A Step Toward Optimization of Cancer Therapeutics [Chronobiological Investigations]
ABSTRACT An integrative physiology model has been designed, which takes into account the cell proliferation at the level of a population of cells by age-structured partial differential equations (PDEs), its control by cell cycle proteins, and the control of these molecular mechanisms by the circadian system, designed as a network of coupled oscillators also described by ODEs. Cancer growth and response to therapy by anticancer drugs have been shown to be dependent on circadian clock inputs. This multiscale modeling framework will provide clinicians with a theoretical tool to bridge the gap between the pharmaceutical clinical control level and the molecular pharmacological hidden level of drug action.
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ABSTRACT: The chronotherapy concept takes advantage of the circadian rhythm of cells physiology in maximising a treatment efficacy on its target while minimising its toxicity on healthy organs. The object of the present paper is to investigate mathematically and numerically optimal strategies in cancer chronotherapy. To this end a mathematical model describing the time evolution of efficiency and toxicity of an oxaliplatin anti-tumour treatment has been derived. We then applied an optimal control technique to search for the best drug infusion laws. The mathematical model is a set of six coupled differential equations governing the time evolution of both the tumour cell population (cells of Glasgow osteosarcoma, a mouse tumour) and the mature jejunal enterocyte population, to be shielded from unwanted side effects during a treatment by oxaliplatin. Starting from known tumour and villi populations, and a time dependent free platinum Pt (the active drug) infusion law being given, the mathematical model allows to compute the time evolution of both tumour and villi populations. The tumour population growth is based on Gompertz law and the Pt anti-tumour efficacy takes into account the circadian rhythm. Similarly the enterocyte population is subject to a circadian toxicity rhythm. The model has been derived using, as far as possible, experimental data. We examine two different optimisation problems. The eradication problem consists in finding the drug infusion law able to minimise the number of tumour cells while preserving a minimal level for the villi population. On the other hand, the containment problem searches for a quasi periodic treatment able to maintain the tumour population at the lowest possible level, while preserving the villi cells. The originality of these approaches is that the objective and constraint functions we use are L∞ criteria. We are able to derive their gradients with respect to the infusion rate and then to implement efficient optimisation algorithms.Mathematical Modelling and Numerical Analysis. 01/2006; 39:1069-1086.
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ABSTRACT: We analyse both theoretically and numerically a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments that is described in [F. Bekkal Brikci, J. Clairambault, B. Ribba, B. Perthame, An age-and-cyclin-structured cell population model with proliferation and quiescence, INRIA Research Report No 5941, 2006]. It is a physiological age and molecule-structured population model for the cell division cycle, which aims at representing both healthy and tumoral tissues. A noticeable feature of this model is to exhibit tissue homeostasis for healthy tissue and unlimited growth for tumoral tissue. In particular, the present paper analyses model parameters for which a tumoral tissue exhibits polynomial growth and not mere exponential growth. Polynomial tumour growth has been recently advocated by several authors, on the basis either of experimental observations or of individual cell-based simulations which take space limitations into account. This model is able to take such polynomial growth behaviour into account without considerations of space, by proposing exchange functions between the proliferative and quiescent compartments.Mathematical and Computer Modelling. 01/2008;
