A general algorithm for distributed treatments of multiple brain metastases

Abstract

Purpose:

Stereotactic radiosurgery (SRS) has become a primary treatment for multiple brain metastases (BM) but may require distribution of BMs over several sessions to make delivery time and radiation toxicity manageable. Contrasting to equal fraction dose in conventional fractionation, distributed SRS delivers full dose to a subset of BMs in each session while avoids adjacent BMs in the same session to reduce toxicity from overlapping radiation. However, current clinical treatment planning for distributed SRS relies on manual BM assignment, which can be tedious and error prone. This work describes a novel approach to automate the distribution of BM in the Gamma Knife (GK) clinical workflow.

Methods:

We represent each BM as an electrostatic field of the same polarity that exerts repulsive forces on other BMs in the same session. This representation naturally leads to separation of close BMs into different sessions to lower the potential energy. Indeed, the BM distribution problem can be formulated as minimization of the total potential energy from all treatment sessions subject to delivery time constraints in mixed-integer quadratic programming (MIQP). We retrospectively studied eight clinical GK cases of multiple BM and compared the automated MIQP solution with clinically used BM distribution to demonstrate the efficacy of the proposed approach.

Results:

With the problem size equal to the number of BMs times the number of sessions, this MIQP can be solved in a minute on a personal workstation. The MIQP solution effectively separated BMs for a given number of treatment sessions and evened out the delivery time distribution among sessions. Compared to the clinically used manual BM distributions in paired t-test for a similar range of delivery time variation, the automated BM distributions had lower energy objectives (range of decrease: [11% 89%]; median: 25%; $P=.073$), more uniformly distributed treatment volumes (range of decrease for the normalized standard deviation of volume distribution: [0.02 0.95]; median: 0.16; $P=.013$), more scattered BMs in each treatment session (range of increase for the mean minimum BM distance: [0 14] mm; median: 6 mm; $P=.008$), and lower overall $V_{12}$ (range of decrease: [0.0 1.6] cc; median: 0.2 cc; $P=.052$). Moreover, without distribution, i.e. with all BMs treated in the same session, $V_{12}$ was substantially larger compared to both manual and automated BM distributions; the increase ranged from 0.1 cc to 16.6 cc with median of 1.3 cc.

Conclusions:

The proposed approach models the clinical practice and provides an efficient solution for optimal selection of BM subsets for distributed SRS. Further evaluations are underway to establish this approach as a tool for improving clinical workflow and to facilitate systematic study on the benefits of distributed SRS treatments.

1. Introduction

Stereotactic radiosurgery (SRS) has become an established treatment option for brain metastases (BM), while for many decades whole brain radiation therapy (WBRT) was the mainstay of treatment for multiple BM (> 3)¹. With rapidly evolving treatment paradigms, SRS provided a suitable alternative to WBRT. Unlike WBRT, SRS does not cause notable neurocognitive decline or compromise survival outcomes^2,3. This is quite beneficial for patients’ quality of life because of the expected longer survival thanks to the advances in systemic cancer management.

SRS delivers large focused radiation dose in single or few treatment sessions to provide effective local control. For patients with a large number of BMs, SRS posed two major hurdles: prolonged delivery time and increased radiation toxicity (relative to treatments with small numbers of BMs). To mitigate these two hurdles, our clinic has adopted the distributed SRS strategy utilizing the Gamma Knife (GK) system (Leksell GammaKnife® PerfexionTM/IconTM, Elekta AB, Stockholm, Sweden) for the treatment of multiple BM⁴. In contrast to equal fraction dose with conventional fractionation, every treatment session of distributed SRS focuses on a subset of BMs while adjacent BMs are treated in different sessions. The advantage of this concept is a reduced risk of radiation toxicity (brain necrosis) by avoiding overlapping spilled dose from adjacent BMs on the day of treatment while giving each BM the full prescription dose in a single session⁴.

