The SSV model is intended to give a first-order estimation of ROI geometry variability using few variables as input. This rough model, dependent on a linear correlation between the ROI centroid locations and volume, is used to guide necessity of finer linear model, i.e., PCA model.
How the SSV model predicts an ROI surface position change due to deformable motions is illustrated in Figure 24. The SSV model relies on a strong correlation (and therefore a simple function) between the ROI-CTVprostate centroid distance (CDROI-P) and the ROI volume (VROI) as
Figure 24. An illustration of how to obtain a new position of surface voxel of a ROI (other than CTVprostate)
(SROI’) based on a SSV model. As CTVprostate-centroid alignment is assumed, the centroid position of
CTVprostate, old as CCTVprostate and new as CCTVprostate’, is always known. By sampling a new position of the
ROI centroid (CROI’), the new centroid distance between ROI and CTVprostate` (CDROI-P’) and ROI centroid
offset (ΔCROI) can be calculated. Assume that ROI volume change (ΔVROI) is a function of CDROI-P’, ΔVROI,
and ΔCROI can be used to estimate the surface point change relative to old position (SROI) so that SROI’ can be
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ROI ROI-P
V a(CD c)b d
(35)
where a b c d, , , are the parameters to fit using correlation and least-square fitting based on information from the patient database. On each image set for each patient, CTVprostate volume
(VCTVprostate), ROI volume (VROI) and the position of ROI surface voxel (SROI), ROI centroid
(CROII), CTVprostate centroid (CCTVprostate) and their difference (CDROI-P) are known and can be
used to determine a b c d, , , . Therefore, the ROI volume change between the reference and the fractional image sets becomes
ROI ROI-P ROI-P
ΔV a[(CD c)b(CD 'c) ]b
(36)
where ' means a new value on the fractional image set. The SSV model also assumes that the ROI radius 3
ROI ROI
r ΔV , so the new ROI surface voxel position SROI' can be estimated by
3 ROI
' v( )u( ΔV )
ROI ROI
ROI ROI ROI ROI
ROI ROI
S ' C '
S S C ' C
| S ' C '| (37)
where u v, are the parameters to fit based on data variation from the patient database. Equation (37) means that SROI' is determined by three components: (1) the old position of ROI
ROI
S , (2) the ROI centroid position change CROI' C ROI, and (3) the radius changes in direction pointing from SROI' to CROI'. Assume the direction of SROI relative to CROI remains unchanged and substitute equation (36) into equation (37), we have
3
ROI-P ROI-P
' v( )u a( [(CD c)b(CD 'c) ])b )
ROI ROI
ROI ROI ROI ROI
ROI ROI
S C
S S C ' C
| S C | (38)
For a CTVprostate-centroid-aligned treatment, the new CTVprostate centroid position (CCTV-centroid') is
always “known”. By sampling ROI centroid position (CROI') from a PDF based on the patient database, the new ROI surface voxel position SROI' can be calculated.
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To test if the SSV is valid for deformable organ motion, the mean and SD of correlation coefficients of different volume and different centroid distance relative to CTVprostate across 19
patients are calculated in Table X. Only the bladder-CTVprostate centroid distance and bladder
volume show strong correlation (> 0.9), when parameter b = 1. Similar coefficient calculations have been done for bladder wall and rectal wall, but none of them showed strong correlations. When b is replaced by 2 and 3, the correlation coefficient does not change significantly for all the structures. For example, patient E has correlation coefficients of 0.947 for bladder volume (VB) and bladder-CTVprostate centroid distance (CDB-P,z), 0.950 VB and CDB-P,z2 and 0.939
VB and CDB-P,z3. Based on the correlation coefficient, the SSV model has limited applications
for prostate cancer modeling and is only potentially useful to predict bladder deformable motion.
Table X: The mean and SD values of the correlation coefficients between volumes and centroid distances of ROI for 19 NKI patients. Highly correlated variables are highlighted.
mean SD VB, CDB-P,z 0.943 0.039 VB, CDB-P 0.923 0.064 VR, CDR-P 0.427 0.447 VR, CDR-P,y 0.183 0.461 VB, CDR-P 0.149 0.425 VR, CDB-R 0.125 0.395 VR, CDR-P,z 0.100 0.426 VB, CDR-P,y 0.075 0.374 VR, VB 0.029 0.454 VB, VP -0.063 0.369 VR, VP -0.164 0.283
Abbreviations: VB: bladder volume; VR: rectum volume; VP: CTVprostate volume; CDB-P,z: centroid distance
between bladder and CTVprostate in z axis; CDB-P: centroid distance between bladder and CTVprostate; CDR-P: centroid
distance between rectum and CTVprostate; CDR-P,y: centroid distance between rectum and CTVprostate in y axis.
For the bladder SSV model, the residual error of bladder volume in equation (35) with fitted parameters based for an individual patient can be significant. In Figure 25 (a), the bladder volume residual error using fitted parameters specific for patient E is about 50 cc for a 200 cc VB
and therefore the surface position error, if comparable to radius difference, is approximately 3mm. For each patient, the residual error as a result of each patient-specific fitting is plotted into
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a population-based histogram and fitted by a normal distribution. The SD of this fitted normal distribution is 27 cc, so the uncertainty for VB is ±54 cc for a 95% confidence interval. Based on
these numbers, the SSV model is oversimplified and not representative for modeling organ deformable surface positions for prostate cancer patients. A higher dimensional model, i.e., PCA model, is needed to represent more realistic organ deformable motions in the prostate cancer studies.