408
8 Morphogenesis
where μ and E are the shear and elastic moduli, respectively. Again, because the
emphasis is on qualitative behavior, all parameters are treated as dimensionless
unless units are specified.
For simplicity, we simulate both growth and contraction using the single strainenergy function of (8.1), with the elastic deformation gradient tensor given by
F
∗
= F · G
−1 ,
where the growth tensor G is generally a specified function of space and time. Thus,
contraction is simulated as negative growth, often accompanied by an increased
modulus. 4 With (3.56), Eqs. (3.69) 1,3 yield
I
∗
1 = tr C
∗
I
∗
3 = det C
∗ ,
where
C
∗
= F
∗T
· F
∗ .
Most finite-element codes operate primarily in global Cartesian coordinates. The
required stress and strain components can be extracted in the usual way.
A Few Additional Notes As discussed in Chap. 1, researchers studying mechanisms of morphogenesis are faced with a number of challenging issues. Thus, it
is best to attack a problem from as many directions and using as many tools as
possible. For modeling purposes, we propose the following approach:
1. Minimize the number of free parameters as much as possible.
2. Determine unknown parameters using experimental data for normal development.
3. Test the model using experimental results for perturbed development.
Many of the examples in this chapter illustrate this strategy. However, we always
should be mindful of the possibility that backup mechanisms may compensate for
perturbations, and a given perturbation may affect more than one mechanism.
For readers unfamiliar with terminology commonly used in anatomy, we try to
keep it relatively simple in this chapter. The following terms are used to indicate
directions: cranial (head); caudal (tail); ventral (front); dorsal (back); medial
(center); and lateral (left and right). In some cases, anterior and posterior may
substitute for cranial and caudal, respectively. Also, combinations are sometimes
used to denote axes, e.g., craniocaudal (head to tail) and mediolateral (middle to
left/right).
4 It is a simple matter to enforce incompressibility of fiber volume during contraction, i.e., set
det G = 1. For simplicity, however, this constraint is sometimes ignored in this chapter.
8 Morphogenesis
where μ and E are the shear and elastic moduli, respectively. Again, because the
emphasis is on qualitative behavior, all parameters are treated as dimensionless
unless units are specified.
For simplicity, we simulate both growth and contraction using the single strainenergy function of (8.1), with the elastic deformation gradient tensor given by
F
∗
= F · G
−1 ,
where the growth tensor G is generally a specified function of space and time. Thus,
contraction is simulated as negative growth, often accompanied by an increased
modulus. 4 With (3.56), Eqs. (3.69) 1,3 yield
I
∗
1 = tr C
∗
I
∗
3 = det C
∗ ,
where
C
∗
= F
∗T
· F
∗ .
Most finite-element codes operate primarily in global Cartesian coordinates. The
required stress and strain components can be extracted in the usual way.
A Few Additional Notes As discussed in Chap. 1, researchers studying mechanisms of morphogenesis are faced with a number of challenging issues. Thus, it
is best to attack a problem from as many directions and using as many tools as
possible. For modeling purposes, we propose the following approach:
1. Minimize the number of free parameters as much as possible.
2. Determine unknown parameters using experimental data for normal development.
3. Test the model using experimental results for perturbed development.
Many of the examples in this chapter illustrate this strategy. However, we always
should be mindful of the possibility that backup mechanisms may compensate for
perturbations, and a given perturbation may affect more than one mechanism.
For readers unfamiliar with terminology commonly used in anatomy, we try to
keep it relatively simple in this chapter. The following terms are used to indicate
directions: cranial (head); caudal (tail); ventral (front); dorsal (back); medial
(center); and lateral (left and right). In some cases, anterior and posterior may
substitute for cranial and caudal, respectively. Also, combinations are sometimes
used to denote axes, e.g., craniocaudal (head to tail) and mediolateral (middle to
left/right).
4 It is a simple matter to enforce incompressibility of fiber volume during contraction, i.e., set
det G = 1. For simplicity, however, this constraint is sometimes ignored in this chapter.