314
6 Growth
The active muscle stress σ zA is the total stress in the contracting muscle, i.e.,
σ zA = σ z . Substituting (6.114) into (6.109) 2 and (6.113) yields two equations to
solve for G c and λ z , with initial condition G c (0) = 1.
Results
This model can be used to simulate muscle growth in response to exercise against a
resistance (the spring). Experience tells us that muscles grow thicker with exercise
over a period of weeks, with the growth rate increasing with the spring stiffness k.
Before the exercise regimen begins, we assume that the time-averaged stress during
normal activity is equal to the active homeostatic stress σ A0 = 1. We choose the
value K = 0.7 for the contraction ratio, so the initial stress during exercise is greater
than σ A0 . The other parameters are set as φ p = 0.2, φ a = 0.8, c p = 1, c a = 10, and
α c = 2.
Illustrative results are shown in Fig. 6.22. Consistent with the expected behavior,
the model predicts that cross-fiber growth G c increases with k (Fig. 6.22b). Moreover, the increasing cross-sectional area reduces the stress toward the homeostatic
value σ A0 = 1 as growth approaches a new equilibrium value (Fig. 6.22b, c).
Muscle shortening decreases as k increases, with k → ∞ corresponding to
isometric contraction. For this case, λ z = 1 and Eq. (6.113) shows that σ z remains
constant, even as the muscle grows thicker. Thus, σ zA remains above its homeostatic
value no matter how large the cross-sectional area becomes, and G c increases
without bound (Fig. 6.22b, dashed line on left). Of course, infinite growth is not
realistic for most people.
This observation suggests that the proposed growth laws are either not correct or
incomplete. Most theories are deficient in some way, but they still can be useful if
limitations are clearly defined. The present theory could be improved by including
metabolic and other factors in the analysis. We also could fix this problem by
limiting growth through Eq. (6.95) or a similar constraint, but the stress would not
return to its homeostatic value.
Another possible solution, as suggested by several experimental studies, is to
include remodeling in the analysis and let the active modulus c a decrease with
growth (Lesch et al. 1968; Rowe 1969; Freeman and Luff 1982; Kandarian and
White 1989). For illustration, we postulate that the modulus evolves according to
the relation
˙
c a = −b(σ zA − σ A0 )c a ,
(6.115)
where b > 0. This remodeling law has the same form as the growth law (6.109) 2 .
The dashed curves labeled fixed/decr c a in Fig. 6.22b, c were computed for isometric
contraction with b = 3.5. The growth rate approaches zero as σ → σ A0 = 1.
If k = 0 (no spring), the muscle contracts without constraint. In this case, the
passive muscle constituents are compressed by φ p σ zp , and the active constituents
are stretched by the equal and opposite tensile stress φ a σ za , making the total axial
6 Growth
The active muscle stress σ zA is the total stress in the contracting muscle, i.e.,
σ zA = σ z . Substituting (6.114) into (6.109) 2 and (6.113) yields two equations to
solve for G c and λ z , with initial condition G c (0) = 1.
Results
This model can be used to simulate muscle growth in response to exercise against a
resistance (the spring). Experience tells us that muscles grow thicker with exercise
over a period of weeks, with the growth rate increasing with the spring stiffness k.
Before the exercise regimen begins, we assume that the time-averaged stress during
normal activity is equal to the active homeostatic stress σ A0 = 1. We choose the
value K = 0.7 for the contraction ratio, so the initial stress during exercise is greater
than σ A0 . The other parameters are set as φ p = 0.2, φ a = 0.8, c p = 1, c a = 10, and
α c = 2.
Illustrative results are shown in Fig. 6.22. Consistent with the expected behavior,
the model predicts that cross-fiber growth G c increases with k (Fig. 6.22b). Moreover, the increasing cross-sectional area reduces the stress toward the homeostatic
value σ A0 = 1 as growth approaches a new equilibrium value (Fig. 6.22b, c).
Muscle shortening decreases as k increases, with k → ∞ corresponding to
isometric contraction. For this case, λ z = 1 and Eq. (6.113) shows that σ z remains
constant, even as the muscle grows thicker. Thus, σ zA remains above its homeostatic
value no matter how large the cross-sectional area becomes, and G c increases
without bound (Fig. 6.22b, dashed line on left). Of course, infinite growth is not
realistic for most people.
This observation suggests that the proposed growth laws are either not correct or
incomplete. Most theories are deficient in some way, but they still can be useful if
limitations are clearly defined. The present theory could be improved by including
metabolic and other factors in the analysis. We also could fix this problem by
limiting growth through Eq. (6.95) or a similar constraint, but the stress would not
return to its homeostatic value.
Another possible solution, as suggested by several experimental studies, is to
include remodeling in the analysis and let the active modulus c a decrease with
growth (Lesch et al. 1968; Rowe 1969; Freeman and Luff 1982; Kandarian and
White 1989). For illustration, we postulate that the modulus evolves according to
the relation
˙
c a = −b(σ zA − σ A0 )c a ,
(6.115)
where b > 0. This remodeling law has the same form as the growth law (6.109) 2 .
The dashed curves labeled fixed/decr c a in Fig. 6.22b, c were computed for isometric
contraction with b = 3.5. The growth rate approaches zero as σ → σ A0 = 1.
If k = 0 (no spring), the muscle contracts without constraint. In this case, the
passive muscle constituents are compressed by φ p σ zp , and the active constituents
are stretched by the equal and opposite tensile stress φ a σ za , making the total axial