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T ansac ions on Compu a ional Biology and Bioin o ma ics
1
A Bioma hema ical Model o Tumo Response o
Radioimmuno he apy wi h αPDL1 and αCTLA4
Isabel Gonz´
alez-C espo, An onio G´
omez-Caama˜
no, ´
Osca L´
opez Pouso, John D. Fenwick,
and Juan Pa do-Mon e o
Abs ac —The e is e idence o syne gy be ween adio he apy and immuno he apy. Radio he apy can inc ease libe a ion o umo
an igens, causing ac i a ion o an i umo T-cells. This e ec can be boos ed wi h immuno he apy. Radioimmuno he apy has po en ial o
inc ease umo con ol a es. Bioma hema ical models o esponse o adioimmuno he apy may help on unde s anding o he
mechanisms a ec ing esponse, and assis clinicians on he design o op imal ea men s a egies. In his wo k we p esen a
bioma hema ical model o umo esponse o adioimmuno he apy. The model uses he linea -quad a ic esponse o umo cells o
adia ion (o a ia ion o i ), and builds on p e ious de elopmen s o include he adia ion-induced immune e ec . We ha e ocused his
s udy on he combined e ec o adio he apy and αPDL1/αCTLA4 he apies. The model can i p eclinical da a o olume dynamics
and con ol ob ained wi h di e en dose ac iona ions and αPDL1/αCTLA4. A bioma hema ical s udy o op imal combina ion s a egies
sugges s ha a good unde s anding o he in ol ed biological delays, he biokine ics o he immuno he apy d ug, and he in e play
be ween hem, may be o pa amoun impo ance o design op imal adioimmuno he apy schedules. Bioma hema ical models like he
one we p esen can help o in e p e expe imen al da a on he syne gy be ween adio he apy and immuno he apy, and o assis in he
design o mo e e ec i e ea men s.
Index Te ms—Radioimmuno he apy, adio he apy, αPDL1, αCTLA4, bioma hema ical modeling.
F
1 INTRODUCTION
CANCER immuno he apy (IT) is a he apeu ic s a egy
agains cance ha aims a boos ing and exploi ing
he na u al immune esponse o con ol and cu e umo s
[1], [2]. Checkpoin inhibi o s a e a ype o IT which is
used o ea men o se e al cance s, including melanoma,
p os a e, NSCLC and leukemia [3], [4]. These inhibi o s
block di e en checkpoin p o eins, like CTLA-4 and PD-
1/PD-L1, which a e well known supp esso s o he immune
esponse agains umo s. Inhibi o s o hese p o eins ha e
shown p omising esul s in p eclinical expe imen s, and
he e a e se e al monoclonal an ibodies agains PD-1/PD-
L1 app o ed o ea di e en ypes o cance [5]. Howe e ,
e icacy o IT as cance ea men is s ill limi ed, bu o
pa icula cases. Fo example, ipilimumab has shown an
imp o emen on he su i al o melanoma pa ien s, bu
esponse a es a e low, in he 10-15% ange [6].
•I. Gonz´alez-C espo is wi h he G oup o Medical Physics and
Bioma hema ics, Ins i u o de In es igaci´on Sani a ia de San iago,
and he Depa men o Applied Ma hema ics, Uni e sidade de San iago
de Compos ela, Spain.
•A. G´omez-Caama˜no is wi h he Depa men o Radia ion Oncology,
Clinical Uni e si y Hospi al o San iago, Spain.
•O. L´opez Pouso is wi h he G oup o Medical Physics and Bioma hema ics,
Ins i u o de In es igaci´on Sani a ia de San iago, and he Depa men o
Applied Ma hema ics, Uni e sidade de San iago de Compos ela, Spain.
•J.D. Fenwick is wi h he Depa men o Molecula and Clinical Cance
Medicine, Ins i u e o T ansla ional Medicine, Uni e si y o Li e pool,
Li e pool, Uni ed Kingdom.