GK is one of the advanced radiotherapy modalities that offer localized dose for SRS⁵. By positioning the treatment target at the system’s isocenter, GK delivers gamma ray beams from 192 Co-60 radiation sources arranged into eight sectors surrounding the skull, all focusing at the isocenter, to create cross-firing radiation called a shot and provides conformal dose with rapid dose fall-off⁶. The clinical GK treatment planning system (GammaPlan® v11.1.1, Elekta AB, Stockholm, Sweden) does not provide any algorithm/interface for multi-session distributed treatment. The current clinical practice thus relies on manually assigning each BM to a treatment session.

Typically, the BMs are listed in the order of shot delivery time and then sequentially assigned, balancing delivery time and avoiding close BM in the same session. A similar strategy has been described in the literature: sorting the BMs by their size from the largest to smallest and sequentially assigning each BM to a session of minimum dose proxy at the BM’s location, where the dose proxy is calculated with the inverse square law based on the already assigned BMs⁷. An opposite strategy, clustering instead of separating BMs, has also been used to reduce delivery time per session⁸. After the assignment, only shots for the session’s BMs are kept in each session and saved as a treatment plan.

Manual assigning the BM distribution, however, is tedious and error prone and may become intractable or result in a suboptimal distribution (less separated BMs than achievable) when the number of BMs is large. Computer sorting followed by sequential assignment provides a heuristic solution but may still be suboptimal. In this work, we proposed a novel approach to model the intuition behind the manual assignment of the BM distribution, allowing the problem to be formulated as a tractable optimization problem.

2. Methods

The BM distribution problem for distributed treatments can be described as follows: given a set of BMs to be treated, a master treatment plan consisting of treatment shots (delivery information) for all BMs in the set, and the number of treatment sessions, find a distribution (session assignment) of the BMs such that 1) each BM is in one and only one session, 2) close BMs are assigned to different sessions with more weight for larger BM volume, and 3) delivery time is approximately the same (without large variation) for all treatment sessions. To solve the BM distribution problem, we casted it in the field potential framework and turned it into an optimization problem for automated solution.

2.1. Distribution of BMs by minimizing the potential energy

Each BM is represented by an electrostatic field of the same polarity that exerts mutual repulsive forces on other BMs in the same session. Distribution of BMs can be modeled as a distribution problem for repulsive fields of electric charges:

Given a set of electric charges $\left\{\left(r_i,Q_i\right)\right\}$ to be divided into $K$ subsets, where $i$, $r_i$, and $Q_i$ denote the charge’s index, position, and magnitude, respectively, what is the optimal distribution that minimizes the total potential energy, assuming there is no potential energy between charges placed in different subsets?

This problem can be formulated as an optimization problem. For two charges, indexed by $i$ and $j$, in the same subset, the electriostatic potential energy $U_{i,j}$ can be expressed as in (1) based on the repulsive force and the inverse-square law, omitting the Coulomb constant.

$$U_{i,j}=\left(Q_i{Q}_j\right)/\Vert r_i-r_j\Vert$$ (1)

Let $x_{i,k}\in \left\{0,1\right\}$ denote the indicator variable: $x_{i,k}$ equals 1 if the ith charge is in the kth session, and 0 otherwise. Then, the total potential energy $\mathcal{U}$ can be expressed as in (2).

$$\mathcal{U}={\sum }_{i,j,k}{U}_{i,j}{x}_{i,k}{x}_{j,k}$$ (2)

Now, the electric charge distribution problem becomes a mixed-integer quadratic programming (MIQP) problem with binary variables $\left\{x_{i,k}\right\}$ to be solved.

$$\begin{gathered} \min \mathcal{U},\hspace{0.17em}\text{s}.\text{t}.\hspace{0.17em}{x}_{i,k}\in \left\{0,1\right\} \\ {\sum }_k{x}_{i,k}=1,\forall i \end{gathered}$$ (3)

The constraints in (3) mean that each charge exists in one and only one of the $K$ subsets. Additional constraints may be imposed as in (4) so that the charges in each subset are bounded by lower and upper limits, denoted by $lb$ and $ub$, respectively.

