This document discusses various mathematical optimization techniques used in intensity modulated proton therapy (IMPT) treatment planning. It covers:
1) IMPT inverse planning formulated as an optimization problem to determine optimal beam weights, with the goal of minimizing an objective function while satisfying dose constraints.
2) Methods for multicriteria optimization that consider tradeoffs between multiple objectives like target dose and organ-at-risk sparing.
3) Robust optimization techniques that account for uncertainties in beam delivery due to factors like setup errors and anatomical changes.
4) Approaches for 4D/temporospatial optimization that incorporate target motion over the treatment period.
2. Content
• Mathematical aspects of treatment planning
• 3D conformal therapy and design of spread-out Bragg peaks, forward planning
• IMPT inverse planning as mathematical optimization problem and possible
solutions
• Multicriteria optimization
• Robust optimization
• Intrafractional motion in treatment planning
• Accounting of radiobiological effects in IMPT
• Other applications optimization in proton therapy
2
4. Optimization of SOBP fields
• 3D conformal proton therapy
• Spread-out Bragg peak – uniform covering of large tumor volumes
• Usually iterative process of optimization
• SOBP is modulated according to the energy – optimization has to follow
different ranges in tissue
• Intensity modulated SOBP field – temporally optimizing of beam current
during the modulation cycle
4
5. Forward planning with SOBP fields
• Manual optimization = current changes
• Disadvantages:
depends on planner´s capabilities
it is not systematic -> not an optimization in mathematical sense
5
choosing of
angle,
direction of
beam
range and
SOBP
modulation
forward
calculation
based on
assumed
beam fluence
adjusting
fluences and
weights of
beams
6. IMPT as an optimization problem
• IMPT fields deliver typically nonuniform dose distribution => it helps
achieve highest conformity of proton distribution
• Most common IMPT technique = 3D-modulation method
• Individually weighted Bragg peaks spots are placed through the target volume
6
7. Important difference between IMRT and IMPT
• Spread-out Bragg peak -> modulation of its brings another degree of
freedom to the proton treatment
• Despite this – optimization is same problem for both methods
7
9. Mathematical approach
• Defining of objectives and constraints
• Volumes of interests (VOI) – include targets and critical organs OAR
• Total dose distribution from IMPT field – sum of contributions from static
pencil beams of various voxels (dose influence matrix Dij)
• Total dose to any voxel:
9
10. 10
• Large number of pencil beams – optimization methods required
• Output of the plan optimization: set of beam weight distributions –
FLUENCE MAP
• IMRT – 2D fluence maps (process uses different angles of irradiation)
• IMPT – separate map for every energy setting
11. Objective function
• Optimization = process of looking for the minimal OF
• Quadratic penalty function:
• In general: Constraints have to be fulfilled in order to make plan acceptable
11
12. General formulation of IMPT optimization
problem
• Soft constraints setting
• On…objectives
αn…weighting factors
Cm…constrains
lm, um…lower and upper bounds
12
13. Solving the optimization problem
• Variables = beam weights x (can have any nonnegative value, often
discretized)
• Objectives and gradient of the objective can be often calculated analytically
• Two types:
• Constrained -
• Unconstrained – no dosimetric hard constraints – all treatment goals defined as
objectives (Newton method, gradient methods)
13
14. Constrained optimization for IMPT
• Large number of variables and large number of voxels
• Linear programming (just linear objectives and constraints) vs. Sequential
quadratic programming
• Convex (constraints and objectives are convex functions) and nonconvex
(for example including radiobiological models; becoming more common in
this time)
14
16. Principle
• Single criterion optimization problem =
ONE objective + ALL OTHER CONDITIONS are constraints
• Main objective = deliver prescribed dose to whole target volume
• Other objectives = to keep dose in OARs minimal
16
17. Biological models
• Hypothetically – single criteria – maximizing TCP to the NTCPs
BUT
• Reality – pacientspecific trade-off: „How big gain do we get in TCP, when we
allow NTCP for some organ in this amount?“
= MULTICRITERIUM!!!
17
18. 18
• Presently – standard commercial systems use single-criterion approach
• Disadvantage:
• Longer process = iteration cycle depending on the treatment planner
• Difficult to set weights and function parameters
20. 2) Pareto surface (PS) approach
• Instead of prioritizing of objectives treats each objective equally
• Yields not a single plan, but a set of optimal plans – trade off objectives in
many ways
• PARETO OPTIMAL PLAN = plan which is feasible and there is not
another plan, which is strictly better for at least one objective and it is not
worse for any other
20
21. Mathematical formulation PS
• X… all beamlet and dose constraints
N… number of objective functions
• Algorithm issues:
• 1) How to compute a reasonable set of diverse PS plans
• 2) How to present resulting information to decision markers
21
22. Main strategies of PS approach in radiotherapy
A) WEIGHTED SUM methods
Combining all the objectives into a
weighted sum => solving of the
resulting scalar optimization problem
22
23. Main strategies of PS approach in radiotherapy
B) CONSTRAINT methods
Use objective functions as constraints
(same as for prioritized optimization)
– varying constraint levels (finding of
different pareto-optimal solutions)
Problem - error measures are not
natural part of algorithm or output
23
24. Navigation on the PS based approach
• How to allow user to select plan from the set of Pareto-optimal soloutions?
1. N sliders, each for one objective
2. Allow to choose N sliders and 2 of N objectives -> picturing of the trade-
off for those objectives
24
27. Comparing Prioritized Optimization and PS-
Based MCO
Prioritized optimization
Programmable procedure – result is
single Pareto-optimal plan
Only one plane presented to user in
the end!
PS based MCO
Result are all optimal options
presented to user
Not useful for routine planning – user
has to decide which one is best
manually from large number of
options
27
29. • Many uncertainties influencing the delivered dose
• The optimal plan needs to be robust
• Small deviations from the planed dose distribution don´t influent the
treatment outcome
29
30. • IMPT – inhomogeneous dose, more proton energies – combinations of dose
distributions
• More variations – mismatches of doses from different fields
• Dose gradients - even bigger sensitivity for setup errors
• Hot and cold spots (dose in critical organs)
• More conformal dose -> more complex fluence map -> more sensitive plan
for uncertainties
30
31. Sensitivity of plan to setup errors
• The same plan, different error setup – big impact on the dose contribution
of the posterior beam
31
32. Robust optimization strategies
• Delivered dose distribution depends on set uncertain parameters
• Models of uncertainties:
• Rigid setup error without rotation – parameter is 3D vector of shifting in space λ
• Pencil beam simple model – overshooting and undershooting uncertainties of all beams
• More complicated pencil beam model – different errors for different pencil beams
32
33. The probabilistic approach = Stochastic
programming approach
• Dose distribution: d(x,λ), where x…beam spot weights to be optimized
λ…uncertain parameter
• Objective function: O(d(x, λ)) … describes dose distribution
• Probability distribution: P(λ) … probability, that error λ occurs
33
34. • We need to minimize the expected value of objective function
• The general goal:
To find the treatment plan, that is good for all possible errors, BUT: larger
weights for scenarios with higher probability to occur, lower weights for
scenarios with lower probability
• Typically – quadratics difference – minimizing of objective function in each
voxel
34
35. The robust approach
CONSTRAINTS
• Constraints have to be satisfied for every realization of the uncertain
parameters
• We have constraints (example less than 50Gy for spinal chord) => robust
approach sets less than that dose for every possible range of setup error!
