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Pulmonary arterial hypertension (PAH) is a rare disorder characterized by elevated blood pressure and pulmonary vascular resistance, often followed by right ventricular hypertrophy and heart failure. The effect of PAH and its treatments on the mechanics, function, and remodelling of the right ventricle (RV) is currently not well understood. To study cardiac biomechanics and functionality as PAH progresses, we implemented a computational model of the heart simulating right ventricular maladaptive remodelling. Our Windkessel-based model, which accounts for direct ventricular interaction and the presence of the pericardium, is utilized to simulate various disease stages of PAH. We find that the pericardium has a larger effect on heart performance than ventricular interaction through the septum.We also examined the effectiveness of two treatments, ventricular assist device (RVAD) and atrial septostomy, on diseased hearts. We show that while both pulsatile and continuous RVADs restore cardiac function, pulsatile RVAD improves cardiac output 29.4% more than continuous RVAD. We also demonstrate that atrial septostomy improves cardiac output by 19.5%. Our model can be further extended by simulating the heart’s response to other treatments such as extracorporeal membrane oxygenation (ECMO), and by incorporating ventricular remodelling growth simulations and finite-element ventricular modelling.

Mathematical models of the cardiovascular system have come a long way since they were first introduced in the early 19th century. Driven by a rapid development of experimental techniques, numerical methods, and computer hardware, detailed models that describe physical scales from the molecular level up to organs and organ systems have been derived and used for physiological research. Mathematical and computational models can be seen as condensed and quantitative formulations of extensive physiological knowledge and are used for formulating and testing hypotheses, interpreting and directing experimental research, and have contributed substantially to our understanding of cardiovascular physiology. However, in spite of the strengths of mathematics to precisely describe complex relationships and the obvious need for the mathematical and computational models to be informed by experimental data, there still exist considerable barriers between experimental and computational physiological research. In this review, we present a historical overview of the development of mathematical and computational models in cardiovascular physiology, including the current state of the art. We further argue why a tighter integration is needed between experimental and computational scientists in physiology, and point out important obstacles and challenges that must be overcome in order to fully realize the synergy of experimental and computational physiological research.

Cardiomyocytes are the functional building blocks of the heart—yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to follow a normal-tangential pattern concentrated along the membrane, with substantial spatial variations both under contraction and stretching. We also examined how it scales to larger size simulations, considering multicellular domains. Our work extends existing continuum models, providing a new geometrical-based framework for exploring complex cell–cell and cell–matrix interactions.

Cardiac resynchronization therapy (CRT) is an effective treatment for a subgroup of heart failure (HF) patients, but more than 30% of those selected do not improve after CRT implantation. Imperfect pre-procedural criteria for patient selection and optimization are the main causes of the high non-response rate. In this study, we evaluated a novel measure for assessing CRT response. We used a computational modeling framework to calculate the regional stress of the left ventricular wall of seven CRT patients and seven healthy controls. The standard deviation of regional wall stress at the time of mitral valve closure (SD_MVC) was used to quantify dyssynchrony and compared between patients and controls and among the patients. The results show that SD_MVC is significantly lower in controls than patients and correlates with long-term response in patients, based on end-diastolic volume reduction. In contrast to our initial hypothesis, patients with lower SD_MVC respond better to therapy. The patient with the highest SD_MVC was the only non-responder in the patient cohort. The distribution of fiber stress at the beginning of the isovolumetric phase seems to correlate with the degree of response and the use of this measurement could potentially improve selection criteria for CRT implantation. Further studies with a larger cohort of patients are needed to validate these results.