$$\begin{gathered} \min \mathcal{U},\hspace{0.17em}\text{s}.\text{t}.\hspace{0.17em}{x}_{i,k}\in \left\{0,1\right\} \\ {\sum }_k{x}_{i,k}=1,\forall i \\ lb\le {\sum }_i{t}_i{x}_{i,k}\le ub,\hspace{0.17em}\forall k \end{gathered}$$ (4)

Here, the weight $t_i$ associated with the charge $i$ is determined a priori. For BM distribution, a charge represents a BM, the position $r$ represents the mass center of the BM, and the magnitude $Q$ represents the volume of the BM. The weight $t_i$ represent the shot delivery time for the ith BM. Thus, the upper and lower bounds in (4) in effect set constraints on delivery time per session. For example, the delivery time may be constrained to be about the same in each session. The MIQP solves for the binary variables $\left\{x_{i,k}\right\}$ and has a problem size of $N\times K$, where $N$ is the total number of BM and $K$ is the number of sessions. For a typical BM distribution problem ($N\le 100$ and $K\le 5$), the MIQP can be solved within a minute using many commercial (e.g. Gurobi, https://www.gurobi.com/products/gurobi-optimizer/) or open source solvers, such as the open-source Matlab solver OPTI^9,10 used in this study.

2.2. Comparison with clinical BM distributions

We retrospectively studied the effect of automated MIQP distribution for eight clinical cases each with a larger number of BMs. The RTPlan, RTStruct and RTDose of each case were exported from our clinical GammaPlan® TPS and subsequently deidentified. Table 1 lists the statistics for each of the eight cases, including the number of treated BMs; the total volumes of all BMs; the minimum, maximum, and median volume per BM; total delivery time; the minimum, maximum, and median delivery time per BM; the prescription dose; and the number of treatment sessions. The delivery time per BM was calculated by summing the time of all shots associated with the BM. The prescription dose was specified in the plan report at an isodose level, ranging from 50% to 80%, due to the requirement of target coverage for various sizes of the BMs.

Table 1.

Statistics of the eight clinical cases in our retrospective study.

Pat #	# of BM	Total vol (mm3)	Vol per BM (mm3)			Total delivery time (min)	Delivery time per BM (min)			Pres. dose (Gy)	# of sessions
			Min	Max	Med		Min	Max	Med
1	30	16423	4	6586	102	326.93	6.23	47.40	9.76	15	4
2	31	2370	3	267	57	258.63	5.99	10.90	8.36	15	2
3	39	7752	16	1251	116	306.82	6.08	14.68	7.40	15	6
4	23	4777	3	868	184	219.20	5.97	21.12	8.69	15	2
5	20	726	14	129	33	182.80	7.34	12.63	9.10	15	4
6	33	1952	11	400	32	248.58	4.78	9.52	7.58	15	5
7	28	1052	4	176	23	177.36	4.48	9.81	5.99	13	5
8	26	593	4	91	17	224.43	6.54	13.26	8.38	15	5

We compared the automated distribution (MIQP) and clinically-used manual distribution (Clinical) for their energy objectives as described in (2), the minimum distance between BMs in each session, the standard deviations of BM’s volume and delivery time per session, and the $V_{12}$ for each patient using paired t-test. $V_{12}$ is the brain volume that receives 12 Gy dose or above, excluding the BMs; though not specified in the optimization objective, $V_{12}$ is affected by the size of BMs and distance between BMs. The automated MIQP solution was constrained by the lower and upper bounds of 75% and 150% the average session time, respectively, in optimization (4). We also compared the $V_{12}$ of MIQP and clinical distribution with that of the single session (Single) non-distributed plan.

