The SofTMech Statistical Emulation and Translation Hub

Lead Research Organisation: University of Glasgow
Department Name: School of Mathematics & Statistics

Abstract

There have recently been impressive developments in the mathematical modelling of physiological processes. As part of a previously EPSRC-funded research centre (SofTMech), we have developed mathematical models for the mechanical and electrophysiological processes of the heart, and the flow in the blood vessel network. This allows us to gain deeper insight into the state of a variety of serious cardiovascular diseases, like hypoxia (a condition in which a region of the body is deprived of adequate oxygen supply), angina (reduced blood flow to the heart), pulmonary hypertension (high blood pressure in the lungs) and myocardial infarction (heart attack). A more recent extension of this work to modelling blood flow in the eye also provides novel indicators to assess the degree of traumatic brain injury.
What all these models have in common is a complex mathematical description of the physiological processes in terms of differential equations that depend on various material parameters, related e.g. to the stiffness of the blood vessels or the contractility of the muscle fibres. While knowledge of these parameters would be of substantial benefit to the clinical practitioner to help them improve their diagnosis of the disease status, most of the parameters cannot be measured in vivo, i.e. in a living patient. For instance, the determination of the stiffness and contractility of the cardiac tissue would require the extraction of the heart from a patient and its inspection in a laboratory, which can only be done in a post mortem autopsy.
It is here that our mathematical models reveal their diagnostic potential. Our equations of the mechanical processes in the heart predict the movement of the heart muscle and how its deformations change in time. These movements can also be observed with magnetic resonance image (MRI) scans, and they depend on the physiological parameters. We can thus compare the predictions from our model with the patterns found in the MRI scans, and search for the parameters that provide the best agreement. In a previous proof-of-concept study we have demonstrated that the physiological parameters identified in this way lead to an improved understanding of the cardiac disease status, which is important for deciding on appropriate treatment options.
Unfortunately, the calibration procedure described above faces enormous computational costs. We typically have a large number of physiological parameters, and an exhaustive search in a high-dimensional parameter space is a challenging problem. In addition, every time we change the parameters, our mathematical equations need to be solved again. This requires the application of complex numerical procedures, which take several minutes to converge. The consequence is that even with a high-performance computer, it takes several weeks to determine the physiological parameters in the way described above. It therefore appears that despite their enormous potential, state of the art mathematical modelling techniques can never be practically applied in the clinical practice, where diagnosis and decisions on alternative treatment option have to be made in real time.
Addressing this difficulty is the objective of our proposed research. The idea is to approximate the computationally expensive mathematical model by a computationally cheap surrogate model called an emulator. To create this emulator, we cover the parameter space with an appropriate design, solve the mathematical equations in parallel numerically for the chosen parameters, and then fit a non-linear statistical regression model to this training set. After this initial computational investment, the emulator thus created gives predictions for new parameter values practically instantaneously, allowing us to carry out the calibration procedure described above in real time. This will open the doors to harnessing the diagnostic potential of state-of-the art mathematical models for improved decision support in the clinic.

Planned Impact

According to the British Heart Foundation (BHF), heart and circulatory diseases cause more than a quarter of all deaths in the UK, that is nearly 170,000 deaths each year, an average of 460 deaths each day or one every three minutes in the UK. There are around 7.4 million people living with heart and circulatory disease in the UK: 3.9 million men and 3.5 million women.

Mathematical modelling in cardiovascular physiology is a topical research area and has in principle paradigm-shifting potential for improving our understanding of a patient's cardio-vascular disease status, elucidating the nature of pathophysiological processes, improving patient-specific disease prognostication, and providing more accurate decision support for alternative treatment options. However, a major obstacle is the exorbitant computational cost of model calibration, as discussed in the "Summary" section. These are typically in the order of several weeks even on a high-performance computer, which currently renders state of the art mathematical models completely for the clinical practice.

