7.4 Examples: Remodeling in 1D
361
causing the bar to immediately shorten slightly (λ x < 1, Fig. 7.7c). If the prestretch is large enough, this elastic response is followed by a prolonged period of
shortening, as old shortened fibers are replaced by new fibers with more tension.
In fact, the model predicts that the bar continues to shorten without bound. The
transition from infinite lengthening to infinite shortening is extremely sensitive to
the value of λ 0 . This unrealistic behavior can be improved by including stressdependent growth (see Sect. 7.6.2) or a constituent (e.g., elastin) that effectively
does not turn over.
7.4.3 Remodeling of Connective Tissue
Connective tissues bind tissues and organs together, effectively keeping our bodies
from falling apart. These tissues are composed almost entirely of extracellular
matrix, usually dominated by collagen. Elastin and other constituents, including
cells, are generally present in considerably lesser amounts. Like other load-bearing
structures, connective tissues remodel in response to changing loads.
The musculoskeletal system includes two main types of connective tissue.
Tendons connect muscle to bone, while ligaments connect two bones. The following
example examines remodeling of a tendon transplanted to a new location, where it
is exposed to new forces.
Problem In his heyday, Fred pitched for the Bedrock Boulders, a local minorleague baseball team. One day, a ligament in his elbow ruptured, potentially dashing
his major-league dreams. Although he had little chance to ever make the majors, he
was delighted to hear about a new surgical technique that had just been developed
to repair this type of injury. This method was the forerunner of today’s Tommy John
surgery, where a damaged ligament is replaced by a tendon taken from another part
of the body. With time, the tendon remodels and acquires mechanical properties
roughly equivalent to those of the ligament it replaced. In fact, Fred threw harder
than he did before the injury, but, unfortunately, he remained one of the least talented
pitchers to ever put on a uniform.
To simulate Tommy John surgery and its aftermath, consider a homogeneous,
rectangular, incompressible bar as a simple model for a tendon. The bar is composed
of collagen, elastin, and ground substance with initial volume fractions φ c
0 , φ e
0 , and
φ
g
0 , respectively. Collagen and elastin are treated as axially aligned, one-dimensional
fibers embedded in the ground substance, which is taken as an isotropic material.
The respective strain-energy density functions are given by
W
c∗
=
c c
α c
e
α c [(λ c∗
x ) 2 −1] 2 − 1
W
e∗
= c e [(λ
e∗
x )
2
− 1]
2
W
g∗
= c g
λ
∗2
x + λ
∗2
y + λ
∗2
z − 3
.
(7.43)
361
causing the bar to immediately shorten slightly (λ x < 1, Fig. 7.7c). If the prestretch is large enough, this elastic response is followed by a prolonged period of
shortening, as old shortened fibers are replaced by new fibers with more tension.
In fact, the model predicts that the bar continues to shorten without bound. The
transition from infinite lengthening to infinite shortening is extremely sensitive to
the value of λ 0 . This unrealistic behavior can be improved by including stressdependent growth (see Sect. 7.6.2) or a constituent (e.g., elastin) that effectively
does not turn over.
7.4.3 Remodeling of Connective Tissue
Connective tissues bind tissues and organs together, effectively keeping our bodies
from falling apart. These tissues are composed almost entirely of extracellular
matrix, usually dominated by collagen. Elastin and other constituents, including
cells, are generally present in considerably lesser amounts. Like other load-bearing
structures, connective tissues remodel in response to changing loads.
The musculoskeletal system includes two main types of connective tissue.
Tendons connect muscle to bone, while ligaments connect two bones. The following
example examines remodeling of a tendon transplanted to a new location, where it
is exposed to new forces.
Problem In his heyday, Fred pitched for the Bedrock Boulders, a local minorleague baseball team. One day, a ligament in his elbow ruptured, potentially dashing
his major-league dreams. Although he had little chance to ever make the majors, he
was delighted to hear about a new surgical technique that had just been developed
to repair this type of injury. This method was the forerunner of today’s Tommy John
surgery, where a damaged ligament is replaced by a tendon taken from another part
of the body. With time, the tendon remodels and acquires mechanical properties
roughly equivalent to those of the ligament it replaced. In fact, Fred threw harder
than he did before the injury, but, unfortunately, he remained one of the least talented
pitchers to ever put on a uniform.
To simulate Tommy John surgery and its aftermath, consider a homogeneous,
rectangular, incompressible bar as a simple model for a tendon. The bar is composed
of collagen, elastin, and ground substance with initial volume fractions φ c
0 , φ e
0 , and
φ
g
0 , respectively. Collagen and elastin are treated as axially aligned, one-dimensional
fibers embedded in the ground substance, which is taken as an isotropic material.
The respective strain-energy density functions are given by
W
c∗
=
c c
α c
e
α c [(λ c∗
x ) 2 −1] 2 − 1
W
e∗
= c e [(λ
e∗
x )
2
− 1]
2
W
g∗
= c g
λ
∗2
x + λ
∗2
y + λ
∗2
z − 3
.
(7.43)