Bayesian inverse problems for soft tissue mechanics

在过去的几十年里，大量的文献集中在连续力学框架内对生物软组织的力学行为进行数学建模。特别关注的是在大应变（非线性）弹性和粘弹性的背景下推导本构方程。许多最广泛使用的模型采用现象学方法（例如，通过提出应变能，它是应变不变量的指数函数，具有可以拟合到实验数据的自由参数）；然而，这种方法不能用于预测微观结构变化对宏观力学的影响。另一种方法是明确地模拟软组织的微观结构，用可测量的参数来描述构成组织的胶原纤维网络的几何形状，以及胶原纤维本身的简单（线性）本构方程。这些线性本构模型可以很好地拟合单个胶原纤维的实验数据；然而，由于难以精确测量纤维的几何形状和作用在纤维上的力，这些纤维的直径可以小到几十纳米，因此报告的本构参数（例如胶原的杨氏模量）的值会发生数量级的变化。这些参数值的不确定性使得用确定性的微观结构模型进行定量预测变得困难；因此，需要一种替代方法来解释不确定性。在这个项目中，我们将调用贝叶斯框架，该框架通过将软组织模型与先验信念和宏观实验数据相结合，将为我们提供观测条件下参数的后验概率分布。这些分布不仅包含有关参数可能值的信息，而且还允许我们量化和评估由它们产生的估计中固有的不确定性。在实践中，这可以通过实现马尔可夫链蒙特卡罗（MCMC）方法来实现，该方法通过构造一个平稳分布等于后验分布的遍历马尔可夫链，允许我们从分布中采样以表征它。这可以通过实现现有的MCMC方法来实现，或者通过设计能够有效地针对由这些逆问题产生的分布类型的新方法来实现。

Over the last few decades, a large body of literature has focussed on mathematically modelling the mechanical behaviour of biological soft tissues within a continuum mechanics framework. Particular attention has focused on deriving constitutive equations within the context of large strain (nonlinear) elasticity and viscoelasticity. Many of the most widely used models take a phenomenological approach (for example, by proposing a strain energy which is an exponential function of the strain invariants, with free parameters that can be fitted to experimental data); however, this approach cannot be used to predict the effects of microstructural changes on macroscale mechanics. An alternative approach explicitly models the microstructure of the soft tissue, with measurable parameters to describe the geometry of the collagen fibre network that makes up the tissue, along with simple (linear) constitutive equations for the collagen fibres themselves. These linear constitutive models fit experimental data on individual collagen fibres well; however, due to the difficulty in precisely measuring the geometries of, and forces acting on, fibres that can be as small as tens of nanometres in diameter, the reported values of the constitutive parameters (e.g. the collagen Young's modulus) vary by orders of magnitude. This uncertainty in the values of these parameters makes it difficult to make quantitative predictions with deterministic, microstructural models; therefore, an alternative approach is needed which accounts for the uncertainty. In this project, we will invoke the Bayesian framework, which, by incorporating the models of soft tissues with prior beliefs and macroscale experimental data, will give us a posterior probability distribution of the parameters conditioned on the observations. These distributions contain not only information about the likely values of the parameters, but also allow us to quantify and assess the uncertainty inherent in the estimates that arise from them. In practice this can be achieved by implementing Markov chain Monte Carlo (MCMC) methods, which through construction of an ergodic Markov chain with stationary distribution equal to the posterior distribution, allows us to sample from the distribution in order to characterise it. This might be done through the implementation of existing MCMC methods, or through the design of new methodologies which are able to efficiently target the types of distributions which arise from these inverse problems.