Assumption of Sufficient Sampling

The convolution model employs a PDF to describe the motion of the target during treatment. As described by Craig et al. [110], in the strict mathematical interpretation of the convolution model the assumption being made is that the dose is being delivered in an infinite number of fractions, each delivering an infinitesimal dose. In the case of interfraction motion, each daily fraction represents a single sample of the PDF describing the motion. A typical fractionated treatment plan will deliver the prescription dose over the course of approximately 30 fractions. As a result, the positions sampled by the anatomy during the delivery of this finite number of fractions may not end up being representative of the PDF used to make the predictions by the convolution model. Craig et al. showed that the average maximum dose error was 11% for the convolution model when the effect of finite fractions was considered. Ultimately the authors warned that the effect of finite fractions appears to have a greater impact on the delivered dose distribution than typical plan evaluation parameters.

When using the convolution model in the context of intrafraction motion the problem of sufficient sampling also must be considered. In this case, the motion of the target during beam

delivery must correspond to the motion of the target used to generate the PDF. In other words, the proportions of the positions sampled by the target while the treatment beam is on must be substantially similar to the proportions of the positions measured a priori. In order to test this assumption a simulation of ‘beam on sampling time’ was developed.

Each recorded breathing trace was analyzed to determine the minimum required time to achieve sufficient sampling. This was done empirically using the Kullback-Leibler divergence as a measure of the similarity between two probability distributions. The sufficient sampling simulation was performed by first selecting a random point in the breathing trace as the starting point. The trace was then broken into 0.1 s subsections beginning with the starting point, with the trace repeating itself as necessary. A ‘sub-PDF’ was then generated from each of the new trace subsections and the sub-PDF was compared to the PDF generated from the full trace using the Kullback-Leibler divergence. The goal is then to determine ‘at what level of Kullback-Leibler divergence between the two PDFs are the resulting blurred distributions substantially similar to one another?’. To determine an empirical answer, each sub-PDF was also convolved against a dose profile to determine the resulting blurred distribution. To evaluate the difference in dose

coverage the D95metric was used. The D95 is the minimum dose received by 95% of the target

volume. A ratio between the D95 of each blurred dose profile from the sub-PDF convolutions

and the D95of the blurred dose profile from the full PDF convolution was calculated. By relating

the Kullback-Leibler divergence of each sub-PDF to the relative D95offered by each sub-PDF,

we can get an understanding of the sampling time required to achieve sufficient sampling. As part of this simulation study, 42,347 sub-trace PDFs were generated and compared to their corresponding full trace PDF. An arbitrary threshold Kullback-Leibler divergence value of 1 was selected to define a sub-trace PDF as ’substantially similar’ to the corresponding full trace

PDF. A histogram of the relative D95 when the Kullback-Leibler divergence is less than 1 is

the expected value after convolution (Figure 4.3). Furthermore, 94.4% of all the patient breathing traces analyzed reached a Kullback-Leibler divergence of less than 1 within the maximum 12 seconds of sampling time used in this simulation. The minimum required sampling time to generate a sub-PDF with Kullback-Leibler divergence less than 1 when compared to the full PDF for each of the patient breathing trace is summarized in Figure 4.4. Although the vast majority of breathing traces met the Kullback-Leibler divergence threshold within 12 seconds or less, the breathing traces which are most likely to fail to meet the Kullback-Leibler divergence threshold are traces with very large standard deviation (≥ 0.8cm).

Figure 4.3: A histogram showing the distribution of relative D95 values when comparing sub-

PDF and full PDF convolutions. All sub-PDFs included in this data had Kullback-Leibler diver- gence less than 1 when compared to the full trace PDF.

Figure 4.4: A histogram showing the distribution of sampling times required to generate a sub- PDF that has a Kullback-Leibler divergence of less than 1 when compared to the full trace PDF.

As can be seen in Figure 4.4, after six seconds of beam on time, 92.8% of the breathing traces had achieved sufficient sampling. By ten seconds of beam on time, over 99% of the breathing traces had achieved sufficient sampling. Beam on times of six seconds or greater are common for many types of radiotherapy treatments offered to lung cancer patients. In the case of a conventional 3D conformal radiation treatment (3DCRT) or four field box, the typical beam will deliver around 80 − 100 monitor units (MU). Using the standard dose rate of 400 M U/min, this corresponds to a beam on time of 12 − 15 s. By decreasing the dose rate, additional positional sampling can be achieved in order to help ensure the sufficient sampling assumption is met. Although modern treatment technology is trending towards higher dose rates, increasing the dose rate has the effect of decreasing the reliability of the convolution model by jeopardizing the sufficient sampling assumption. Additional care must be taken when using high dose rates (> 400 M U/min) if the convolution model is being employed. Regardless of the dose rate being

employed for treatment, sufficient beam on time must be used to ensure accurate application of the convolution model. For example, a modern flattening- filter-free (FFF) treatment beam may be operated at a dose rate of 1200 M U/min. To ensure sufficient sampling with a 10 s beam on time, a minimum of 200 M U should be delivered by the treatment beam.

The assumption of sufficient sampling may also be jeopardized in the case of an IMRT treat- ment. IMRT treatments have beams which are subdivided into many segments with different beam shapes. Each beam segment may account for as little as 10 M U , and in this case it is very likely that the sufficient sampling assumption will not be strictly met. The problem of small beam segments interacting with targets in motion is known as the ‘interplay effect’ and it has been an- alyzed by several authors [112, 113, 114]. These authors report that in most cases the interplay effect results in a negligible difference between the planned and delivered dose distributions over the course of a conventionally fractionated treatment plan. This fact makes the application of the convolution model to IMRT treatment plans possible, however special care should be taken to avoid small beam segments, and ensure that a large number (∼ 30) of fractions are used, in order to satisfy the sufficient sampling criteria.