35
36. OBJECTIVES
= Worst-case optimization problem
• Result: as good as possible treatment plan for the worst case, that can occur
• Minimizing of maximum dose, which can be delivered for any possible range
or setup error
36
37. Optimization of the Worst-Case Dose
Distribution
• Hypothetical
• Unphysical – every voxel is considered independently
• Principle: for every voxel the dose is defined as he worst dose value that can
be realized for any error in the uncertainty model
• Primary objective function is done by sum of objective function in case of
no errors and objective function in the worst-case
37
38. Example of robust optimization
a) Conventional calculated plan
optimized without accounting an
uncertainty
b) Plan optimized for range and
setup uncertainty using the
probabilistic approach
(DVH for CTV for spinal cord)
38
39. Dose distribution of
individual beams
(A) Conventional IMPT
(B) Robust IMPT - Range uncertainty only
(C) Robust IMPT - Setup uncertainty only
(D)Robust IMPT - Considering both range
and setup uncertainty
39
41. Temporospatial (4D) Optimization
• What affect precision of IMPT?
• Changes in setup
• Motions of target (respiration, peristaltic movements, gravity…)
• Degrading of gradients, increasing irradiation of healthy tissues
• IMPT requires to consider possible changes in radiological depth to target
41
42. Avoiding to impact of intrafractional motions
• Many methods: compensator
expansion, beam gating, field
rescanning…
• Intrafractional motion as a part of
beam weights optimization:
• Requiring of geometrical variation of
patient anatomy = 4D CT for
recording breathing cycle,
42
43. Optimization based on a known motion
probability density function
• Based on 4D CT scanning
• Delivering of inhomogeneous dose distribution to a static geometry
• Edge-enhancement = dose boosting at the edges of the target (most
important part of target which is influenced by motions)
• PDF-based is strongly influent by reproducibility of target motions (motion
deviates form expectation, significant deviation may occur)
43
45. Performation of optimization
1) RBE
For IMPT usually constant RBE -> treatment plan ptimization based on
physical dose
2) LET
Higher LET -> increasing of radiation-induced cell-kills (at the end of proton
beams range)
Objective function formulated in the terms of LET and RBE (f.e. linear-
quadratic model) 45
47. Scan path optimization
• In reality of 3D scanning, large number of beam spots have zero weight
• Spot scanning – steering of the beam in the zigzag pattern over entire grid
including positions corresponding with zero weight
• Scan path optimization – avoiding regions with zero weight spots (simulated
annealing principle)
47
48. Beam current optimization for coninuous
scanning
• Spot scanning: dose is delivered according to the optimized spot weight
• Continuous scanning: beam is constantly moving according to predefined
pattern
• Optimization methods applied in the step of converting optimized spot
positions at discrete positions to beam-current modulation with the same
fluence
48
49. References
• H Paganetti: Proton Therapy Physics, CRC Press, 2012
• F. Albertini: Planning and Optimizing Treatment Plans for Actively Scanned
Proton Therapy: evaluating and estimating the effect of uncertainties,
Disertation, ETH Zurich, 2011
49
This chapter describes various applications of mathematical optimization
techniques in treatment planning for proton therapy. The most prominent
example is the optimization of beam weights for intensity-modulated proton
therapy (IMPT). The conceptual and practical aspects of IMPT have
been introduced in previous chapters (see primarily Chapter 11). Here, we
focus on the mathematical aspects of treatment planning. First, in Section
15.1, we briefly recapitulate the basics of three-dimensional (3D) conformal
therapy, including the design of spread-out Bragg peaks (SOBPs), and
forward planning. In Section 15.2, we illustrate how IMPT inverse planning
is formulated as a mathematical optimization problem and comment on
methods to solve this problem. Then, we discuss advanced optimization
techniques for proton therapy: multicriteria optimization (Section 15.3),
robust optimization methods for handling range and setup uncertainty
(Section 15.4), incorporation of intrafractional motion in treatment planning
(Section 15.5), and consideration of radiobiological effects in IMPT
optimization (Section 15.6). Finally, in Section 15.7, we review other
applications of mathematical optimization in proton therapy, such as the
optimization of beam current modulation and scan path for continuous
scanning.
15.1 Optimization of SOBP Fields
An SOBP is the foundation of forward treatment planning for 3D-conformal
proton therapy. It is used to achieve a longitudinal conformality of the
required dose to the target. In his seminal paper on therapeutic use of protons,
Dr. Robert Wilson recognized the need for optimization of proton dose
distribution for clinical treatments, by pointing out that Bragg peaks need
to spread out to uniformly cover large tumor volumes. In his assessment
this could be “easily accomplished by interposing a rotating wheel of various
thickness” in the beam path (1), the method of modulation that is now
widely used for proton therapy (see Chapter 5). By using the word “easily,”
Dr. Wilson, perhaps, anticipated the fact that the problem of optimization of
modulation of SOBP would be relatively easy compared to the optimization
problems yet to arise in proton therapy.
15.1.1 Optimization of Field Flatness
To create a clinically relevant SOBP of the desired flatness in a passive beam
scattering system, a variety of components must operate in conjunction to
produce the desired beam parameters. Koehler et al. (2) described one of
the earliest examples of design of flat SOBPs using computer-based optimization.
Based on the input values of range and modulation width, the
Treatment-Planning Optimization 463
code written in Fortran IV iteratively searched for the set of amplitudes of
shifted pristine peaks, and spacings between them (in other words, relative
width, and thickness of the wheel steps), which realized the desired SOBP
(see Figure 15.1A). Notably, because the shape of the Bragg peak curve varies
with the beam energy, the weights of individual peaks in the SOBP need to
be optimized separately for different ranges in tissue, to avoid sloping in the
SOBP, as shown in Figure 15.1B and C.
In the early days of proton therapy, the wide variety of clinically required
combinations of range and SOBP modulation required a large number of
premanufactured wheels, with separate wheels required for shallow and
deep tumors, one wheel for a close set of modulation width (the smallest
steps of the propeller could be added or removed to allow for some variation
in the total modulation width). A more flexible modern solution uses
a beam current modulation system, with a limited number of wheel tracks
(see Chapter 5). The pulled-back Bragg peaks can be individually controlled
to produce uniform dose plateaus for a large range of treatment depths using
only a small number of modulator wheels (3–5).
