Implications of Clinical Target Distribution Weighted Radiotherapy Optimization Ivar BengtssonAnders ForsgrenAlbin Fredriksson

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Implications of Clinical Target Distribution

Weighted Radiotherapy Optimization

Ivar Bengtsson∗† Anders Forsgren∗Albin Fredriksson†

Abstract

Delineating and planning with respect to regions suspected to contain

microscopic tumor cells is an inherently uncertain task in radiotherapy.

The recently proposed clinical target distribution (CTD) is an alternative

to the conventional clinical target volume (CTV), with initial promise.

Previously, using the CTD in planning has primarily been evaluated in

comparison to a conventionally deﬁned CTV. We propose to compare the

CTD approach against CTV margins of various sizes, dependent on the

threshold at which the tumor inﬁltration probability is considered rele-

vant. First, a theoretical framework is presented, concerned with optimiz-

ing the trade-oﬀ between the probability of suﬃcient target coverage and

the penalties associated with high dose. From this framework we derive

conventional CTV-based planning and contrast it with the CTD approach.

The approaches are contextualized further by comparison with established

methods for managing geometric uncertainties. Second, for both a one-

and a three-dimensional phantom, we compare a set of CTD plans cre-

ated by varying the target objective function weight against a set of plans

created by varying both the target weight and the CTV margin size. The

results show that CTD-based planning gives slightly ineﬃcient trade-oﬀs

between the evaluation criteria for a case in which near-minimum target

dose is the highest priority. However, in a case when sparing a proxi-

mal organ at risk is critical, the CTD is better at maintaining suﬃciently

high dose toward the center of the target. We conclude that CTD-based

planning is a computationally eﬃcient method for planning with respect

to delineation uncertainties, but that the inevitable eﬀects on the dose

distribution should not be disregarded.

Keywords: Radiotherapy optimization, clinical target distribution, target delineation

uncertainty.

∗Optimization and Systems Theory, Department of Mathematics, KTH Royal Institute of

Technology, SE-100 44 Stockholm, Sweden (ivarben@kth.se,andersf@kth.se).

†RaySearch Laboratories AB, SE-104 30 Stockholm, Sweden (afred@raysearchlabs.com).

arXiv:2210.06049v1 [physics.med-ph] 12 Oct 2022

1 Introduction

Accurate delineation of regions of interest (ROIs) is a vital part of radiation therapy

treatment planning. Conventionally, delineation is performed by an experienced radia-

tion oncologist in a manual and time-consuming process. Yet, it is sometimes referred

to as the weakest link in the radiotherapy chain, and many studies suggest that there

is considerable inter-observer variability between clinicians, especially with regard to

the clinical target volume (CTV) [1, 2, 3]. The CTV is deﬁned in ICRU report 50

[4] as the volume suspected to contain microscopic tumour inﬁltration with clinically

relevant probability. A CTV delineated by a clinician will thus depend largely on two

factors: the perceived probability distribution of the tumorous volume and the thresh-

old at which probability of tumor presence is considered relevant. The investigations

in this work address the ambiguity associated with the second factor. We will thus

assume a known probability distribution over the potential target shapes and assess

the merit of a recently proposed approach of moving away from the threshold-based,

binary deﬁnition of the CTV.

This approach, which accounts for CTV delineation uncertainties explicitly in plan-

ning, is to use what is known as the clinical target distribution (CTD). The CTD

is a distribution over the potentially tumorous voxels that for each voxel speciﬁes a

probability of tumor presence. In treatment planning, one may then use the CTD as

voxel-wise weights in the optimization functions, to give higher priority to high-risk

voxels. This approach was proposed by Shusharina et al. [5] and has since then been

explored further by Ferjancic et al. [6] and applied in a robust optimization context

by Buti et al. [7]. The idea of including voxel-wise probabilities of ROI occupation in

optimization was proposed by Baum et al. [8] for managing overlapping margins in

prostate treatments. Unkelbach and Oelfke then demonstrated that the approach was

equivalent to minimizing the expected value of certain objective functions with respect

to the ROI-delineation uncertainty [9]. Ideally, CTD-weighted optimization will assign

dose even to low-probability regions if there is little conﬂict with sparing organs at risk

(OARs), while balancing OAR sparing and target coverage based on the probabilities

in regions where the objectives conﬂict. Compared to more advanced approaches, e.g.

the tumor control probability maximization by Bortfeld et al. [10], this method has

the advantage that the scaling of voxel weights in optimization preserves convexity

and does not introduce any additional computational complexity.

In the present paper, we investigate the implications of CTD optimization compared to

using some optimally chosen margin with respect to the underlying tumor inﬁltration

model, the dose deliverability conditions, and the evaluation criteria. The comparison

is based on the trade-oﬀs between a target coverage criterion and penalties associ-

ated with dose to healthy tissue. The primary target coverage criterion considered is

formulated as the probability of (almost) all parts of the target receiving (almost) the

prescribed dose. We have suspected that the low voxel-weight toward the edge of the

CTD would result in plans with sub-optimal trade-oﬀs between this criterion and the

conﬂicting objectives, and that there rather exists some margin which is more eﬃcient

in the described sense. For cases when a proximal, critical OAR does not allow satis-

factory values of the primary target coverage criterion, we consider additional criteria.

