Bayesian optimisation for efficient parameter inference in a cardiac mechanics model of the left ventricle

Abstract

We consider parameter inference in cardio‐mechanic models of the left ventricle, in particular the one based on the Holtzapfel‐Ogden (HO) constitutive law, using clinical in vivo data. The equations underlying these models do not admit closed form solutions and hence need to be solved numerically. These numerical procedures are computationally expensive making computational run times associated with numerical optimisation or sampling excessive for the uptake of the models in the clinical practice. To address this issue, we adopt the framework of Bayesian optimisation (BO), which is an efficient statistical technique of global optimisation. BO seeks the optimum of an unknown black‐box function by sequentially training a statistical surrogate‐model and using it to select the next query point by leveraging the associated exploration‐exploitation trade‐off. To guarantee that the estimates based on the in vivo data are realistic also for high‐pressures, unobservable in vivo, we include a penalty term based on a previously published empirical law developed using ex vivo data. Two case studies based on real data demonstrate that the proposed BO procedure outperforms the state‐of‐the‐art inference algorithm for the HO constitutive law.

Image-based estimation of the left ventricular cavity volume using deep learning and Gaussian process with cardio-mechanical applications

“…At the beginning of diastole, it is assumed that blood pressure is zero, the unloaded state, while as we approach the end of diastole, the ventricular pressure increases and the myocardium is passively stretched to balance the applied pressure (Wang et al, 2013). Stiffness of the myocardium has been considered a potential biomarker to characterize diastolic dysfunction, while inferring myocardial stiffness non-invasively is still a challenging research problem (Gao et al, 2017b;Borowska et al, 2022). In this study, an incompressible material model derived from the constitutive model proposed by Holzapfel and Ogden (2009) is used, that is…”

“…( 4) by matching the model predictions to the measured data, as we observe from CMR scans (Gao et al, 2015). The search for optimal parameters can be computationally intensive due to the multi-modality of the inverse problem and the high computational costs of iterative numerical solutions of the LV model using the finite element method (Borowska et al, 2022). To avoid this issue, a statistical emulator can be trained to replace the computationally expensive LV model, reducing computational costs by several orders of magnitude; see Noè et al (2019).…”

“…Our objective function (𝑓 𝑜 ) to be optimized is adopted from Klotz et al (2006), Borowska et al (2022) and Lazarus et al (2022b), where a Klotz term is included to allow for the high stretch behavior of the tissue:…”

“…As has been discussed extensively in Lazarus et al (2022b), including the Klotz curve (Klotz et al, 2006) in the optimization process effectively introduces a constraint that reduce the degrees of freedom. In other words, the cardio-mechanical model should not only predict the correct enddiastolic in vivo volume at 8 mmHg, but also the extrapolated volume at 30 mmHg based on the Klotz curve (Borowska et al, 2022). However, considering the high-dimensional search space for material parameters and limited measured data, a simple reparameterization approach was adopted here assuming 𝑎 = 𝑎 f and 𝑏 = 𝑏 f .…”

Efficient approximation of cardiac mechanics through reduced order modeling with deep learning-based operator approximation

“…For instance, in [19,20] surrogate models were generated via the polynomial chaos expansion approach to accelerate uncertainty quantification (UQ) studies and sensitivity analysis of left ventricular mechanics. Many machine learning-based models have been proposed as real-time cardiac mechanics simulators [21,22,23], while statistical emulators, such as Gaussian processes, have been used to speed-up parameter inference [24,25] or to reduce the complexity of parametric searches for high-fidelity models [26]. Despite being well suited for the rapid and repeated evaluation of the input-output map, this models may lack of accuracy when dealing with a single, patient-specific forward simulation of the cardiac activity.…”