3. Results

The MIQP and clinical BM distributions were summarized per patient and presented in bar graph for clarity. The numerical details of these distributions per treatment session can be found in

Table 2 in Appendix. Figure 1 (a) shows the ratio of the energy objective of MIQP distribution to that of clinical distribution for each patient. The ratios were less than one for all patients: the ratio ranged from .15 to .89 with the median ratio of .75 (equivalently, 11% to 85 % reduction in energy objective by MIQP with median reduction of 25%), indicating the MIQP distributions achieved lower energy objectives than the clinical distributions for all studied cases. Figure 1 (b) shows the mean of minimum distances between BMs in each session for each patient. The mean minimum distances were almost always increased by the MIQP distribution: the increase ranged from 0 mm to 14 mm with median increase of 6 mm, indicating that the BMs were more separated on average with MIQP distribution than with clinical distribution. Figure 1 (c) shows the normalized standard deviation of BM’s volume per session (standard deviation divided by mean); the values were decreased with MIQP distribution: the decrease ranged from 0.02 to 0.95 with median decrease of 0.16, indicating the MIQP had more evenly distributed treatment volumes than the clinical distributions. Figure 1 (d) shows the normalized standard deviation of delivery time per session; these values were within a 30% range for MIQP distributions due to the time bound of 75% to 150% the average delivery time as optimization constraints, while they varied with a larger range, up to around 40%, for clinical distributions. Figure 2 shows the bar graph of the aggregate of $V_{12}$ from all treatment sessions for each patient, comparing clinical distribution, MIQP distribution, and single treatment session (non-distributed). Compared to the clinical distribution, the MIQP distribution had slightly lower $V_{12}$, ranging from virtually no difference to a decrease of 1551 $m{m}^3$ with median decrease of 153 $m{m}^3$; the decrease in $V_{12}$ can be anticipated from the larger minimum distance between BMs with MIQP distribution. Moreover, the $V_{12}$ for non-distributed treatments was substantially larger than that for distributed treatments by the amount ranging from 140 $m{m}^3$ to 16552 $m{m}^3$ with a median increase of 1259 $m{m}^3$. This large variation can be attributed to the size of BMs and distance between the BMs. Indeed, the improvement in $V_{12}$ with distributed SRS was more pronounced in those patient cases with larger maximum BM volumes (pat # 1, 3, 4, and 6 in Table 1).

Table 2.

Numerical details of clinical and MIQP distributions.

	Clinical distribution						MIQP distribution
Pat #	BM	Energy	Min dist. (mm)	Vol (mm3)	Time (min)	V12 (mm3)	BM	Energy	Min dist. (mm)	Vol (mm3)	Time (min)	V12 (mm3)
1	5	5416.16	41.08	6450	58.20	3270	3	2614.82	41.08	6623	61.68	1700
	10	10094.45	20.74	1095	100.21	3288	6	7288.70	38.32	6456	64.24	1708
	5	16516.58	18.93	6674	77.65	1792	11	17646.66	22.43	1649	108.70	3325
	10	29793.02	19.39	2204	90.88	1450	10	18285.64	31.30	1695	92.31	2855
2	11	4053.79	13.19	697	89.99	1609	16	8980.56	16.00	1209	134.07	1046
	20	23234.34	9.16	1673	168.64	1031	15	9001.15	13.83	1161	124.56	1316
3	4	53.47	39.92	110	36.98	1169	7	4974.96	27.74	1618	60.96	1822
	4	58.59	33.39	109	31.57	2163	7	5041.14	52.54	1174	55.44	1812
	8	6558.24	31.05	1107	58.44	6546	6	5041.31	39.32	1044	44.62	2945
	7	6822.25	29.48	1111	54.82	2600	6	5980.63	59.26	1280	49.23	5081
	9	12174.05	15.21	1324	79.38	1235	7	6381.26	18.41	1356	54.30	2201
	7	96687.41	17.53	3991	45.63	3314	6	7442.33	38.89	1280	42.27	1615
4	11	26189.53	13.95	2189	110.93	2535	12	28700.18	27.25	2383	113.67	2060
	12	44541.20	29.25	2588	108.28	1592	11	34350.19	15.67	2394	105.53	1974
5	5	104.01	32.45	143	50.57	486	4	112.44	57.03	193	35.97	222
	5	181.81	32.95	192	43.39	221	5	145.89	40.31	169	45.70	227
	5	230.81	20.85	165	41.18	332	6	158.87	39.94	189	53.72	590
	5	261.03	39.57	226	47.67	283	5	168.96	40.96	175	47.41	275
6	7	175.33	21.23	168	52.93	2964	6	359.51	21.89	405	41.81	908
	7	206.28	12.60	175	52.75	709	6	426.36	29.45	494	44.69	2145
	5	229.87	42.53	246	37.38	488	8	543.91	45.51	314	58.15	785
	7	2185.76	7.69	407	51.09	677	8	544.73	16.29	367	62.56	452
	7	13553.49	10.88	956	54.43	575	5	590.14	53.83	372	41.37	496
7	6	96.89	32.57	134	38.64	165	6	131.48	33.47	232	32.95	267
	5	124.55	37.84	177	36.84	195	5	159.38	40.57	199	35.90	276
	5	197.72	42.41	243	30.09	260	5	166.61	30.03	200	30.69	134
	6	252.94	24.34	217	36.41	311	6	181.55	33.49	228	35.35	187
	6	449.85	29.21	281	35.38	188	6	225.13	40.75	193	42.47	234
8	5	33.59	23.91	99	38.69	203	5	48.10	33.89	134	45.41	204
	5	45.47	45.86	92	43.13	208	4	49.09	31.43	127	33.75	354
	5	57.15	56.66	141	44.95	254	6	49.96	58.37	103	50.74	259
	5	81.86	18.42	132	45.19	417	5	56.77	20.88	116	43.64	246
	6	126.11	23.02	129	52.48	211	6	60.79	22.43	113	50.89	226

