. (6). In a sensitivity study, the effect of the degree of myofiber coupling in the transmural and apex-to-base directions on the final cardiac structure and function was determined.

## METHODS

In a numerical model, we simulated adaptation of global LV geometry and myofiber architecture to local mechanical tissue load. Mechanical quantities computed at the beginning and end of ejection, using a model of LV mechanics, act as stimuli for adaptation of tissue mass, tissue shape (cavity volume), and myofiber orientations. Subsequently, the adapted myocardial structure and geometry are used in the end-diastolic state of the next cardiac cycle, thus completing an adaptation cycle. The process of adaptation was simulated as a sequence of adaptation cycles.

### Model of LV Mechanics

#### Kinematics.

The LV geometry is represented by a rotational symmetric cylinder. To allow for axial variation in mechanics and structure, the cylindrical model is divided in a number of ring-shaped wall segments (*N*_{ring}) stacked from the apex to base (see Fig. 1). Deformation of such a segment is assumed to be a combination of axial stretch (λ_{z}), torsion (circumferential-axial shear strain) (τ), and contraction as quantified by the ejection volume (V_{ej}).

As a consequence of the deformation field, a reference point with cylindrical coordinates (*r*_{0},φ_{0},*z*_{0}) translates to point (*r*,φ,*z*): (1) where V_{cav,0} and *r*_{i,0} are the cavity volume and internal radius in the reference configuration, respectively. The ejection volume V_{ej} is prescribed. For a derivation of the function *f*, we refer to the appendix.

#### Constitutive laws.

The myocardial tissue is considered to be incompressible and to consist of myofibers with variable orientation embedded in an isotropic soft tissue matrix. Total Cauchy stress in the tissue is considered to be the sum of hydrostatic pressure (p) and the Cauchy stress (σ_{f}) actively generated by the myofibers. In the ejection phase, passive tissue stresses are neglected, as follows: (2) We define the myofiber direction (**e**_{f}) by the angle α between the circumferential direction (**e**_{φ}) and the apex-to-base direction (**e**_{z}), as follows: (3) The relation between deformed fiber angle α and reference fiber angle α_{0} is given in the appendix. The active stress (σ_{f}) generated by the myofibers during ejection depends linearly on their stretch ratio (λ_{f}), elastance (*E*), and normalized contractility (*C*), as follows: (4) where λ_{f,zs} represents the myofiber stretch ratio at zero myofiber stress, *l*_{f} is the actual myofiber length, and *l*_{f,0} is a reference myofiber length. Given the reference geometry and myofiber angle α_{0}, myofiber stretch λ_{f} is derived as a function of ejection volume V_{ej}, torsion τ, and axial stretch ratio λ_{z} (see the appendix).

Normalized contractility *C* of the myofibers represents the relative change in elastance *E* required to reach an ejection pressure (p_{cav,ej,pref}) assumed to be preferred by the body. A value of *C* = 1 means that *E* is sufficient to generate an ejection cavity pressure that is equal to the preferred pressure.

#### Balance laws.

Within the wall, stresses are restricted by the balances of axial force, radial force, and axial momentum (e.g., Ref. 10). Assuming zero epicardial pressure, (5) (6) (7) where p_{cav} is the cavity pressure and *r*_{i} and *r*_{o} are the inner and outer radii of the ring, respectively. The components of the stress tensor (σ_{rr}, σ_{φφ}, σ_{zz}, and σ_{φz}) are given by *Eq. 2*.

#### Numerical implementation.

Kinematics, constitutive laws, and balance laws are applied to each ring. Thus, each ring is attributed a torsion τ and an axial extension λ_{z}. Ejection volume V_{ej} of each ring is prescribed by the total ejection volume V_{ej,tot} of the LV, i.e., V_{ej} = V_{ej,tot}/*N*_{ring}. The elastance *E* is constant throughout the wall, indicating synchronous mechanical activity. The contractility *C* is constant in each ring but may vary in the axial direction between ring segments.

The torsion τ and axial extension λ_{z} for the rings are iteratively determined from the balance laws in *Eqs. 5* and *6* by means of the conjugate gradient method for a given reference geometry and structure. To evaluate the integrals in *Eqs. 5*–*7*, each ring is radially divided into a number of shells (*N*_{sh}) that all have the same volume (V_{sh}) (see Fig. 1). Consequently, the wall mass of the ring (V_{w}) is given by V_{w} = V_{sh} × *N*_{sh}. As a result, the continuous integrals over the radius *r* become summations over the discrete number of shells (for details, see Ref. 2). Myofiber stretch ratio λ_{f} may vary between all shells in the transmural direction as well as in the axial direction.

