Coverage probability is an important and unique dosimetric concept that directly reveals the relationship of ROI dose and GUs. This metric is important for the evaluation/comparison of plans under the influence of GUs. A coverage probability value is the probability that a realized target or OAR dose metric Dv exceeds the dose of interest (Rx, tolerance or other dose) when
treatment planning and delivery uncertainties are taken into account. (Gordon and Siebers 2008) For evaluation of a static plan (one free of GUs), coverage probability reduces to boolean (i.e., 0% or 100%) values. However, realistically, a static patient geometry is not possible in multi- fractioned radiation therapy. Coverage probability values of targets and OARs often differ considerably from the value implied by the static plans (Gordon and Siebers 2008, Xu et al. 2011). Therefore, coverage probability evaluations are essential for evaluating plans under the influence of GUs.
Two methods have been used to estimate coverage probability. One is using DVCM and the other is dosimetric margin distribution (DMD). The DVCM method estimates coverage probability by simulating possible treatment outcome while the DMD method calculates coverage probability by a formula under the condition that the relationship of GUs PDF and coverage probability is known. In fact, the DMD method is a simplified version of the DVCM method when a simple GUs PDF (e.g., Gaussian) is considered. Compared to the DMD method which estimates coverage probability values for only a single dose-volume metric, the DVCM method (mentioned in section 2.1) is more general and permits simultaneous analysis of multiple dose-volume metrics. Therefore, the DVCM method is used to estimate coverage probabilities for this dissertation. Both methods will be discussed in the following paragraphs.
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3.2.1 Coverage probability estimation method: DVCM
One way to estimate coverage probability is by constructing a DVCM, a 2D dose volume coverage map that contains coverage probability for each dose-volume metric for an ROI. (Gordon et al. 2010) It has been mentioned briefly in the workflow of pDVH and COP in Figure 6 (page 20) and will be described in detail (Figure 11) here. To get a DVCM, a large number (ntx) of virtual treatment courses, each with multiple fractions (nfrac) are simulated. For each
virtual treatment course, different GUs sampled from PDF of the constructed GU model result in different ROI DVHs as a result of the accumulated dose distribution of all the nfrac fractions. The
axes of dose and volume of DVH are divided into small increment, 0.1Gy and 1% respectively, to create a 2D grid map. For the 1st DVH of the 1st virtual treatment course, the grid squares below or left to the DVH curve are assigned value 1.0 (=100%) and the others are 0.0. Then for each DVH, increment all grid squares lying below or to the left of the DVH by 1.0. All the grid values are divided by total number of DVHs (=ntx) to finalize this map, which is so-called
DVCM. In an ROI DVCM, each grid square (corresponding dose d and volume v) contains the probability that, in an individual treatment course, Dv ≥ d can be achieved for the ROI when
GU(s) are considered. For the example of a ROI DVCM in Figure 11, Dv with probability 1.0
can be achieved 100% based on the 2 simulated virtual treatment courses. The probability of each grid square is called the coverage probability corresponding to the metric Dv and can be
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3.2.2 Coverage probability estimation method: DMD
Another way to estimate coverage probability is via a DMD, the distribution of dosimetric margins (DMs) over 3D directions. DMD was inspired by the fact that ROI coverage probability with respect to dose d is a function of the distance between the ROI and the volume enclosed by the critical isodose surface of dose d. (Gordon and Siebers 2008) Such distance in a specific direction (φ, θ) is called DM and denoted as Mv,d(φ, θ). DM is the safety margin that the
ROI can be offset while still satisfying a dose constraints Dv ≥ d for CTV or Dv ≤ d for OAR.
For example Mmin,70(φ, θ) (=M100,70(φ, θ)) denotes the maximum distance the CTV (or OAR) can
be offset in the direction (φ, θ) before its Dmin (=D100) falls below (exceeds) 70 Gy. For an
isodose surface that is within or crosses the structure, DM = 0. Examples of DMD for a target CTV and a bladder for prostate cancer treatment are shown in Figure 12. A type I (type II) ROI indicates that the static plan meets (violates) the specified dose-volume criteria. A type II ROI is non-standard and ends up with low coverage for targets and high coverage for OARs. Please refer to Appendix II.b for detailed distinction of type I and type II structures.
Figure 11. Example of how a ROI DVCM is generated for a simple case with only two virtual treatment courses. Get DVH (grey lines) with different GU(s) for each treatment course. For each DVH, increment all grid squares lying below or to the left of the DVH by 1.0. Then normalize the whole grid values.
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When DM Mv d, ( , ) and geometric uncertainty parameter(s) such as SD of systematic setup errors Σ are known, the corresponding coverage probability Q( , , ) in this specific direction can be estimated using the following function,
,
( , , ) ( v d( , ), )
Q f M
(24)
where f() denotes a coverage function whose form depends on the PDF of the geometric uncertainty. Note here we use Q (not q) to represent the coverage probability that is obtained via DMD method (not DVCM method). The overall coverage Q( ) is
, , ( ) ( , , ) Q W Q
(25)where W , is a weighting factor equal to the fraction of 4π sr covered by the ray in the direction
(φ, θ). An example of how Q( ) of a CTV and an OAR varies with Σ is shown in Figure 13, where random uncertainties are accounted for through fluence convolution. Each coverage curve corresponds to a single dose-volume metric Dv of the ROI.
Figure 12. Examples of DMD sampled using fixed angular increment method for (a) CTV Dmin, 79.2(type I),
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For coverage estimations using DMD method, it is important to note that a sufficient sample of DM is required to ensure a representative DMD and accurate Q. The earlier study (Xu et al. 2011) proved that DMD sampling with angular increment ω (fixed angular increment method) or ωeff (isotropic sampling method) = 10° and δ = 0.5 mm should be adequate for
planning purposes.