High Resolution Medical Models and Geometric Reasoning starting from
CT/MR Images
K. Subburaj and B. Ravi
OrthoCAD Network Research Facility, Department of Mechanical Engineering,
Indian Institute of Technology Bombay, Mumbai - 400 076, India
[email protected], [email protected]
Abstract
The ubiquitous availability of high power
computers has opened up the possibility of handling
large (high resolution) volumetric data to accurately
represent medical models, and performing geometric
reasoning for various applications. In this paper, we
present an efficient protocol to reconstruct accurate
medical models from CT/MR images having equal or
unequal values of slice thickness, inter slice distance,
and pixel size. It involves modifying the slice thickness
while leaving the in-slice resolution intact; issues such
as slice overlap and inter-slice gap are handled using
slice based interpolation. Noise reduction and better
delineation of object boundaries and segmentation are
performed in voxel space. Geometric analysis of
reconstructed volumetric data is performed to generate
internal thickness mapping, useful for pre-operative
planning and custom implant design. A test case of
pelvic model reconstruction from CT slices is
described to illustrate the algorithms.
1. Introduction
Medical model reconstruction involves creating a
volumetric model from a contiguous set of 2D slices
acquired from medical imaging modalities such as
Computed Tomography (CT), and Magnetic
Resonance Imaging (MRI) [1]. The reconstructed
model is useful for several applications, including
visualisation, diagnosis, pre-operative planning [2][3],
designing patient specific prosthesis [3][4], virtual
manipulation of anatomical elements, education, and
training.
In CT/MR, the ratio of pixel resolution in the image
plane (x,y), to resolution along the normal (z) axis is
1:5 (sometimes even 1:10). Various attempts have
been made to approximate these anisotropic voxels
into isotropic voxels by linear and non-linear
interpolation [6][7]. Region or object of interest is
segmented by classifying the voxels based on intensity
values (using manual, seeding, or semi-automatic
methods) [8]. The segmented volume data are rendered
directly using ray tracing or transfer functions, or
indirectly by fitting surface using Marching Cubes [9],
Delaunay triangulation [10], or surface tiling.
Geometrical
operations,
such
as
distance
transformation mapping, region identification, shape
extraction, thickness mapping [11], and positioning
analysis [2], are important in medical applications [4].
In summary, methods that simply join the contours
in object based interpolation encounter problems
because of the absence of any concrete a priori
knowledge about the imaged structure and noise.
Surface tiling and Delaunay triangulation based
reconstructed models may be good for visualisation
but not accurate enough for downstream applications
and also generates a huge number of triangles
(computationally expensive). Segmentation using
seeding may not lead to accurate reconstruction due to
presence of noise and discontinuity or missing regions.
This work mainly focuses on building high
resolution medical models with accuracy closer to the
original images, and geometric reasoning of these
models, with near-real time user interaction. The next
section presents the detailed steps of reconstruction
and rendering, followed by thickness analysis of
segmented object. The implementation of the software
program and its testing are presented next. Finally,
concludes with the further research work.
2. Virtual 3D reconstruction
Reconstruction of 3D models from a series of axial
CT/ MRI slices comprises several stages. The acquired
CT image data is stacked and interpolated to convert
contiguous volumetric data before pre-processing them
to reduce the noise. Then 3D segmentation is
978 -1-4244-1579-3/07/$25.00 © 2007 IEEE
performed to segregate the objects of interest by
thresholding of volumetric data. The objects are
visualised using a special rendering algorithm.
2.1. Image data
The standard DICOM (Digital Imaging and
Communications in Medicine) format is chosen as
input of 2D images for carrying out 3D reconstruction.
In CT, each pixel [f(x, y)] is assigned a numerical
value (average of all the attenuation values contained
within the corresponding voxel), which is compared to
the attenuation value of water (zero) and displayed on
a normalised scale of Hounsfield units (HU). Each
number represents a shade of grey on a range of 2000
HU wide, with +1000 (white) and –1000 (black) at
either end of the spectrum.
2.2. Interpolation
CT/MRI scan slices are stacked (figure 1) in the
direction as they are acquired and re-sampled into the
thickness of pixel size (T) to get an accurate, highresolution, isotropic medical model. Once the voxel
size is decided, the interpolation between adjacent
slices is carried out to generate the missing information
between slices.
