Isodose feature‐preserving voxelization (IFPV) for radiation therapy treatment planning

Abstract

Purpose

Inverse planning involves iterative optimization of a large number of parameters and is known to be a labor‐intensive procedure. To reduce the scale of computation and improve characterization of isodose plan, this paper presents an isodose feature‐preserving voxelization (IFPV) framework for radiation therapy applications and demonstrates an implementation of inverse planning in the IFPV domain.

Methods

A dose distribution in IFPV scheme is characterized by partitioning the voxels into subgroups according to their geometric and dosimetric values. Computationally, the isodose feature‐preserving (IFP) clustering combines the conventional voxels that are spatially and dosimetrically close into physically meaningful clusters. A K‐means algorithm and support vector machine (SVM) runs sequentially to group the voxels into IFP clusters. The former generates initial clusters according to the geometric and dosimetric information of the voxels and SVM is invoked to improve the connectivity of the IFP clusters. To illustrate the utility of the formalism, an inverse planning framework in the IFPV domain is implemented, and the resultant plans of three prostate IMRT and one head‐and‐neck cases are compared quantitatively with that obtained using conventional inverse planning technique.

Results

The IFPV generates models with significant dimensionality reduction without compromising the spatial resolution seen in traditional downsampling schemes. The implementation of inverse planning in IFPV domain is demonstrated. In addition to the improved computational efficiency, it is found that, for the cases studied here, the IFPV‐domain inverse planning yields better treatment plans than that of DVH‐based planning, primarily because of more effective use of both geometric and dose information of the system during plan optimization.

Conclusions

The proposed IFPV provides a low parametric representation of isodose plan without compromising the essential characteristics of the plan, thus providing a practically valuable framework for various applications in radiation therapy.

1. Introduction

How to extract essential information of a treatment plan for concise description of the plan and other applications in radiation oncology is a fundamental problem. A treatment plan is generally given in the form of voxelized dose distribution in the context of 3D anatomy of the patient. The information is, from the perspective of practical usage, heavily oversampled in most regions. Dose‐volume histogram (DVH) is commonly employed to condense the isodose data to characterize a treatment plan¹ and many inverse planning models have been discussed to optimize a plan in terms of DVH‐based criteria.^{2, 3, 4, 5, 6, 7, 8} However, the DVH provides only fractional volumes of an organ receiving certain doses without any spatial information of the doses. Furthermore, the DVHs can be obtained only after an isodose plan is computed, prohibiting its use during the process of dose optimization.^{9, 10, 11} Less commonly used are some other aggregated quantities to aid treatment planning, including equivalent uniform dose (EUD), tumor control probability (TCP), and normal tissue complication probability (NTCP).^{12, 13, 14}

Another way to consolidate information of a treatment plan is to downsample the voxels through discarding voxels by, for instance, naïve random selection or pyramidal method. Despite the adverse impact of downsampling to the plan quality, naïve downsampling approaches have been commonly adopted for fast planning.^{15, 16, 17, 18, 19, 20} Nonetheless, it is intuitively evident that the downsampling can lead to reduced resolution and cause a loss of planning accuracy. Indeed, according to Matuszak,²¹ by universally downsampling the voxels, only a noticeable improvement of computational cost can be achieved at the cost of a deteriorated plan quality. Naïve downsampling can cause an undesirable underestimate of doses on normal tissues. Ungun et al.²² proposed voxel and beam clustering‐based downsampling schemes, which segment and consolidate rows or columns of dose‐influence matrix to speed up the planning process without explicitly accounting for dosemetric, spatial, and anatomical information.

