Biomedical Engineering Reference
In-Depth Information
To date, the two primary vascular applications of the continuum approach have
been to compute pressure and velocity fields in blood flow (i.e., hemodynamics)
and to compute stress and strain fields within the vascular wall (i.e., wall
mechanics), each of which requires explicit solution of mass and linear momentum
balance. Although cells cannot sense continuum metrics such as stress and strain,
these quantities have proven useful in correlating mechanobiological responses by
cells to diverse loads [
31
]. For example, simple parallel plate flow experiments
demonstrate that ECs are very responsive to changes in wall shear stress, which is
calculated using the continuum approach; simple organ culture experiments on
straight segments of arteries and arterioles demonstrate that SMCs are very
responsive to changes in pressure and extension, which induce intramural changes
in stress and strain. Given the complexity of the microstructure of cells and tissues
down to the level of molecular interactions, it is inconceivable that one would
attempt to use a purely molecular dynamics simulation to study problems that
manifest at a physiological or clinical scale. That is, continuum biomechanics is
much more appropriate to study problems involving, for example, changes in the
structural stiffness of the arterial wall in hypertension, the effects of evolving
vascular diseases such as atherosclerosis or aneurysms, or the design of novel
interventional devices such as intravascular stents, heart valves, or left ventricular
assist devices.
Notwithstanding past successes, until recently continuum biomechanics had
focused primarily on material behaviors at a particular time, not how they evolve.
Moreover, most studies had assumed that the tissue (or cell) is materially uniform.
Yet, all tissues are materially non-uniform, consisting of different types of cells
and matrix that turnover, and so too cells consist of different organelles and
cytoskeletal proteins that change over time. In an attempt to address these com-
plexities, Humphrey and Rajagopal [
33
] proposed a Constrained Mixture Model
(CMM) that allows one to track evolving changes in the properties, turnover rates,
and natural (i.e., stress-free) configurations of individual structural constituents
that comprise a tissue or cell. Computations have shown that this approach can
capture salient features of diverse vascular adaptations and disease processes (cf.
[
6
,
61
]). Briefly, a CMM of arterial growth and remodeling consists of full mixture
equations for mass balance plus a single equation for overall linear momentum
balance that is solved for the net stress field. The associated two classes of con-
stituents are: structurally insignificant but soluble constituents, such as vasoactive
molecules, growth factors, cytokines, and proteases, and structurally significant
but insoluble constituents, such as elastin, collagen, and muscle. Linear momen-
tum balance is solved via a rule-of-mixtures constitutive relation for the struc-
turally significant constituents. The need for but a single linear momentum
equation stems from the assumption that negligible momentum exchanges exist
between structurally significant constituents, which appear to deform together with
the mixture. Because inertial loads are often negligible in the calculation of arterial
wall stresses, even during transient loading, we assume further that structurally
significant constituents experience quasi-static loading. The primary constitutive
relations thus reduce to equations for the production and removal of structurally
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