Impact of cardiac patch alignment on restoring post-infarct ventricular function.

Acute myocardial infarction (MI) leads to a loss of cardiac function which, following adverse ventricular remodeling (AVR), can ultimately result in heart failure. Tissue-engineered contractile patches placed over the infarct offer potential for restoring cardiac function and reducing AVR. In this computational study, we investigate how improvement of pump function depends on the orientation of the cardiac patch and the fibers therein relative to the left ventricle (LV). Additionally, we examine how model outcome depends on the choice of material properties for healthy and infarct tissue. In a finite element model of LV mechanics, an infarction was induced by eliminating active stress generation and increasing passive tissue stiffness in a region comprising 15% of the LV wall volume. The cardiac patch was modeled as a rectangular piece of healthy myocardium with a volume of 25% of the infarcted tissue. The orientation of the patch was varied from 0 to 150 ∘ relative to the circumferential plane. The infarct reduced stroke work by 34% compared to the healthy heart. Optimal patch support was achieved when the patch was oriented parallel to the subepicardial fiber direction, restoring 9% of lost functionality. Typically, about one-third of the total recovery was attributed to the patch, while the remainder resulted from restored functionality in native myocardium adjacent to the infarct. The patch contributes to cardiac function through two mechanisms. A contribution of tissue in the patch and an increased contribution of native tissue, due to favorable changes in mechanical boundary conditions.

Strain-controlled electrophysiological wave propagation alters in silico scar-based substrate for ventricular tachycardia.

Introduction: Assessing a patient's risk of scar-based ventricular tachycardia (VT) after myocardial infarction is a challenging task. It can take months to years after infarction for VT to occur. Also, if selected for ablation therapy, success rates are low. Methods: Computational ventricular models have been presented previously to support VT risk assessment and to provide ablation guidance. In this study, an extension to such virtual-heart models is proposed to phenomenologically incorporate tissue remodeling driven by mechanical load. Strain amplitudes in the heart muscle are obtained from simulations of mechanics and are used to adjust the electrical conductivity. Results: The mechanics-driven adaptation of electrophysiology resulted in a more heterogeneous distribution of propagation velocities than that of standard models, which adapt electrophysiology in the structural substrate from medical images only. Moreover, conduction slowing was not only present in such a structural substrate, but extended in the adjacent functional border zone with impaired mechanics. This enlarged the volumes with high repolarization time gradients (≥10 ms/mm). However, maximum gradient values were not significantly affected. The enlarged volumes were localized along the structural substrate border, which lengthened the line of conduction block. The prolonged reentry pathways together with conduction slowing in functional regions increased VT cycle time, such that VT was easier to induce, and the number of recommended ablation sites increased from 3 to 5 locations. Discussion: Sensitivity testing showed an accurate model of strain-dependency to be critical for low ranges of conductivity. The model extension with mechanics-driven tissue remodeling is a potential approach to capture the evolution of the functional substrate and may offer insight into the progression of VT risk over time.

Stimulus-effect relations for left ventricular growth obtained with a simple multi-scale model: the influence of hemodynamic feedback.

Cardiac growth is an important mechanism for the human body to respond to changes in blood flow demand. Being able to predict the development of chronic growth is clinically relevant, but so far models to predict growth have not reached consensus on the stimulus-effect relation. In a previously published study, we modeled cardiac and hemodynamic function through a lumped parameter approach. We evaluated cardiac growth in response to valve disease using various stimulus-effect relations and observed an unphysiological decline pump function. Here we extend that model with a model of hemodynamic feedback that maintains mean arterial pressure and cardiac output through adaptation of peripheral resistance and circulatory unstressed volume. With the combined model, we obtain stable growth and restoration of pump function for most growth laws. We conclude that a mixed combination of stress and strain stimuli to drive cardiac growth is most promising since it (1) reproduces clinical observations on cardiac growth well, (2) requires only a small, clinically realistic adaptation of the properties of the circulatory system and (3) is robust in the sense that results were fairly insensitive to the exact choice of the chosen mechanics loading measure. This finding may be used to guide the choice of growth laws in more complex finite element models of cardiac growth, suitable for predicting the response to spatially varying changes in tissue load. Eventually, the current model may form a basis for a tool to predict patient-specific growth in response to spatially homogeneous changes in tissue load, since it is computationally inexpensive.

Evaluation of stimulus-effect relations in left ventricular growth using a simple multiscale model.

