Voxel-based automatic multi-criteria optimization for intensity modulated radiation therapy

The proposed automatic multi-criteria optimization framework includes two main loops (Fig. 1), as follows: an outer loop (blue box), to prospect the most appropriate constraints; and an inner loop (black box), to explore the global optimum. The outer loop first adjusts the constraints automatically based on the optimization results from a previous iteration. It relaxes constraints if they are not satisfied, but otherwise tightens them. Whenever the constraints are set, the voxel-based FMO is immediately launched to obtain a Pareto optimal solution under these constraints. The loop is repeated until all constraints cannot be relaxed or tightened further. The inner loop is the mentioned voxel-based FMO. The loop iterates between two parts. One is the automatic adjustment of voxel-based parameters, and the other is the solver of FMO problem for given voxel-based parameters. Loops will stop when all given constraints in the outer loop are met or the iteration reaches its maximum number. Details are stated carefully in the following parts.

Normally, for IMRT optimization, the primary step is to obtain optimal intensity maps for beams that penetrate from assuming known and fixed angles. Here, we implement a voxel-based quadratic model formula below for fluence map optimization [22].

Notation f is the beamlet intensity vector with all nonzero elements. H is the dose deposition coefficient matrix pre-calculated using the quadrant infinite beam (QIB) [23, 24] algorithm implemented in a computational environment for radiotherapy research (CERR) platform [25]. The dose distribution d is linear to beamlet intensity, and the calculation can be quickly performed by d=Hf during optimization.

s(f) denotes the objective function of FMO problem, including two terms. The former term is the dose difference between the received dose Hf and the reference dose \( {d}_v^p \) (prescription dose for target, a reasonable low dose for OAR). The latter term is a regularization term that ensures a smooth fluence map. κ is the fluence map smooth weighting factor. M is an operator equal to Δf/f, Δf is the discrete Laplacian operator to fluence. Two types of voxel related parameters are incorporated in the former part of Eq. (1), namely, the regular organ-based weight ξ_v and the refined voxel-dependent parameter w_j, which is the element of diagonal matrix \( {\tilde{w}}_v \) with its dimension equals to the number of voxels selected in optimization, j. j ∈ v is the voxel index belongs to ROI v. ξ_v is the weight assigned for different ROIs v ∈ V. D_x% ≥ (≤)d₁ and V_x ≤ v₁ are the commonly used dose-volume constraints. When all parameters in Eq. (1) are set, a bound constrained convex quadratic problems (BOXCQP) algorithm [26] is called to solve the optimization problem, thereby achieving a feasible solution in rapid convergence.

When sets of constraints were updated in the outer loop of the framework, voxel-based parameters should be tuned consequently to satisfy these constraints. Considering that the voxels involved in optimization are enormous, typically ~ 10⁷, an automatic voxel-based parameter adjustment method should therefore be preferred over traditional manual tuning. The principle of adjusting voxel-based parameter is intuitive. We find the voxel-violated constraints and increase their corresponding values.

Two main types of constraints are considered in this study, namely, dose and dose-volume constraints. These constraints are commonly used in clinics. Detailed selections for these two situations are shown in Fig. 2. The shaded area is the selected voxel region.

Figure 2 (1) (2) show the selection when constraints are a maximum dose (for PTV and OARs) and minimum dose (for PTV), respectively. D_c is the critical dose level. For maximum dose constraints, voxels with received dose ≥D_c are selected and their assigned parameters are increased. Similarly, for minimum dose constraints, voxels with received dose ≤D_c are selected.

For dose-volume constraints, reducing or increasing doses for voxels with their received dose just over or under the critical dose level is more efficient [27]. Figure 2 (3) (4) show the selection of voxels for maximum dose-volume constraints (for PTV and OARs) and minimum dose-volume constraints (for PTV), D_c is the critical dose level, V_c is the critical volume, D is the dose level with a volume equals to V_c. For the maximum dose-volume constraint, the voxels with dose value between D_c and D (D > D_c) are selected. For the minimum dose-volume constraint, the voxels with dose values between D and D_c (D < D_c) are selected.