The general impact of the proposed research hub is the fact that methodological improvements in statistical emulation will provide a decisive stepping stone towards enabling the use of state-of-the-art soft-tissue, electro-physiological and fluid-dynamic models for real-time decision making in the clinic and thereby harness their enormous potential for patient-specific disease prognostication. The emulation of soft-tissue mechanical models of the left ventricle of the heart will help assess the risk and treatment options for myocardial infarction (heart attack). The emulation of cardio-electrophysiological models will allow the monitoring of post-infarction scars to prevent sudden cardiac death. The emulation of fluid dynamic models for the pulmonary circulation system linked to the right ventricle of the heart will enable the non-invasive diagnosis of pulmonary hypertension, which is a major risk factor for stroke, heart failure and coronary artery disease. Endovascular drug delivery will be made more effective by emulating the patient-specific device-tissue-fluid interactions. And an extension of the cardiovascular modelling to the emulation of fluid dynamics in the human eye will allow the fast identification of traumatic brain injury, which will provide e.g. a clinical indicator for the "shaken baby syndrome".

To make specific progress towards these objectives, we will closely engage with the Scottish Pulmonary Vascular Unit at the Golden Jubilee Hospital in Clydebank, with the Cardiology Department at Queen Elizabeth University Hospital in Glasgow, and with NHS Scotland, as described in more detail in the "Pathways to Impact" section of this proposal.

The proposed research will also be relevant to companies that aim to deliver realistic simulation applications to explore real-world behaviour of complex systems particularly related to physiology, in that it will allow them to substantially reduce the computational complexity of inference and uncertainty quantification and thereby make their simulation systems applicable to decision-making in real time. A particular example is Dassault Systems, with whom the proposed research hub will closely engage. Moreover, the proposed research is relevant to companies that manufacture endovascular devices, like stents and drug-coated balloons, in that mathematical models of device-tissue-fluid interactions allow improvement of device design, and emulation is critical for fast patient-specific decisions. As a specific first step, the proposed hub will establish a collaboration with Terumo Aortic.

Statistical emulation is not only relevant to healthcare, but to the mathematical modelling of complex systems for safety-critical situations more generally. This includes e.g. early warning systems for tsunamis and volcanic activities, which will benefit from the methodological advancements made in the proposed research.

Publications

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Borowska A (2022) Bayesian optimisation for efficient parameter inference in a cardiac mechanics model of the left ventricle. in International journal for numerical methods in biomedical engineering

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Cruz-González O (2021) Effective behavior of long and short fiber-reinforced viscoelastic composites in Applications in Engineering Science

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Escuer J (2022) Mathematical modelling of endovascular drug delivery: Balloons versus stents. in International journal of pharmaceutics

 
Description 1 year EPSRC Fellowship at UoG 2022/23 A. Lazarus
Amount £83,330 (GBP)
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 02/2022 
End 02/2023
 
Title Cardiac Modelling with dispersed myofibre and collagen structures 
Description It is the accompanying dataset and model the paper "modelling of fibre dispersion and its effects on cardiac mechanics from diastole to systole", accepted in the Journal of Engineering Mathematics. It implements two different fibre dispersion models within two ventricular finite element models: a bi-ventricular rabbit heart and a human left ventricular model. 
Type Of Material Computer model/algorithm 
Year Produced 2021 
Provided To Others? Yes  
Impact This study highlights the importance of fibre dispersion in cardiac mechanics, and for the first time to investigate how to incorporate a complex fibre dispersion distribution into a cardiac mechanics model. This work has been presented in the Living Heart Project Seminar, and we are working with the Virtual Human Team from Dassault System to implement it in the Living Heart Project. 
URL https://github.com/HaoGao/DispersedFibresMyocardiumModelling
 
