Visualization and quantification of tumour biology for personalized radiotherapy
Personalized Radiotherapy Design for Glioblastoma ...
9
0278-0062 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2019.2902044, IEEE Transactions on Medical Imaging 1 Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models, Multimodal Scans and Bayesian Inference Jana Lipkov´ a 1,2,12 , Panagiotis Angelikopoulos 3 , Stephen Wu 4 , Esther Alberts 2 , Benedikt Wiestler 2 , Christian Diehl 2 , Christine Preibisch 5 , Thomas Pyka 6 , Stephanie Combs 7 , Panagiotis Hadjidoukas 8,10 , Koen Van Leemput 9 , Petros Koumoutsakos 10 , John Lowengrub 11 , Bjoern Menze 1,12 Abstract—Glioblastoma is a highly invasive brain tumor, whose cells infiltrate surrounding normal brain tissue beyond the lesion outlines visible in the current medical scans. These infiltrative cells are treated mainly by radiotherapy. Existing radiotherapy plans for brain tumors derive from population studies and scarcely account for patient-specific conditions. Here we provide a Bayesian machine learning framework for the rational design of improved, personalized radiotherapy plans using mathematical modeling and patient multimodal medical scans. Our method, for the first time, integrates complementary information from high resolution MRI scans and highly specific FET-PET metabolic maps to infer tumor cell density in glioblastoma patients. The Bayesian framework quantifies imaging and modeling uncer- tainties and predicts patient-specific tumor cell density with credible intervals. The proposed methodology relies only on data acquired at a single time point and thus is applicable to standard clinical settings. An initial clinical population study shows that the radiotherapy plans generated from the inferred tumor cell infiltration maps spare more healthy tissue thereby reducing radiation toxicity while yielding comparable accuracy with standard radiotherapy protocols. Moreover, the inferred regions of high tumor cell densities coincide with the tumor radioresistant areas, providing guidance for personalized dose- escalation. The proposed integration of multimodal scans and mathematical modeling provides a robust, non-invasive tool to assist personalized radiotherapy design. Index Terms—Glioblastoma, radiotherapy planning, Bayesian inference, FET-PET, multimodal medical scans. 1 Dept. of Informatics, Technical University Munich (TUM) Germany 2 Dept. of Neuroradiolog, Klinikum Rechts der Isar, TUM, Germany 3 D.E. Shaw Research, L.L.C, USA 4 Institute of Statistical Mathematics, Tokyo, Japan 5 Dept. of Diagnostic and Interventional Neuroradiology & Neuroimaging Center & Clinic for Neurology, Klinikum Rechts der Isar, TUM, Germany 6 Dept. of Nuclear Medicine, Klinikum Rechts der Isar, TUM, Germany 7 Dept. of Radiation Oncology, Klinikum Rechts der Isar, TUM & Institute of Innovative Radiotherapy, Helmholtz Zentrum Munich & Deutsches Konsortium f¨ ur Translationale Krebsforschung, Germany 8 IBM Research - Zurich, Switzerland. (IBM, the IBM logo, and ibm.com are trademarks or registered trademarks of International Business Machines Corporation in the United States, other countries, or both. Other product and service names might be trademarks of IBM or other companies.) 9 Harvard Medical School, Boston, USA & Dept. of Applied Mathematics and Computer Science, TU Denmark, Denmark. 10 Computational Science and Engineering Lab, ETH Z¨ urich, Switzerland 11 Dept. of Mathematics, Biomedical Engineering, Chemical Engineering and Materials Science & Center for Complex Biological Systems & Chao Family Comprehensive Cancer Center, UC, Irvine, USA 12 Institute for Advanced Study, TUM, Germany The supplementary materials are available at http://ieeexplore.ieee.org I. I NTRODUCTION G LIOBLASTOMA (GBM) is the most aggressive and most common type of primary brain tumor, with a me- dian survival of only 15 months despite intensive treatment [1]. The standard treatment consists of immediate tumor resection, followed by combined radio- and chemotherapy targeting the residual tumor. All treatment procedures are guided by mag- netic resonance imaging (MRI). In contrast to most tumors, GBM infiltrates surrounding tissue, instead of forming a tumor with a well-defined boundary. The central tumor, which is visible on medical scans, is commonly resected. However, the distribution of the infiltrating residual tumor cells in the nearby healthy-appearing tissue, which are likely to contribute to tumor recurrence, is not known. Current radiotherapy (RT) planning handles these uncertainties in a rather rudimentary fashion. Guided by population-level studies, standard-of-care RT plans uniformly irradiate the volume of the visible tumor extended by a uniform margin [1]–[3], which is referred as the clinical target volume (CTV). However, the extent of this margin varies by few centimeters even across the official RT guidelines [4]. Moreover, GBM infiltration is anisotropic and thus a uniform margin very likely does not provide an optimal dose distribution. In addition, GBM invasiveness is highly patient-specific, and thus not all patients benefit equally from the same margin, which impairs comparison and advancement of RT protocols. Despite treatment almost all GBMs recur [5]. Biopsies [6] and post-mortem studies [7] show that tumor cells can invade be- yond the CTV, which reduces RT efficiency, and is a possible cause of recurrence. At the same time radioresistance of tumor cells inside the CTV can also reduce RT efficiency. Radioresis- tance tends to occur in regions with complex microenviroment and hypoxia [8], both of which are commonly encountered in areas of high tumor cellularity. To address tumor radioresis- tance, several studies have suggested local dose-escalations [8]–[10]. In these approaches, a boosted dose is delivered into a single or multiple co-centered regions defined by adding uniform margins to the tumor outlines visible in MRI scans [11]. The Radiation Therapy Oncology Group (RTOG) phase- I-trial [10] showed an increase in median survival of 8 months with dose-escalation. However, no benefit in progression- Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
Transcript of Personalized Radiotherapy Design for Glioblastoma ...
0278-0062 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2019.2902044, IEEETransactions on Medical Imaging
1
Personalized Radiotherapy Design for Glioblastoma:Integrating Mathematical Tumor Models,Multimodal Scans and Bayesian Inference
Jana Lipkova1,2,12, Panagiotis Angelikopoulos3, Stephen Wu4, Esther Alberts2, Benedikt Wiestler2, ChristianDiehl2, Christine Preibisch5, Thomas Pyka6, Stephanie Combs7, Panagiotis Hadjidoukas8,10, Koen Van Leemput9,
Petros Koumoutsakos10, John Lowengrub11, Bjoern Menze1,12
Abstract—Glioblastoma is a highly invasive brain tumor, whosecells infiltrate surrounding normal brain tissue beyond the lesionoutlines visible in the current medical scans. These infiltrativecells are treated mainly by radiotherapy. Existing radiotherapyplans for brain tumors derive from population studies andscarcely account for patient-specific conditions. Here we providea Bayesian machine learning framework for the rational design ofimproved, personalized radiotherapy plans using mathematicalmodeling and patient multimodal medical scans. Our method, forthe first time, integrates complementary information from highresolution MRI scans and highly specific FET-PET metabolicmaps to infer tumor cell density in glioblastoma patients. TheBayesian framework quantifies imaging and modeling uncer-tainties and predicts patient-specific tumor cell density withcredible intervals. The proposed methodology relies only ondata acquired at a single time point and thus is applicable tostandard clinical settings. An initial clinical population studyshows that the radiotherapy plans generated from the inferredtumor cell infiltration maps spare more healthy tissue therebyreducing radiation toxicity while yielding comparable accuracywith standard radiotherapy protocols. Moreover, the inferredregions of high tumor cell densities coincide with the tumorradioresistant areas, providing guidance for personalized dose-escalation. The proposed integration of multimodal scans andmathematical modeling provides a robust, non-invasive tool toassist personalized radiotherapy design.
Index Terms—Glioblastoma, radiotherapy planning, Bayesianinference, FET-PET, multimodal medical scans.
1 Dept. of Informatics, Technical University Munich (TUM) Germany2 Dept. of Neuroradiolog, Klinikum Rechts der Isar, TUM, Germany3 D.E. Shaw Research, L.L.C, USA4 Institute of Statistical Mathematics, Tokyo, Japan5 Dept. of Diagnostic and Interventional Neuroradiology & Neuroimaging
Center & Clinic for Neurology, Klinikum Rechts der Isar, TUM, Germany6 Dept. of Nuclear Medicine, Klinikum Rechts der Isar, TUM, Germany7 Dept. of Radiation Oncology, Klinikum Rechts der Isar, TUM &
Institute of Innovative Radiotherapy, Helmholtz Zentrum Munich & DeutschesKonsortium fur Translationale Krebsforschung, Germany
8 IBM Research - Zurich, Switzerland. (IBM, the IBM logo, and ibm.comare trademarks or registered trademarks of International Business MachinesCorporation in the United States, other countries, or both. Other product andservice names might be trademarks of IBM or other companies.)
9 Harvard Medical School, Boston, USA & Dept. of Applied Mathematicsand Computer Science, TU Denmark, Denmark.
10 Computational Science and Engineering Lab, ETH Zurich, Switzerland11 Dept. of Mathematics, Biomedical Engineering, Chemical Engineering
and Materials Science & Center for Complex Biological Systems & ChaoFamily Comprehensive Cancer Center, UC, Irvine, USA
12 Institute for Advanced Study, TUM, GermanyThe supplementary materials are available at http://ieeexplore.ieee.org
I. INTRODUCTION
GLIOBLASTOMA (GBM) is the most aggressive andmost common type of primary brain tumor, with a me-
dian survival of only 15 months despite intensive treatment [1].The standard treatment consists of immediate tumor resection,followed by combined radio- and chemotherapy targeting theresidual tumor. All treatment procedures are guided by mag-netic resonance imaging (MRI). In contrast to most tumors,GBM infiltrates surrounding tissue, instead of forming a tumorwith a well-defined boundary. The central tumor, which isvisible on medical scans, is commonly resected. However,the distribution of the infiltrating residual tumor cells in thenearby healthy-appearing tissue, which are likely to contributeto tumor recurrence, is not known.Current radiotherapy (RT) planning handles these uncertaintiesin a rather rudimentary fashion. Guided by population-levelstudies, standard-of-care RT plans uniformly irradiate thevolume of the visible tumor extended by a uniform margin[1]–[3], which is referred as the clinical target volume (CTV).However, the extent of this margin varies by few centimeterseven across the official RT guidelines [4]. Moreover, GBMinfiltration is anisotropic and thus a uniform margin very likelydoes not provide an optimal dose distribution. In addition,GBM invasiveness is highly patient-specific, and thus not allpatients benefit equally from the same margin, which impairscomparison and advancement of RT protocols.Despite treatment almost all GBMs recur [5]. Biopsies [6] andpost-mortem studies [7] show that tumor cells can invade be-yond the CTV, which reduces RT efficiency, and is a possiblecause of recurrence. At the same time radioresistance of tumorcells inside the CTV can also reduce RT efficiency. Radioresis-tance tends to occur in regions with complex microenviromentand hypoxia [8], both of which are commonly encountered inareas of high tumor cellularity. To address tumor radioresis-tance, several studies have suggested local dose-escalations[8]–[10]. In these approaches, a boosted dose is delivered intoa single or multiple co-centered regions defined by addinguniform margins to the tumor outlines visible in MRI scans[11]. The Radiation Therapy Oncology Group (RTOG) phase-I-trial [10] showed an increase in median survival of 8 monthswith dose-escalation. However, no benefit in progression-
Copyright (c) 2019 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained fromthe IEEE by sending a request to [email protected].