After the selection of penalized voxels, their corresponding parameters are increased accordingly. For minimum dose and dose-volume constraints, voxel-based parameters are updated by \( {w}_j^{k+1}={w}_j^k{\left[\left({D}_c+\alpha \right)/{D}_j^k\right]}^{\mu },\left({D}_j^k\le {D}_c\right) \). For maximum dose and dose-volume constraints, voxel-based parameters are updated by\( {w}_j^{k+1}={w}_j^k{\left[\left({D}_j^k+\alpha \right)/{D}_c\right]}^{\mu },\left({D}_j^k\ge {D}_c\right) \), where \( {D}_j^k \) is the received dose for voxel j at k iteration. \( {w}_j^k \) is the voxel-based parameter for voxel j at previous iteration k. \( {w}_j^{k+1} \) is the updated voxel-based parameter for voxel j at current iteration k + 1. Parameters μ and α(α > 0) are the user-defined enhancement factors. Their effect on the convergence speed are discussed in this paper.

Normally, the optimized dose distribution will meet or become close to the constraints after several iterations of voxel-based parameter update and FMO optimization. When all constraints are met, the iteration is immediately terminated. However, in certain cases, the constraints may be set too tight for this particular patient and are thus hard to meet. Then, the algorithm must force itself to stop by a reasonable maximum iteration time.

The promising solution to an efficient and high-quality plan generation is to feed the optimization engine with appropriate constraints based on clinical planning experience. When the given constraints are too tight, the optimization engine cannot easily find a satisfying solution. By contrast, if the constraints are too loose, the achieved plan could be sub-optimal. This situation is the uppermost exact reason for the current trial-and-error planning procedure until the dosimetrists set constraints that are sufficiently tight. Therefore, attempting to automatically adjust these constraints should be addressed (the outer loop in Fig. 1). Prior to the automatic constraints adjustments, a user-defined constraint priority list with some initial values should be given first.

Constraint priority list is a pivotal conversion from the clinical bias to the scientific trade-offs. The list is generated by classifying and sequencing the planning endpoints (normally from clinically adopted protocols) following their clinical importance. Endpoints with their constraints are arranged at different levels, with a corresponding number that indicates its priority. Moreover, constraints are divided into two classes, namely, hard and soft constraints. With a priority number 0, hard constraints cannot be violated and are forbidden to relax or tighten during optimization. By contrast, soft constraints have nonzero priority numbers, and their priorities are decreased gradually with their increasing number. Soft constraints can be relaxed and tightened, and they may be promoted to hard constraints during optimization.

Given an example of priority list set on patients who received IMRT with GYN cancer (Table 1), the optimization requiring dosimetric endpoints and constraints are accordingly derived from a clinical protocol, i.e. the International of Radiotherapy Technology Effectiveness in Cervical Cancer (“INTERTECC”) in this study. Among these, the first imperative criteria would be given to D_min (the minimum dose), D_max (the maximum dose), D_97% (the minimum percentage dose of 97%) of the PTV, and D_max of the Body-(PTV + 1) (1 cm) for considering the PTV coverage and dose homogeneity and basic normal tissue protection. Thus, they are set as hard constraints with the “highest” priority number of 0. Here “Body-(PTV + 1)” is considered rather than “Body-PTV,” because a dose transition region between high-dose PTV and low dose normal tissue is usually needed to make the optimization problem easier. Other planning-related OAR endpoints are subsequently considered soft constraints, such as endpoints of rectum, bladder, bowel, femoral heads, and bone. Their priorities are co-determined based on both planning experience and physician’s preference. Details are listed as in Table 1.

