Research Papers

A Mean-field Model of Ventricular Muscle Tissue

[+] Author and Article Information
Jagir R. Hussan1

Mark L. Trew

Peter J. Hunter

Auckland Bioengineering Institute,  University of Auckland, Auckland 1010, New Zealandp.hunter@auckland.ac.nz


Corresponding author.

J Biomech Eng 134(7), 071003 (Jul 09, 2012) (13 pages) doi:10.1115/1.4006850 History: Received March 26, 2012; Revised April 03, 2012; Posted May 18, 2012; Published July 09, 2012; Online July 09, 2012

A theoretical model of the cross-linking topology of ventricular muscle tissue is developed. Using parameter estimation the terms of the theoretical model are estimated for normal and pathological conditions. The model represents the anisotropic structure of the tissue, reproduces published experimental data and characterizes the role of different tissue components in the observed macroscopic behavior. Changes in the material parameters are consistent with expected structural changes and the model is extended to reproduce force-Calcium relationships. Model results are invoked to argue that semisoft behavior and the material axis anisotropy arise from the constraints on the extracellular matrix cross-linking topology.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 2

Organization, components and simplification of ventricular muscle tissue. Myofibers are strongly bound along a planar direction and form a sheet like structure. (a), Myocytes (M) are bound together by collagen to form myofibrils; they also contain supporting structures like capillaries (C). (b), Schematic of a segment (size of a half sarcomere ~1.4 μm) of the myocyte. The sarcomere (and to some extent the cytoskeleton) are active force producing units. Force production is regulated by proteins and ions (like Ca) through intricate regulatory pathways. (c), A stack of horizontally laid sheet structures with myofibers branching across sheets. (d), Schematic of homogenization of myocardial tissue. Myofibrils are discretized into elastically homogeneous cylindrical units. These units are tightly bound by collagen. (e), ECM collagen is homogenized to be described by p-chain units that mesh together myofibrils and sheets. (f), Schematic of local myocyte-p-chain interaction. The domain’s deformation mechanics with respect to the myocyte axis (solid arrow) is described using the p-chain step-length parallel ℓ∥ and perpendicular ℓ⊥ to the myocyte axis. (g), Myofiber degrees of freedom- splay, twist and bend. Arrows indicate the direction of freedom. (h), Myofiber orientation estimated by eigen-analysis of structure tensors at varying frequencies of the width of the domain of influence in rat ventricular tissue [11]. Smoothing to lower frequency gradually eliminates the fiber splay and the helical orientation dominates Fig. (a) reproduced from Fig. 7 of LeGrice [3] Fig. (c) reproduced from Fig. 3 of Zhao [34] Fig. (h) reproduced from Fig. S1 of Rutherford [11].

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Figure 3

Schematic of myocyte director, the principal axis along which myofibrils are correlated. (a) and (b) the helical orientation of the fibers (cylinders) across a portion of the ventricular wall and the coordinate system attached to the director, n, used to describe the myocyte director at a spatial position in the tissue. (a), Representation of the undeformed configuration, Ω0, of myofibrils. (b), Representation of the deformed (contracted) configuration, Ω, of myofibrils. The tissue block lengths between undeformed (l) and deformed (L) configurations satisfy the relationship l<L. The expansion/contraction of the myofibril in the plane perpendicular to the fiber axis is modeled as a change in angle ψ. (c), Schematic of ψ for the undeformed myocyte. (d), Schematic of ψ for the contracted myocyte.

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Figure 4

Shear experimental data. (a), Schematic of the deformation modes and their alignment with material axes. (b), Shear strain-energy profile of ventricular myocardial tissue, determined from the experimental data of Dokos [15]. (c), Strain-Energy profile for equibiaxial fiber (FF) and sheet (SS) extensions, determined from biaxial experimental results reported in Smaill [16] Fig. (a) reproduced from Fig. 4 of Holzapfel [35].

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Figure 5

Model versus experimental data comparisons. (a), Difference between model predicted elastic energy and experiment. (b), Difference between model and equibiaxial fiber and sheet extensions energies. (c), Difference between Costa Law (constitutive relationship) elastic energy and experiment

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Figure 6

Change in geometric parameters for each deformation mode. (a), φ variation for six shear modes. (b), φ variation for equibiaxial fiber and sheet extensions. (c), ψ variation for six shear modes

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Figure 7

Characteristic elements and tissue organization in ventricular edema. (a), Schematic diagram shows myocytes connected by collagen struts, large perimysial collagen fibers aligned with the myocytes, and proteoglycans associated with the collagen. (b), Under passive uniaxial stretch, the perimysial collagen fibers uncoil and once straightened resists further extension, protecting myocytes from overstretch. (c), Loss of hydration balance leads to mechanical changes in tissue behavior (termed edema). Here, increased hydration of the proteoglycans prestresses the collagen network causing a differential behavior (Note the change in parameter ℓ⊥ between (b) and (c)). (d), Prestressed collagen network resists deformation at much lower strains, shifting the uniaxial stress strain curve to the left. Reproduced from Fig. 1 of Fomovsky [25].

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Figure 8

Results of fitting the model to papillary tissue extension data. The profile for edema set P3 is plotted against the secondary y-axis. (a), Published strain-energy profile for control and progressively edematic (P 1→ P2 → P3) tissue. (b), Difference between model versus published data.

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Figure 9

Force-Calcium relationships. (a), Experimentally measured F-pCa relationship in rat cardiac-muscle tissue(skinned). The sarcomere length was maintained at 2.05μm. The horizontal-axis is the log of activator [Ca] in μM, the vertical-axis is the normalized force. The mean measured force is plotted as diamonds, the error bars show the standard error. The solid line is the Hill function fit. (b), The F-pCa profile predicted by the proposed model (FF mode) using Eq. 18 to model ψ variation (solid line) as function of [Ca], the normalized predicted force is given by the broken line. The FF mode is the most similar setup to the experiment. (c), Overlap of Hill function fit in (a) (solid line), with model predicted Force (broken line) for the range of [Ca] values. Fig. (a) reproduced from Fig. 1 of Rice [36].

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Figure 1

Schematic of soft deformation due to myocyte contraction (cylinder). Fixed polymer chain cross-linking is represented by the shaded rectangle. Energy is not stored (a) and (b) until the myocyte configuration deforms the cross-linking (c), increasing strain energy and producing observed elastic behavior.




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