Title Cardiovascular Modelling Subject to Medical Interventions 
Description This GitHub repository contains data and code to reproduce the results reported in the paper 'Inference in Cardiovascular Modelling Subject to Medical Interventions' by L. Mihaela Paun, Agnieszka Borowska, Mitchel J. Colebank, Mette S. Olufsen and Dirk Husmeier, published in the Proceedings of ICSTA 2021. 
Type Of Material Computer model/algorithm 
Year Produced 2021 
Provided To Others? Yes  
Impact The software allows the results of the above-mentioned paper to be reproduced, and it can be used more widely to correct closed-loop effects in cardiovascular modelling and inference subject to medical interventions. 
URL https://github.com/LMihaelaP/Cardio_Vasodilation.git
 
Title Code for Estimation of Parameters for an Archetypal Model of Cardiomyocyte Membrane Potentials 
Description This is a suite of MATLAB/Octave functions for numerical solution of and for estimation of the parameter values of the cardiomyocyte membrane potential model of Biktashev et al. (Bull Math Biol, 70(2), 2008,doi:10.1007/s11538-007-9267-0) - "the archetypal model". In particular, the code can be used determine parameter values for the archetypal model such that its solutions approximate the action potential traces and the action potential duration restitution curves of (a) other electrophysiologically detailed mathematical models of the transmembrane ionic currents of single cardiac myocytes - "target models", as well as (b) traces and curves measured experimentally - "target data". Data and functions for several detailed ionic models from the CellML physiological model repository (cellml.org) are included as examples of usage. 
Type Of Material Database/Collection of data 
Year Produced 2021 
Provided To Others? Yes  
Impact N/A 
URL https://zenodo.org/record/4568662
 
Title Fibre Dispersion Myocardial Mechanics 
Description It contains the computational models for the following two papers 1. Guan, D., Mei, Y., Xu, L., Cai, L., Luo, X., & Gao, H. (2022). Effects of dispersed fibres in myocardial mechanics, Part I: passive response. Mathematical Biosciences and Engineering, 19(4), 3972-3993. 2. Guan, D., Wang, Y., Xu, L., Cai, L., Luo, X., & Gao, H. (2022). Effects of dispersed fibres in myocardial mechanics, Part II: active response. Mathematical Biosciences and Engineering, 19(4), 4101-4119. Published Year: 2022 
Type Of Material Computer model/algorithm 
Year Produced 2022 
Provided To Others? Yes  
Impact This work has attracted interest from the Virtual Human Team from Dassault System. It will further improve the cardiac modelling by including detailed fibre dispersion, in particular in fibrosis modelling. 
URL https://github.com/HaoGao/FibreDispersionMyocardialMechanics
 
Title Improving cardio-mechanic inference by combining in vivo strain data with ex vivo volume-pressure data 
Description GitHub repository with the software and data needed to reproduce the results of the following article: Alan Lazarus, Hao Gao, Xiaoyu Luo and Dirk Husmeier (2022): ``Improving cardio-mechanic inference by combining in vivo strain data with ex vivo volume-pressure data", Journal of the Royal Statistical Society, Series C, accepted for publication. 
Type Of Material Computer model/algorithm 
Year Produced 2022 
Provided To Others? Yes  
Impact The software and data allow the user to reproduce the results of the above paper. The user can use the code for cardiac mechanics applications more generally, by systematically integrating in vivo strain data, extracted from cardiac magnetic resonance images, with ex-vivo volume-pressure data. 
URL https://github.com/lazarusal/klotz-codes
 
