تجزیه و تحلیل حساسیت بیزی از یک مدل دریچه آئورت

Understanding the mechanics of the aortic valve has been a focus of attention for many years in the biomechanics literature, with the aim of improving the longevity of prosthetic replacements. Finite element models have been extensively used to investigate stresses and deformations in the valve in considerable detail. However, the effect of uncertainties in loading, material properties and model dimensions has remained uninvestigated. This paper presents a formal statistical consideration of a selected set of uncertainties on a fluid-driven finite element model of the aortic valve and examines the magnitudes of the resulting output uncertainties. Furthermore, the importance of each parameter is investigated by means of a global sensitivity analysis. To reduce computational cost, a Bayesian emulator-based approach is adopted whereby a Gaussian process is fitted to a small set of training data and then used to infer detailed sensitivity analysis information. From the set of uncertain parameters considered, it was found that output standard deviations were as high as 44% of the mean. It was also found that the material properties of the sinus and aorta were considerably more important in determining leaflet stress than the material properties of the leaflets themselves.

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 and 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).

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 the working of the model by exploring how parameter variations affect certain outputs; for example the extent of valve opening has been shown to be highly dependent on the stiffness of the aorta/sinus, reinforcing the supposition that the expansion of the aorta is intrinsic in the opening mechanism of the valve. A surprising conclusion was that some of the chosen leaflet material parameters (the transition stretch and post-transition modulus) were not found to be very influential, being overshadowed by the effect of the expansion of the sinus on the buckling of the leaflet. It is not claimed here that an exhaustive uncertainty analysis has been performed, since the number of uncertain parameters can be extended almost without limit. However, a practical means of investigating selected uncertainties has been presented. Furthermore it has demonstrated the necessity for consideration of uncertainties in AV models, since the dispersion of model outputs is significant. This necessity can presumably be extended to apply to biomechanical models in general, since they are prone to uncertainties for similar reasons to the AV.