Built-in Wavelet-Induced Smoothness to Reduce Plan Complexity in Intensity Modulated Radiation Therapy (IMRT)

Abstract

Objective:

Reducing plan complexity in intensity modulated radiation therapy (IMRT) to ensure dosimetric accuracy and delivery efficiency of the radiation treatment plans. We propose a novel approach by representing the beamlet intensities using an incomplete wavelet basis that explicitly excludes fluctuating intensity maps from the decision space (explicit hard constraint). This technique provides a built-in wavelet-induced smoothness that improves both dosimetric plan quality and delivery efficiency

Approach:

The beamlet intensity maps need to be especially smooth in the leaf travel direction (referred to as the X-direction). We treat the intensity map of each beam as a 2D image and represent it using the wavelets corresponding to low-frequency changes in the X-direction (i.e., approximation and horizontal). The absence of wavelets corresponding to high-frequency changes (i.e., vertical and diagonal) induces built-in smoothness. We still utilize a regularization term in the objective function to promote smoothness in the Y-direction (perpendicular to the X-direction) and further possible smoothness in the X-direction. This technique has been tested on three patient cases of different disease sites (paraspinal, lung, prostate) and all final evaluations and comparisons have been performed on an FDA-approved commercial treatment planning system (Varian Eclipse^™).

Main results:

Wavelet-induced smoothness reduced monitor units by about 10%, 45%, and 14% for paraspinal, lung, and prostate cases, respectively. It also improved organ at risk sparing, especially on the complex paraspinal case where it resulted in about 7%, 13%, and 14% less mean dose to esophagus, lung, and cord, respectively. Moreover, built-in wavelet-induced smoothness desensitizes the results to changing the weight associated to the regularization term, and thereby mitigates the weight fine-tuning difficulty.

Significance:

Fluence smoothness is often achieved by smoothing the beamlet intensity maps using a proper regularization term in the objective function aiming at disincentivizing fluctuation in the beamlet intensities (implicit soft constraint). This work reports a novel application of wavelets in imposing an explicit smoothness hard constraint in the search space using an incomplete wavelet basis. This idea has been successfully applied to exclude complex and clinically irrelevant radiation plans from the search space. The code and pertained models along with a sample dataset are released on our LowDimRT GitHub (https://github.com/PortPy-Project/LowDimRT).

1. Introduction

Modulated intensity maps improve dosimetric plan quality by creating dose distributions that conform well to the shape of the tumor. Complex intensity maps, on the other hand, could lead to dosimetric inaccuracies and inefficient plan delivery [1–4]. Therefore, it is of great importance to balance the tradeoffs between the plan complexity and the dosimetric plan quality. Various methods have been proposed in the literature to reduce plan complexity by smoothing the intensity maps, often through reducing the variations in the intensity of the neighboring beamlets. In these methods, the smoothing can happen: 1) after the entire optimization process, 2) after each iteration of the optimization, or 3) inside the objective function of the optimization problem. The first and second group of techniques are easier to implement and allow using more sophisticated smoothing filters without any convexity/differentiability considerations [5]. However, they do not take the dosimetric quality impacts of smoothing into consideration and could substantially degrade dosimetric plan quality [6]. This problem has been addressed by incorporating complexity in the objective function (the third group) and adding an extra regularization term to measure the plan complexity beside the existing objective function that measures the dosimetric plan quality. The regularization term often measures the variations in the beamlet intensities as a complexity metric. Different regularization terms have been proposed based on the $l_1$-norm [7, 8], $l_2$-norm [5], Savitzky-Golay filter [6], and diffusion equation [4]. Among all the regularization terms, the $l_2$-norm term has been arguably the most popular one that has been successfully implemented in the clinic [5, 9, 10].

