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In order to provide a solution to this problem, we implemented a linear programming LP formulation in our planning process [ 18 ]. LP based optimization allows to keep the molecular information from the functional image throughout the planning process, as demonstrated in a recent study [ 19 ]. The mathematical expression of the new algorithm is presented below 1.
This algorithm tries to minimize the objective function OF , where the overdose and underdose in each voxel i of the target x i and y i , respectively are penalized by the factors P T , max and P T , min respectively, and the overdose in each voxel i of the OARs x i is penalized by the factor P OAR , max. N t is the total number of voxels of the target and N is the total number of voxels of the problem.
Although the size of the CT grid is x pixels per axial slice, the usual size for the dose calculation grid in the commercial TPSs is x pixels, or even 64 x 64 pixels per axial slice in order to minimize the computational time. Our proposal maintains a commitment to precision so, to account for the expected high heterogeneity, a grid size of x voxels was used in the dose calculation.
In order to test the more adequate co-registration for the grid of dose calculation, three different interpolation methods of the image dataset were analyzed with CARMEN platform: three-dimensional linear, nearest-neighbor and cubic spline. Fig 2 presents the experimental setup for this co-registration study. At the left side, a picture of the anthropomorphic phantom CIRS model is shown, which contains tissue simulating resins that mimic the X-ray attenuation properties of human tissue in head and neck.
Lesions with different volumes and uptake values were simulated to perform realistic SUV segmentations with the highest possible resolution. The left side of Fig 2 also shows the Eppendorf tube V1 of 0.
On the right, the corresponding segmentation of volumes for 3D visualization in the CT grid. Due to the lack of background SUV, it was possible to study the active volumes with the two image reconstructions methods and their correspondence with the known actual volumes. The direct comparison between segmented and actual volumes clearly shows that the EARL protocol led to wrong volume reconstruction, as it was expected.
According the data in Table 1 , the interpolation method selected for the PET and CT fusion images was cubic spline interpolation, and nearest-neighbor interpolation for resampling these data in the dose calculation grid size.
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The choice was taken by considering the lower deviation for all volumes in the different scenarios for each stage along the process. Nevertheless, these small volumes were selected for this test, in order to assess the procedure under an adequate 'stress-proof'.
Imaging in Medical Diagnosis and Therapy
As it was commented before, BT definition by means of volumes implies the inclusion of uncertainties along the planning process and, therefore, biases the correct ulterior monitoring of disease. In response to this, DP has been proposed under two approaches: DP by contour DPBC , where intensity thresholding of subvolumes within the conventional target is made and DP by number DPBN , where the dose is prescribed directly to the values numbers in the functional image. Actually, DPBN approach could be considered as "true dose painting" in the sense that imaging can be really used for prescribing biological distributions and be implemented in the treatment planning algorithms for theranostics purpose.
Unfortunately, as far as we know, there is no commercial TPS in which the numbers are directly managed. Traditionally, DPBN has been only used for specific research studies where the information was discretized by clustering of voxels, which is a halfway solution of the real challenge to incorporate the functional information in the optimization algorithm for the treatment planning. Clustering method remains an optimization based on dose to volumes so, the prescribed dose is done to a whole region as usual, by keeping the dose to healthy tissue under restrictions obeying toxicity levels imposed by population evidence-based medicine.
On the other side, basing the optimization on the biological information provided by the numbers from the functional image supposes a change of paradigm that generates lack of confidence in the physicians.
It would be desirable to have a tool for delineation and planning optimization at the voxel level, but which was also able to allow the traditional evaluation of the treatment based on dose to volumes. This solution would allow to carry out the true DPBN approach and to provide, at the same time, a directly transferable solution to the scenario as a result of using volumes. This joint evaluation was only possible by means of the optimization carried out through our LP formulation, since it is possible to prescribe the dose to each voxel, although this prescription can be the same for all the voxels corresponding to a specific clustering, what is defined for DPBC option.
Now, in the algorithm, each voxel of the targets must fulfill the restrictions of maximum and minimum dose D i , max and D i , min , where each one of these limits is sometimes imposed by the dose prescription map corresponding to DPBC and other times by the corresponding to DPBN.
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These dose prescription maps were imposed voxel by voxel through a linear relationship with SUV, what is usually applied for [ 18 F]FDG [ 28 ]. If another radiotracer is used, such as an indicator of hypoxia or cell proliferation, other relations to impose the prescription map could be considered, but this methodology could be directly applicable. ART is the most important effort in RT for patient-tailored planning, and DP is the way to achieve the most ambitious adaptation of the treatment to the evolution of the disease.
