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Jennifer J. Young |
| Research Project Descriptions Modeling of Intestinal Edema: Intestinal edema is a medical condition referring to the excess build-up of fluid in the interstitial spaces of the intestinal tissue. Normally, the volume of fluid in your body is maintained at a fairly constant value by the even exchange of fluids between your circulatory and lymphatic systems. Blood capillaries leak fluid into the intersitium, and the lymphatics pump it out, either back to the circulatory system or to your kidneys. If the rate at which fluid is entering the interstitium exceeds the lymphatic pumping rate, fluid can build up in the spaces between tissue cells. When this build-up occurs in between the muscle cells of the intestinal wall, it causes a condition known as ileus, which is a disruption in the propulsive activity of the muscle cells in the intestine. The muscle layer experiences decreased contractile activity, which can lead to a blocked intestine. Our goal is to develop a computational model of the intestine during edema formation to try to determine a mechanical relationship between fluid bluid-up and decreased muscle contractility. Thus far we have developed a two-dimesional poroelastic model of the intestinal wall. This model contains several subdomains that represent the three main layers of the intestinal wall: the mucosa, submucosa, and muscle layer. The Biot model of poroelasticity is used to set up the main equations. Fluid is added and removed from the system using the Starling-Landis and Drake-Laine models of microvascular and lymphatic flow. We compare the results of our model against the experimental results of the lab of Cox Jr. et. al. at the University of Texas-Housotn Medical School. The development of the model and the initial results are presented in the two publications listed below. Future work includes adding a domain to model the lumen, the main intestinal cavity. This cavity does not contain solid material, and thus we would model fluid flow in this domain with the Navier-Stokes equations. We would also like to include forces from muscle contraction. Lastly, this current model simulates a segment of intestine as edema forms. We know from experimental results that this excess fluid triggers some unkwon chain reaction of mechanical and biochemical events that leads to decreased muscle cell contracitility. We would like to create a multi-scale model of the intestine, from the organ level down to the muscle cell level to try to understand edema's effect on gastrointestinal smooth muscle contraction. Publications: J. Young, B. Riviere, C. S. Cox Jr. and K. Uray. A mathematical model of intestinal edema to explore mechanical triggers of ileus. Submitted to Biophysical Journal. August 2011. J. Young and B. Riviere. The development of a computational, poroelastic model of intestinal edema. Proceedings of the ECCOMAS Thematic International Conference on Simulation and Modeling of Biological Flows (SIMBIO 2011) September 21--23, 2011, VUB, Brussels, Belgium Movies: Top movie is the finite element domain as it deforms during edema formation. The three layers pictured are the mucosa (blue), submucosa (green), and muscle layer (red). This is cross-sectional slice of the intestinal tube. Bottom movie is of the same domain, but shows the pressure contours over time. The scale of the colorbar is in mmHg. Pressure builds up highest in the submucosa because it has the highest stiffness. The mucus deforms the most because it has the lowest stiffness and is also the region where most fluid is added during edema. A Particle-Based Model of the Cytoskeleton: The cytoskeleton is a complex network of cross-linked fibers that gives the animal cell mechanical support. The cytoskeleton is dynamic, in that it can break, rearrange, and reform cross-links in its structure to perform various tasks within the cell. A common method for modeling the cytoskeleton is as a network of cross-linked springs or beams (as mentioned in the above C-M model). I am currently working on a new, particle model of the cytoskeleton as an alternative to the spring model. A "particle" in the system is an actin fiber, and this point particle contains the following information about the fiber: center of mass position (x,y), velocity (u,v), orientation angle (z), and angular velocity (w). Cross-links are represented in this model by distance-based potential functions, meaning if two fibers are within a certain threshold distance (rthres) of each other then a virtual spring (cross-link) appears between them. A force of the form: K(rij-r0) will be applied to both fibers, where K is the cross-link spring constant, rij is the current distance between fiber i and j and r0 is the equilibrium length of a cross-link. The breaking and reforming of cross-links is easily accounted for in this model when rij becomes greater or less than rthres. An advantage of this model is that it is not necessary to store (in memory) which fiber is connected to which other fibers. The motion of this system of particles can be described by 6 ordinary differential equations per particle: 3 velocity equations for x, y, z and 3 Newton's law equations (F=ma) for u, v, w. The Newton's law equations contain terms representing forces from cytoplasm fluid drag, elastic forces from cross-links with neighboring fibers, and frictional forces from neighbors as well. Preliminary results