0278-0062 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2019.2902044, IEEETransactions on Medical Imaging
2
free survival was observed, indicating a complex relationshipbetween true progression of the underlying disease and thetumor extent visible in MRI scans.Alternatively, positron emission tomography (PET) scans,which map tumor metabolic activities targeted by specifictracers, can be used to identify radioresistant regions. Apromising tracer used in GBM imaging is 18F-fluoro-ethyl-tyrosine (FET) [12], whose uptake values have been shownto be proportional to tumor cell density, although the con-stant of proportionality is unknown and patient-specific [13],[14]. A prospective phase-II-study [5] demonstrated that dose-escalation based on FET-PET enhancement delineates thetumor structure better than uniform margins, thus leading tolower radiation toxicity. Still, FET-PET based dose-escalationdid not increase progression-free survival. One possible expla-nation is that PET enhances mainly the tumor core, which isusually resected, while the PET uptake values in the remainingtumor periphery coincide with the baseline signal from thehealthy tissue. This, together with a rather low resolution ofPET scans, limits their ability to fully target radioresistanttumor residuals. This is also consistent with our results.Standard RT plans can be improved by incorporating in-formation from computational tumor models. These models,calibrated against patient medical scans, provide estimatesof tumor infiltration that extend the information availablein medical images and can guide personalized RT design.Despite extensive development of tumor growth models [15]–[20] and calibration strategies [21]–[28], their translation intoclinical practice remains very limited. We postulate that thereare (at least) three translational weakness: 1) Most modelcalibrations rely on data not commonly available in clinicalpractice. For example, in [22]–[28] medical scans with visibletumor progression from at least two time points are used forthe model calibration. However, for GBM patients only scansacquired at single preoperative time point are available. 2)Models are based on simplified assumptions motivated byin-vitro studies. For instance, it is frequently assumed thatthe tumor cell density is constant along the tumor bordersvisible on MRI scans (e.g., [21]–[27]). However, the tumorcell density varies significantly along the visible lesion bordersdue to anatomical restrictions and anisotropic tumor growth.3) Even if advanced calibration techniques as in [28] are used,it is not clear how robust the model predictions are and whatbenefits they offer over the standard treatment protocols.Here, we address these translational issues and provide clin-ically relevant patient-specific tumor predictions to improvepersonalized RT design. We present a Bayesian machinelearning framework to calibrate tumor growth models frommultimodal medical scans. We show that an integration of in-formation from complementary structural MRI and functionalFET-PET metabolic maps enables the robust inference of thetumor cell densities from scans acquired at single time point.To the best of our knowledge, this is the first study makingjoint use of FET-PET and MRI scans for the patient-specificcalibration of a tumor growth model. Our Bayesian approachinfers modeling and imaging parameters under uncertaintiesarising from measurement and modeling errors. We propagatethese uncertainties through the computational tumor model
to obtain robust estimates of the tumor cell density togetherwith credible intervals that can be used for personalized RTdesign. The patient-specific tumor estimates offer an advantagein determining margins of CTV as well as regions for dose-escalation. A clinical study is used to assess benefits of thepersonalized RT design over standard treatment protocols.In the remainder of the paper, Section II introduces theBayesian framework for model calibration, including the tu-mor growth and imaging models. The results are presented inSection III where the framework is applied to synthetic andclinical data, followed by a personalized RT study. Conclu-sions are presented in Section IV. Additional technical detailsare given in the Supplementary Materials (SM) available inhttp://ieeexplore.ieee.org.
II. BAYESIAN MODEL CALIBRATION
The Bayesian framework we develop combines a determin-istic model Mu for tumor growth with a stochastic imagingmodel MI relating model predictions with tumor observationsavailable from patient medical scans. Bayes theorem is usedto estimate the probability distribution of the unknown pa-rameters of both models, accounting for modeling and mea-surement uncertainties. Identified parametric uncertainties arepropagated to obtain robust patient-specific tumor predictions.An overview of the framework is given in Fig. 1.
A. Tumor growth model
Many tumor growth models are based on the Fisher-Kolmogrov (FK) equation [23], which captures the main tumorbehaviour: proliferation and infiltration. We use FK equationto describe the tumor model Mu. The equation is solved in apatient-specific brain anatomy reconstructed from MRI scans,where each voxel corresponds to one simulation grid point. LetΩ ∈ R3 be the brain anatomy consisting of white and greymatter and ui(t) ∈ [0, 1] be normalized tumor cell densityat time t and voxel i at location (ix, iy, iz) ∈ Ω, wherei = 1, · · · , N is index across all voxels. The dynamics ofthe tumor cell density u ··= ui(t)Ni=1 is modeled as:
∂u
∂t= ∇ · (D∇u) + ρu (1− u) in Ω, (1)
∇u · ~n = 0 in ∂Ω. (2)
The term ρ [1/day] denotes proliferation rate. The tumorinfiltration into the surrounding tissues is modeled by thetensor D = Di IN
i=1 where I is a 3× 3 identity matrix and
Di =
pwiDw + pgiDg if i ∈ Ω
0 if i /∈ Ω.(3)
The terms pwi and pgi denote percentage of white andgrey matter at voxel i, while Dw and Dg stand for tu-mor infiltration in the corresponding matter. We assumeDw = 10Dg [mm2/day] [28]. The skull and ventricles arenot infiltrated by the tumor cells and act as a domain boundarywith an imposed no-flux boundary condition Eq. (2), where ~nis the outward unit normal to ∂Ω. The tumor is initializedat voxel (icx, icy, icz) and its growth is modeled from timet = 0 until detection time t = T [day]. The parameters
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2019.2902044, IEEETransactions on Medical Imaging
3
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· ·=M
u,M
F1,···,M
FS
are
assu
med
tobe
unkn
own.
A.
Par
amet
ers
estim
atio
nan
dun
cert
aint
ypr
opag
atio
n.A
prio
rpr
obab
ility
dist
ribut
ion
func
tion
(PD
F)P(
◊|M
)is
used
toin
corp
orat
ean
ypr
ior
info
rmat
ion
abou
t◊.
Bay
esia
nm
odel
calib
ratio
nup
date
sth
ispr
ior
info
rmat
ion
base
don
the
avai
labl
eda
taD
.T
heup
date
dpo
ster
ior
isco
mpu
ted
byth
eB
ayes
theo
rem
:P(
◊|D,M
)ÃP(D|◊,M
)·P(
◊|M
),[2
]
whe
reP(D|◊,M
)ist
helik
elih
ood
ofob
serv
ing
data
Dfro
mth
em
odelM
for
agi
ven
valu
eof
◊.
Inm
ost
case
san
anal
ytic
alex
pres
sion
forE
q.(2
)isn
otav
aila
ble
and
sam
plin
gal
gorit
hms
are
used
toob
tain
sam
ple
◊(l
) ,l
œ1,···,N
fro
mth
epo
ste-
rior
P(◊|D,M
).W
eus
eda
Tran
sitio
nalM
arko
vC
hain
Mon
teC
arlo
(TM
CM
C)
sam
plin
gal
gorit
hm(2
3),w
hich
itera
tivel
yco
nstr
ucts
ase
ries
ofin
term
edia
tePD
Fs:
P j(◊|D,M
)≥P(D|◊,M
)pj·P
(◊|M
),[3
]
whe
re0
=p0<p1<
···<
pm
=1
andj
=1,···,m
is
age
nera
tion
inde
x.T
hete
rmpj
cont
rols
the
conv
erge
nce
ofth
esa
mpl
ing
proc
edur
ean
dis
com
pute
dau
tom
atic
ally
byth
eT
MC
MC
algo
rithm
.T
MC
MC
cons
truc
tsa
larg
enu
mbe
rof
inde
pend
ent
chai
nsth
atex
plor
epa
ram
eter
spac
em
ore
eci
ently
than
trad
ition
alsa
mpl
ing
met
hods
(23)
and
allo
wpa
ralle
lexe
cutio
n.A
high
lypa
ralle
lim
plem
enta
tion
ofth
eT
MC
MC
algo
rithm
ispr
ovid
edby
the
4U
fram
ewor
k(2
4).
The
infe
rred
para
met
ricun
cert
aint
ies
are
prop
agat
edth
roug
hth
em
odelM
toob
tain
robu
stpo
ster
ior
pred
ictio
nsab
outu,
give
nby
the
PDF:
P(u|
D,M
)=⁄
P(u|
◊,M
)·P
(◊|D,M
)d◊,
[4]
orby
simpl
ified
mea
sure
ssuc
has
them
eanµu
=E
[u(◊
)]©m
1an
dva
rianc
e‡
2 u=E
[u2 (
◊)]
≠m
2 1©m
2≠m
2 1,de
rived
from
the
first
two
mom
entsm
k,k
=1,
2:
mk
=⁄
! u(◊|M
)" k·P
(◊|D,M
)d◊
¥1 N
N ÿ l=1
! u(◊(l
) |M)" k
,[5
]
whe
re
deno
tes
the
spac
eof
allu
ncer
tain
para
met
ers.
B.
Tum
orgr
owth
mod
el.T
umor
mod
els
ofva
rious
com
plex
ityca
nbe
pers
onal
ized
usin
gth
eB
ayes
ian
appr
oach
desc
ribed
abov
e.H
ere,
we
mod
elu(
◊ u|M
u)
with
the
FKm
odel
whi
chde
scrib
esan
isotr
opic
tum
orgr
owth
inth
ebr
ain
œ
R3
as:
ˆu ˆt
=Ò
·(D
Òu)
+flu
(1≠
u),
in
[6]
Òu·n
=0,
inˆ
,
[7]
whe
reu
=u
i(t
)#
Vox
els
i=1
andui(t
)œ
[0,1
]is
ano
rmal
ized
tum
orce
llde
nsity
attim
et
and
voxe
li=
(ix,iy,iz)œ
.
The
term
fl[1/day]d
enot
estu
mor
prol
ifera
tion
rate
.T
hetu
mor
infil
trat
ion
into
the
surr
ound
ing
tissu
esis
mod
eled
byth
ete
nsor
D=
DiI
#Vox
els
i=1
whe
reI
isa
3◊
3id
entit
ym
atrix
and
Di=
IDg
ifi
œgray,
Dw
ifi
œwhite,
0ifi/œ
gray
fiwhite.
[8]
The
term
sDw
andDg
deno
tetu
mor
infil
trat
ion
inw
hite
and
gray
mat
ter
and
weas
sum
eDw
=10Dg[m
m2 /day](
12).
The
2|
ww
w.p
nas.
org/
cgi/
doi/
10.1
073/
pnas
.XX
XX
XX
XX
XX
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
Lipk
ová
etal
.
Patie
nt-s
peci
fic p
redi
ctio
ns:
std
MA
PM
ean
P(D
|,M
)=
P y
T1G
d|,
M P
yF
LA
IR|,
M P
yF
ET|,
M
I) M
edic
al Im
ages
A)Sa
gitt
al
Coro
nal
FET-
PET
T1G
dFL
AIR
C)
B)
ObservationsModalities
u
=D
,,T
,ic x
,ic y
,ic z
CSFWhite matter Grey matter A)
Imag
ing
para
met
ers:
I=
uT
1G
dc
,uF
LA
IR
c,
↵,b
,
P(y
s|
,M)
=
N Y i=1
P(y
s i|
,ui)
=
N Y i=1
↵y
s ii
·(1↵
i)1
ys i.
Stru
ctur
al s
cans
: s=
T1G
d, F
LAIR
P(y
FE
T|,
M)
=
N Y i=1
P(y
FE
Ti
|,u
i)=
N Y i=1
N(u
i
by
FE
Ti
,2)
Met
abol
ic P
ET s
can:
E) F)D)
M=
Mu,M
I
=
u,
I
A)B)
Dose distribution Dose escalationC)
D)
Recu
rren
ceTr
eatm
ent p
lans
Pers
onal
ized
A)
Stan
dard
B)
T1GdFLAIR F)E)
Para
met
er e
stim
atio
n:
INFI
LTRA
TIO
NPR
OLI
FERA
TIO
N
DRAFT
totu
mor
cell
dens
ity(1
1).
Apr
ospe
ctiv
eph
ase
IIst
udy
(3)
dem
onst
rate
dth
atdo
sees
cala
tion
base
don
FET
-PE
Ten
-ha
ncem
entd
elin
eate
sthe
tum
orst
ruct
ure
bett
erth
anun
iform
mar
gins
,thu
sle
adin
gto
low
erra
diat
ion
toxi
city
.St
ill,i
tdi
dno
tin
crea
seth
epr
ogre
ssio
n-fre
esu
rviv
al.
One
poss
ible
expl
a-na
tion
isth
atFE
Tsc
ansh
owsa
base
line
signa
lalso
inno
rmal
tissu
e,w
hich
toge
ther
with
the
rath
erpo
orre
solu
tion
ofPE
Tlim
itsits
abili
tyto
dist
ingu
ishbe
twee
nhe
alth
yan
dno
rmal
tissu
ein
regi
ons
oflo
win
filtr
atio
nin
the
tum
orpe
riphe
ry.
The
stan
dard
RTpl
ansc
anbe
impr
oved
byin
corp
orat
ing
info
r-m
atio
nfro
mco
mpu
tatio
nalt
umor
mod
els(
12,1
3).M
ostb
rain
tum
orgr
owth
mod
els
are
base
don
the
Fish
er-K
olm
ogor
ov(F
K)
equa
tion
(14)
.T
hese
mod
els,
calib
rate
dag
ains
tpa
tient
med
ical
scan
s,pr
ovid
ees
timat
esof
tum
orin
filtr
atio
nth
atex
tend
the
info
rmat
ion
avai
labl
ein
med
ical
imag
esan
dca
ngu
ide
the
pers
onal
ized
RTde
signs
.D
espi
tegr
eat
adva
nces
inth
ede
sign
oftu
mor
grow
thm
odel
s(1
4–18
)cr
itica
lque
stio
nsre
mai
nsfo
rth
eap
plic
abili
tyof
thes
em
odel
s,su
chas
how
tolin
km
odel
feat
ures
(e.g
.tu
mor
cell
dens
ity)
tocl
inic
alim
age
obse
rvat
ions
.C
urre
ntap
proa
ches
forc
alib
ratin
ggl
iom
am
odel
sco
nsid
erva
rious
dete
rmin
istic
(13,
14,
19,
20)
and
prob
abili
stic
Bay
esia
n(2
1,22
)m
etho
dsfo
res
tabl
ishin
gth
islin
k.In
(13,
14,1
9–21
),au
thor
sas
sum
ea
fixed
tum
orce
llde
nsity
atth
etu
mor
bord
ersv
isibl
ein
T1G
dan
dFL
AIR
scan
san
dth
em
odel
para
met
ers
are
estim
ated
byfit
ting
the
volu
me
(19,
20)
orsh
ape
(13,
21)
ofth
evi
sible
lesio
n.H
owev
er,t
hetu
mor
cell
dens
ityva
ries
alon
gth
evi
sible
tum
orou
tline
sdu
eto
anat
omic
alre
stric
tions
and
inho
mog
eneo
usgr
owth
patt
ern.