150
A B
C D
100
50
Relative dose [%] Relative dose [%]
Relative dose [%] Relative dose [%]
0 0
Depth [cm] Depth [cm]
Proton SOBP
Pristine peaks
Proton SOBP
Pristine peaks
Proton SOBP
Pristine peaks
Proton SOBP
Pristine peaks
5 10 15 20 0 5 10 15 20
0
Depth [cm] Depth [cm]
5 10 15 20 0 5 10 15 20
150
100
50
150
100
50
0
150
100
50
FIGURE 15.1
Depth–dose profile of a spread-out Bragg peak (SOBP), and constituent pristine peaks: optimization
of pristine peak weights leads to (A) uniform SOBP dose, while variation in the pristine
peak dose profile may introduce a (B) raising or (C) falling slope in SOBP. In principle, arbitrary
profiles of the peak dose can be achieved by optimization, for example, (D) a profile with the
integrated dose boost of 10% to the middle part of the SOBP.
464 Proton Therapy Physics
In principle, by temporally optimizing the beam current during the modulation
cycle, one can create SOBPs with arbitrary depth–dose profiles. This
includes “intensity-modulated” fields according to the common definition,
namely, dose distributions, that are inhomogeneous by design. Notably, the
beam current modulation literally constitutes intensity modulation of the
beam, regardless of whether the resulting distribution is inhomogeneous or
not. An example of inhomogeneous dose achievable with range modulation
is the SOBP including a simultaneous integrated dose boost delivered to a
subsection of the target, as in Figure 15.1D. It should be noted though that
this technique allows for intensity modulation only in depth, whereas the
beam intensity is homogeneous laterally.
15.1.2 Forward Planning with SOBP Fields
Procedures of forward planning for 3D-conformal proton therapy have
been well described by Bussière and Adams (6), as well as in Chapter 10 of
this book. Figure 15.2 illustrates how a “manual optimization” of a treatment
plan might be undertaken. The search for a satisfactory solution does
involve iterative adjustment; however, it is rather subjective (e.g., depends on
the planner’s training, habit, and judgment) and is not systematic (e.g., iterations
do not always lead toward a more preferable solution). Thus the process
cannot be termed optimization in the strictly mathematical sense.
First, the irradiation directions are selected as well as the range and SOBP
modulation width necessary to cover the target. Range compensators are
designed to conform the dose to the distal aspect of the target, and accommodations
are made to prevent underdosing of the target in case of misalignment
of treatment field and tissue heterogeneities, for example, using the
technique of compensator expansion, or “smearing” (7) (also see Chapter 10).
Once these steps are completed, a forward calculation is performed to determine
the dose from the given field, based on the assumed beam fluence. The
task of the planner is then to iteratively adjust the fluences, or “weights,” of
multiple beams and to combine their doses so that the resulting distribution
suits a particular set of requirements. For example, in the case illustrated
in Figure 15.2, irradiating the spinal cord up to the tolerance (Figure 15.2, D
and G) may be considered acceptable in a certain situation, because this configuration
minimizes the integral dose and the main irradiation direction is
least affected by internal motion (e.g., of liver with respiration for the rightanterior
beam) or variations in the stomach and bowel filling (for the left
beam lateral). In other situations, such as repeat treatments, cord tolerance
may be reduced, and other directions have to be used. In those cases, the
clinically optimal balance, between irradiation of various structures, needs
to be selected (compare, e.g., Figure 15.2, H vs. I).
15.2 IMPT as an Optimization Problem
Intensity-modulation methods allow one to achieve highest conformality of
proton dose distributions to the target volume and best sparing of healthy
tissue. Unlike 3D-conformal treatments, in which each SOBP field delivers
a uniform (within a few percent) dose to the whole target volume, individual
IMPT fields typically deliver nonuniform dose distributions (e.g.,
see Figure 15.3). Similar to IMRT with photons, these nonuniform field contributions
combine to produce the desired therapeutic dose distribution,
which may be shaped to conform to the clinical prescription. An important
difference from photon intensity-modulated radiation therapy (IMRT) is
that the Bragg peak of the proton depth dose distribution introduces an
additional degree of freedom in modulation of the dose in depth along the
beam axis, in addition to the modulation in the transverse plane, which is
available in both IMRT and IMPT. Despite this difference, IMRT and IMPT
are very similar regarding the mathematical formulation of the treatmentplanning
problem.
466 Proton Therapy Physics
To take full advantage of the possibility to sculpt the dose in depth,
IMPT treatments use narrow proton pencil beams, which can be scanned
across the transverse plane while changing energy and intensity to control
the dose to a point. The most common and versatile IMPT technique is the
3D-modulation method, in which individually weighted Bragg peak “spots”
are placed throughout the target volume (8). The examples in this chapter
use 3D-modulation; however, most optimization methods described below
are equally applicable to other techniques, such as single-field uniform dose
(SFUD) treatments or the distal edge tracking (DET). SFUD treatments also
use weighted pencil beams distributed in three dimensions, but aim at delivering
a homogeneous dose to the target from every individual field direction.
In the DET method, Bragg peaks are placed only at the distal surface of
the tumor (9).
IMPT plan for a paraspinal tumor.
CT scan showing outlines of the tumor and the spinal cord,
dose distribution from a 3D IMPT treatment plan, using three beam directions. => we can observe isodose levels
Dose contributions from individual beams are shown for (C) right-posterior oblique, (D) posterior, and (E) left-posterior oblique fields. (The conventional IMPT plan did
not include any consideration of delivery uncertainties.)
ROBUST ALGORITHMS
As an illustration, consider a conventional IMPT plan for a case of paraspinal
tumor, shown in Figure 15.3. The target entirely surrounds the spinal
cord, which is to be spared. Total IMPT dose distribution was optimized
using a quadratic objective function, thus aiming at a homogeneous target
dose. As is characteristic of IMPT, the homogeneous dose distribution in the
target is achieved through a superposition of highly inhomogeneous contributions
delivered from three beam directions.
5.2.1 Setup of the IMPT Optimization Problem
To apply general optimization methods to radiation therapy planning, technical
limitations and treatment goals need to be formulated mathematically
as objectives and constraints. For that purpose, the patient image data are
partitioned into volumes of interest (VOI), which could include targets, critical
organs at risk of undesired side effects (organs at risk [OAR]), and other
tissue volumes. VOI are further divided into basic geometric elements called
voxels.
The total dose distribution from an IMPT field delivered with a scanned
beam can be calculated as the sum of contributions from “static” pencil
beams fixed at various positions along the scan path. The dose from individual
pencil beams to various voxels of interest can be represented in the
form of the dose influence matrix Dij , where i is the voxel index, and j is the
beam index. The total dose to any voxel is then calculated as follows:
d= x D i j ij
j
Σ ⋅
(15.1)
where xj is the relative “weight” of the beam j, which is proportional to the
total number of protons delivered at the given spot, that is the position of
Bragg peak. The weights xj are the optimization variables that need to be
determined in treatment planning.