#### The cardiac cycle.

Within the cardiac cycle, we discriminate between *1*) the end of diastole, which acts as the reference (as denoted by subscript “0”); *2*) the beginning of ejection (as denoted by subscript “be”); and *3*) the end of ejection (as denoted by subscript “ee”). During the isovolumic contraction phase, the myofibers start to develop force as modeled by an increase in elastance *E* to 80 kPa at the beginning of ejection. Normalized contractility *C* is set to 1 in all rings. During the transition from the end of diastole to the beginning of ejection, the LV cavity volume remains constant by specifying V_{ej,tot} = 0. To satisfy balance *Eqs. 5* and *6*, the individual rings will deform to their configuration at the beginning of ejection, exhibiting a torsion τ_{be} and an axial extension λ_{z,be}. At the new equilibrium, *Eq. 7* gives the cavity pressures for each of the rings (denoted p_{cav,be}). During the ejection phase, the volume change of the total LV (V_{ej,tot}) is set to 80 ml, whereas the elastance *E* in all rings remains at 80 kPa. Again, the balance laws in *Eqs. 5* and *6* provide the torsion τ_{ee} and axial extension λ_{z,ee}, and *Eq. 7* gives the cavity pressure p_{cav,ee} for all rings at the end of ejection.

Subsequently, the contractility *C* of the myofibers (*Eq. 4*) in each ring adapts so that a preferred cavity pressure (p_{cav,ej,pref}) is reached. We relate the contractility to the mean of the LV cavity pressure at the beginning and end of ejection. Consequently, for *ring j*, the contractility is given by the following: (8) Next, a new cardiac cycle is simulated with the new contractility values (*C*_{j}). The mechanical load quantities calculated at the beginning and end of ejection in this cycle are used as stimuli for adaptation.

### Rules for Adaptation of Geometry and Myofiber Architecture

#### Stimuli.

Wall mass, tissue shape (end-diastolic cavity volume), and myofiber orientations of each ring are locally adapted based on five load stimuli (*s*) during the cardiac cycle. At the beginning of ejection, we define the myofiber stretch stimulus (*s*_{f,be}) for *shell i* in *ring j* as follows: (9) Similarly, we define stimuli due to deviations from preferred values of myofiber shortening during ejection (*s*_{f,ej}) and normalized contractility (*s*_{C}) as follows: (10) (11) In addition, we define stimuli at the beginning of ejection due to myofiber misalignment in the transmural (radial) direction (*s*_{r}) and the apex-to-base (axial) direction (*s*_{z}) as follows: (12) (13) The modulus is added since angles at +90° and −90° essentially describe the same myofiber orientation.

It is noted that, during adaptation, sarcomere properties, as described by *Eq. 4*, are assumed to be unaffected.

#### Tissue mass.

Local tissue mass was considered to change in response to myofiber stretch ratio at the beginning of ejection λ_{f,be} and normalized contractility *C* as follows: (14) where g_{i,j} specifies the amount of growth. However, the calculation of mechanics requires a discrete number of shells with the same volume. Therefore, to implement growth, the shells were either duplicated and placed on the outside of the original shell (positive growth) or removed (negative growth). For this purpose, a probability function G is computed from g as follows: (15) The value of δ_{g}, randomly taken between −β and +β (flat histogram), is added to translate a change of wall mass (g) into a probability (G) of shell duplication or removal. To duplicate or remove the shell, (16) We found that β = 0.5 yielded a sufficiently fast and stable growth response.

#### Tissue shape.

In case the myocardial tissue is subjected to large deformations as induced by increased filling, the cavity pressure-volume relationship has been shown to shift to higher volumes (8). Increased filling logically translates into end-diastolic myofiber strain. However, due to an absence of the filling phase in our model, we used ejection myofiber strain as an estimate for end-diastolic strain. Consequently, to model the shift in the pressure-volume relation, we change the end-diastolic ring cavity volume (V_{cav,ed}) before adaptation to (V*_{cav,ed}) after adaptation in response to the transmural average stimulus (*s*_{f,ej}), as follows: (17) Each ring is assumed to contribute equally to the total end-diastolic cavity volume (V_{cav,ed}), i.e., V_{cav,ed} = V_{cav,ed,tot}/*N*_{ring}.

