Abstract
Blood vessels exhibit a remarkable ability to adapt throughout life that depends upon genetic programming and well-orchestrated biochemical processes. Findings over the past four decades demonstrate, however, that the mechanical environment experienced by these vessels similarly plays a critical role in governing their adaptive responses. This article briefly reviews, as illustrative examples, six cases of tissue level growth and remodeling, and then reviews general observations at cell-matrix, cellular, and sub-cellular levels, which collectively point to the existence of a “mechanical homeostasis” across multiple length and time scales that is mediated primarily by endothelial cells, vascular smooth muscle cells, and fibroblasts. In particular, responses to altered blood flow, blood pressure, and axial extension, disease processes such as cerebral aneurysms and vasospasm, and diverse experimental manipulations and clinical treatments suggest that arteries seek to maintain constant a preferred (homeostatic) mechanical state. Experiments on isolated microvessels, cell-seeded collagen gels, and adherent cells isolated in culture suggest that vascular cells and sub-cellular structures such as stress fibers and focal adhesions likewise seek to maintain constant a preferred mechanical state. Although much is known about mechanical homeostasis in the vasculature, there remains a pressing need for more quantitative data that will enable the formulation of an integrative mathematical theory that describes and eventually predicts vascular adaptations in response to diverse stimuli. Such a theory promises to deepen our understanding of vascular biology as well as to enable the design of improved clinical interventions and implantable medical devices.
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Notes
Residual stresses can be visualized by inverting a short segment of elastomeric tubing; the resulting inner part is under compression and the outer part is under tension, even though the segment is not acted on by external loads.
Prestress can be appreciated by holding a rubber band taut (prestress) before loading it transversely.
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Acknowledgments
This work was supported, in part, by grants from the NIH (HL-64372, HL-76319, HL-80415). I would also like to acknowledge general contributions to this work by former and current students, S. Baek, R. Gleason, S. Na, and A. Valentin, and excellent artwork by Nathalia de la Hoz.
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Appendix
Appendix
For details on mathematical modeling relevant to continuum biomechanics, the interested reader is referred to Holzapfel [147], Humphrey [18], and Cowin and Doty [47] for both an introduction and an extended list of references. Here, however, let us note a few basics. Whereas the 1D Hooke’s law for linearly elastic behavior can be written as σ = Eε, a generalized 3D Hooke’s law can be written as σ ij = C ijkl ε kl (using the Einstein summation convention, whereby repeated indices are summed 1 to 3 unless otherwise indicated and free indices can take values from 1 to 3, thereby yielding the different components of stress, stiffness, and strain). Of course, σ ij ≡σ ji (i≠j) by angular momentum balance (one of the two basic postulates that restricts constitutive relations). Recalling that material stiffness is obtained via ∂σ ij /∂ε kl , C ijkl represents 81 components of the “stiffness matrix,” in which 21 components are non-zero and independent for general anisotropy, with 9, 5, and 2 components representing orthotropy (three preferred directions), transverse isotropy (a single preferred direction), and isotropy (i.e., the same behavior in all directions, which can be accounted for via a single Young’s modulus and Poisson’s ratio). Note, too, that the entropy inequality (the other basic postulate that restricts constitutive relations) implies the existence of a “stored energy function” for elastic processes; it can be shown for Hooke’s law that this energy is given by W = 0.5C ijkl ε ij ε kl , whereby σ ij = ∂W/∂ε ij (with C ijkl = C klij by general symmetry arguments) in this hyperelastic approach. As noted above, although the linearized theory of elasticity (both a linearized stress–stress relation and a linearized strain measure) is mathematically elegant, it has long been believed [57] and continually confirmed [44] that it is not appropriate for most problems of vascular mechanics.
The reason for noting these few results from linear elasticity is to emphasize the need to generalize stress–strain relations to 3D (cf. Fig. 5), to account for anisotropy in behavior (e.g., behaviors differ in radial, circumferential, and axial directions in arteries; [18, 44]), and to emphasize that the entropy inequality restricts the form of the constitutive relation in terms of a stored energy function W. As it turns out, a stored energy function can play a basic role in extending elastic descriptors to viscoelastic (cf. [199]) and growth and remodeling [34] models. Whereas the linearized strain is determined from the deformation gradient tensor F (the fundamental measure of motion, including deformation and rigid body rotations) via ε = 0.5(F + F T−2I), where I is the identity tensor, an exact nonlinear measure of strain is the Green strain E = 0.5(F T · F−I). It is easy to show, therefore, that the linearized measure is sensitive to rigid body motions, whereas the nonlinear measure is properly insensitive to rigid body motions (because F = R · U, where R is rigid body motion and U describes deformation). It is also easy to show in a simple 1D case that the deformation gradient tensor reduces to a single component of stretch, namely F → λ, that the 1D ε = λ − 1 and E = 0.5(λ 2 − 1). Of course, this E can also be written as E = 0.5(λ + 1)(λ − 1), whereby for small strains, λ∼1 and thus E = 0.5(∼2)(λ − 1) ∼ λ − 1, or ε∼E, hence justifying the use of the linearized measure in cases of small deformations (F ∼ I). Recall, however, that the axial prestretch alone often ranges from 1.2 (aortic arch) to 1.7 (rat carotid), and the prestretch of cytoskeletal proteins appears to be ∼1.2 even in the basal state. The linearized strain introduces a 9% error at a stretch of 1.2 and larger errors at higher stretches.