Toward physically more meaningful extraction of a treatment plan information and to facilitate many specific tasks in radiation therapy, we introduce a new framework, called the isodose feature‐preserving voxelization (IFPV), for a sparse representation of treatment plan. In this representation, a dose distribution is characterized by a collection of clustered voxels that preserving a certain feature(s) of a spatial dose distribution. The clusters in IFPV are defined by preserving important isodose features with respect to anatomy (or ROIs) under physical and heuristic considerations. The central idea of the strategy is to divide the space into, instead of regular voxels, irregularly shaped and clustered voxels according predefined feature extraction criteria. In this way, a treatment plan can be characterized by a substantially reduced number of the isodose feature‐preserving (IFP) voxel clusters which carry dosimetric, spatial, and anatomical information. Therefore, the proposed way of information extraction is physically more meaningful than the existing downsampling, such as the beam and voxel clustering by Ungun et al.²² Furthermore, from the perspective of practical applications, the IFPV provides a high‐fidelity sparse representation of a radiotherapy treatment plan, thus can potentially be used for many other applications that requires a concise description of a treatment plan (e.g., plan data compression, fast transmission of treatment plan, and characterization and analysis of plan quality assurance data). Consequently, our proposed IFPV downsampling scheme is fundamentally different from the existing downsampling schemes. In the following, we will provide a detailed description of IFPV scheme and show that the proposed strategy provides a computationally valuable representation for dose planning and related applications with little compromise in spatial resolution.

2. Methods

2.A. IFPV clustering of voxels

IFPV combines the conventional voxels in an isodose plan that are spatially and dosimetrically close into physically more meaningful clusters. By removing the redundant dosimetric information of voxels within homogeneous volumes, the IFPV enables the use of significantly reduced number of spatial parameters to describe the dose distribution, enabling a more compact representation of isodose plan with little loss of information. In practice, there are many ways to cluster voxels. Our clustering algorithm is realized by a combination of K‐means algorithm and support vector machine (SVM). These two modules run sequentially. The K‐means algorithm generates initial clusters based on both geometric and dosimetric information. A flowchart of the proposed IFPV clustering is presented in Fig. 1. For initialization, one can either start with an arbitrary dose distribution or with a mapped dose plan from prior plan(s) of similar disease site. To generate IFPV clusters for a given plan, we invoke Algorithms 2, 1, and 3, sequentially for only once. To generate a completely new plan, the IFPV needs to be updated in each iteration. In this case, an inverse planning model will be invoked following the IFPV clustering and update the dose distribution. The iteration terminates when a user‐specified maximal number of repetitions is reached. More details regarding this K‐means algorithm is given in Section 2.A.1. The output clusters of the K‐means algorithms may include clusters with geometrically disjoint cluster elements. To ensure that all elements within the same cluster are geometrically connected, a SVM‐based smoothing algorithm is invoked after the K‐means algorithm. The SVM step will be discussed in Section 2.A.2.

Figure 1.

Flowchart for IFPV clustering.

2.A.1. Preclustering via K‐means

K‐means algorithm was first employed to cluster voxels with similar dose and close distance. The goal here is to divide the entire isodose region into IFPV voxels.

The ith voxel is denoted by v _i . Each voxel is associated with a tuple of information T _i  = [x _i , y _i , z _i , d _i ]^⊤, where (x _i , y _i , z _i ) and d _i are the coordinate and dose, respectively. The K‐means clustering algorithm groups the voxels into multiple clusters of a predefined number of clusters K. Denote by $T_k^c={[x_k^c,y_k^c,z_k^c,d_k^c]}^{\top }$ the centroid of the kth cluster. The distance between voxel i and centroid k is calculated according to

$${\delta }_{\mathit{ik}}=\sqrt{\omega ·[{(x_i-x_k)}^2+{(y_i-y_k)}^2+{(z_i-z_k)}^2]+{(d_i-d_k)}^2}$$ (1)

where ω > 0 is a weighting parameter to control the compactness of the clusters, with a larger ω for more regularly shaped IFPV voxels. With the above definition, we elect to employ the K‐means algorithm²³ with “K‐means++” initialization²⁴ for the task of clustering. Specifically, the following provide the pseudo‐code for K‐means.

Algorithm 1. k‐means clustering²³ .

1.

Step 0. Randomly generate an initial set of k‐many centroids $\{T_k^c\}$ through K‐means++. Let iteration count t: = 0.

Step 1. For all i, assign T _i to the cluster k*(i) whose centroid is the closest to the T _i . Specifically, $k^{\ast }(i):=argmin\{||T_i-T_k^c||:1\le k\le K\}$.