AB
ayes
ian
appr
oach
acco
untin
gfo
rvar
ying
tum
orce
llde
nsity
was
prop
osed
in(2
2).
All
thes
eca
libra
tions
howe
ver
use
MR
Isc
ans
from
atle
ast
two
time
poin
ts,m
akin
gth
emun
suita
ble
for
real
clin
ical
sett
ing,
whe
reon
lysc
ans
acqu
ired
atsin
gle
preo
pera
tive
time
poin
tar
eav
aila
ble.
Inth
ispa
per
wepr
esen
ta
syst
emat
icB
ayes
ian
fram
ewor
kto
deve
lop
robu
sttu
mor
grow
thm
odel
sin
form
edfr
omcl
inic
alim
agin
gfe
atur
es.
Our
appr
oach
over
com
eslim
itatio
nssp
ecifi
cto
die
rent
imag
ing
mod
aliti
esan
din
turn
inte
grat
esth
eir
info
rmat
ion
toca
libra
teth
etu
mor
grow
thm
odel
sfo
rth
eRT
plan
ning
.W
eco
mbi
neco
mpl
emen
tary
info
rmat
ion
from
stru
c-tu
ralM
RIa
ndfu
nctio
nalF
ET-P
ETm
etab
olic
map
s,ac
quire
dat
asin
gle
time
poin
t,to
iden
tify
the
patie
nt-s
peci
fictu
mor
cell
dens
ity.
Our
Bay
esia
nap
proa
chin
fers
mod
elin
gan
dim
ag-
ing
para
met
ers
unde
run
cert
aint
ies
arisi
ngfro
mm
easu
rem
ent
and
mod
elin
ger
rors
.W
epr
opag
ate
thes
eun
cert
aint
ies
tom
odel
pred
ictio
nsto
obta
inro
bust
estim
ates
ofth
etu
mor
cell
dens
ityto
geth
erw
ithco
nfide
nce
inte
rval
sth
atca
nbe
used
for
pers
onal
ized
RTpl
anni
ng.
The
estim
ated
patie
nt-s
peci
fictu
mor
cell
dens
ityo
ers
anad
vant
age
inde
term
inin
gm
argi
nsof
the
CT
Vas
wel
las
regi
ons
for
dose
esca
latio
n.A
stud
yco
nduc
ted
ona
smal
lclin
ical
popu
latio
nis
used
toas
sess
the
bene
fits
ofth
epe
rson
aliz
edRT
plan
sov
erst
anda
rdpr
otoc
ols.
1.B
ayes
ian
mod
elpe
rson
aliz
atio
n
The
tum
orce
llde
nsity
u(◊ u|M
u)
isco
mpu
ted
bya
mod
elM
u
with
para
met
ers
◊ u.
The
para
met
ers
◊ uar
epa
tient
-spe
cific
and
apr
obab
ility
dist
ribut
ion
ofth
eir
plau
sible
valu
esis
in-
ferr
edfro
mtu
mor
obse
rvat
ions
(dat
a)D
· ·=y
kS k
=1
obta
ined
fromS
med
ical
scan
s.W
epos
tula
teth
atth
emod
elpr
edic
tions
uan
dea
chim
age
obse
rvat
ion
ykar
ere
late
dby
ast
ocha
stic
imag
ing
mod
elFk
with
para
met
ers
◊ Fk
asyk
=Fk! u! ◊ u
|Mu
" ,Á! ◊ F
k
"",
[1]
whe
reÁ
isa
pred
ictio
ner
rora
ccou
ntin
gfo
rmod
ellin
gan
dm
ea-
sure
men
tun
cert
aint
ies.
Para
met
ers
◊· ·=
◊u,◊F
1,···,◊F
S
ofth
em
odelM
· ·=M
u,M
F1,···,M
FS
are
assu
med
tobe
unkn
own.
A.
Par
amet
ers
estim
atio
nan
dun
cert
aint
ypr
opag
atio
n.A
prio
rpr
obab
ility
dist
ribut
ion
func
tion
(PD
F)P(
◊|M
)is
used
toin
corp
orat
ean
ypr
ior
info
rmat
ion
abou
t◊.
Bay
esia
nm
odel
calib
ratio
nup
date
sth
ispr
ior
info
rmat
ion
base
don
the
avai
labl
eda
taD
.T
heup
date
dpo
ster
ior
isco
mpu
ted
byth
eB
ayes
theo
rem
:P(
◊|D,M
)ÃP(D|◊,M
)·P(
◊|M
),[2
]
whe
reP(D|◊,M
)ist
helik
elih
ood
ofob
serv
ing
data
Dfro
mth
em
odelM
for
agi
ven
valu
eof
◊.
Inm
ost
case
san
anal
ytic
alex
pres
sion
forE
q.(2
)isn
otav
aila
ble
and
sam
plin
gal
gorit
hms
are
used
toob
tain
sam
ple
◊(l
) ,l
œ1,···,N
fro
mth
epo
ste-
rior
P(◊|D,M
).W
eus
eda
Tran
sitio
nalM
arko
vC
hain
Mon
teC
arlo
(TM
CM
C)
sam
plin
gal
gorit
hm(2
3),w
hich
itera
tivel
yco
nstr
ucts
ase
ries
ofin
term
edia
tePD
Fs:
P j(◊|D,M
)≥P(D|◊,M
)pj·P
(◊|M
),[3
]
whe
re0
=p0<p1<
···<
pm
=1
andj
=1,···,m
is
age
nera
tion
inde
x.T
hete
rmpj
cont
rols
the
conv
erge
nce
ofth
esa
mpl
ing
proc
edur
ean
dis
com
pute
dau
tom
atic
ally
byth
eT
MC
MC
algo
rithm
.T
MC
MC
cons
truc
tsa
larg
enu
mbe
rof
inde
pend
ent
chai
nsth
atex
plor
epa
ram
eter
spac
em
ore
eci
ently
than
trad
ition
alsa
mpl
ing
met
hods
(23)
and
allo
wpa
ralle
lexe
cutio
n.A
high
lypa
ralle
lim
plem
enta
tion
ofth
eT
MC
MC
algo
rithm
ispr
ovid
edby
the
4U
fram
ewor
k(2
4).
The
infe
rred
para
met
ricun
cert
aint
ies
are
prop
agat
edth
roug
hth
em
odelM
toob
tain
robu
stpo
ster
ior
pred
ictio
nsab
outu,
give
nby
the
PDF:
P(u|
D,M
)=⁄
P(u|
◊,M
)·P
(◊|D,M
)d◊,
[4]
orby
simpl
ified
mea
sure
ssuc
has
them
eanµu
=E
[u(◊
)]©m
1an
dva
rianc
e‡
2 u=E
[u2 (
◊)]
≠m
2 1©m
2≠m
2 1,de
rived
from
the
first
two
mom
entsm
k,k
=1,
2:
mk
=⁄
! u(◊|M
)" k·P
(◊|D,M
)d◊
¥1 N
N ÿ l=1
! u(◊(l
) |M)" k
,[5
]
whe
re
deno
tes
the
spac
eof
allu
ncer
tain
para
met
ers.
B.
Tum
orgr
owth
mod
el.T
umor
mod
els
ofva
rious
com
plex
ityca
nbe
pers
onal
ized
usin
gth
eB
ayes
ian
appr
oach
desc
ribed
abov
e.H
ere,
we
mod
elu(
◊ u|M
u)
with
the
FKm
odel
whi
chde
scrib
esan
isotr
opic
tum
orgr
owth
inth
ebr
ain
œ
R3
as:
ˆu ˆt
=Ò
·(D
Òu)
+flu
(1≠
u),
in
[6]
Òu·n
=0,
inˆ
,
[7]
whe
reu
=u
i(t
)#
Vox
els
i=1
andui(t
)œ
[0,1
]is
ano
rmal
ized
tum
orce
llde
nsity
attim
et
and
voxe
li=
(ix,iy,iz)œ
.
The
term
fl[1/day]d
enot
estu
mor
prol
ifera
tion
rate
.T
hetu
mor
infil
trat
ion
into
the
surr
ound
ing
tissu
esis
mod
eled
byth
ete
nsor
D=
DiI
#Vox
els
i=1
whe
reI
isa
3◊
3id
entit
ym
atrix
and
Di=
IDg
ifi
œgray,
Dw
ifi
œwhite,
0ifi/œ
gray
fiwhite.
[8]
The
term
sDw
andDg
deno
tetu
mor
infil
trat
ion
inw
hite
and
gray
mat
ter
and
weas
sum
eDw
=10Dg[m
m2 /day](
12).
The
2|
ww
w.p
nas.
org/
cgi/
doi/
10.1
073/
pnas
.XX
XX
XX
XX
XX
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
Lipk
ová
etal
.
Di=
(p
wiD
w+
pg
iD
gif
i2
0if
i/2
.
FLA
IRFE
T-PE
TT1
Gd
Observations
b
u
Ref
eren
ces
1.St
upp,
R.e
tal.
Hig
h-gr
ade
glio
ma:
Esm
ocl
inic
alpr
actic
egu
idel
ines
ford
iagn
osis
,tre
atm
enta
ndfo
llow
-up.
Anna
lsO
ncol
.(2
014)
.
2.B
urne
t,N
.G.,
Tho
mas
,S.J
.,B
urto
n,K
.E.&
Jeff
erie
s,S.
J.D
efini
ngth
etu
mou
ran
dta
rget
volu
mes
for
radi
othe
rapy
.C
ance
rIm
agin
g4,
153–
161
(200
4).
3.St
upp,
R.e
tal.
Rad
ioth
erap
ypl
usco
ncom
itant
and
adju
vant
tem
ozol
omid
efo
rglio
blas
tom
a.N
ewE
ngl.
J.M
edic
ine
352,
987–
996
(200
5).
4.Pa
ulss
on,A
.K.e
tal.
Lim
ited
mar
gins
usin
gm
oder
nra
diot
hera
pyte
chni
ques
does
noti
ncre
ase
mar
gina
lfai
lure
rate
ofgl
iobl
asto
ma.
Am
.jou
rnal
clin
ical
onco
logy
37,1
77(2
014)
.
5.R
ockw
ell,
S.,D
obru
cki,
I.T.
,Kim
,E.Y
.,M
arris
on,S
.T.&
Vu,
V.T.
Hyp
oxia
and
radi
atio
nth
erap
y:pa
sthi
stor
y,on
goin
gre
sear
ch,a
ndfu
ture
prom
ise.
Cur
r.m
olec
ular
med
icin
e9
(200
9).
2/2
x
Ref
eren
ces
1.St
upp,
R.e
tal.
Hig
h-gr
ade
glio
ma:
Esm
ocl
inic
alpr
actic
egu
idel
ines
ford
iagn
osis
,tre
atm
enta
ndfo
llow
-up.
Ann
als
Onc
ol.
(201
4).
2.B
urne
t,N
.G.,
Tho
mas
,S.J
.,B
urto
n,K
.E.&
Jeff
erie
s,S.
J.D
efini
ngth
etu
mou
ran
dta
rget
volu
mes
for
radi
othe
rapy
.C
ance
rIm
agin
g4,
153–
161
(200
4).
3.St
upp,
R.e
tal.
Rad
ioth
erap
ypl
usco
ncom
itant
and
adju
vant
tem
ozol
omid
efo
rglio
blas
tom
a.N
ewE
ngl.
J.M
edic
ine
352,
987–
996
(200
5).
4.Pa
ulss
on,A
.K.e
tal.
Lim
ited
mar
gins
usin
gm
oder
nra
diot
hera
pyte
chni
ques
does
noti
ncre
ase
mar
gina
lfai
lure
rate
ofgl
iobl
asto
ma.
Am
.jou
rnal
clin
ical
onco
logy
37,1
77(2
014)
.
5.R
ockw
ell,
S.,D
obru
cki,
I.T.
,Kim
,E.Y
.,M
arri
son,
S.T.
&V
u,V.
T.H
ypox
iaan
dra
diat
ion
ther
apy:
past
hist
ory,
ongo
ing
rese
arch
,and
futu
repr
omis
e.C
urr.
mol
ecul
arm
edic
ine
9(2
009)
.
2/2
u
Ref
eren
ces
1.St
upp,
R.e
tal.
Hig
h-gr
ade
glio
ma:
Esm
ocl
inic
alpr
actic
egu
idel
ines
ford
iagn
osis
,tre
atm
enta
ndfo
llow
-up.
