An Inverse Model for Estimating Elasticity of the Arterial Wall using Immersed Boundary Method - Gadkari, Tushar (My University, 2007)
Atherosclerosis generally occurs near the branching in the arteries where there tends to be flow irregularities. A build up of fatty deposits (plaque) occurs in the blood vessel in such regions making it to lose its elasticity. Such hardening of the arteries and the narrowing of the lumen can cause severe atheromas and even high blood pressure and blockage of the vessels. It is observed in North America that nearly 47% of the deaths are caused due to cardiovascular diseases and hence determination of such regions becomes very critical and can be very beneficial if done at an earlier stage. In this thesis, we present: an approach to model the pulsating flow of blood through such an atherosclerosis affected region of the artery using finite element method and further discuss the statistical model used to implement the optimization techniques to estimate the region of maximum rigidity. Here within we present a numerical and non-invasive approach to predict such regions. The computational modeling is carried out under two categories: a. The mathematical model and b. The statistical model.
The mathematical model which is the forward model, comprises of the artery and the cardiac muscle as hyperelastic material modeled with the neo-hookean model and the three dimensional Navier-Stokes equations solve for the blood flowing through it. We perform fluid dynamic analysis for the blood flowing through the vessel to compute the velocity at different time instances and mechanical analysis to compute the deformation of the artery which is a function of the elasticity of the vessel. The two models are interconnected to each other by boundary conditions as the normal component of the surface force provides the coupling between the two models. The shear modulus represents a measure of the elasticity of the vessel. We use linear spatial basis functions to model the shear modulus which spatially varies along the geometry of the vessel thus we have a region of atherosclerosis and the geometry shows the stenosis. The change in the shear modulus affects the velocity of blood through the vessel.
In the statistical model, we propose an inverse computational model for estimating the elasticity profile of the arterial wall where we implement the inverse modeling approach to estimate the maximum shear modulus which helps us to predict the region of atherosclerosis. The velocity and the deformation obtained for a particular shear modulus from our COMSOL forward model provide the realistic simulated measurements that are made noisy by introduction of white Gaussian noise with different SNR and we try to estimate the shear modulus that minimizes the error-function. We use COMSOL with MATLAB for simultaneous iterative computations of velocity and deformation measurements by running the optimization code. We estimate these unknown parameters using optimization algorithm that minimizes the cost function of our model. For our estimation we use the least squares estimator and we derive the maximum likelihood estimator. The unconstrained optimization is carried out with Neider Mead Simplex Method and the Trust Region Method which uses only the function evaluations to find the minimum: making it a very robust algorithm and very efficient for problems that are nonlinear or have a number of discontinuities. Our preliminary results demonstrate significant change in velocity of the blood and occurrence of vortices in the region of less elasticity and the tendency of the artery to deform minimum in the hardened less elastic region. Our estimation results show that the parameters are identifiable. The mean square error of the estimate as a function of SNR shows accuracy of the estimation.
Keywords: Inverse Model; Elasticity; Arterial Wall; Boundary Method