This work takes a different approach to reducing plan complexity by employing wavelets and representing beamlet intensities in the wavelet domain. Wavelets are particularly well-known in the fields of signal processing and imaging, due mainly to their ability to provide a sparse representation of signals/images, which has allowed numerous applications (e.g, image/signal compression, denoising, shortened medical imaging scanning) [11]. The sparsity, however, does not enforce smoothness in the decision space. While the smoothness of natural images is believed to be associated with their sparsity in the wavelet domain, the converse is not necessarily true and sparsity does not imply smoothness. Moreover, the sparsity can only be promoted in the objective function as a soft constraint, similar to the regularization approach. To induce smoothness, we remove the wavelets corresponding to high-frequency changes from the complete wavelet basis and then represent the beamlet intensities using the resultant incomplete wavelet basis. This amounts to adding a set of linear constraints to the optimization problem which, can be handled efficiently without adding significant computational overhead. Wavelets induce inherently different smoothness compared to the regularization terms and our experiments show that wavelet-induced smoothness not only correlates better with the plan complexity but also has smaller impacts on the dosimetric plan quality. In other words, wavelets reduce plan complexity at a smaller cost to the dosimetric plan quality compared to the regularization approach.

Another advantage of the proposed approach is that it explicitly removes unnecessary complex intensity maps from the decision space (search space) and serves as a hard constraint for smoothing, as opposed to the regularization approach which serves as a soft constraint by implicitly making the complex intensity maps unattractive choices to the objective function. This could in turn alleviate the difficulty of balancing the tradeoffs between the plan complexity and the dosimetric plan quality, which is often handled by manually fine-tuning the regularization weight in the objective function. Given the fact that intensity maps need to be particularly smooth in the leaf travel direction (X-direction), compared to the direction perpendicular to the leaf travel (Y-direction), we only apply wavelets to the X-direction and still employ a regularization term for smoothing in the Y-direction as well as further possible smoothing in the X-direction.

To the best of our knowledge, this is the first application of wavelets to impose smoothness in the decision space. This is especially a new application of wavelets in radiotherapy treatment planning.

2. Methods

2.1. Mathematical Formulation and Regularization-Induced Smoothness: The Existing

For modeling purposes, the patient’s body is discretized into small 3D voxels. The radiation dose delivered to each voxel $j(j=1,\dots ,J)$ at the unit intensity of each beamlet $i(i=1,\dots ,I)$, denoted by $a_{ji}$, is calculated using a dose calculation algorithm. The values $a_{ji}$ define the dose influence matrix $A\in R^{J\times I}$, with rows and columns corresponding to the voxels and beamlets, respectively (i.e., beamlets from all the candidate beams are concatenated). The goal is to optimize the beamlet intensities $x_i(i=1,\dots ,I)$ to get desirable radiation doses $d_j(j=1,\dots ,J)$ delivered to the patient’s body, where $d=Ax$. The general IMRT problem takes the following form:

$$\begin{array}{ll}\text{minimize}& f\left(x\right)+\lambda \times h\left(x\right)\\ \text{subject}\text{to}& x\in \chi ,\end{array}$$ (1)

where $f\left(x\right)$ and $h\left(x\right)$ account for the dosimetric plan quality and the plan complexity, respectively, and $\lambda$ is a weight hyper-parameter that needs to be tuned to balance the tradeoffs between the dosimetric quality and the deliverability.

The feasible set $\chi$ is defined by a non-negativity constraint for beamlet intensities $(x\ge 0)$ and typically some linear maximum dose and mean dose constraints on the healthy tissues and tumor as:

$$\begin{array}{ll}\text{max}\left(A^{s}x\right)\le d_s^{\mathit{\text{max}}},& s\in \text{ Structures with max. constraints, }\\ \text{mean} \left(A^{s}x\right)\le d_s^{\mathit{\text{mean}}},& s\in \text{ Structures with mean constraints, }\\ x\ge 0,& \text{ Non-negativity constraint, }\end{array}$$ (2)

where $A^s$ represents the rows of the influence matrix $A$ whose corresponding voxels belonging to the structure $s(s=1,\dots ,S)$. Constraints 2 are often referred to as hard constraints which are guaranteed to fulfill and further improvements beyond them is incentivised in the objective function (i.e., closer doses to the prescription dose for the PTV and lower doses for the OARs).