#### Fiber orientation.

The end-diastolic (reference) myocardial structure, defined by angles α_{0,i,j}, adapts via reorientation of the myofibers. We hypothesize that myofibers reorient to achieve a shortening during ejection of 15% (λ_{f,ej,pref} = 0.85). In addition, we hypothesize that myofibers reorient to form straight pathways to relieve internal stresses arising in the extracellular matrix due to myofiber contraction.

Therefore, in the model, a local objective function (*O*_{i,j}) was defined as follows: (18) The weight factors *w*_{r} and *w*_{z} scale contributions to *O*_{i,j} of the misalignment in the transmural (radial) and apex-to-base (axial) direction, respectively. Higher values of *w*_{r} and *w*_{z} indicate a higher degree of myofiber alignment imposed on the myocardial structure.

To determine a more preferred myocardial structure, in each adaptation cycle, proposed end-diastolic (reference) myofiber angles α*_{0} are determined by adding values δ_{α} randomly selected between −2.5° and +2.5° to the existing myofiber angles α_{0} as follows: (19) Next, a cardiac cycle is simulated with the proposed myofiber angles α*_{0}, and the values of objective function *O** are determined. The proposed angles are accepted if the corresponding value of *O**_{i,j} is less than the original value of *O*_{i,j}; if not, myofiber angles remain unaltered.

### Simulations Performed

#### Adaptation in the single-ring model of the LV.

A multiring model of the LV requires boundary conditions at the apex and base. To obtain these boundary conditions, we first performed a set of simulations with a single ring (*N*_{ring} = 1), starting with random myofiber orientations. Total ejection volume was set at V_{ej} = 80 ml. Initial cavity volume was arbitrarily set to V_{cav,ed,init} = 90 ml, and the wall initially consisted of *N*_{sh,init} = 10 shells of V_{sh} = 1 ml each. Imposed myofiber alignment within the ring in the transmural direction was given by *w*_{r} = 1.0. For a complete overview of the parameter values used in the model of cardiac mechanics and in the adaptation rules, see Table 1. Adaptation was simulated over 600 cycles.

In the simulations, after 600 cycles, either one of two stable myocardial structures was found (see Fig. 2*A*). In the NORM structure, myofiber angles gradually changed from positive at the endocardium to negative at the epicardium. The MIRROR structure was the mirror image of the NORM structure.

To asses the sensitivity of the final myocardial structure on λ_{f,ej,pref}, p_{cav,ej,pref}, *E*, and λ_{f,zs}, a set of single-ring simulations was performed in which those parameters were individually increased and decreased. The resulting change of the myocardial structure was quantified by the root mean square (RMS) in the end-diastolic (reference) myofiber angle, as follows: (20) with α_{0} as the angles that are obtained with the default value in Table 1, and α̂_{0} as the myofiber angles resulting from the parameter change.

#### Adaptation in the multiring model of the LV.

Subsequently, adaptation is simulated with a model consisting of 30 axially stacked rings (the multiring model). Boundary conditions at the base and apex determined whether a normal LV or a SIT LV is simulated. For the normal LV, the NORM structure was used at the apical boundary as well as the basal boundary. For the SIT LV, the NORM structure was used at the apical boundary, whereas the MIRROR structure was taken at the basal boundary. Boundary structures remained unaltered during the adaptation process.

Between the apex and base, myofiber orientations, end-diastolic cavity volume, and wall volume were adapted. At the start, myofibers were randomly orientated while the end-diastolic cavity volumes for each ring were obtained by normalizing the resulting volumes in the single-ring simulation with respect to the number of rings. Shell volume was also normalized (equal to 1/*N*_{ring}; in ml).

The imposed alignment between myofibers in the transmural and axial direction was specified by *w*_{r} = 1.0 and *w*_{z} = 1.0, respectively. The sensitivity of the final SIT structure on values of *w*_{r} and *w*_{z} was determined via simulations using combinations of threefold increased or threefold decreased values of *w*_{r} and *w*_{z}.

## RESULTS

### Adaptation in the Single-Ring Model of the LV

Figure 2 shows the resulting transmural course of end-diastolic (reference) myofiber angle α_{0} and the mechanical loads λ_{f,be} and λ_{f,ej} for the single-ring simulations.