Because of the large number (thousands
or tens of thousands) of such pencil beams involved, IMPT treatment planning
requires mathematical optimization methods (10, 11). The output of the
plan optimization is a set of beam weight distributions, often called intensity
or fluence maps. Unlike in IMRT, where a single two-dimensional (2D) fluence
map characterizes a field, in IMPT, many beam energies may be used
to irradiate the target from the same direction, and optimization will yield
separate maps for every energy setting.
Dosimetric or other planning objectives may be defined for volumes
or individual voxels. The planning objectives and their priorities can be
expressed in the objective function (OF). The term optimization, in the context
of treatment planning, typically signifies the search for a set of plan
parameters that minimize the value of the OF, subject to a set of constraints
that have to be fulfilled.
A widely used objective function that aims at minimizing the volume,
within a given OAR n, that exceeds the maximum tolerance dose Dmax is
given by the quadratic penalty function:
Od Hd D d D n i i
i OARn
( ) = ( − )( − )
∈
Σ max max 2
(15.2)
where H(d) is the Heavyside step function. Similarly, one can define a quadratic
function that aims to reduce volumes of the tumor, which receive less
than the minimum dose Dmin. Objective functions may also include the generalized
equivalent uniform dose (12):
O d
N
d n
n
i
p
i OARn
p
( )= ( ) ⎛
⎝ ⎜⎜
⎞
⎠ ⎟⎟
∈
1 Σ
1/
(15.3)
468 Proton Therapy Physics
where Nn is the number of voxels in the VOI n, and p is an organ-specific
parameter.
In addition, there may be constraints on the dose in a VOI that have to
be fulfilled in order to make the treatment plan acceptable. For example,
one can request that the dose in every voxel belonging to the tumor should
be between a minimum dose Dmin and a maximum dose Dmax. This would
result in the hard constraint
D d D i VOI i
min ≤ ≤ max " ∈ . (15.4)
In clinical situations, treatment objectives often directly conflict each other:
for example, a target may not be completely irradiated to the prescribed level
if a dose-sensitive critical structure is immediately adjacent to it. In this case,
hard dosimetric constraints have to be used with care, and it is often necessary
to reformulate a constraint as an objective. For example, it may be
necessary to minimize the dose to an OAR that exceeds the tolerance dose
through a quadratic objective, rather than enforcing the dose to be below
the maximum dose in every voxel through a constraint. Such an objective
is often referred to as a “soft constraint” in the medical physics literature.
Multicriteria optimization methods, discussed in Section 15.3, address such
inherent treatment-planning contradictions.
Thus, one can formulate the general IMPT optimization problem as follows:
minimize (with respect to the beam weights x):
αn n
n
Σ ⋅O (d)
subject to the constraints:
d= x D i j ij
j
Σ ⋅
l C d u m m m ≤ ( )≤
xj 0. (15.5)
In the above formulation the different objectives On are multiplied by respective
weighting factors αn and are added together to form a single composite objective.
By selecting and adjusting the weighting factors, the treatment planner
can prioritize different objectives and control the trade-off between them. The
approach of a weighted sum of objectives is pursued in most current treatmentplanning
systems. (An alternative to this standard approach is multicriteria
optimization.) The functions Cm denote a general constraint function, and lm
and um are upper and lower bounds. An example of a simple constraint function
is the minimum or maximum dose constraint mentioned above. Alternatively,
constraints on equivalent uniform dose (EUD) can be imposed (12).
Treatment-Planning Optimization 469
A number of additional parameters often need to be specified before optimization
of beam weights is performed. These include, for example, the
choice of the algorithm for placement of Bragg peaks (8), as well as the volume
used for placement (which may be larger than the target), the spacing of
peaks and layers in depth (13), and the size of the pencil beam used for delivery
of therapy (14). These additional treatment parameters, or hyperparameters,
affect the outcome of optimization; however, they are not determined
through an optimization algorithm in the mathematical sense. Instead they
are chosen based on experience, planning studies, and physical or theoretical
considerations.
s
15.2.2 Solving the Optimization Problem
IMPT represents a textbook example of a large-scale optimization
problem, especially if convex objectives and constraints are used. The
variables, the beam weights x, are continuous, that is, can take any nonnegative
value (although these are often discretized, when sequenced for
delivery).
The objectives can typically be formulated in closed form as a
function of the optimization variables, and also the gradient of the objective
can be calculated analytically. Therefore a large variety of algorithms
can be applied. Those can be categorized into constrained and unconstrained
methods. In the case of unconstrained methods, no dosimetric
hard constraints are applied, that is, all treatment goals are formulated as
objectives. The only constraints that always need to be fulfilled to yield
a physically meaningful plan are the variable bound constraints xj ≥ 0.
However, those can be treated through relatively simple methods such as
gradient projection methods. Most current treatment-planning systems
use unconstrained optimization methods. In this case, improved gradient
methods are used such as quasi-Newton methods or the diagonalized
Newton method.
Constrained optimization for IMPT is still challenging because of the large
number of variables (103 to 105) and the large number of voxels (105 to 107).
If only linear objectives and constraints are used, the linear programming
framework can be applied. In the nonlinear case, sequential quadratic programming
methods have been used (RayStation; Raysearch Laboratories,
Stockholm, Sweden) as well as barrier-penalty methods (Monaco; Elekta,
Stockholm, Sweden).
Optimization problems may further be classified as convex or nonconvex.
In a convex optimization problem, all of the constraints as well as the minimized
objective function are convex functions. For example, linear functions,
and therefore, linear programming problems are convex. The feasible
region (i.e., all sets of spot weights x that fulfill the constraints) is then also
convex, being the intersection of convex constraint functions. With convex
objectives and a convex feasible region a local optimal solution is also a
global optimal solution. Thus, optimization would either yield the globally
470 Proton Therapy Physics
optimal solution or demonstrate that there is no feasible solution. All the
objectives and constraints described above (e.g., quadratic function, EUD)
are convex.
Conversely, a nonconvex optimization problem is any problem where
the objective is nonconvex or nonconvex constraint functions give rise to a
nonconvex feasible region. In this setting, multiple local optimal solutions
are possible and, in practice, considering the large number of variables in
IMPT, it is typically not possible to guarantee that an algorithm used to
solve the optimization problem indeed converges to the globally optimal
solution.
Nonconvex constraints are becoming increasingly common in optimization.
Examples of nonconvex objectives include typical radiobiological models
of tumor control and normal tissue complication probabilities. Another
example is the dose-volume constraints, which can be conveniently defined
to specify the desired shape of the dose-volume histogram (DVH) directly
(15); for example, “the fraction of the volume of a specific OAR irradiated to
40 Gy is not to exceed 30%.”
15.3 Multicriteria Optimization
Optimization theory is built up around the single criterion optimization
problem, where there is one objective and other problem considerations are
included as constraints.