The adjustment of constraints is a four-stage process that has been discussed thoroughly in Breedveld et al. [27]. The first is the preparatory stage. In this stage, we generate an initial dose distribution by applying an in-house developed voxel-dependent parameter optimization. The initial organ-dependent parameters, voxel-dependent parameter (all start from 1), and priority list are the preliminary inputs and are set empirically. Similar to clinical circumstances, constraints set in this stage are consistently too tight. Thus, a solution that can satisfy all the constraints cannot easily be obtained. In this case, partial constraints must be relaxed. This situation leads to the second stage, with an attempt to relax the constraints until each of them can be satisfied. The relaxation begins from low priority to high priority, cluster by cluster. For example, if one or more soft constraints are violated, the lowest priority ones are simultaneously selected for relaxation. A simple but effective means for relaxation is to increase current constraint values with an appropriate interval, such as 0.5% for most cases based on experience. Results generated in this stage have a high possibility to improve. Thus, in the following third and fourth stages, the potential loose constraints should be tightened until the improvement of any single constraint is accompanied by at least one other constraint violation. By contrast, in the tightening round, constraints are adjusted from high-priority to low-priority, endpoint by endpoint. Each stage calls the automated voxel-dependent FMO with a maximum number of iterations. The soft constraints relaxed in the second stage are tightened in the third stage, and the remaining soft constraints are tightened in the fourth stage. If all constraints are met, including the tightened constraints, the optimization system would continue to tighten the remaining ones until it reaches the maximum iteration. Otherwise, the ongoing constraint is reset to its previous value and becomes promoted as a hard constraint. Similarly, a constraint is tightened by subtracting an appropriate interval such as 0.5%.

The feasibility and efficiency of this proposed automatic multi-criteria optimization framework are evaluated by 10 GYN IMRT cases, with a prescription dose of 45 Gy to the PTV and treated with seven beams (150°, 100°, 50°, 0°, 310°, 260°, and 210°). All these IMRT plans are originally exported from the commercial Eclipse TPS (Varian Medical Systems, Palo Alto, CA). We re-optimized each IMRT plan using this in-house developed automatic multi-criteria optimization framework (called an optimized plan below) and compared it with the original one (called the original plan) in terms of DVH curves and particular endpoints. Beam set-ups are maintained unchanged.

To investigate synoptically the plan quality improvement, a paired t-test was performed on each plan dosimetric endpoint for all evaluated cases between the optimized plan and the original plan. Furthermore, we recorded the stage-status to observe the transition tendency along with the four-stage optimization proceeded by examining their constraint-settings and subsequent optimized values for each dosimetric endpoint.

Other parameters were set by experience, as follows: organ-based weights 100, 1, 2, 0.1, 0.1, 0.1, 0.1 and 0.1 for PTV, bladder, rectum, bowel, femoral heads, bone, Body-(PTV + 1), and body, respectively. The initial voxel-based weight was set to 1, and the initial constraint priority list is shown in Table 1. The reference dose \( {d}_v^p \) in Eq. (1) was set to a prescription dose for PTV and empirically for each OAR. Maximum voxel-based parameter updating time was set to 30, 10, 30, and 30 for each stage. Here, 30 is determined based on the convergence of the optimization cost function value as with voxel-weight updating time (Fig. 3). Voxel-dependent parameter auto-tuning influence factors μ and α were 20 and 0.5, based on the investigation to the convergence when μ and α ranged from 5 to 25 and 0.001–1, respectively, based on experience (as shown in Fig. 4). Several of these cases couldn’t generate a solution when μ was during 20 to 25 owing to an ill-condition Hessian matrix. The same situation occurred when μ was equal to 20 and when α changed from 0.5 to 1. This whole framework was implemented with Matlab 2013a and installed on a 3.4 GHz Intel Core4 computer running Windows 7.

Both DVH curves and particular endpoint comparisons show plan quality improvement for our optimized plans. The DVH comparison of the optimized plans and the original plans for the evaluated 10 GYN IMRT cases is shown in Fig. 5. The optimized plans not only improved the PTV dose homogeneity (with decreased PTV hotspot) but also provided a better OAR dose sparing for most cases. By contrast, Table 2 further illustrates specific DVH endpoint comparison for case 2 as an example. Almost all the endpoints in both plans satisfy the protocol well, except for one soft constraint, i.e., V₃₀ of the rectum, which was violated in both plans because of the large overlapping area with PTV. The other two soft constraints, namely, V₁₀ and V₂₀ of bone, also failed in the original plan but were satisfied in the optimized one. For PTV, comparable D_97% and its cold spot D₉₇are also maintained in the optimized plan, but hot spots decrease, thereby indicating improved PTV coverage and dose homogeneity. Moreover, for OARs, most endpoints lower in the optimized process.