In the NORM structure, the myofiber angle shows a gradual decrease from +60° at the subendocardium to about −30° at 92% of wall thickness, at which a large gradient occurs toward −90° at the subepicardium (Fig. 2*A*). The MIRROR structure shows the mirrored transmural course from −60° at the subendocardium to about −90° at the subepicardium. Associated stretch ratios of the myofibers at the beginning and end of ejection for the NORM simulation as well as the MIRROR simulation have obtained their reference values in the major part of the wall except for a 2% deviation in shortening near 92% of the wall thickness. This is the transmural location where the large gradient in the myofiber angle occurs (Fig. 2*B*).

Figure 2, *C*–*E*, shows the evolution of average ejection pressure p_{cav,ej}, wall volume V_{w}, and end-diastolic cavity volume V_{cav,ed} during the adaptation process, respectively. Table 2 shows that the global parameters in the single-ring LV have converged to stable values after 600 adaptation cycles. The obtained pressure value p_{cav,ej} is at the preferred level p_{cav,ej,pref} of 13 kPa, indicating that the increase in wall mass has fully compensated for the initial low ejection pressure (see Fig. 2, *C* and *D*). Except for the sign of torsion at end of ejection, the adapted global parameters were not significantly different between the NORM and MIRROR simulations.

Table 3 shows the RMS of the myofiber angles in response to changes in preferred myofiber stretch during ejection λ_{f,ej,pref}, preferred ejection pressure p_{cav,ej,pref}, myofiber elastance *E*, and myofiber stretch at zero stress λ_{f,zs}.

### Adaptation in the Multiring Model of the LV

After 1,000 adaptation cycles, myofiber orientations in the multiring models of the normal and SIT LV have converged. For the multiring model of the normal LV, the resulting cavity volume, wall volume, torsion, and fiber orientations were very similar to those obtained with the single-ring model in the NORM simulation (see Table 2 and Figs. 2 and 3). Hence, for the analysis and discussion of the multiring simulations, we will focus on the SIT LV only.

The SIT LV developed a transition zone in between the normal orientations at the apex and the mirrored orientations at the base. In this transition zone, especially in the endocardial layers, more myofibers appear to have adopted an orientation parallel to the apex-to-base axis. Figure 3, *C* and *D*, shows that the myofiber stretch ratio at the beginning of ejection is homogeneously distributed over the wall. However, within the transition zone, the myofiber stretch ratio during ejection is heterogeneously distributed, ranging from 0.75 to 0.95.

After adaptation reached a stable result, cavity pressures within all rings of the SIT LV obtained a value of 13 kPa. Cavity and wall volume for the SIT LV were 118.1 ± 0.3 and 177.7 ± 0.3, respectively. Torsion values at different levels between the apex and base are shown in Fig. 3*F*. In the SIT LV, torsion gradually changed sign over the transition zone from +0.136 rad at the base to −0.135 rad at the apex.

Figure 4 shows the sensitivity of myofiber stretch ratios during ejection and torsion at end of ejection to different values of *w*_{z} and *w*_{r} for the SIT LV. With increasing axial coupling factor *w*_{z}, the transition zone widened, as reflected by the widening of the zone in which the myofiber stretch ratio during ejection is distributed inhomogeneously. Furthermore, torsion at the end of ejection changed more gradually from the apex to base. Decreasing *w*_{z} showed opposite results in myofiber shortening and torsion. Alterations in imposed myofiber coupling in the transmural direction by changing *w*_{r} did not significantly affect the myofiber stretch ratio and torsion patterns. Parameter *w*_{r} could be reduced to zero without a significant change in myofiber stretch ratio and torsion patterns.

## DISCUSSION

In the present study, we investigate whether, based on adaptation to mechanical loading, we could estimate the myocardial torsion and structure of the SIT LV (6). We followed the approach as adopted by Arts et al. (2) in which, in a model of cardiac mechanics, parameters of shape, mass, and internal myocardial architecture are simultaneously adapted based on mechanical stimuli. Stimuli were linked to adaptation responses of the tissue using adaptation rules that phenomenologically describe results from in vivo and in vitro experiments.