Ann
als
Onc
ol.
(201
4).
2.B
urne
t,N
.G.,
Tho
mas
,S.J
.,B
urto
n,K
.E.&
Jeff
erie
s,S.
J.D
efini
ngth
etu
mou
ran
dta
rget
volu
mes
for
radi
othe
rapy
.C
ance
rIm
agin
g4,
153–
161
(200
4).
3.St
upp,
R.e
tal.
Rad
ioth
erap
ypl
usco
ncom
itant
and
adju
vant
tem
ozol
omid
efo
rglio
blas
tom
a.N
ewE
ngl.
J.M
edic
ine
352,
987–
996
(200
5).
4.Pa
ulss
on,A
.K.e
tal.
Lim
ited
mar
gins
usin
gm
oder
nra
diot
hera
pyte
chni
ques
does
noti
ncre
ase
mar
gina
lfai
lure
rate
ofgl
iobl
asto
ma.
Am
.jou
rnal
clin
ical
onco
logy
37,1
77(2
014)
.
5.R
ockw
ell,
S.,D
obru
cki,
I.T.
,Kim
,E.Y
.,M
arri
son,
S.T.
&V
u,V.
T.H
ypox
iaan
dra
diat
ion
ther
apy:
past
hist
ory,
ongo
ing
rese
arch
,and
futu
repr
omis
e.C
urr.
mol
ecul
arm
edic
ine
9(2
009)
.
2/2
x
Ref
eren
ces
1.St
upp,
R.e
tal.
Hig
h-gr
ade
glio
ma:
Esm
ocl
inic
alpr
actic
egu
idel
ines
ford
iagn
osis
,tre
atm
enta
ndfo
llow
-up.
Anna
lsO
ncol
.(2
014)
.
2.B
urne
t,N
.G.,
Tho
mas
,S.J
.,B
urto
n,K
.E.&
Jeff
erie
s,S.
J.D
efini
ngth
etu
mou
ran
dta
rget
volu
mes
for
radi
othe
rapy
.C
ance
rIm
agin
g4,
153–
161
(200
4).
3.St
upp,
R.e
tal.
Rad
ioth
erap
ypl
usco
ncom
itant
and
adju
vant
tem
ozol
omid
efo
rglio
blas
tom
a.N
ewE
ngl.
J.M
edic
ine
352,
987–
996
(200
5).
4.Pa
ulss
on,A
.K.e
tal.
Lim
ited
mar
gins
usin
gm
oder
nra
diot
hera
pyte
chni
ques
does
noti
ncre
ase
mar
gina
lfai
lure
rate
ofgl
iobl
asto
ma.
Am
.jou
rnal
clin
ical
onco
logy
37,1
77(2
014)
.
5.R
ockw
ell,
S.,D
obru
cki,
I.T.
,Kim
,E.Y
.,M
arris
on,S
.T.&
Vu,
V.T.
Hyp
oxia
and
radi
atio
nth
erap
y:pa
sthi
stor
y,on
goin
gre
sear
ch,a
ndfu
ture
prom
ise.
Cur
r.m
olec
ular
med
icin
e9
(200
9).
2/2
JOU
RN
AL
OF
LAT E
XC
LASS
FILE
S,V
OL.
14,N
O.8
,AU
GU
ST20
153
AB
CD
T1G
dFL
AIR
FET
-PE
T
b
uT c
uF c
u
xx
xx
uu
u
uF c
Figu
re 1
; Fin
al p
lot
uT1G
dc
uFLA
IRc
Fig.
1.Sc
hem
atic
illus
tratin
gth
ere
latio
nbe
twee
nth
eim
age
obse
rvat
ions
D(to
p)an
dth
etu
mor
cell
dens
ityu
(bot
tom
).A
:A
ctua
lFL
AIR
scan
and
sche
mat
icof
the
tum
orce
llde
nsity
alon
gth
elin
ein
dica
ted
byth
ebl
ack
arro
w.
B,C
:B
inar
yse
gmen
tatio
nsof
the
visi
ble
tum
oran
dth
eir
rela
tion
toth
eun
know
nth
resh
old
valu
esu
T1G
dc
,uF
LAIR
c.D
:FE
T-PE
Tm
etab
olic
map
and
the
unkn
own
cons
tant
ofpr
opor
tiona
lity
b.
uT1
Gd
c
uF
LAIR
c B.
Mul
timod
alim
agin
gm
odel
We
extra
ctth
etu
mor
info
rmat
ion
from
T1G
dan
dFL
AIR
MR
Ian
dm
etab
olic
FET-
PET
map
s.Th
eM
RI
scan
spr
ovid
em
orph
olog
ical
info
rmat
ion
abou
tth
evi
sibl
etu
mor
inth
efo
rmof
bina
ryse
gmen
tatio
ns.T
heFE
Tsi
gnal
ispr
opor
tiona
lto
tum
orce
llde
nsity
with
anun
know
nco
nsta
ntof
prop
or-
tiona
lity
[11]
.A
rela
tion
betw
een
the
imag
eob
serv
atio
nsD
=y
T1G
d,y
FLA
IR,y
FE
T
and
tum
orce
llde
nsity
uis
sket
ched
inFi
g.1.
Sinc
eea
chof
the
scan
sca
ptur
ea
diff
eren
tphy
siol
ogic
alpr
oces
s,th
eob
serv
atio
nsar
eas
sum
edin
depe
nden
tan
dth
elik
elih
ood
can
beex
pres
sas
:
P(D
|,M
)=
P(y
T1G
d|,
M)P
(yF
LAIR|,
M)P
(yF
ET|,
M).
Ast
ocha
stic
imag
ing
mod
elF
kis
used
tode
fine
the
likel
ihoo
dfu
nctio
nfo
rea
chm
odal
ityyk
.Th
ebi
nary
segm
enta
tions
ys,s
2T
1Gd,
FLA
IR,
assi
gna
labe
ly
s i=
1to
each
voxe
lw
ithvi
sibl
etu
mor
and
ys i
=0
othe
rwis
e.Th
epr
obab
ility
ofob
serv
ing
ase
gmen
tatio
nys
with
asi
mul
ated
tum
orce
llde
nsity
uis
assu
med
asa
Ber
noul
lidi
strib
utio
n[2
2]:
P(ys
|,M
)=
Nd ys
Y i=1
P(y
s i|
,ui)
=
Nd ys
Y i=1
↵y
s ii
·(1
↵i)1
y
s i.
(4)
Her
e↵
iis
the
prob
abili
tyof
obse
rvin
gth
etu
mor
inth
eM
RI
scan
and
itis
assu
med
asa
doub
lelo
gist
icsi
gmoi
d:
↵i(u
i,u
s c)
=0.5
+0.5
·sig
n(u
i
us c)
1
e(u
i
us c)2
2 ↵
!,
(5)
whe
reu
s cde
note
san
unkn
own
cell
dens
ityth
resh
old
be-
low
whi
chtu
mor
indu
ced
chan
ges
are
not
visi
ble
inth
eM
RI
scan
,w
hile
the
term
2 ↵
repr
esen
tsun
certa
inty
inu
s c.
The
para
met
erN
d ys
deno
tes
the
set
ofvo
xels
used
for
the
likel
ihoo
dco
mpu
tatio
n.Th
eM
RI
scan
sw
ere
acqu
ired
atun
iform
1m
mre
solu
tion.
Toco
pew
ithpo
tent
ial
corr
elat
ions
betw
een
neig
hbou
ring
voxe
ls,
only
ever
yse
cond
voxe
lis
incl
uded
inN
d ys,
intro
duci
nga
corr
elat
ion
leng
thd
=2
mm
.Th
ebi
nary
segm
enta
tions
have
thre
eun
know
npa
ram
eter
s
FT1
Gd,
FF
LAIR
=u
T1G
dc
,uF
LAIR
c,
2 ↵.A
norm
aliz
edF
ET
sign
al
yFE
Tha
sco
ntin
uous
valu
esy
FE
Ti
2[0
,1]
whi
chca
nbe
rela
ted
with
the
tum
orde
nsity
ui
bya
Gau
ssia
ndi
strib
utio
n:
P(yF
ET|,
M)
=
Nd y
FE
TY i=
1
P(y
FE
Ti
|,u
i)=
Nd y
FE
TY i=
1
N(u
i
by
FE
Ti
,
2),
whe
reb
isan
unkn
own
cons
tant
ofpr
opor
tiona
lity
and
de
scrib
esth
eno
ise
inth
eFE
Tsc
an.
Toel
imin
ate
sign
alfr
oma
heal
thy
and
necr
otic
tissu
e,N
d yF
ET
cons
ists
ofth
evo
xels
incl
uded
inth
eT1
Gd
and
FLA
IRtu
mor
segm
enta
tions
,ex
clud
ing
the
tum
orne
crot
icco
re.
The
FET
scan
,ac
quire
dat
4.3
mm
reso
lutio
n,is
inte
rpol
ated
to1
mm
reso
lutio
n,le
adin
gto
aco
rrel
atio
nle
ngth
d=
4 .3
mm
.Th
eFE
Tsc
anin
trodu
ces
two
addi
tiona
lunk
now
npa
ram
eter
s F
FE
T=
b,
.Th
eun
know
npa
ram
eter
s
=
u,
FT1
Gd,
FF
LAIR,
FF
ET
are
assu
med
inde
pend
entw
ithun
iform
prio
rdi
strib
utio
nP
(|M
)gi
ven
inSu
pple
men
tary
Info
rmat
ion
(SI)
.The
tum
orce
llde
n-si
tyu( u
|Mu)
isco
mpu
ted
bya
mod
elM
uw
ithpa
ram
eter
s u
.Th
epa
ram
eter
s u
are
patie
nt-s
peci
fican
da
prob
abili
tydi
strib
utio
nof
thei
rpl
ausi
ble
valu
esis
infe
rred
from
tum
orob
serv
atio
ns(d
ata)
D· ·=
ykS k
=1
obta
ined
from
Sm
edic
alsc
ans.
We
post
ulat
eth
atth
em
odel
pred
ictio
nsu
and
each
imag
eob
serv
atio
nyk
are
rela
ted
bya
stoc
hast
icim
agin
gm
odel
Fk
with
para
met
ers F
kas
yk=
Fk u u
|Mu
," F
k
,
(6)
whe
re"
isa
pred
ictio
ner
ror
acco
untin
gfo
rm
od-
ellin
gan
dm
easu
rem
ent
unce
rtain
ties.
Para
met
ers
· ·=
u,
F1,·
··,
FSo
fthe
mod
elM
· ·=M
u,M
F1,·
··,M
FS
are
assu
med
tobe
unkn
own.
C.
Para
met
ers
estim
atio
nan
dun
cert
aint
ypr
opag
atio
n
Apr
ior
prob
abili
tydi
strib
utio
nfu
nctio
n(P
DF)
P(|M
)is
used
toin
corp
orat
ean
ypr
ior
info
rmat
ion
abou
t.
Bay
esia
nm
odel
calib
ratio
nup
date
sth
ispr
ior
info
rmat
ion
base
don
the
avai
labl
eda
taD
.The
upda
ted
post
erio
rPD
Fis
com
pute
dby
the
Bay
esth
eore
m:
P(|
D,M
)/
P(D
|,M
)·P
(|M
),(7
)
whe
reP(
D|,
M)
isth
elik
elih
ood
ofob
serv
ing
data
Dfr
omth
em
odel
Mfo
ra
give
nva
lue
of.
Inm
ost
case
san
anal
ytic
alex
pres
sion
for
Eq.(
7)is
nota
vaila
ble
and
sam
plin
gal
gorit
hms
are
used
toob
tain
sam
ple(
l),
l2
1,·
··,N
fr
omth
epo
ster
iorP
(|D
,M).
We
used
aTr
ansi
tiona
lMar
kov
Cha
inM
onte
Car
lo(T
MC
MC
)sam
plin
gal
gorit
hm[2
3],w
hich
itera
tivel
yco
nstru
cts
ase
ries
ofin
term
edia
tePD
Fs:
P j(
|D,M
)
P(D
|,M
)pj
·P(
|M),
(8)
whe
re0
=p 0
<p 1
<··
·<p m
=1
and
j=
1,·
··,m
is
age
nera
tion
inde
x.Th
ete
rmp j
cont
rols
the
conv
erge
nce
ofth
esa
mpl
ing
proc
edur
ean
dis
com
pute
dau
tom
atic
ally
byth
eTM
CM
Cal
gorit
hm.
TMC
MC
cons
truct
sa
larg
enu
mbe
rof
inde
pend
ent
chai
nsth
atex
plor
epa
ram
eter
spac
em
ore
effic
ient
lyth
antra
ditio
nal
sam
plin
gm
etho
ds[2
3]an
dal
low
para
llel
exec
utio
n.A
high
lypa
ralle
lim
plem
enta
tion
ofth
eTM
CM
Cal
gorit
hmis
prov
ided
byth
e
4Ufr
amew
ork
[24]
.Th
ein
ferr
edpa
ram
etric
unce
rtain
ties
are
prop
agat
edth
roug
h
u
Ref
eren
ces
1.St
upp,
R.e
tal.
