Biomedical Engineering Reference
In-Depth Information
As discussed in [19] and [15], the stress rate experienced by the constituents might
also affect their mechanical behaviour. This effect can be modelled by letting the ma-
terial parameters
c
i
, involved in the definition of the strain energy densities
W
k
,be
either functions of instantaneous values of suitable stress-rate measures or function-
als of the time history of such measures over the G&R time interval
[
τ
,
s
]
(assuming
that the constituent was deposited at time
τ
), that is
c
i
or
c
i
c
i
(
k
c
i
(
τ
,
k
s
)=
σ
(
s
)
s
)=
σ
([
τ
,
s
])
.
(7.26)
These functional dependencies take into account, for example, the possibility that
the stiffness of the newly deposited fibres (i.e., collagen) can depend on the current
stress rate of the tissue and that some constituents (i.e., elastin) might manifest early
aging or fatigue/damage induced by severe increase in the stress rate over the cardiac
cycle. These might be regarded as remodelling phenomena driven by the stress rate
in the vessel and thus, ultimately, by the hemodynamics.
Constitutive assumptions (7.22)-(7.26) introduce the dependence of G&R, on
time scale
s
, on the rate of phenomena happening on the cardiac cycle time scale
t
, explaining the coupling effect depicted as an upward arrow on the far right side of
Fig. 7.3. Conversely, the hemodynamics on the cardiac cycle time scale
t
depends on
the wall geometry and material properties at time
s
and this explains the downward
arrow in Fig. 7.3. Therefore the two time scales are fully coupled in this multiscale
formulation of the G&R problem.
7.3 Results
Thin-walled arterial models of the class described in Sect. 7.2.1 have been used
to describe arterial adaptation to several diseases or hemodynamic perturbations.
In [5, 21] the time course of a cerebral vasospasm was successfully described by
a thin-wall model including endothelial damage and recovery. The model suggests
that chemical and mechanical mediators of cellular and extracellular matrix turnover
can differentially dominate the progression and resolution of vasospasm. Arterial
adaptation to hypertension and perturbed blood flow rate were modelled in [37].
The model is able to predict arterial wall stiffening in hypertensive human basilar
arteries and highlighted the complementary effects of active contraction and matrix
turnover in response to severe perturbations in the mean blood flow rate. Adaptation
to changes in the axial stretch was considered in [38]; the model predicts a marked
increase in fibrous constituent production, leading to a compensatory lengthening
that restores original mechanical behaviour after a step increase in axial length.
Nevertheless, all of these models are based on a quasi-static version of the one
presented here, that is G&R is only driven by mean values on the cardiac cycle and
not by the rate measures. Such models may fail in the accurate description of clinical
scenarios in which dynamical features of the hemodynamics, like the pulsatility of
blood flow rate and pressure, undergo significant perturbations. The case of aortic
arch banding model of hypertension is one of these scenarios [15]. Indeed the aor-
