Multilevel Monte Carlo for Uncertainty Quantification in Brain Simulations

Background

The motivation for this project is the analysis of the symptoms and causes of brain diseases. More specifically, we study the anomalies in the fluid dynamics of the brain that are believed to be related to dementia (which includes Alzheimer’s) and hydrocephalus. The movement of fluids within the brain can be simulated with the use of mathematical models. However, many of the parameter values in these models are subject to uncertainty and measurement errors. Can we quantify this uncertainty and discern how much this affects our model predictions? Which physiological parameters do we need to measure more accurately to obtain sensible predictions and useful insights?

The models we use are the Biot equations and the multiple-network poroelasticity model with random coefficients. We use finite element techniques to solve these models and multilevel Monte Carlo methods to quantify the uncertainty.

Progress

Numerical methods

To account for the uncertainty in our model, we replace the parameters subject to errors with random variables or Gaussian stochastic fields (random functions whose value at any point is given by a Gaussian random variable). Monte Carlo methods require the sampling of these parameters from their probability distributions. However, sampling Gaussian fields is non-trivial and computationally expensive.

The sampling problem can be recast as the solution of a partial differential equation with a random forcing term called white noise. White noise is a very rough and non-smooth object (see Figure 1). The lack of smoothness of white noise makes theoretical analysis of the finite element method (FEM) we are using to solve this equation complicated. In addition, when continuous Lagrange finite elements are used to solve the white noise equation, the sampling of white noise becomes expensive and approximations are generally used in practice. Despite these difficulties, we have shown that the FEM converges with optimal order and we presented a new white noise sampling technique that is cheap and efficient. It works for any family of finite elements and is compatible with the multilevel Monte Carlo method. Once the white noise equation can be solved, Gaussian fields of arbitrary smoothness can be sampled (see Figure 1).

Figure 1: Realisations of different random fields. Top-left is highly non-smooth white noise, the rest are Gaussian fields of increasing smoothness.

Uncertainty quantification in mouse brain simulations

We applied the techniques described above to quantify the uncertainty in the Biot model when used to simulate the fluid movement in a mouse brain slice (see Figure 2). We were able to compute approximate high confidence intervals for various quantities of interest (e.g. the maximum interstitial fluid flux over time in a specific region). These quantities are important as there is active medical research trying to determine whether anomalies in these values are indicators of specific brain diseases.

Figure 2: Expected behaviour of the mouse brain fluid dynamics. From left to right: expected displacement of the brain matter, expected paravascular fluid pressure and velocity.

Future Work

We are planning to extend the above numerical techniques to more efficient sampling methods such as quasi Monte Carlo and multilevel quasi Monte Carlo. This will be complemented by extending the work on the 2D mouse brain model to a full 3D scale simulation of the human brain.