Joint Longitudinal Modeling of Brain Appearance in Multimodal MRI for the Characterization of Early Brain Developmental Processes - Longitudinal Registration and Shape Modeling - page 52

instant to gradient direction in the adult-like scan. Consider the regional SIR
value for the image scanned at a time point t when image appearance is adult-
like. If SIR of the adult-like image satisfies the inequality SIR i,m ( t )

1 >
0, then contrast values at all time points which satisfy the same inequality
SIR i,m ( t )

−

1 > 0 are assigned a positive sign. The contrast values which do
not satisfy the same inequality as the adult-like image are assigned a negative
sign. The same rule is applicable if the SIR of the adult-like image satisfies the
inequality SIR i,m ( t )

−

1 < 0. The direction of contrast for region R , for a modal-
ity m scan belonging to subject i , taken at time instant t , relative to adult-like
contrast observed at time t is denoted by CONTDIR and defined using the
signum function:

−

SIR i,m ( t )

−

1

CONTDIR i,m ( t )= signum

. (5)

SIR i,m ( t ) − 1

To summarize, the SIR encodes the actual direction of relative white-gray
matter intensity gradient, while the CONTDIR helps determine if reversal
in gradient direction takes place. For multimodal mixed effect analysis, the
SIR-based directional attributes are modeled independently from contrast. The
CONTDIR value, however, is used to provide a sign to the contrast measure
defined in the earlier section in all subsequent analysis. Therefore only reversals
in contrast are characterized in the HD-based contrast analysis by adding a sign
using CONTDIR. The actual direction of the intensity gradient is independently
modeled using the SIR with LME (Linear Mixed Effects) analysis. If the SIR-
based directional attributes were used to add a sign to the contrast measure,
contrast values of images belonging to different modalities would have opposite
signs and their ranges would not be comparable.

2.4 Nonlinear Mixed Effects Modeling of Contrast
Our study data is longitudinal, i.e. repeated images of each subject are obtained
over time. Taking into account correlations of repeated measures, different time
spacing and varying number of timepoints per subject, as well as resistance to
noise, statistics is offering the methodology of mixed-effect modeling. Unlike
regression of the set of measures assuming independence, mixed effect modeling
correctly includes intra-subject correlations and estimates temporal trajecto-
ries of the whole group (fixed effect) and of each individual (random effects).
Accounting for nonlinear temporal changes of contrast, we apply a nonlinear
mixed-effects modeling technique (NLME) [ 9 , 15 ]. The NLME framework we use
is well established and has several advantages including robustness to noise and
outliers, and the ability to work with datasets that include missing and unevenly
spaced data. The NLME model uses mixed effects parameters consisting of a
linear combination of population-based fixed effects and subject-specific random
effects to estimate growth trajectories. The observation of the i th individual at
the j th time point t i,j is hence modeled using NLME as :
y ij = f (

ˆ i , t ij )+ e ij .

(6)