Step 2. For all k, update the centroid of the kth cluster $T_k^c$ by the average of all the voxel tuples that is assigned to cluster k. Namely,

$$T_k^c=\frac{1}{|\{i:k^{\ast }(i)=k\}|}\sum _{i:k^{\ast }(i)=k}{T}_i$$

Step 3. If the Euclidean distance between cluster centroids of two consecutive iterations are greater than the tolerance level specified as 0.1, go to Step 1. Otherwise, algorithm terminates and outputs the clustering results and the centroids.

The initialization algorithm is provided as following:

Algorithm 2. K‐means++ initialization²⁴ .

1.

Step 0. Let centroid count k: = 1. Let S = {i:1 ≤ i ≤ N}, where N is the number of voxels, and let S ^c  = ∅.

Step 1. Randomly select voxel tuples i _r by a uniform distribution and assign it to be the first centroid $T_1^c$. Update that S: = S − i _r and S ^c  = S ^c  ∪ i _r .

Step 2. For each voxel tuple T _i , compute

$$\begin{array}{cc}\hfill D_{\mathit{ik}}=& {min}_{i^{\prime }\in S^c}\hfill \\ & \sqrt{\omega ·[{(x_i-x_{i^{\prime }})}^2+{(y_i-y_{i^{\prime }})}^2+{(z_i-z_{i^{\prime }})}^2]+{(d_i-d_{i^{\prime }})}^2}.\hfill \end{array}$$

Step 3. Randomly select a voxel tuple i _r from S following a discrete distribution given as

$$P[\text{select}{T}_i\text{to be}(k+1)\text{th centroid}]=\left\{\begin{array}{cc}0& \text{if}i\notin S\\ \frac{D_{\mathit{ik}}^2}{{\sum }_{i^{\prime }\in S}{D}_{i^{\prime }k}^2}& \text{if}i\in S\end{array}\right.$$

Step 4. If k = K or if S = ∅, algorithm terminates. Otherwise, update that S: = S − i _r , S ^c  = S ^c  ∪ i _r , and k: = k + 1. Go to Step 2.

2.A.2. Support vector machine‐based projection for continuous 3D clusters

The preclustering algorithm may generate geometrically partitioned clusters because both dose and geometric information were considered as one of the features for clustering. Essentially, the resulting clusters are in four‐dimensions. There are many ways to improve the connectivity of the clusters. Here, we propose to approximate the four‐dimensional (4D) clusters by spatially continuous three‐dimensional (3D) clusters via projecting the 4D clusters onto 3D clusters without geometric partitions. To this end, we employ an SVM‐based approach. The idea of this approach is to generate 3D curves that can best differentiate the original 4D clusters. The details of the SVM‐based approach are provided in the following.

Algorithm 3. SVM‐based projection algorithm.

1.

Step 0. Let centroid count k: = 1. Let S = {i:1 ≤ i ≤ N}, where N is the number of voxels, and let S ^c  = ∅.

Step 1. For all clusters k, solve the following support vector machine problem to generate cluster boundaries:

$$\begin{array}{cc}& (v^{k\ast },b^{k\ast },{\xi }^{k\ast })=\underset{{\alpha }^k,b^k,{\xi }^k}{argmin}\frac{1}{2}{(v^k)}^{\top }{v}^k+\rho \sum _{i=1}^K{\xi }_i^k\hfill \\ & \mathit{s}.t.{(v)}^{\top }\varphi (x_i,y_i,z_i)+b^k\ge 1-{\xi }_i^k\text{if}i\in C_k\hfill \\ & {(v^k)}^{\top }\varphi (x_i,y_i,z_i)+b^k\le -1+{\xi }_i^k\text{if}i\notin C_k\hfill \\ & {\xi }_i^m\ge 0,i=1,\dots ,K\hfill \end{array}$$

where (υ ^k *, b ^k *, ξ ^k *) is the optimal solution to the above problem and ρ > 0 is a user‐specified scalar.

