SofTMech Work Package WP5 Progress Report

Luo, H. Gao, R. Simitev, N.A. Hill, R. Ogden, S. Heath Richardson, G. Smith,

P. Mortensen, J. Mackinzie, M. Hifzhudin, D. Guan, LY. Feng, X. Zhuan.

July 2021

5.1 Poroelastic and coronary circulation modelling for heart perfusion

We developed a poroelastic ventricular framework accounting for three-phase interaction, which is used to build a left ventricle model.  We published interesting new results and extensive validation using this framework. We found that poroelastic left ventricle model behaves differently from the hyperelastic left ventricle model. For example, accounting for perfusion results in a smaller diastolic chamber volume, agreeing well with the well-known wall-stiffening effect under perfusion reported previously. Meanwhile differences in systolic function, such as fibre strain in the basal and middle ventricle, are found to be comparatively minor.   We also started to develop an immerse interface model for both hyperelastic and poroelastic material. 


Richardson SI, Gao H, Janiczek R, Cox J, Griffith BG, Berry C, Luo XY, Modelling cardiac poroelastic fluid-structure interaction using a hybrid immersed boundary/finite element framework, International Journal for Numerical Methods in Biomedical Engineering,  37 (5), e3446, 2021

5.2 Constitutive modelling and fluid-structure interaction in the heart

Using the AIC-based model reduction for the general Holzapfel-Ogden myocardial constitutive law, we showed how to select material parameters based on various different experiments and demonstrated the importance of shear effects in bi-axial tests.   We developed a coupled agent-based and hyperelastic modelling of the left ventricle post-myocardial infarction. We comprehensively studied how fibre dispersion affects ventricular pump function at both diastole and systole by using a non-rotationally symmetric dispersion model and identified the critical dispersion degree which will significantly affect ventricular mechanics. A discrete fibre-bundle dispersion model for the myocardial passive response with tension-compression switch and a general structure tensor for active tension is developed for the first time. We further developed a novel myocardial active contraction model based on the general Hill muscle model by combing both the active strain and active stress approaches. 

Growth and Remodelling: We published a new model for estimating the residual stress of the heart using multiple opening angle cuts.   The key result is that multiple cuts must be in a combination of radial and circumferential cuts and that the residual stress could be twenty times greater compared with existing one-cut opening angle methods.  We estimated growth tensors of the heart from human longitudinal data and identified three different growth patterns:  shrinkage, dilation, and no-change.   We revealed that the growth tensor can be used to indicate a longer-term growth trend of the heart.   We also developed a new volumetric growth theory, which is able to model tissue growth from the current loaded configuration.  This work has been submitted for reviews.

Heart Valve:    We developed a coupled fluid‐structure interaction model of the left atrium and mitral valve and showed how valve-heart interaction changes the pressure waves and flow field in both physiological and pathological cases.  By performing the mechanical and morphometric study of mitral valve chordae tendineae and related papillary muscle, we found the chordae associated with the posteromedial papillary muscle are much longer and stiffer than those associated with the anterolateral papillary muscle, this can be further explained by the higher collagen core ratio and larger collagen fibril density of chordae between the two groups. We further demonstrated that the Ogden and reduced Holzapfel-Ogden strain energy function can well characterize the experimental data.  Based on the experimental data, we published a modelling study on some effects of different constitutive laws on simulating aortic valve dynamics with fluid-structure interaction, and we found that a single-family of fibres based exponential form of invariants-based strain energy function can both fit the experimental data well and consistent hemodynamics and structural dynamics of the AV compared to experimental and clinical observations.


  1. Zhuan X, Luo XY, Stress Estimates from Multi-cut Opening Angles of the Left Ventricle, Cardiovascular Engineering and Technology, 11, 381-393, 2020.
  2. Li WG, Gao H, Mangion K, Berry C, Luo XY, Apparent growth tensor of left ventricular post myocardial infarction–In human first natural history study, Computers in Biology and Medicine, 129, 104168, 2021
  3. Feng LY, Gao H, Qi N, Danton, M, Hill NA and Luo XY, Fluid-structure interaction in a fully coupled three-dimensional mitral-atrium-pulmonary model, Biomechanics and Modelling in Mechanobiology, 2021
  4. Guan D, Yao J, Luo XY, and Gao, H. Effect of myofibre architecture on ventricular pump function by using a neonatal porcine heart model: from DT-MRI to rule-based methods, Royal Society Open Science, 7, 191655, 2020
  5. Guan DB, Zhuan X, Holmes W, Luo XY, Gao H, Modelling of fibre dispersion and its effects on cardiac mechanics from diastole to systole, Journal of Engineering Mathematics, 128, 1, 2021
  6. Cai L, Zhang RH, Li YQ, Zhu GY, Ma XS, Wang YH, Luo XY, and Gao H, The Comparison of Different Constitutive Laws and Fiber Architectures for the Aortic Valve on Fluid–Structure Interaction Simulation, Frontiers in Physiology, section Computational Physiology and Medicine, 682893, 12, 2021
  7. Chen, S., Sari, C. R., Gao, H., Lei, Y., Segers, P., De Beule, M., ... & Ma, X. (2020). Mechanical and morphometric study of mitral valve chordae tendineae and related papillary muscle. journal of the mechanical behavior of biomedical materials111, 104011.
  8. Guan, X. Luo, and H. Gao. Constitutive modelling of soft biological tissue from ex vivo to in vivo: Myocardium as an example. In JSPS2020, volume Accepted, 2021
  9. Guan, H. Gao, L. Cai and X. Luo. A new active contraction model for myocardium using a modified hill model. Submitted to Biophysical Journal5.1 Electrophysiology:

