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public 01:29:48

Jake Taylor-King : Generalized Jump Processes and Osteocyte Network Formation

  -   Mathematical Biology ( 139 Views )

My talk will have two parts. PART I, From Birds to Bacteria: Generalised Velocity Jump Processes. There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long distance movement state and a resting or local movement state. In this talk, I will give a mathematical description of this process, adapted from Friedrich et. al. (2006). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of partial integro-differential equations are tumultuous and therefore it is necessary to both simplify and derive summary statistics. We derive an expression for the mean squared displacement, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally a large time diffusive approximation is considered via a Cattaneo approximation (Hillen, 2004). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution. We also consider the Levy regime where the variance of the running distribution tends to infinity. This leads to a fractional diffusion equation for superdiffusive Levy walks and can be solved analytically. Our theory opens up new perspectives both for the systematic derivation of such equations, and for experimental data analysis of intermittent motion. I will also briefly discuss recent developments (by other researchers) within the field of velocity jump processes. PART II: Modelling Osteocyte Network Formation: Healthy and Cancerous Environments. Advanced prostate, breast, and lung cancer can metastasize to bone. In pathological bone, the highly regulated bone remodeling signaling pathway is disrupted. Within bone dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone formation. In the presence of a cancerous microenvironment, the morphology of this network changes. Commonly osteocytes appear to be either overdifferentiated (i.e., there are more dendrites) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, preliminary studies measuring the number of osteocytes per unit area using pathology slides show that the number density of osteocytes change from healthy to metastatic prostate and breast cancer xenografted mice. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and number density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We consider variations of our model to test specific physiological hypotheses related to osteoblast differentiation; for example we can predict how changing measurable biological parameters, such as rates of bone secretion, rates of dendrite growth and rates of osteoblast differentiation can allow for qualitatively different network morphologies, and vice versa. We thenuse our model to hypothesize reasons for the limited efficacy of zoledronate therapy on metastatic breast cancer.

public 01:14:42

Hans Othmer : A hybrid model of tumor-stromal interactions in breast cancer

  -   Mathematical Biology ( 137 Views )

Ductal carcinoma in situ (DCIS) is an early stage non-invasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. We will discuss a mathematical model that incorporates the cross-talk between stromal and tumor cells, and which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-$\beta$ signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation, and describe a hybrid model for the interaction of cells with the tumor microenvironment. The analysis sheds light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggests how epigenetic changes in transformed cells affect tumor progression.

public 01:29:52

Anita Layton : Myogenic Response to Systolic Pressure in the Afferent Arteriole

  -   Mathematical Biology ( 131 Views )

Elevations in systolic blood pressure are believed to be closely linked to the pathogenesis and progression of renal diseases. It has been hypothesized that the afferent arteriole (AA) protects the glomerulus from the damaging effects of hypertension by sensing increases in systolic blood pressure and responding with a compensatory vasoconstriction. To investigate this hypothesis, we developed a mathematical model of the myogenic response of an AA smooth muscle cell, based on an arteriole model by Gonzalez-Fernandez and Ermentrout (Math Biosci 1994). renal hemodynamic regulation. The model incorporates ionic transport, cell membrane potential, contraction of the AA smooth muscle cell, and the mechanics of a thick-walled cylinder. The model represents a myogenic response based on a pressure-induced shift in the voltage dependence of calcium channel openings: with increasing transmural pressure, model vessel diameter decreases; and with decreasing pressure, vessel diameter increases. Further, the model myogenic mechanism includes a rate-sensitive component that yields constriction and dilation kinetics similar to behaviors observed in vitro. A parameter set is identified based on physical dimensions of an AA in a rat kidney. Model results suggest that the interaction of Ca2+ and K+ fluxes mediated by voltage-gated and voltage-calcium-gated channels, respectively, gives rise to periodicity in the transport of the two ions. This results in a time-periodic cytoplasmic calcium concentration, myosin light chains phosphorylation, and crossbridges formation with the attending muscle stress. Further, the model predicts myogenic responses that agree with experimental observations, most notably those which demonstrate that the renal AA constricts in response to increases in both steady and systolic blood pressures. The myogenic model captures these essential functions of the renal AA, and it may prove useful as a fundamental component in a multi-scale model of the renal microvasculature suitable for investigations of the pathogenesis of hypertensive renal diseases.