Heterogeneous mechanical dyskinesis has been implicated in many arrhythmogenic phenotypes. Strain-dependent perturbations to cardiomyocyte electrophysiology may contribute to this arrhythmogenesis through processes referred to as mechanoelectric feedback. Although the role of stretch-activated ion currents has been investigated using computational models, experimental studies suggest that mechanical strain may also promote arrhythmia by facilitating calcium wave propagation. To investigate whether strain-dependent changes in calcium affinity to the myofilament may promote arrhythmogenic intracellular calcium waves, we modified a mathematical model of rabbit excitation-contraction coupling coupled to a model of myofilament activation and force development. In a one-dimensional compartmental analysis, we bidirectionally coupled 50 sarcomere models in series to model calcium diffusion and stress transfer between adjacent sarcomeres. These considerations enabled the model to capture 1) the effects of mechanical feedback on calcium homeostasis at the sarcomeric level and 2) the combined effects of mechanical and calcium heterogeneities at the cellular level. The results suggest that in conditions of calcium overload, the vulnerable window of stretch-release to trigger suprathreshold delayed afterdepolarizations can be affected by heterogeneity in sarcomere length. Furthermore, stretch and sarcomere heterogeneity may modulate the susceptibility threshold for delayed afterdepolarizations and the aftercontraction wave propagation velocity.

Computational models of the heart have reached a level of maturity that enables sophisticated patient-specific simulations and hold potential for important applications in diagnosis and therapy planning. However, such clinical use puts strict demands on the reliability and accuracy of the models and requires the sensitivity of the model predictions due to errors and uncertainty in the model inputs to be quantified. The models typically contain a large number of parameters, which are difficult to measure and therefore associated with considerable uncertainty. Additionally, patient-specific geometries are usually constructed by semi-manual processing of medical images and must be assumed to be a potential source of model uncertainty. In this paper, we assess the model accuracy by considering the impact of geometrical uncertainties, which typically occur in image-based computational geometries. An approach based on 17 AHA segments diagram is used to consider uncertainties in wall thickness and also in the material properties and fiber orientation, and we perform a comprehensive uncertainty quantification and sensitivity analysis based on polynomial chaos expansions. The quantities considered include stress, strain and global deformation parameters of the left ventricle. The results indicate that important quantities of interest may be more affected by wall thickness, and highlight the need for accurate geometry reconstructions in patient-specific cardiac mechanics models.

Cardiac stress (load) and strain (stretch) are widely studied indicators of cardiac function and outcome, but are difficult or impossible to directly measure in relation to the cardiac microstructure. An alternative approach is to estimate these states using computer methods and image-based measurements, but this still requires knowledge of the tissue material properties and the unloaded state, both of which are difficult to determine. In this work, we tested the sensitivity of these two interdependent unknowns (reference geometry and material parameters) on stress and strain calculations in cardiac tissue. Our study used a finite element model of the human ventricle, with a hyperelastic passive material model, and was driven by a cell model mediated active contraction. We evaluated 21 different published parameter sets for the five parameters of the passive material model, and for each set we optimised the corresponding unloaded geometry and contractility parameter to model a single pressure-volume loop. The resulting mechanics were compared, and calculated systolic stresses were largely insensitive to the chosen parameter set when an unloading algorithm was used. Meanwhile, material strain calculations varied substantially depending on the choice of material parameters. These results indicate that determining the correct material and unloaded configuration may be highly important to understand strain driven processes, but less so for calculating stress estimates.

In atrial cardiomyocytes without a well-developed T-tubule system, calcium diffuses from the periphery towards the center creating a centripetal wave pattern. During atrial fibrillation, rapid activation of atrial myocytes induces complex remodelling in diffusion properties that result in failure of calcium to propagate in a fully regenerative manner towards the center; a phenomenon termed "calcium silencing''. This has been observed in rabbit atrial myocytes after exposure to prolonged rapid pacing. Although experimental studies have pointed to possible mechanisms underlying calcium silencing, their individual effects and relative importance remain largely unknown. In this study we used computational modelling of the rabbit atrial cardiomyocyte to query the individual effects and combined effects of the proposed mechanisms leading to calcium silencing and abnormal calcium wave propagation. We employed a population of models obtained from a newly developed model of the rabbit atrial myocyte with spatial representation of intracellular calcium handling. We selected parameters in the model that represent experimentally observed cellular remodelling which have been implicated in calcium silencing, and scaled their values in the population to match experimental observations. In particular, we changed the maximum conductances of I_CaL, I_NCX, and I_NaK, RyR open probability, RyR density, Serca2a density, and calcium buffering strength. We incorporated remodelling in a population of 16 models by independently varying parameters that reproduce experimentally observed cellular remodelling, and quantified the resulting alterations in calcium dynamics and wave propagation patterns. The results show a strong effect of I_CaL in driving calcium silencing, with I_NCX, INaK, and RyR density also resulting in calcium silencing in some models. Calcium alternans was observed in some models where I_NCX and Serca2a density had been changed. Simultaneously incorporating changes in all remodelled parameters resulted in calcium silencing in all models, indicating the predominant role of decreasing I_CaL in the population phenotype.