Title Neural Network-Based Left Ventricle Geometry Prediction from Cardiac Magnetic Resonance Images 
Description Github repository including the software and data for the paper by Lukasz Romaszko, Agnieszka Borowska, Alan Lazarus, David Dalton, Colin Berry, Xiaoyu Luo, Dirk Husmeier and Hao Gao (2021): ``Neural network-based left ventricle geometry prediction from CMR images with application in biomechanics", Artificial Intelligence in Medicine, Volume 119, September 2021, 102140, doi: https://doi.org/10.1016/j.artmed.2021.102140 In particular, the Github repository contains pre-processed data (a subset of re-scaled and cropped original CMR images as well as segmented images and LV geometries), as well as the code (two-stage CNN: segmentation network and geometry prediction network). 
Type Of Material Computer model/algorithm 
Year Produced 2021 
Provided To Others? Yes  
Impact The Github repository allows readers to reproduce the results reported in the above paper and to use the software to automatically predict the shape of the left ventricle of the heart from their own cardiac magnetic resonance images. This is a prerequisite for any subsequent cardiac mechanic modelling. 
URL https://github.com/aborowska/LVgeometry-prediction
 
Title Parameter estimation and uncertainty quantification in differential equation models 
Description Github repository containing the software and the data for the article by L. Mihaela Paun and Dirk Husmeier (2022): Emulation-accelerated Hamiltonian Monte Carlo algorithms for parameter estimation and uncertainty quantification in differential equation models,Statistics and Computing, volume 32, Article number: 1. https://link.springer.com/article/10.1007/s11222-021-10060-4 
Type Of Material Computer model/algorithm 
Year Produced 2022 
Provided To Others? Yes  
Impact The software and the data of the Github repository allow the user to reproduce the results reported in the above paper. The software can be applied to other differential equation models to infer the model parameters and quantify the estimation uncertainty. 
URL https://github.com/LMihaelaP/Hamiltonian-Monte-Carlo-with-emulation.git
 
Title Sensitivity Analysis and Inverse Uncertainty Quantification for the Left Ventricular Passive Mechanic 
Description It contains the codes and data for the paper 'Sensitivity Analysis and Inverse Uncertainty Quantification for the Left Ventricular Passive Mechanics'. This work performs structural identifiability and practical identifiability analysis for a widely used constitutive law of passive myocardium (the Holzapfel-Ogden model), using global sensitivity analysis to assess structural identifiability, and inverse-uncertainty quantification to assess practical identifiability. 
Type Of Material Computer model/algorithm 
Year Produced 2022 
Provided To Others? Yes  
Impact It elucidates the dependence of parameter identifiability on external factors for the first time in a nonlinear cardiomechanic model, with a particular focus on the H-O myocardial model. 
URL https://github.com/HaoGao/ho-uncertainty-quantification
 
Title Sensitivity Analysis and Inverse Uncertainty Quantification for the Left Ventricular Passive Mechanics 
Description GitHub repository with the software and the data needed to reproduce the results reported in the following article: Alan Lazarus, David Dalton, Dirk Husmeier, Hao Gao (2022), ``Sensitivity Analysis and Inverse Uncertainty Quantification for the Left Ventricular Passive Mechanics", Biomechanics and Modelling in Mechanobiology Accepted for publication 
Type Of Material Computer model/algorithm 
Year Produced 2022 
Provided To Others? Yes  
Impact The software and data allow the user to reproduce the results reported in the above paper. The user can adapt the software to use it for global sensitivity analysis and uncertainty quantification more generally. 
URL https://github.com/HaoGao/ho-uncertainty-quantification
 
Title GlasgowHeart 
Description GlasgowHeart platform for personalized modelling of the human heart. It is organized into 4 modules, and each can be run separately. Currently, MatLab is the main programming language and using scripts for run, this will require certain knowledge of Matlab. In the future, we will develop a GUI package for easy use. The four modules are image processing, biomechanics modelling, personalization, and parameter inference of left ventricular (LV) mechanics and statistical emulation. 
Type Of Technology Software 
Year Produced 2021 
Open Source License? Yes  
Impact The package is actively being used by the researchers from the SofTMech Centre and supports a few cardiac research projects. 
 