Step 2. Then output the following functions as boundaries of the clusters:

$$g(x,y,z):={(v^{k\ast })}^{\top }\varphi (x,y,z)+b^{k\ast }+{\xi }_i^{k\ast }-1,\text{for all}k$$

where (x, y, z) is the coordinate of a voxel.

Step 3. Calculate and output the centroids as in Step 2 of Algorithm 1:

$$T_k^c=\frac{1}{|\{i:k^{\ast }(i)=k\}|}\sum _{i:k^{\ast }(i)=k}{T}_i$$

2.B. Implementation of inverse planning with isodose feature‐preserving downsampling

As a specific application of the IFPV, we consider a general inverse treatment planning problem. Given a dose‐influence matrix, A ∈ ℜ ^N×m , where N and m are the number of voxels and the number of beamlets, we have

$$\begin{array}{cc}\hfill \underset{w}{\mathit{min}}& F(w;A)\hfill \\ \hfill s.t.& w\in W(A)\hfill \end{array}$$ (2)

in which F( · ; A) and W(A) ⊂ ℜ ^m are a user‐specified criterion that measures the performance of a treatment plan and the feasible region that consists of all the constraints, respectively, for the beamlet intensities w. The corresponding downsampled inverse planning model is formulated as

$$\begin{array}{cc}\hfill \underset{w}{\mathit{min}}& F(w;A_C)\hfill \\ \hfill s.t.& w\in W(A_C)\hfill \end{array}$$ (3)

where $A_C\in {\Re }^{N_c\times m}$ , with N _c  ∈ {1, …, N} being the number of clusters, is a submatrix of A that consists only the rows corresponding to the centroids of the voxel clusters. For instance, suppose that F(w) is in the form of the quadratic one‐sided penalties and W(A) is a set of linear constraints that require intensities to be non‐negative and dose levels upper bounded by D _max . Problem (2) and (3) becomes:

$$\begin{array}{cc}\hfill minF(w;A):=& \frac{1}{N}\sum _{n=1}^N{\alpha }_n{(max\{0,T_n-d_n(w)\})}^2\hfill \\ & +{\beta }_n{(max\{0,d_n(w)-T_n\})}^2\hfill \\ & s.t.w\in W(A)=\{w\ge 0:Aw\le D_{\mathit{max}}\}\hfill \end{array}$$ (4)

and

$$\begin{array}{cc}\hfill minF(w;A_C):=& \frac{1}{K}\sum _{n\in V_C}{\alpha }_n{(max\{0,T_n-d_n(w)\})}^2\hfill \\ & +{\beta }_n{(max\{0,d_n(w)-T_n\})}^2\hfill \\ & s.t.w\in W(A_C)=\{w\ge 0:A_{C}w\le D_{\mathit{max}}\}\hfill \end{array}$$ (5)

where V _C denotes the set of all centroids and K is the number of centroids. We would like to emphasize that our proposed IFPV scheme is general enough to apply to other choices of objective function F and the constraints W(A), instead of being merely applicable to the above model. Since Models (4) and (5), as well as the subproblem in Algorithm 3, are convex, they can be solved efficiently using commercial solvers.

Notice that the IFPV depends on the dose distribution, which is unavailable for a new patient before planning. To proceed, one can either start with an initialization of zero dose or with a mapped dose plan from prior plan(s) of similar disease site. In either case, the IFPV map is updated iteratively using the flowchart in Fig. 1.

2.C. Evaluation

We implemented Algorithms 1–3 using Matlab, with Models (4) and (5) and the subproblem in Algorithm 3 solved using the Mosek.²⁵ The approach was tested using three coplanar prostate IMRT cases with eight 6 MV photon beams and the first head‐and‐neck case of the TROTS dataset.²⁶ The detailed implementation of the platform has been described previously.^{27, 28} For prostate cases, 1 × 1 cm beamlet size and 3 × 3 × 2.5 mm voxel size were used during the calculation. The dose‐influence matrix is calculated using pencil beam algorithm using the computational environment for radiotherapy research (CERR).²⁹ For the head‐and‐neck case, detailed information was provided by Breedveld and Heijmen.²⁶ The voxel size was 0.9766 × 0.9766 × 2.5 mm. Algorithm parameter ω (as in Algorithm 2) is fixed as 2. Note that, for providing a high‐fidelity plan representation, the IFPV is required to respect the boundaries of the involved structures. Thus, the clustering is done separately for each structure. The k‐means clustering can also be done independent of structures, but with the structural boundaries included as constraints to respect during the clustering process. In all the test cases, the IFPV started with zero dose.

