## WP4

**SofTMech Work Package WP4 Progress Report**

D. Husmeier, A. Borowska, M. Paun, A. Lazarus, L. Romaszko, D. Dalton

July 2021

**4.1 ****Parameter inference in cardio-mechanic models using statistical emulation**

** **Agnieszka Borowska, PhD students Alan Lazarus and David Dalton)

**Background**

A central topic of WP4 is statistical inference for soft-tissue mechanical models in cardiovascular physiology, and to estimate the biophysical parameters that determine the mechanical properties of the tissues and fibres non-invasively from magnetic resonance images (MRI). In principle this is achieved by comparing strains extracted from the MRI scans with those predicted from the mathematical model, and applying multivariate optimization algorithms to find the parameters that minimize this mismatch. Unfortunately, this method does not lead to a decision support tool for the clinical practice. The reason is that the soft-tissue mechanical equations have no closed-form solution and require a numerical integration of the underlying partial differential equations with finite element discretization. This procedure has to be repeated hundreds or thousands of times during the iterative optimization of the material parameters, leading to computational costs of several weeks even on a high-performance computing cluster.

**Work done so far**

We have addressed these difficulties with statistical emulation. The idea is to simulate a large number of samples at different parameter specifications in advance and use these simulations combined with a flexible multivariate interpolation methods to replace the computationally expensive simulator in the optimization procedure with a statistical surrogate model called an emulator. Following up on the work described in last year’s report, we have carried out work on automating the left ventricle shape extraction from magnetic resonance image scans, a comparative evaluation of different emulation methods, the methodological improvement of state-of-the-art methods for cardio-mechanic emulation, and improving the accuracy of cardio-mechanic parameter estimation by systematically integrating in-vivo strain with ex-vivo volume-pressure data.

**Results**

We have developed a novel machine learning method based on convolutional neural networks that automatically extracts a mesh representation of the left ventricle (LV) of the heart from cardiac magnetic resonance image scans. The results are reported in Romaszko et al. (2021). This LV mesh is an essential input to a cardiac mechanics emulator.

We have carried out a comparative evaluation of different emulation methods for cardiac mechanics. The results are reported in Dalton, Lazarus and Husmeier (2020). This article won the best paper award at ICSTA 2020.

We have developed novel improved methods for cardio-mechanics emulation. The improvement in performance gained in this way is discussed in detail in Dalton & Husmeier (2020).

To improve the accuracy of cardio-mechanic parameter estimation even further, we have developed a novel Bayesian statistical model that systematically combines in-vivo strain data, extracted from cardio-mechanic images, with ex-vivo volume-pressure data. This leads to a substantial boost in parameter estimation accuracy, as described in detail in Lazarus et al. (2021).

We have also successfully applied state-of-the art statistical inference and machine learning techniques to analyse cardiac amyloidosis disease progression, leading to novel model-based clinical markers. See Li et al. (2020) for details.

** 4.2** **Uncertainty quantification in haemodynamic models**

** ** (PhD student Mihaela Paun)

**Background**

Parameter estimation and uncertainty quantification in physiology and pathophysiology is a vital step towards personalized medicine. The work in WP4 aims to develop novel sampling methods for a complex partial differential equation (PDE) system of the haemodynamics in the pulmonary circulation system, which is relevant for disease diagnosis (pulmonary hypertension) from limited noisy blood flow and pressure data. The data likelihood is analytically intractable and computational expensive to evaluate numerically as it requires numerically solving a system of PDEs in every step of the inference procedure. Our work is focused on approximate inference that best trades off accuracy and computational efficiency.

**Work done so far**

We have developed novel emulation techniques for the pulmonary circulation system, improved their computational efficiency, allowed for model mismatch, improved the model by allowing for vessel-specific stiffness parameters, and corrected the closed-loop effects that need to be taken into consideration when applying our method in the clinical practice.

