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Absolute dose reconstruction in proton therapy using PET imaging modality: feasibility study

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IOP P
UBLISHING
P
HYSICS IN
M
EDICINE AND
B
IOLOGY
Phys. Med. Biol.
54
(2009) N217–N228
doi:10.1088/0031-9155/54/11/N02
NOTE
Absolute dose reconstruction in proton therapy usingPET imaging modality: feasibility study
E Fourkal, J Fan and I Veltchev
Department of Radiation Oncology, Fox Chase Cancer Center, 333 Cottman Avenue,Philadelphia, PA, USA
Received 3 December 2008, in ﬁnal form 8 April 2009Published 13 May 2009Online at stacks.iop.org/PMB/54/N217
Abstract
A simple analytical model is developed that allows efﬁcient absolute dosereconstruction in patients undergoing radiation treatments using proton beams.Themodelisbasedonthesolutionoftheinverseproblemofdoserecoveryfromthe 3D information contained in the PET signal, obtained immediately after thetreatment. The core of the proposed model lies in the analytical calculation of the introduced positron emitters’ species matrix (PESM) or kernel, facilitatedby previously developed theoretical calculations of the proton energy ﬂuencedistribution. Once the PESM is known, the absolute dose distribution in apatient can be found from the deconvolution of the 3D activity distributionobtained from the PET scanner with the calculated species matrix. As anexample, we have used FLUKA Monte Carlo code to simulate the delivery of the radiation dose to a tissue phantom irradiated by a parallel-opposed beamarrangement and calculated the resultant total activity. Deconvolution of thecalculated activity with the PESM leads to the reconstructed dose being within2% of that delivered.(Some ﬁgures in this article are in colour only in the electronic version)
1. Introduction
Radiation treatment using proton beams is accompanied by the production of small amountsof positron-emitting isotopes along the beam paths. These short-lived radioisotopes, mainly
11
C,
13
N and
15
O, allow the imaging of three-dimensional
in vivo
activity distribution usingpositron emission tomography (PET) (Oelfke
et al
1996, Parodi
et al
2002, Vynckier
et al
1993), which can be ultimately implemented into quality assurance and treatment veriﬁcationafter proton therapy (Knopf
et al
2008, Parodi
et al
2007). The proposed PET imaging fortreatment veriﬁcation however, only exploits the possibility of using information containedin the spatial activity distribution to verify the ﬁeld positioning and to gain insight into thebeam penetration depth. The natural extension of these ideas would be to use the three-dimensional activity to reconstruct the dose delivered to a patient. Since there is no direct
0031-9155/09/110217+12$30.00 © 2009 Institute of Physics and Engineering in Medicine Printed in the UK N217
N218 E Fourkal
et al
correlation between the absorbed dose and activity (which is intrinsically due to the differencein the underlying physics behind the energy deposition process in a case of absorbed dose andnuclear processes involved in
β
+
activation in the case of induced activity), one cannot readilyobtain one from the other once the treatment has been completed. Therefore, in some previousresearch endeavors, the authors used an indirect way to verify particle (carbon) beam dosedelivery (Enghardt
et al
1999). The method is based on simulating the PET isotope activationdistribution and image formation on a computer and subsequently comparing it to a measuredPET image. The result of this comparison gives an indication about the successfulness of thedelivery process. In one form or the other this indirect comparison method was used in manyrecent investigations concerning this issue (Hishikawa
et al
2002, Nishio
et al
2005, Paansand Schippers 1993).Even though the indirect method is appealing and relatively easy to develop, it does notprovide direct comparison between the planned and delivered doses, thus always bringingsome uncertain (or unsatisfactory) aspect associated with it. This is why the development of the dose reconstruction method from the information stored in the spatial activity distributionpresents a more desirable solution. It should be noted that the problem of dose reconstructionfrom the PET signal presents quite a challenging endeavor with many individual elements thatmay inﬂuence the resultant dose