Among the three prostate cases, in Section 3.A, the first case was used to test the impact of the proposed downsampling to inverse planning using the same planning model. Specifically, we first generated a clinically acceptable plan with full resolution using Model (3) [the model in Eq. (3)], which is one of the most common models for inverse planning, after trial‐and‐error parameter tweaking. Then, with the same set of weighting parameters, we reduce Model (3) to its downsampled version, Model (4). The downsampling approach with Algorithm 1 was compared with the pyramidal downsampling method as a built‐in of CERR. Both IFPV and pyramidal schemes are downsampling approaches. After downsampling with the approaches, we compared the same inverse planning model using different downsampled datasets to see which downsampling strategy better preserves the necessary information to maintain plan quality. In Section 3.B, two other prostate cases were used to compare the performance of the inverse planning model with IFPV and those with different planning schemes. A regularization term in the form of L‐1 norm³⁰ in IFPV domain is incorporated into an inverse planning model to penalize the difference between the new plan and a formerly approved treatment case. Specifically, we conducted IFPV clustering to the reference plan and added the following regularization term to the objective function of Model (5):

$$R(w):=\gamma \sum _{n\in V_C}|d_n(w)-d_n^{\mathit{ref}}|$$

where γ > 0 is a user‐specified tuning parameter and $d_n^{\mathit{ref}}$ is the dose level of the nth reference IFPV centroid. Here, the IFPV centroid in the reference plan that has the smallest difference in coordinates was chosen as the nth reference IFPV centroids. In Section 3.C on a head‐and‐neck case, Model (5) with IFPV downsampling approach is also compared with the solution reported by TROTS and a solution generated with by Model (5) with organ‐specific uniform downsampling. The downsampling was done separately in a structure specific fashion. For each of the structures, 2,800 voxels are maintained (which is approximately 55% of the total number of PTV voxels). For a fair comparison, we ensured that the number of voxels considered in each organ in the organ‐specific uniform downsampling scheme is the same as that in the IFPV‐based approach. In all these experiments, we terminated the IFPV clustering after four iterations.

Involved as a benchmark scheme was the DVH‐guided optimization approach based on Wang et al.⁴ Both schemes used the same reference plan. Nonetheless, for the latter, we allowed only the DVH information to be incorporated through the DVH‐based constraints. All experiments were conducted using Matlab in a single thread on a PC with Intel Core i7 3.40 GHz CPU and 16 GB RAM.

3. Results

3.A. IFPV downsampling preserves plan quality

This subsection compares the proposed IFPV‐based downsampling and the pyramidal downsampling. For the pyramidal approach, two levels of downsampling rates, 4 and 16, were considered and referred to as pyramidal medium resolution (MR) and pyramidal low resolution (LR). The number of voxels after downsampling is given in Table 1. From the table, it is seen that a significant portion of the voxels were disregarded after the IFPV downsampling in comparison with the original data (HR) and with the pyramidal approaches, MR, and LR. In Fig. 2, we present the clustering results, where IFPVs are defined by blue contours in Subplot (b). In Fig. 3, we compare the DVHs generated by solving Model (5) with different downsampling schemes. As can be seen from the figure, the dose coverage generated by the original model and the model after the IFPV downsampling were competitive against each other, despite that the number of voxels considered by the latter model was substantially less than that of the former. Specifically, HR achieved a better PTV coverage, whereas the IFPV downsampled model resulted in lower doses on rectum and bladder. The DVHs generated by MR are noticeably worse than both HR and IFPV, and a dramatic decay of quality was observed in the DVHs generated by LR. The computational time was 20 s for solving Model (5) after IFPV‐based downsampling, and four iterations were involved to obtain the final plan. The total computational time was about 2 min. In contrast, it took 13 min to solve the model without downsampling. Note that voxels where x‐ray intensity is zero (i.e., the corresponding row in the dose‐influence matrix contain only zero entries) are not accounted for. The numbers of beamlets were 969, 839, and 653, respectively, for HR, MR, and LR, while the beamlet number after IFPV downsampling was 969.