•J. Pa do-Mon e o is wi h he G oup o Medical Physics and Bioma he-
ma ics, Ins i u o de In es igaci´on Sani a ia de San iago, Spain. E-mail:
juan.pa do.mon e o@se gas.es
Many p eclinical s udies ha e shown ha he combina-
ion o adio he apy (RT) and IT, in pa icula inhibi o s o
CTLA-4, and PD-1/PD-L1, is signi ican ly mo e e ec i e
han RT and IT alone [7], [8], [9], [10], [11]. The dominan
biological cell killing mechanism behind he e ec o RT
is he gene a ion o double s and b eaks in he DNA
by ionizing pa icles [12]. Howe e , he e is e idence ha
adia ion can igge o he cell killing mechanisms, pa i-
cula ly when deli e ed a high-doses pe ac ion, which
may be impo an o he syne gy wi h immuno he apy.
High-doses o adia ion can damage he umo ascula
sys em, e en ually igge ing cell dea h [13], and also lead
o an immune esponse agains su i ing umo cells [10],
[14], [15], inc easing he likelihood o umo con ol. The
mechanisms behind hese induced immune e ec s seem o
be ela ed o he inc eased libe a ion o umo an igens,
which cause he ac i a ion o an i umo T-cells, and he
modi ica ion o umo mic oen i onmen , killing immune
down- egula o s like T eg and MDSC cells, and acili a ing
T-cell in il a ion in he umo [7], [16], [17].
Despi e he po en ial o IT and RT, how he bes com-
bina ion o bo h he apies can be achie ed is s ill a ma e
o s udy. In his ega d, alida ed bioma hema ical models
o RT+IT (in silico umo models) would be e y use ul.
Bioma hema ical models, based on solid expe imen al and
clinical da a a e o high impo ance in o de o in e p e
esul s, and may also assis on he design o op imal
he apeu ic s a egies, po en ially guiding clinicians in he
selec ion o op imal ea men s. Modeling he esponse o
umo s o IT has been add essed in he bioma hema ical
li e a u e, ollowing bo h phenomenological and sys ems
biology app oaches [18], [19], [20], [21], [22], [23], [24]. On
he o he hand, modeling he syne gis ic combina ion o IT
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model can be shown in a simple way as:
dynamics C=p oli e a ion − adia ion dea h
−immune dea h
dynamics ˆ
A=na u al elease +RT-media ed elease
−na u al elimina ion −T-cell ac i a ion
dynamics Ta=ac i a ion/in il a ion − adia ion dea h
−immune dea h −na u al elimina ion
The model is buil ollowing a mechanis ic app oach
a he han a pu ely phenomenological app oach. I in-
cludes many simpli ica ions, hough, in o de o p esen a
ac able p oblem. Among he mos ele an simpli ica ions,
we should ci e: i) We do no include spa ial coo dina es in
ou mul i-compa men al model; ii) The p ocess o immune
dea h is o e ly simpli ied. In pa icula , we do no include
o he cell ypes ha pa icipa e in his p ocess, ei he a o-
ing immuni y o ac ing as supp esso s, like na u al kille s
o T- egs; iii) We mos ly ely on he linea -quad a ic model
(LQ) o accoun o adia ion cell dea h, bu depa u es om
he LQ model a high-doses a e s udied and discussed; i )
The p ocess o T-cell in il a ion in he umo is no included
in he model.
2.2 Di ec Cell Dea h and Kine ics
Dose deli e y is modeled as ins an aneous (i seems like a
good app oxima ion, as in ypical ac iona ed ea men s
dose deli e y akes minu es, while he ypical imes o ou
model a e days). Radia ion umo cell dea h is ypically
modeled wi h he linea -quad a ic (LQ) model:
log SF =−αd −βd2(1)
whe e SF is he su i ing ac ion o a popula ion o cells
a e being i adia ed o a adia ion dose, d, and αand βa e
he LQ linea and quad a ic pa ame e s.
