2.7 Applications of mSDM
2.7.3 Mapping for a surface with boundaries
2.7 Applications of mSDM 45
(a) (b) (c) (d)
Figure 2.15: Parametric surfaces of (a)-(b) a vertebra and (c)-(d) a pelvis. (a) and (c):
top side view; (b) and (d): left side view.
2.7 Applications of mSDM 46
(a) (b) (c)
Figure 2.16: Example of mapping for a surface with boundaries: (a) surface model of the brain block, (b) target surface, (c) mapping result.
One application using the mapping method is to analyze a specific region of an organ model in detail by mapping the regions of different individuals onto a common target surface, such as the functional mapping of a brain.
Table 2.5: The mean and standard deviation of final energiesEf,Eb, and computational times in the edge length-preserving mapping results of a brain block.
Vertices Control points Ef[×10−2mm] Eb Computational time
Brain block 1,553 2,033 2.66±0.02 0 1.25±0.11
The mapping algorithm for the surface model with boundaries is applied to the sur-face model of the block of a brain shown in Figure 2.16 (a) which has two boundaries.
This block is obtained by cutting a cadaver brain by a cutting machine. Figure 2.16 (c) shows the mapping result of the brain block onto a ring-shaped target surface with one hole shown in Figure 2.16 (b). Specifically, the target surface is constructed by cut-ting a spherical surface by two different planes. Two boundaries of the target surface are specified as the corresponding lines of the two boundaries of the brain block. To evaluate the mapping accuracy of the boundaries, I calculated the distance between the boundary of the mapping result and its corresponding boundary of the target surface.
Practically, to represent the shape of the boundary of the target surface as a simple function, an elliptical line is fitted to the control points on the boundary of the target surface. Then, it is performed to calculate the distance between the elliptical line and each vertex on the boundary of the mapping result. The mean distances of the outer
2.7 Applications of mSDM 47
and inner boundaries of the mapping result are 3.53×10−1[mm]and 4.41×10−1[mm].
Since the values are sufficiently small compared with the bounding box size of the target surface, 91.3×93.9×42.3[mm], the brain block is mapped onto the target sur-face while controlling the mapping positions of the two boundaries. Ef, Eb and the computational time in the length-preserving mapping result are shown in Tables 2.5.
The mean values of earea,eangle, andelength for the three types of geometrical feature preserving mapping are 7.08×10−5, 2.36×10−1, and 4.07×10−1, respectively.
Consequently, mSDM achieves one-to-one mapping of the surface model onto the target surface without foldovers while locating the landmarks at their target positions accurately and preserving the geometrical features stably. The proposed mapping method of surfaces with boundaries using mSDM has no limitation about the number of boundaries included in a surface model. Therefore, the proposed mapping method is also applicable to map the surface model with three or more boundaries onto its tar-get surface although it is necessary to manually determine the correspondences of the boundaries between the surface model and its target surface.
volumetric Self-organizing 3
Deformable Model (vSDM)
Among the organs in a human body, there are many parenchymatous organs which have internal structures including inner organs and/or blood vessels within the external surface of the organ. Especially, a human brain, one of parenchymatous organs, has layered cell structures and many inner organs including basal ganglia and thalamus.
Considering the complex structures of the parenchymatous organs, the SSM for the parenchymatous organ should describe the shape variations of both the surface and each inner organ of the parenchymatous organ. In addition, it is important for a safe and efficient treatment of parenchymatous organs that the SSM describes the structure of the parenchymatous organ including the spatial relationship among the inner organs.
For example, neurosurgeons want to know the brain structure of a patient to accurately approach a target inner organ while precluding damage to its surrounding organs.