Table 1.

Number of voxels after downsampling with different schemes

Downsampling scheme	Original (HR)	IFPV	Pyramidal medium resolution (MR)	Pyramidal low resolution (LR)
Number of voxels after downsampling	1.18 × 107	3669	2.97 × 106	7.41 × 105

Figure 2.

The voxelization result for Case 1. Subplot (a) is the original CT scan. Subplot (b) presents the new voxels defined by the additional boundaries. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 3.

Comparison of DVHs of (a) PTV and body, (b) bladder, and (c) rectum for plans generated after different downsampling schemes. [Color figure can be viewed at wileyonlinelibrary.com]

3.B. Inverse planning in IFPV domain for prostate IMRT

This subsection compares the performance of inverse planning obtained using the IFPV domain and the conventional DVH‐based scheme using two prostate IMRT cases (Cases 1 and 2). In the latter approach, no downsampling was involved. For both cases, the two approaches achieved competitive performance against each other in terms of the DVHs as shown in Fig. 4. However, the performance in the resultant dose distributions was different. The comparisons of the 3D doses in Fig. 5 indicated undesirable dose distributions generated by the DVH‐based approach, as some of the healthy regions were exposed to undesirable and unnecessary high doses. In contrast, the IFPV‐based scheme achieved better dose distributions. The numbers of beamlets were 791, 651, and 445, respectively, for HR, MR, and LR while the beamlet number after IFPV downsampling was 791 in Case 1. In Case 2, the beamlet numbers were 1041, 893, and 692, respectively, for HR, MR, and LR, and 1041 for the IFPV scheme. The total computational time of the IFPV‐based scheme for Cases 1 and 2 were both less than 3 min. In contrast, the HR spent approximately 11 and 15 min for Cases 1 and 2, respectively.

Figure 4.

Comparisons of DVHs for (a) Case 1 and (b) Case 2. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 5.

Comparisons of 3D doses for Case 1 [(a)–(f)] and Case 2 [(g)–(l)] generated by the IFPV‐based scheme [(a)–(c) for Case 1 and (g)–(i) for Case 2] and by the DVH‐based scheme [(d)–(f) for Case 1, and (j)–(l) for Case 2]. [Color figure can be viewed at wileyonlinelibrary.com]

3.C. Inverse planning in IFPV domain for head‐and‐neck IMRT

This subsection compares the proposed IFPV‐based downsampling with a uniform downsampling approach. We ensured that the number of voxels after downsampling were the same for different schemes. We also compared the solution generated by the downsampled model with the optimal solution reported by TROTS. The results are provided in Fig. 6. In subplots (b), (d), and (f), it can be seen that IFPV‐based downsampling generates a better plan than the conventional downsampling approach. In particular, the maximal dose of the patient is undesirably higher than that of the PTV in plan generated by the conventional approach. In contrast, the maximal dose of the patient in the IFPV‐based plan is acceptable. Furthermore, subplots (a), (c), and (e) compare the IFPV‐based approach with the optimal solution reported by TROTS.²⁶ As is shown in the figure, the IFPV approach generated a plan with clinically more favorable trade‐offs than the alternative solution using only downsampled information. For the two approaches, the downsampled models both had 9977 beamlets and they both took approximately 4 min to compute. The IFPV was done for four iterations, resulting in 16 min of total computational time. Nonetheless, the resulting higher quality solution can well compensate for the comparatively more computational effort than the uniform downsampling‐based planning.

Figure 6.

