# Bishesh Khanal, Marco Lorenzi, Nicholas Ayache and Xavier Pennec

Abstract

This paper proposes a framework to simulate patient specific structural Magnetic Resonance Images (MRIs) from the available time-points of Alzheimer’s Disease(AD) subjects. We use a biophysical model of brain deformation due to atrophy that can generate biologically plausible deformation for any given desired volume changes at the voxel level of the brain MRI. Large number of brain regions are segmented in 45 AD patients and the atrophy rates per year are estimated in these regions from two extremal available scans. Assuming linear progression of atrophy, the volume changes in scans closest to the middle time-point images from the baseline scans are computed. These atrophy maps are prescribed to the baseline images to simulate the middle time-point images by using the biophysical model of brain deformation. The volume changes from the baseline image to the real middle time-point are compared to the volume changes in the simulated middle time-point images. This present framework also allows to introduce desired atrophy patterns at different time-points to simulate non-linear progression of atrophy. This opens a way to use a biophysical model of brain deformation to evaluate methods that study the temporal progression and spatial relationships of atrophy evolution in AD.

Keywords: Alzheimer’s disease, biophysical modeling, biomechanical simulation

This is a pre-print of the following published article: Bishesh Khanal, Marco Lorenzi, Nicholas Ayache, Xavier Pennec. Simulating Patient Specific Multiple Time-point MRIs From a Biophysical Model of Brain Deformation in Alzheimer’s Disease. Workshop on Computational Biomechanics for Medicine - X, Oct 2015, Munich, France. 2015.

# Introduction

\label{sec:introduction} Alzheimer’s Disease (AD) is one of the most common types of dementia. It is a neurodegenerative disease that progresses gradually over several years with the accumulation of neurofibrillary tangles (NFTs) and amyloid-$$\beta$$ (A-$$\beta$$) plaques (Braak 1991). These microscopic neurobiological changes are followed by the progressive neuronal damage that leads to the atrophy of the brain tissue. The atrophy or the volume changes of brain tissue is a macroscopic change that structural Magnetic Resonance Imaging (MRI) can estimate in different brain regions. Many different methods have been proposed to estimate atrophy in some particular regions of brain that are known to be affected in AD (Frisoni 2010).

In addition to estimating specific brain structures with atrophy, longitudinal imaging data could also potentially be used to study the temporal inter-relationship of atrophy in different structures. For instance in (Carmichael 2013), authors estimate per-individual rates of atrophy in $$34$$ cortical regions and in hippocampus. Then they study the groupings of these structures based on the correlation of the atrophy rates. In (Fonteijn 2012), authors define AD progression as a series of discrete events. Atrophy in different parts of the brain are taken as different events along with clinical events. Without any prior to their ordering, the model finds most probable order for these events from the data itself. They use Bayesian statistical algorithms for fitting in the event-based disease progression model. The objective of these kinds of studies is to understand how different regions of brain interact during the neurodegeneration and find its trajectory. Such studies can benefit with large number of longitudinal images of AD patients. In this context, a model that can simulate many time-point images from a few available longitudinal images can be a valuable tool.

Atrophy simulators (Karaçali 2006)(Pieperhoff 2008)(Smith 2003)(Camara 2006) have been proposed in the literature and used mostly for the validation of registration or segmentation methods (Camara 2007)(Sharma 2010), or to estimate uncertainty in the measured atrophy (Sharma 2013). The simulators in (Karaçali 2006)(Pieperhoff 2008)(Sharma 2010) use a Jacobian based methods where the desired level of atrophy is set at each voxel, and the deformation that best approximates the desired level of atrophy is found. Regularization is used in the optimization to enforce certain desired conditions such as topology preservation. The advantage of these methods is the ability to define atrophy maps at the voxel level. However regularization parameters used to enforce topology preservation are generally difficult to relate to a plausible biophysical process of AD and can create difficulties in simulating opening of certain structures such as sulci. It is not trivial to consider different tissue behaviors in such approaches. In (Smith 2003)(Camara 2006), authors proposed a model of brain deformation based on thermoelasticity. The authors defined the volume changes in particular structures and tissues of a meshed brain by assigning different thermal coefficients. Simulation of the images is done by first solving the thermoelastic model of tissue deformation with Finite Element Method (FEM), and then by interpolating the obtained displacement field from the mesh to the image. FEM involves moving back and forth from voxels to meshes which creates numerical difficulties and inaccuracies in the model personalization.

In (Khanal 2014) we proposed a biophysical model of brain deformation due to atrophy in AD. The mechanisms of neuronal deaths and its evolution are not well known for AD and are likely to be primarily guided by complex physiological processes. However we believe that the biomechanics of brain tissue might play an important role in determining the consequence of the neuronal deaths on brain shape changes. The model we proposed in (Khanal 2014) builds upon the assumptions that we relate to the biophysical process of tissue shape changes as the consequence of local volume loss. This model can be used to simulate time-series MRIs starting from a real input baseline MRI.

In this work we use our biophysical model developed in (Khanal 2014) to present a framework that allows to interpolate or extrapolate patient specific unseen time-point images from at least two available time-point images of the subject and to assess how closely these simulated trajectories follow real patient trajectories. We also improve the implementation of the boundary condition of the model by imposing zero deformation in the skull and all the regions outside of the skull. In (Khanal 2014) the zero deformation was imposed at the image boundaries and not at the brain-skull boundary.

The following section briefly explains the assumptions and implementation of the biophysical model we presented in (Khanal 2014), and in section \ref{sec:experiments} we present how we interpolate new images between two acquisition time points.