We will i da a wi h di e en ac iona ion schedules. I
is well known ha cell dea h can depa om he s anda d
LQ-model, especially a high doses pe ac ion. Se e al
e ec s con ibu e o his, including e-oxygena ion [31],
sa u a ion [32], o ascula damage [13]. In o de o in-
es iga e possible depa u es om he LQ model, we will
check whe he o he models p o ide be e i s. In pa icula ,
we will in es iga e a simple ad hoc modi ica ion o he
quad a ic e m o he LQ-model [33]:
β→β01 + c√d(2)
whe e cis a ee pa ame e and dis he dose pe ac ion.
We will also use he Linea -Quad a ic-Linea (LQL)
model [34], which ob ains he su i ing ac ion as:
log SF =−αd −2βxd −1 + e−xd
x2(3)
whe e xis an ex a pa ame e ha modula es he slope o
he cu e.
We will use (1), (2) and (3) o model adia ion-induced
umo cell dea h. T-cell dea h will be desc ibed using he
LQ-model.
Cells a ally damaged by adia ion (doomed) do no die
ins an ly, bu ollow a gi en kine ics, gene ally a pause
(mi o ic delay) ollowed by a p og essi e dea h as le hally
damaged cells en e mi osis and su e mi o ic ca as ophe.
We will model his by conside ing a mi o ic delay ollowed
by an exponen ial dea h [35].
Viable umo cells can p oli e a e, and we desc ibe his
by using he logis ic o malism [36]. On he o he hand,
while doomed cells may ca y some p oli e a i e capaci y
(abo i e di isions [37]), i should be limi ed and does no
con ibu e o he long- e m cell popula ion. The e o e, we
will igno e i .
F om he abo e conside a ions we can w i e he ollo-
wing equa ion o iable umo cells:
dC
d ( ) = λ1C( )[1 −λ2C o ( )] −KC( )(4)
whe e λ1and λ2a e cons an s and C o ( ) = C( ) + Cd( )
is he o al numbe o umo cells a ime .KC( )is an
impulse e m accoun ing o he e ec o he adia ion dose:
KC( ) = (1 −SF C(d( )))C( )X
i
δ( − i)(5)
whe e { i}is he ec o o adia ion deli e y imes, {di}
a e he doses deli e ed a he imes { i}, and SF Cis he
su i ing ac ion o umo cells gi en by (1), (2) o (3).
δ(x)is he Di ac del a unc ion.
We conside ha each adia ion ac ion c ea es new
doomed cells, bu does no in e e e wi h he adia ion
kine ics o exis ing ones. The e o e, we can spli he com-
pa men o doomed cells in o ncompa men s c ea ed by
ndose ac ions:
dCd,i
d ( ) = KC( )−φω(¯
i)Cd,i( )(6)
Cd( ) = X
i
Cd,i (7)
He e, Cd,i( )deno es doomed cells c ea ed by he adia ion
dose ac ion di, iis he deli e y ime o ha ac ion, and
¯
i= − i. No ice ha Cd,i( )is de ined as ze o o < i.
The pa ame e φis he dea h a e, and ωmodels he mi o ic
delay and p og essi e inco po a ion o damaged cells o cell
dea h kine ics a e a adia ion ac ion:
ω(¯
) =
0, o ¯
≤τd1
¯
−τd1
τd2−τd1
, o τd1<¯
≤τd1
1, o ¯
>τd2
(8)
Radia ion also kills T-cells p esen in he umo as:
dTa
d ( ) = −KT( )(9)
whe e KT( )is he impulse e m o T-cells, which has he
same o m o (5), bu wi h SF Tas he su i ing ac ion
o T-cells. I is assumed ha adia ion-damaged T-cells die
ins an ly, and so he e a e no kine ic e ms associa ed o such
p ocess.