A very common choice for the function $f$ is a two-sided quadratic function that measures the deviation of the delivered radiation dose from the ideal prescription dose $p$ using the associated weights $w^{+}/w^{-}$ for over-dose/under-dose as:

$$f\left(x\right):=\sum _{s=1}^S w_s^{+}\text{max}{\left(A^{s}x-p,0\right)}^2+w_s^{-}\text{min}{\left(A^{s}x-p,0\right)}^2.$$ (3)

For the tumor voxels, the prescription dose $p$ is defined by a radiation oncologist, and for the healthy voxels, the prescription dose is often zero.

A typical choice for the regularization function $h$ is a quadratic function that measures the variation of the intensity of each beamlet $i$ with respect to the intensity of its right neighbor beamlet $R_i$ and lower neighbor beamlet $L_i$ as:

$$h\left(x\right):=\sum _{b=1}^B \sum _{i\in I_b} \alpha {\left(x_i-x_{R_i}\right)}^2+\beta {\left(x_i-x_{L_i}\right)}^2,$$ (4)

where $I_b$ is a set of beamlets belong to beam $b$, and $\alpha$ and $\beta$ promote smoothness in the X-direction (leaf travel direction) and the Y-direction, respectively $(\alpha >\beta )$.

3.1. The Wavelet-Induced Smoothness: The Proposed Method

The beamlet intensity $x$ admits any non-negative value in the feasible region $\chi$ (i.e., $x \in R_{+}^I$) in Problem 1. The central idea here is to define and characterize the subspace of smooth beamlet intensities, $\psi \subseteq R^I$, and ensure that $x$ lies in that subspace. In other words, we need a constraint of the following form:

$$\left\{x\in \psi \mid \psi \subseteq R^I,\text{ and }\psi \text{ contains only smooth intensity maps. }\right\}$$

We show in the remainder of this section that $\psi$ can be defined as a subspace spanned by an incomplete wavelet basis.

For each beam angle, we can treat the beamlet intensity map as a 2D image. This allows us to represent the beamlet intensities in an alternate wavelet space as:

$$\left\{x\mid x=Wz:z\in R^I\right\},$$

where $W$ is the discrete wavelet transform matrix. Matrix $W$ can be decomposed into four sub-matrices:

$$W=\left[W^A,W^H,W^V,W^D\right],$$

known as approximation $\left(W^A\right)$, horizontal $\left(W^H\right)$, vertical $\left(W^V\right)$, and diagonal $\left(W^D\right)$ matrices. The approximation/horizontal matrices can also be interpreted as applying low-pass filter on rows and low-pass/high-pass filter on the columns of a 2D image, where vertical/diagonal matrices can be interpreted as applying high-pass filter on rows and low-pass/high-pass filter on columns. This means the vertical and diagonal matrices correspond to the high-frequency variations in the rows which, in this application, means undesirable sharp changes in the beamlet intensities along the X-direction. Therefore, we can induce smoothness by not including the wavelets corresponding to these matrices, which are responsible for high fluctuations in the beamlet intensities. Thus, set $\psi$ containing only smooth intensity maps can be characterized using the incomplete wavelet basis $\hat{W}=\left[W^A,W^H\right]$ as:

$$\psi =\left\{x\mid x=\hat{W}y:y\in R^K\right\}.$$ (5)

One point to consider is that some of the beamlets on the boundary of the beam, which are outside and far from the tumor’s beam-eye-view (BEV), contribute insignificant radiation dose to the tumor. Therefore, it makes sense to set their values to zero in advance (in a pre-processing manner) so they are not involved in the optimization process. This improves the computational efficiency and also rules out the possibility of assigning non-zero values to these beamlets due to the smoothing effects or modeling inaccuracies (e.g., approximation/numerical errors). Fig. 1 shows a simple example where a beam is divided into 48 small beamlets (6-by-8 rectangle of beamlets) with the blue beamlets representing the beamlets inside or within few millimeters (typically 1–5 millimetres) of tumor’s BEV. We would like to obtain a wavelet basis to represent the intensity of those beamlets inside the blue area. One can in principle derive such a wavelet basis using the mathematical definitions of wavelets, but it would be extremely cumbersome! We instead use the inverse wavelet transform, shown in Fig. 1, to obtain the wavelet basis.

