Research Papers

Biokinetic Mechanisms Linked With Musculoskeletal Health Disparities: Stochastic Models Applying Tikhonov’s Theorem to Biomolecule Homeostasis

[+] Author and Article Information
Asit K. Saha

Center for Allaying Health Disparities through Research and Education (CADRE), Department of Mathematics & Computer Science, Central State University, Wilberforce, OH 45384asaha@centralstate.edu

Yu Liang

Center for Allaying Health Disparities through Research and Education (CADRE), Department of Mathematics & Computer Science, Central State University, Wilberforce, OH 45384yliang@centralstate.edu

Sean S. Kohles1

Regenerative Bioengineering Laboratory, Department of Mechanical & Materials Engineering, Portland State University, Portland, OR 97207; Department of Surgery, Oregon Health & Science University, Portland, OR 97239kohles@cecs.pdx.edu


Corresponding author.

J. Nanotechnol. Eng. Med 2(2), 021004 (May 13, 2011) (9 pages) doi:10.1115/1.4003876 History: Received March 13, 2011; Revised March 23, 2011; Published May 13, 2011; Online May 13, 2011

Multiscale technology and advanced mathematical models have been developed to control and characterize physicochemical interactions, respectively, enhancing cellular and molecular engineering progress. Ongoing tissue engineering development studies have provided experimental input for biokinetic models examining the influence of static or dynamic mechanical stimuli (Saha, A. K., and Kohles, S. S., 2010, “A Distinct Catabolic to Anabolic Threshold Due to Single-Cell Nanomechanical Stimulation in a Cartilage Biokinetics Model,” J. Nanotechnol. Eng. Med., 1(3) p. 031005; 2010, “Periodic Nanomechanical Stimulation in a Biokinetics Model Identifying Anabolic and Catabolic Pathways Associated With Cartilage Matrix Homeostasis,” J. Nanotechnol. Eng. Med., 1(4), p. 041001). In the current study, molecular regulatory thresholds associated with specific disease disparities are further examined through applications of stochastic mechanical stimuli. The results indicate that chondrocyte bioregulation initiates the catabolic pathway as a secondary response to control anabolic processes. In addition, high magnitude loading produced as a result of stochastic input creates a destabilized balance in homeostasis. This latter modeled result may be reflective of an injurious state or disease progression. These mathematical constructs provide a framework for single-cell mechanotransduction and may characterize transitions between healthy and disease states.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 3

Phase-plane diagrams demonstrating the interactive dynamics of cytokines and growth factors. The stochastic function defines varying mechanical load magnitudes as the applied stimulus. A harmonic balance with a clockwise rotating limit cycle results from relative loads of (a) 0.001 with centering at the steady state value of (C¯, G¯)T≈(18.03, 20.95)T, (b) 0.01 with centering at the steady state value of (C¯, G¯)T≈(17.49, 20.37)T, and (c) 0.1 with centering at the steady state value of (C¯, G¯)T≈(16.51, 19.62)T. All concentration units are in dimensionless form.

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Figure 2

The pivotal region of the threshold parameters Ω1 and Ω2 are shown in the shaded area bounded by the upper bound as defined by the lower left intercept=(15.489,19.161) and the upper right intercept=(16.489,21.161), where the lower bound is (C0,G0)=(16.489,19.161)(7). These relationships satisfy that described in Eq. 14.

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Figure 1

The schematic diagram of the growth factor-cytokine interaction network as they influence the structural molecules, collagen, and proteoglycans, within the ECM. The three different planes represent the three different time scales.

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Figure 4

The phase-plane diagram indicating a disruption in the cyclic nature of the cytokine and growth factor dynamics due to relative loads greater than 0.5 as produced by the stochastic function. The nonoscillatory steady state now has a trajectory toward a singular value at (C¯, G¯)T≈(15.04, 19.73)T.

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Figure 5

A Monte Carlo computation of the standard deviations associated with cytokine and growth factor concentrations at different dimensionless random load levels. The standard deviations appear to overlap at ∼5 relative load units. The critical transition zone is identified at 1.0≤load≤5.0 (see inset). Persistence in the stability of the oscillatory behavior of the anabolic and catabolic pathways toward homeostasis is possible only when random loading is less than unity. Random loading greater than 5.0 appears to help destabilize the oscillatory nature of the system, thus enhancing the growth factors to promote anabolic pathways (synthesis) over catabolism (degradation), a biologically relevant transition for tissue remodelling.

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Figure 6

As stochastic loading increases to over 10 relative units, the harmonic balance between cytokine and growth factor activities is not visible, rather a chaotic nature is observed with a lower concentration of cytokines and a higher concentration of growth factors. The steady state value, calculated from 50,000 sample points, is given by (C¯, G¯)T≈(5.23, 51.44)T.

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Figure 7

As the stochastic loading is increase above 60 relative units, a chaotic nature in the phase-plant diagram is much more aggressive with the steady state value defined at (C¯, G¯)T≈(3.91, 276.70)T.

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Figure 8

The mean dimensionless accumulation of GAG and collagen molecules within ECM as the dimensionless random mechanical loading increases from 0.1 to 65. A numerical transition in accumulation is indicated at >5.0 relative units.



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