The CircAdapt computational model of the human heart and circulation is a closed-loop lumped-parameter model that simulates beat-to-beat hemodynamics and mechanics of the heart and blood vessels (Arts et al., 2004; Lumens et al., 2009). The model uses a simplified ventricular geometry, where cardiac walls are represented by thick-walled spherical shells consisting of myofibers. The TriSeg module allows for interventricular interaction by coupling the left (LV) and right ventricular (RV) walls through the interventricular septum (Lumens et al., 2009). Walls can be subdivided into patches using the MultiPatch module (Walmsley et al., 2015), which enables heterogeneity of myocardial tissue properties within the walls. Myofiber active and passive stress generation is modeled using a three-element Hill model (Arts et al., 2003). In brief, the series contractile element with the time-dependent length L s i ( t ) generates an active stress σ f , a c t ( t ) which represents the force developed through cross-bridge formation between actin and myosin filaments. The zero-active stress contractile element length L s i 0 , a c t defines the length at which no active stress is developed. The series elastic element with the time-dependent length L s e ( t ) describes the intrinsic elasticity of the sarcomere, and a reference length of L s e , i s o is used to define the length of the series elastic element at the onset of isovolumetric contraction. Furthermore, the density of cross-bridge formation is described by a time-dependent contractility state variable C ( t ,   L s i ( t ) ) , while active stress generation is scaled by a parameter S f , a c t . Taken together, active stress is calculated as follows: σ f , a c t ( t ) =   S f , a c t . C ( t ,   L s i ( t ) ) . ( L s i ( t ) − L s i , a c t ) . L s e ( t ) L s e , i s o (1)

Passive myocardial tissue behavior is described by the parallel elastic element, which captures both extracellular and intracellular structures. Therefore, total passive stress σ f , p a s ( t ) is the sum of the extracellular matrix stress σ f , E C M ( t ) and intracellular stress, that is, titin stress σ f , t i t ( t ) : σ f , p a s ( t ) =   σ f , E C M ( t ) +   σ f , t i t ( t ) (2)

The extracellular matrix stress σ f , p a s ( t ) depends as follows on the total time-dependent sarcomere length L s ( t ) , which is the sum of L s i ( t ) and L s e ( t ) : σ f , E C M ( t ) =   S f , p a s . ( ( L s ( t ) L s 0 , p a s ) k E C M − 1 ) (3)

Here, S f , p a s is a parameter scaling passive stress development, while L s 0 , p a s is the zero-passive stress sarcomere length. The parameter k E C M determines the nonlinearity of the relation. The titin stress σ f , t i t ( t ) depends on L s ( t ) , S f , a c t , L s 0 , p a s , and two parameters which scale the development of titin stress ( k 1 , t i t ) and the nonlinearity of the relation ( k 2 , t i t ): σ f , t i t ( t ) =   S f , a c t . k 1 , t i t . ( ( L s ( t ) L s 0 , p a s ) k 2 , t i t − 1 ) (4)

In a combined experimental and modeling study, Sunagawa et al. (1982) showed that the LV systolic function during regional ischemia was best captured by a two-compartment model including an active and a passive compartment. The rationale behind this two-compartment model is that the active compartment lumps all of the normally functioning tissues within the nonischemic and ischemic region, while the passive compartment lumps all of the nonfunctional parts of the ischemic region. As mentioned before, we hypothesized that we could use a similar modeling approach not only for simulating global LV pump mechanics during regional ischemia but also to simulate average regional LV tissue mechanics (i.e., the myofiber stress and strain) in ischemic myocardial segments, which are defined by deformation imaging techniques, for example, speckle tracking echocardiography (STE). In the CircAdapt model, tissue properties are defined per myocardial patch, and by default, each myocardial wall includes one myocardial patch. The active and passive stress of a patch is calculated using Eqs 1–4. To simulate reduced patch contractility, one approach is to reduce parameter S f , a c t (Eq. 1). Since this approach does not involve subdivision of the myocardial patch into multiple patches (i.e., compartments), we term this as a one-compartment modeling approach for simulating contractile dysfunction. For the two-compartment model of ischemia, the CircAdapt MultiPatch module (Walmsley et al., 2015) is applied. The MultiPatch module has previously been used to subdivide myocardial walls into different patches, each having their own unique tissue properties (Walmsley et al., 2015; Lumens et al., 2015; Voigt et al., 2015). In this study, we applied the MultiPatch module to further subdivide an ischemic myocardial patch with volume V p a t c h into a contractile and a noncontractile patch (i.e., compartment) with volumes V C and V N C , respectively (Figure 1). Compared to the implementation of Sunagawa et al. (1982), where the subdivided volume V p a t c h equaled the volume of the entire LV wall, we additionally applied the two-compartment model on a regional level, where V p a t c h equals the volume of an echocardiographic segment by approximation. Mechanics of V C are described by Eqs 1–4, while for V N C it holds S f , a c t = 0 ; therefore, the mechanics of V N C are fully described by Eq. 3. The degree of patch contractile dysfunction is determined by the noncontractile volume fraction ( N C V F ), defined as the relative volume of the noncontractile compartment: N C V F =   V N C V p a t c h (5)

The calculated strain ε ( t ) during the cardiac cycle represented the engineering strain based on the sarcomere length L s ( t ) : ε ( t ) = ( L s ( t ) L s , r e f − 1 ) .100 % (6) where L s , r e f is a reference sarcomere length used for strain calculations.