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ABSTRACT: At the tissue level, there is experimental and clinical data to suggest a cytokinetic coordination of the cell cycle with a greater proportion of cycling cells entering S-phase and mitosis at specific times of the day. The association of certain cell-cycle proteins with defined events in the cell cycle is well established and may be used to study the timing of cell-cycle phases over 24 hours. In this study oral mucosal biopsies were obtained from six normal human volunteers at 4-hour intervals, six times over 24 hours. Using immunohistochemistry, the number of positive cells expressing the proteins p53, cyclin-E, cyclin-A, cyclin-B1, and Ki-67 was determined for each biopsy and expressed as the number of positive cells per mm of basement membrane. We found a statistically significant circadian variation in the nuclear expression of all of these proteins with the high point of expression for p53 at 10:56 hours, cyclin-E at 14:59 hours, cyclin-A at 16:09 hours, cyclin-B1 at 21:13 hours, and Ki-67 at 02:50 hours. The circadian variation in the nuclear expression of cyclins-E (G1/S phase), -A (G2-phase), and -B1 (M-phase) with a normal physiological progression over time suggests a statistically significant circadian variation in oral epithelial cell proliferation. The finding of a circadian variation in the nuclear expression of p53 protein corresponding to late G1 is novel. This information has clinical implications regarding the timing of chemotherapy and radiotherapy.American Journal Of Pathology 03/1999; 154(2):613-22. · 4.52 Impact Factor
A Step Toward Optimization
of Cancer Therapeutics
mic clock – and this is most likely related to cell cycle
molecular circadian clock and cell cycle timings? How should
anticancer therapeutics be designed so as to induce efficient
recovering of such control mechanisms? To answer these ques-
tions, an integrative physiology model has been designed,
which takes into account the cell proliferation at the level of a
population of cells by age-structured partial differential equa-
nary differential equations [ODEs]), and the control of these
molecular mechanisms by the circadian system, designed as a
ancer growth and response to therapy byanticancer
clock inputs , . Indeed, tumor growth is
enhanced by perturbations of the central hypothala-
Cell Proliferation: Populations of Cells
In tissues subject to renewal, in particular in tumors, but also in
fast renewing healthy tissues such as gut, skin, and bone mar-
row, the individual cell is the fundamental level of description
for the interacting molecular mechanisms at stake in cell cycle
progression and circadian clock timing systems. But cancer
growth and healthy tissue-controlled proliferation are a matter
of cell population dynamics. This leads us to consider age-
structured cell populations, which are described, for each phase
of the cell division cycle, classically divided in phases G1,
S?G2, and M, by evolution equations for cell densities depend-
ent on time and age spent in each phase. Cells in renewing tis-
sues are either in the quiescent, G0, phase (where no growth
occurs),orintheproliferating phasesG1,S (for DNAsynthesis,
i.e., genome duplication), G2, and M (for mitosis, or actual cell
division). Exchanges of cells between phases G0and G1may
point has not been reached; afterwards, a proliferating cell is
is stopped, e.g., by anticancer drugs, in its progression at cell
occur at the G1=S and G2=M transitions. The transition rates
from one phase to the following one (hereafter noted Ki!iþ1in
the model) are the main targets of the circadian (and pharmaco-
An Age-Structured PDE Model of the Cell Cycle
The model chosen – is linear and of the Von Foerster-
McKendrick type. For each phase i (1 ? i ? I) of thecell divi-
sion cycle, ni¼ ni(t,a) denotes the density of cells of age a in
phase i at time t. Inside each phase, the variation of the density
of cells is either due to a spontaneous death rate di¼ di(t,a) or
to a transition rate Ki!iþ1¼ Ki!iþ1(t,a) from phase i to phase
i þ 1 (withthe conventionI þ 1 ¼ 1):
við0Þniðt,a ¼ 0Þ ¼
2 ? i ? I
n1ðt,a ¼ 0Þ ¼ 2
@a½viðaÞni? þ ½diþ Ki!iþ1?ni¼ 0,
Ki?1!iðt,aÞ ni?1ðt,aÞ da,
KI!1ðt,aÞ nIðt,aÞ da,
The transition rates are chosen as wi(t)1fa?aig(a): a minimum
age aimust be spent in phase i, and a dynamic control t7!wi(t)
isexertedonthetransitionfromphaseitophasei þ 1.Thefunc-
tions wiintegrate physiological control (hormonal or circadian)
as well as pharmacological influences. Another possibility to
take circadian control into account would be by an action on the
death rates di, an apparently natural representation since anti-
cancer drugs increase death rates in cell populations. But it can
be shown that at least by itself such control representation is not
compatible with observations from laboratory experiments on
tumor growth involving circadian clock disruption , . The
mainoutput of thismodelisa (positive) growthexponent,k, the
first eigenvalue of an underlying differential operator, which
determines its asymptotic behavior; its existence is granted by
a?0ni(t,a)da ¼ 1,vi(a) denotes a speed function
of age a in phase i with respect to time t, and Ki!iþ1(t,a) ¼