### Model Results Compared With Experimental Data

First, similar to Arts et al. (2), adaptation was simulated in a circumferentially symmetric model, representing a LV with no apex-to-base variation in myocardial structure (the single-ring model in the present study). As a result of adaptation, a random initial architecture converged to either one of two stable architectures that were mirror imaged with respect to one another (the NORM and MIRROR myofiber angle distributions in Fig. 2*A*). Geometrical parameters in both populations were identical (see Table 2 and Fig. 2, *C*–*E*). End-diastolic volumes, ejection fractions, and wall mass were realistic at ∼125 ml, 64%, and 178 ml, respectively. Myofiber shortening during ejection was homogeneous across the wall.

As shown in Fig. 5, myofiber orientation in the NORM structure is within the range of experimental data of the normal LV. In addition, the torsion value of −0.135 rad at the end of ejection was in the range of experimental observations of the normal LV (6). The single-ring simulations typically show a large gradient in myofiber angle near the epicardium (see Fig. 2).

As shown in Fig. 5 and as also reported by Arts et al. (2), experimental data have shown a similar increase in transmural gradient in myofiber angle near the epicardium (2, 9, 21, 16). An explanation for this increased gradient has been provided previously (1). In short, near the epicardium, the myofiber direction is similar to the principle direction of tissue shortening (3). Consequently, the amount of shortening near the epicardium becomes relatively insensitive to the myofiber direction. Conversely, from the argument that the myofiber orientation adapts to achieve a certain amount of shortening, a wide range of myofiber directions are possible near the epicardium. Apparently, to achieve a balance of forces in our model, this local degree of freedom in myofiber orientation results in a steep gradient to more axial orientations.

Our model suggests that the mirror image of the normal structure is stable as well. However, to the best of our knowledge, a LV with a completely mirrored structure has never been reported. Instead, the SIT LV was found to have a dualistic myocardial structure in which a normal structure at the apex appears to transit into a mirrored structure at the base (6) (Fig. 3). To allow for the axial variation in the SIT myocardial structure, a multiring model was created. Preliminary simulations with the multiring model showed that when all ring were allowed to adapt, eventually the structure exhibited no transition zone, i.e., the structure became either fully normal (and equal to the results obtained with the single-ring model) or fully mirrored. Therefore, to enforce a transition zone, we fixed the ring at the basal boundary in the mirrored structure and the ring at the apical boundary in the normal structure as obtained after convergence in the single-ring model (Fig. 2). Starting with a random intermediate structure, the local adaptation resulted in a stable geometry and structure (see Table 2 and Fig. 3). The final myocardial architecture exhibited a small transition zone between the apex and base with predominantly axially and circumferentially oriented myofibers (see Fig. 3*B*). Myofiber shortening during ejection was locally heterogeneous (see Fig. 3*D*). Currently, the experimental data presented or referred to by Ref. 6 are all that is available on SIT hearts. Quantification of the SIT myocardial structure with magnetic resonance diffusion tensor imaging (MR-DTI) could be used to assess in more detail whether the structure estimated by the model is realistic. However, existing SIT heart specimens have been preserved in formalin or ethanol-glycerine for a long time, which makes them unsuitable for MR-DTI. Still, qualitatively, the model-estimated myofiber orientations shown in Fig. 3*B* are similar to the experimental data shown in Fig. 3*A*. Furthermore, the transmural course in myofiber angle is reflected by the torsion. Although there is some spread in the data, the experiments show a reversal in sign of torsion from the apex to base, where absolute values of apical and basal torsion are about the same. The torsion as estimated by the model exhibits the same characteristics (see Fig. 3*F*).

The model estimates some differences between the normal and SIT LV. Estimated end-diastolic cavity volume is slightly decreased by 5% with respect to that of the normal LV. Since stroke volume was used as a fixed input in the model, ejection fraction is somewhat elevated by 4% (see Table 2). However, these differences are small. In addition, wall mass and ejection pressures were similar to those in the normal LV. This indicates that for both hearts, the globally generated external work (area pressure-volume loop) per unit of mass is equal, despite local heterogeneity in SIT myofiber shortening.

### Sensitivity Analysis

In the model, we used a preferred myofiber shortening and average cavity pressure during ejection as target values for the adaptation. Quantification of normal LV deformation with magnetic resonance tagging revealed myofiber shortening during ejection in the real normal LV to be ∼15% (14). In addition, cavity pressure during ejection is roughly 13 kPa. Apparently, in reality, the normal LV geometry and structure have adapted to achieve ∼15% myofiber shortening during ejection at an average pressure of 13 kPa. To investigate the sensitivity of the myocardial structure on the preferred values of myofiber shortening or ejection pressure, we performed a parameter variation in the single-ring model (see Table 3). The simulations revealed that the RMS value in the end-diastolic myofiber angle was no more than 2°, indicating that changes in the myocardial structure were minor. Cavity volume, wall mass, and deformation (torsion and axial shortening) appeared to have adapted in such a way that the preferred fiber shortening was achieved with the same myocardial structure.