For the two-compartment model, the sarcomere length L s ( t ) of V p a t c h was calculated as the volume-weighted average of the sarcomere lengths of the contractile and noncontractile compartments, respectively, L s , C ( t ) and L s , N C ( t ) : L s ( t ) = ( 1 − N C V F ) . L s , C ( t ) +   N C V F . L s , N C ( t ) (7)

Constructed ESPVRs for the one- and two-compartment simulations of LV acute regional ischemia are shown in Figure 3. The associated values of V 0 and E m a x are provided in Table 1. Compared to the baseline simulation, E m a x (2.07 mmHg/ml) was moderately reduced for the two-compartment simulation (1.21 mmHg/ml), but it was severely reduced for the one-compartment simulation (0.45 mmHg/ml) of regional ischemia. Furthermore, with respect to the baseline V 0 (13.2 ml), V 0 was increased for the two-compartment simulation (46.7 ml), consistent with a rightward shift of the ESPVR, while the one-compartment simulation demonstrated a reduction of V 0 to −102.4 ml.

Simulated LV pressure-segment length loops and corresponding strain patterns at the baseline ( N C V F = 0) and during acute ischemia ( N C V F = 1) were successfully calibrated using the experimental measurements (Figures 4, 5). The calibrated parameter values are provided in Table 2, and an increase in L s 0 , p a s and k E C M during acute ischemia with respect to the baseline is demonstrated. Starting from this simulation of acute ischemia, a reduction of the N C V F shifted the LV pressure-segment length loop leftward, that is, back toward the baseline loop, while the loop area increased. The loops and strain patterns for N C V F values of 0.4 and 0.2 were similar to the measured loops and strain patterns after 15 min and 3 h of reperfusion, respectively (Figures 4, 5).

When starting from the calibrated simulation of acute ischemia, increasing k E C M led to a reduced amplitude of the segmental systolic stretch, while also shifting the LV pressure-segment length loop leftward (Figures 6, 7). Calibration of parameters L s 0 , p a s and k E C M demonstrated that a further increase of both L s 0 , p a s and k E C M was needed to reproduce the measured loop after 4 h of ischemia (Table 3), while a reduction of L s 0 , p a s combined with a further increase of k E C M was needed to reproduce the data after 15 min of subsequent reperfusion.

Patient characteristics are summarized in Table 4. Results of the patient-specific estimation of the N C V F for two different patients (patients 1 and 2) are shown in Figures 8, 9, respectively. For all four other patients, results are shown in Supplementary Section S11.3. In both patients 1 and 2, it can be noted that estimated N C V F values were increased in segments with increased systolic stretch or reduced peak systolic strain, both at the baseline and 6-month follow-up. In the acute phase after MI, patient 1 (Figure 8) demonstrated a localized area of increased N C V F in the anterior, anteroseptal, and inferoseptal segments, extending from the base all the way into the apex. At 6-month follow-up, LGE revealed the presence of scar tissue within the same area, especially in the apical septal and mid-ventricular anterior and anteroseptal regions. While increased segmental N C V F values remained after 6 months, the N C V F in the anterior, anteroseptal, and inferoseptal segments were lower than at the baseline. This functional improvement over time was also reflected by an increase of LVEF from 30% at the baseline to 49% at 6-month follow-up. In patient 2 (Figure 9), increased values of N C V F at the baseline were found, especially in the apical anterior and septal segments, but it was extended into the mid-ventricular and basal segments. At 6-month follow-up, while the size of the region with increased N C V F was reduced, severe contractile dysfunction remained in the apex and mid-ventricular anteroseptal region. LGE revealed the presence of a large, transmural scar in this area.

Analysis of all 22 myocardial segments with LGE>25% at 6-month follow-up revealed that in most segments (68%), the N C V F was higher at the baseline than after 6 months (Figure 10). At the baseline, 21 (95%) segments had N C V F >25% and 12 (55%) had N C V F >50%, while at 6-month follow-up, 15 (68%) segments had N C V F >25% and 6 (27%) had N C V F >50%. Average segmental Δ N C V F between the baseline and 6-month follow-up was −16.1 ± 21.8%.