In this case, in order to provide a non-uniform radiation dose distribution to the intra-tumor heterogeneity, the most adequate technique is intensity modulated RT IMRT , where variable radiation intensity is generated across multiple beams shaped with MLC sequenced during radiation delivery.
Usually, a mathematical solution is achieved by means of an inverse method to obtain the map of intensity values for each beam, previously subdivided into virtual beamlets with an individual intensity. Segmentation and sequencing processes are used for finding the required geometrical MLC apertures able to provide the corresponding intensity maps. Therefore, the solution of the optimization process is a set of apertures for each incidence angle with different beam intensities. Unfortunately, searching the MLC apertures from intensity maps demands to start the replanning process from the beginning by considering the data image as a new case, what is a time-consuming process.
To overcome the inadequate optimization method based on the traditional intensity maps when biological considerations have to be taken into account, we developed the BIOMAP algorithm [ 18 ], already implemented in CARMEN, which is based on a specific direct aperture optimization assumed to be more efficient for ART [ 29 ].
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This algorithm takes into account both the morphological and the functional information by means of the generation of ray-tracing projections for each incident beam, which are managed as matrices under Boolean combinations. Therefore, the sequencing process to obtain apertures is performed on maps with biophysical information instead of on intensity maps [ 18 ]. Because the apertures generated by BIOMAP are only based on the image information, possible changes in an updated image from IGRT or from follow-up PET image during the treatment could be easily incorporated in the planning, giving the mentioned patient-tailored treatment.
This solution was integrated in our DP approach based on LP to manage the optimization at the voxel level. No morphological changes were included, but it did not affect the presented methodology since our algorithm can consider each voxel for dose calculation with the corresponding new physical density and SUV for dose prescription. Initially, this case presented a BT composed by three separate lesions to receive a prescription dose of 70 Gy during 30 sessions and a dose escalation up to 82 Gy inside the follow-up SUV heterogeneity.enter site
AAPM VL-Dual-Energy CT Imaging in Diagnostic Imaging and Radiation Therapy
The cumulative dose volume histogram DVH represents the relative volume of the structures receiving doses greater than or equal to values discretized in bins. We used the quality index Q [ 28 ], which is defined as the ratio between the planned dose and the prescribed dose in each voxel and thus, the unity is considered as the ideal value.
Usually, this index is visualized as a histogram QVH to represent the results in a way closer to the traditional evaluation. Another criterion is the quality factor QF , defined as the average absolute deviation of Q from 1 within BTs. Histogram methodology does not provide any spatial information, so the dose evaluation should include the visualization of isodose lines over the image to show the relative dose distribution covering the targets and avoiding OARs as far as possible.
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In this work we did an effort to present results based on numbers in the way that the planning solutions based on volumes are usually evaluated, in order to provide a fair comparison. P3 is a compromise plan able to satisfy both prescription maps, simultaneously. None of the plans did reach the toxicity levels to OARs. Isodose lines are presented for three representative axial slices to meet both prescriptions represented on the CT images. The third and fourth rows show the robust solution fulfilling both prescriptions, simultaneously. The corresponding isodose lines show the relative dose distributions as a percentage of the minimum prescription value 70 Gy for the three plans.
The isolines are visualized over both prescription approaches on CT image. DVHs and QVHs are presented at the left for each planning solving the corresponding dose prescription.
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Quality evaluation results for the same three plan solutions are given in Table 3 , regarding the Q, QF, and also the CI for the dose to volumes evaluation. Considering that isolines distribution is a relative dose evaluation method suited to volumes but not so much for numbers, as it is evident in Fig 3 , we decided also to show in Fig 4 the Q index maps corresponding to the same slices for both, the initial planning to Phase I, and the adaptive planning to Phase II. Recurrent green color in these voxels maps corresponds to Q index equal to one. The shown isolines correspond to several percentages of the minimum prescription dose of 70 Gy and they covered the heterogeneous prescription according to both SUV maps.
Q index representation is also included. Table 4 presents Q index under different tolerance criteria and QF values for the two obtained adaptive plan solutions along treatment time. Number of necessary MLC apertures to solve the adaptive planning was also included. The results presented on Table 2 revealed that when the dose prescription map evaluated was the one based on BIOGRAPH protocol, P1 solution was always of superior quality, showing larger percentage of voxels within the tolerance for Q, as expected.