In the multiring model, alignment of myofibers was imposed by coupling neighboring myofibers in the transmural and apex-to-base direction. To asses the influence of the degree of myofiber coupling on the final myofiber architecture, a parameter sensitivity analysis was conducted. A change in transmural myofiber coupling (factor *w*_{r}) from the reference value did not significantly alter final ejectional shortening and torsion (see Fig. 4*A*). In fact, even in the absence of coupling (*w*_{r} = 0), shortening and torsion did not change significantly. Since myofiber shortening is very sensitive to myofiber orientations (5), it is likely that the final myocardial structure did not change significantly as well. In contrast, shortening and torsion changed significantly in response to a variation in parameter *w*_{z}, controlling coupling between myofibers in the axial (apex-to-base) direction (see Fig. 4*B*). This indicates that myofiber orientations have changed as well. The size of the transition zone increased, as indicated by an increase of the zone in which myofiber shortening is heterogeneously distributed and by a more gradual change in torsion from the apical to basal value. Simulations with *w*_{z} < 0.3 showed that no transition zone developed between the apical and basal structure, i.e., the structure in the upper half of the LV was fully mirrored, whereas the structure in the bottom half of the LV was fully normal. In the case of *w*_{z} > 3.0, the transition zone extended to the apical and basal boundaries. In that case, the prescribed apical and basal boundary conditions, rather than the adaptation rules, dominated the architecture in the transition zone. Our model suggests that for the development and maintenance of the transmural variation in myofiber orientation, imposed myofiber coupling in the transmural direction is not required, i.e., the mechanical load (fiber shortening during ejection) sensed by each individual cell appears to be sufficient.

### Assumptions and Limitations

#### Adaptation rules.

The adaptation rules used in this study are based on the rules proposed by Arts et al. (2). We added an apex-to-base component in the adaptation rules to study adaptation in SIT. In our model, *1*) increased diastolic myofiber stretch and increased contractility are linked to an increase in wall mass, *2*) increased strain excursions induces cavity dilation, and *3*) myofibers align with their environment, attempting to achieve a preferred amount of systolic shortening. These rules are substantiated by more recent in vitro and in vivo data and remain up to date. For instance, strain remains one of the primary correlates to changes in wall mass and cavity volume (7, 8, 11). Nonetheless, experiments have not yet ruled out other mechanical stimuli such as stress or strain rate (12, 22).

There is still some debate on whether increased contractility is a stimulus to hypertrophy. Cathecholamine release in response to low systemic pressure is known to induce an acute increase in contractility via adrenergic receptors, chronically leading to an increase in wall mass in vivo (4, 20). However, in an in vivo volume overload model, Holmes (11) did not find a high correlation between observed contractility [as quantified by a maximum rate of pressure increase (dp/d*t*_{max})] and the increase in wall mass.

It is noted that, in the adult heart, myofiber reorientation was never confirmed in vivo or ex vivo (9). Therefore, it may seem odd that we propose fiber reorientation as a means for the tissue to adapt to mechanical loading. However, the high sensitivity of load distribution to myofiber orientation (5) and the apparent absence of heterogeneities in loading in the real LV (25, 27) advocate for the existence of a myofiber orientation-controlling mechanism (2). In other words, mechanical load redistribution may be achieved by a very small amount of myofiber reorientation. This amount may well fall within the measurement accuracy of ±6° for MR-DTI, which is currently one of the most accurate techniques to measure myofiber orientations (9, 19). Thus, the lack of in vivo or ex vivo confirmation of fiber reorientation does not exclude its existence.

In reality, adaptation is likely to be a continuous process. However, the time scale at which changes in geometry and structure occur are orders of magnitude larger than the time scale at which changes in mechanics occur during the cardiac cycle. We found that, provided the adaptation yielded a stable solution, this solution was insensitive to the rate at which the adaptation occurs (see also Ref. 2). Therefore, the time scale of the simulated adaptation becomes arbitrary. In our model, we have chosen to adapt the geometry and structure in a discrete fashion after each cardiac cycle to improve computational efficiency.