In radiotherapy, the main objective—to cover the
target with the prescription dose—is in direct conflict with the other objectives
of keeping the dose to the healthy organs to a minimum.
If biological response models such as tumor control probability (TCP) and normal
tissue complication probability (NTCP) were reliable, one might be able
to solve radiotherapy optimization well in a single criterion mode: maximize
TCP subject to the NTCPs of the relevant organs at risk being below
acceptable levels. However, even in this setting, depending on the patientspecific
trade-off (for the treatment plan under consideration, how much
gain in TCP is there if you allow NTCP for some organ to increase by some
amount), were there a tool to easily explore other options, a physician might
choose a different plan than the plan returned from the single-criterion
optimization.
Presently, the standard commercial systems available for treatment optimization
still attempt to solve the radiotherapy optimization problem with a
single-criterion approach, and this leads to a lengthy optimization iteration
cycle, where treatment planners try to find the set of weights and function
parameters that give a plan that best matches the physician’s goals for treatment.
The problem is, it is very difficult to guess those weights and function
parameters to get a good plan, and as the number of organs to consider
Treatment-Planning Optimization 471
increases, this task becomes increasingly more difficult. Several groups are
at work to bring multicriteria optimization (MCO) into routine clinical usage
(16–22).
There are two main approaches to MCO for radiotherapy treatment planning:
prioritized optimization and the Pareto surface (PS) approach. Below,
we describe the two approaches, show how they are related, and discuss
their pros and cons.
1915.3.1 Prioritized Optimization
Prioritized optimization, or lexicographic ordering, as it is sometimes called
in the literature, is a natural approach for dealing with multiple objectives
when the objectives can be ranked in terms of importance (23, 24). Letting O1
denote the highest priority objective, O2 the second highest, etc., prioritized
optimization solves the following sequent of optimization problems for k
priority levels:
(1) minimize O1(x) subject to x ∈ X ;
(2) minimize O2(x) subject to x ∈ X , and O1(x) ≤ O1
* · (1 + ε);
…
(k) minimize Ok(x) subject to x ∈ X, and, for all i < k, Oi(x) ≤ Oi
* · (1 + ε), (15.6)
where x is a set of the decision variables, X is a constraint set that represents
constraints on the beamlet fluences (upper and lower bounds) and is also
used to denote hard dosimetric constraints, such as voxel dose, organ mean
dose, or EUD that must be met by every considered solution. O1
* is the optimal
objective value from the first optimization (i.e., the fluence values in the
case of IMPT), and ε is a small positive slip factor. Multiplication by (1 + ε)
allows a small degradation in the value of the first optimization, thus hopefully
permitting the second priority objective to achieve a good value, and
so forth.
The result of the final optimization is the single result of the prioritized
optimization approach. The choice of ε (and whether it is the same for each
step) and the priority ordering of the objectives will influence the final result.
15.3.2 PS Approach
The PS approach does not prioritize the objectives, but instead treats every
objective equally. Unlike prioritized optimization, the PS approach yields
not a single plan, but a set of optimal plans that trade off the objectives in a
variety of ways. Given a set of objectives and constraints, a plan is considered
Pareto-optimal if it is feasible and if there does not exist another feasible
472 Proton Therapy Physics
plan that is strictly better with respect to one or more objectives and that is at
least as good for the rest. Assuming that the objectives are chosen correctly,
Pareto-optimal plans are the plans of interest to planners and doctors. The
set of all Pareto-optimal plans comprises the PS.
The PS-based MCO problem can be formulated as follows:
minimize [O1(x), O2(x), … , ON(x)] subject to x ∈ X (15.7)
where X is used, as before, to represent all beamlet and dose constraints,
and N is the number of objective functions. The algorithmic decisions to
be made for this approach are as follows: (1) how to compute a reasonable
set of diverse Pareto-optimal plans and (2) how to present the resulting
information to the decision makers. Radiotherapy seems to be one of the
first fields, if not the first, to fully address the question of populating PSs
for N ≥ 3.
Two main types of strategies populating the PS have been put forward
for the radiotherapy problem: weighted sum methods and constraint
methods.
Weighted sum methods are based on combining all the objectives into a
weighted sum and solving the resulting scalar optimization problem. By
solving the problem for a variety of weights, a variety of different Paretooptimal
plans are found. If the underlying objectives and constraint set
are convex, every Pareto-optimal point can be found by some weighted
sum. Several publications describe methods to choose the weights appropriately,
to produce a small set of plans that covers the PS sufficiently
well (18, 21, 25). These methods intrinsically take into account convex
combinations of calculated PS points when evaluating the goodness of
a set of Pareto plans. All of these methods get bogged down when the
number of objectives is large (e.g., >8). Fortunately, on a practical level,
even as few as N + 1 PS plans are often sufficient to determine good treatment
plans (26, 27).
Constraint methods use the objective functions as constraints (as in
prioritized optimization), and by varying the constraint levels, different
Pareto-optimal solutions are found. The state-of-the-art of constraint-based
method is the improved normalized normal constraint (NNC) method (28).
The main deficit of constraint-based methods is that error measures, which
give the quality of the PS approximation, are not a natural part of the algorithm
or output, as they are in the methods of Craft et al. (18) and Rennen
et al. (25).
Weighted sum and constraint methods are graphically depicted for 2D PSs
in Figure 15.4.
15.3.3 Navigation of the PS
The final task in a PS-based approach to treatment planning is to allow the
user to select a plan from the PS. Because the PS is represented by a finite set
Treatment-Planning Optimization 473
of Pareto-optimal treatment plans, there are two natural approaches to plan
selection. The easiest way is simply to allow the treatment planner to select
one of the computed Pareto-optimal treatment plans. In the case of IMPT,
where treatment plans can be weighted and combined to form other valid
treatment plans, it makes sense to allow users to smoothly transition between
the computed solutions. When navigating across convex combinations of the
database plans, either forcing Pareto optimality or not, the standard method
is to present N sliders, one for each objective, and the underlying algorithmic
task is to determine how to move in the objective space in response to a slider
movement (21, 29).
An alternative to presenting the users with N sliders is to allow them
to select two of the N objectives and then display a 2D trade-off for those
two objectives. For the other N – 2 objectives, the user can impose upper
bounds, influencing the 2D tradeoff surface being evaluated. The benefit of
this method is that it allows the user to visualize a 2D slice of PS, which may
yield intuition into the problem at hand. Figure 15.5 shows what this might
look like for examining the trade-off between sparing the lung and controlling
hot spots within a target.
A) Weighted sum method B) e-Constraint method
Normalized normal
C) constraint method
O2 O2
w = (.6,.4)
w = (.2,.8)
O2
O1 O1 O1
FIGURE 15.4
Methods to compute a database of Pareto surface points. (A) Weighted sum, (B) e-constraint,
and (C) normalized normal constraint method.
Target dose
homogeneity
OAR Sparing
FIGURE 15.5
(See color insert.) Illustration of two Pareto-optimal plans, showing trade-offs in OAR sparing
vs. target dose homogeneity.