Comparisons of IFPV‐based scheme with TROTS [(a), (c), and (e)] and with uniform downsampling [(b), (d), and (f)]. [Color figure can be viewed at wileyonlinelibrary.com]

4. Discussion and conclusion

Downsampling by aggregating adjacent voxels is a commonly used strategy to reduce computational burden in science and engineering. While useful, the scheme does not take into account of the content information of the individual voxels, leading to a less intelligent voxelization scheme and thus compromising the resultant spatial resolution. This paper introduces a novel concept of IFPV and reports a treatment planning framework enabled by IFPV‐based downsampling. The introduction of IFPV reduces the dimensionality of the problem and allows us to simplify the planning and plan evaluation. It was shown that an accurate surrogate to the original inverse planning model is attainable with only a small number of IFP clusters. Specifically, in Section 3.A, we compared the calculations of Model (5) with different downsampling approaches to that without downsampling. In all these cases, Model (5) was solved using the same commercial solver. Our results show that, with IFPV‐based downsampling, computational efficiency is improved, even if the formulation, solver, and algorithm remain unchanged.

It is interesting to point out that, despite of substantial reduction in dimensionality of the inverse planning model, the results in IFPV domain seems to be very favorable as compared to that obtained using the conventional DVH‐based inverse planning. In addition to the improved computational efficiency, the IFPV‐domain inverse planning outperforms the traditional DVH‐based planning, primarily because of effective use of both geometric and dosimetric information during the plan optimization. In reality, there have been several versions of automate inverse planning by referring to DVHs from prior knowledge or plan(s), such as the references by Xing et al.,^{11, 31} Wang et al.,⁴ Zarepisheh et al.,⁶ Appenzoller et al.,³² and Wu et al.³³ The numerical results here suggest the need and possibility of an automated inverse planning in IFPV domain. As presented in Section 3.B, the DVH‐based scheme, as an alternative approach than the proposed IFPV, generated a reasonable DVH but the dose distribution is not always clinically acceptable. Note that our emphasis is that conventional approach alone is not sufficient to consolidate all the essential information of a treatment to facilitate the subsequent planning procedures. The proposed IFPV‐based approach, without the aid of any ring structure, seems to be suitable for the purpose. It is believed that the underlying reason for this is the degeneracy of DVHs in characterizing treatment plans and its lack of spatial information. In contrast, the IFPV‐based scheme is designed to incorporate both doses, anatomy and geometric information, leading to a more desirable performance.

It is worth mentioning that an anticipated limitation of the IFPV scheme is its dependence to the initial solution. In the experiments presented herein, all started with a zero dose. It seems that such a simple initialization works reasonably well in all four numerical cases. Nonetheless, we think it possible that a better initial solution potentially may improve the plan quality. It is also possible that by initializing the plans with a historically similar case would facilitate/hinder planning. A more comprehensive study regarding the correlation between initial points and the final output should be conducted in future.

Another potential complexity in using the IFPV scheme is the determination of parameter ω. Due to (1), a higher value of ω will render the formation of clusters less sensitive to the dose inhomogeneity. In our experiment, ω is fixed to be 2. An extensive set of numerical experiments are necessary to fully explicate the impact of different choices of the parameter to the IFPV and the resulting planning quality.

In summary, a novel IFPV is introduced as a sparse representation of isodose plan, which may be useful for many applications in RT such as plan optimization, plan data compression, and concise presentation of quality assurance data. By assembling voxels that are spatially and dosimetrically close into physically more meaningful clusters, the IFPV greatly reduces the dimensionality of the plan representation problem with little compromise in spatial resolution. We have demonstrated that it is conceptually straightforward to implement an inverse planning system in the IFPV domain. Our study suggests that inverse planning in IFPV domain seems to outperform existing DVH‐based optimization techniques in both computational efficiency and plan quality. Given the fundamental roles of the voxelization in therapeutic planning, the proposed IFPV framework should find useful applications in many other applications in radiation therapy. In particular, because of the enormous reduction in dimensionality while conserving the chief features of isodose distribution, the formalism opens a new vista for IFPV‐domain machine learning and/or other data‐driven autonomous planning.

Conflicts of Interest

The authors have no relevant conflicts of interest to disclose.