Cardiac growth is the natural capability of the heart to change size in response to changes in blood flow demand of the growing body. Cardiac diseases can trigger the same process leading to an abnormal type of growth. Prediction of cardiac growth would be clinically valuable, but so far published models on cardiac growth differ with respect to the stimulus-effect relation and constraints used for maximum growth. In this study, we use a zero-dimensional, multiscale model of the left ventricle to evaluate cardiac growth in response to three valve diseases, aortic and mitral regurgitation along with aortic stenosis. We investigate how different combinations of stress- and strain-based stimuli affect growth in terms of cavity volume and wall volume and hemodynamic performance. All of our simulations are able to reach a converged state without any growth constraint, with the most promising results obtained while considering at least one stress-based stimulus. With this study, we demonstrate how a simple model of left ventricular mechanics can be used to have a first evaluation on a designed growth law.

A mathematical model to investigate the effects of intravenous fluid administration and fluid loss.

The optimal fluid administration protocol for critically ill perioperative patients is hard to estimate due to the lack of tools to directly measure the patient fluid status. This results in the suboptimal clinical outcome of interventions. Previously developed predictive mathematical models focus on describing the fluid exchange over time but they lack clinical applicability, since they do not allow prediction of clinically measurable indices. The aim of this study is to make a first step towards a model predictive clinical decision support system for fluid administration, by extending the current fluid exchange models with a regulated cardiovascular circulation, to allow prediction of these indices. The parameters of the model were tuned to correctly reproduce experimentally measured changes in arterial pressure and heart rate, observed during infusion of normal saline in healthy volunteers. With the resulting tuned model, a different experiment including blood loss and infusion could be reproduced as well. These results show the potential of using this model as a basis for a decision support tool in a clinical setting.

Simulation of fetal heart rate variability with a mathematical model.

In the clinic, the cardiotocogram (CTG), the combined registration of fetal heart rate (FHR) and uterine contractions, is used to predict fetal well-being. Amongst others, fetal heart rate variability (FHRV) is an important indicator of fetal distress. In this study we add FHRV to our previously developed CTG simulation model, in order to improve its use as a research and educational tool. We implemented three sources of variability by applying either 1/f or white noise to the peripheral vascular resistance, baroreceptor output, or efferent vagal signal. Simulated FHR tracings were evaluated by visual inspection and spectral analysis. All power spectra showed a 1/f character, irrespective of noise type and source. The clinically observed peak near 0.1 Hz was only obtained by applying white noise to the different sources of variability. Similar power spectra were found when peripheral vascular resistance or baroreceptor output was used as source of variability. Sympathetic control predominantly influenced the low frequency power, while vagal control influenced both low and high frequency power. In contrast to clinical data, model results did not show an increase of FHRV during FHR decelerations. Still, addition of FHRV improves the applicability of the model as an educational and research tool.

Determinants of biventricular cardiac function: a mathematical model study on geometry and myofiber orientation.

In patient-specific mathematical models of cardiac electromechanics, usually a patient-specific geometry and a generic myofiber orientation field are used as input, upon which myocardial tissue properties are tuned to clinical data. It remains unclear to what extent deviations in myofiber orientation and geometry between model and patient influence model predictions on cardiac function. Therefore, we evaluated the sensitivity of cardiac function for geometry and myofiber orientation in a biventricular (BiV) finite element model of cardiac mechanics. Starting out from a reference geometry in which myofiber orientation had no transmural component, two new geometries were defined with either a 27 % decrease in LV short- to long-axis ratio, or a 16 % decrease of RV length, but identical LV and RV cavity and wall volumes. These variations in geometry caused differences in both local myofiber and global pump work below 6 %. Variation of fiber orientation was induced through adaptive myofiber reorientation that caused an average change in fiber orientation of [Formula: see text] predominantly through the formation of a component in transmural direction. Reorientation caused a considerable increase in local myofiber work [Formula: see text] and in global pump work [Formula: see text] in all three geometries, while differences between geometries were below 5 %. The findings suggest that implementing a realistic myofiber orientation is at least as important as defining a patient-specific geometry. The model for remodeling of myofiber orientation seems a useful approach to estimate myofiber orientation in the absence of accurate patient-specific information.

New insights from a computational model on the relation between pacing site and CRT response.