Description Effective parameter inference for a mathematical model of the left ventricle 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Invited talk at ASCE Engineering Mechanics Institute International Conference 2021 (EMI 2021)
Year(s) Of Engagement Activity 2021
URL https://emi2020-ic.webspace.durham.ac.uk
 
Description Inference in Cardiovascular Modelling Subject to Medical Interventions 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Talk at the International Conference on Statistics: Theory and Applications (ICSTA 2021), given by Dirk Husmeier
Year(s) Of Engagement Activity 2021
URL https://avestia.com/ICSTA2021_Proceedings/files/papers.html
 
Description Invited lecture 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Invited lecture for ECMINT 4.2 Cerebral aneurysms (European Course of Minimal Invasive Therapy). Intense theoretical course (University of Oxford) contributing to neuroradiology training and education, focusing on neuroendovascular therapy/repair (https://www.esmint.eu/ecmint/)
Year(s) Of Engagement Activity 2021
 
Description Invited presentation at the Isaac Newton Institute 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Presentation entitled "Probabilistic Calibration of Personalised Heart Models from Sparse and Noisy Measurements" invited as part of work programme on "The Role of Uncertainty in Mathematical Modelling of Pandemics" at the Isaac Newton Institute in Cambridge.
Year(s) Of Engagement Activity 2022
URL https://gateway.newton.ac.uk/event/tgm110/programme
 
Description Invited talk "A Novel Excitation-Contraction Model Based on Classical Hill Model" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Invited talk on "A Novel Excitation-Contraction Model Based on Classical Hill Model", given by Hao Gao at the fourth international meeting on computational cardiology, NPU, XiAn China.
Year(s) Of Engagement Activity 2021
 
Description Invited talk ``Constrained Mixture Based-cardiac Growth and Remodelling" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Invited talk on " Constrained Mixture Based-cardiac Growth and Remodelling", given by Hao Gao at the7th international symposium: virtual twin of human & living heart, organized by Dassault Systemes.
Year(s) Of Engagement Activity 2021
URL https://events.3ds.com/sites/default/files/international-symposium-2021-agenda.pdf
 
Description Invited talk at the University of Edinburgh which was live-streamed and recorded 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact Talk entitled "Gaussian process enhanced semi-automatic ABC for inference in a stochastic differential equation system for chemotaxis" by Dirk Husmeier's post-doc Agnieszka Borowska at a Statistics seminar of the University of Edinburgh on the 8th of November 2021. The talk questions afterwards, including a follow-up discussion on gather.town. The recording of the talk can be accessed here:
https://ed-ac-uk.zoom.us/rec/play/f-leD9akIBpTJCiF9ZIlKdq2QEYByLsP79lV4ZwPtaXA3iv0L12TYBLI8cRgoJiWvuIowPSt_Necyekm.YwhHusq0ucFvlT0S?continueMode=true&_x_zm_rtaid=9RE7lTR9SviipSXtp58ZrQ.1637266969453.752b91c756c02a3426d923e4580c6e85&_x_zm_rhtaid=330
Year(s) Of Engagement Activity 2021
URL https://www.maths.ed.ac.uk/school-of-mathematics/events/statistics
 
Description Mini-symposium "Progress and Trends in Mathematical Modelling of Cardiac Function" at BMC BAMC 2021 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact Mini-symposium "Progress and Trends in Mathematical Modelling of Cardiac Function" at BMC BAMC 2021 organised by Hao Gao
Year(s) Of Engagement Activity 2021
URL https://sites.google.com/view/bmcbamc2021/home
 
Description Mini-symposium "Stochastic models in biology informed by data" at BMC BAMC 2021 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact Talk entitled "Parameter estimation and uncertainty quantification in a stochastic differential equation model of cell movement and chemotaxis" given by Dirk Husmeier's post-doc Agnieszka Borowska at the Mini-symposium "Stochastic models in biology informed by data" at the British Applied Mathematics Colloquium on the 6th of April 2021, which sparked questions and discussion afterwards, including follow-up emails.
Year(s) Of Engagement Activity 2021
URL https://sites.google.com/view/bmcbamc2021/home
 