The purpose of the present study was to compare the effects of short-term high-frequency failure vs non-failure blood flow–restricted resistance exercise (BFRRE) on changes in satellite cells (SCs), myonuclei, muscle size, and strength. Seventeen untrained men performed four sets of BFRRE to failure (Failure) with one leg and not to failure (Non-failure; 30-15-15-15 repetitions) with the other leg using knee-extensions at 20% of one repetition maximum (1RM). Fourteen sessions were distributed over two 5-day blocks, separated by a 10-day rest period. Muscle samples obtained before, at mid-training, and 10-day post-intervention (Post10) were analyzed for muscle fiber area (MFA), myonuclei, and SC. Muscle size and echo intensity of m.rectus femoris (RF) and m.vastus lateralis (VL) were measured by ultrasonography, and knee extension strength with 1RM and maximal isometric contraction (MVC) up until Post24. Both protocols increased myonuclear numbers in type-1 (12%–17%) and type-2 fibers (20%–23%), and SC in type-1 (92%–134%) and type-2 fibers (23%–48%) at Post10 (p < 0.05). RF and VL size increased by 5%–10% in both legs at Post10 to Post24, whereas the MFA of type-1 fibers in Failure was decreased at Post10 (−10 ± 16%; p = 0.02). Echo intensity increased by ~20% in both legs during Block1 (p < 0.001) and was ~8 to 11% below baseline at Post24 (p = 0.001–0.002). MVC and 1RM decreased by 5%–10% after Block1, but increased in both legs by 6%–11% at Post24 (p < 0.05). In conclusion, both short-term high-frequency failure and non-failure BFRRE induced increases in SCs, in myonuclei content, muscle size, and strength, concomitant with decreased echo intensity. Intriguingly, the responses were delayed and peaked 10–24 days after the training intervention. Our findings may shed light on the mechanisms involved in resistance exercise-induced overreaching and supercompensation. Paid Open Access UNIT agreement

Global longitudinal strain and circumferential strain are found to be reduced in HFpEF, which some have interpreted that the global left ventricular (LV) contractility is impaired. This finding is, however, contradicted by a preserved ejection fraction (EF) and confounded by changes in LV geometry and afterload resistance that may also affect the global strains. To reconcile these issues, we used a validated computational framework consisting of a finite element LV model to isolate the effects of HFpEF features in affecting systolic function metrics. Simulations were performed to quantify the effects on myocardial strains due to changes in LV geometry, active tension developed by the tissue, and afterload. We found that only a reduction in myocardial contractility and an increase in afterload can simultaneously reproduce the blood pressures, EF and strains measured in HFpEF patients. This finding suggests that it is likely that the myocardial contractility is reduced in HFpEF patients. Graphical abstract.

We aim to extend existing models of single-cell mechanics to theEMIframework, to define spatially resolved mechanical models of cardiac myocytes embedded in a passive extracellular space. The models introduced here will be pure mechanics models employing fairly simple constitutive laws for active and passive mechanics. Future extensions of the models may include a coupling to the electrophysiology and electro-diffusion models described in the other chapters, to study the impact of spatially heterogeneous ion concentrations on the cell and tissue mechanics.

