  Deformable registration of spatiotemporal Cone-Beam Computed Tomography
(CBCT) images taken sequentially during the radiation treatment course yields a
deformation field for a pair of images. The Jacobian of this field at any voxel
provides a measure of the expansion or contraction of a unit volume. We analyze
the Jacobian at different sections of the tumor volumes obtained from
delineation done by radiation oncologists for lung cancer patients. The
delineations across the temporal sequence are compared post registration to
compute tumor areas namely, unchanged (U), newly grown (G), and reduced (R)
that have undergone changes. These three regions of the tumor are considered
for statistical analysis. In addition, statistics of non-tumor (N) regions are
taken into consideration. Sequential CBCT images of 29 patients were used in
studying the distribution of Jacobian in these four different regions, along
with a test set of 16 patients. Statistical tests performed over the dataset
consisting of first three weeks of treatment suggest that, means of the
Jacobian in the regions follow a particular order. Although, this observation
is apparent when applied to the distribution over the whole population, it is
found that the ordering deviates for many individual cases. We propose a
hypothesis to classify patients who have had partial response (PR). Early
prediction of the response was studied using only three weeks of data. The
early prediction of response of treatment was supported by a Fisher's test with
odds ratio of 5.13 and a p-value of 0.043.