Figure 1.

A sketch illustrating the inverse wavelet transform process for a beam. LoR/HiR stands for low-pass/high-pass filter reconstruction. $\left[\begin{array}{ll}1& \uparrow \end{array}2\right]/\left[\begin{array}{ll}2\uparrow & 1\end{array}\right]$ stands for row/column up-sampling. The blue area represents the beamlets that are within, or in the vicinity, of the tumor’s beam-eye-view. An incomplete wavelet basis can be generated by iteratively calling the inverse wavelet transform, and at each iteration, assigning the value one to only one of the blue entries in the approximation/horizontal matrices.

Using the inverse discrete wavelet transform function idwt2(cA, cH, cV, cD, wname) in Matlab, where cA/cH/cV/cD is the approximation/horizontal/vertical/diagonal matrix and wname is the name of the wavelet family, the wavelet basis can be obtained by iteratively calling idwt2. We can reconstruct $W^A/W^H$ by assigning the value one to only one of the entries in cA/cH and zero to others. The resultant matrix from idwt2, at each iteration, needs to be vectorized and added as an additional column to the basis matrix. To take into account the tumor’s beam-eye-view, we can assign the value one to only the corresponding entries (blue areas in Fig. 1) in the approximation/horizontal matrices (these matrices can be obtained by removing the even rows and columns from the original $6\times 8$ matrix)

Algorithm 1 provides a pseudocode for generating an incomplete wavelet basis for a patient.

The number of the wavelets in the incomplete wavelet bases $\left[W^A,W^H\right]$ is about half of the size of the beamlets (would be exactly half if all the beams have an even number of rows and we incorporate all the beamlets within each beam). This means that by adding the linear constraint $x=\hat{W}y$ in Problem 1, we are restraining the beamlet intensity values by halving their degree of freedom so that they cannot form a complex intensity map.

Putting together Problem 1 with Equations 2,5, the optimization problem with the wavelet-induced smoothness reads as follows:

$$\begin{array}{ll}\text{minimize}\hfill & \sum _{s=1}^S\left(w_s^{+}\text{max}{\left(A^{s}x-,0\right)}^2+w_s^{-}\text{min}{\left(A^{s}x-p,0\right)}^2\right)+\lambda \times {\sum }_{b=1}^B{\sum }_{i\in I_b}\left(\alpha {\left(x_i-x_{R_i}\right)}^2+\beta {\left(x_i-x_{L_i}\right)}^2\right)\hfill \\ \text{subject}\text{to}\hfill & \hfill \\ \text{max}\left(A^{s}x\right)\le d_s^{\mathit{\text{max}}},\hfill & s\in \text{Structures}\text{with}\text{max}.\text{constraints},\hfill \\ \text{mean}\left(A^{s}x\right)\le d_s^{\mathit{\text{mean}}},\hfill & s\in \text{Structures}\text{with}\text{mean}\text{constraints},\hfill \\ x=\hat{W}y,\hfill & \text{Wavelet}-\text{induced}\text{smoothness},\hfill \\ x\ge 0,\hfill & \text{Non}-\text{negativity}\text{constraint}.\hfill \end{array}$$ (6)

2.3. Data Preparation and Computational Platform

We use three patients of different disease sites (paraspinal, lung, prostate). For each patient, the beams are selected by an expert physicist. The dose influence matrix $\left(A\right)$ is precalculated using Eclipse^™ API 15.5 (application programming interface) with its Analytical Anisotropic Algorithm (AAA, version 15.5.11) 12]. Tables 1 and 2 summarize the patients’ data and the clinical dose constraints used for the optimization, respectively. PTV (planning target volume) in these tables refers to the target area. For each patient, the dimension of the dose influence matrix $A$ is $(\text{number}\text{of}\text{voxels}\times \text{number}\text{of}\text{beamlets})$ and the dimension of the wavelet basis matrix $\hat{W}$ is $(\text{number}\text{of}\text{beamlets}\times \text{number}\text{of}\text{wavelet}\text{basis}\text{vectors})$. To make the comparisons easier, in all the experiments, the plans are normalized according to the normalization value provided in Table 1.

