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Abstract
Patient-specific modelling of haemodynamics in arterial networks has so far relied on parameter estimation for inexpensive or small-scale models. We describe here a Bayesian uncertainty quantification framework which makes two major advances: an efficient parallel implementation, allowing parameter estimation for more complex forward models, and a system for practical model selection, allowing evidence-based comparison between distinct physical models. We demonstrate the proposed methodology by generating simulated noisy flow velocity data from a branching arterial tree model in which a structural defect is introduced at an unknown location; our approach is shown to accurately locate the abnormality and estimate its physical properties even in the presence of significant observational and systemic error. As the method readily admits real data, it shows great potential in patient-specific parameter fitting for haemodynamical flow models.
Highlights
Mathematical models for haemodynamics trace back to the work of Euler, who described a one-dimensional treatment of blood flow through an arterial network with rigid tubes [1,2]; more sophisticated one-dimensional models are still used to study a variety of physio-pathological phenomena [3,4,5,6,7,8,9,10]
If a uniform prior is assumed on models, this posterior is directly proportional to the evidence ρ(D | Mi), and so model selection is ‘free’ when the evidence is already calculated for parameter estimation [29,38,39,40]
While there exist many approaches to solving the proposed Bayesian inverse problem (e.g. [41,42,43]), few are constrained by the main computational barrier in this application: the complex forward problem g which appears in the fitness J(θ, D | M )
Summary
Mathematical models for haemodynamics trace back to the work of Euler, who described a one-dimensional treatment of blood flow through an arterial network with rigid tubes [1,2]; more sophisticated one-dimensional models are still used to study a variety of physio-pathological phenomena [3,4,5,6,7,8,9,10]. Several approaches exist for parameter estimation and uncertainty quantification for these models. The chief contribution of this work is to introduce a Bayesian framework for uncertainty quantification in a bifurcating network of one-dimensional extensible arteries. The advantages of the approach are twofold It uses transitional Markov chain Monte Carlo (TMCMC), a highly parallelizable algorithm for approximate sampling which allows practical uncertainty quantification even for large arterial networks [27,28,29]; our high-performance implementation Π4U will be shown to simultaneously and efficiently estimate several unknown parameters in this setting. We use Bayesian model selection to probabilistically locate the defect within the network and accurately recover its structural properties, showing the approach to be effective even when parameters are corrupted with Gaussian noise. As the method readily admits clinical blood flow data, which have been shown to be measurable with non-invasive procedures [30,31,32,33], it shows great potential in diagnosing patient-specific structural issues in the circulatory system
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