2.3 An igen Release and T-cell Ac i a ion
An igens a e conside ed o be eleased bo h na u ally ( a e
p opo ional o he numbe o umo cells, bo h iable and
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doomed), and du ing adia ion-induced cell dea h (p opo -
ional o he a e o cell dea h). We also include a e m des-
c ibing na u al elimina ion. A biological delay, τ1, be ween
an igen elease and T-cell ac i a ion is included (which may
be in e p e ed as he ime ha APCs ake o collec an igens
and ca y hem o he ac i a ion si es, Fig. 1(A)):
dˆ
A
d ( ) = ρC o ( −τ1) + ψφω( −τ1)Cd( −τ1)
−σˆ
A( )(10)
The ac i a ion o T-cells agains umo cells is modeled
h ough ou bilinea equa ions which desc ibe he gene a-
ion o ac i a ed T-cells ( ˆ
Ta) o blocked T-cells ( ˆ
Tb) ( h ough
he CTLA-4 ecep o ) om a pool o blank T-cells ( ˆ
T):
dˆ
A
d ( ) = −aˆ
A( )ˆ
T( )−bˆ
A( )ˆ
T( )(11)
dˆ
T
d ( ) = −aˆ
A( )ˆ
T( )−bˆ
A( )ˆ
T( ) + h(12)
dˆ
Ta
d ( ) = aˆ
A( )ˆ
T( )(13)
dˆ
Tb
d ( ) = bˆ
A( )ˆ
T( )(14)
The cons an s aand b( = 1 + b/a) desc ibe he a ini ies
o ac i a ion/inac i a ion, espec i ely [25]. The pool o
blank T-cells s a s om ˆ
T(0) = T0, which is assumed o
be he ca ying capaci y o T-cells. I can be deple ed due
o ac i a ion/inac i a ion, and in ha si ua ion i can enew
a cons an a e (due o ma u a ion o new T-cells), h. A
cons ain is imposed o a oid he T-cell compa men om
exceeding he ca ying capaci y: ˆ
T( )≤T0. Ac i e T-cells,
ˆ
Ta, mig a e and in il a e in he umo (wi h a biological de-
lay τ2, which phenomenologically models he ime needed
by ac i e T-cells o ac on umo cells) whe e hey become
pa o he compa men Ta(no e he ha no a ion):
dTa
d ( ) = dˆ
Ta
d ( −τ2) = aˆ
A( −τ2)ˆ
T( −τ2)(15)
We ha e es ed he hypo hesis ha ascula damage a
high adia ion doses [13] may educe he e ec i eness o
adioimmuno he apy by limi ing he in il a ion o T-cells in
he umo . The e o e, we include a dose and ime dependen
T-cell in il a ing pa ame e o accoun o ascula damage
and eco e y. Inspi ed by [13] and [38], we conside c i ical
ascula damage o doses beyond 15 Gy, and a p og essi e
eco e y o ascula unc ion as,
( ) = min{0.05 , 1}(16)
whe e he ime pos -i adia ion, , is measu ed in days. This
e m ep esen s he ac ion o ac i e T-cells eaching he
umo , and mul iplies (15).
2.4 T-cell Media ed Tumo Cell Dea h
In e ac ion be ween ac i e T-cells and umo cells esul s in
he pa ial deple ion o bo h. Following he wo k o de Pillis
e al. [18], we model his in e ac ion wi h a bilinea e m o
he compa men Ta, in addi ion o an exponen ial na u al
elimina ion:
dTa
d ( ) = −ιTa( )C o ( )−ηTa( )(17)
On he o he hand, T-cell media ed umo cell dea h is
modeled wi h he ollowing e m [18]:
dC
d ( ) = −p(Ta( )/C o ( ))q
s+ (Ta( )/C o ( ))qC( )(18)
The same exp ession holds o Cd( ). No e ha hese e ms
a e coupled o he ea lie equa ions.
2.5 The E ec o αPDL1 and αCTLA4
The concen a ion biokine ics (in a bi a y uni s) o αPDL1
(p1) and αCTLA4 (c4) is modeled as an ins an aneous sou ce
e m a injec ion imes ({ p1}and { c4}, espec i ely) and a
con inuous exponen ial elimina ion:
dc4
d ( ) = ic4( )δ( −{ c4})−νc4( )(19)
dp1
d ( ) = ip1( )δ( −{ p1})−µp1( )(20)
The p ecise pha macokine ic modeling o hese d ugs
is beyond he scope o he p esen a icle. Howe e , his
seems a good app oxima ion o he kine ics o αCTLA4
as Selby e al. [39] in es iga ed di e en αCTLA4 d ugs
biokine ics, inding ha hey ollow linea o ms like (19).