Hig
h-gr
ade
glio
ma:
Esm
ocl
inic
alpr
actic
egu
idel
ines
ford
iagn
osis
,tre
atm
enta
ndfo
llow
-up.
Anna
lsO
ncol
.(2
014)
.
2.B
urne
t,N
.G.,
Tho
mas
,S.J
.,B
urto
n,K
.E.&
Jeff
erie
s,S.
J.D
efini
ngth
etu
mou
ran
dta
rget
volu
mes
for
radi
othe
rapy
.C
ance
rIm
agin
g4,
153–
161
(200
4).
3.St
upp,
R.e
tal.
Rad
ioth
erap
ypl
usco
ncom
itant
and
adju
vant
tem
ozol
omid
efo
rglio
blas
tom
a.N
ewE
ngl.
J.M
edic
ine
352,
987–
996
(200
5).
4.Pa
ulss
on,A
.K.e
tal.
Lim
ited
mar
gins
usin
gm
oder
nra
diot
hera
pyte
chni
ques
does
noti
ncre
ase
mar
gina
lfai
lure
rate
ofgl
iobl
asto
ma.
Am
.jou
rnal
clin
ical
onco
logy
37,1
77(2
014)
.
5.R
ockw
ell,
S.,D
obru
cki,
I.T.
,Kim
,E.Y
.,M
arris
on,S
.T.&
Vu,
V.T.
Hyp
oxia
and
radi
atio
nth
erap
y:pa
sthi
stor
y,on
goin
gre
sear
ch,a
ndfu
ture
prom
ise.
Cur
r.m
olec
ular
med
icin
e9
(200
9).
2/2
x
Ref
eren
ces
1.St
upp,
R.e
tal.
Hig
h-gr
ade
glio
ma:
Esm
ocl
inic
alpr
actic
egu
idel
ines
ford
iagn
osis
,tre
atm
enta
ndfo
llow
-up.
Ann
als
Onc
ol.
(201
4).
2.B
urne
t,N
.G.,
Tho
mas
,S.J
.,B
urto
n,K
.E.&
Jeff
erie
s,S.
J.D
efini
ngth
etu
mou
ran
dta
rget
volu
mes
for
radi
othe
rapy
.C
ance
rIm
agin
g4,
153–
161
(200
4).
3.St
upp,
R.e
tal.
Rad
ioth
erap
ypl
usco
ncom
itant
and
adju
vant
tem
ozol
omid
efo
rglio
blas
tom
a.N
ewE
ngl.
J.M
edic
ine
352,
987–
996
(200
5).
4.Pa
ulss
on,A
.K.e
tal.
Lim
ited
mar
gins
usin
gm
oder
nra
diot
hera
pyte
chni
ques
does
noti
ncre
ase
mar
gina
lfai
lure
rate
ofgl
iobl
asto
ma.
Am
.jou
rnal
clin
ical
onco
logy
37,1
77(2
014)
.
5.R
ockw
ell,
S.,D
obru
cki,
I.T.
,Kim
,E.Y
.,M
arri
son,
S.T.
&V
u,V.
T.H
ypox
iaan
dra
diat
ion
ther
apy:
past
hist
ory,
ongo
ing
rese
arch
,and
futu
repr
omis
e.C
urr.
mol
ecul
arm
edic
ine
9(2
009)
.
2/2
JOU
RN
AL
OF
LAT E
XC
LA
SSFI
LE
S,V
OL
.14,
NO
.8,A
UG
UST
2015
3
AB
CD
T1G
dFL
AIR
FET
-PE
T
b
uT c
uF c
u
xx
xx
uu
u
uF c
Figu
re 1
; Fin
al p
lot
uT1G
dc
uFLA
IRc
Fig.
1.Sc
hem
atic
illus
trat
ing
the
rela
tion
betw
een
the
imag
eob
serv
atio
nsD
(top
)an
dth
etu
mor
cell
dens
ityu
(bot
tom
).A
:A
ctua
lFL
AIR
scan
and
sche
mat
icof
the
tum
orce
llde
nsity
alon
gth
elin
ein
dica
ted
byth
ebl
ack
arro
w.
B,C
:B
inar
yse
gmen
tatio
nsof
the
visi
ble
tum
oran
dth
eir
rela
tion
toth
eun
know
nth
resh
old
valu
esu
T1G
dc
,uF
LAIR
c.D
:FE
T-PE
Tm
etab
olic
map
and
the
unkn
own
cons
tant
ofpr
opor
tiona
lity
b.
uT1
Gd
c
uF
LAIR
c B.
Mul
timod
alim
agin
gm
odel
We
extr
act
the
tum
orin
form
atio
nfr
omT
1Gd
and
FLA
IRM
RI
and
met
abol
icFE
T-PE
Tm
aps.
The
MR
Isc
ans
prov
ide
mor
phol
ogic
alin
form
atio
nab
out
the
visi
ble
tum
orin
the
form
ofbi
nary
segm
enta
tions
.The
FET
sign
alis
prop
ortio
nal
totu
mor
cell
dens
ityw
ithan
unkn
own
cons
tant
ofpr
opor
-tio
nalit
y[1
1].
Are
latio
nbe
twee
nth
eim
age
obse
rvat
ions
D=
yT1
Gd,y
FLA
IR,y
FE
T
and
tum
orce
llde
nsity
uis
sket
ched
inFi
g.1.
Sinc
eea
chof
the
scan
sca
ptur
ea
diff
eren
tphy
siol
ogic
alpr
oces
s,th
eob
serv
atio
nsar
eas
sum
edin
depe
nden
tan
dth
elik
elih
ood
can
beex
pres
sas
:
P(D
|,M
)=
P(y
T1G
d|,
M)P
(yF
LAIR|,
M)P
(yF
ET|,
M).
Ast
ocha
stic
imag
ing
mod
elF
kis
used
tode
fine
the
likel
ihoo
dfu
nctio
nfo
rea
chm
odal
ityyk
.Th
ebi
nary
segm
enta
tions
ys,s
2T
1Gd,
FLA
IR,
assi
gna
labe
ly
s i=
1to
each
voxe
lw
ithvi
sibl
etu
mor
and
ys i
=0
othe
rwis
e.T
hepr
obab
ility
ofob
serv
ing
ase
gmen
tatio
nys
with
asi
mul
ated
tum
orce
llde
nsity
uis
assu
med
asa
Ber
noul
lidi
stri
butio
n[2
2]:
P(ys
|,M
)=
Nd ys
Y i=1
P(y
s i|
,ui)
=
Nd ys
Y i=1
↵y
s ii
·(1
↵i)1
y
s i.
(4)
Her
e↵
iis
the
prob
abili
tyof
obse
rvin
gth
etu
mor
inth
eM
RI
scan
and
itis
assu
med
asa
doub
lelo
gist
icsi
gmoi
d:
↵i(u
i,u
s c)
=0.5
+0.5
·sig
n(u
i
us c)
1
e(u
i
us c)2
2 ↵
!,
(5)
whe
reu
s cde
note
san
unkn
own
cell
dens
ityth
resh
old
be-
low
whi
chtu
mor
indu
ced
chan
ges
are
not
visi
ble
inth
eM
RI
scan
,w
hile
the
term
2 ↵
repr
esen
tsun
cert
aint
yin
us c.
The
para
met
erN
d ys
deno
tes
the
set
ofvo
xels
used
for
the
likel
ihoo
dco
mpu
tatio
n.T
heM
RI
scan
sw
ere
acqu
ired
atun
ifor
m1
mm
reso
lutio
n.To
cope
with
pote
ntia
lco
rrel
atio
nsbe
twee
nne
ighb
ouri
ngvo
xels
,on
lyev
ery
seco
ndvo
xel
isin
clud
edin
Nd ys,
intr
oduc
ing
aco
rrel
atio
nle
ngth
d=
2m
m.
The
bina
ryse
gmen
tatio
nsha
veth
ree
unkn
own
para
met
ers
F
T1G
d,
FF
LAIR
=u
T1G
dc
,uF
LAIR
c,
2 ↵.A
norm
aliz
edF
ET
sign
al
yFE
Tha
sco
ntin
uous
valu
esy
FE
Ti
2[0
,1]
whi
chca
nbe
rela
ted
with
the
tum
orde
nsity
ui
bya
Gau
ssia
ndi
stri
butio
n:
P(yF
ET|,
M)
=
Nd y
FE
TY i=
1
P(y
FE
Ti
|,u
i)=
Nd y
FE
TY i=
1
N(u
i
by
FE
Ti
,
2),
whe
reb
isan
unkn
own
cons
tant
ofpr
opor
tiona
lity
and
de
scri
bes
the
nois
ein
the
FET
scan
.To
elim
inat
esi
gnal
from
ahe
alth
yan
dne
crot
ictis
sue,
Nd y
FE
Tco
nsis
tsof
the
voxe
lsin
clud
edin
the
T1G
dan
dFL
AIR
tum
orse
gmen
tatio
ns,
excl
udin
gth
etu
mor
necr
otic
core
.T
heFE
Tsc
an,
acqu
ired
at4.
3m
mre
solu
tion,
isin
terp
olat
edto
1m
mre
solu
tion,
lead
ing
toa
corr
elat
ion
leng
thd
=4 .
3m
m.
The
FET
scan
intr
oduc
estw
oad
ditio
nalu
nkno
wn
para
met
ers F
FE
T=
b,
.T
heun
know
npa
ram
eter
s
=
u,
FT1
Gd,
FF
LAIR,
FF
ET
are
assu
med
inde
pend
entw
ithun
ifor
mpr
ior
dist
ribu
tion
P(
|M)
give
nin
Supp
lem
enta
ryIn
form
atio
n(S
I).T
hetu
mor
cell
den-
sity
u( u
|Mu)
isco
mpu
ted
bya
mod
elM
uw
ithpa
ram
eter
s u
.T
hepa
ram
eter
s u
are
patie
nt-s
peci
fican
da
prob
abili
tydi
stri
butio
nof
thei
rpl
ausi
ble
valu
esis
infe
rred
from
tum
orob
serv
atio
ns(d
ata)
D· ·=
ykS k
=1
obta
ined
from
Sm
edic
alsc
ans.
We
post
ulat
eth
atth
em
odel
pred
ictio
nsu
and
each
imag
eob
serv
atio
nyk
are
rela
ted
bya
stoc
hast
icim
agin
gm
odel
Fk
with
para
met
ers F
kas
yk=
Fk u u
|Mu
," F
k
,
(6)
whe
re"
isa
pred
ictio
ner
ror
acco
untin
gfo
rm
od-
ellin
gan
dm
easu
rem
ent
unce
rtai
ntie
s.Pa
ram
eter
s
· ·=
u,
F1,·
··,
FSo
fthe
mod
elM
· ·=M
u,M
F1,·
··,M
FS
are
assu
med
tobe
unkn
own.
C.
Para
met
ers
estim
atio
nan
dun
cert
aint
ypr
opag
atio
n
Apr
ior
prob
abili
tydi
stri
butio
nfu
nctio
n(P
DF)
P(|M
)is
used
toin
corp
orat
ean
ypr
ior
info
rmat
ion
abou
t.
Bay
esia
nm
odel
calib
ratio
nup
date
sth
ispr
ior
info
rmat
ion
base
don
the
avai
labl
eda
taD
.The
upda
ted
post
erio
rPD
Fis
com
pute
dby
the
Bay
esth
eore
m:
P(|D
,M
)/
P(D
|,M
)·P
(|M
),(7
)
whe
reP(
D|,
M)
isth
elik
elih
ood
ofob
serv
ing
data
Dfr
omth
em
odel
Mfo
ra
give
nva
lue
of.
Inm
ost
case
san
anal
ytic
alex
pres
sion
for
Eq.
(7)
isno
tava
ilabl
ean
dsa
mpl
ing
algo
rith
ms
are
used
toob
tain
sam
ple(
l),
l2
1,·
··,N
fr
omth
epo
ster
ior
P(|D
,M).
We
used
aTr
ansi
tiona
lMar
kov
Cha
inM
onte
Car
lo(T
MC
MC
)sam
plin
gal
gori
thm
[23]
,whi
chite
rativ
ely
cons
truc
tsa
seri
esof
inte
rmed
iate
PDFs
:
P j(
|D,M
)
P(D
|,M
)pj
·P(
|M),
(8)
whe
re0
=p 0
<p 1
<··
·<p m
=1
and
j=
1,·
··,m
is
age
nera
tion
inde
x.T
hete
rmp j
cont
rols
the
conv
erge
nce
ofth
esa
mpl
ing
proc
edur
ean
dis
com
pute
dau
tom
atic
ally
byth
eT
MC
MC
algo
rith
m.