Figure 1.

Bar graphs comparing the results of clinical distribution and MIQP distribution for the eight studied patient cases. (a) The ratio of MIQP distribution’s energy objective to clinical distribution’s. (b) The mean minimum distance between BMs in each session. (c) The normalized standard deviation (SD) of treatment volume per session. (d) The normalized standard deviation (SD) of delivery time per session.

Figure 2.

The aggregated $V_{12}$ from all treatment sessions of each patient for clinical distribution, MIQP distribution, and single session (non-distributed).

Summarizing in paired $t$-test, the automated MIQP distribution compared to clinically used manual distribution had a lower energy objective $\left(P=.073\right)$, larger minimum distance between BMs $\left(P=.008\right)$, lower standard deviations of BM volume per session $\left(P=.013\right)$; insignificant difference in delivery time distribution $\left(P=.282\right)$; and lower aggregated $V_{12}$ from all treatment sessions of each patient $\left(P=.052\right)$. Note that these statistics were based on eight patient cases.

4. Discussions

For brain metastases, SRS has the benefit of local control, reduced delivery time, and reduced cognitive impairment compared to the whole brain radiotherapy⁴. However, SRS is associated with increased risk of radiation necrosis when adjacent targets are treated simultaneously and has a prolonged delivery time when the number of BMs is large. To address these issues, our clinic has adopted the distributed treatment strategy. Unlike conventional fractionation, the distributed SRS aims to fractionate the OAR dose but not, or minimally, the target dose⁷. Such effect can be seen in $V_{12}$ of the studied cases (Figure 2). The effect was more pronounced for the cases with larger maximum BMs. While for small BMs the benefit of reducing toxicity may not be distinct, the distributed SRS could still be used to even out the treatment time for patient comfort.

The current distributed SRS, however, is implemented with manual BM distribution, which is time-consuming and error prone. We proposed a novel approach to model the intuition behind the strategy adopted in the clinic and used several metrics to demonstrate the quality of the automated BM distribution. With this approach, the automated distribution is expected to have lower energy objectives than manual distribution if both have the same range of delivery time per session, since the automated distribution is the optimized solution (4). The minimum distance between BM in each session is an indication for the BM’s spread, which tends to increase with optimization because the larger the distance the lower the energy (2). The standard deviations of BM volume and delivery time per treatment session are used to compare how evenly distributed the treatment volume and delivery time are with the two distributions. The optimization tends to find a more even distribution than an extreme one since the former generally corresponds to a lower energy. The quantity $V_{12}$ is of clinical interest and reducing $V_{12}$ is a motivation for distributed treatment.