Averaged plan dosimetric endpoints for all evaluated 10 GYN IMRT cases are also investigated, as shown in Fig. 6. For PTV, the D_97% and V_115% are maintained, whereas the average D_99% changes from 43.26Gy (±0.41Gy) to 43.24Gy (±0.26Gy), and the V_110% decreased from 0.15% (±0.18%) to 0.06% (±0.10%), for the original and optimized plans, respectively. The average maximum dose of bowel, V₄₅ of rectum and bladder, V₃₀ of rectum and femoral heads, V₂₀ and V₁₀ of bone are decreased from 48.83Gy (±1.21Gy) to 48.72 Gy (±1.00Gy), 46.38% (±0.17%) to 43.81% (±0.21%), 47.23% (±0.11%) to 35.96% (±0.13%), 97.73% (±0.02%) to 88.47% (±0.11%), 14.53% (±0.10%) to 8.27% (±0.05%), 72.05% (±0.02%) to 59.02% (±0.04%), and 87.85% (±0.01%) to 79.18% (±0.01%), for the original and optimized plans, respectively. With slightly different trade-offs, the average maximum dose of rectum, bladder and femoral heads increase from 47.57Gy (±0.96 Gy) to 49.4 Gy (±1.29 Gy), 47.95Gy (±1.11 Gy) to 48.42Gy (±1.18 Gy), and 41.24 Gy (±3.73 Gy) to 41.52 Gy (±3.42 Gy), respectively. However, these values effectively satisfy the corresponding constraints.

Results on difference significance analysis between the optimized plan and the original plan on all 10 cases are listed in Table 3. Among 13 observed endpoints, 6 are shown with significant differences between the original and optimized plans (P < 0.05), such as D_{max_rec}, V_{30_rec}, V_{45_bla}, V_{10_bm}, V_{20_bm}, and V_{30_fh}.

As mentioned above, four stages were designed in this multi-criteria optimization framework, namely, the first preparatory initial stage, the second violated constraint relaxing stage, the third backward relaxed constraint tightening stage, and the last constraint further tightening stage. Plan quality was inspected in the entire time with progression of the stages, and constraints changed during optimization.

Figure 7 shows an example of the clinical-relevant dostrimetric endpoint value changing process, with respect to its corresponding pre-defined constraints. The first stage ((a)–(b)) contains 30 voxel tuning iterations and no constraint adjustment. Normally, at the end of this stage, we cannot easily find a plan that meets all constraints, thereby resulting in the second stage, namely, relaxing violated constraints. This second stage begins at the violation of V_{40_bla} and relaxing it from 65 to 65.5%and ends at the plan V_{40_bla} was increased from 65.12 to 65.36% after optimization, requiring another 74 iterations for this case ((b)–(c))).In the third stage, we attempted to undo the constraint relaxations of stage 2 begins at 104th iterations. Between (c) and (d), the algorithm minimizes the dose-volume constraints for the bladder, because the dose-volume constraint for the bladder is violated and relaxed previously. After optimization, the plan V_{40_bla} was decreased from 65.36 to 64.34% at the end of stage 3. At stage 4, first, the V_{40_rec} constraint is tightened, and until the algorithm failed to find a solution ((d)–(e)). Then, the V_{30_rec} is considered. It undergoes a considerable reduction ((e)–(f)). Finally, the V_{40_bowel} ((f)–(g)), V_{30_f} ((g)–(h)), V_{10_bm} ((h)–(i)) andV_{20_bm} ((i)–(j)) are tightened in turn. The entire optimization requires 855 iterations of voxel-based FMO optimization, which requires approximately 30 min to complete.

Table 4 and Fig. 8 illustrate an example of specific dosimetric endpoint values and its corresponding DVH curves at the end of each stage. Both of the dosimetric endpoint values and their corresponding DVH curves gradually change as optimization proceeds. Relative drastic variations normally occur at the end of stages 1 and 4; for example, the V₃₀ of rectum decreases to 71.31 and 61.30%, respectively, compared with the original (93.51%).