Biomedical Engineering Reference
In-Depth Information
59, 104, 105, 118, 121, 146]. The estimated deformation field should basically
obey the rule of the Navier equation:
2
u
+
(
λ
+
µ
)
∇
(
di
v
(
u
))
+
F
=
0
,
µ
∇
where
u
is the deformation field to estimate,
λ
and
µ
are the Lame coef-
ficients and
F
is the sum of forces that are applied on the system. The
problem is to specify the forces
F
that will lead to a correct registration.
Bajcsy proposes to compute these forces so as to match the contours [5].
Davatzikos [39] and Peckar [104] do not compute any forces but segment the
brain surface and the ventricles using two different methods. The matching
of these surfaces provide boundary conditions that make it possible to solve
the problem. These two approaches are therefore sensitive to segmentation
errors.
The use of elastic methods raises the following questions:
What should be the values of Lame coefficients? The choice of these coef-
ficients influence the deformation. Earliest work proposed that
λ
=
0 but
it appears nowadays to be a limitation.
This modeling cannot handle large deformations. As a matter of fact, the
equation of Navier is only valid for small displacements. To solve this
problem, two kind of approaches can be used. A rigid registration can
provide a good initialization (Bajcsy [5] uses principal inertia axes and
Davatzikos [38] uses the stereotaxic space). Another way [104] is to solve
the problem iteratively using a multiresolution approach.
The topology of present structures will be preserved. This may be inter-
esting in some applications but more questionable when matching brains
of different subjects. Ono [103] has shown that cortical structures are not
topologically equivalent among subjects indeed.
8.2.3.3
Fluid Models
Following the same inspiration as elastic models, Christensen and Miller [27]
propose to compute a deformation that obeys the rule of fluid mechanics (equa-
tion of Navier-Stokes). The major difference with the elastic modeling is the
fact that the fluid continuously “forgets” about its initial position. Large displace-
ments and complex motions are therefore much easier to handle. The equation