/
activity correlation (e.g. inﬂuence of organ motion, presenceof density
/
material inhomogeneities, image blurring due to positron transport, presence of perfusion and washout effects, etc). However, the core mathematical formulation of it isrelatively simple and straightforward. The inﬂuence of effects related to image formation andacquisition, as well as perfusion and washout processes can be subsequently built into themain model of dose reconstruction.Although the underlying physical processes responsible for dose deposition and isotopeproduction are different, they both depend on certain physical characteristic—the protonﬂuence differential in energy (the absorbed dose is the integral of the product of the protoncollision stopping power and the energy ﬂuence spectrum while the activity is proportionalto the integral of the product of the nuclear activation cross section and the energy ﬂuencespectrum). Theexistenceofthiscorrelationbetweenbothquantitiesmaysigniﬁcantlysimplifytheproblemofdosereconstruction,whichmayeventuallyleadtothedevelopmentofapractical
in vivo
dose veriﬁcation system after the proton radiotherapy.Inthefollowing, wewillofferamathematical formulationthatallows dosereconstructionfrom the activity distribution. It is based on the solution of the inverse problem of activityde-convolution with the so-called positron emitters’ species matrix or PESM. We also offeran analytical method for ﬁnding the species matrix, which signiﬁcantly decreases the totalcomputation time needed in the reconstruction process. It is based on the solution of theBoltzmannkineticequationfortheprotonenergydistributionfunctioninthemedium. Finally,the developed model is applied to study the test case of two parallel-opposed proton beamsincident on a tissue phantom.
2. Mathematical model for dose reconstruction
It is a well known fact that proton as well as other heavy ion therapies allows dose modulationin the direction of beam propagation, also known as the spread out Bragg peak (SOBP) effect.The SOBP depth dose distribution is practically achieved by using the modulator wheel asdone in the conventional passive scattering technique, or inserting different thickness low
Z
material into the beam line as done in a spot scanning technique used at Paul Scherrer Institutein Switzerland. Regardless of the method by which the depth proﬁling is achieved, its maineffect lies in special shaping of the energy spectrum (Bortfeld and Schlegel 1996, Fourkal
Absolute dose reconstruction in proton therapy using PET imaging modality N219
et al
2007) of protons incident on a patient. This energy spectrum ensures the constancy of the dose along the desired depth extension. With that in mind, expressions for the delivereddose and activity distributions have the following form:
A(
r
,t)
=
E
max
E
min
f
SOBP
(E)A(E,
r
,t)
d
E
≈
E
i
w(E
i
)A(E
i
,
r
,t)D(
r
)
=
E
max
E
min
f
SOBP
(E)d(E,
r
)
d
E
≈
E
i
w(E
i
)d(E
i
,
r
),
(1
.
1)where
A(
r
,t)
and
D(
r
)
are the total ‘in-patient’ activity and dose distributions,
E
is theproton energy before entering the phantom,
f
SOBP
(E)
or its discretized representation
w(E
i
)
is the proton energy spectrum that delivers SOBP depth–dose in the desired spatial extent,
t
is time,
A(E
i
,
r
,t)
and
d(E
i
,
r
)
are the corresponding activity and dose distribution kernelsfor a given
E
i
. They can be either simulated using the Monte Carlo technique or calculatedanalytically, provided that the proton ﬂuence differential in energy (also a function of spatialposition
r
) and nuclear activation cross sections are known. Equation (1
.
1) can be extendedto account for multiple beams
/
beamlets to give
A(
r
,t)
=
i,j
˜
w
i,j
k
˜
N
k
(
r
,E
i
)(
1
−
e
−
λ
k
t
irr
,j
)t
irr
,j
e
−
λ
k
(t
−
t
end
,j
)
D(
r
)
=
i,j
˜
w
i,j
˜
d
j
(E
i
,
r
),
(1
.
