Bayesian sensitivity analysis of a model of the aortic valve
Introduction
The Aortic Valve (AV) has attracted much attention in the biomechanics community due to its remarkable durability—typically experiencing 3.7 billion cycles in its lifetime (Thubrikar, 1990), usually without failure. Prosthetic replacements are necessary when the natural valve becomes diseased, yet both mechanical replacements and bioprosthetics have significant drawbacks (Silberman et al., 2008) and cannot perform with the same reliability as the natural counterpart. Understanding the biomechanics of the natural valve is a key requirement in improving prosthetic design, therefore Finite Element (FE) models have been used extensively to better understand the AV. Recent simulations have been of high complexity, with fluid structure interaction (FSI) included (see e.g. De Hart and Peters, 2003, Carmody and Burriesci, 2006), and encompassing multi-scale approaches (Weinberg and Mofrad, 2007).
A difficulty which is rarely acknowledged however is the problem of dealing with model uncertainties. The AV is typical of a biological system; many model inputs are often quoted over fairly wide ranges—valve dimensions, material properties and loading vary significantly from one individual to the next. To ignore the uncertainty in these parameters can only place the validity of the model in doubt. Some work has been performed (Ranga et al., 2004) to investigate uncertainty in the material properties of the aortic root, although this was not a formal statistical analysis. This paper aims to perform a detailed Uncertainty Analysis (UA) specifically on an AV model, and on a broader scale to highlight the importance (and feasibility) of UA in modelling biological systems in general. A furtherance of UA, known as Sensitivity Analysis, measures the sensitivity of the model output to particular (subsets of) inputs. This can provide a deeper insight into the model itself and suggest approaches for reducing the uncertainty in the output. A full discussion of SA is given by Saltelli et al. (2000).
Section snippets
The finite element model
The FE model was built in LS-Dyna, an explicit solver that has been shown to be capable of modelling the transient behaviour of the AV—some examples include (Howard and Patterson, 2003, Carmody and Burriesci, 2006, Weinberg and Mofrad, 2007). A one-sixth section of the full valve (shown in Fig. 1) was created, illustrated in Fig. 2 with appropriate symmetry conditions. The initial geometry of the valve was taken to be in what is assumed to be the unstressed state (as used for example in
Material properties
The material of the AV leaflet is known to be highly nonlinear and anisotropic. Collagen and elastin fibres generally run in the circumferential direction, and are “crimped” in the relaxed state of the valve; as such, at low strains they are not under tension and resistive force is due purely to the connective tissue. However, once the fibres are un-crimped, they provide a strong (and virtually linear-elastic (Billiar and Sacks, 2000)) resistance—this is responsible for the hyperelastic, or
Uncertainty analysis
A classic approach to UA/SA is to perform a Monte Carlo (MC) simulation (Shreider, 1964), which involves running the model at a large number of points over the range of input space. This is however very impractical in the case of a large model such as the AV model. The approach adopted here is known as an emulator-based approach, where a small number of model samples is used to build an emulator or metamodel which mimics the behaviour of the real model, for a greatly reduced computational cost.
Results and discussion
Before the uncertainty analysis, the model was run at mean parameter values (see Table 1) in order to test whether the model output agreed with experimental data. Fig. 4 shows the opening of the valve over a simulation time of 20 ms with approximate leaflet-base angles. Thubrikar (1990) reports that the opening of the aortic valve takes between 17 and 20 ms, which is demonstrated here. Additionally the leaflet-base angle agrees with the ranges given in Thubrikar (around 20° in diastole to 99° in
Conclusions
This paper has followed an efficient statistical method for analysing uncertainties and sensitivities in an aortic valve model. Uncertainties in biomechanical models are often overlooked or informally dealt with, but Bayesian uncertainty analysis allows them to be considered and investigated in a thorough statistical fashion, yielding detailed information about the robustness of the model. Additionally, by examining sensitivity to input parameters it is possible to gain a deeper insight into
Conflict of interest statement
None declared.
Acknowledgement
The authors would like to thank Ove Arup for supplying LS-Dyna and LS-Opt. Additional thanks go to Marc Kennedy of the Probability and Statistics Department at The University of Food and Environment Research Agency, UK, for the use of Gem-SA (Kennedy, 2009). This work is funded by a grant from the EPSRC.
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