TM
CM
Cco
nstr
ucts
ala
rge
num
ber
ofin
depe
nden
tch
ains
that
expl
ore
para
met
ersp
ace
mor
eef
ficie
ntly
than
trad
ition
alsa
mpl
ing
met
hods
[23]
and
allo
wpa
ralle
lex
ecut
ion.
Ahi
ghly
para
llel
impl
emen
tatio
nof
the
TM
CM
Cal
gori
thm
ispr
ovid
edby
the
4Ufr
amew
ork
[24]
.T
hein
ferr
edpa
ram
etri
cun
cert
aint
ies
are
prop
agat
edth
roug
h
POST
ERIO
RLI
KELI
HO
OD
PRIO
R
DRAFTto
tum
orce
llde
nsity
(11)
.A
pros
pect
ive
phas
eII
stud
y(3
)de
mon
stra
ted
that
dose
esca
latio
nba
sed
onFE
T-P
ET
en-
hanc
emen
tdel
inea
test
hetu
mor
stru
ctur
ebe
tter
than
unifo
rmm
argi
ns,t
hus
lead
ing
tolo
wer
radi
atio
nto
xici
ty.
Still
,it
did
not
incr
ease
the
prog
ress
ion-
free
surv
ival
.O
nepo
ssib
leex
pla-
natio
nis
that
FET
scan
show
saba
selin
esig
nala
lsoin
norm
altis
sue,
whi
chto
geth
erw
ithth
era
ther
poor
reso
lutio
nof
PET
limits
itsab
ility
todi
stin
guish
betw
een
heal
thy
and
norm
altis
sue
inre
gion
sof
low
infil
trat
ion
inth
etu
mor
perip
hery
.T
hest
anda
rdRT
plan
scan
beim
prov
edby
inco
rpor
atin
gin
for-
mat
ion
from
com
puta
tiona
ltum
orm
odel
s(12
,13)
.Mos
tbra
intu
mor
grow
thm
odel
sar
eba
sed
onth
eFi
sher
-Kol
mog
orov
(FK
)eq
uatio
n(1
4).
The
sem
odel
s,ca
libra
ted
agai
nst
patie
ntm
edic
alsc
ans,
prov
ide
estim
ates
oftu
mor
infil
trat
ion
that
exte
ndth
ein
form
atio
nav
aila
ble
inm
edic
alim
ages
and
can
guid
eth
epe
rson
aliz
edRT
desig
ns.
Des
pite
grea
tad
vanc
esin
the
desig
nof
tum
orgr
owth
mod
els
(14–
18)
criti
calq
uest
ions
rem
ains
for
the
appl
icab
ility
ofth
ese
mod
els,
such
asho
wto
link
mod
elfe
atur
es(e
.g.
tum
orce
llde
nsity
)to
clin
ical
imag
eob
serv
atio
ns.
Cur
rent
appr
oach
esfo
rcal
ibra
ting
glio
ma
mod
els
cons
ider
vario
usde
term
inist
ic(1
3,14
,19
,20
)an
dpr
obab
ilist
icB
ayes
ian
(21,
22)
met
hods
for
esta
blish
ing
this
link.
In(1
3,14
,19–
21),
auth
ors
assu
me
afix
edtu
mor
cell
dens
ityat
the
tum
orbo
rder
svisi
ble
inT
1Gd
and
FLA
IRsc
ans
and
the
mod
elpa
ram
eter
sar
ees
timat
edby
fittin
gth
evo
lum
e(1
9,20
)or
shap
e(1
3,21
)of
the
visib
lele
sion.
How
ever
,the
tum
orce
llde
nsity
varie
sal
ong
the
visib
letu
mor
outli
nes
due
toan
atom
ical
rest
rictio
nsan
din
hom
ogen
eous
grow
thpa
tter
n.A
Bay
esia
nap
proa
chac
coun
ting
forv
aryi
ngtu
mor
cell
dens
itywa
spr
opos
edin
(22)
.A
llth
ese
calib
ratio
nsho
weve
rus
eM
RI
scan
sfr
omat
leas
ttw
otim
epo
ints
,mak
ing
them
unsu
itabl
efo
rre
alcl
inic
alse
ttin
g,w
here
only
scan
sac
quire
dat
singl
epr
eope
rativ
etim
epo
int
are
avai
labl
e.In
this
pape
rwe
pres
ent
asy
stem
atic
Bay
esia
nfr
amew
ork
tode
velo
pro
bust
tum
orgr
owth
mod
els
info
rmed
from
clin
ical
imag
ing
feat
ures
.O
urap
proa
chov
erco
mes
limita
tions
spec
ific
todi
ere
ntim
agin
gm
odal
ities
and
intu
rnin
tegr
ates
thei
rin
form
atio
nto
calib
rate
the
tum
orgr
owth
mod
els
for
the
RTpl
anni
ng.
We
com
bine
com
plem
enta
ryin
form
atio
nfro
mst
ruc-
tura
lMR
Iand
func
tiona
lFET
-PET
met
abol
icm
aps,
acqu
ired
ata
singl
etim
epo
int,
toid
entif
yth
epa
tient
-spe
cific
tum
orce
llde
nsity
.O
urB
ayes
ian
appr
oach
infe
rsm
odel
ing
and
imag
-in
gpa
ram
eter
sun
der
unce
rtai
ntie
sar
ising
from
mea
sure
men
tan
dm
odel
ing
erro
rs.
We
prop
agat
eth
ese
unce
rtai
ntie
sto
mod
elpr
edic
tions
toob
tain
robu
stes
timat
esof
the
tum
orce
llde
nsity
toge
ther
with
confi
denc
ein
terv
als
that
can
beus
edfo
rpe
rson
aliz
edRT
plan
ning
.T
hees
timat
edpa
tient
-spe
cific
tum
orce
llde
nsity
oer
san
adva
ntag
ein
dete
rmin
ing
mar
gins
ofth
eC
TV
asw
ella
sre
gion
sfo
rdo
sees
cala
tion.
Ast
udy
cond
ucte
don
asm
allc
linic
alpo
pula
tion
isus
edto
asse
ssth
ebe
nefit
sof
the
pers
onal
ized
RTpl
ans
over
stan
dard
prot
ocol
s.
1.B
ayes
ian
mod
elpe
rson
aliz
atio
n
The
tum
orce
llde
nsity
u(◊ u|M
u)
isco
mpu
ted
bya
mod
elM
u
with
para
met
ers
◊ u.
The
para
met
ers
◊ uar
epa
tient
-spe
cific
and
apr
obab
ility
dist
ribut
ion
ofth
eir
plau
sible
valu
esis
in-
ferr
edfro
mtu
mor
obse
rvat
ions
(dat
a)D
· ·=y
kS k
=1
obta
ined
fromS
med
ical
scan
s.W
epos
tula
teth
atth
emod
elpr
edic
tions
uan
dea
chim
age
obse
rvat
ion
ykar
ere
late
dby
ast
ocha
stic
imag
ing
mod
elFk
with
para
met
ers
◊ Fk
asyk
=Fk! u! ◊ u
|Mu
" ,Á! ◊ F
k
"",
[1]
whe
reÁ
isa
pred
ictio
ner
rora
ccou
ntin
gfo
rmod
ellin
gan
dm
ea-
sure
men
tun
cert
aint
ies.
Para
met
ers
◊· ·=
◊u,◊F
1,···,◊F
S
ofth
em
odelM
· ·=M
u,M
F1,···,M
FS
are
assu
med
tobe
unkn
own.
A.
Par
amet
ers
estim
atio
nan
dun
cert
aint
ypr
opag
atio
n.A
prio
rpr
obab
ility
dist
ribut
ion
func
tion
(PD
F)P(
◊|M
)is
used
toin
corp
orat
ean
ypr
ior
info
rmat
ion
abou
t◊.
Bay
esia
nm
odel
calib
ratio
nup
date
sth
ispr
ior
info
rmat
ion
base
don
the
avai
labl
eda
taD
.T
heup
date
dpo
ster
ior
isco
mpu
ted
byth
eB
ayes
theo
rem
:P(
◊|D,M
)ÃP(D|◊,M
)·P(
◊|M
),[2
]
whe
reP(D|◊,M
)ist
helik
elih
ood
ofob
serv
ing
data
Dfro
mth
em
odelM
for
agi
ven
valu
eof
◊.
Inm
ost
case
san
anal
ytic
alex
pres
sion
forE
q.(2
)isn
otav
aila
ble
and
sam
plin
gal
gorit
hms
are
used
toob
tain
sam
ple
◊(l
) ,l
œ1,···,N
fro
mth
epo
ste-
rior
P(◊|D,M
).W
eus
eda
Tran
sitio
nalM
arko
vC
hain
Mon
teC
arlo
(TM
CM
C)
sam
plin
gal
gorit
hm(2
3),w
hich
itera
tivel
yco
nstr
ucts
ase
ries
ofin
term
edia
tePD
Fs:
P j(◊|D,M
)≥P(D|◊,M
)pj·P
(◊|M
),[3
]
whe
re0
=p0<p1<
···<
pm
=1
andj
=1,···,m
is
age
nera
tion
inde
x.T
hete
rmpj
cont
rols
the
conv
erge
nce
ofth
esa
mpl
ing
proc
edur
ean
dis
com
pute
dau
tom
atic
ally
byth
eT
MC
MC
algo
rithm
.T
MC
MC
cons
truc
tsa
larg
enu
mbe
rof
inde
pend
ent
chai
nsth
atex
plor
epa
ram
eter
spac
em
ore
eci
ently
than
trad
ition
alsa
mpl
ing
met
hods
(23)
and
allo
wpa
ralle
lexe
cutio
n.A
high
lypa
ralle
lim
plem
enta
tion
ofth
eT
MC
MC
algo
rithm
ispr
ovid
edby
the
4U
fram
ewor
k(2
4).
The
infe
rred
para
met
ricun
cert
aint
ies
are
prop
agat
edth
roug
hth
em
odelM
toob
tain
robu
stpo
ster
ior
pred
ictio
nsab
outu,
give
nby
the
PDF:
P(u|
D,M
)=⁄
P(u|
◊,M
)·P
(◊|D,M
)d◊,
[4]
orby
simpl
ified
mea
sure
ssuc
has
them
eanµu
=E
[u(◊
)]©m
1an
dva
rianc
e‡
2 u=E
[u2 (
◊)]
≠m
2 1©m
2≠m
2 1,de
rived
from
the
first
two
mom
entsm
k,k
=1,
2:
mk
=⁄
! u(◊|M
)" k·P
(◊|D,M
)d◊
¥1 N
N ÿ l=1
! u(◊(l
) |M)" k
,[5
]
whe
re
deno
tes
the
spac
eof
allu
ncer
tain
para
met
ers.
B.
Tum
orgr
owth
mod
el.T
umor
mod
els
ofva
rious
com
plex
ityca
nbe
pers
onal
ized
usin
gth
eB
ayes
ian
appr
oach
desc
ribed
abov
e.H
ere,
we
mod
elu(
◊ u|M
u)
with
the
FKm
odel
whi
chde
scrib
esan
isotr
opic
tum
orgr
owth
inth
ebr
ain
œ
R3
as:
ˆu ˆt
=Ò
·(D
Òu)
+flu
(1≠
u),
in
[6]
Òu·n
=0,
inˆ
,
[7]
whe
reu
=u
i(t
)#
Vox
els
i=1
andui(t
)œ
[0,1
]is
ano
rmal
ized
tum
orce
llde
nsity
attim
et
and
voxe
li=
(ix,iy,iz)œ
.
The
term
fl[1/day]d
enot
estu
mor
prol
ifera
tion
rate
.T
hetu
mor
infil
trat
ion
into
the
surr
ound
ing
tissu
esis
mod
eled
byth
ete
nsor
D=
DiI
#Vox
els
i=1
whe
reI
isa
3◊
3id
entit
ym
atrix
and
Di=
IDg
ifi
œgray,
Dw
ifi
œwhite,
0ifi/œ
gray
fiwhite.
[8]
The
term
sDw
andDg
deno
tetu
mor
infil
trat
ion
inw
hite
and
gray
mat
ter
and
weas
sum
eDw
=10Dg[m
m2 /day](
12).
The
2|
ww
w.p
nas.
org/
cgi/
doi/
10.1
073/
pnas
.XX
XX
XX
XX
XX
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
Lipk
ová
etal
.
Fig.
1.O
verv
iew
ofth
ein
fere
nce
fram
ewor
k.I)
Med
ical
imag
essh
owpr
eope
rativ
epa
tient
scan
s:(I
.A)
3Dre
cons
truc
tion
ofT
1Gd
imag
esan
d(I
.B)
slic
esac
ross
all
mod
aliti
es.
Tum
orob
serv
atio
ns(e
.g.,
segm
enta
tions
)ex
trac
ted
from
each
mod
ality
are
illus
trat
edin
(I.C
).II
)C
ompu
tatio
nal
med
icin
ein
clud
esa
tum
orgr
owth
mod
elMu
(II.B
),w
hich
sim
ulat
estu
mor
evol
utio
nin
the
patie
ntan
atom
y(I
I.A),
and
imag
ing
mod
elMI
(II.E
)w
hich
rela
tes
the
imag
eob
serv
atio
nsw
ithth
em
odel
edtu
mor
cell
dens
ityu.