Tumor Cells: Unlimited Growth
is bound to show exponential growth. The first eigenvalue, or
BY JEAN CLAIRAMBAULT
Digital Object Identifier 10.1109/MEMB.2007.907363
Physiologically Based Modeling of Circadian
Control on Cell Proliferation
IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE0739-5175/08/$25.00©2008IEEE JANUARY/FEBRUARY 2008
Malthus exponent k, of the underlying differential operator is
positive and governs the asymptotic behavior (in time t) of the
solutions ni. This means that for all i, the normalized solutions
t7!e?ktni(t) are bounded, asymptotically constant if w is con-
stant, and asymptotically periodic if the wiare periodic, with
the same common period. This Malthus exponent k may be
assessed in the case of solid tumors by in vivo tissue growth
measurements (only approximately, if one neglects the pres-
ence of necrotic material) by k ¼ (ln2)=Td, if Tdis the appa-
Healthy Cells: Tissue Homeostasis
Inthecase ofhealthy cells, therecanbe noexponentialgrowth
and tissue homeostasis must be ensured, in the sense that for
healthy renewing tissues cell loss must be made up for, and
with no excess, by influx from newly formed cells. This can
be modeled by the introduction of a G0(or quiescent) phase
exchanging cells with the G1phase, taking into account in the
control of these exchange mechanisms a limitation by cell
density dependent inhibition. This has been done in an
extended version of this model including cyclin concentration
as a structure variable . In this case, the Malthus exponent
ulation of cells is evolution to stationarity or periodicity, and
there is no exponential growth, except when the total cell pop-
ulation is small, but on the contrary convergence toward a
steady state when thetotal cellpopulation grows.
An interest in simultaneously representing a tumor and a
healthy cell population is, in the perspective of therapeutic
optimization, to control unwanted toxicity on healthy tissues,
a constant side effect in anticancer therapies. These unwanted
effects on the healthy cell division cycle are also dependent on
the control of cell cycle phase transitions by the circadian
clock, but possibly with phase differences, by comparison
with tumors: a phase delay of 12 h between circadian peaks
of anti-tumor therapeutic efficacy and circadian peaks of
unwanted toxicity to healthy tissues is usually observed, what-
ever the drug . The therapeutic control objective may be
seen as to obtain a negative Malthus exponent for tumor cells
and zero growth forhealthy cells.
Cell Proliferation Control: Cyclins
and Cyclin-Dependent Kinases
Control of Cell Cycle Phase Transitions
by Cyclins and Cyclin-Dependent Kinases
Cyclins and their activating kinases (cyclin-dependent kinases
[CDKs]) are the proteins that are the most important determi-
nants for the processing through cell cycle phases and transi-
tions between phases. Cyclin E associated with CDK2
controls G1=S transition, and its dimerization is dependent on
protein p21, and Cyclin B associated with CDK1 is essential
for G2=M transition, and the dimerization is controlled by the
clock-controlled kinase Wee1 ,, .
Circadian Control on Cyclins and CDKs
Of these two mechanisms, the most known is the inhibition of
G2=M transition by the circadian clock-controlled kinase
Wee1, which deactivates CDK1, blocking cells in G2. It has
been modeled by Goldbeter in  with an ODE system in
which Wee1 is a parameter. This CDK system oscillates tran-
siently with an intrinsic frequency that depends on its parame-
ters, but it may be entrained at a 24-h period by Wee1 if Wee1
is an output of the molecular circadian clock (in fact, the clock
protein Bmal1 plays this role whereas the control on G1=S
through p21 is exerted by PER). The entrainment of the nor-
malized M-cell population t7!e?ktRþ1
the population, by a 24 h-entrained CDK1 as function w1in
shown inFigure 1.
This means that for t? ! þ 1 the total population in phase
i, NiðtÞ ¼Rþ1
being the sameforall phases1 ? i ? I.
n2(t,a) da, in a two-
phase model (n1¼ cells in G1? S ? G2, n2¼ cells in M) of
ni(t,a) da is of the form Ni(t) ¼ ektWi(t),
where Wiis a periodic function with the same period as wi, k
CDK1 (Lower Trace)
M Cells (Upper Trace)
Fig. 1. Entrainment of G2=M transition in a two-phase cell
cycle population model (n1: G1? S ? G2cells; n2: M cells)
by the circadian clock: CDK1 kinase (lower trace), taken as
function w1in the cell cycle model and entrained by Wee1
(here a square wave, not shown, of 4 h duration with 24 h
period) in Goldbeter’s mitotic oscillator model, and normal-
ized population of cells in M phase (upper trace) (abscissae:
tens of hours; ordinates: arbitrary units).
