Value of intermediate imaging in adaptive robust radiotherapy planning to manage radioresistance

Abstract

In radiotherapy, uncertainties in tumor radioresistance and its progression can degrade the efficacy of deterministic treatments. While a robust methodology can overcome this, it often produces overly conservative or suboptimal decisions, especially when there are changes in time. We aim to develop an adaptive radiotherapy planning framework that can reduce over-conservatism yet remain robust to the uncertainties in radioresistance. Specifically, intermediate imaging is used to update the uncertainty at each stage and curb over-conservatism. While additional imaging reduces uncertainty, it accrues costs such as extra radiation to organs, which deters continuous imaging. We probe this trade-off in uncertainty and cost of observation by computing and comparing results from two-stage, three-stage, and four-stage robust models. The three robust models are also compared to two currently practiced deterministic methods, one that does not account for radioresistance and one that assumes a constant radioresistance. All five models are evaluated on a clinical prostate case. The three robust models improve control of the tumor compared to the deterministic model ignoring radioresistance, at comparable radiation dose to critical organs. The robust models also reduce tumor overdose and organ dose compared to the deterministic model assuming a constant radioresistance. Increasing the number of intermediate imaging leads to further improvements, especially on tumor dose criteria under best-case and nominal scenarios. Under the worst-case, intermediate images provide no additional benefit as robust optimization inherently protects against the worst-case. The proposed method is generic and can include additional sources of uncertainties that reduce the effect of radiation.

Appendices

Appendix A: reformulation of constraint (5)

Constraint (5) can be expressed as

$$\begin{aligned} \sum _{i = 1}^{{\mathcal {I}}} d_{i,j}(t) x^{t}_{i}(k_m) \ge \max _{\rho _j(t)\in {{\mathcal {U}}_{\rho _j}}(t,k_m)} \underline{{\hat{\theta }}}_j \rho _j(t). \end{aligned}$$

(10)

Then, by extending the reformulations provided in Nohadani & Roy (2017) and Bertsimas et al. (2004) to a multi-stage setting, we define ${\mathbf {y}}_m = \frac{{\rho }_j(t)-({\rho }_j^{k_m}(\tau _m)+\mathbf {\eta }(t-\tau _m))}{{\varGamma }(t-\tau _m)}$. Then we can express the worst-case radioresistance factor for constraint j as

$$\begin{aligned} \max _{\{\rho _j(t) \in {\mathcal {U}}_{\rho _j}(t,k_m)\}} \underline{{\hat{\theta }}}_j \rho _j(t)&= \underline{{\hat{\theta }}}_j \max _{\{{\mathbf {y}}_m :\; |{\mathbf {y}}_m|\le 1\}} \left( {\varGamma }(t-\tau _m) {\mathbf {y}}_m + {\rho }_j^{k_m}(\tau _m) + \mathbf {\eta }(t - \tau _m)\right) \\&= \underline{{\hat{\theta }}}_j (\rho ^{k_m}_j(\tau _m)+\eta (t-\tau _m)+{\varGamma }(t-\tau _m)). \end{aligned}$$

Appendix B: proof of proposition 1

Model (4) is equivalent to

(11)

where

and

(12)

For each $m=1,...,n-1$, the set $S_m$ contains finitely many scenarios. Hence, the adaptive policy for weekly dose $x_{i}^{t}$ in the week $t\in [\tau _m,\tau _{m+1}-1)$ is a finite policy each of which corresponds to a scenario in $S_m$. Therefore, (7) is equivalent to (11), which is equivalent to (4). $\square$

Appendix C: spatial distribution of tumor control probability

Voxel-wise tumor control probability (TCP) is computed for the planning target volume (PTV), i.e. tumor, under best- (see Fig. 4) and worst-case (see Fig. 5) radioresistance conditions.