Description Oral talk "A hybrid active contraction for myocardium" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Oral talk by Hao Gao on "A hybrid active contraction for myocardium" at the 26th Congress of the European Society of Biomechanics, Milano, Italy.
Year(s) Of Engagement Activity 2021
URL https://esbiomech.org/conference/esb2021/
 
Description Oral talk "Constitutive Modelling of Soft Biological tissue from ex vivo to in vivo" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Oral talk by Hao Gao on "Constitutive Modelling of Soft Biological tissue from ex vivo to in vivo" at 25th International Congress of Theoretical and Applied Mechanics, Milano, Italy.
Year(s) Of Engagement Activity 2021
URL https://ima.org.uk/15361/25th-international-congress-of-theoretical-and-applied-mechanics/#:~:text=T...
 
Description SofTMech Soft Tissue workshop 2021 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact The 5th Soft Tissue workshop took place from 1st-3rd June 2021. The workshop focused on the most recent advances in the field of soft tissue mechanics, with a clear vision of the landscape of multiscale soft tissue modelling and both fundamental and translational research.
Year(s) Of Engagement Activity 2021
 
Description SofTMech Training Workshop Scientific Computation 2022 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Postgraduate students
Results and Impact Training day for PhD students on Scientific Computation, held on 28th January 2022. The event attracted more than 40 participants from Maths-in-Healthcare centres from around the UK. Sparked questions and discussion afterwards.
Year(s) Of Engagement Activity 2022
 
Description Special Interest Group on the Fluid Mechanics of the Eye 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Since the COVID lockdown, Peter Stewart has organised virtual study groups on the fluid mechanics of the eye. The first event took place on 30th November 2020 and there have been 6 subsequent meetings across 2021 and 2022. These events have gathered clinicians and modellers (including some PhD students) to derive mathematical models pertinent to the eye. In particular, we have modelled:
(1) uveoscleral flow as a drug delivery platform to the macula
(2) formation of macular holes in aging eyes.
Several publications are in preparation.
We are hosting a follow up conference in Bath in June 2022, funded by the Macular Society.
Year(s) Of Engagement Activity 2020,2021,2022
URL https://eyefluidssig.wordpress.com/history/
 
Description Talk at the "ML in PL 2021" conference 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Talk entitled "Neural network-based left ventricle geometry prediction from CMR images with application in biomechanics" given by Dirk Husmeier's post-doc Agnieszka Borowska at the "ML in PL 2021" conference (online) on the 7th of November 2021. The talk sparked questions afterwards, including requests for further information.
Year(s) Of Engagement Activity 2021
URL https://conference2021.mlinpl.org/
 
Description Talk at the Newton Gateway workshop on ``The Role of Uncertainty in Mathematical Modelling of Pandemics" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Richard Clayton giving a talk on ``Probabilistic Calibration of Personalised Heart Models from Sparse and Noisy Measurements" which has been recorded and is publicly available: https://gateway.newton.ac.uk/presentation/2022-02-09/34658
Year(s) Of Engagement Activity 2022
URL https://gateway.newton.ac.uk/event/tgm110/programme
 
Description Talk on Graph Neural Network Emulation of Cardiac Mechanics 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Contributed talk given by David Dalton at the 3rd International Conference on Statistics: Theory and Applications (ICSTA'21).
The outcome was making the larger community aware of the potential of graph-based emulation methods in soft tissue mechanics.
Year(s) Of Engagement Activity 2021
URL https://avestia.com/ICSTA2021_Proceedings/
 
Description invited talk "An initial experience of constrained mixture based cardiac growth and remodelling" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Invited talk "An initial experience of constrained mixture based cardiac growth and remodelling", given by Hao Gao at SIAM MS21 "Multi-scale modelling in Biomechanics".
Year(s) Of Engagement Activity 2021
URL https://wp.bcamath.org/siamms21/