AIMS: Cardiac resynchronization therapy (CRT) produces clinical benefits in chronic heart failure patients with left bundle-branch block (LBBB). The position of the pacing site on the left ventricle (LV) is considered an important determinant of CRT response, but the mechanism how the LV pacing site determines CRT response is not completely understood. The objective of this study is to investigate the relation between LV pacing site during biventricular (BiV) pacing and cardiac function. METHODS AND RESULTS: We used a finite element model of BiV electromechanics. Cardiac function, assessed as LV dp/dtmax and stroke work, was evaluated during normal electrical activation, typical LBBB, fascicular blocks and BiV pacing with different LV pacing sites. The model replicated clinical observations such as increase of LV dp/dtmax and stroke work, and the disappearance of a septal flash during BiV pacing. The largest hemodynamic response was achieved when BiV pacing led to best resynchronization of LV electrical activation but this did not coincide with reduction in total BiV activation time (∼ QRS duration). Maximum response was achieved when pacing the mid-basal lateral wall and this was close to the latest activated region during intrinsic activation in the typical LBBB, but not in the fascicular block simulations. CONCLUSIONS: In these model simulations, the best cardiac function was obtained when pacing the mid-basal LV lateral wall, because of fastest recruitment of LV activation. This study illustrates how computer modeling can shed new light on optimizing pacing therapies for CRT. The results from this study may help to design new clinical studies to further investigate the importance of the pacing site for CRT response.

A mathematical model to simulate the cardiotocogram during labor. Part A: Model setup and simulation of late decelerations.

The cardiotocogram (CTG) is commonly used to monitor fetal well-being during labor and delivery. It shows the input (uterine contractions) and output (fetal heart rate, FHR) of a complex chain of events including hemodynamics, oxygenation and regulation. Previously we developed a mathematical model to obtain better understanding of the relation between CTG signals and vital, but clinically unavailable signals such as fetal blood pressure and oxygenation. The aim of this study is to improve this model by reducing complexity of submodels where parameter estimation is complicated (e.g. regulation) or where less detailed model output is sufficient (e.g. cardiac function), and by using a more realistic physical basis for the description of other submodels (e.g. vessel compression). Evaluation of the new model is performed by simulating the effect of uterine contractions on FHR as initiated by reduction of uterine blood flow, mediated by changes in oxygen and blood pressure, and effected by the chemoreflex and baroreflex. Furthermore the ability of the model to simulate uterine artery occlusion experiments in sheep is investigated. With the new model a more realistic FHR decrease is obtained during contraction-induced reduction of uterine blood flow, while the reduced complexity and improved physical basis facilitate interpretation of model results and thereby make the model more suitable for use as a research and educational tool.

A mathematical model to simulate the cardiotocogram during labor. Part B: Parameter estimation and simulation of variable decelerations.

During labor and delivery the cardiotocogram (CTG), the combined registration of fetal heart rate (FHR) and uterine contractions, is used to monitor fetal well-being. In part A of our study we introduced a new mathematical computer model for CTG simulation in order to gain insight into the complex relation between these signals. By reducing model complexity and by using physically more realistic descriptions, this model was improved with respect to our previous model. Aim of part B of this study is to gain insight into the cascade of events from uterine contractions causing combined uterine flow reduction and umbilical cord compression, resulting in blood and oxygen pressure variations, which lead to changes in FHR via the baro- and chemoreflex. In addition, we extensively describe and discuss the estimation of model parameter values. Simulation results are in good agreement with sheep data and show the ability of the model to describe variable decelerations. Despite reduced model complexity, parameter estimation still remains difficult due to limited clinical data.

Assessment and comparison of left ventricular shear in normal and situs inversus totalis hearts by means of magnetic resonance tagging.

Situs inversus totalis (SIT) is characterized by complete mirroring of gross cardiac anatomy and position combined with an incompletely mirrored myofiber arrangement, being normal at the apex but inverted at the base of the left ventricle (LV). This study relates myocardial structure to mechanical function by analyzing and comparing myocardial deformation patterns of normal and SIT subjects, focusing especially on circumferential-radial shear. In nine control and nine SIT normotensive human subjects, myocardial deformation was assessed from magnetic resonance tagging (MRT) image sequences of five LV short-axis slices. During ejection, no significant difference in either circumferential shortening (εcc) or its axial gradient (Δεcc) is found between corresponding LV levels in control and SIT hearts. Circumferential-radial shear (εcr) has a clear linear trend from apex-to-base in controls, while in SIT it hovers close to zero at all levels. Torsion as well as axial change in εcr (Δεcr) is as in controls in apical sections of SIT hearts but deviates significantly towards the base, changing sign close to the LV equator. Interindividual variability in torsion and Δεcr values is higher in SIT than in controls. Apex-to-base trends of torsion and Δεcr in SIT, changing sign near the LV equator, further substantiate a structural transition in myofiber arrangement close to the LV equator itself. Invariance of εcc and Δεcc patterns between controls and SIT subjects shows that normal LV pump function is achieved in SIT despite partial mirroring of myocardial structure leading to torsional and shear patterns that are far from normality.