Al hough Deng e al. [40] in es iga ed he biokine ics o
αPDL1, inding a mo e complex non-linea beha iou .
Ra he han conside ing a complex kine ics model o
he cha ac e iza ion o he e ec o αCTLA4 on he de-
inhibi ion o T-cells, we model i as a simple dependence
on he pa ame e bin (11), (12) and (14) on c4, simila ly o
[25]:
b→b
1 + c4( )(21)
On he o he hand, he e ec o αPDL1 on he immune-
dea h o umo cells is modeled by in oducing a depen-
dence on he pa ame e pin (18) as:
p→p(1 + p1( )) (22)
2.6 The Comple e Model
The assembled model is he ollowing o C,Cd,ˆ
Aand Ta:
dC
d ( ) = λ1C( ) (1 −λ2C o ( )) −KC( )
−p(1 + p1( )) (Ta( )/C o ( ))q
s+ (Ta( )/C o ( ))qC( )
dCd,i
d ( ) = KC( )−φω(¯
i)Cd,i( )
−p(1 + p1( )) (Ta( )/C o ( ))q
s+ (Ta( )/C o ( ))qCd,i( )
dTa
d ( ) = −KT( ) + aˆ
A( −τ2)ˆ
T( −τ2)(23)
−ιTa( )C o ( )−ηTa( )
dˆ
A
d ( ) = ρC o ( −τ1) + ψφ X
i
ω(¯
i−τ1)Cd,i( −τ1)
−σˆ
A( )−aˆ
A( )ˆ
T( )−b
1 + c4( )ˆ
A( )ˆ
T( )
In addi ion, (12)-(14) con ol he ac i a ion o T-cells agains
umo cells, and (19, 20) con ol he biokine ics o αPDL1
and αCTLA4.
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2.7 Modeling Tumo Con ol P obabili y: Ma ko model
We ha e employed he clonogenic cell hypo hesis [41] o ob ain
umo con ol p obabli ies (TCP) om ou model. I s a es
ha in o de o con ol he umo , all cells wi h p oli e a i e
capaci y, which we iden i y wi h he compa men C, need
o be elimina ed. As de ined in he p e ious sec ion, he
model is con inuous and de e minis ic. In o de o calcula e
TCPs we need a disc e e model (numbe s o cells) and
s ochas ici y. The e o e, o low numbe s o cells (C < 1000
cells), he model is con e ed o a Ma ko bi h/dea h
s ochas ic p ocess [42] by in e p e ing e ms in he di e en-
ial equa ions as bi h/dea h p obabili ies. In a simula ion,
he umo is conside ed con olled i C eaches 0.
In addi ion o he s ochas ici y o he Ma ko model,
o ob ain popula ional TCPs we also implemen ed andom
pe u ba ions o he model pa ame e s o simula e he he-
e ogenei y o a popula ion. Fo he popula ion (a gi en
numbe o simula ions), TCP is compu ed as:
TCP =numbe o con ols
numbe o simula ions (24)
Mo e de ails abou he TCP calcula ion and implemen a-
ion a e p o ided in he Supplemen a y Ma e ial.
2.8 Expe imen al Da a and Model Fi ing
In [8] he au ho s s udied he esponse o umo s in mice
o RT (di e en ac iona ions) and αCTLA4, ei he as
mono he apies o in combina ion. Tumo cells (TSA b eas
ca cinoma cells) we e plan ed on he side o mice and le
g ow o 12 days, when hey eached a olume o ∼32 mm3.