Cancer growth and response to therapy by
anticancer drugs have been shown to be
dependent on circadian clock inputs.
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Pharmacological Control on Cyclins and CDKs
Cell cycle control proteins such as p53 act both at the G1=S
and G2=M checkpoints as sensors of DNA damage and effec-
tors of cell cycle arrest by inhibiting the formation of Cyc
E/CDK2 and Cyc B/CDK1 complexes. Anticancer drugs such
as alkylating agents act by damaging DNA and thus provoke
cell cycle arrestby triggering p53 and its effects on phase tran-
sitions at G1=S and G2=M (). It has been shown that p53,
on the one hand, and many cellular enzymatic drug detoxifica-
tion mechanisms (such as reduced glutathione), on the other
hand, are dependent on the cell circadian clock, showing 24-h
periodicity in their gene expression (). Hence, the molecu-
lar circadian clock exerts its control on cell cycle transitions
both physiologically and when an external pharmacological
control is applied. In this modeling frame, the target of both
controls will be on the time-dependent transition functions
t7!wiðtÞ introduced earlier.
The Circadian System: A Network of Oscillators
This circadian clock control on the cell division cycle is not
independent of individual or environmental factors. Although
this clock is endowed with an intrinsic circa 24-h (circa
diem) period given by a hypothalamic pacemaker, the supra-
chiasmatic nuclei (SCN), it is dependent on photic inputs
(entrainment by the light/dark cycle through the retinohypo-
thalamic tract) and also on the disruptive input of circulating
molecules such as cytokines, often elevated in cancers,
which may perturb its amplitude so deeply that any physio-
logical body circadian rhythm (temperature, cortisol, rest/
activity, etc.) becomes undetectable. Indeed, it has been
shown that patients with cancer showing disrupted circadian
rhythms are less responsive to chemotherapy and have
poorer prognosis than do patients with preserved circadian
rhythmicity , .
The Network and Its Constituents
The body circadian clock , , as far as its control on
peripheral cell proliferation is concerned, may be seen as an
orchestra consisting in the same basic individual oscillators:
cell molecular circadian clocks, since the molecular mecha-
nisms are the same whatever the cells, its peripheral constitu-
ents (the musicians) being slaves, not communicating
together, to the central pacemaker located in the SCN (the
conductor), located inthe hypothalamus.
Various physiological ODE models of individual circadian
clocks have been published in the last ten years. They rely on
transcriptional regulation, a mechanism possibly yielding limit
periodic behavior , , . The simplest such physio-
logical model of circadian clock is the three-dimensional ODE
model of FRQ (or PER) protein regulation in Neurospora
Crassa , and it was chosen for the individual cell clock
Knþ Zn? Vm
Kdþ PER? k1PER þ k2Z
¼ ksRNAm? Vd
dt¼ k1PER ? k2Z,
where PER and mRNA stand for the PER cytoplasmic protein
and its messenger RNA concentrations and Z for a transcrip-
tional retroinhibition factor linkedto PER synthesis.
The Conductor: The SCN
Diffusive Neuronal Coupling in the SCN
From this individual clock model,anetwork of circadianoscil-
latorsresultsby diffusivecoupling of PER. Such couplingmay
be physiologically achieved through gap junction connections
between suprachiasmatic neurons, through local release in the
intercellular space of neurotransmitters such as vasointestinal
peptide, abundant in the ventrolateral part of these nuclei,
where the main pacemaker is supposedto be located, andbind-
ing to VPAC2membrane receptors on these same neurons, or
even through electrical signaling with glial participation .
This coupling is thus not necessarily instantaneous, possibly
relying mainly on slow tissue diffusion through gap junctions
or ligand-receptor connections, but nevertheless may be con-
sidered as fast at a time scale of 24 h, relevant for cell division
Knþ ZðiÞn? VmðiÞ
Kdþ PERðiÞ? k1PERðiÞ
¼ ksmRNAðiÞ ? Vd
þ k2ZðiÞ þ Ke
½PERðjÞ ? PERðiÞ?