Layer-specific radiofrequency ultrasound-based strain analysis in a porcine model of ischemic cardiomyopathy validated by a geometric model.

Local layer-specific myocardial deformation after myocardial infarction (MI) has not been studied extensively although the sub-endocardium is more vulnerable to ischemia and interstitial fibrosis deposition. Radiofrequency (RF) ultrasound-based analysis could provide superior layer-specific radial strain estimation compared with clinically available deformation imaging techniques. In this study, we used RF-based myocardial deformation measurements to investigate layer-specific differences between healthy and damaged myocardium in a porcine model of chronic MI. RF data were acquired epicardially in healthy (n = 21) and infarcted (n = 5) regions of a porcine chronic MI model 12 wk post-MI. Radial and longitudinal strains were estimated in the sub-endocardial, mid-wall and sub-epicardial layers of the left ventricle. Collagen content was quantified in three layers of healthy and infarcted regions in five pigs. An analytical geometric model of the left ventricle was used to theoretically underpin the radial deformation estimated in different myocardial layers. Means ± standard errors of the peak radial and longitudinal strain estimates of the sub-endocardial, mid-wall and sub-epicardial layers of the healthy and infarcted tissue were: 82.7 ± 5.2% versus 39.9 ± 10.8% (p = 0.002), 63.6 ± 3.3% versus 38.8 ± 7.7% (p = 0.004) and 34.3 ± 3.0% versus 35.1 ± 5.2% (p = 0.9), respectively. The radial strain gradient between the sub-endocardium and the sub-epicardium had decreased 12 wk after MI, and histologic examination revealed the greatest increases in collagen in the sub-endocardial and mid-wall layers. Comparable normal peak radial strain values were found by geometric modeling when input values were derived from the in vivo measurements and literature. In conclusion, the estimated strain values are realistic and indicate that sub-endocardial radial strain in healthy tissue can amount to 80%. This high value can be explained by the cardiac geometry, as was illustrated by geometric modeling. After MI, strain values were decreased and collagen content was increased in the sub-endocardial and mid-wall layers. Layer-specific peak radial strain can be assessed by RF strain estimation and clearly differs between healthy and infarcted tissue. Although the relationship between tissue stiffness and tissue strain is not strictly local, this novel technique provides a valuable way to assess layer-specific regional cardiac function in a variety of myocardial diseases.

Effects of activation pattern and active stress development on myocardial shear in a model with adaptive myofiber reorientation.

It has been hypothesized that myofiber orientation adapts to achieve a preferred mechanical loading state in the myocardial tissue. Earlier studies tested this hypothesis in a combined model of left ventricular (LV) mechanics and remodeling of myofiber orientation in response to fiber cross-fiber shear, assuming synchronous timing of activation and uniaxial active stress development. Differences between computed and measured patterns of circumferential-radial shear strain E(cr) were assumed to be caused by limitations in either the LV mechanics model or the myofiber reorientation model. Therefore, we extended the LV mechanics model with a physiological transmural and longitudinal gradient in activation pattern and with triaxial active stress development. We investigated the effects on myofiber reorientation, LV function, and deformation. The effect on the developed pattern of the transverse fiber angle α(t,0) and the effect on global pump function were minor. Triaxial active stress development decreased amplitudes of E(cr) towards values within the experimental range and resulted in a similar base-to-apex gradient during ejection in model computed and measured E(cr). The physiological pattern of mechanical activation resulted in better agreement between computed and measured strain in myofiber direction, especially during isovolumic contraction phase and first half of ejection. In addition, remodeling was favorable for LV pump and myofiber function. In conclusion, the outcome of the combined model of LV mechanics and remodeling of myofiber orientation is found to become more physiologic by extending the mechanics model with triaxial active stress development and physiological activation pattern.

Simulation of changes in myocardial tissue properties during left ventricular assistance with a rotary blood pump.