Table 1.

Data summary for the three patients used in the study

	Patient
	1	2	3
Disease site	Paraspinal	Lung	Prostate
Prescription (p in Gray)	24	60	25
Number of fractions	1	15	5
PTV volume (cm3)	90	172	158
BEV margin (mm)	3	5	5
Number of beams	9	7	9
Number of voxels (J)	152,598	133,479	148,497
Number of beamlets (I)	5,568	3,173	4,492
Number of wavelets (K)	3,286	1,863	2,573
Normalization (V100%(PTV))	90%	90%	95%

Table 2.

Clinical max/mean dose constraints (in Gray) used in the optimization.

Disease site	Structure	$d_s^{\mathit{max}}$	$d_s^{\mathit{\text{mean}}}$
Paraspinal	PTV	31.2
	Esophagus	25.2
	Lungs	26.4
	Cord	14
Lung	PTV	66
	Esophagus	60
	Lungs	66
	Heart	63
Prostate	PTV	27.5
	Rectum	26.5	11
	Bladder	26.5	16.5
	Femurs	25

The proposed algorithm is implemented in MATLAB 2021b (The MathWorks, Inc., Natick, MA) and run on a PC with 2.4 GHz Intel Xeon CPU and 128 GB RAM. The convex constrained optimization problems are solved using the CVX toolbox [13] and the optimization engine MOSEK [14]. The resultant optimal beamlet intensity values $\left(x\right)$ are imported to Eclipse^™ (Varian Medical Systems, Inc., Palo Alto, California) for the final leaf sequencing and dose calculation.

3. Results

In the next sections, we solve Problem 6 with the max/mean dose constraints defined according to Table 2. The importance weights $w^{+}/w^{-}$ are empirically adjusted for each patient to generate a clinically reasonable plan. For the regularization term, we use $\lambda =10,\alpha =0.6$, and $\beta =0.4$. The wavelet basis $\hat{W}$ is generated based on Algorithm 1 and using sym4 as the wavelet family. In Section 3.1, we solve Problem 6 with and without the linear wavelet constraint $x=\hat{W}y$. In Section 3.2, we investigate the impact of using different values for the regularization weight $\lambda$, and in Section 3.3, we study the performance of different wavelet families.

3.1. Improved Monitor Units and Plan Quality

Fig. 2 compares the plans obtained with and without the wavelet-induced smoothness constraint $(x=\hat{W}y)$ for the paraspinal case. Figs. 2-a and 2-b show the dose distribution of a representative slice of the plans without and with wavelet-induced smoothness, respectively. Improvements in 50% dose-color-wash (blue area), especially around esophagus and cord, can be seen in the plan with wavelet smoothness (Fig. 2-b). 100% dose-color-wash (green area) also conforms better to the shape of the PTV in the plan with wavelet smoothness. Fig. 2-c compares the plans with (dashed line) and without (solid line) wavelet smoothness in terms of DVHs. It can be seen that the plan with wavelet smoothness (dashed line) delivers less radiation dose to the healthy tissues (esophagus, cord, lungs), while providing similar PTV coverage. Fig. 2-d provides a comparison with respect to the duty cycle (defined as MU divided by the prescription dose per fraction in centigray (cGy)), showing that the wavelet-induced smoothness improves the duty cycle by about 10%.

Figure 2. (Paraspinal Case):

Comparison of the plans generated with/without the wavelet-induced smoothness in terms of: (a),(b) dose distribution, (c) DVHs, and (d) duty cycle (MU/cGy). Plans are normalized to $V_{100\%}\left(PTV\right)=90\%$.

Figs. 3 and 4 provide similar comparisons for the lung and prostate cases, respectively. For the lung case, modest improvement in the DVHs and substantial improvement in duty cycle are observed using the wavelet-induced smoothness. For the prostate case, although wavelet-induced smoothness provides a marginal improvement in dose distribution (the 100% dose line conforms better to the PTV shape and has fewer holes), the two plans can be considered dosimetrically equivalent. However, the duty cycle is improved by (14%).