The algorithm for the slice interpolation is:
if(ST>ISD){
OL=ST-ISD;MST=ST-OL;H=(ST-OL)*N;}
if(ST<ISD){
GAP=ISD-ST;MST=ST+GAP; H=(ST+GAP)*N;}
if(ST=ISD) H=ST*N;
MN=H/T
if(ST t ISD) { i=0;
For v
1 to MN {
VP
v V ; IP
i MST;
if(VP IP) I[v]
I[i];
else if((VP ! IP)&((VP V) IP))
I[v]
I[i] I[i 1]/ 2 ;
else { i ; I[v]
I[i]; }
}}
If(ST<ISD){i=0;
1 to MN {
For v
VP
v V ; IP
i MST;
if(VP IP) I[v]
I[i];
else if((VP ! IP)&(VP IP GAP) &
((VP V) IP)) {
(IP VP V) / V ; b
1 a;
a
I[v]
a I[i] b I[i 1] }
else if((VP ! IP) & (VP IP GAP)) {
(VP IP) / GAP; b
1 a;
a
I[v]
a I[i] b I[i 1]; }
else if(VP ! IP GAP) { i ; I[v]
}}
I[i]; }
Where, ST=Slice Thickness; MST=Modified Slice
Thickness; ISD=Inter Slice Distance; OL=OverLap;
GAP=Gap; N=Number of slices; MN=Modified
Number of slices; H=Height; VP=Voxel position;
IP=Image position; I[v] = voxel intensity, I[i] = pixel
intensity;
Figure 1: Stacking of CT images for 3D reconstruction
2.3. Pre-processing
Gradient anisotropic diffusion, a non-linear
smoothing filter, is used to remove the noise and
texture. It produces a Gaussian smoothed image, which
is the solution to the heat equation, with a variable
conductance (k) term (function of the gradient
magnitude of the image) to limit smoothing at edges.
The non-linear smoothening expression is
’I
2
wI
2
div>c’I @ , c( ’I ) e 2 k ,
wt
where, div is the divergence operator, ’, is the image
gradient, and c is the diffusion coefficient. Allowing
the diffusion coefficient to vary with respect to the
local image gradient produces anisotropic diffusion
(quality is mainly tied with the method). This filter
requires three parameters: the number of iterations, the
time step, and the conductance parameter.
2.4. Voxel based segmentation
The object or region of interest in the volume data is
extracted using threshold based segmentation. Each
slice may consist of N objects representing distinct
tissues: tumour, invasion, marrow, bone, fat, muscle
and others. Each of these objects is characterised by a
range of intensity values, or threshold range (Imin and
Imax). The object flags are set in a single pass, using the
following criteria
­1 if I min f ( x, y , z ) I max
v( x, y, z ) ®
0 otherwise
¯
where, Imin and Imax are the minimum and maximum
values, respectively, in the threshold range for the
object of interest.
2.5. Volume rendering
A realistic display of volumetric data is difficult owing
to lack of information about surfaces and their normal
data. In this work, the surface normal of the boundary
voxels are estimated by a bit coding method of
neighbourhood configuration. The method of index
coding is shown in figure 2 along with an example
index code computation; the arrow indicates the
direction of surface normal.
Figure 2: Neighbourhood configuration driven normal
generation with an example
3. Geometric reasoning: thickness analysis
Precise measures of the thickness and their
distribution in different regions are vital in custom
implant design, especially hip joint in which the
mating surfaces of implant and the resected portion
must align perfectly to reduce the failure rate.
Thickness distribution visualisation also helps in
predicting the disease at early stage (for example knee
arthritics). A definition of thickness suitable for highly
intricate objects is taken from our earlier work [11];
the methodology to evaluate the same is described here
(figure 3).
3
Let p, s  Z , and ȡ is a sequence of moves from p to
s. The number of moves in face, edge, and vertex is
noted as f, g, and h respectively. The weighted distance
(d) can be calculated as
l (U) fa gb hc
d ( p ) min la ,b,c (U)
The values a, b, and c are the weights assigned to
voxels corresponding to the movement. The actual
distance of any cell from the closest boundary can be
calculated by the following equation.
l1,
2, 3
(U) V ( f 2 g 3 h )
The computed weights of each cell in a slice after
first and second pass are shown in figure 4.a and 4.b.
