X-ray Tomography. A superficial introduction, but sufficient enough to get us started in surgical navigation.

Transcription

1 X-ray Tomography A superficial introduction, but sufficient enough to get us started in surgical navigation.

2 X-ray absorption in homogeneous tissue I o I o / I d m = density I I=I o e -kdm k= constant re. bean energy/hardness d ln(i) ln(i o ) =-kdm ln(i o ) ln(i) / kd = m

3 X-ray absorption in the body I o I o I o d d I b I a I o I o I a I b

4 X-ray absorption in the body single beam I 1 =I o e -kdm 1 I o I 2 =I 1 e -kdm 2 I 3 =I 2 e -kdm 3 I 2 =I o e -kdm 1 e -kdm 2=I o e -kd(m 1 +m 2 ) I 3 =I o e -kdm 1 e -kdm 2 e -kdm 3e -kd(m 1 +m 2 +m 3 ) d, m 1 d, m 2 I n =I n-1 e -kdm n I n =I o e -kd(m 1 +m 2 +m 3 +m n ) d, m i ln(i o )- ln(i n ) = kd(m 1 +m m n ) d, m n (ln(i o )- ln(i n ) )/kd = m 1 +m m n I o If we consider the m densities as unknown variables, this is a linear equation.

5 X-ray absorption in the body multiple beams (ln(i o )- ln(i n ) )/kd = sum(m L1 ) sum along L 1 (ln(i o )- ln(i n ) )/kd = sum(m L2 ) sum along L 2 (ln(i o )- ln(i n ) )/kd = sum(m L3 ) sum along L 3 (ln(i o )- ln(i n ) )/kd = sum(m Ln ) sum along L n Lots of linear equations Number of variables = as many little voxels I am using 10cmx10cmx10cm tissue at 0.1mm resolution = 10 9 I o L 1 L 2 L 3 L n Number of equations = as many as beams I can differentiate in the detector. Example: 20cm x 20cm detector, 0.05 mm pixel size 4000 x 4000 = 16 x 10 6 beams Trouble: each voxel is only hit by maximum one beam. i.e. each variable appears only once in the whole equation system.

6 X-ray absorption in the body even more beams (ln(i o )- ln(i n ) )/kd = sum(m L1 ) sum along L 1 (ln(i o )- ln(i n ) )/kd = sum(m L2 ) sum along L 2 (ln(i o )- ln(i n ) )/kd = sum(m L3 ) sum along L 3 (ln(i o )- ln(i n ) )/kd = sum(m Ln ) sum along L n Lots of linear equations Number of variables = as many little voxels I am using 10cmx10cmx10cm tissue at 0.1mm resolution = 10 9 Number of equations = as many as beams I can differentiate in the detector. Example: 20cm x 20cm detector, 0.05 mm pixel size 4000 x 4000 = 16 x 10 6 beams Trouble: each voxel is only hit by maximum one beam. i.e. each variable appears only once in the whole equation system. Must take many images from many poses

7 Ta dah! Computer Tomography (ln(i o )- ln(i n ) )/kd = sum(m L1 ) sum along L 1 (ln(i o )- ln(i n ) )/kd = sum(m L2 ) sum along L 2 (ln(i o )- ln(i n ) )/kd = sum(m L3 ) sum along L 3 (ln(i o )- ln(i n ) )/kd = sum(m Ln ) sum along L n (ln(i o )- ln(i 1 ) )/kd (ln(i o )- ln(i 2 ) )/kd (ln(i o )- ln(i 3 ) )/kd (ln(i o )- ln(i 3 ) )/kd (ln(i o )- ln(i i ) )/kd = H n x m matrix with 0s and 1s huge Very sparse m 1 m 2 m 1 m 1 m 1 m 1 m 1 m 3 H is numerically invertible yielding the m vector: material density for each voxel (ln(i o )- ln(i n ) )/kd m m

8 CT = density map

9 Concept of CT Scanner Planar fan beam Type1 Type2

10 CT beam configurations

11 Spiral CT, multi-detector CTs SINGLE SLICE SPIRAL MULTISLICE SPIRAL Spiral path with 4,8,16,32,64, 128, 256, 512 rows of detectors

12 Recent CT scanners (2008)

13 CT gantry inside

14 Modern CT scanners (2018) GE Phillips

15 CT catered to us as series of slices

16 CT coordinate system pixel Y Field of view X Slice thickness Slice index

17 Navigation between CT slices k -- slice index thk = slice thickness (cm or mm)

18 j pixel (row index) NY number of pixels (typically 512) FOVY size of the body captured (in cm) dy = FOVY / NY (pixel size, in cm) Navigation in a CT Slice i pixel (column index) NX number of pixels (typically 512) FOVX size of the body captured (in cm) dx = FOVX / NX (pixel size, in cm)

19 Conversion between pixel and metric coordinates in CT imaging Simple scaling: P(x,y,z) = P(i*dx, j*dy, k*thk) Where: dx = FOVX / NX dy = FOVY / NY & FOVX, FOVY, NX, NY, thk are usually printed on the CT image header S = d x d y thk

20 Tilted CT gantry y z Ouch! No longer a Cartesian coordinate system

21 Cone beam CT reconstruction

22 Cone beam CT - truncation

23 Cone beam CT metal artifacts F Edward Boas & Dominik Fleischmann *by

24 Cone beam CT yields isotropic volume

25 Cone beam CT reconstruction examples

26 Organs can be contoured

27 Organs reconstructed from contours

28 CT/X-ray fiducial markers (donut)

29 CT/X-ray fiducial markers (BB)

30 Next tracked navigation A C B v 3 v 1 v 2

31 Tracked Navigation F Trac The target anatomy is known in F CT The head markers were found in F CT The head markers are tracked in F trac The tool is tracked in F trac The tool (tip/axis/etc) is known in F Tool F Tool FMar Trac F Trac Tool model of tool Computer F Tool real tool C F A Mar Tool F Tool Tracked navigation = paint the model of the tool over the CT on the computer screen as the real tool is moving over/on/inside the patient in the operating room F CT Mar B v 3 v 1 v 2 F CT F Mar Operating room For this we need to compute the transformation from Tool to CT coordinates F CT Tool = F CT Mar * F Mar Trac * F Trac Tool