¼ k1PERðiÞ ? k2ZðiÞ,
where 1?i,j?N, N being the number of neurons connected
in the pacemaker network. The degradation rate Vmof the
messenger RNA is supposed to differ from one neuron to the
other (with random distribution around a central value) and
thus holds variability in individual PER period. The output of
this pacemaker network, transmitted to the periphery, is an
In the case of healthy cells,
there can be no exponential growth and
tissue homeostasis must be ensured.
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higher amplitude in its periodic variations (i.e., good synchro-
nization) asthe couplingstrength Keisstronger.
i¼1PER(i) between neurons, which shows as
Synchronizing Photic Inputs
It is also known that light, through the retinohypothalamic
tract, modulates, equally for all neurons, the transcription rate
Vs, which is thus supposed to be Vs¼ Vs01 þ Lcos(2pt)=24
This modulation actually entrains the suprachiasmatic pace-
maker, initially endowed with a period dependent on the
parameters, e.g., 21 h 30 min with our parameter set, to a
forced 24-h period, provided that the entrainment by light L
is strong enough to overcome the spontaneous period result-
ing from the coupling between neurons inside the pacemaker.
A higher risk of cancer is related with circadian disruption
induced by light/dark rhythm perturbations in shiftwork (see
review in , and the present model aims at taking such
perturbations into account.
Disruptive Inputs from Cytokines and Drugs
Circulating molecules such as cytokines (interferon, interleu-
kins), either secreted by the immune system in the presence of
cancer cells or delivered by therapy, are known to have a dis-
ruptive effect on the circadian clock, most likely at the central
pacemaker level , and elevated levels of cytokines have
been shown to be correlated with fatigue , asymptom con-
stantly found in cancer, which presents close relationships
with the jet-lag of transmeridian flights. This has also been
found with anticancer drugs and is supposed to be a result of
drug toxicity on the SCN. It may be taken into account in the
model by an additive effect on variable Z, which represents
transcriptional inhibition,in the central clocks.
Transmission Pathways from the Center
to Peripheral Cells
These pathways are not completely known but the autonomic
nervous system, and neurohormonal ways using the hypothal-
amocorticosurrenal axis, is a likely candidate to this role .
The messengers may be represented by supplementary
evolution equations describing in a simple way a chain
between messengers: intercentral, hormonal, such as adreno-
corticotropic hormone (ACTH), and peripheral tissue terminal
(such as cortisol)
dt¼ k3PERðSCNÞ ? k4U
dt¼ k4U ? k5V
b þ V? cW:
The Musicians: Peripheral Cell Circadian Clocks
This terminal control is supposed, as in the case of cytokines
in the pacemaker, to be exerted at the transcriptional level
through a modification of variable Z in individual peripheral
will act itself on each peripheral circadian clock by additively
modifying to Z þ rW the feedback variable Z, where r is a
parameter. If the central pacemaker rhythm is strong enough,
i.e., constituted of enough synchronized neuronal oscillators, it
will entrain each peripheral individual clock at a 24 h-rhythm;
but there is no intercellular communication between peripheral
cells unlike between neurons in the central pacemaker. In this
setting, cell cycle phase transition functions wiin peripheral
tissues will be supposed to be tissue-averaged versions of local
PER proteins., i.e., wi¼ kiPER, where ki are constants and
PER isa time-dependent function, averaged ofPER(j) over the
cells (1 ? j ? N, each j cell having its own period, distributed
around a central value) of a peripheral tissue assembly. They
can also be CDKs, as in Figure 1, controlled by a protein such
as Wee1, itself resulting from an averaging of PER (or rather
Bmal1 in the case of Wee1, but PER and Bmal1 are in anti-
phase, i.e., phase delayed by 12 h in their circadian rhythm).