show that this model exhibits many of the experimentally shown cytoskeletal behaviors such as strain hardening, network rupture and reformation, and energy transfer through the network. Movies: Top is a movie showing cytoskeletal rupture as forces are applied to the left and right sides of the network. The red dots are fiber centers and the blue lines are cross-links. You see that as the network is stretched out, many of the cross-links break. The next movie is of cytoskeletal reformation. Two independent patches of cytoskeleton are pushed towards each other. New cross-links (shown in green) begin to form between the two networks. The last movie depicts a set of fibers that are initially organized into columns. These columns are spaced just far enough apart so that there is no interaction between fibers in two different columns. The first column at the far left is then pushed over very close to the second column. This causes the fibers in the two columns to begin to interact, and repel each other since they are now TOO close together. The secondn column starts to move away from the first, and subsequently runs into the third column. The point of this movie is to show how an energy wave could propagate in a cytoskeletal network. The column set-up was chosen to make the motion of the wave clear. Continuum-Microscopic Modeling with Statistical Sampling Creating accurate, macroscopic scale models of microscopically heterogeneous media is computationally challenging. The difficulty is increased for materials with time-varying microstructures. Common modeling approaches for heterogeneous media include purely continuum-based models and homogenization methods. However, these methods tend to blur the inhomogeneities of the material that can influence local mechanical properties. Continuum-microscopic (CM) models are a class of methods that incorporate microscopic information into faster macroscopic models of the medium. CM methods have been used for heterogeneous media with static microstructures. This research project focuses on creating an extension of a basic CM algorithm to model heterogeneous media with time-varying microstructures. Fibrous media are chosen as a class of materials upon which to test the algorithm. Information from the material's microstructure is saved over time in the form of probability distribution functions (PDFs). These PDFs are then extrapolated forward in time to predict what the microstructure will look like in the future. Keeping track of the microstructure over time allows for accurate computation of the local mechanical parameters used in the continuum-level equations. The model was tested on a generic fibrous material with randomly oriented and crosslinked fibers. Results show that the mechanical parameters computed with this algorithm are similar to those computed with a fully-microscopic model. Errors for continuum level variables in the 5-10\% range are a good trade-off for the large savings in computational expense offered by this method. Publications: J. Young and S. Mitran. A Continuum-Microscopic Algorithm for Modeling Fibrous, Heterogeneous Media with Dynamic Microstructures. Multiscale Modeling and Simulation. Volume 9, Issue 1. (2011) p. 241-257. Blebbing Cell Project Animal cells are
composed of organelles, cytoplasm, a cytoskeleton and an encasing
plasma
membrane. A “bleb” is a balloon-like
protrusion of the plasma membrane that forms when the membrane
separates from
the underlying cytoskeletal network of actin filaments, and is pushed
outward
by flowing cytoplasm. Blebs are one of a
number of cell motility mechanisms and they also play a key role in
apoptosis
and mitosis. The
physics behind bleb formation is
not yet clearly understood. We propose a
mathematical model based on the following assumptions: Once
the
membrane
and
cytoskeleton
have
separated,
the
creation
of blebs is driven by pressure gradients in the
flowing
cytoplasm. As the bleb grows, actin monomers
are swept into the protrusion and begin to form a new actin cortex
within the bleb. The protrusion begins to retract when this new
actin mesh contracts, pulling the escaped membrane inward for
reattachment to the cytoskeleton.This two-dimensional
model
includes the motion of the actin filaments, the actin and myosin
monomer
concentration, the plasma membrane, the
cytoplasm,
and their interactions. The filaments
and membrane are modeled by elasticity equations while the cytoplasm is
modeled
by the Stokes equation. The protein concentrations
are modeled with an advection-diffusion equation. A
volume
constraint is also included in the model to maintain the overall cell
volume at
a constant value. These components of
the model interact with one another through external forces and
boundary
conditions.
Movies:
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INTESTINAL EDEMA PROJECT  Domain Deformationn during Edema Formation Mucosa (blue), Submucosa (green), Muscle (red)  Pressure Contours (mmHg) during Edema Formation CYTOSKELETON PARTICLE MODEL PROJECT  Cytoskeletal Rupture  Cytoskeletal Reformation  Energy Wave through a Network CONTINUUM-MICROSCOPIC MODELING PROJECT  Microscopic Model (left) and Continuum Model (right) BLEBBING CELL PROJECT  Cytoskeleton (thin lines) and membrane (thick line) as a bleb forms  Fluid velocity vectors (arrows) in the full cell during blebbing  Actin concentration contours in blebbing regin of cell  Pressure contours in blebbing region of cell |