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MATHEMATICAL ASPECTS OF RADIATION THERAPY TREATMENT PLANNING: CONTINUOUS INVERSION
MATHEMATICAL ASPECTS OF RADIATION THERAPY
TREATMENT PLANNING: CONTINUOUS INVERSION
VERSUS FULL DISCRETIZATION AND OPTIMIZATION
VERSUS FEASIBILITY
YAIR CENSOR
Abstract.
A mathematical formulation of the radiation therapy problem consists
of a pair of forward and inverse problems. The inverse problem is to determine external
radiation beams, along with their locations, pro les, and intensities, that will provide a
given dose distribution within the irradiated object. We discuss the inverse problem in
its fully discretized formulation.
1. Introduction.
This paper deals with radiation
teletherapy
where
beams of penetrating radiation are directed at the lesion (tumor) from an
external source. The other radiation delivery mode which involves direct
implantation of radioactive sources inside the lesion, called
brachytherapy
,
is not included in our discussion. Chapter 11 of the book by Censor and
Zenios 9] and Brahme's special issue 6] and references therein, as well as
the tutorial review of Altschuler, Censor, and Powlis 2], can be used as
introductory material to this area.
Based on understanding of the physics and biology of the situation,
there are two principal aspects of radiation teletherapy that call for math-
ematical modelling. The rst is the calculation of the
radiation dose
which
is a measure of the actual energy absorbed per unit mass everywhere in the
irradiated tissue. In dose calculation, termed
dosimetry
, the relevant phys-
ical and biological characteristics of the irradiated object and the relevant
information about the radiation source (geometry, physical nature, inten-
sity, etc.) serve as input data. The result (output) of the calculation is a
dose function
whose values are the dose absorbed as a function of location
inside the irradiated body.
The second aspect is the
mathematical inverse problem
of the rst. In
addition to all physical and biological parameters of the irradiated object
we assume here that the relevant information about the capabilities and
speci cations of the available
treatment machine
(i.e., radiation source) is
given. Based on medical diagnosis, knowledge, and experience, the physi-
cian prescribes a
desired dose function
to the case. The output of this prob-
lem should be a
radiation intensity function
whose values are the radiation
intensity at the source as a function of source location, that would result
in a dose function which is identical to the desired one. To be of practical
value, this resulting radiation intensity function must be implementable, in
a clinically acceptable form, on the available treatment machine.
Department of Mathematics, University of Haifa, Mt. Carmel, Haifa 31905, Israel.
E-mail: yair@mathcs2.haifa.ac.il
1 2
YAIR CENSOR
In what follows we discuss, from a mathematician's point of view, two
main modelling dilemmas: (i) continuous inversion versus full discretiza-
tion, and (ii) optimization versus feasibility.
Much of current
radiation therapy treatment planning
(RTTP) is still
done in two dimensions where only a single plane through the center of the
target is considered. RTTP is also still done mostly in a trial-and-error
fashion by picking a machine setup that gives rise to a certain external
radiation intensity eld (function) and then using a forward-problem-solver
software package to determine the resulting dose function, see Figure 1. If
the discrepancy between this dose function and the prescribed dose function
is unacceptable then some changes are made to the machine setup and the
process is repeated until the physician and dosimetrist are satis ed with the
resulting dose function. Only then actual patient treatment is performed.
External radiation field (u,w) organ
Dose distribution
D(r, ) level
contours Target
2D crosssection
Fig. 1
.
2D{R
TTP,
an
external
r
adiation
eld
(
u
w
)
r
esults
in
a
dose
distribution
D
(
r
)
.
Such 2D{RTTP has achieved success due to accumulated experience
and also because of the ever increasing quality and speed of forward-
problem-solvers. MATHEMATICAL ASPECTS OF RADIATION THERAPY TREATMENT
3
Automated solution of the inverse problem of RTTP should be useful
in handling di cult planning cases, particularly in 3D{RTTP, see Figure 2.
There, it would be much more di cult to reach an acceptable plan by trial-
and-error because of the multitude of potential directions from which the
3D object can be irradiated. Nonetheless, even a 2D discussion, as given
here, is enough to expose the nature of the dilemmas that we consider.