Abstractly, the electrostatic field potential approach considers the charge’s mass, distance between charges, and the force type, which dictates the distribution to be charge separation or clustering, and can thus be easily modified to address different treatment requirement. For instance, some regions in the brain may be more vulnerable to radiation and benefit more with spatial temporal distributed treatments. This can be accommodated by imposing weights manifested as increased magnitude for BM’s that is adjacent to the sensitive regions and demands being more isolated than others. In fact, this approach allows versatile modeling. The definition of potential energy may be varied in this framework to simulate different clinical emphases, such as maximizing between-shot distance, absolute dose gradients, or radiobiological effectiveness.

More concretely, this field potential approach can also be applied for staged treatments of large arteriovenous malformations (AVM)¹¹. Like distributed treatments, staged treatments reduce delivery time and radiation toxicity, but unlike distributed treatments, they have a different treatment sub-volume criterion, which requires a connected partial volume in each session. This can be modeled as shot clustering in the proposed framework with the attraction force; that is, the potential energy in (1) is changed to $U_{i,j}=-\left(Q_i{Q}_j\right)/\Vert r_i-r_j\Vert$. The negative potential energy objective will lead to clustering of the shots in each staged plan. Another potential application of attractive filed for clustering is automatic isocenter selection for LINAC-based SRS treatments¹². In this scenario, the goal is to group treatment targets so that each group (cluster) uses a single isocenter to minimize dosimetric uncertainty and reduce the number of isocenters for treatment efficiency.

The effect of the repulsive/attractive field potential approach in separation/clustering for BMs or shots can be illustrated in 2D simulations (Figure 3). The first row, from left to right, illustrates distributed treatments in 2–5 sessions with the circle indicating BM and color indicating session; the second row with the same BM configuration as in the first row illustrates BM clustering; the third row illustrates shot clustering for the staged treatment of AVM.

Figure 3.

Illustration for the effect of automated BM distributions, BM clustering, and shot clustering. First row simulates distributed treatment for multiple BM (indicated by circles) in 2–5 sessions (color-coded), from left to right. Second row simulates BM clustering with the same configuration as the first row. Third row simulates staged treatments for large AVM in 2–5 sessions, from left to right, with the circle indicating the shot.

Treatments have been ever spatially localized with improvements in machine flexibility, beam quality, and optimization algorithms. Another route is to exploit the difference of radiation response in tumor and normal tissues via fractionation to gain the therapeutic ratio^13,14. More generally, the search of target dose localization may be expanded to include both space and time dimensions as spatiotemporal optimization¹⁵. For multiple BM or large AVM, however, the additional constraints that require each BM or a connected partial volume to be treated in only one single session further complicate the already hard-to-solve spatiotemporal optimization problem. The field potential approach provides a workaround spatiotemporal solution, and despite not fully spatiotemporally optimized, it can be intuitively evaluated. Besides complex optimization, solving spatiotemporal optimized plans are challenged by the uncertainty of radiobiological parameters such as the $\alpha$, $\beta$ values. As future projects, the proposed automatic BM distribution approach will be evaluated extensively on clinical plans for its efficacy, and spatiotemporal optimized plans will be investigated and compared.

5. Conclusions

We proposed a novel approach using the electrostatic field and potential energy to model BM distribution for distributed SRS treatments, extending the benefits of SRS for patients with a large number of BMs. This model is effective and efficient in finding solutions, allowing automated BM distribution with consistent quality and reducing tedious manual processing for improved clinical workflow.

Appendix

The numerical details of BM distributions are shown in Table 2. Each column is for an attribute, and each row is for a treatment session. For example, the first patient has four treatment sessions with BM numbers $\left\{5,10,5,10\right\}$ and $\left\{3,6,11,10\right\}$ for clinical and MIQP distributions, respectively, and the first row corresponds to the result of the first session with five BMs in clinical distribution and with three BMs in MIQP distribution. Here, the sequence of BM numbers is sorted with increasing energy for the purpose of comparing the distributions.