Figure 3. (Lung Case):

Comparison of the plans generated with/without the wavelet-induced smoothness in terms of: (a), (b) dose distribution, (c) DVHs, and (d) duty cycle (MU/cGy). Plans are normalized to $V_{100\%}\left(PTV\right)=90\%$.

Figure 4. (Prostate Case):

Comparison of the plans generated with/without the wavelet-induced smoothness in terms of: (a), (b) dose distribution, (c) DVHs, and (d) duty cycle (MU/cGy). Plans are normalized to $V_{100\%}\left(PTV\right)=95\%$.

Fig. 5 summarizes the relative improvement obtained in MU (the same as the relative improvement in duty cycle), Paddic conformity index (CI), the max/mean dose of each healthy tissue, and the total variation (TV) in beamlet intensities along the leaf travel direction (absolute variations in the neighboring beamlet intensities) by using the wavelet-induced smoothness (a positive value on the y-axis means that using wavelets has improved the metric). The wavelet-induced smoothness does not seem to make clinically meaningful changes in the max/mean/min dose values of PTV, and therefore these metrics are not included. According to Fig. 5, wavelet-induced smoothness improves both the dosimetric plan quality and plan complexity for the paraspinal and lung cases, and it reduces the plan complexity for the prostate case. It is however worth mentioning that the clinical metrics provided in this figure are not of equal clinical importance and especially for cord the max dose is often used in the clinic not the mean dose.

Figure 5.

Relative improvement in some plan quality metrics obtained using the wavelet-induced smoothness (compared to the plans without wavelets) for three different cases.

3.2. Less Sensitivity to the Regularization Weight

As mentioned before, we still include the regularization term $h$ to provide smoothness in the Y-direction and also further possible smoothness, beyond the wavelet-induced smoothness, in the X-direction. However, as this section will confirm using some experimental results, the embedded smoothness provided by wavelets desensitizes/robustifies the model against the changes in the regularization weight $\lambda$.

Fig. 6 compares the plans generated with/without wavelet smoothness and using three different regularization weights $(\lambda =0,1,10)$ for the paraspinal case. The plans are compared in terms of the 2D beamlet intensity map of a representative beam (plots in the first and second rows), the 1D beamlet intensity profile of a representative row in the same beam (third row), and the DVHs (fourth row). The plots in the first column of Fig. 6, do the comparison in the absence of the regularization term $(\lambda =0)$. By comparing 6-a and 6-d, the built-in smoothness can be observed in the beamlet intensity map in 6-d as there is less fluctuation in the beamlet intensities in the leaf movement direction (this is more pronounced in Fig. 8 of the prostate case). This is even more obvious for a representative row demonstrated in Fig. 6-g with the zigzagging and smooth behavior of the solid and dashed lines, respectively. It means the beamlet intensity map is already smooth by using wavelets and without any attempt from the regularization term.

Figure 6.

(Paraspinal Case): Comparison of the plans generated with/without wavelets using different regularization weights $(\lambda =0,1,10)$. (a)-(f): Beamlet intensity map of a representative beam for different weights. (g)-(i): Beamlet intensity profile of a representative row (j)-(l): DVHs of the plans with (dashed line) and without (solid line) wavelet smoothness.

Figure 8. (Prostate Case):

Comparison of the plans generated with/without wavelets using different regularization weights $(\lambda =\mathrm{0,1},10)$. (a)-(f): Beamlet intensity map of a representative beam for different weights. (g)-(i): Beamlet intensity profile of a representative row (j)-(l): DVHs of the plans with (dashed line) and without (solid line) wavelet smoothness.

The dashed line in Fig. 6-j represents the plan with wavelets which delivers less radiation to the healthy tissues at a small cost on PTV dose homogeneity. While increasing the regularization weight to $\lambda =\mathrm{1, 10}$ improves the smoothness for both of the plans with/without wavelets, the impact is significantly less in the presence of wavelet-induced smoothness. Figs. 7 and 8 illustrate similar phenomena for the lung and prostate cases, respectively.

