384
7 Remodeling
More generally, the average elastic stretch ratio for a fiber of the nth family is
given by
λ n∗ (t) = λ n
0
λ(t)
n (t)
,
(7.89)
where λ n
0 is the deposition stretch. Because this theory does not keep track of
deposition times, the above relation can be obtained by setting τ = 0 in the 1D
form of Eq. (7.20), along with λ(0) = 1 and G n (t)/G n (0) = n (t).
We assume that the recruitment stretch evolves according to the remodeling law
˙
n
= A(λ
n∗
− λ
n
0 ))
n ,
(7.90)
which yields λ n∗ = λ n
0 [and n = λ by (7.89)] during homeostatic equilibrium
( ˙
n = 0). Thus, n evolves until the elastic stretch ratio of a fiber reaches its
deposition stretch ratio, as required to maintain a homeostatic state according to the
Humphrey-Rajagopal theory. In addition, changes in fiber volume can be included
through the relation (Watton et al. 2004)
˙
J
n
= B(λ
n∗
− λ
n
0 )J
n .
(7.91)
The corresponding growth ratios G n can be computed using equations like (7.23)
and (7.24). Since this theory does not involve hereditary integrals, the computations
parallel those used for volumetric growth. Recently, Latorre and Humphrey (2019)
proposed a rate-based formulation of Humphrey-Rajagopal remodeling theory that
offers similar advantages.
Example 7.2 An incompressible rectangular bar consists of passive muscle cells
and collagen fibers oriented in the axial (x) direction. The bar is free of all external
loads and constraints, and the muscle grows longer by the specified growth ratio
G
m
x = G
m
x (0)(1 + at),
(7.92)
relative to the initial homeostatic configuration. Here, a is a constant and G m
x (0) =
(φ m
0 ) 1/3 by (7.55) 1 with φ m
0 ≡ φ m (0). The collagen remodels according to
Eq. (7.90), but its volume remains unchanged. Material properties for the muscle
and collagen, respectively, are defined by
W
m∗
= c m
λ
∗2
x + λ
∗2
y + λ
∗2
z − 3
W
c∗
=
c c
α c
e
α c [(λ c∗
x ) 2 −1] 2 − 1
.
(7.93)
Determine the total stretch ratio λ x and the constituent stresses as functions of time.
7 Remodeling
More generally, the average elastic stretch ratio for a fiber of the nth family is
given by
λ n∗ (t) = λ n
0
λ(t)
n (t)
,
(7.89)
where λ n
0 is the deposition stretch. Because this theory does not keep track of
deposition times, the above relation can be obtained by setting τ = 0 in the 1D
form of Eq. (7.20), along with λ(0) = 1 and G n (t)/G n (0) = n (t).
We assume that the recruitment stretch evolves according to the remodeling law
˙
n
= A(λ
n∗
− λ
n
0 ))
n ,
(7.90)
which yields λ n∗ = λ n
0 [and n = λ by (7.89)] during homeostatic equilibrium
( ˙
n = 0). Thus, n evolves until the elastic stretch ratio of a fiber reaches its
deposition stretch ratio, as required to maintain a homeostatic state according to the
Humphrey-Rajagopal theory. In addition, changes in fiber volume can be included
through the relation (Watton et al. 2004)
˙
J
n
= B(λ
n∗
− λ
n
0 )J
n .
(7.91)
The corresponding growth ratios G n can be computed using equations like (7.23)
and (7.24). Since this theory does not involve hereditary integrals, the computations
parallel those used for volumetric growth. Recently, Latorre and Humphrey (2019)
proposed a rate-based formulation of Humphrey-Rajagopal remodeling theory that
offers similar advantages.
Example 7.2 An incompressible rectangular bar consists of passive muscle cells
and collagen fibers oriented in the axial (x) direction. The bar is free of all external
loads and constraints, and the muscle grows longer by the specified growth ratio
G
m
x = G
m
x (0)(1 + at),
(7.92)
relative to the initial homeostatic configuration. Here, a is a constant and G m
x (0) =
(φ m
0 ) 1/3 by (7.55) 1 with φ m
0 ≡ φ m (0). The collagen remodels according to
Eq. (7.90), but its volume remains unchanged. Material properties for the muscle
and collagen, respectively, are defined by
W
m∗
= c m
λ
∗2
x + λ
∗2
y + λ
∗2
z − 3
W
c∗
=
c c
α c
e
α c [(λ c∗
x ) 2 −1] 2 − 1
.
(7.93)
Determine the total stretch ratio λ x and the constituent stresses as functions of time.