344
7 Remodeling
manner, they also increase transverse stiffness, altering tissue anisotropy. Crosslinks, therefore, play a major role in regulating matrix properties.
Before delving into the theory, we need to define what we call a “fiber.” The main
constituents of a tissue include cells, extracellular matrix, and water. While muscle
cells are often called fibers, this term is herein reserved for solid components of the
matrix and cytoskeleton (including sarcomeres). A fiber family is a group of fibers
of one type oriented in the same direction.
7.2 Fundamental Remodeling Mechanics
At the microscopic level, remodeling involves reorganization of tissue fibers. In the
extracellular matrix, as well as inside cells, either existing fibers reorganize, or new
fibers are created with different structure. Under homeostatic conditions, new fibers
are nominally identical to the old fibers. Immediately after an injury, however, the
new fibers may differ considerably, e.g., during the formation of scar tissue.
Here, we focus on a theory for remodeling developed by Humphrey and
Rajagopal (Humphrey 1999; Humphrey and Rajagopal 2002, 2003) that is based
on an approach proposed by Rajagopal and Wineman (1992) for polymer network
mechanics. The theory also includes growth via net addition or subtraction of constituents and, therefore, represents a mechanistic theory for growth and remodeling
(G&R). This ground-breaking Humphrey-Rajagopal theory has led to important
advances in vascular mechanics, such as providing new insight into the formation
and progression of aneurysms (Baek et al. 2006; Kroon and Holzapfel 2007).
In this theory, matrix remodeling occurs through turnover, as old fibers degrade
and are replaced by newly synthesized fibers. The turnover rate can vary widely
between different constituents. In arteries, for example, the half-life of collagen
is about 60–70 days, whereas elastin is much more stable, with a half-life of 25–
70 years. Both of these rates can increase markedly, however, in injury or disease
(Cyron and Humphrey 2017). In cells, the actin cytoskeleton can remodel within
minutes (Humphrey 2008; Coravos et al. 2017).
Before tackling the mathematical aspects of the theory, it is useful to examine
basic ideas from a qualitative perspective. For comparison purposes, we first
consider growth of a single fiber. Suppose axial growth is governed by the equation
˙
G = a(σ − σ 0 )G,
(7.1)
where a > 0 and σ 0 is the homeostatic stress corresponding to the elastic stretch
ratio λ ∗ = λ 0 . As shown in Fig. 7.3, the unloaded (old) fiber is first stretched by
λ 0 (A) to establish a homeostatic state. Then, further stretch by λ 1 /λ 0 (total stretch
ratio λ 1 ; B) triggers growth by an amount that effectively cancels out the additional
stretch, i.e., G = λ 1 /λ 0 (C), lowering the stress back to its homeostatic value. When
unloaded (D), the old fiber has grown by G into a “new” longer fiber.
7 Remodeling
manner, they also increase transverse stiffness, altering tissue anisotropy. Crosslinks, therefore, play a major role in regulating matrix properties.
Before delving into the theory, we need to define what we call a “fiber.” The main
constituents of a tissue include cells, extracellular matrix, and water. While muscle
cells are often called fibers, this term is herein reserved for solid components of the
matrix and cytoskeleton (including sarcomeres). A fiber family is a group of fibers
of one type oriented in the same direction.
7.2 Fundamental Remodeling Mechanics
At the microscopic level, remodeling involves reorganization of tissue fibers. In the
extracellular matrix, as well as inside cells, either existing fibers reorganize, or new
fibers are created with different structure. Under homeostatic conditions, new fibers
are nominally identical to the old fibers. Immediately after an injury, however, the
new fibers may differ considerably, e.g., during the formation of scar tissue.
Here, we focus on a theory for remodeling developed by Humphrey and
Rajagopal (Humphrey 1999; Humphrey and Rajagopal 2002, 2003) that is based
on an approach proposed by Rajagopal and Wineman (1992) for polymer network
mechanics. The theory also includes growth via net addition or subtraction of constituents and, therefore, represents a mechanistic theory for growth and remodeling
(G&R). This ground-breaking Humphrey-Rajagopal theory has led to important
advances in vascular mechanics, such as providing new insight into the formation
and progression of aneurysms (Baek et al. 2006; Kroon and Holzapfel 2007).
In this theory, matrix remodeling occurs through turnover, as old fibers degrade
and are replaced by newly synthesized fibers. The turnover rate can vary widely
between different constituents. In arteries, for example, the half-life of collagen
is about 60–70 days, whereas elastin is much more stable, with a half-life of 25–
70 years. Both of these rates can increase markedly, however, in injury or disease
(Cyron and Humphrey 2017). In cells, the actin cytoskeleton can remodel within
minutes (Humphrey 2008; Coravos et al. 2017).
Before tackling the mathematical aspects of the theory, it is useful to examine
basic ideas from a qualitative perspective. For comparison purposes, we first
consider growth of a single fiber. Suppose axial growth is governed by the equation
˙
G = a(σ − σ 0 )G,
(7.1)
where a > 0 and σ 0 is the homeostatic stress corresponding to the elastic stretch
ratio λ ∗ = λ 0 . As shown in Fig. 7.3, the unloaded (old) fiber is first stretched by
λ 0 (A) to establish a homeostatic state. Then, further stretch by λ 1 /λ 0 (total stretch
ratio λ 1 ; B) triggers growth by an amount that effectively cancels out the additional
stretch, i.e., G = λ 1 /λ 0 (C), lowering the stress back to its homeostatic value. When
unloaded (D), the old fiber has grown by G into a “new” longer fiber.