A modelagem da mecânica cardiaca tem levado a descobertas interessantes, porem este continua sendo um problema complexo e de alta demanda computacional, especialmente em modelos eletromecânicos fortemente acoplados. O tecido cardiaco e geralmente considerado como um material hiperelastico, quase incompressivel e ortotropico, fatores que dificultam a solucao numerica do modelo. Neste trabalho, melhorias foram realizadas em um simulador da mecânica cardiaca para tratar tais problemas numericos de forma mais eficiente. Com este simulador mais eficiente foi possivel tratar problemas que demandam de um maior esfoco computacional, como as analises de sensibilidade e quantificacao de incertezas, onde varias simulacoes precisam ser realizadas. Este tipo de analise tem sido topico de interesse cientifico para avaliar a possibilidade de usar simulacoes personalizadas por paciente em aplicacoes clinicas. Porem, estas simulacoes ainda sao problemas desafiadores, por causa da grande variabilidade biologica entre pacientes e das incertezas em medidas experimentais e em representacoes geometricas do coracao. Devido a estas incertezas em entradas do modelo, e dificil definir um modelo confiavel que possa ser usado em aplicacoes clinicas. Estudos recentes tem se voltado a investigacao de como estas incertezas podem influenciar no resultado de simulacoes e, consequentemente, descobrir como tornar os modelos mais confiaveis. Entao, o presente trabalho quantifica incertezas nas geometrias usadas nas simulacoes para investigar como quantidades de interesse da mecânica cardiaca podem ser afetadas. A abordagem do polinomio caos e utilizada para a quantificacao de incertezas em geometrias do ventriculo esquerdo submetidas a simulacoes da mecânica cardiaca. Inicialmente, as analises foram realizadas usando geometrias simplificadas em simulacoes da fase de preenchimento ventricular e, posteriormente, analises de quantificacao de incertezas em geometrias mais realisticas submetidas a simulacoes do ciclo cardiaco completo sao realizadas.

Mechanically-induced alterations in cardiac electrophysiology are referred to as mechano-electric feedback (MEF), and play an important role in electrical regulation of cardiac performance. The influence of mechanical stress and strain on electrophysiology has been investigated at all levels, however the role of MEF in arrhythmia remains poorly understood. During the normal contraction of the heart, mechano-sensitive processes are an implicit component of cardiac activity. Under abnormal mechanical events, stretch-activated mechanisms may contribute to local or global changes in electrophysiology (EP). While such mechanisms have been hypothesised to be involved in mechanically-initiated arrhythmias, the details of these mechanisms and their importance remain elusive. We assess the theoretical role of stretch mechanisms using coupled models of cellular electrophysiology and sarcomere contraction dynamics. Using models of single ventricular myocytes, we first investigated the potential MEF contributions of stretch-activated currents (SAC), and stretch-induced myofilament calcium release, to test how strain and fibrosis may alter cellular electrophysiology. For all models investigated, SACs were alone not sufficient to create a pro-arrhythmic perturbation of the action potential with stretch. However, when combined with stretch-induced myofilament calcium release, the action potential could be shortened depending on the timing of the strain. This effect was highly model dependent, with a canine epicardial EP model being the most sensitive. These model results suggest that known mechanisms of mechano-electric coupling in cardiac myocyte may be sufficient to be pro-arrhythmic, but only in combination and under specific strain patterns.

Cardiac resynchronization therapy (CRT) is a frequently effective treatment modality for dyssynchronous heart failure, however, 30% of patients do not respond, usually due to suboptimal activation of the left ventricle (LV). Multisite pacing (MSP) may increase the response rate, but its effect in the presence of myocardial scars is not fully understood. We use a computational model to study the outcome of MSP in an LV with scars in two different locations and of two different sizes. The LV was stimulated from anterior, posterior and lateral locations individually and in pairs, while a septal stimulation site represented right ventricular (RV) pacing. Intraventricular pressures were measured, and outcomes evaluated in terms of maximum LV pressure gradient (dP/dtmax)- change compared to isolated RV pacing. The best result obtained using various LV pacing locations included a combination of sites remote from scars and the septum. The highest dP/dtmax increase was achieved, regardless of scar size, using MSP with one pacing site located on the LV free wall opposite to the scar and one site opposite to the septum. These in silico modelling results suggest that making placement of pacing electrodes dependent on location of scarring, may alter acute haemodynamics and that such modelling may contribute to future CRT optimization.

In this chapter we will mainly focus on two data types: dictionaries and strings. Dictionaries can be considered a generalization of the list data type, where the indices are not required to be integers. We have already used strings multiple times in the previous chapters, but we will revisit them here to introduce a number of new and useful functions.

In this chapter, programming starts to become useful. The concepts introduced in the previous chapter are essential building blocks in all computer programs, but our example programs only performed a few calculations, which we could easily do with a regular calculator. In this chapter, we will introduce the concept of loops, which can be used to automate repetitive and tedious operations.