Figure 7. (Lung Case):

Comparison of the plans generated with/without wavelets using different regularization weights $(\lambda =\mathrm{0, 1},10)$. (a)-(f): Beamlet intensity map of a representative beam for different weights. (g)-(i): Beamlet intensity profile of a representative row (j)-(l): DVHs of the plans with (dashed line) and without (solid line) wavelet smoothness.

Fig. 9 illustrates duty cycle values for different regularization weights for models with and without wavelets. The narrower boxes with smaller values reveal that the wavelet-induced smoothness not only reduces the duty cycle but also robustifies the model against the changes in the regularization weight, thereby easing the weight adjustment. The wavelet-induced smoothness improved the mean/variance of the duty cycle for different values of $\lambda (\lambda =\mathrm{0, 1},10)$ by 22%/91%, 38%/78%, and 41%/85% for the paraspinal, lung, and prostate cases, respectively.

Figure 9.

Duty cycle of the plans generated with/without wavelets using different regularization weights $(\lambda =\mathrm{0, 1},10)$. The bottom/middle/top part of each box represents the duty cycle corresponding to $\lambda =10/1/0$.

3.3. Performance of Different Wavelet Families

We have thus far employed the sym4 wavelet family to generate the wavelet basis, and the question that naturally arises here is: How about other wavelet families? While answering this question is beyond the scope of this work and requires a comprehensive study using a big cohort of patients, we try to get a sense of the potential impacts of using different wavelet families in this section using the lung case. Fig. 10 compares the plan generated using sym4 and the plans generated using different wavelet families for the lung case by a heat map. Except for the PTV minimum dose metric for which the higher value is considered better, the lower value is considered better for all other metrics.

Figure 10.

A heat map comparing the performance of different wavelet families, compared to sym4, for generating the wavelet basis for the lung case.

Although there does not seem to be a clear winner among the wavelet families, the orthogonal and more symmetric families of symlet and coiflet seem to have a better overall performance, and in particular, sym3 , sym4, sym6, sym7, and coif2 perform well. We would like to mention that using the simple Haar wavelet is equivalent to using the coarser beamlet resolution in the X-direction (i.e., merging every two consecutive beamlets), which obviously degrades the plan quality and is therefore not included in our comparison.

4. Conclusion and discussion

Wavelets have had profound impacts on many areas of science and engineering [11] (e.g., imaging, data science, compressed sensing) and have been employed to perform several fundamental tasks (e.g., compression, denoising). However, this is the first application, to the best of our knowledge, where wavelets are used to dictate smoothness in the decision space. The idea has been validated for smoothing the intensity maps in IMRT.

The increase in plan complexity in IMRT has adverse effects on treatment delivery and quality assurance (QA) workload [3, 15]. This is particularly of great importance in online adaptive IMRT treatment planning using nowadays MR-Linac machines [16, 17] where planning and QA are performed under mounting time pressure. This issue has been mainly addressed in the literature by adding some regularization terms in fluence map optimization, aiming at reducing the variations in the intensity of the neighboring beamlets. This work proposed an alternative approach by representing beamlet intensities using an incomplete wavelet basis, containing only low-frequency wavelets in the direction of the leaf travel (X-direction). Removing high-frequency wavelets from the basis essentially excludes those dispensable complex plans with unnecessary modulation. In other words, the slowly changing wavelets, unable to form complex maps, are the building blocks of the intensity map in the new approach. In the regularization approach, in contrast, the intensity map comprises the intensity of the individual beamlets with penalty for sharp variations. The main distinction between the proposed wavelet approach and the existing regularization techniques is that the wavelet approach explicitly removes the highly fluctuated beamlet profiles from the search space while the regularization approaches discourage those beamlet profiles in the cost function.

