Mechanical modelling of living systems: from cancer modelling to control in sports
Creators: Hall, Meghan
Publisher: name: University of Alberta Library
Year: 2022

<p>From the mechanical processes that produce the convolutions in the human brain needed for complex thought, to the precise and controlled movements derived by intuitive calculations of body position in figure skating, mechanics plays a role in everything we do. In this thesis, we apply and examine the role of mechanics in models of glioma spread and figure skating. </p>

<pre><code>In Chapter 1, we introduce the topic of glioma modelling and present our glioma spread model in 3D-a continuum model for the density of tumor cells coupled to a general momentum balance equation for the mechanical properties of the glioma and brain tissue. Glioma cells are highly invasive, with the tumors often having diffuse and irregular boundaries. It has been well established that tissue heterogeneity significantly affects glioma spread and tumor cell behavior. Recent work exploring the effects of mechanical properties has strengthened the idea that mechanics plays a large role in determining glioma spread patterns and invasiveness. We focus on modelling two aspects that affect glioma spread: Structural effects on tumor cell migration and the impact of mechanical interactions on tumor spread. The model includes the glioma population as a cell density that can proliferate, spread via fully anisotropic diffusion, and that is advected by the velocity generated by the growing tumor mass. The momentum balance equation determines the deformation caused by the growing tumor mass, with this deformation causing an advection velocity. Not only is this advection velocity coupled to the tumor cells, but it is also applied to material properties, which can include diffusion tensors and elasticity parameters (i.e. shear modulus, bulk modulus). With the exact nature of brain tissue mechanics still being an open question, the form of the momentum balance equation is not specified for the 3D glioma model in Chapter 1, but rather is left in a general form allowing the 3D model to be used a framework for modelling glioma spread which can be used for any description of brain tissue mechanics. 

In Chapter 2, we analyze a 1D version of the glioma model to identify, characterize, and simulate travelling waves. Within the 1D model, we consider three biological scenarios representing different stages of glioma development, with each biological scenario characterized by which material properties (elasticity parameters and or diffusion), vary over space and time. In addition to the biological scenarios, we also consider multiple mechanical models, including linear elasticity and the nonlinear one-term Ogden elasticity model, as well as viscoelastic versions of both. For each biological scenario, we compare how these mechanical models affect glioma spread, as well as the resulting deformation and stress. For every mechanical model, we found that travelling waves existed with the same minimal wave speed. However, the deformation and stress associated with each mechanical model differs significantly. The Ogden model results in deformation and stresses two and three orders of magnitude less than the linear model, respectively. We also present wave speed analysis for a generalized elasticity model, finding that the analytically determined wave speed is indeed conserved among such elasticity models.

In Chapter 3, we present a 2D version of the 3D glioma spread model with linear elasticity. We develop and implement a numerical framework for simulating the 2D model, which integrates imaging data for both the simulation domain and diffusion tensors. We employ ExploreDTI to access and extract imaging data, including diffusion tensors and medical images. Diffusion tensor data is translated to cancer cell diffusion tensors and used to initialize the diffusion tensor in the glioma model simulations. Medical images are processed and used to define the simulation domain. Finally, the model is simulated using a finite element method. Through simulations, we are able to produce simulations with realistic rates of glioma spread and deformation levels. We also explore the effects of the model parameters using simulations. We show that the parameter scaling the body forces significantly affects the rate and shape of glioma spread, making it a desirable target for parameter fitting and further exploration.

Finally, in Chapter 4, we discuss the application of the Chaplygin sleigh as a model for a figure skate. A classical element in the sport of figure skating is reproducing specific patterns on the ice through very detailed, precise control of the skater&#39;s movements. We formalize this process using the Chaplygin sleigh as a model of the figure skate, with an added mass representing the skater&#39;s moving center of mass and acting as a control parameter for the system. Using a previous result on approximating piecewise curves with arcs, we present a modified form of the Chaplygin sleigh which is limited to producing circular arcs. Finally, we present a control algorithm based on minimization of the energy of control mass which successfully reproduces a prescribed pattern.
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