To illustrate this, Figure 2 shows variables mRNA, PER, and Z
of the model circadian clock averaged in a population of
peripheral cells subject to entrainment by the central pace-
The behavior of the cell cycle model, which is presented here
in simple form, completely linear, limits its applicability to
the early stages of tumor growth. They may be considered as
the most important because even when a tumor is discovered
at an advanced stage of its growth, the first administered anti-
cancer treatments, when they do not eradicate it, may shrink it
to cell population conditions putting it in such early stages.
Besides, an extended version of this model with nonlinear
feedback by cyclin concentration and total cell population
number ,  opens it to the representation of different types
of tumor behavior, namely, polynomial in time, as has been
experimentally observed .
The circadian system representation, with its central and
peripheral components, is meant to be able to account for
disrupting inputs to it from tumor-produced cytokines or anti-
cancer drugs, and their possible overcoming by shielding mol-
ecules such as epidermal growth factor receptor (EGFR)
antagonists. An analysis of its effects onto cell cycle phase
transitions (by functions wiin this model) remains to be done.
More detailed representations of the molecular circadian
clock, as recent ones by Leloup and Goldbeter , with
0 100200300 400500 600700
Fig. 2. Three epochs of 240 h for variables mRNA, PER and Z
of a peripheral circadian clock (average of local circadian
clocks): a) without entrainment by light (L ¼ 0); b) with
entrainment (L ¼ 1); c) without (L ¼ 0) (abscissae: hours;
ordinates: arbitrary units).
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Bmal1, Clock, Per, Cry, and other genes, rather than the sim-
plifiedmodel usedhere may also be used.
This article presents a modeling frame for the circadian control
of cell and tissue proliferation. It aims at providing a rationale
for therapeutic optimization of cancer chemotherapies, that is,
maximization of tumor cell kill while shielding healthy renew-
ing tissues from unwanted toxic side effects, by using synchro-
nization of the drug delivery schedule and its cell processing
with intrinsic cell cycle timing. Such synchronization naturally
relies on the circadian system and it has been in use in clinical
,  and in theoretical ,  settings in oncology with
macroscopic modeling for drug delivery regimen and therapeu-
tic efficacy and unwanted toxicity representation. Anticancer
drugs are delivered at the whole organism level but act at the
cell and tissue level on cell cycle control mechanisms. This
multiscale modeling framework will provide clinicians with a
theoretical tool, still under construction, to bridge the gap
between the pharmaceutical clinical control level and the
molecular pharmacological hidden level of drug action; opti-
mal design of pharmacological control on the cell cycle thus
may relyon the knowledge of the natural synchronizing control
of both cell proliferation and cell drug processing mechanisms
by the circadian system, which may be routinely assessed by
noninvasive measurements in the clinic . Future modeling
developments will aim at designing practical clinical rules for
dynamic drug delivery schedule optimization based on molec-
ular pharmacokinetic-pharmacodynamic data for the drugs in
use and drug enzymatic metabolism and circadian profiles for
patients under anticancer treatment, as recently suggested .
This article is the extended version of a paper first published
under the same title, pp. 176–179 in the proceedings of the
28th Annual International IEEE-EMBS Conference, New
York, 2006. I thank Benoı ˆt Perthame and Fadia Bekkal Brikci
for their contributions to cell cycle modeling, Be ´atrice
Laroche for helping deriving the circadian system oscillator
network model, and Sonia Montanes for helpful information
on synchronization in the SCN.
This work was supported by the EU Network of Excellence
BIOSIM, contract 005137.
Jean Clairambault received a Ph.D. in
mathematics from Universite ´ Paris VII-
Denis Diderot in 1978 and an M.D. from
Universite ´ Paris VI-Pierre-et-Marie-Curie
in 1989. He is a researcher at Institut
National de Recherche en Informatique et
en Automatique (INRIA), Rocquencourt,
France. His research interests include mo-
lecular pharmacokinetics-pharmacodynamics of anticancer
drugs, the cell cycle and its control by physiological and
pharmacological inputs for cancer therapeutics, the modula-
tion of these effects by the body circadian clock, and anti-
cancer therapeutic optimization.
Address for Correspondence: Jean Clairambault, Institut
National de Recherche en Informatique et en Automatique
(INRIA), Domaine de Voluceau, BP 105, F78153 Rocquen-
court,France. E-mail: firstname.lastname@example.org.
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