Constructed LV ESPVRs at baseline and during acute regional ischemia using both a one- and two-compartment modeling approach demonstrated that E m a x was much less affected when a two-compartment model was used (Figure 3; Table 1). This result was expected and is in line with the observations of Sunagawa et al. (1982). In their study, they found no reduction in E m a x for an ischemic region size of 20%; however, our simulation demonstrated a reduction of E m a x from 2.07 mmHg/ml to 1.21 mmHg/ml. This lower value of E m a x in our simulation was likely caused by the relatively compliant passive tissue constitutive behavior, determined by the value of parameter k E C M (Eq. 3). When increasing k E C M , the nonlinearity of the passive stress-sarcomere length relation increases, which causes stiffer passive tissue behavior, thereby increasing E m a x . This effect of infarct stiffness on the LV systolic function was also previously demonstrated in a modeling study by Fomovsky et al. (2011). It is noted that, at the same time, as a result of the parameterization of the passive tissue behavior, the change in V 0 with respect to the baseline (+33.5 ml) was more pronounced in our two-compartment model than in the experimental observations, where it was approximately +10 ml for an ischemic region size of 20% (Sunagawa et al., 1982). Qualitatively, however, a similar rightward shift of the ESPVR was observed. The one-compartment simulation of acute regional ischemia demonstrated a severe reduction of V 0 to a value of −102.4 ml, which is not consistent with experimental observations.

Simulated strain patterns and LV pressure-segment length loops for N C V F values of 1, 0.4, and 0.2 qualitatively agreed with the experimental data on acute ischemia followed by 15 min and 3 h of reperfusion, respectively (Figures 4, 5). While parameters L s 0 , p a s and k E C M were calibrated for the simulation of N C V F = 1 to match the reported end-diastolic segment length and systolic stretch amplitude during acute ischemia, contractile recovery was simulated by reducing N C V F without further modifying L s 0 , p a s and k E C M (Table 2). This result supports N C V F as a parameter quantifying the degree of contractile dysfunction. Calibrated parameter values for the simulation of acute ischemia demonstrated that L s 0 , p a s and k E C M were increased with respect to their baseline values. Increases in L s 0 , p a s during acute ischemia have been reported in previous studies, which have found that ischemic tissue dimensions were larger at matched pressures (Richardson et al., 2015). Adding to the discussion of the previous paragraph, the relatively large increase in k E C M during acute ischemia with respect to the baseline may indicate an inadequate parameterization of k E C M in the baseline simulation, causing the myocardium to be too compliant. While immediate increases in tissue stiffness during acute ischemia have been reported (Theroux et al., 1977), an acute effect of ischemia on tissue passive constitutive behavior has not clearly been established (Holmes et al., 2005). The data showed that after 3 h of reperfusion following acute ischemia, contractile recovery was still incomplete, which was confirmed by our simulations which demonstrated a good agreement with the data for N C V F = 0.2. This persistent dysfunction which follows reperfusion after a short-term period of ischemia has been attributed to myocardial stunning (Braunwald and Kloner, 1982). While it is known that stunning involves complex abnormalities at the cellular level including decreased calcium responsiveness (Gao et al., 1995), which are not explicitly described in our model, the parameter N C V F captured the mechanics during stunning relatively well.

Our simulations of prolonged ischemia and subsequent reperfusion demonstrated that k E C M was increased between 15 min and 4 h of ischemia and was even further increased after 15 min of subsequent reperfusion (Figures 6, 7; Table 3). This increased stiffness during prolonged ischemia, and reperfusion is in agreement with measurements of compliance performed in the experiment and was linked to tissue edema as reflected by the increased myocardial water content (Lyseggen et al., 2005). The occurrence of interstitial edema and associated increases of myocardial stiffness during the necrotic phase have also been suggested in other studies (Holmes et al., 2005). At the same time, L s 0 , p a s was increased between our simulations of 15 min and 4 h of ischemia, which may indicate early infarct expansion (Holmes et al., 2005). However, for the simulation of subsequent reperfusion, L s 0 , p a s was again reduced, potentially indicating compaction of the necrotic region during the infarct healing process.