Subp
lot
(II.D
)sh
ows
asc
hem
atic
repr
esen
tatio
nof
the
actu
altu
mor
cell
dens
ityal
ong
the
dash
edlin
esh
own
in(I
.C)
and
itsre
latio
nto
each
tum
orob
serv
atio
nav
aila
ble
from
the
med
ical
scan
s.T
heun
know
n,pa
tient
-spe
cific
para
met
ers
for
each
mod
elar
elis
ted
inTa
bles
(II.C
,F).
III)
Bay
esia
nin
fere
nce
isus
edto
iden
tify
the
prob
abili
tydi
stri
butio
nsof
the
unkn
own
para
met
ers,
acco
untin
gfo
rth
em
odel
ing
and
mea
sure
men
tun
cert
aint
ies.
Para
met
ric
unce
rtai
ntie
sar
eth
enpr
opag
ated
toob
tain
robu
stpr
edic
tions
abou
tpa
tient
-spe
cific
tum
orce
llde
nsiti
es.
The
mos
tpr
obab
letu
mor
cell
dens
ity,
give
nby
the
max
imum
apo
ster
ior
(MA
P)es
timat
e,is
show
nin
(III
.A,B
),w
hile
the
mea
nan
dst
anda
rdde
viat
ion
(std
)ar
esh
own
in(I
II.B
).IV
)Pe
rson
aliz
edR
adio
ther
apy
uses
the
patie
nt-s
peci
ficpr
edic
tions
toim
prov
edo
se-d
istr
ibut
ion
and
esca
latio
nde
sign
.Aco
mpa
riso
nof
ast
anda
rdan
dpe
rson
aliz
eddo
sedi
stri
butio
nsis
show
nin
(IV.
A-C
).T
here
gion
sof
estim
ated
high
tum
orce
llde
nsiti
esar
em
arke
dby
the
oran
geis
ocon
tour
in(I
V.D
,F).
Obs
erve
dtu
mor
recu
rren
ces
inT
1Gd
and
FLA
IR,m
arke
dby
the
pink
and
yello
wcu
rves
in(I
V.E
,F),
are
used
toco
mpa
reth
etr
eatm
ent
plan
s.T
hepe
rson
aliz
edpl
ansp
ares
mor
ehe
alth
ytis
sue,
whi
leac
hiev
ing
tum
orco
vera
ges
com
para
ble
toth
est
anda
rdpr
otoc
olan
dgu
ides
the
desi
gnof
pers
onal
ized
dose
esca
latio
npl
ans.
0278-0062 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2019.2902044, IEEETransactions on Medical Imaging
4
θu = Dw, ρ, T, icx, icy, icz are considered unknown andpatient-specific. The model is implemented in a 3D extensionof the multi-resolution adapted grid solver [29] with a typicalsimulation time of 1-3 minutes using 2 cores. An overview ofthe model Mu and its parameters is shown in Fig. 1 (II. A-C).
B. Multimodal imaging model
We consider routinely acquired T1 gadolinium enhanced(T1Gd) and fluid attenuation inversion recovery (FLAIR)MRI scans in combination with FET-PET maps. A stochasticimaging model MI is designed to relate model predictionsof the tumor cell density u and the tumor observationsD = yT1Gd, yFLAIR, yFET available from the medical scans.Here, y denotes a vector of tumor observations obtained from acertain image modality, yi is an entry in y and i enumerates allvoxels in the given image. A voxel i corresponds to the samelocation (ix, iy, iz) in each scan and the simulation domain Ω.See Fig. 1 for an overview of all the imaging modalities (I.B),corresponding tumor observations (I.C) and their relation totumor cell density u (II.D).The MRI scans provide morphological information aboutthe visible tumor in the form of binary segmentations. Thesegmentation ys, s ∈ T1Gd, FLAIR assigns a label ysi = 1to each voxel with visible tumor and ysi = 0 otherwise. Theprobability of observing a segmentation ys with a simulated tu-mor cell density u is modeled by a Bernoulli distribution [28]:
P(ys|θ,M) =
N∏
i=1
P(ysi |θ, ui) =
N∏
i=1
αysii · (1− αi)1−ysi . (4)
Here αi is the probability of observing the tumor in the MRIscan and it is assumed to be a double logistic sigmoid:
αi(ui, usc) = 0.5 + 0.5 · sign(ui − usc)
(1− e−
(ui−usc)2
σ2α
), (5)
where usc denotes an unknown cell density threshold belowwhich tumor cells are not visible in the MRI scan, whilethe term σ2
α represents uncertainty in usc. The parametersθIT1Gd , θIFLAIR = uT1Gd
c , uFLAIRc , σ2
α are assumed unknown andpatient-specific.The FET-PET signal is proportional to tumor cell density withan unknown constant of proportionality [13], [14]. Let yFET bethe normalized FET-PET signal after subtracting the patient-specific baseline signal from healthy tissue i.e., yFET
i ∈ [0, 1]and b the corresponding constant of proportionality. We as-sume that yFET
i can be related with the modeled tumor celldensity ui as
yFETi =
1
bui + ε, (6)
where ε is prediction error accounting for modeling andmeasurement uncertainties. Because of the noisy nature of thePET scan, the error term is assumed to be a normal distributionε ∼ N (0, σ2). The probability of observing the PET signalyFET with the simulated tumor cell density u is then modeled as
P(yFET |θ,M) =
N∏
i=1
P(yFETi |θ, ui) =
N∏
i=1
N(yFETi −
1
bui, σ
2
). (7)
The PET scan, acquired at 4mm resolution, is registered tothe MRI scans with 1mm resolution. To justify the product in
Eq. (7) only voxels separated by distance 4mm are used. Theparameters θIFET = b, σ are unknown and patient-specific.An overview of the imaging model MI and its parameters isshown Fig. 1 (II. D-F).
C. Parameters estimation and uncertainty propagation
The parameters θ = θu, θI of the model M = Mu,MIwhere I = IT1Gd, IFLAIR, IPET, are assumed unknown and aprobability distribution function (PDF) is used to quantify theirplausible values. A prior PDF P(θ|M) is used to incorporateany prior information about θ. Bayesian model calibrationupdates this prior information based on the available data D.The updated posterior PDF is computed by the Bayes theorem:
P(θ| D, M) ∝ P(D| θ,M) · P(θ|M), (8)
where P(D|θ,M) is the likelihood of observing data D fromthe model M for a given value of θ. Since each of the medicalscans captures a different physiological process, the tumorobservations are assumed independent and the likelihood func-tion can be expressed as:
P(D|θ,M) = P (yT1Gd|θ,M) · P (yFLAIR|θ,M) · P (yFET |θ,M) .
The prior PDF is assumed uniform with details specified inthe SM. Since an analytical expression for Eq. (8) is notavailable, sampling algorithms are used to obtain samplesθ(l), l ∈ 1, · · · , S from the posterior P(θ|D,M). We useTransitional Markov Chain Monte Carlo (TMCMC) algorithm[30] which iteratively constructs series of intermediate PDFs:
Pj(θ|D,M) ∼ P(D|θ,M)pj · P(θ|M), (9)
where 0 = p0 < p1 < · · · < pm = 1 and j = 1, · · · ,m isa generation index. The term pj controls the convergence ofthe sampling procedure and is computed automatically by theTMCMC algorithm. TMCMC method constructs a large num-ber of independent chains that explore parameter space moreefficiently than traditional sampling methods [30] and allowparallel execution. We use a highly parallel implementation ofthe TMCMC algorithm provided by the Π4U framework [31].The inferred parametric uncertainties are propagated throughthe model M to obtain robust predictions about u given by:
P(u| D, M) =
∫
Θ
P(u| θ,M) · P(θ|D, M) dθ, (10)
or by simplified measures such as the mean µu = E[u(θ)] ≡m1 and variance σ2
u = E[u2(θ)] − m21 ≡ m2 − m2
1 derivedfrom the first two moments mk, k = 1, 2:
mk =
∫
Θ
(u(θ|M)
)k · P(θ|D,M) dθ ≈ 1
S
S∑
l=1
(u(θ(l)|M)
)k,
where Θ is the space of all unknown parameters. The mostprobable tumor cell density estimate, is given by the maximuma posteriori (MAP) defined as uMAP = argmaxθ P(u| D, M).
0278-0062 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMI.2019.2902044, IEEETransactions on Medical Imaging
5
MAP Mean stdGT
Synthetic 01
MAP Mean std
MR
IM
RI+PET
INPU
T
Observations
A)
B)
C) D) E)
F) G) H)— T1Gd — FLAIR
Fig. 2. Synthetic test case. Orange box: A 2D slice of the synthetic groundtruth (GT) tumor cell density (A) and corresponding image observations (B):the normalized FET-PET signal with additive noise (red-blue color scale)and the outlines of the T1Gd (yellow) and FLAIR (pink) binary tumorsegmentations. Blue box: Results of the Bayesian calibration with multimodaldata. The results are in close agreement with GT data. Green box: Calibrationresults using only the MRI data, which do not provide enough information torecover the tumor cell density profile correctly.
III. RESULTS
The Bayesian framework described in the previous section isfirst applied to synthetic data to test sensitivity of the inferenceand to show the role of multimodal image information on themodel calibration. Afterwards, clinical data are used to inferpatient-specific tumor cell densities and to design personalizedRT plans. Tumor recurrence patterns are used to compare theproposed and standard RT plans. The software and data usedin this paper are publicly available1.
A. Sensitivity study
The model Mu is used to generate a 3D synthetic tumorin a brain anatomy obtained from [32] using the parametersreported in Table I. A 2D slice of the simulated ’ground-truth’(GT) tumor cell density is shown in Fig. 2 (A). The syntheticT1Gd and FLAIR tumor segmentations are constructed bythresholding the GT tumor cell density at uT1Gd
c = 0.7 anduFLAIRc = 0.25. The FET-PET signal is designed by taking the
GT tumor cell density within the T1Gd and FLAIR segmen-tations, adding Gaussian noise with zero mean and standarddeviation (std) σ, and normalizing the result. The value of σis chosen as average std of the FET signal from the healthybrain tissue. The generated synthetic image observations areshown in Fig. 2 (B).A sensitivity study for the number of samples is performed,indicating that 6000 samples is adequate for the model. Themanifold of the inferred probability distribution is presentedin Fig. 3 and the calibrated parameters are given in Table I.As seen from the probability distribution manifold, tumorobservations from a single time point do not contain enoughinformation to infer time dependent parameters (Dw, ρ, T )exactly, since different combinations of these parameters can
1https://github.com/JanaLipkova/GliomaSolver
Synthetic Small All
data
Dw Tρ σ
T
σ
ρ
Dw
ρ
T
Fig. 3. The results of the Bayesian calibration for the synthetic case. Abovethe diagonal: Projection of the TMCMC samples of the posterior distributionP(θ|D,M) in 2D space of the indicated parameters. The colors indicatelikelihood values of the samples. The number in each plot shows the Pearsoncorrelation coefficient between the parameter pairs. The colored triangles markthe four selected parameters used in Fig. 4. Diagonal: Marginal distributionsobtained with Gaussian kernel estimates. Boxplot whiskers demarcate the 95%percentiles. Below the diagonal: Projected densities in 2D parameter spaceconstructed by 2D Gaussian kernel estimates. The black dots mark the valuesused to generate the synthetic data.
(0.095, 0.022, 364)(0.16, 0.036, 218)(0.32, 0.069, 111) (0.049, 0.01, 761)
Fig. 4. Insensitivity of the tumor cell density to the speed of the growth.Shown are slices of the tumor cell densities computed with different combi-nations of parameters (Dw, ρ, T ) as listed at the bottom of each plot. Thesecorrespond to the colored triangles in Fig. 3. Despite significant variationin the parameter values, all combinations lead to very similarly-appearingtumors. In the absence of temporal information, the time dependent parametersare not identifiable, since the model calibration cannot distinguish betweencompensating effects among the parameters that affect the dynamics. Asshown here, tumors with similar Dw/ρ and Tρ values appear very similar toone another (here Dw/ρ ≈ 4.5, Tρ ≈ 7.7). Hence, the Bayesian calibrationidentifies the probability distribution of all the plausible values.
generate the same tumor cell density as shown in Fig. 4.The lack of identifiability of (Dw, ρ, T ) poses a challengefor calibration approaches searching only for a single valueof θ. Instead, Bayesian calibration provides fairer estimate;the inferred probability distribution shows a strong correla-tion between the parameters (Dw, ρ, T ), while the high stdvalues imply low confidence in these parameters. On theother hand, parameters that affect the tumor spatial pattern,e.g. (icx, icy, icz), are identified with high accuracy, whichis reflected by their low std. The image related parameters
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TABLE IRESULTS OF THE BAYESIAN CALIBRATION FOR THE SYNTHETIC CASE GENERATED WITH THE GROUND TRUTH (GT) VALUES. REPORTED IS MAXIMUM A
POSTERIORI (MAP), MEAN AND STANDARD DEVIATION (STD). THE UNITS ARE Dw ∼ mm2/day; ρ ∼ 1/day; T ∼ day; AND icx , icy , icz ∼ mm.