2)where
t
irr,
j
is the delivery time for the beam number
j
,
t
end,
j
is the time at which the beam
j
hasended its dose delivery,
λ
k
is the decay constant for species
k
and
˜
N
k
(
r
,E
i
)
is the numberof isotopes of species
k
produced during the beam
j
delivery time
t
irr,
j
normalized per incidentproton, ˜
d
j
(E
i
,
r
)
is the dose distribution kernel normalized per incident proton and ˜
w
i,j
arethe proton energy spectra for different beamlets
j,
multiplied by the number of protons in thebeamlet
j
and the energy sampling size
E
. In equation (1
.
2),
A(
r
,t)
is the three-dimensionaldistributionofactivityobtainedfromthePETscannerattime
t
andthepositronemitters’matrix
˜
N
k
(
r
,E
i
)
canbefoundeitherbyusingtheMonteCarlosimulationsorviaenergyintegrationof the given nuclear activation cross section for species
k
with the proton ﬂuence spectrum atposition
r
. The system of equations (1
.
2) is the main mathematical model that allows solvingtheproblemofdosereconstructionfromthemeasuredactivitydistribution. Itspresentstructurehowever does not include the blurring effects related to image formation and acquisition (e.g.
β
+
decay and positron transport in a patient, propagation of annihilation photons and theirdetection); they can be subsequently incorporated into the model by convoluting the emitters’matrix with the 3D Gaussian point spread function (Parodi
et al
2007). Moreover, one wouldalso have to correct it by an additional temporal factor, which accounts for a ﬁnite acquisitiontime by the PET imager. These additional correction factors are not going to be addressed inthe present work, since they are related to the intrinsic characteristics of the given PET imagerand represent a ﬁne-tuning modiﬁcation to the main mathematical formulation consideredhere. Instead, we will concentrate on ﬁnding a time-efﬁcient technique to solve the system of equations (1
.
2), which in itself also requires developing an efﬁcient method for the calculationof the positron emitters’ matrix. As mentioned earlier, this matrix can be calculated using theMonte Carlo method. The drawback of this approach however lies in its time inefﬁciency,since one would need to generate the PESM for all possible energies and store them externallyin ﬁles, which in turn would dramatically slow down the reconstruction process. Therefore, inthe following section we will describe an analytical method for the calculation of the speciesmatrix
˜
N
k
(
r
,E
i
)
.
N220 E Fourkal
et al
Once the activity distribution
A(
r
,t)
and the species matrix
˜
N
k
(
r
,E
i
)
are known, onecan solve the upper equation in (1
.
2) for the unknown weights ˜
w
i,j
, subsequently substitutingthem into the lower equation in (1
.
2) to ﬁnd the absorbed dose. It is worth noting here thatthe reconstructed weights serve a dual purpose in this approach. First, they allow obtainingthe dose delivered to the patient from the actual activity distribution. Second, they provide aquality assurance function in which the reconstructed weights should be compared to thosegiven by the treatment planning system. Any difference between them would point to adiscrepancy between the delivered and planned doses, warranting further investigations.In the current work, the unknown weights ˜
w
i,j
are found from the solution of the systemof linear inhomogeneous equations using the ‘random creep’ algorithm (Deng
et al
2001). Itis also interesting to note that the solution to the system (1
.
1) or (1
.
2) in terms of unknownweights ˜
w
i,j
is unique. This can be seen from the following argument. Suppose that therewere two different solutions
w
(
1
)
and
w
(
2
)
which both give rise to the same
A(
r
,t)
or
D(
r
)
.Then the difference between these solutions
w
would satisfy the condition
i
w(E
i
)A(E
i
,
r
j
,t)
=
0
,
i
w(E
i
)d(E
i
,
r
j
)
=
0
,
which is the system of linear homogeneous equations. It is well known that such systemalways has the trivial solution in which all
w
are equal to zero. It may also have an inﬁnitenumber of solutions, provided that the determinant det[
A(E
i
,
r
j
,t)
] or det[
d(E
i
,
r
j
)
] is zero.The requirement of zero determinant means that at least two rows or columns in the matrixare identical, which in turn would require that activity
A(E
i
,
r
j
,t)
or dose
d(E
i
,
r
j
)
kernelsbe identical for two different proton energies
E
i
and
E
j
(i
=
j)
throughout the spatial region.Such degeneracy however is not present in the dosimetric characteristics of the proton beam,thus det[
d(E
i
,
r
j
)
]
=
0 or det[
A(E
i
,
r
j
,t)
]
=
0
,
and the only solution satisfying the aboveconstraint is the trivial one. This in turn veriﬁes the uniqueness conjecture stated earlier.
