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Inverse planning is a technique for designing a radiotherapy treatment plan. A radiation oncologist defines a patient's critical organs and tumour and gives target doses and importance factors for each. Then, an optimization algorithm is used to determine a suitable set of treatment parameter values that best match all the input criteria.

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Inverse Planning Approaches

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Quadratic Objective Functions

The simplest inverse planning algorithms use quadratic objective functions, incorporating minimum / maximum specifications for dose to targets and organs-at-risk (OARs). The resulting algorithms can be readily implemented using standard gradient-based algorithms, such as steepest descent or conjugate gradient.

Goals for minimum / maximum dose are simple to define, but they lack the ability to reflect more complex clinical concerns. For instance, some OARs can be partially irradiated without significant side effects. If a simple maximum constraint is specified for the OAR, it may reduce dose to the target even though some parts of the OAR could have received additional dose without risking side effects.

For this reason, when evaluating the planned dose distribution, clinical users typically use the dose-volume histogram (DVH) in addition to basic statistics. DVHs are often expressed in cumulative form, so that the DVH value is at 100% of the structure volume at its minimum dose and reaches 0% at the maximum dose. This form of the DVH curve quickly provides information about the fraction of a particular structure receiving dose at or above a certain level.

Quadratic objective functions can be enhanced to represent some aspects of the shape of the desired DVH curves. These are usually incorporated as partial volume constraints, which are discrete points along the cumulative DVH curve specifying both maximum dose and % volume to be satisfied. Partial volume constraints add additional degrees of freedom that address some of the limitations of minimum / maximum constraints.



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DVH-Matching Objective Functions

The partial volume constraint approach can be generalized even further to allow the user to specify an entire DVH curve to be matched. Carol et. al. (2000, U.S. Patent 6,038,283) use an objective function based on a direct match of the cumulative DVH to a goal curve within a simulated annealing optimization. Though simulated annealing is believed to arrive at a global optimum more reliably than gradient-based optimization, it provides no opportunity to take advantage of performance improvements from explicitly computed gradient information. Additionally, because it is a stochastic method, it may have characteristics that make it less steerable than gradient-based methods.

Hristov et. al. (2002) demonstrates that, in theory, it is possible to directly evaluate the gradient of an objective function based on a direct matching of DVH curves. To do this, they build up an analytical expression for the objective function using the Heaviside function (unit step function) and then apply the chain rule to calculate the partial derivatives. The use of the Heaviside function follows from their objective function that matches the cumulative form of the DVH, as with the Carol method.

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Matching Dose PDFs using Metrics from Information Theory

For a clinical user, the cumulative form of the DVH may be the most familiar for evaluating the dose distribution and for specifying the goal DVH curve to be matched. However, there are possible advantages to using the DVH in other forms, at least for the internal objective function calculation.

The differential (non-cumulative) DVH, when normalized, is a suitable approximation to the dose probability density function (PDF). Formulating the objective function as a matching of dose PDFs opens the opportunity to utilize a number of methods from the information theory toolbox. The Pheonix algorithm uses one metric from information theory, Kullback-Liebler (K-L) divergence, to compare the actual PDF to the goal PDF.


The K-L divergence is zero if both PDFs match exactly, and becomes larger as they become less similar.

Using the K-L divergence within the context of optimization confers the advantage of “focusing” the optimization on the parts of the PDF for which changes can be the most profitable. In the figure, the PDF region with the highest K-L divergence term is the region where the actual PDF has significant density that differs substantially from the goal PDF. This is the part of the actual PDF where the optimization should concentrate in order to best meet the goal.

For more information about this approach see Pheonix Theory of Operation.

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References

• Hristov et. al. 2000
• Carol et. al. 2000