We considered a mathematical model to investigate changes in geometric and hemodynamic indices of left ventricular function in response to changes in myofiber contractility and myocardial tissue stiffness during rotary blood pump support. Left ventricular assistance with a rotary blood pump was simulated based on a previously published biventricular model of the assisted heart and circulation. The ventricles in this model were based on the one-fiber model that relates ventricular function to myofiber contractility and myocardial tissue stiffness. The simulations showed that indices of ventricular geometry, left ventricular shortening fraction, and ejection fraction had the same response to variations in myofiber contractility and myocardial tissue stiffness. Hemodynamic measures showed an inverse relation compared with geometric measures. Particularly, pulse pressure and arterial dP/dtmax increased when myofiber contractility increased, whereas increasing myocardial tissue stiffness decreased these measures. Similarly, the lowest pump speed at which the aortic valve remained closed increased when myofiber contractility increased and decreased when myocardial tissue stiffness increased. Therefore, simultaneous monitoring of hemodynamic parameters and ventricular geometry indirectly reflects the status of the myocardial tissue. The appropriateness of this strategy will be evaluated in the future, based on in vivo studies.

Insight into variable fetal heart rate decelerations from a mathematical model.

During labor and delivery, variable decelerations in the fetal heart rate (FHR) are commonly seen on the cardiotocogram (CTG) that is used to monitor fetal welfare. These decelerations are often induced by umbilical cord compression from uterine contractions. Via changes in oxygenation and blood pressure, umbilical cord compression activates the chemo- and baroreceptor reflex, and thus affects FHR. Since the relation between the CTG and fetal oxygenation is complex, assessment of fetal welfare from the CTG is difficult. We investigated umbilical cord compression-induced variable decelerations with a mathematical model. For this purpose, we extended our model for decelerations originating from caput compression and reduced uterine blood flow with the possibility to induce umbilical venous, arterial and total cord occlusion. Model response during total occlusion is evaluated for varying contractions (duration and amplitude) and sensitivity of the umbilical resistance to the uterine pressure. A clinical scenario is used to simulate a labor CTG with variable decelerations. Simulation results show that fetal mean arterial pressure increases during umbilical cord occlusion, while fetal oxygenation drops. There is a clear relation between these signals and the resulting FHR. The extent of umbilical compression and thus FHR deceleration is positively related to increased contraction duration and amplitude, and increased sensitivity of the umbilical resistance to uterine pressure. No relation is found between contraction interval and FHR response, which can probably be ascribed to the lack of catecholamines in the model. The simulation model provides insight into the complex relation between uterine pressure, umbilical cord compression, fetal oxygenation, blood pressure and heart rate. The model can be used for individual learning, and incorporated in a simulation mannequin, be used to enhance obstetric team training.

Simulation of reflex late decelerations in labor with a mathematical model.

Fetal welfare during labor and delivery is commonly monitored through the cardiotocogram (CTG), the combined registration of uterus contractions and fetal heart rate (FHR). From the CTG, the fetal oxygen state is estimated as the main indicator of the fetal condition, but this estimate is difficult to make, due to the complex relation between CTG and oxygen state. Mathematical models can be used to assist in the interpretation of the CTG, since they enable quantitative modeling of the flow of events through which uterine contractions affect fetal oxygenation and FHR. We propose a mathematical model to simulate reflex 'late decelerations', i.e. variations in FHR originating from uteroplacental flow reduction during uterine contractions and mediated by the baroreflex and the chemoreflex. Results for the uncompromised fetus show that partial oxygen pressures reduce in relation to the strength and duration of the contraction. Above a certain threshold, hypoxemia will evoke a late deceleration. Results for uteroplacental insufficiency, simulated by reduced uterine blood supply or reduced placental diffusion capacity, demonstrated lower baseline FHR and smaller decelerations during contraction. Reduced uteroplacental blood volume was found to lead to deeper decelerations only. The model response in several nerve blocking simulations was similar to experimental findings by Martin et al. [18], indicating a correct balance between vagal and sympathetic reflex pathways.

Why SIT works: normal function despite typical myofiber pattern in Situs Inversus Totalis (SIT) hearts derived by shear-induced myofiber reorientation.