To construct the SSM of parenchymatous organs satisfying these requirements,
48
49
(a) (b) (c) (d) (e)
Figure 3.1: (a) The surface of the brain volume model; (b) The brain volume model cut by a virtual plane for the interior visualization; (c) A cross section of the OMS (pink), IMS1 (blue), and LMS2 (green); (d) The surface of the target volume; (e) A cross section of the OTS (light blue), ITS1 (yellow), and ITS2 (orange).
by extending Fast mSDM, I propose a method, called volumetric Self-organizing De-formable Model (vSDM), for mapping the volume model of a human organ to a target volume with simple shape. In vSDM, both the volume model of a human organ and the target volume consist of a set of tetrahedra. vSDM achieves the volume organ model mapping by fitting the surface of the organ to that of the target volume while mapping each inner organ of the organ onto its corresponding inner target in the target volume. In addition, vSDM mapping preserves geometrical features in the original volume model.
As stated in Chapter 1.1, when applying to determine the model correspondence, the mapping methods need to meet the four constraints: one-to-one mapping, map-ping location control of characteristic regions on the target surface, geometrical fea-ture preservation, and applicability to target space with various shapes and topologies.
Although conventional mapping methods [49; 40; 17; 28; 1] satisfy two of the four constraints at most, no method satisfies the three or more constraints simultaneously as shown in Table 1.2. On the contrary, unlike the conventional methods for surface model mapping, Fast mSDM achieves an organ surface model mapping satisfying the four constraints. Utilizing the advantages of Fast mSDM, vSDM has the potential for the volume organ model mapping considering the four constraints. Moreover, the vol-ume organ models obtained by the vSDM mapping describes the spatial relationship among the whole organ and its inner organs. Therefore, the use of the volume model obtained by the vSDM mapping enables to obtain a reliable correspondences of both the volume models and their inner organs among individuals.
3.1 Definitions 50
3.1 Definitions
In vSDM, a tetrahedral volume modelMvof a human organ is used as an initial vSDM.
The external surface ofMvis regarded as the outer model surface (OMS) of the vSDM.
vSDM contains the inner volume models of the inner organs inside the OMS. When some of the inner organs are selected to analyze their shapes, the surfaces of the se-lected inner organs are used as the inner model surfaces (IMSs) of the vSDM. As example of the initial vSDM, a brain volume model is shown in Figures 3.1 (a)-(c).
The brain volume model consists of brain surface (the pink part in Figure 3.1 (c)) and two inner organs: the set of lateral ventricle and 3rd ventricle (the blue part), and 4th ventricle (the green part). In this example, the brain surface is used as the OMS while the surfaces of the set of lateral ventricle and 3rd ventricle, and the 4th ventricle are used as the IMSs.
The vertices in Mv are classified into three types. The vertices on the OMS and IMSs, respectively, are named as the OMS and IMS vertices while the rest vertices are regarded as the inner vertices. For each vertex except the OMS vertices, its 1-ball region is the set of the tetrahedra containing the vertex as shown in Figure 3.2 (a).
When vSDM is mapped onto a target volumeTv, the external surface of the target volume, called the outer target surface (OTS), is the mapping destination of the OMS.
The target volume includes inner targets within the OTS. Each IMS is mapped onto its corresponding inner target surface (ITS). Here, the position of the initial vSDM is determined so that the OMS and the OTS overlap each other as largely as possible.
As an example, a target volume used in my experiment is a volume model with an average shape of brain surfaces (the light blue region in Figures 3.1 (d) and (e)). In the target volume, there are two inner targets: a volume model with an average shape of the sets of lateral ventricle and 3rd ventricle (the yellow regions in Figure 3.1 (e)) and an ellipsoid (the orange regions in Figure 3.1 (e)). In this case, the OTS is the average surface of the brain surfaces while the ITSs are the ellipsoidal surface, and the average surface of the sets of lateral ventricle and 3rd ventricle. Here, the average shapes of the brain surface and the set of lateral ventricle and 3rd ventricle are obtained by mSDM-based remeshing method described in Chapter 2.7.1. Then, the position of each inner target is determined manually for each vSDM so that the IMS and ITS overlap each other as largely as possible.