Dw ρ T icx icy icz σ b uT1Gdc uFLAIR
c σ2α
GT 0.1300 0.0250 302.00 80.64 171.52 128.00 0.0230 0.8800 0.7000 0.2500 -MAP 0.1226 0.0296 271.13 80.61 171.55 127.92 0.0280 0.9884 0.8440 0.4874 0.0502mean 0.1543 0.0361 253.85 80.64 171.47 127.80 0.0294 0.9775 0.8325 0.4841 0.0505std 0.0530 0.0125 107.59 0.1280 0.1024 0.2048 0.0022 0.0139 0.0132 0.0076 0.0004
(uT1Gdc , uFLAIR
c , b, σ) are slightly overestimated due to the as-sumed correlation length and the effect of the complex brainanatomy. The role of the anatomy is discussed further in theSM.The inferred parametric uncertainties are propagated to obtainrobust posterior predictions about the tumor cell density shownin Fig. 2 (C-E). Despite the large parametric uncertainties,the MAP and mean tumor cell density estimates are almostindistinguishable from the GT tumor. The low std valuesimply that, using our Bayesian formulation, the informationcontained in multimodal data is sufficient to infer tumor celldensity from single time point scans.For comparison, if the model calibration is performed onlywith the MRI data, i.e. D = yT1Gd, yFLAIR, the estimatedtumor cell densities shown in Fig. 2 (F-G) deviate from theGT tumor mainly in the central part of the lesion, whichis also consistent with the regions of high std shown inFig. 2 (H). Nonetheless, the outlines of the predicted tumorare similar to those of the GT tumor. This is because thetumor morphology is mainly constrained by the MRI data,since the FET-PET signal coincides with the baseline signalof the healthy tissue in the regions of lower tumor infiltration.On the other hand, the FET-PET signal constrains the tumorcell density profile in the regions of high tumor infiltration.This highlights the importance of integrating structural andfunctional image information for the model calibration whendealing with single time point data.
B. Patient study
A retrospective clinical study is conducted on 8 patientsdiagnosed with GBM. Scans of the patients P1-P8 are shownFig. 5 and the details about acquisition protocols and imageprocessing are reported in the SM. All patients received thestandard treatment, surgery followed by combined radio- andchemotherapy [2]. There was no visible tumor after the treat-ment and patients were regularly monitored for recurrence.The preoperative scans shown in Fig. 5 (A-D) are used for theBayesian inference. The calibrated parameters are reported inTable 1 in the SM and the posterior, patient-specific predictionsfor the tumor cell densities are shown in Fig. 5 (E-G).These patient-specific predictions provide estimates about thepossible tumor cell migration pathways in the surrounding ofthe visible tumor, constrained by the patient anatomy and theavailable tumor observations. The predicted tumor infiltrationpathways can be validated by the patterns of the first detectedtumor recurrence shown in Fig. 5 (H), where the outlines of thepredicted infiltrations (blue) and recurrence tumors (pink) are
depicted. For patients P5,P7,P8 the model accurately predictstumor infiltration also inside the healthy-appearing collateralhemisphere, whereas for cases P1-P4 the tumor predictions arecorrectly restricted only to one hemisphere. Moreover, despitea similar appearance of the preoperative tumors in patientsP1 and P2, the model correctly predicts more infiltrativebehaviour for the patient P1, which is consistent with therecurrence pattern. The high confidence in the predictions isreflected by low std shown in Fig. 5 (F).
C. Personalized radiotherapy design
The patient-specific tumor cell density predictions canbe used to design margins of the CTV and to identifyhigh cellularity regions that could mark areas of increasedradioresistance. The personalized RT plan can be based eitheron the most probable scenario given by MAP estimate or theworst case scenario given as a sum of the mean and std of thetumor cell density. Since in the present study, the mean andMAP estimates are very similar, and the std values are small,the MAP estimates are used. An overview of the proposedpersonalized RT design is shown in Fig. 1 (IV), while thedetails are described in the following subsections. The tumorsrecurrence patterns are used to assess the benefits of theproposed RT plan over the standard treatment protocol. Forevaluation purposes, all recurrence scans are registered to thepreoperative anatomy. To prevent registration errors arisingfrom mapping the anatomy with the resection cavity to thepreoperative brain, rigid registration is used. This provides asufficient mapping for most cases, however it cannot capturethe post-treatment tissue displacement around the ventriclesin patients P7-P8, making the mapping less accurate inthese regions. The design of methods that provide robustregistration between pre- and post-operative brain anatomiesis still an open problem.
1) Dose distribution: An ideal CTV covers all the residualtumor, including infiltrating tumor cells that are invisible onthe pretreatment imaging scans, while sparing healthy tissue.We use the tumor recurrence pattern to evaluate the efficiency(ηCTV ) of the CTV, defined here as the relative volume of therecurrence tumor (V REC) contained within the CTV:
ηCTV =|V REC ∩ CTV |
V REC× 100%. (11)
Figure 5 (H) shows the FLAIR scans with the first detectedtumor recurrence outlined by the pink lines. The margin ofthe administered CTV RTOG, designed by the standard RTOGprotocol with a 2 cm margin around the visible tumor, is
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Preo
pera
tive
Infe
renc
eR
ecur
renc
eT1
Gd
FLAI
RSe
gm.
FET-PE
TMAP
std
3DT1
Gd
FLAI
RT1
Gd+
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P1 P2
60 63 69 72 Gy 60 63 69 72 Gy 60 63 69 72 Gy 60 63 69 72 Gy 60 63 69 72 Gy 60 63 69 72 Gy 60 63 69 72 Gy 60 63 69 72 Gy
P4 P5 P8P3 P6A)
B)
C)
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P7FLAIR
tumour
MAP dose escalation
T1G
d tumour
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FET dose escalation
Fig. 5. Results of the Bayesian calibration for patients P1-P8. Orange box: Preoperative scans showing (A) T1Gd; (B) FLAIR; (C) Tumor segmentations:T1Gd (yellow) and FLAIR (pink); (D) FET-PET. Blue box: (E) MAP and (F) std of the inferred tumor cell densities shown in the preoperative FLAIR scans.The CTV RTOG margin is shown as the green curves. (G) The 3D reconstructions show outlines of the MAP tumor (blue) together with tumor extent visibleon the FET-PET scans (orange) in the preoperative anatomy (white). Green box: Scans of the first detected tumor recurrence. (H) FLAIR tumor recurrence(pink), CTV RTOG(green), and CTV MAP(blue) margins. (I) T1Gd tumor recurrence (yellow) and the dose-escalation outlines proposed by FET enhancement(magenta) and MAP estimates (orange). (J) Multilevel dose-escalation designed by MAP estimates.
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Figure 4; Final plot
P1 P2 P3 P4 P5 P6 P7 P8pID
0
500
1000C
TV [c
m3 ]
RT 2cmRT MAP
P1 P2 P3 P4 P5 P6 P7 P8pID
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VCTV
[cm
3 ]
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P1 P2 P3 P4 P5 P6 P7 P8pID
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P1 P2 P3 P4 P5 P6 P7 P8pID
0
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ηC
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]
patients
Irradiated volume
Radiation efficiency
patients
Fig. 6. A comparison of the RT plan based on the RTOG protocol (green)and MAP estimates (blue). (A) The overall irradiated volume V CTV and (B)the corresponding efficiency ηCTV . The CTV MAP uses a smaller irradiationvolume while having a comparable efficiency as the CTV RTOG.
marked by the green lines in (E,F,H). The personalized CTV,referred as CTV MAP, is constructed by thresholding the MAPtumor cell density at u = 0.1% for all patients. (This valuewas chosen so that the efficiency of the CTV MAP is comparableto that of the CTV RTOG ). The outlines of the proposedCTV MAP are shown as the blue isocontours in Fig. 5 (H-I).A visual comparison of the CTVs shown in Fig. 5 (H) andFig. 1 (IV), and a quantitative comparison presented in Fig. 6,show that the proposed personalized plans spare more healthytissue, hence reducing radiation toxicity, while maintainingthe efficiency of the standard RTOG protocol. Both plansshow reduced efficiency for patients P7-P8, mainly around theventricles, which may be caused by misalignment between thepreoperative and recurrence anatomies.These preliminary results imply that the regions predicted as atumor-free by the model, remain tumor-free and thus the modelpredictions have potential to guide personalized CTV design.The standard or hospital-specific protocols can be updated bythe model predictions to spare brain tissue not infiltrated bythe tumor. This can lead to significant savings in the healthytissue, especially in the cases of large lesions or lesions closeto hemispheres separation and other anatomical constraints.
2) Dose-escalation: No dose-escalation plan for GBM pa-tients has been yet approved by phase-III-clinical trials. Here,we present a theoretical comparison of two escalation planstargeting high tumor cellularity regions identified by: 1) FET-PET enhancement as proposed in [5] and 2) MAP estimates.We evaluate the efficiency of an escalation plan by its capabil-ity of targeting T1Gd-enhanced tumor recurrence regions. Inthese regions, the recurrent tumor has high cellularity, despitehaving received the full radiation dose, suggesting tumor ra-dioresistance. Figure 5 (I) shows the T1Gd scans with the firstdetected tumor recurrence. The margins of the T1Gd-enhancedtumor recurrence are marked by the yellow lines, while theoutlines of the dose-escalation plans designed by the FET-PETenhancements are shown in magenta. The FET enhancementsdo not fully cover the T1Gd recurrent tumor in patients P4-P7, providing a possible explanation for why improvements
in progression-free survival have not been observed in [5]. Incomparison, the MAP estimates, calibrated by the FET-PETsignal, extend the information about the tumor cell densityin the periphery of the visible lesion. Figure 5 shows twopossible dose-escalation plans based on the inferred MAPtumor cell density: (I) a single-level dose-escalation basedon the thresholded MAP solution with threshold u = 30%marked by the orange lines and (J) a cascaded four-levelescalation plan constructed by thresholding the MAP tumorcell density at u = [0.1, 25, 50, 75]%. The optimal designof a personalized dose-escalation plan would require moreextensive studies. However, these preliminary results show thatthe inferred high cellularity regions coincide with the areasof tumor recurrence better than those suggested by the FET-PET enhancement alone. The Bayesian inference frameworkdeveloped here thus provide a promising tool for a rationaldose-escalation design.
IV. CONCLUSION
We have demonstrated that patient-specific, data-drivenmodeling can extend the capabilities of personalized RTdesign for infiltrative brain lesions. We combined patientstructural and metabolic scans from a single time point witha computational tumor growth model through a Bayesianinference framework and predicted the tumor distributionbeyond the outlines visible in medical scans. The patient-specific tumor estimates can be used to design personalized RTplans, targeting shortcomings of standard RT protocols. Thesoftware and data used in this work are publicly released2
to facilitate translation to clinical practice and to encouragefuture improvements. In the future, the Bayesian frameworkdeveloped here could also be extended to predict individualpatient responses to RT by incorporating data obtained duringthe course of treatment as done in [33], in which non-spatial tumor models were used. In this way, the treatmentcan be further improved by adaptively refining the RT plansbased on the predicted patient responses. Moreover, the basicFK tumor growth model could be replaced by a Fokker-Planck diffusion model [16], which would not increase thenumber of unknown parameters or affect the computationalcomplexity significantly, but might provide a better descriptionof biological diffusion. Future work could also incorporatemore advanced models, such as [34], that account for cancerstem cells, their progeny and nonlinear coupling between thetumor and the neovascular network. However, it remains tobe seen whether scans acquired at a single time point wouldprovide enough information to calibrate the advanced modelssufficiently. If not, then simpler, well-calibrated models mayprove to be more informative. Finally, in future studies, thecomputational framework developed here will be tested ona larger patient cohort and prospective clinical trials will beperformed. In summary, the results presented here provide aproof-of-concept that multimodal Bayesian model calibrationholds a great promise to assist the development of personalizedRT protocols.
2https://github.com/JanaLipkova/GliomaSolver
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ACKNOWLEDGMENT
JL acknowledges partial funding from the National Sci-ence Foundation-Division of Mathematical Sciences (NSF-DMS) through grant DMS-1714973 and the Center for Mul-tiscale Cell Fate Research at UC Irvine, which is supportedby NSF-DMS (DMS-1763272) and the Simons Foundation(594598, QN). JL additionally acknowledges partial fundingfrom the National Institutes of Health (NIH) through grant1U54CA217378-01A1 for a National Center in Cancer Sys-tems Biology at the University of California, Irvine, andNIH grant P30CA062203 for the Chao Comprehensive CancerCenter at the University of California, Irvine.JL acknowledges the partial funding from the Bavaria Califor-nia Technology Center (BaCaTec) through grant 6090142.
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