474 Proton Therapy Physics
15.3.4 Comparing Prioritized Optimization and PS-Based MCO
Prioritized optimization and PS MCO are compared graphically in Figure
15.6. It is important to note that both methods rely on optimization with hard
constraints. In the prioritized approach, this is obvious because objectives
move into the constraint section. In the PS method, constraints are important
in the problem formulation, to restrict the domain of the PS to a useful
one. For example, it makes sense to put an absolute lower bound on target
doses normally, even if a user is interested in exploring some underdosing
of the tumor to improve OAR sparing (otherwise, anchor plans for OAR
will be “all 0” dose plans, which are not helpful for planning). Similarly, a
hard upper maximum dose on all voxels is useful. Therefore, MCO methods
in general are best used when a constrained solver is at hand. Solvers
implemented in RayStation (RaySearch Laboratories), Pinnacle (Philips
Healthcare, Andover, MA), Monaco (Electa), UMPlan/UMOpt (University
of Michigan, Ann Arbor, MI), and Astroid (Massachusetts General Hospital,
Boston, MA) are examples of solvers that allow true hard constraints (as
opposed to those that handle constraints approximately by using a penalty
function with a high weight).
The advantage of the prioritized approach is that it is a programmable
procedure that results in a single Pareto-optimal plan, but the disadvantage
is there is only one plan presented to the user at the end of the process.
PS methods on the other hand present all optimal options to the user, but
might be considered overwhelming for routine planning because the user
has to decide on selecting a single plan manually from the large number
of options on PS. However, plan selection in standard cases may be fast,
even with many options, because sliding with navigation sliders is much
more efficient than the reoptimization iteration loop. Notably, because the
navigation process is user-driven, it is not as reproducible as the prioritized
approach.
From the delivery point of view, an optimal plan needs also to be “robust,” that is, designed in such a way that slight deviations from the plan due to various uncertainties during treatment delivery will not affect the quality of treatment outcome. In other words, a robust treatment plan will deliver a clinically acceptable dose distribution as long as the deviations from the planned do not exceed the assumed levels.
15.4.1 IMPT Dose in the Presence of Uncertainties
Doses delivered from different directions in IMPT are typically inhomogeneous
and require the use of a number of proton energies. For this reason,
variations in the target setup and penetration depth during delivery can lead
to misalignment and mismatch of doses from individual fields, and, consequently,
alter the combined dose distribution.
To satisfy the requirement of dose conformity to the target, steep dose gradients
are often delivered at the target border. Such steep dose gradients
in the dose contributions of individual beams make IMPT plans yet more
sensitive to both range and setup errors. In particular, dose gradients in the
beam direction make the treatment plan vulnerable to range errors, because
an error in the range of the proton beams corresponds to a relative shift of
these dose contributions longitudinally inside the patient. As a consequence,
the dose within the target may not add up to a homogeneous dose as desired.
Hot and cold spots may arise. Moreover, dose may be shifted into critical
organs. Generally, the more conformal the combined IMPT dose is, the more
complex the fluence maps per field are and the more sensitive the plans are
to the delivery uncertainties.
The dose distribution that results from a range overshoot of all pencil
beams in this plan (i.e., protons penetrate further into the patient than anticipated
during planning) would lead to a higher dose to the spinal cord, as
shown in Figure 15.7A. Sensitivity of the same plan to setup errors is illustrated
in Figure 15.7B, which shows the dose distribution resulting from a
3.5-mm setup error posteriorly (upwards in the picture). This shift has no
impact on the dose contribution of the posterior beam. However, the oblique
beams hit the patient surface at a different point. For a posterior shift, the
dose contributions of the oblique beams are effectively shifted apart, which
results in the cold spots around the spinal cord.
From this illustration, it is evident that, unlike in conventional x-ray
therapy, plan degradation in the presence of range and setup uncertainties
in IMPT cannot be prevented, to a satisfying degree, with safety
margins. Expanding the irradiated area around the target with margins
could potentially reduce underdosage at the edge of the target in the presence
of an error. However, the general problem of misaligning the dose
contributions of different fields, which leads to dose uncertainties in all
of the target volume, cannot be solved through margins. This problem
instead relates to steep dose gradient in the dose contributions of individual
fields.
15.4.2 Robust Optimization Strategies
The methods presented in this section have been described largely in three
publications (32–34) that deal specifically with range and setup uncertainty
in IMPT. In addition, a number of earlier publications investigate the handling
of uncertainty and motion in IMRT with x-rays. Some of that work
could also be applied to IMPT. For a review of developments in handling of
motion and uncertainty in IMRT, see Orton et al. (35). Although this section
illustrates robust optimization techniques in the context of range and setup
errors, the methodology is also applicable to other types of uncertainty, for
example, irregular breathing motion or uncertainty in the biological effectiveness
of radiation (36).
A B
FIGURE 15.7
(See color insert.) Estimated dose distribution from the plan in Figure 16.3, assuming (A) a 5-mm
range overshoot of all pencil beams, and (B) a systematic 3.5-mm setup error (posterior shift).
Treatment-Planning Optimization 477
Several approaches that apply either the concepts of stochastic programming
or robust optimization have been suggested for incorporating uncertainty
into IMPT optimization. The common feature of these approaches is
that the delivered dose distribution depends on a set of uncertain parameters.
In the case of a rigid setup error without rotation, the set of uncertain
parameters would be a 3D vector describing the patient shift in space.
A simple model of range uncertainty, where it is assumed that all pencil
beams simultaneously overshoot or undershoot, would have one uncertain
variable that describes the range error of all beams. A more complicated
model of range uncertainty could allow for different range errors for different
pencil beams.
Below, we denote the set of uncertain parameters by a vector λ. The dose
distribution d(x, λ) delivered to the patient depends on the beam spot weights
x to be optimized, and the values of the uncertain parameters λ. The objective
function used for treatment planning O(d(x, λ)) is a function of the dose
distribution.
15.4.2.1 The Probabilistic Approach
In the probabilistic or stochastic programming approach (34), a probability
distribution P(λ), reflecting the probability for a given error to occur, is
assigned to the set of uncertain parameters. Treatment plan optimization is
performed by optimizing the expected value of the objective function:
minimize E[O] = ∫O(d(x,λ))P(λ)dλ. (15.8)
This composite objective function can be interpreted in a multicriteria view:
The composite objective is a sum of objectives for every possible error scenario
weighted with the probability of that error to occur. The general goal is
to find a treatment plan that is good for all possible errors, but larger weights
are assigned to those scenarios that are likely to occur, and lower weights to
large errors that are less likely to happen.