#### Mechanics model.

The presented mechanics model provides a suitable initial step toward simulation of load-induced adaptation. In this mechanics model, the cardiac LV is represented by a rotationally symmetric cylinder. The real LV basal geometry is, by good approximation, cylindrical, whereas the apical geometry is rather spherical. Also, the internal myocardial structure is not entirely symmetric, exhibiting markable differences between the septum and free wall (e.g., Ref. 9). In addition, the right ventricle (RV) affects mechanics of the LV through direct mechanical coupling via the septum. Neglecting the contribution of the RV to LV mechanics may affect the adaptation and, as a result, the final geometry and structure of the LV. In future studies, finite element models may be used to include more complex asymmetries in cardiac geometry and structure to investigate the influence of the RV and a closed apex on the final structure.

Furthermore, the fluid-fiber model of the cardiac tissue behavior during ejection neglects contributions of the passive matrix as well as transmural components in the myofiber orientation. During systole, stresses passively induced due to matrix deformation are usually much smaller than the actively developed stress (13). Therefore, exclusion of passive stress is not likely to affect the model results on myocardial structure and torsion. Similarly, although a transmural component in myofiber orientation may affect the torsion (24), the torsion pattern from the apex to base is not likely to be significantly affected. In the SIT LV model, a dualistic pattern will still be seen.

In addition, sarcomere elastance and myofiber stretch ratio at zero active stress did not change during adaptation. Experimental data are unclear as to whether sarcomere properties change during adaptation or not (22). The stress-strain relation, as given by *Eq. 4*, was obtained from data presented by Donker et al. (7). They found that during ejection, the stress-strain relation is approximately linear. Although variation of sarcomere elastance and myofiber stretch ratio at zero active stress resulted in a change in wall mass, cavity volume, and cavity pressure, they did not yield significant changes in myofiber orientation (see table 3).

### Summary and Conclusions

In the present model study, we tested hypothesized mechanical stimuli and responses in cardiac adaptation to mechanical load on their ability to estimate a realistic myocardial structure of the normal and SIT LV. Local changes in tissue altered cavity volume, wall mass, and myocardial architecture at the organ level. The model was able to generate a realistic LV geometry and structure. In addition, the model was also able to simulate the instigating situation in the rare SIT LV with mirrored movements and transmural courses in myofiber direction between the apex and base (6). These results substantiate the importance of mechanical load in the formation and maintenance of cardiac structure and geometry. Furthermore, adapted myocardial architecture was found to be insensitive to fiber misalignment in the transmural direction, i.e., myofiber strain during ejection was sufficient to generate a realistic transmural variation in myofiber orientation. In addition, the model estimates that the normal and SIT LV have similar global pump work and mass, despite differences in structure.

## APPENDIX

During deformation, the material point displaces from cylindrical coordinates (*r*_{0},φ_{0},*z*_{0}) to coordinates (*r*,φ,*z*). Globally, the deformation is parameterized by ejection volume V_{ej}, axial stretch λ_{z}, and torsion τ. Assuming the tissue to be incompressible, (A1) where *r*_{i} and *r*_{i,0} are the inner radius of the cylinder in the deformed and reference configuration, respectively. The ejection fraction related *r*_{i} and *r*_{i,0} as follows: (A2) where V_{cav,0} is the cavity volume enclosed by the cylinder. Rewriting *Eq. A2* gives the following: (A3) The combination of *Eqs. A1* and *A3* results in the following: (A4) (A5) From continuum mechanics, the deformation field is described by the deformation gradient tensor **F**. Tensor components with respect to the cylindrical coordinate base are stored in matrix F, which is given by: (A6) with According to *Eq. 3* for the components of reference myofiber direction (A7) where α_{0} is the myofiber angle defined in the reference configuration. The components of deformed myofiber direction is given by the following: (A8) with myofiber stretch ratio λ_{f}: (A9) The combination of *Eqs. A6*, *A7*, and *A9* results in the following: (A10) Furthermore, fiber angle α in the deformed configuration is given by the following: (A11)

## GRANTS

This work was supported by The Netherlands Heart Foundation Grant 2000T036 and University Hospital Maastricht Grant PF155. T. Delhaas is a Clinical Fellow of The Netherlands Heart Foundation (Dr. E. Dekker Fund).

## Footnotes

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- Copyright © 2008 by the American Physiological Society