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Research Papers

Parameter Sensitivity Study of a Constrained Mixture Model of Arterial Growth and Remodeling

[+] Author and Article Information
A. Valentín

Department of Biomedical Engineering, Texas A&M University, 337 Zachry Engineering Center, 3120 TAMU, College Station, TX 77843-3120

J. D. Humphrey1

Department of Biomedical Engineering, Texas A&M University, 337 Zachry Engineering Center, 3120 TAMU, College Station, TX 77843-3120jhumphrey@tamu.edu

1

Corresponding author.

J Biomech Eng 131(10), 101006 (Sep 02, 2009) (11 pages) doi:10.1115/1.3192144 History: Received December 08, 2008; Revised May 25, 2009; Published September 02, 2009

Computational models of arterial growth and remodeling promise to increase our understanding of basic biological processes, such as development, tissue maintenance, and aging, the biomechanics of functional adaptation, the progression and treatment of disease, responses to injuries, and even the design of improved replacement vessels and implanted medical devices. Ensuring reliability of and confidence in such models requires appropriate attention to verification and validation, including parameter sensitivity studies. In this paper, we classify different types of parameters within a constrained mixture model of arterial growth and remodeling; we then evaluate the sensitivity of model predictions to parameter values that are not known directly from experiments for cases of modest sustained alterations in blood flow and pressure as well as increased axial extension. Particular attention is directed toward complementary roles of smooth muscle vasoactivity and matrix turnover, with an emphasis on mechanosensitive changes in the rates of turnover of intramural fibrillar collagen and smooth muscle in maturity. It is shown that vasoactive changes influence the rapid change in caliber that is needed to maintain wall shear stress near its homeostatic level and the longer term changes in wall thickness that are needed to maintain circumferential wall stress near its homeostatic target. Moreover, it is shown that competing effects of intramural and wall shear stress-regulated rates of turnover can develop complex coupled responses. Finally, results demonstrate that the sensitivity to parameter values depends upon the type of perturbation from normalcy, with changes in axial stretch being most sensitive consistent with empirical reports.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic representation of the iterative process of the modern scientific method, as applied to G&R biomechanics. Computational models enable time- and cost-efficient simulations that can both serve an important role in the refinement of hypotheses/theories and motivation of experiments and their design. For example, simulations enable one to evaluate competing hypotheses, thereby focusing the experimental need.

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Figure 2

Active stress-stretch muscle responses for indicated basal values of constrictor to dilator ratio CB at time s=0 (cf. Eqs. 6,7). All other parameters are as listed in Table 2. Each curve represents a functionally different artery. The abscissa “normalized muscle fiber stretch” is expressed as a range of values for λθm(act)(0) because no G&R takes place. Note the inverse parabolic behavior (cf. Ref. 17).

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Figure 3

Active muscle responses for indicated basal values of constrictor to dilator ratio CB. All other parameters are as listed in Table 2. Increasing values for CB result in increasing vasoactive responsiveness to changes in τw. Note the near sigmoidal behavior.

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Figure 4

Normalized target inner radius (panel a) and active muscle responses (panel b) as functions of changing constrictor to dilator ratio C (shown here as the percentage of ΔC=C−CB) for indicated values of CS. Flowrate Q is constant at the homeostatic value. Higher values of CS allow the artery to better maintain the target inner radius over a wider range of C. Lower values of CS cause the artery to generate peak active stresses for smaller changes in C. By increasing CS, the artery is able to accommodate larger changes in constrictor concentration (see Eq. 6). Note the generally sigmoidal behavior (cf. Ref. 17).

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Figure 5

Comparison of intramural and wall shear stress regulation of circumferential collagen production rates. Values given as functions of changes in the scalar measure of stress borne by circumferential collagen (panels a, c, and e) and changes in constrictor to dilator ratio C (panels b, d, and f) for G&R in response to a sustained 30% decrease in flow from days 0 to 30 (solid) and from days 30 to 1000 (dashed). Arrows indicate advancing time. Note the direct relationship between changes in σ and mass production when Kσk=1 and Kτwk=0 (panel a) and the direct relationship between changes in C and mass production when Kσk=0 and Kτwk=1 (panel d), each linear as postulated separately (Table 1). In contrast, note the biphasic progression of G&R when Kσk=Kτwk=1 (panels e and f), that is, when production rates depend on changes in both intramural and wall shear stresses, which reveals potentially competing effects.

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Figure 6

Comparison of intramural and wall shear stress regulation of axial collagen production rates. Values given as functions of changes in the scalar measure of stress borne by circumferential collagen (panels a, c, and e) and changes in constrictor to dilator ratio C (panels b, d, and f) for G&R in response to a sustained 2% increase in axial length from days 0 to 40 (solid) and from days 40 to 1000 (dashed). Arrows indicate advancing time. Note the biphasic progression when Kσk=Kτwk=1 (panels e and f). From days 0 to 40, mass production is inversely related to σ while directly related to C. After day 40, mass production is directly related to σ while inversely related to C.

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Figure 7

Percent changes in inner radius (panel a), thickness (panel b), unloaded inner radius (panel c), and unloaded length (panel d) for a sustained 50% increase in pressure; results shown at days 1, 7, 14, and 100 of G&R with the arrows denoting advancing time. The model predicts small changes in inner radius as the vessel restores τw toward τwh. Wall thickness increased by 50% as it should, with higher values of Kik accelerating the process. Unloaded length increases with increased collagen deposition, thus resisting the recoiling effects of elastin. Note the near singular behavior at (Kσk,Kτwk)=(0,0), consistent with prior findings that such values are unrealistic biologically (6).

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Figure 9

Percent changes in inner radius (panel a), thickness (panel b), unloaded inner radius (panel c), and unloaded length (panel d) for a sustained 2% increase in axial length; results shown at days 1, 7, 14, and 100 of G&R with the arrows denoting advancing time. As in the case of increasing pressure, the inner radius remains nearly constant, given the constant flow. Unloaded length increases, as expected, due to increasing deposition of axial collagen at its preferred value.

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Figure 8

Percent changes in inner radius (panel a), thickness (panel b), unloaded inner radius (panel c), and unloaded length (panel d) for a sustained 30% decrease in flow; results shown at days 1, 30, 100, and 1000 of G&R with the arrows denoting advancing time. The inner radius ultimately decreases by the predicted amount (0.71/3=0.88(79)) as it should. Note the large changes in unloaded inner radius as the artery remodels around its new constricted state. Unloaded axial length decreases as the artery’s collagen to elastin ratio decreases and the thickness decreases, thereby allowing the elastin to retract the artery further when unloaded.

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