The parameter N C V F is defined in the model as the relative volume of the noncontractile compartment, and increasing N C V F , therefore, leads to increased contractile dysfunction. In the experimental dataset of acute ischemia and subsequent reperfusion, contractile recovery occurred, which was reflected by a reduction of N C V F in the model. However, in the dataset of prolonged ischemia and reperfusion, there was no contractile recovery, and contractile dysfunction remained at the same level ( N C V F = 1). Similarly, the patient-specific N C V F estimations demonstrated that this parameter was increased in segments with systolic stretching or reduced peak systolic strain. Interestingly, in this patient cohort, among all segments with LGE>25% 6 months after MI, N C V F values decreased substantially in most segments from the baseline to 6-month follow-up (Figure 10), suggesting partial recovery of myocardial contractile function over time. A possible explanation for this observed contractile recovery in the majority of segments is that most segments had nontransmural infarction at 6-month follow-up (LGE<50%), and nontransmural infarcts have been associated with functional recovery following revascularization (Ugander et al., 2010). Increased values of N C V F were found especially in segments with LGE>25% after 6 months but were not limited to the LGE-positive segments. It is likely that nonscarred segments with increased N C V F in the acute phase after MI were dysfunctional due to myocardial stunning (Braunwald and Kloner, 1982). Increased N C V F values in nonscarred regions at 6-month follow-up could potentially be related to adverse myocardial remodeling. When we compared our obtained strain data and N C V F estimations with other modeling (Leong et al., 2017) and clinical studies (Zhang et al., 2005; Chan et al., 2006; Becker et al., 2009; Kihlberg et al., 2015; Huttin et al., 2016) of strain in MI, we noticed a common finding that transmural MI causes significant transmural strain abnormalities, including systolic stretching, a reduction of peak systolic strain, and post-systolic shortening. For subendocardial MI, Leong et al. (2017) found in their model that strain was affected in the subendocardial layers, but mid-myocardial and subepicardial strains were preserved. A similar conclusion was reached in a clinical study by Becker et al. (2009), who showed that nontransmural infarction led to greater functional impairment of the endocardial layer than of the epicardial layer. While this transmural heterogeneity in function may exist, multiple studies using strain or strain rate imaging have successfully differentiated subendocardial MI from normal myocardium (Chan et al., 2006; Zhang et al., 2005) and thereby suggest that abnormal strains can be found in segments with subendocardial MI. The same suggestion holds for segments adjacent to an MI region, which have been shown to have a lower peak strain than segments remote from MI (McComb et al., 2015). From these and our own observations, we conclude that the N C V F is a functional parameter, which effectively indicates the degree of ischemia-induced contractile dysfunction; therefore, its use could enhance current diagnostics in patients with cardiovascular diseases. However, at the same time, N C V F alone seems to be unable to differentiate between underlying types of ischemic dysfunction; therefore, it is not suitable to replace LGE. Rather, it could provide useful diagnostic information on its own, or it could be used in addition to LGE.

The experimental measurements of Lyseggen et al. (2005) used in our study were acquired by ultrasonic crystals which were aligned parallel to the LV long axis, thereby closely resembling the LV longitudinal strain. However, in our model, we used the one-fiber model to simulate the myofiber strain within a spherical LV geometry; therefore, there can be a systematic discrepancy between simulated and experimental strain values. In the patient-specific optimizations, we corrected for this model discrepancy by normalization of measured and simulated strains to global strain amplitude, but global strain values were not measured in the experiment; hence, no correction was performed. However, the temporal behavior of the experimental and simulated strains was similar; therefore, we expected that this potential mismatch could have had consequences for the quantitative values of calibrated parameters, but the qualitative results remained unaffected. We also assumed that the measurements were obtained from the anterior portion of the LV free wall, implying that we compared these data to simulated strains in the ischemic region located within the LV free wall, thereby disregarding septal deformation. This assumption, if not fully valid, could additionally have caused a minor mismatch in the strain data between the model and experiment. Furthermore, a sensitivity analysis (data not shown) demonstrated that correlations between the N C V F and passive material properties ( L s 0 , p a s and k E C M ) are very small or even absent and that N C V F is the parameter that most significantly determines the contractile function. Since L s 0 , p a s and k E C M   are parameters which are modified in the noncontractile compartment only, they merely modify the end-diastolic sarcomere length and amplitude of systolic stretch, respectively, of the noncontractile compartment, thereby not impacting intrinsic myocardial contractility. Further research is needed to investigate whether and how L s 0 , p a s and k E C M   are identifiable from regional strain measurements. The patient data used for this study were retrospectively obtained, and no arterial pressure measurements were available. We assumed that none of the patients had hypertension, and we set pressures in the model to default values. Since strain is known to depend on pre- and afterload (Skulstad et al., 2002; Ferferieva et al., 2012; P. Reant, 2016), this assumption on the patients’ blood pressures could have influenced the strain signals. Furthermore, in our optimization protocol, the number of estimated parameters was limited. We did not estimate, for example, the timing of mechanical activation, which potentially could also differ regionally in patients with MI due to slower electrical conduction in the scar tissue (Richardson et al., 2015). Further research is required to determine whether the set of estimated parameters was sufficient or could potentially be improved. In addition, we only used systolic and early diastolic strains for optimization in this study since we considered the late diastolic strain to be relatively inaccurate due to the effects of, for example, drift compensation. While we believe that the strain interval used is large enough for estimating the relevant parameters in this study, future studies could evaluate whether the late diastolic strain adds relevant information to the objective function.