3. Calculation of the positron emitters’ species matrix
The positron emitters’ species matrix can be calculated using the following expression:
˜
N
k
(
r
,E
i
)
=
j
N
j
N
0
E
σ
kj
(E)
d
(E,E
i
,
r
)
d
E
d
E
(1
.
3)where
N
j
is the number of atoms of species
j
(oxygen, nitrogen, carbon, etc),
σ
k j
is the crosssection for the production of the isotope
k
from species
j
in nuclear interaction, d
/
d
E
is theprotonenergyﬂuencespectrum,
E
i
and
N
0
aretheinitialenergyandnumberofprotonsenteringthe medium. As one can see, the species matrix
˜
N
k
(
r
,E
i
)
is completely determined by theactivationcrosssectionandtheprotonenergyﬂuence. Thenuclearactivationcrosssectionsareknown functions of proton energy (either found semi-empirically from the emission spectraof recoils (Beebe-Wang
et al
2002) or obtained theoretically
/
experimentally (InternationalCommission on Radiation Units and Measurements 2000)). Their energy dependence forthe production of
11
C,
13
N and
15
O (the three main tracers considered here) is shown inﬁgures 1(a), (b) and (c) (obtained from the Fluka simulation code). As one can see, the crosssections exhibit non-monotonous behavior in the proton energy of up to 100 MeV range, afterwhich they settle into very week energy dependence.The next step in the development of the analytical model lies in ﬁnding an expression forthe proton energy ﬂuence spectrum. With the help of earlier published work (Fourkal
et al
2007)wheretheprotonenergyspectrumneededindepthproﬁlingcalculationswasconsidered,an expression for the proton energy ﬂuence spectrum will be obtained. The starting point in
Absolute dose reconstruction in proton therapy using PET imaging modality N221
0 50 100 150 200E/MeV00.00010.00020.00030.0004
n o r m a l i z e d c r o s s s e c t i o n ( a r b . u n i t s )
12
C(p,pn)
11
C
14
N(p,2p2n)
11
C
16
O(p,3p3n)
11
C0 50 100 150 200E/MeV00.00010.00020.00030.0004
n o r m a l i z e d c r o s s s e c t i o n ( a r b . u n i t s )
14
N(p,pn)
13
N
16
O(p,2p2n)
13
N0 50 100 150 200E/MeV00.00010.00020.00030.0004
n o r m a l i z e d c r o s s s e c t i o n ( a r b . u n i t s )
16
O(p,pn)
15
O
(a)(b)(c)
Figure 1.
(a) The nuclear activation cross sections for the production of
11
C (
12
C(p,np)
11
C,
14
N(p,2p2n)
11
C and
16
O(p,3p3n)
11
C) as functions of proton energy. (b) The nuclear activationcrosssectionsfortheproductionof
13
N(
16
O(p,2p2n)
13
N,
14
N(p,pn)
13
N)and(c)
15
O(
16
O(p,pn)
15
O)as functions of proton energy
E
. The cross sections are normalized by the inverse of the atomicdensities
˜
σ
=
σ
AwS N Aρ
where
A
w
and
ρ
are the atomic mass of the element and its mass density,
N
A
is the Avogadro’s number and
S
is the area of the proton ﬁeld.

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