For a pure quadratic objective function, O d D i i i
= Σ ( − pres )2 , the expected
value of the objective function is
E O = E d D +E d E d i i
pres
i i
i
[ ] [ ]− ( ) − [ ] ( ) ⎡⎣
⎤⎦
Σ( ) 2 2
(15.9)
which is the sum of two terms: the first term is the quadratic difference of
the expected dose E[d] and the prescribed dose, and the second term is the
variance of the dose. Hence, minimizing the expected value of the quadratic
objective function aims at bringing the expected dose close to the prescribed
dose in every voxel and simultaneously minimizes the variance of the dose
in every voxel such that the expected dose is approximately realized even if
an error occurs.
478 Proton Therapy Physics
15.4.2.2 The Robust Approach
In robust optimization (32), the values of uncertain parameters are assumed
to be within some interval called the uncertainty set. Treatment planning
is performed by solving the robust counterpart of the conventional IMPT
optimization problem. For an introduction to robust optimization, see
Ben-Tal and Nemirovski (37). Typically, this means that the constraints of
the optimization problem have to be satisfied for every realization of the
uncertain parameters. For example, if the original problem constrained that
the maximum dose to the spinal cord be less than 50 Gray (Gy), the robust
counterpart would demand that the maximum spinal cord dose is less than
50 Gy for every possible range and setup error within the uncertainty set.
For objectives, this formulation of the robust counterpart results in a worst-case
optimization problem: that is, if the objective was to minimize the maximum
dose to the spinal cord, then the robust counterpart would minimize the
maximum spinal cord dose that can happen for any possible range or setup
error. Hence, the aim is to find a treatment plan, which is as good as possible
for the worst case that can occur.
15.4.2.3 Optimization of the Worst-Case Dose Distribution
Yet another approach to robust IMPT planning utilizes the concept of a
worst-case dose distribution (33). This hypothetical dose distribution is
defined voxel by voxel as the worst dose value that can be realized for any
error anticipated in the uncertainty model. For every target voxel, the worst
dose value is the minimum dose, whereas for nontarget voxels it is the maximum
dose. The worst-case dose distribution is unphysical because every
voxel is considered independently. Whereas in one voxel the worst case may
correspond to a patient shift anteriorly, the worst case in another voxel may
correspond to a patient shift posteriorly.
Hence the worst-case dose distribution
cannot be realized. However, it can be considered as a lower bound for
the quality of a treatment plan. The method optimizes the weighted sum of
the objective function evaluated for the nominal case dnom (no errors) and the
objective function evaluated for the worst-case dose distribution dwc. If O is
the primary objective function, then the composite objective to be optimized
is given by Ocomp =O(dnom) +wO(dwc ).
15.4.3 Examples of Robust Optimization
Incorporating uncertainty in IMPT optimization yields increasingly robust
treatment plans. Consider two treatment plans: a conventional plan optimized
without accounting for uncertainty, and a plan optimized for range
and setup uncertainty using the probabilistic approach (i.e., the setup and
range uncertainties modeled with a Gaussian distribution). Figure 15.8
shows the DVHs corresponding to dose distributions calculated for range
Treatment-Planning Optimization 479
and setup errors randomly sampled from these Gaussian distributions. For
the conventional plan, target coverage is strongly degraded in many cases,
and the dose to the spinal cord can be very high for some scenarios. The
variation in the DVHs of the robust plan is greatly reduced, ensuring better
target coverage and lower spinal cord doses.
To gain some insight into how this robustness is achieved, let us consider
the dose contributions of individual beams. Figure 15.9 compares four treatment
plans: the conventional plan, a plan optimized for range uncertainty
only, a plan optimized for setup uncertainty only, and a plan incorporating
both types of errors. The conventional plan is characterized by steep dose
100
80
60
40
20
00 10 20 30 40
Conventional
Robust
50 60 70 80 90
FIGURE 15.8
DVH comparison between a conventional and a robust IMPT plan. DVHs for the CTV and the
spinal cord are shown for randomly sampled range and setup errors.
A B
C D
FIGURE 15.9
For the case illustrated in Figure 15.3, dose contributions from the posterior beam from four differently
optimized plans. (A) Conventional IMPT, robust IMPT incorporating (B) range uncertainty
only, (C) setup uncertainty only, and (D) considering both range and setup uncertainty.
480 Proton Therapy Physics
gradients both in beam direction and laterally, especially around the spinal
cord. The plan optimized for range uncertainty shows reduced dose gradients
in beam direction and avoids placing a steep distal falloff of a Bragg peak
in front of the spinal cord. The lateral falloff is used instead of the distal falloff
to shape the dose distribution around the spinal cord. The plan optimized
for setup errors only shows reduced dose gradients in the lateral direction,
but it does not avoid placing a distal Bragg peak falloff in front of the critical
structure and therefore does not provide robustness against range errors
per se. The plan optimized for both range and setup errors shows reduced
dose gradients both longitudinally, in the beam direction, and laterally.
In summary, robustness is achieved through a redistribution of dose
contributions among the beam directions and through avoiding unfavorable
dose gradients. For our sample paraspinal case, the price of robustness
is a higher dose to the spinal cord for the nominal case. In a conventional
plan, the steep distal Bragg peak falloff is utilized, which allows for optimal
sparing of the spinal cord. If range errors are to be accounted for, the shallower
lateral falloff is used, leading to a more shallow dose gradient between
tumor and spinal cord for the nominal case. Publications by Pflugfelder et al.
(33) and Unkelbach et al. (34) provide a more detailed analysis. In the experience
of the authors, all of the methods to account for uncertainty, described
above, lead to similar treatment plans and may be equally suited to account
for systematic uncertainties.
15.5 Temporospatial (4D) Optimization
Precision of therapy delivery can be affected not only by the changes in
setup and patient anatomy between treatment fractions, but also by the
intrafractional motion of the target, which could be due to respiration, peristalsis,
or organ settling due to gravity (see Chapter 14 for more details). If
no action is taken, there is always a risk that parts of the target may move
outside of the treatment field, resulting in a loss of dose coverage. Even in
cases where treatment-planning margins are generous enough to cover the
full amplitude of motion, intrafractional motion would degrade dose gradients
and increase irradiation of surrounding healthy tissues.
An important
difference from x-ray therapy is that, in particle therapy, because of the limited
range, the use of margin expansions, such as internal target volumes,
requires explicit consideration of possible changes in radiological depth to
target, because these are often affected by organ motion (38).
Additionally, as with x-rays, in dynamically delivered intensity-modulated
therapy, certain patterns of superposition of motion of the target and the
scanned beam, or so-called “motion interplay,” can have a severe impact on
the delivered dose (e.g., 39).
Treatment-Planning Optimization 481
Numerous ideas have been put forward that aim to mitigate the impact
of intrafractional motion: these include recommendations for selection
of planning image set, compensator expansion, internal margins (40, 41),
delivery methods using beam gating (42), field rescanning (43), and target
tracking (44). In this section, we review approaches to incorporate intrafractional
motion into the optimization of beam weights in IMPT. Those methods
have been investigated primarily in the context of IMRT with photons.