**Results**

We have developed novel statistical emulation methods for the pulmonary circulation system, and we have combined them with powerful Hamiltonian Monte Carlo samplers to improve computational efficiency. The results are described in Paun & Husmeier (2021). The first author won the gold medal in the category ``mathematical sciences” at the STEM for Britain competition in Westminster in February 2020 for this work.

We have made the noise model more realistic. Our previous work, reported in last year’s report, was based on the assumption of independent and identically distributed additive Gaussian noise. We have generalized this to explicitly model the correlation structure of the noise, as well as to allow for model discrepancy, i.e. the mismatch between the mathematical model and the real physiological system. We have also improved our physiological model to allow for vessel-specific stiffness parameters, and to infer them in the context of a hierarchical Bayesian model to avoid overfitting. The results are reported in Paun et al. (2020).

When connecting our mathematical model predictions and statistical inference to the clinical decision process, new challenges arise. We have identified and discussed the complications that potentially result from *closed-loop *effects (Paun & Husmeier, 2020a), and the model extensions that are required to reduce the ensuing bias (Paun & Husmeier 2020b). We have carried out a series of cardiovascular simulations to assess the reliability of cardiovascular physiological parameter estimation in the presence of medical interventions. Our principal result is that if the closed-loop effect of medical interventions is accounted for, the model calibration provides accurate parameter estimates. This finding has important implications for the applicability of cardio-physiological modelling in the clinical practice (Paun et al. 2021).

**4.3 ****Multi-scale modelling of cancer**

(RA Agnieszka Borowska)

**Background**

Multiscale modelling of cancer starts with the individual cell and needs to understand the process of chemotaxis. Chemotaxis is a type of cell movement in response to a chemical stimulus which plays a key role in multiple biophysical processes, such as embryogenesis and wound healing, and which is crucial for understanding metastasis in cancer research. In the literature, chemotaxis has been modelled using biophysical models based on systems of nonlinear stochastic partial differential equations (NSPDEs), which are known to be challenging for statistical inference due to the intractability of the associated likelihood and the high computational costs of their numerical integration. Therefore, data analysis in this context has been limited to comparing predictions from NSPDE models to laboratory data using simple descriptive statistics.

**Work done so far**

We have developed a statistically rigorous framework for parameter estimation in complex biophysical systems described by NSPDEs such as the one of chemotaxis. We have adopted a likelihood-free approach based on approximate Bayesian computations with sequential Monte Carlo (ABC-SMC) which allows circumventing the intractability of the likelihood. To find informative summary statistics, crucial for the performance of ABC, we have developed a novel Gaussian process (GP) regression model. The interpolation provided by the GP regression turns out useful on its own merits: it relatively accurately estimates the parameters of the NSPDE model and allows for uncertainty quantification, at a very low computational cost.

**Results**

Our proposed methodology allows a considerable part of computations to be completed before having observed any data, providing a practical toolbox to experimental scientists whose modes of operation frequently involve experiments and inference taking place at distinct points in time. In an application to externally provided synthetic data we have demonstrated that the correction provided by ABC-SMC is essential for accurate estimation of the NSPDE model parameters and for more flexible uncertainty quantification. The results are reported in Borowska et al. (2021).

**Publications**

A. Lazarus, H. Gao, X. Luo and D. Husmeier (2021)

Improving cardio-mechanic inference by combining in-vivo strain data with ex-vivo volume-pressure data

*Journal of the Royal Statistical Society*, Series C, under review.

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*, under review

L. Mihaela Paun, Agnieszka Borowska, Mitchel J. Colebank, Mette S. Olufsen and Dirk Husmeier (2021)

Inference in Cardiovascular Modelling Subject to Medical Interventions

Proceedings ICSTA 2021, DOI: 10.11159/icsta21.109

Borowska, A., Giurghita, D. and Husmeier, D. (2021)

Gaussian process enhanced semi-automatic approximate Bayesian computation: parameter inference in a stochastic differential equation system for chemotaxis. *Journal of Computational Physics*, 429, 109999. (doi: 10.1016/j.jcp.2020.109999)

