40 research outputs found
Coupled Systems of Differential-Algebraic and Kinetic Equations with Application to the Mathematical Modelling of Muscle Tissue
We consider a coupled system composed of a linear differential-algebraic
equation (DAE) and a linear large-scale system of ordinary differential
equations where the latter stands for the dynamics of numerous identical
particles. Replacing the discrete particles by a kinetic equation for a
particle density, we obtain in the mean-field limit the new class of partially
kinetic systems. We investigate the influence of constraints on the kinetic
theory of those systems and present necessary adjustments.
We adapt the mean-field limit to the DAE model and show that index reduction
and the mean-field limit commute. As a main result, we prove Dobrushin's
stability estimate for linear systems. The estimate implies convergence of the
mean-field limit and provides a rigorous link between the particle dynamics and
their kinetic description.
Our research is inspired by mathematical models for muscle tissue where the
macroscopic behaviour is governed by the equations of continuum mechanics,
often discretised by the finite element method, and the microscopic muscle
contraction process is described by Huxley's sliding filament theory. The
latter represents a kinetic equation that characterises the state of the
actin-myosin bindings in the muscle filaments. Linear partially kinetic systems
are a simplified version of such models, with focus on the constraints.Comment: 32 pages, 18 figure
Diffusion of MMPs on the Surface of Collagen Fibrils: The Mobile Cell Surface – Collagen Substratum Interface
Remodeling of the extracellular matrix catalyzed by MMPs is central to morphogenetic phenomena during development and wound healing as well as in numerous pathologic conditions such as fibrosis and cancer. We have previously demonstrated that secreted MMP-2 is tethered to the cell surface and activated by MT1-MMP/TIMP-2-dependent mechanism. The resulting cell-surface collagenolytic complex (MT1-MMP)2/TIMP-2/MMP-2 can initiate (MT1-MMP) and complete (MMP-2) degradation of an underlying collagen fibril. The following question remained: What is the mechanism of substrate recognition involving the two structures of relatively restricted mobility, the cell surface enzymatic complex and a collagen fibril embedded in the ECM? Here we demonstrate that all the components of the complex are capable of processive movement on a surface of the collagen fibril. The mechanism of MT1-MMP movement is a biased diffusion with the bias component dependent on the proteolysis of its substrate, not adenosine triphosphate (ATP) hydrolysis. It is similar to that of the MMP-1 Brownian ratchet we described earlier. In addition, both MMP-2 and MMP-9 as well as their respective complexes with TIMP-1 and -2 are capable of Brownian diffusion on the surface of native collagen fibrils without noticeable dissociation while the dimerization of MMP-9 renders the enzyme immobile. Most instructive is the finding that the inactivation of the enzymatic activity of MT1-MMP has a detectable negative effect on the cell force developed in miniaturized 3D tissue constructs. We propose that the collagenolytic complex (MT1-MMP)2/TIMP-2/MMP-2 represents a Mobile Cell Surface – Collagen Substratum Interface. The biological implications of MT1-MMP acting as a molecular ratchet tethered to the cell surface in complex with MMP-2 suggest a new mechanism for the role of spatially regulated peri-cellular proteolysis in cell-matrix interactions
Development of a mathematical model for predicting electrically elicited quadriceps femoris muscle forces during isovelocity knee joint motion
<p>Abstract</p> <p>Background</p> <p>Direct electrical activation of skeletal muscles of patients with upper motor neuron lesions can restore functional movements, such as standing or walking. Because responses to electrical stimulation are highly nonlinear and time varying, accurate control of muscles to produce functional movements is very difficult. Accurate and predictive mathematical models can facilitate the design of stimulation patterns and control strategies that will produce the desired force and motion. In the present study, we build upon our previous isometric model to capture the effects of constant angular velocity on the forces produced during electrically elicited concentric contractions of healthy human quadriceps femoris muscle. Modelling the isovelocity condition is important because it will enable us to understand how our model behaves under the relatively simple condition of constant velocity and will enable us to better understand the interactions of muscle length, limb velocity, and stimulation pattern on the force produced by the muscle.</p> <p>Methods</p> <p>An additional term was introduced into our previous isometric model to predict the force responses during constant velocity limb motion. Ten healthy subjects were recruited for the study. Using a KinCom dynamometer, isometric and isovelocity force data were collected from the human quadriceps femoris muscle in response to a wide range of stimulation frequencies and patterns. % error, linear regression trend lines, and paired t-tests were used to test how well the model predicted the experimental forces. In addition, sensitivity analysis was performed using Fourier Amplitude Sensitivity Test to obtain a measure of the sensitivity of our model's output to changes in model parameters.</p> <p>Results</p> <p>Percentage RMS errors between modelled and experimental forces determined for each subject at each stimulation pattern and velocity showed that the errors were in general less than 20%. The coefficients of determination between the measured and predicted forces show that the model accounted for ~86% and ~85% of the variances in the measured force-time integrals and peak forces, respectively.</p> <p>Conclusion</p> <p>The range of predictive abilities of the isovelocity model in response to changes in muscle length, velocity, and stimulation frequency for each individual make it ideal for dynamic applications like FES cycling.</p
Measurement of contractile forces generated by individual fibroblasts on self-standing fiber scaffolds
# The Author(s) 2010. This article is published with open access at Springerlink.com Abstract Contractility of cells in wound site is important to understand pathological wound healing and develop therapeutic strategies. In particular, contractile force generated by cells is a basic element for designing artificial three-dimensional cell culture scaffolds. Direct assessment of deformation of three-dimensional structured materials has been used to calculate contractile forces by averaging total forces with respect to the cell population number. However, macroscopic methods have offered only lower bounds of contractility due to experimental assumptions and the large variance of the spatial and temporal cell response. In the present study, cell contractility was examined microscopically in order to measure contractile forces generated by individual cells on self-standing fiber scaffolds that were fabricated via femtosecond laser-induced two-photon polymerization. Experimental assumptions and calculation errors that arose in previous studies of macroscopic and microscopic contractile force measurements could be reduced by adopting a columnar buckling model on individual, standing fiber scaffolds. Via quantifying eccentric critical loads for the buckling of fibers with various diameters, contractile forces of single cells were calculated in the range between 30–116 nN. In the present study, a force magnitude of approximately 200 nN is suggested as upper bound of the contractile force exerted by single cells. In addition, contractile forces by multiple cells on a single fiber were calculated in the range between 241–709 nN