Figure 4: WDT: (a) after 1st pass (b) after 2nd pass
4. Implementation and results
Figure 3: Internal thickness at a point P
Definition: Interior thickness at a point Pi inside an
object is defined as the minimum distance of Pi from
the nearest surface Sj of the object.
Interior thickness Tint (Pi) =min (dist (Pi, Sj))
The locus of points with the highest local internal
thickness values defines the object skeleton. The
thickness definition can be computed using distance
transformation.
Computation: Let a binary model consists of an
object O and its background Oƍ. A distance
transformation produces the distance map D, in which
the value for any voxel is the distance to the nearest
voxel of O’. Complexity is proportional to number of
voxels. A generic way to approximate Euclidean
distance (EDT) is assigning weights (3-4-5) for a move
in different directions (face, edge, and vertex) to a
minimum neighbour (N26 configuration) against the
actual distance (1-¥2-¥3). This is termed as weighted
distance transformation (WDT), which is two-pass
algorithm, faster than and not as complex as EDT.
The methodology for virtual 3D reconstruction was
implemented using Microsoft Visual C++ 8.0 and
OpenGL in Windows XP environment. The results are
presented here for pelvic bone reconstruction from 105
CT axial slices in DICOM format. The images were
acquired with the following parameters: pixel
width=0.77 mm, slice thickness=2.10 mm, and inter
slice distance=3.0 mm. Thus, gap=1 mm. The voxel
size was set to pixel size. According to modified slice
thickness, the total number of slices increased to 394
with a resolution of 0.7735 mm. The total time taken
for importing, stacking, pre-processing, segmenting,
and rendering was 26.3 sec and memory occupied was
564 MB using a 2 GHz CPU with 2 GB RAM. The
final volume model is displayed with three sectional
views of sagittal, coronal, and axial images as cutting
plane passes through the middle of model in all three
directions as shown in figure 5.
Internal thickness was calculated and displayed for
one half of segmented pelvic model (figure 6), which
took 9.7 sec on the same computer mentioned above.
In the figure, black denotes minimum thickness (4.1
mm) and white denotes maximum thickness (11.3
mm).
provides valuable information for various applications
including clinical (surgery planning) and engineering
(custom implants). The successful testing of the
algorithms on a standard Windows computer has
demonstrated the potential and benefits of high
resolution models in medical applications.
7. Acknowledgement
Figure 5: Pelvic bone reconstruction
This work is supported by the Office of the
Principal Scientific Advisor to the Government of
India, New Delhi, under the project titled ‘OrthoCAD
Network Research Facility for Development of Mega
Endo Prostheses’. The authors acknowledge the
valuable inputs and medical images provided by Dr.
Manish Agarwal, Orthopaedic Surgeon, Tata
Memorial Hospital, Mumbai.
8. References
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Cambridge University Press, Cambridge, UK, 2002.
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[1]
Figure 6: Pelvic bone section thickness mapping
5. Further work
This work is a part of an ongoing project for
developing a complete computer aided surgery
planning and customised implant design system driven
by geometric reasoning at OrthoCAD Network
Research Facility, IIT Bombay, India. This work is
being further extended by including: (i) automated
tumour identification and quantification for joint
salvage surgery, (ii) extraction of prosthesis design
parameters, and (iii) virtual interaction tools.
6. Conclusion
An efficient approach to reconstruct highresolution, accurate volumetric medical models starting
from CT/MRI slices has been presented. It can handle
variable slice thickness and inter slice distance
(common in older machines), as well as contiguous
slices (produced by spiral CT machines). Voxel-based
noise reduction and segmentation has enabled faster
and computationally inexpensive results than the
earlier image-based operations. The volume rendering
using a look-up table based on neighbourhood
configurations enabled faster realistic visualization. All
these led to higher speed (near real time on standard
computers), accuracy (finer resolution), and robust
medical models (in a single pass algorithm). The
geometric analysis and mapping of internal thickness