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Register a new userModeling the Mitral Valve
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This work is concerned with modeling and simulation of the mitral valve, one of the four valves in the human heart. The valve is composed of leaflets, the free edges of which are supported by a system of chordae, which themselves are anchored to the papillary muscles inside the left ventricle. First, we examine valve anatomy and present the results of original dissections. These display the gross anatomy and information on fiber structure of the mitral valve. Next, we build a model valve following a design-based methodology, meaning that we derive the model geometry and the forces that are needed to support a given load, and construct the model accordingly. We incorporate information from the dissections to specify the fiber topology of this model. We assume the valve achieves mechanical equilibrium while supporting a static pressure load. The solution to the resulting differential equations determines the pressurized configuration of the valve model. To complete the model we then specify a constitutive law based on a stress-strain relation consistent with experimental data that achieves the necessary forces computed in previous steps. Finally, using the immersed boundary method, we simulate the model valve in fluid in a computer test chamber. The model opens easily and closes without leak when driven by physiological pressures over multiple beats. Further, its closure is robust to driving pressures that lack atrial systole or are much lower or higher than normal.
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This paper presents a new method for modeling the mechanics of the aortic valve, and simulates its interaction with blood. As much as possible, the model construction is based on first principles, but such that the model is consistent with experimental observations. We require that tension in the leaflets must support a pressure, then derive a system of partial differential equations governing its mechanical equilibrium. The solution to these differential equations is referred to as the predicted loaded configuration; it includes the loaded leaflet geometry, fiber orientations and tensions needed to support the prescribed load. From this configuration, we derive a reference configuration and constitutive law. In fluid-structure interaction simulations with the immersed boundary method, the model seals reliably under physiological pressures, and opens freely over multiple cardiac cycles. Further, model closure is robust to extreme hypo- and hypertensive pressures. Then, exploiting the unique features of this model construction, we conduct experiments on reference configurations, constitutive laws, and gross morphology. These experiments suggest the following conclusions, which are directly applicable to the design of prosthetic aortic valves. (i) The loaded geometry, tensions and tangent moduli primarily determine model function. (ii) Alterations to the reference configuration have little effect if the predicted loaded configuration is identical. (iii) The leaflets must have sufficiently nonlinear material response to function over a variety of pressures. (iv) Valve performance is highly sensitive to free edge length and leaflet height. For future use, our aortic valve modeling framework offers flexibility in patient-specific models of cardiac flow.
Mathematical modeling in cancer has been growing in popularity and impact since its inception in 1932. The first theoretical mathematical modeling in cancer research was focused on understanding tumor growth laws and has grown to include the competition between healthy and normal tissue, carcinogenesis, therapy and metastasis. It is the latter topic, metastasis, on which we will focus this short review, specifically discussing various computational and mathematical models of different portions of the metastatic process, including: the emergence of the metastatic phenotype, the timing and size distribution of metastases, the factors that influence the dormancy of micrometastases and patterns of spread from a given primary tumor.
In this article, we review the mathematical modeling for the vascular system.
While it is well-understood what a normal human placenta should look like, a deviation from the norm can take many possible shapes. In this paper we propose a mechanism for this variability based on the change in the structure of the vascular tree.
Multiple myeloma (MM), a plasma cell cancer, is associated with many health challenges, including damage to the kidney by tubulointerstitial fibrosis. We develop a mathematical model which captures the qualitative behavior of the cell and protein populations involved. Specifically, we model the interaction between cells in the proximal tubule of the kidney, free light chains, renal fibroblasts, and myeloma cells. We analyze the model for steady-state solutions to find a mathematically and biologically relevant stable steady-state solution. This foundational model provides a representation of dynamics between key populations in tubulointerstitial fibrosis that demonstrates how these populations interact to affect patient prognosis in patients with MM and renal impairment.
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