Increasing plan complexity and monitor units leads to a greater susceptibility to a wide spectrum of uncertainties in inverse treatment planning including: dose calculation accuracy, leaf positional accuracy, interleaf leakage and transmission, and intra-fractional motions [3]. This is of particular concern for more complex geometries that often require more modulated intensity maps and higher duty cycles. Paraspinal and lung cases are usually considered more difficult plans than prostate cases due to the anatomical complexities and the sensitive healthy tissues nearby (e.g., esophagus, cord, lungs) with demanding dose constraints (max/mean) relative to the prescription dose. The wavelet-induced smoothness seems to benefit these difficult cases the most. We expect the reduced complexity achieved by wavelets would reduce the discrepancy between the planed and delivered doses especially for complex anatomies and we are planning to perform measurement-based QA in the near future to check this hypothesis.

In our experiments, which we have not reported in this work, we noticed that removing the high-frequency wavelets for the Y-direction could potentially result in over-smoothness (in Y-direction) and degradation in plan quality. This does not seem to be the case for X-direction. Our experiments using complex anatomical structures with demanding dose constraints (paraspinal and lung) confirm that shrinking the search space using wavelets does not exclude the modulation required for these complex cases, and in fact there is still room for further smoothing using the regularization term. A more comprehensive study is needed to see whether this holds true for other complex geometries like some head-and-neck cases, which would determine whether this idea can be applied to all or some disease sites.

There exists a wide variety of IMRT optimization frameworks including robust optimization [18], hierarchical optimization [9, 10], multi-criteria optimization [19], and knowledge-based optimization [20] in research and clinic with supports for a variety of objective functions and constraints including dose volume histogram (DVH) [21, 22], generalized equivalent uniform dose (gEUD) [23], tumor control probability (TCP) [24], and normal tissue complication probability (NTCP) [24–26]. One of the advantages of the proposed wavelet-based technique is that it is agnostic to the optimization framework and can be easily integrated with any framework by simply adding a set of linear constraints. In this study, we employed a commonly used constrained optimization framework with quadratic objective function and linear max/mean constraints and demonstrated the impact of adding linear constraints to induce wavelet-based smoothness. However, one can incorporate this technique in a more sophisticated optimization framework.

The complexity reduction techniques in treatment planning are judged by their ability to effectively simplify the plans at a reasonable cost for dosimetric plan quality. Our computational experiments show that wavelet-induced smoothness outperforms the regularization-induced smoothness by generating plans with lower MUs and better dosimetric quality (dose distribution, DVHs, mean/max dose values). The dosimetric improvements in plans using wavelets can also be explained by the fact that smoother fluences can be reproduced more faithfully in the leaf sequencing process compared to the fluences with more fluctuations. The computational burden of adding the linear constraints corresponding to the wavelet basis was also insignificant (< 5%). One limitation of the proposed approach is that it only applies to the fluence map optimization (FMO) techniques and not to the direct aperture optimization (DAO) techniques. The definition of plan complexity for DAO is completely different and is based on the regularity of the resulting aperture shapes [27], and it does not seem obvious how this work can be expanded to DAO. However, many DAO techniques rely on FMO for a good warm-start solution and the wavelet-based approach may benefit DAO by providing a better warm-start solution.

Among the well-known wavelet families, our limited computational experiments show that orthogonal and more symmetric wavelet families such as sym4 and coif2 have good performances, although more comprehensive future research is needed to draw a solid conclusion. It is also worth mentioning that given the possible abrupt changes in beamlet intensities (Figs. 6, 7, 8), wavelet transforms seem to be better choices than the alternative Fourier transforms for this application. We also believe multi-resolution wavelet transforms are not applicable in this application since we need all the approximation components of the wavelets and multi-resolution essentially allows breaking the approximation components into subcomponents.

Finally, we would like to mention that various complexity metrics have been proposed in the literature (e.g., monitor units, modulation complexity score, edge metric, leaf travel, modulation index) [3], but unfortunately, there is no consensus on which complexity measure correlates better with treatment accuracy. We have employed monitor units as a complexity metric in this study, which has been arguably the most common metric used in the clinic, but the gold standard to measure the dosimetric accuracy is to perform patient-specific measurement-based QA. This evaluation, along with further validation of the proposed technique using a large and diverse patient cohort, and also expansion to direct aperture optimization (DAO) techniques including volumetric modulated arc therapy (VMAT) [27], defines natural directions for future research.