Although the methodology can be transferred to IMPT, those approaches
have not been validated in detail regarding the specific challenges mentioned
above, that is, interplay effects and sensitivity to changes in radiological
path length.
Methods to incorporate intrafractional motion in plan optimization require
a characterization of the geometrical variation of the patient’s anatomy. For
respiratory motion, this can be obtained from respiratory-correlated computed
tomography (CT) (often called 4D CT), which provides the geometry
of the patient in several phases of the breathing cycle (45). The task of evaluating
the actual dose distribution delivered to a moving target requires first
calculating instantaneous dose to all phases of the 4D CT. Figure 15.10 illustrates
variation in the proton dose distribution delivered to a changing anatomy,
throughout the respiratory cycle. Such instantaneous doses can then
be mapped onto a reference anatomical set, by using the correspondence
established between the voxels of different CT sets, obtained through elastic
image registration. The mapped dose can be subsequently added along with
contributions from all instances of variable anatomy, to yield the dose accumulated
throughout the respiratory cycle (46, 47).
Instantaneous dose
on phase-specific CT
Full inhalation
Full
A
B C
D E
exhalation
Mid-ventilation
Instantaneous dose
mapped to exhalation CT
FIGURE 15.10
Dosimetric evaluation of a treatment plan for a tumor in the liver, using respiratory-correlated
CT. Estimated instantaneous dose delivered during (A) the full exhalation, (B) full inhalation,
and (C) mid-ventilation phases of the respiratory cycle. To estimate the total dose, contributions
from various instances of the anatomy have to be mapped onto the reference CT set, for
example, for (D) full inhalation dose (dose “B” mapped onto the full exhalation CT “A”), and
(E) mid-ventilation (dose “C” mapped onto CT “A”). (CT images courtesy of Dr. S. Mori (NIRS).
With permission.)
15.5.1 Plan Optimization Based on a Known Motion
Probability Density Function
In a simple approach to include motion, it is assumed that motion is sufficiently
well described by the reconstructed phases of a 4D CT and that the
dose delivered to a voxel i is obtained by summation of the dose contributions
from all phases:
d= p d = p x D i
r
r
i
r r
r
j ij
r
j
Σ ( ) ( ) Σ ( )Σ ⋅ ( ) .
(15.10)
Here, r is an index to the instance of geometry, and the voxel index i refers
to an anatomical voxel defined in the reference phase. Dij
(r) is the dose influence
matrix for phase r. Its calculation requires elastic registration of the CT
of phase r with the reference phase. The parameters p(r) are probabilities that
the patient is in phase r and are referred to as the motion probability density
function (PDF), which can be estimated from a recorded breathing signal.
Treatment planning can be performed by optimizing the beam weights
xj based on objectives and constraints evaluated with the cumulative dose
from Equation 15.10 above (48).
The general idea is that, rather than passively letting the motion deteriorate
the original plan, one should anticipate it, and, in fact, actively engage it
in shaping the desired dose distribution. The resulting treatment plan would
deliver an inhomogeneous dose distribution to a static geometry. However,
the inhomogeneities are designed such that, after accumulating dose over
the whole breathing cycle, the desired dose distribution is obtained. Because
one of the most manifest effects of motion on the dose is the smoothing or
washout of gradients both within the target and at its borders, the logical
way to counter this effect is dose boosting at the edges of the target, in what
is termed “edge-enhancement” (49).
The exact pattern of optimum inhomogeneity enhancement is determined
by the form of motion PDF. Generally, the effect of motion on the dose may
be approximated as convolution of the dose with the PDF; thus, the desired
motion-compensated plan can be roughly approximated with the inverse
process: deconvolution. However, this is constrained by the requirement that
the fluences delivered at all pencil beam spots are physical; thus, if negative
values arise from deconvolution or during optimization, those need to be
reset to zero (or the minimum should be allowed, if the beam cannot be completely
turned off, e.g., in a continuous scan).
Because the PDF does not depend on time, the use of probabilistic planning
does not require complex technical delivery modifications to ensure
synchronization of the beam with the motion cycle, and thus delivery of such
fields can be relatively easily implemented in practice. However, PDF-based
optimization methods rely on the reproducibility of target motion patterns
during delivery, and sufficient sampling of the motion PDF. When motion
deviates from the expectation, a significant dosimetric deviation may occur.
Treatment planning for proton therapy usually uses a constant relative biological
effectiveness (RBE) factor of 1.1 for the conversion of physical dose di
to “biological” dose (see Chapter 19). The biological effective dose is defined
as the photon dose from a 60Co source that would produce the same cell-kill
in the tumor. Under the assumption of a constant RBE, treatment plan optimization
can be performed based on the physical dose alone as described
in the preceding sections of this chapter. In other words, the physical dose
is the only measure that is needed to characterize the radiation field and to
assess the quality of the treatment plan. However, this may be an oversimplification
and a second quantity may be needed to characterize the radiation
field and its radiobiological effectiveness. This second measure is the linear
energy transfer (LET) (see Chapters 2 and 19).
Radiobiological experiments suggest that the amount of radiation-induced
cell-kill increases with higher LET, and consequently at the end of range of
the proton beams. To directly incorporate effects of varying RBE in IMPT
planning, the objective function needs to be formulated in terms of physical
dose and LET, instead of dose alone. One approach has been suggested
by Wilkens and Oelfke (52), who formulate their objective function based
on the linear-quadratic cell survival model, where the α-parameter depends
484 Proton Therapy Physics
linearly on LET. Recently, Grassberger et al. (53) have demonstrated that it is
feasible in IMPT optimization to influence the distribution of LET without
significantly altering the physical dose distribution.
15.7.1 Scan Path Optimization
In 3D spot scanning, beam spots are typically placed on a regular grid over
the tumor region. In practice though, a large number of beam spots will be
assigned zero weight in the optimization of the treatment plan. Nevertheless,
in a naïve implementation of spot scanning, the beam would be steered in
a zigzag pattern over the entire grid, including the spot positions that correspond
to zero weight. Kang et al. (54) investigated the optimization of
the scan path of the beam in order to avoid regions with zero weight spots.
The problem corresponds to a “traveling salesman” problem and simulated
annealing has been applied to solve the problem.
15.7.2 Beam Current Optimization for Continuous Scanning
There are different ways to perform pencil beam scanning. In spot scanning,
the proton beam is steered to one desired position on the grid, delivers
dose according to the optimized spot weight, is switched off, and is moved
to the next grid point. In continuous scanning, the beam is constantly moving
according to a predefined pattern. The intensity-modulated field is delivered
by modulating the beam current in time while the beam is repeatedly
scanned over the tumor volume. In this case, an additional computational
step is needed that converts the optimized spot weights defined at discrete
positions to the beam-current modulation that approximately delivers the
same fluence. For this step, optimization methods have been applied (14);
however, this optimization can be performed in fluence space. It does not
require dose calculation in the patient and is therefore easier to solve than
the optimization of spot weights.
