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7 Remodeling
During surgery, the replacement tendon is stretched and attached to the bones.
A brace is then used to immobilize the elbow for 10 days to keep the tissue from
deforming as it begins to heal. During this period, the tendon remodels into a
reference state with all constituents stretched to their homeostatic stretch ratios, i.e.,
λ c∗ = λ c
0 , λ e∗ = λ e
0 , and λ g∗ = 1. The end of the 10-day period is defined as model
time t = 0, with the total stretch ratio for the mixture being λ x (0) = 1.
After the brace is removed, the collagen and elastin turn over with pre-stretches
λ n
0 (n = c, e). Their deposition rates and survival functions are
˙
J
n + = k
n + J
n
0
q
n (t, τ ) = e
−k n − (t−τ ) ,
(7.44)
where J n
0 = J n (0). Assume the ground substance undergoes no further growth or
remodeling.
Under Wilma’s supervision, several months of exercises follow to strengthen the
graft and increase its range of motion. These exercises produce a constant timeaveraged stretch ratio ˆ
λ relative to the reference state, giving
λ x (t) =
1, t < 0
ˆ
λ, t ≥ 0.
Simulate the remodeling that occurs during the exercise period (t ≥ 0). Determine
the total and partial stresses as functions of time and explore how the stress-strain
behavior of the tendon evolves as it heals.
Analysis The analysis closely follows those in the previous two examples; we just
need to add two more constituents. For t ≥ 0, λ x (t) = λ x (τ ) = ˆ
λ, and Eq. (7.20)
gives
λ
n∗
x (t, τ ) = λ
n
0
G n
x (τ )
G n
x (t)
(7.45)
for the collagen and elastin fibers. Since the ground substance does not undergo
G&R, λ
g∗
x = ˆ
λ.
Substituting (7.44) into (7.13) and integrating yield Eq. (7.15), i.e.,
J
n (t) = J
n
0
e
−k n − t
+
k n +
k n −
1 − e
−k n − t
,
(7.46)
where J c
0 = φ c
0 and J e
0 = φ e
0 . Because the ground substance does not grow, J g (t) =
dV g /dV 0 = φ
g
0 = 1−φ c
0 −φ e
0 for all t ≥ 0, where φ n
0 ≡ φ n (0). For isotropic growth
(γ = 1), Eqs. (7.6) 1 and (7.24) give
J = J
c
+ J
e
+ J
g
7 Remodeling
During surgery, the replacement tendon is stretched and attached to the bones.
A brace is then used to immobilize the elbow for 10 days to keep the tissue from
deforming as it begins to heal. During this period, the tendon remodels into a
reference state with all constituents stretched to their homeostatic stretch ratios, i.e.,
λ c∗ = λ c
0 , λ e∗ = λ e
0 , and λ g∗ = 1. The end of the 10-day period is defined as model
time t = 0, with the total stretch ratio for the mixture being λ x (0) = 1.
After the brace is removed, the collagen and elastin turn over with pre-stretches
λ n
0 (n = c, e). Their deposition rates and survival functions are
˙
J
n + = k
n + J
n
0
q
n (t, τ ) = e
−k n − (t−τ ) ,
(7.44)
where J n
0 = J n (0). Assume the ground substance undergoes no further growth or
remodeling.
Under Wilma’s supervision, several months of exercises follow to strengthen the
graft and increase its range of motion. These exercises produce a constant timeaveraged stretch ratio ˆ
λ relative to the reference state, giving
λ x (t) =
1, t < 0
ˆ
λ, t ≥ 0.
Simulate the remodeling that occurs during the exercise period (t ≥ 0). Determine
the total and partial stresses as functions of time and explore how the stress-strain
behavior of the tendon evolves as it heals.
Analysis The analysis closely follows those in the previous two examples; we just
need to add two more constituents. For t ≥ 0, λ x (t) = λ x (τ ) = ˆ
λ, and Eq. (7.20)
gives
λ
n∗
x (t, τ ) = λ
n
0
G n
x (τ )
G n
x (t)
(7.45)
for the collagen and elastin fibers. Since the ground substance does not undergo
G&R, λ
g∗
x = ˆ
λ.
Substituting (7.44) into (7.13) and integrating yield Eq. (7.15), i.e.,
J
n (t) = J
n
0
e
−k n − t
+
k n +
k n −
1 − e
−k n − t
,
(7.46)
where J c
0 = φ c
0 and J e
0 = φ e
0 . Because the ground substance does not grow, J g (t) =
dV g /dV 0 = φ
g
0 = 1−φ c
0 −φ e
0 for all t ≥ 0, where φ n
0 ≡ φ n (0). For isotropic growth
(γ = 1), Eqs. (7.6) 1 and (7.24) give
J = J
c
+ J
e
+ J
g