T ea men s s a ed a ha ime, and e olu ion o umo
olumes we e moni o ed e e y 3 days. They s udied umo
esponse o di e en combina ions o RT+IT. In pa icula :
i) no ea men ; ii) RT alone, 20 Gy single- ac ion; iii) RT
alone, 3 ac ions o 8 Gy (days 12, 13 and 14); i ) RT
alone, 5 ac ions o 6 Gy (days 12, 13, 14, 15, and 16); ) IT
alone, deli e ed in 3 ac ions (days 14, 17, 20); i) combined
RT+IT, 20 Gy + 3 ac ions o IT (ii+ ); ii) combined RT+IT,
(6 Gy×5) + 3 ac ions o IT (i + ); iii) combined RT+IT,
(8 Gy×3) + 3 ac ions o IT (iii+ ); ix) combined RT+IT, (8
Gy×3) + 3 ac ions o IT a days 12, 15, 18; x) combined
RT+IT, (8 Gy×3) + 3 ac ions o IT a days 16, 18, 20. The
s udy also epo ed he ac ion o animals whe e umo
con ol was achie ed (no e idence o umo a he ime o
eu hanasia).
In [9] he au ho s p esen ed esponses o umo s o
adio he apy and αPDL1. Cance cells (TUBO b eas ca -
cinoma cells) we e implan ed in mice, and umo s we e
allowed o g ow o 14 days, when hey had olumes
a ound 120 mm3. A ha ime, ea men s s a ed and
umo olumes we e moni o ed up o day 35. The di e en
expe imen al a ms we e: i) no ea men ; ii) single dose o
12 Gy; iii) ou ac ions o αPDL1 a days 14, 17, 20 and 23;
i ) 12 Gy + αPDL1 (ii+iii).
In o de o i he epo ed e olu ion o (popula ion-
a e aged) umo olumes, we used he con inuous model.
Fi s ly, we le he modeled umo s o eely g ow un il hey
eached he ele an p e- ea men olumes epo ed in [8]
and [9]. Tha ime was de ined as e e ence (day 0), and
ea men imes a e de ined ela i e o i . No ice ha he
ime o each hose olumes di e s om he expe imen al
esul s, as we s a wi h di e en numbe s o cells han
hose expe imen ally injec ed and ou model does no aim
o desc ibe he p ocess o umo g ow h, which may be
domina ed by di e en mechanisms han umo esponse.
Tumo olumes in ou model a e compu ed by conside -
ing he popula ions o bo h umo cells ( iable and doomed)
and T-cells loca ed in he umo :
Vmodel( ) = C o ( )VC+Ta( )VT(25)
whe e VCand VTa e he olumes o indi idual umo cells
and T-cells, espec i ely.
A simula ed annealing me hod [43] was implemen ed
o ind bes i ing pa ame e s. The objec i e unc ion o
be minimized is he weigh ed sum o squa e di e ences
be ween model and expe imen al alues:
F=X
cu es X
poin s
(Vmodel −Vexp)2
u2(26)
whe e Vexp and Vmodel a e he expe imen al and model
esul s, and ua e he expe imen al unce ain ies. The op-
imiza ion me hod has been applied o all esponse cu es
o each s udy a once (i.e. he sum abo e uns o e di e en
ime poin s and di e en combina ions o adio he apy and
immuno he apy), o a oid di e en bes - i ing pa ame e s
o each cu e.
2.9 E alua ion o goodness-o - i
We ha e used he Akaike In o ma ion C i e ion (AIC) wi h
sample size co ec ion [44] o compa e bes i s ob ained
wi h di e en di ec damage e ms (1), (2) and (3). This
me hodology anks models acco ding o he likelihood o
he i , L, and numbe o ee pa ame e s o he model, k:
AIC =−2 log (L)+2k+2k(k+ 1)
N−k−1(27)
whe e Nis he numbe o expe imen al da a poin s, and Lis
he maximum likelihood (o he bes i ), calcula ed assum-
ing a no mal dis ibu ion o he expe imen al poin s and
unce ain ies. The model wi h he lowes AIC is conside ed
he bes model.
2.10 Biologically E ec i e Dose
The biologically e ec i e dose (BED) [45] was used o design
di e en adiobiologically iso-e ec i e ac iona ions. The
BED o a schedule deli e ing a o al dose Din ac ions
o dose dis gi en by:
BED =D1 + d
α/β (28)
whe e α/β is he a io o he linea and quad a ic e ms in
he LQ model.