Organ
3D external
radiation field
3D cross-section
Target
3D dose distribution level
contours
Fig. 2
.
3D{R
TTP,
ful
ly
3D
cr
oss
se
ction,
external
r
adiation
eld
and
dose
distri-
bution.
In addition to the references given in the sequel we recommend also
Mackie et al. 16], Raphael 18], Webb 19], and Xing and Chen 20].
2. Problem de nition and the continuous model.
Let
D
(
r
) be
a real-valued nonnegative function, of the polar coordinates
r
and , whose
value is the dose absorbed at a point in the patient's planar cross-section
coincident with the plane of the machine's gantry motion. This is the
dose
function
, or dose distribution. A
ray
is a directed line along which radiated
energy travels away from the
source
, i.e., the
teletherapy source
. Rays are
parametrized by variables
u
and
w
in some well-de ned way and the real-
valued nonnegative function (
u
w
) represents the
radiation intensity
along
the ray (
u
w
) due to a point source on the gantry circle. The continuous
forward problem of RTTP is the following. Assume that the cross section
of the patient and its radiation absorption characteristics are known. Given 4
YAIR CENSOR
a radiation intensity function (
u
w
) for 0
u
<
2 and
;W
w
W
,
nd the dose function
D
(
r
) for all (
r
)
2
from the formula
D
(
r
) =
(
u
w
)](
r
)
(2.1)
where is the
dose operator
. This operator relates the dose function to
the radiation intensity function. See, e.g., 8] or 9, Chapter 11], for a
description of the speci c coordinate system.
In other words, the forward problem amounts to the calculation of the
total dose absorbed at each point of a patient section when all parameters of
each radiation beam are speci ed and the description of the patient section
is known. The di culties associated with the forward problem stem from
the fact that there exists no closed-form analytic representation of the dose
operator that will enable us to use equation (2.1) for the calculation of
D
(
r
). Although the interaction between radiation and tissue is measured
and understood at the atomic level, the situation is so complex that, to
solve the forward problem in practice, a good state-of-the-art computer
program, which represents a
computational approximation
of the operator
and which enables reasonably good dose calculations, must be used.
Let us elaborate on what we mean by stating \there exists no close-
form analytic representation of the dose operator ." We actually mean the
following: If drasticallysimplifyingassumptions are made about the physics
of the model as well as the particulars of the desired dose distribution,
then it is sometimes possible to express the dose operator in a closed-form
analytic formula. This has been done rst by Brahme, Roos and Lax 4]
and extended by Cormack and co-workers, consult the review paper of
Cormack and Quinto 12] for further references. See also Brahme's recent
review 5] and Goitein's editorial 13].
In current practice of RTTP, when dose calculations are performed to
verify the dose that will result from a proposed treatment plan, the goal is
to obtain results that are as accurate as possible. To achieve this, various
empirical data, which are often condensed in look-up tables, are incorpo-
rated into the forward calculation. Thus, the true forward calculation, or
true dose operator, is not represented by a closed-form analytic relation be-
tween the radiation intensity function (
u
w
) and the dose function
D
(
r
),
but by a software package that calculates
D
(
r
) from (
u
w
). Thus, what
we really mean by saying that there is no closed-form analytic expression
for is that we choose to adhere to the software representation rather
than compromise by allowing simplifying assumptions that might lead to
a closed-form analytic mathematical formula.
The
inverse problem
of radiation therapy is the treatment planning
problem:
Given a description of the patient section, the dose prescribed for the
target, and the maximum permissible doses to the target, critical organs,
and other tissues, calculate the external con guration and relative inten- MATHEMATICAL ASPECTS OF RADIATION THERAPY TREATMENT
5
sities of radiation sources (i.e., the radiation eld) that will deliver the
speci ed radiation doses (or some acceptable approximation thereof).
Assuming that the cross section of the patient and its radiation
absorption characteristics are known, and given a prescribed dose function
D
(
r
), the problem is to nd a radiation intensity function (
u
w
) such
that equation (2.1) holds, or (
u
w
) =
;1
D
(
r
)] where
;1
is the
inverse operator of . This is the inversion problem that we want to
solve, in a computationally tractab