Dose Calculations: Where and How to Calculate Dose. Allen Holder Trinity University.

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1 Dose Calculations: Where and How to Calculate Dose Trinity University R. Acosta, W. Brick, A. Hanna, D. Lara, G. McQuilen, D. Nevin, P. Uhlig and B. Slater

2 Dose Calculations - Why are they so important? What is a Dose Model? A dose model estimates the rate at which energy, fluence, is deposited into the anatomy. Dependencies The dose model depends on the patient anatomy, the shape and energy of the beam, and the pathway through the patient. Error Propagation Small errors propagate through the design/decision process, and medical physics standards require small error limits (especially for the final analysis). Discrete Approximation The anatomical dose is continuous but is discretely approximated. Data The dose model defines the data used to decide all design questions, without good data, there is no reason to trust the design.

3 Monte Carlo Models Particles are accelerated and pass through a flattening filter, a collimator, and then through various densities of tissue. The simulation tracks the path and interactions of each particle to build a dose profile.

4 Small Pencil Beam Models Idea The idea is to sub-divide a beam into sub-beams and estimate the dose from each sub-beam with a deterministic model. Shortcomings This method has problems. Small beams don t aggregate precisely to measure large beam geometries. Tissue inhomogeneities can be accounted for on average but not locally. Requires machine specific values (table look-ups). Advantages There are advantages: The data works well with optimization models. Accuracy is about 98% in water based models. They are commercially used (BrainLab). They are fast.

5 Mathematical Model Notation Let D (p,a,i) be the rate at which dose point p gains radiation from sub-beam i in beam a. Model We use the following model (due to Nizin) D (p,a,i) = 8 >< >: Interpretation P 0 e µ(d M) (1 e γr ) + rdα d r+m ISF O, d M ` 0.4d M P 0 (1 e γr ) + rmα M r+m ISF O, 0 d < M. P 0 e µ(d M) is the primary dose contribution (1 e γr ) is the amount of open-field radiation in the sub-beam (rdα d )/(r + M) measures scatter contributions from surrounding sub-beams ISF is the inverse square factor (square of the ratio between the distances from the source to the isocenter and from the source to p) O is an off axis calculation

6 Geometry

7 Discrete Approximation of Continuous Dose Accuracy vs Size The underlying optimization problems are large, and every dose-point adds constraints and variables (for dose-volume constraints). Over Optimization If dose-points are spaced too far apart or the sub-beams are too narrow, we find that optimized treatments often use beams that pass through what appear to be vacuums in the patient. Positioning Even with a fine mesh of dose-points, a small shift can make plans look different on standard evaluative measures. Volume Size Computed volumetric calculations depend on the mesh of dose points.

8 An Iterative Approach to Problem Size Standard Approach Conventional treatment design asks the designer to select beams, which limits the variables and allows a fine dose mesh. Our Approach RAD was designed to select beams, which means we need to limit the dose points since initial solves use many beams. A Treatment Swath We define a treatment swath by (realistically) assuming that the only tissue we need for the design process lies along the planes defined by the various couch positions. Iterative Approach We place dose meshes in the target and restricted regions and continue as follows. Design a treatment with no dose points in the remaining tissue. Search for 1cm 3 regions which are likely to lead to hot spots and place dose points in these regions. Repeat until no hot spots are likely.

9 Problem Size Upper Limits Similar problems report the need for 600 Gigabytes of space (Gamma Knife). Swath Reductions The swath alone reduces the size on simple problems from 100s of Gigabytes to 10s of Gigabytes (single couch angle and 2mm grid). Iterative Reductions The iterative approach usually reduces the problem to about 1 Gigabyte but requires several (no more than 5) solves. However, this makes the problem approachable with standard solvers such as CPLEX.

10 Standard Example Approximate dose matrix sizes for a 20cm 3 region with 3 couch angles around a single isocenter in the middle of the patient volume. A 2mm 3D grid spacing is assumed. Each swath is 1cm in width (6 dose points wide), is parallel to one of the axes, and passes through the center of the patient volume. The example assumes that 10, 000 dose point in the treatment volume are not normal and that 20, 000 additional dose points outside the treatment volume are needed to describe the critical structures. Each beam is assumed to have a grid of sub-beams, of which 4 are assumed to strike the target. The final treatment has 10 angles. Size Number of Rows Number of Columns Size of A Patient Volume Sequential Solves Final Solve

11 Dose Resolution and Beam Selection Dose Estimation Affects Beam Selection We have mathematically studied beam selectors and one of the issues left is to understand how dose measurements alter standard beam selectors. An automated selection process was used on the example below, but the dose point mesh changes from 5mm to 3mm

12 Some Goals Stability Models, solution procedures, and beam selectors should not only be able to give trustworthy results, but they should also be somewhat stable with regards to the resolution of the dose. This allows the treatment to be designed on smaller problems that accurately represent the more accurate estimations. It also validates that the discrete approximation matches the continuous dose. Warning Even with small dose meshes, a final comparison between the treatment evaluated on the design mesh versus a fine mesh shows that evaluation itself can adjust our perspective of the same treatment. There is much work to be done.

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