2.11 Quali a i e Sensi i i y Analysis
We ha e pe o med a quali a i e local pa ame ic sensi i i y
analysis. In o de o do so, we ha e e alua ed he sensi i i y
o he cos unc ion o pa ame e pe u ba ions a ound bes -
i ing alues as:
Si=|F(x+∆xi)−F(x)|(29)
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T ansac ions on Compu a ional Biology and Bioin o ma ics
6
whe e Fis he cos unc ion (26), x he se o bes - i ing
pa ame e s and ∆xi= (0, ..., 0,0.1xi,0, ..., 0) (se ing a 10%
pe u ba ion wi h espec o he bes - i ing alue xi).
2.12 Implemen a ion and Pa ame e s
The model was implemen ed using di e en unc ions in
Ma lab (The Ma wo ks, Na ick, MA). The model is sol ed
by employing an explici Eule me hod [46], wi h a ime
s ep o 0.05 days (de ails abou he beha io o he me hod
a e a ailable in he Supplemen a y Ma e ial). The main
unc ions and he da a used o model i ing a e a ailable
om he Da a e se eposi o y [47].
No all model pa ame e s we e ee du ing he i
o expe imen al da a. Cell olumes in (25) we e se o
VC= 10−6mm3and VT= 2 ×10−7mm3[48]. Fo he
adiosensi i i y o T-cells we ha e ixed he LQ pa ame e s
o αT= 0.1and βT=αT/10. The biokine ic elimina ion
a es o αPDL1 and αCTLA4 we e se o µ= 0.5days−1
and ν= 0.1days−1 espec i ely, om i s o da a epo ed
in [39], [40]. The pa ame e s cha ac e izing he mi o ic de-
lay o adia ion-damaged cells we e se o τd1= 1 days,
τd2= 1.5days, which is in he ange o epo ed mi o ic de-
lays. The pa ame e , ela i e o he ac i a ion/inac i a ion
a e o T-cells is se o 5, as in [25].
On he o he hand, alues o bes - i ing pa ame e s
we e cons ained o quali a i e easonable bounding in e -
als when deemed necessa y, in o de o a oid unphysi-
cal/un easonable alues ( o example, α > 0 and β > 0
in he LQ-model, o λ > 0 o umo p oli e a ion). The
bounding in e als a e shown in Supplemen a y Table 4.
3 RESULTS
3.1 The Model Can Fi P e-clinical Da a o Tumo
Response o Combined The apies o Radia ion and
αPDL1/αCTLA4
In Fig. 2 we epo bes i s o ou model o olume dynam-
ics da a p esen ed in [8], when employing he modi ied LQ-
model o di ec adia ion dea h (2). In his da ase , when no
ea men is deli e ed, umo olumes g ow exponen ially,
αCTLA4 alone has no signi ican e ec on umo esponse,
RT alone causes a mode a e umo esponse, and he com-
bina ion o RT and αCTLA4 leads o an impo an umo
esponse, achie ing umo con ol in some cases (mos ly o
8 Gy×3). Ou model ep oduce hese p og ession pa e ns,
as shown in Fig. 2. Bes - i ing pa ame e s a e p esen ed
in Supplemen a y Table 1. We highligh he mos ele an
pa ame e s associa ed wi h p oli e a ion (λ1≃0.14 day−1),
adia ion damage (αC≃0.02 Gy−1,βC≃0.007 Gy−2,
c≃ −0.2 Gy−1/2) and he immune e ec on umo cells
(p≃24.4 day−1) and T-cells (ι≃2×10−8day−1).
Da a i ing poin o a dec ease in ela i e adiosensi-
i i y wi h inc easing dose (nega i e pa ame e cin (2)).