Finally, analogous to the result from the entropy inequality for Hooke’s law, a general stress–strain relation for nonlinear material behavior and finite strains is
where we note that the Cauchy stress tensor σ is defined independent of the specific material behavior or the range of strains and is thus common to both theories. Of course, in the case of small strains, F ∼ I (e.g., F iA ∼δ iA , the Kronecker delta) and ε ∼ E, whereby the linear elastic result can be recovered as a special case. Again, however, the assumption of small strains is not appropriate for blood vessels and the material behavior is nonlinear (cf. Eq. 1), hence necessitating the use of the nonlinear theory. Although fewer exact solutions are available in the nonlinear theory (cf. [200]) than in the linear theory (cf. [53]), once one is familiar with the nonlinear theory it is conceptually as simple as the linear theory and yet not subject to the same limitations.
Finally, as noted above, given an appropriate nonlinearly elastic framework, extension to nonlinearly viscoelastic and general growth and remodeling mechanics can be straightforward. For example, for nonlinear viscoelasticity we can write ([199])
where G(t − τ) is a stress relaxation (viscoelastic) function. Finally, for a growth and remodeling theory, wherein individual constituents can be produced and removed at different rates as well as have different stiffnesses, orientations, and stress-free states, we can write [34]
where the superscript k (= 1,2,...,n) denotes the kth structurally significant constituent in the tissue or cell. Employing a rule of mixtures, we have suggested that
Here, ρ k(t) is the mass density of constituent k, ρ = ∑ρ k(t) is the overall mass density of the tissue or cell (often constant), Q k(t) ∈ [0,1] is a non-dimensional function that reveals what fraction of the material produced at and before time t = 0 survives to time t, with Q k(0)≡1, and \( \hat{W}^{k} \) is the energy stored elastically in constituent k, which depends on \( {\mathbf{E}}^{k}_{{n(\tau )}} \), the Green strain tensor associated with each constituent relative to its individual natural configuration n(τ). To account for the production and loss of individual constituents at different times, m k(τ) is a mass density production and q k(t − τ) is an associated survival function that ensures that a constituent contributes to load bearing only until it is degraded, which because of the convolution integral is reminiscent of viscoelasticity, wherein a constituent contributes more the more recently it was produced. Although this theory is a straightforward extension of classical ideas, the primary challenge is to determine appropriate functional forms for the mass density production, survival function, and stored energy for each constituent, with production and removal depending on mechanotransduction mechanisms. Again, therefore, there is a clear need for more data, particularly on the time course of cell and matrix turnover.
Given the generality of the constrained mixture approach, it is useful to consider a few specific cases to build intuition. If the altered loading after time t = 0 is constant, caused only by a step change in the current configuration, then using exponential decays for the survival functions recovers simple cases as considered in Humphrey [201] and Gleason et al. [82]. Finally, note that at t = 0, prior to a perturbation in loading that elicits a growth and remodeling response, one recovers a classical rule of mixtures relation for the stored energy \( W = {\sum {\phi ^{k} \hat{W}^{k} } } \) (with \( \phi ^{k} = \rho ^{k} /\rho \) the usual mass fractions), which yields the classical relation for stress, which in turn yields the classical single constituent relation when k = 1 (even Hooke’s law if so desired). In other words, this framework recovers multiple special cases as it should. See Humphrey [40] for more on a theory of growth and remodeling for tissues and cells, including a discussion of parallel reaction–diffusion equations for non-structurally significant constituents in the mixture, knowing of course that such modeling is in its infancy and will require continuing advances as more data become available. The main point here is that “the general framework of continuum mechanics is sufficiently broad” [177] to incorporate many of the complexities that arise in cell biology, thus we should not restrict ourselves to often more familiar linearized theories for materially uniform bodies or simple mechanical analog models unless more sophisticated approaches suggest that such simplifications capture the salient features.
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Humphrey, J.D. Vascular Adaptation and Mechanical Homeostasis at Tissue, Cellular, and Sub-cellular Levels. Cell Biochem Biophys 50, 53–78 (2008). https://doi.org/10.1007/s12013-007-9002-3
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DOI: https://doi.org/10.1007/s12013-007-9002-3