Joint Longitudinal Modeling of Brain Appearance in Multimodal MRI for the Characterization of Early Brain Developmental Processes - Longitudinal Registration and Shape Modeling - page 53

Here i =1 , ..., N ind refers to subject indices and j =1 , ..., T ind are indices of time
points of scan. The function f is the nonlinear growth function of choice that is
used to model the contrast change trajectory. This function is dependent on the
covariate vector t ij

as well as the mixed effect parameter vector ˆ i . The error

term e ij refers to the residual i.i.d error which follows the normal distribution

N (0 ,˃ 2 ). The parameter vector e ij ∼ ˆ i which has fixed and random effect

components can be written as :

ˆ i = A i ʲ + B i b i , where b i ∼ N (0 , ˈ ) . (7)

The vector of fixed effects is given by and the vector of random effects by

b i . The design matrices associated with fixed effects and random effects vectors
are given by

ʲ

A i and B i respectively. The random effects which contribute to

parameter ˆ i are assumed to be normally distributed with variance-covariance

matrix over all subjects.

Since we want to model the highly nonlinear trends seen in contrast change
a parametric growth function is adopted for NLME modeling [ 5 ]. Parametric
growth models provide concise description of the data and show greater flexi-
bility compared with linear models. After testing various choices for parametric
functions with low number of parameters based on the Akaike Information Cri-
terion (AIC), we decided on using the logistic growth model. We use the four
parameter logistic growth model defined by the parameters ( ˆ 1 ,ˆ 2 ,ˆ 3 ,ˆ 4 )as:

ˈ

ˆ 2
1+ exp ˆ 3 −t

f ( ˆ ,t )= ˆ 1 + . (8)

ˆ 4

The parameters of the logistic model can be interpreted as follows : (i) ˆ 1 is
the left horizontal asymptotic parameter which is the value taken by the model
for very small values of input t , (ii) ˆ 2 is the right horizontal asymptotic param-
eter at which the model saturates for large values of input t , (iii) ˆ 3 is the
inflection point parameter which indicates the time taken to reach half the dif-
ference between left and right asymptotic values, and (iv) ˆ 4 is a rate parameter
denoting a scaling function on the time axis which indicates the curvature of the
model at the inflection point.
To generate an individual i 's trajectory using NLME modeling with the logis-
tic function, mixed effects parameters

ˆ i consisting of the sum of fixed effect ʲ
and subject-specific random effect b i are used (by setting values of design matri-
ces appropriately). The response y ij

A and B for a region R and subject i at

the j th time instant t ij can be written as :

R i 2

ʲ R 2 + b R i 2

ˆ

y ij =

R

+ e ij = ʲ R 1 + b R i 1 + ˆ

i 1 +

+ e ij . (9)

φ R i 3 −t ij
φ R i 4

ʲ R 3 + b R i 3 −t ij
ʲ R 4 + b R i 4

1+ exp 1+ exp

Since we lack information about contrast at the time of birth, the first parameter
ˆ R i 1 is set to 0 in our analysis. Based on study of variability across subjects and
information criteria, we assume that the right-asymptotic parameter ˆ 2 and
inflection point parameter ˆ 3 have non-zero random effects, while the remaining
parameters don't have a random effects component.