In addition, we view the methods in the light of previously developed frameworks for

managing geometric uncertainties, to better understand and compare the methods.

2 Method

2.1 Notation

Any treatment planning problem in the present paper is treated as an optimization

problem based on a discretization of the patient geometry into a grid of mvoxels.

The set of ROIs, denoted by R, is partitioned into the set of OARs and the set of

targets, denoted by Oand T, respectively. The voxel index set of any ROI r,r∈R, is

then denoted by Rror more speciﬁcally by Oror Trif the type of the ROI is known.

Any voxel index i∈ Rrhas a corresponding relative volume vr,i which is the ratio

between the volume of voxel ithat belongs to r, and the volume of r. It follows that

Pi∈Rrvr,i = 1. For the purposes of the present paper, it is useful to also deﬁne the

absolute volumes ˜vfor which it holds that vr,i = ˜vr,i/Vr,∀r∈R, i ∈ Rr,where Vris

the volume of r.

2.2 Preliminaries

The dose-at-volume (DaV) will be used to evaluate target coverage. For an ROI r, it

is deﬁned as the greatest dose that is received by at least the fraction vof its volume:

DaVr

v(d) = max{d0∈R:X

i∈R:di≥d0

vr,i ≥v},(1)

where the dose vector d∈Rmis a function of the optimization variables x∈ X and

X ⊂ Rnis the feasible set, dependent on the delivery technique. DaV is non-convex

and inherently hard to optimize, and will in this work thus only be used as a target

coverage evaluation criterion. Instead, the optimization is performed with respect to

presumably correlated surrogate functions. In the present paper, the standard notation

D100vis used for DaVvwhen presenting results. To limit dose to some OAR r, we

employ maximum dose functions of the form

fr:=X

i∈Or

˜vr,i(di−ˆ

dr)2

+,(2)

where ˆ

drdenotes a reference dose level ideally not to be exceeded and (·)+is the

positive part operator. Function (2) is convex and thus suitable for optimization.

Since the focus in this work is on implications on target coverage, maximum dose

functions are not only used in optimization but also for evaluation of over dosage.

In what follows, analogous notation is used for dose-promoting functions in target

volumes.

2.3 Problem Setting

We consider patient phantoms with a known GTV and OAR. The CTD extends away

from the edge of the GTV and toward the OAR, and is the region at risk of microscopic

tumor inﬁltration. The External ROI is deﬁned to comprise the full patient volume.

Both a 1D and a 3D setup are considered to ﬁrst display the basic properties of the

methods in the simplest setting, and then show the implications in a setting which is

more realistic. The 1D setup and a 2D slice of the 3D setup are illustrated in Figures

6 12 18

100

120

GTV

CTD

OAR

(a) 1D

GTV

OAR

CTD

(b) 3D

Figure 1: The phantom geometries under consideration.

1a and 1b, respectively. To avoid ambiguity when assessing resulting plans, we do

not account for overlap between the conﬂicting regions. Instead, we assume that the

OAR acts as an impenetrable barrier for the tumor inﬁltration, and should ideally not

receive any dose.

2.3.1 Tumor Expansion Model

To model microscopic tumor spread we consider an isotropic inﬁltration model away

from the GTV, proposed in [7]. Therein, the tumor spread can be characterized by

a single random variable Swith probability density function ρ(s). The probability of

the tumor to have spread beyond a certain distance sfrom the GTV is then given by

P(S≥s) = Z∞

ρ(x)dx. (3)

2.3.2 Discretization into Scenarios

For computational purposes, the distribution of Sis discretized into a scenario set S.

For distributions with negligible tail, this discretization involves setting a cut-oﬀ value

to be considered as the worst case, and renormalizing the scenario probabilities, which

are then denoted by qs,∀s∈ S. In the following, each scenario smaps to a voxel index

set Tsand the corresponding relative volumes. In 3D, this mapping is deﬁned by the

ROI algebra functionality in RayStation (RaySearch Laboratories AB, Stockholm).

2.4 Establishing and Managing Conﬂicting Objectives

Given the unknown extent of the true target tS, one must deﬁne appropriate criteria

by which to evaluate a given treatment plan. In the present paper, we assume that any

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ImplicationsofClinicalTargetDistributionWeightedRadiotherapyOptimizationIvarBengtsson*AndersForsgren*AlbinFredrikssonAbstractDelineatingandplanningwithrespecttoregionssuspectedtocontainmicroscopictumorcellsisaninherentlyuncertaintaskinradiotherapy.Therecentlyproposedclinicaltargetdistribution(CTD)...

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Implications of Clinical Target Distribution Weighted Radiotherapy Optimization Ivar BengtssonAnders ForsgrenAlbin Fredriksson

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