The left ventricle (LV) of mammals with Situs Solitus (SS, normal organ arrangement) displays hardly any interindividual variation in myofiber pattern and experimentally determined torsion. SS LV myofiber pattern has been suggested to result from adaptive myofiber reorientation, in turn leading to efficient pump and myofiber function. Limited data from the Situs Inversus Totalis (SIT, a complete mirror image of organ anatomy and position) LV demonstrated an essential different myofiber pattern, being normal at the apex but mirrored at the base. Considerable differences in torsion patterns in between human SIT LVs even suggest variation in myofiber pattern among SIT LVs themselves. We addressed whether different myofiber patterns in the SIT LV can be predicted by adaptive myofiber reorientation and whether they yield similar pump and myofiber function as in the SS LV. With a mathematical model of LV mechanics including shear induced myofiber reorientation, we predicted myofiber patterns of one SS and three different SIT LVs. Initial conditions for SIT were based on scarce information on the helix angle. The transverse angle was set to zero. During reorientation, a non-zero transverse angle developed, pump function increased, and myofiber function increased and became more homogeneous. Three continuous SIT structures emerged with a different location of transition between normal and mirrored myofiber orientation pattern. Predicted SIT torsion patterns matched experimentally determined ones. Pump and myofiber function in SIT and SS LVs are similar, despite essential differences in myocardial structure. SS and SIT LV structure and function may originate from same processes of adaptive myofiber reorientation.

Modeling cardiac electromechanics and mechanoelectrical coupling in dyssynchronous and failing hearts: insight from adaptive computer models.

Computer models have become more and more a research tool to obtain mechanistic insight in the effects of dyssynchrony and heart failure. Increasing computational power in combination with increasing amounts of experimental and clinical data enables the development of mathematical models that describe electrical and mechanical behavior of the heart. By combining models based on data at the molecular and cellular level with models that describe organ function, so-called multi-scale models are created that describe heart function at different length and time scales. In this review, we describe basic modules that can be identified in multi-scale models of cardiac electromechanics. These modules simulate ionic membrane currents, calcium handling, excitation-contraction coupling, action potential propagation, and cardiac mechanics and hemodynamics. In addition, we discuss adaptive modeling approaches that aim to address long-term effects of diseases and therapy on growth, changes in fiber orientation, ionic membrane currents, and calcium handling. Finally, we discuss the first developments in patient-specific modeling. While current models still have shortcomings, well-chosen applications show promising results on some ultimate goals: understanding mechanisms of dyssynchronous heart failure and tuning pacing strategy to a particular patient, even before starting the therapy.

A mathematical model for simulation of early decelerations in the cardiotocogram during labor.

Fetal welfare during labor and delivery is commonly monitored through the cardiotocogram (CTG), the combined registration of uterus contractions and fetal heart rate (FHR). The CTG gives an indication of the main determinant of the acute fetal condition, namely its oxygen state. However, interpretation is complicated by the complex relationship between the two. Mathematical models can be used to assist with the interpretation of the CTG, since they enable quantitative modeling of the cascade of events through which uterine contractions affect fetal oxygenation and FHR. We developed a mathematical model to simulate 'early decelerations', i.e. variations in FHR originating from caput compression during uterine contractions, as mediated by cerebral flow reduction, cerebral hypoxia and a vagal nerve response to hypoxia. Simulation results show a realistic response, both for fetal and maternal hemodynamics at term, as for FHR variation during early decelerations. The model is intended to be used as a training tool for gynaecologists. Therefore 6 clinical experts were asked to rate 5 real and 5 model-generated CTG tracings on overall realism and realism of selected aspects. Results show no significant differences between real and computer-generated CTG tracings.

Modeling of cardiac growth and remodeling of myofiber orientation.

The heart has the ability to respond to long-term changes in its environment through changes in mass (growth), shape (morphogenesis) and tissue properties (remodeling). For improved quantitative understanding of cardiac growth and remodeling (G&R) experimental studies need to be complemented by mathematical models. This paper reviews models for cardiac growth and remodeling of myofiber orientation, as induced by mechanical stimuli. A distinction is made between optimization models, that focus on the end stage of G&R, and adaptation models, that aim to more closely describe the mechanistic relation between stimulus and effect. While many models demonstrate qualitatively promising results, a lot of questions remain, e.g. with respect to the choice of the stimulus for G&R or the long-term stability of the outcome of the model. A continued effort combining information on mechanotransduction at the cellular level, experimental observations on G&R at organ level, and testing of hypotheses on stimulus-effect relations in mathematical models is needed to answer these questions on cardiac G&R. Ultimately, models of cardiac G&R seem indispensable for patient-specific modeling, both to reconstruct the actual state of the heart and to assess the long-term effect of potential interventions.