Such beha io can be desc ibed by he LQL model. The e-
o e, we ha e pe o med he same i wi h he LQL model
ins ead (Fig. 3), ob aining simila esul s. Bes - i ing pa a-
me e s a e epo ed in Supplemen a y Table 2, including
he mos ele an pa ame e s associa ed wi h p oli e a ion
(λ1≃0.14 day−1), adia ion damage (αC≃0.04 Gy−1,
βC≃0.017 Gy−2,x≃8.4 Gy−1) and he immune
e ec on umo cells (p≃23.3 day−1) and T-cells (ι≃
2×10−8day−1). The bes - i ing alue o he cos unc ion
is F= 29.56 and F= 42.18 o he modi ied LQ and LQL
model, espec i ely.
In Fig. 4 we epo bes i s o ou model o da a p e-
sen ed in [9], which shows umo esponses o adio he apy
and αPDL1. This da ase p esen s simila pa e ns o umo
esponse: αPDL1 alone has no signi ican e ec on umo
esponse, RT alone causes a mode a e umo esponse, and
he combina ion o RT+αPDL1 p esen s syne gy and leads
o an impo an umo esponse.
To i he da a o Fig. 4 we ha e kep ixed mos o
he bes - i ing pa ame e s ob ained when i ing Fig. 2:
only pa ame e s ela ed o he dose o αPDL1, umo cell
p oli e a ion (λ1≃0.12 day−1) and umo cell adiosen-
si i i y (αC≃0.03 Gy−1) we e allowed o a y. Because
his da ase only includes one dose pe ac ion, we ha e
used he LQ model wi h αC/βC= 10 Gy (i.e. only αCis
a ee pa ame e ). While he e a e di e ences in he clones
and umo s ha could jus i y using di e en hos - ela ed
and umo - ela ed pa ame e s, we hink ha imposing such
cons ain s on he op imiza ion poses a se ious es o ou
model, and a oids eaching good i s by o e - i ing. Bes -
i ing pa ame e s a e epo ed in Supplemen a y Table 1.
3.2 Vascula Damage May Limi he E ec i eness o
Radioimmuno he apy
La ge adia ion doses can se iously damage umo ascu-
la u e, which migh limi he in il a ion o ac i e T-cells
in he umo . This migh also explain he poo e esul s
ob ained wi h he 20 Gy single- ac ion i adia ion in [8].
In o de o es he hypo hesis ha ascula damage may
a ec he e ec i eness o adioimmuno he apy, we emo ed
he dependence o he umo cells β- e m on he adia ion
dose (see Sec ion 2.2), and we included a dose and ime
dependen T-cell in il a ing pa ame e in ou model (16), o
accoun o ascula damage and eco e y. Inspi ed by [13],
[38], we conside c i ical ascula damage o i adia ion
abo e 15 Gy, ollowed by a p og essi e eco e y o ascula
unc ion. This ac o ep esen s he ac ion o ac i e T-cells
eaching he umo .
In Fig. 5 we show bes i s o ou model o da a p esen ed
in [8]. The model p o ides a good i o he expe imen al
alues. The goodness o he i is sligh ly be e han hose
epo ed in Fig. 2 and Fig. 3: AIC = 436.24 (k= 19,
F= 25.33,N= 50), e sus AIC = 446.02 (k= 20,
F= 29.59) and AIC = 458.75 (k= 20,F= 42.18)
ob ained wi h he modi ied LQ and LQL models, espec-
i ely. Bes - i ing model pa ame e s a e shown in Supple-
men a y Table 3, including he mos ele an pa ame e s
associa ed wi h p oli e a ion (λ1≃0.14 day−1), adia ion
damage (αC≃0.02 Gy−1,βC≃0.002 Gy−2) and he
immune e ec on umo cells (p≃24.9 day−1) and T-cells
(ι≃6×10−9day−1).
In Table 1 we ank he sensi i i y o he cos unc ion o
model pa ame e s. The model is mos sensi i e o pa ame-
e s desc ibing umo cell p oli e a ion, adiosensi i i y and
immune-media ed umo cell killing.
Volumes p esen ed in p e ious igu es include umo
cells and T-cells. Ce ainly, we do no wan he model o e-
p oduce umo olumes by including low ac ions o umo
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