Fig. 4

Spatial TCP distribution for a $x^{equal}$, b $x^{escal}$, c $x^{2-rob}$, d $x^{3-rob}$, and e $x^{4-rob}$ for the best-case scenario

For the fractionated plan $x^{equal}$ shown in Fig. 4a under best-case conditions, the radioresistant subvolume and a small portion of the radiosensitive subvolume shows poor TCP $=0$. The escalated plan $x^{escal}$ in Fig. 4b shows far superior control for the entire tumor. In fact, there is no visual difference between the TCP distributions for the radiosensitive and radioresistance portions. The three robust plans in Fig. 4c, d, and e are comparable to $x^{escal}$ with marginal loss of control within a portion of the radioresistant subvolume. As the number of intermediate imaging increases for the robust plans, the control decreases. Similar observations are made in the worst-case shown in Fig. 5, with reduced control across all methods.

Fig. 5

Spatial TCP distribution for a $x^{equal}$, b $x^{escal}$, c $x^{2-rob}$, d $x^{3-rob}$, and e $x^{4-rob}$ for the worst-case scenario

Appendix D: heterogeneous tumor radioresistance

Tumor radioresistance was simulated uniformly across the tumor voxels in Model (7), even though voxel-wise constraints were used for each radioresistant tumor voxel $j \in {\mathcal {R}}$. With the availability of weekly PET imaging data, the radioresistance could potentially vary among tumor regions over time. To allow for spatial heterogeneity, (7) can be extended to consider radioresistant subregions (i.e., set of voxels) ${\mathcal {R}}_1, {\mathcal {R}}_2, \ldots , {\mathcal {R}}_p \subseteq {\mathcal {R}}$. Then, the radioresistance $\rho _j$ and its uncertainty parameters $\varvec{\eta }$ and ${\varGamma }$ can be varied across these subregions.

We delineate the tumor into three subregions with sizes $|{\mathcal {R}}_1| = 450$ voxels, $|{\mathcal {R}}_2| = 450$ voxels, and $|{\mathcal {R}}_3| = 437$ voxels. Further, we employ $\varvec{\eta }_1 = -0.0458$ and ${\varGamma }_1 = 0.0375$ for ${\mathcal {R}}_1$, $\varvec{\eta }_2= -0.0625$ and ${\varGamma }_2 = 0.0208$ for ${\mathcal {R}}_2$, and $\varvec{\eta }_3 = -0.0792$ and ${\varGamma }_3 = 0.0042$ for ${\mathcal {R}}_3$. Correspondingly, three nominal scenarios were simulated with $\rho _1^{T,k^{nom}} = 1.225$, $\rho _2^{T,k^{nom}} = 1.125$, and $\rho _3^{T,k^{nom}} = 1.025$ for ${\mathcal {R}}_1$, ${\mathcal {R}}_2$, and ${\mathcal {R}}_3$, respectively, with all three subregions starting with $\rho ^{1} = 1.5$. Thus, subregion 1 is the most radioresistant, followed by subregion 2, and then subregion 3.

Table 3 Dose criteria for the full tumor, bladder, and rectum under nominal heterogeneous tumor radioresistance

Table 3 shows the treatment criteria for the full tumor region, bladder, and rectum for the nominal radioresistance trajectories simulated on the tumor subregions. Compared to the case with spatially uniform radioresistance, tumor $D_{50}$ was slightly reduced across all methods. This is because the mean tumor dose reduced as the uncertainty was lesser in subregions 2 and 3, allowing for lesser dose escalations. Towards the tail of the dose distribution, $D_{95}$ was less sensitive to the change in modeling, since subregion 1 that contained a significant number of voxels still needed the high dose escalations. Indirectly, bladder and rectum dose criteria reduced on average, which was possible due to the reduced uncertainty set size in subregions 2 and 3. In summary, we see that modeling voxel-wise or subregion-specific tumor radioresistance is straightforward using Model (7). This offers greater flexibility when including functional imaging data or in vivo measurements into radiotherapy planning amid radioresistance.