Paun, L. M. and Husmeier, D. (2021) Markov chain Monte Carlo with Gaussian processes for fast parameter estimation and uncertainty quantification in a 1D fluid‐dynamics model of the pulmonary circulation. *International Journal for Numerical Methods in Biomedical Engineering*, 37(2), e3421. (doi: 10.1002/cnm.3421) (PMID:33249755) (PMCID:PMC7901000)

Paun, L. M., Colebank, M. J., Olufsen, M. S., Hill, N. A. and Husmeier, D. (2020) Assessing model mismatch and model selection in a Bayesian uncertainty quantification analysis of a fluid-dynamics model of pulmonary blood circulation. *Journal of the Royal Society: Interface,* 17(173), 20200886. (doi: 10.1098/rsif.2020.0886) (PMID:33353505)

Dalton, D., Lazarus, A. and Husmeier, D. (2020) Comparative evaluation of different emulators for cardiac mechanics. In: Ladde, G. and Noelle, S. (eds.) *Proceedings of the 2nd International Conference on Statistics: Theory and Applications (ICSTA'20).* Avestia Publishing: Ottawa, Canada, p. 126. ISBN 9781927877685 (doi:10.11159/icsta20.126)

Husmeier, D. and Paun, L. M. (2020a) Closed-loop effects in cardiovascular clinical decision support. In: Ladde, G. and Samia, N. (eds.) *Proceedings of the 2nd International Conference on Statistics: Theory and Applications (ICSTA'20)*. Avestia Publishing: Ottawa, Canada, p. 128. ISBN 9781927877685 (doi:10.11159/icsta20.128)

Dalton, D. and Husmeier, D. (2020) Improved statistical emulation for a soft-tissue cardiac mechanical model. In: Irigoien, I., Lee, D.-J., Martínez-Minaya, J. and Rodríguez-Álvarez, M. X. (eds.) *Proceedings of the 35th International Workshop on Statistical Modelling*. Servicio Editorial de la Universidad del País Vasco: Bilbao, Spain, pp. 55-60. ISBN 9788413192673

Husmeier, D. and Paun, L. M. (2020b) Closed-loop effects in coupling cardiac physiological models to clinical interventions. In: Irigoien, I., Lee, D.-J., Martínez-Minaya, J. and Rodríguez-Álvarez, M. X. (eds.) Proceedings of the 35th International Workshop on Statistical Modelling. Servicio Editorial de la Universidad del País Vasco: Bilbao, Spain, pp. 120-125. ISBN 9788413192673

Li, W. et al. (2020) Analysis of cardiac amyloidosis progression using model-based markers. *Frontiers in Physiology*, 11, 324. (doi: 10.3389/fphys.2020.00324) (PMID:32425806) (PMCID:PM

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**Parameter inference and model selection (Update May 2018)**

The mathematical models described in the various WPs depend on several biophysical parameters. However parameters cannot be directly measured in the laboratory, or where patient specific variation needs to be accounted for in a drive towards personalised medicine. We face the challenging task to learn (or “infer”) them within the context of the mathematical model itself, based on a systematic comparison of the outputs from computer simulations and experimental observations. In this work package, we will address this problem with state-of-the-art computer-intensive statistical inference

**Project 1:** parameter optimization in an approximate maximum likelihood sense, or sample them from the posterior distribution with Monto Carlo methods.

**Project2:** Design of a surrogate objective function using a metric based on a set of carefully selected summary statistics for the stochastic agent-based models.

**Project3:** systematically comparing variational verse sequential methods of "data assimilation”.

**Project4:** we will pursue model selection within a sound statistical framework, for example MCMC-based techniques, or lower-order approximations based on the Laplace method or BIC.

**Project 5:** we will upscale these developed inference methods to account for interactions with the extracellular matrix fibres within tissues, as required in other work packages.

**Team:** Prof. Husmeier (team leader), Prof. Ogden, Dr. Yin, Prof. Luo, Prof. Berry, Prof. Chaplain, Prof. Insall, Prof. Smith, PDRA4, PhD5

### Project 1.