    Contemporary realistic mathematical models of single-cell cardiac electrical excitation are immensely detailed. Model complexity leads to parameter uncertainty, high computational cost and barriers to mechanistic understanding. There is a need for reduced models that are conceptually and mathematically simple but physiologically accurate. In (Aziz and SImitev 2021, under review), we consider an archetypal model of single-cell cardiac excitation that replicates the phase-space geometry of detailed cardiac models, but at the same time has a simple piecewise-linear form and a relatively low-dimensional configuration space. In order to make this archetypal model practically applicable, we develop and report a robust method for estimation of its parameter values from the morphology of single-stimulus action potentials derived from detailed ionic current models and from experimental myocyte measurements. The procedure is applied to five significant test cases and an excellent agreement with target biomarkers is achieved. Action potential duration restitution curves are also computed and compared to those of the target test models and data, demonstrating conservation of dynamical pacing behaviour by the fine-tuned archetypal model. An archetypal model that accurately reproduces a variety of wet-lab and synthetic electrophysiology data offers a number of specific advantages such as computational efficiency, as also demonstrated in the study. Open-source numerical code of the models and methods used is provided.

Cardiac electrophysiological heterogeneity manifests in three ways: (i) regional         differences in action potential (AP) waveform, (ii) AP differences in cells    isolated  from a single region, (iii) differences in individual conductances in cells with similar APs. Little is known about (ii) and (iii). In (Lachaud et al, 2021, under review) we address the folowing objectives: Quantify (ii) via inter-cell heterogeneity of AP duration (APD) and (iii) via responses to ion channel block in rabbit ventricular cells and generate a population of mathematical models to investigate the mechanisms underlying both. Methods and Results: APD in >100 isolated cells from subregions of the LV free wall of rabbit hearts were measured using a voltage-sensitive dye. When stimulated at 2Hz, average APD90 value in cells from the basal epicardial region was 254±25ms (mean±SD) in 17 hearts with a mean inter-quartile range (IQR) of 53±17ms. Endoepicardial and apical-basal APD90 differences accounted for ~10% of the IQR value. Highly variable changes in APD occurred after IK(r) or ICa(L) block that included a sub-population of cells (HR) with an exaggerated (hyper) response to IK(r) inhibition. A set of 4,471 AP models matching the experimental APD90 distribution was generated from a larger population of models created by random variation of the maximum conductances (Gmax) of 8 key ion channels/exchangers/pumps. This set reproduced the pattern of cell-specific responses to ICa(L) and IK(r) block, including the HR subpopulation. The models exhibited a wide range of Gmax values with constrained relationships linking ICa(L) with IK(r), ICl, INCX and INaK. Conclusions: Modelling the measured range of inter-cell APDs required a larger (3- 4x) range of key Gmax values indicating that ventricular tissue has considerable intercell variation in channel/pump/exchanger expression. AP morphology is retained by relationships linking specific ionic conductances. These interrelationships are necessary for stable repolarisation despite large inter-cell variation of individual conductances and this explains the variable sensitivity to ion channel block.

The immature electrophysiology of human-induced pluripotent stem cell-derived cardiomyocytes (hiCMs) complicates their use for therapeutic and pharmacological purposes. An insufficient inward rectifying current (IK1) and the presence of a funny current (if) cause spontaneous electrical activity. In (Costa et al., 2021) we tested the hypothesis that the co-culturing of hiCMs with a human embryonic kidney (HEK) cell-line expressing the Kir2.1 channel (HEK-IK1) can generate an electrical syncytium with an adult-like cardiac electrophysiology. The mechanical activity of co-cultures using different HEK-IK1:hiCM ratios was compared with co-cultures using wildtype (HEK–WT:hiCM) or hiCM alone on days 3–8 after plating. Only ratios of 1:3 and 1:1 showed a significant reduction in spontaneous rate at days 4 and 6, suggesting that IK1 was influencing the electrophysiology. Detailed analysis at day 4 revealed an increased incidence of quiescent wells or sub-areas. Electrical activity showed a decreased action potential duration (APD) at 20% and 50%, but not at 90%, alongside a reduced amplitude of the aggregate AP signal. A computational model of the 1:1 co-culture replicates the electrophysiological effects of HEK–WT. The addition of the IK1 conductance reduced the spontaneous rate and APD20, 50 and 90, and minor variation in the intercellular conductance caused quiescence. In conclusion, a 1:1 co-culture HEK-IK1:hiCM caused changes in electrophysiology and spontaneous activity consistent with the integration of IK1 into the electrical syncytium. However, the additional electrical effects of the HEK cell at 1:1 increased the possibility of electrical quiescence before sufficient IK1 was integrated into the syncytium.

