ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

modulation models like the proposed by Stokes et al [7] and Carter et al [10]. Carter et
al [10] described an isotropic growth tensor in terms of the hydrostatic and deviatoric
stresses, while Stokes et al [7] described the longitudinal growth as a function of the
tensile or compressive axial stresses. These models were compared by Lin et al [11],
revealing that the model of Carter et al [10] is less able to predict the growth retardation
under compressive loads than the model proposed by Stokes et al [7]. Both models
describe the ‘baseline’ growth rate as its mechanical modulation from a
phenomenological point of view; therefore the bone elongation is simulated using
equivalent thermal deformations [11].
The main objective of this work is to develop a finite element model that describes the
growth plate behavior and its mechanical modulation from a biological point of view.
We used a two-dimensional domain inspired on the quantitative histology of a human
growth plate [1]. The model was implemented on an ABAQUS User Element
subroutine (UEL) and it was used to simulate the growth for a 2-years old, male human
distal femur.
3. MATEMATHICAL MODEL
The mathematical model is based on the following assumptions:
a) Given that the PTHrP-Ihh regulatory loop is highly stable in time [9], it is assumed
that its mechanism of expression is insensitive to mechanical loads [13]. This
explains, to some extent, the “in vitro” results reported by Villemure et al [14].
b) The mathematical description of the cell distribution and concentration in the
growth plate is performed by second-order transversely isotropic tensors [12],
defined as
R PC
R HC
CPC
= 3
PC 1
r
PC
( 1 + ( r − ) n ⊗ n)
CHC
= 3
HC 1
r
HC
( 1 + ( r − ) n ⊗ n)
, (1)
, (2)
where R PC and R HC are the proliferative and hypertrophic distribution tensors,
respectively, r PC and r HC are the ratios of the number of proliferative and
hypertrophic cells in the growth preferential direction n to the number of cells in
the orthogonal direction, C PC and C HC are the concentrations of proliferative and
hypertrophic chondrocytes and 1 is the unit second order tensor.
c) According to Stokes et al [7] it is assumed that, in the growth direction, no
extracellular matrix exists between the hypertrophic chondrocytes. In addition, it is
supposed that all the new cells produced complete their maduration from the
proliferative to the hypertrophic state. Therefore, the growth per day can be
expressed as
G n p h × = , (3)
max