In (Mortensen et al, 2021b), the analytical theory of our earlier study (Mortensen et al., 2021a, Math. Med. Biol., 38, 106–131) is extended to address the outstanding cases of fibroblast barrier distribution and myocyte strait distribution. In particular, closed-form approximations to the resting membrane potential and to the critical parameter values for propagation are derived for these two non-uniform fibroblast distributions and are in good agreement with numerical estimates. We further developed a ghost structure finite difference method for a fractional FitzHugh-Nagumo monodomain model on moving irregulart domain


  • Huethorst E, Mortensen PB, Simitev, R., Gao H, Pohjolainen L, Talman V, Ruskoaho H, Burton FL, Gadegaard N, Smith GL, Conventional rigid 2D substrates cause complex contractile signals in monolayers of human induced pluripotent stem cell derived cardiomyocytes, Biomaterials, [Submitted 2021-06-25].
  • Aziz MHN., Simitev, R., Estimation of parameters for an archetypal model of cardiomyocyte membrane potentials, International Journal Bioautomation, arXiv:2105.06853, [Submitted 2021-05-14].
  • Lachaud Q., Aziz MHN., Burton F., Myles R., Simitev, R., Smith G.L., Electrophysiological heterogeneity in large populations of rabbit ventricular cardiomyocytes, Cardiovascular Research, [Submitted 2021-02-10].
  • Costa A.S., Mortensen P., Hortigon-Vinagre M.P., van der Heyden M., Burton F., Gao H., Simitev, R., Smith G.L., Electrophysiology of hiPSC-cardiomyocytes co-cultured with HEK cells expressing the inward rectifier channel, International Journal of Molecular Sciences, 22(12), 6621, doi.org/10.3390/ijms22126621, 2021.
  • Mortensen, P., Gao, H., Smith, G., Simitev, R., Addendum: Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling, Mathematical Medicine & Biology, doi.org/10.1093/imammb/dqab005, 2021b.
  • Wang YH, Cai L, Feng, XB, Luo XY, Gao H, A ghost structure finite difference method for a fractional FitzHugh-Nagumo monodomain model on moving irregular domain, Journal of Computational Physics, 428, 110081, 2021

 5.4 Clinical applications



Machine learning: we compared several surrogate models based on machine learning methods for parameter estimation of left ventricle myocardium by learning the filling phase of one left ventricle model, we found that XGBoost is a good candidate surrogate models for predicting the LV diastolic dynamics and estimating passive parameters than the K-nearest neighbour and multi-layer perceptron emulator.  Furthermore, we studied the closed-loop effects in cardiovascular decision support.  We made comparative evaluation of different emulators for cardiac Mechanics. More details are reported in WP4.

Biomarkers:  using our model applied the longitudinal clinical data of seven cardiac amyloidosis patients, we found the progression of cardiac amyloidosis can be effectively predicted using multiple biomarkers.  This was a joint work with GSK.

  1. Li WG, Lazarus A, Gao H, Martinez-Naharro A, Fontana M, Hawkins P, Biswas S, Janiczek R, Cox J, Berry C, Husmeier D, Luo XY, Analysis of cardiac amyloidosis progression using model-based markers, Frontiers in Physiology, 11:324, 2020
  2. Morrow, AJ, ... Luo XY, ... Berry C, Rationale and design of the Medical Research Council Precision medicine with Zibotentan in microvascular angina (PRIZE) trial, American Heart Journal, 229, 70-80, 2020.
  3. Cai, L., Ren, L., Wang, Y., Xie, W., Zhu, G., & Gao, H. (2021). Surrogate models based on machine learning methods for parameter estimation of left ventricular myocardium. Royal Society open science8(1), 201121.


Multiscale heart modelling - Myocardial Infarction (MI) Update: May 2018

Acute MI and remodelling in the heart is a highly complex process, involving pathophysiology, electrophysiology and multiphysics. Developing a multiscale heart model to cover all the aspects is unrealistic, hence we will provide key components with detailed myocardium mechanical models that are linked to subcellular organelle remodelling, as well as developing an extensible multiscale framework. We  will  significantly  extend  the  open-source  immersed  boundary  /finite  element  (IB/FE)  fluid-structure interaction (FSI) framework (note: add a link under IB/FE to https://github.com/IBAMR/IBAMR).

Project 1: IB/FE multiscale rabbit acute-MI model

Project 2: IB/FE with electromechanical coupling

Project 3: Multi-scale Mechanobiological Model of MI

Project 4: Influence of angiogenesis on MI remodelling

Project 5: A multiscale human MI model

Project 6: Translation to healthcare Technologies  by working closely with industrial partners.

Team: Prof. Luo, Prof. Ogden, Dr. McDougall, Prof. Hill, Prof. Smith, Prof. Berry, Dr. Watton, Dr. Radostin Simitev, Prof. Chaplain, Prof. Husmeier, Dr. Gao, PhD6, PhD7