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Xavier Dupuis
To cite this version:
Xavier Dupuis. Optimal control of leukemic cell population dynamics. Mathematical Modelling of Natural Phenomena, EDP Sciences, 2014, Issue dedicated to Michael Mackey, 9 (1), pp.4-26.
�10.1051/mmnp/20149102�. �hal-00858208�
0249-6399ISRNINRIA/RR--8356--FR+ENG
RESEARCH REPORT N° 8356
August 2013
Optimal control of
leukemic cell population dynamics
Xavier Dupuis
RESEARCH CENTRE SACLAY – ÎLE-DE-FRANCE 1 rue Honoré d’Estienne d’Orves Bâtiment Alan Turing
XavierDupuis
∗
Projet-TeamsCOMMANDS
ResearhReport n° 8356August201325pages
Abstrat: We areinterestedin optimizingthe o-administrationoftwodrugsfor someaute
myeloidleukemias(AML),andwearelookingforinvitroprotoolsasarststep. Thisissuean
beformulatedasanoptimalontrolproblem. Thedynamisofleukemiellpopulationsinulture
isgivenbyage-struturedpartialdierentialequations,whihanbereduedtoasystemofdelay
dierential equations, and where the ontrols represent the ation of the drugs. The objetive
funtion relies on eigenelementsof the unontrolled model and on generalrelativeentropy, with
theideatomaximizetheeienyoftheprotools. Theonstraintstakeintoaountthetoxiity
ofthedrugs. Wepresentinthis paperthemodelingaspets,aswellastheoretialandnumerial
resultsontheoptimalontrolproblemthatweget.
Key-words: Aute myeloidleukemia,optimal ontrol,delay dierential equations, population
dynamis,generalrelativeentropy,invitrotherapeuti optimization
Thisworkispartof theDIMLSC projet ALMAon theAnalysisofAuteMyeloidLeukemia. Itbrings
together liniians andbiologistsof the Insermteam18ofUMRS872 (Jean-PierreMarie,Pierre Hirsh,Ruo-
PingTang, FannyFava,AnnabelleBallesta, FatenMerhi)andmathematiiansof theInria teamsBang(Jean
Clairambault,AnnabelleBallesta),Diso(CatherineBonnet,JoséLuisAvila)andCommands(J.FrédériBon-
nans,XavierDupuis).TheauthoralsothankFabienCrausteandThomasLepoutrefortheirusefulinsightsand
advie.
∗
CMAP,EolePolytehnique&InriaSalay,91128Palaiseau,Frane.
E-mail:xavier.dupuismap.polytehnique.fr
Résumé: Noussommesintéressésparoptimiserlao-administrationdedeuxmédiaments
pourertainesleuémiesaigüesmyéloïdes(LAM)etnousherhonsdesprotoolesd'administra-
tion in vitro dans un premier temps. Cela peut se formuler omme un problème de ontrle
optimal. Ladynamiquedepopulationsdeellulesleuémiquesenultureestdonnéepardeséqua-
tionsauxdérivéespartiellesstruturéesenâge,quipeuventserameneràunsystèmed'équations
diérentielles àretards, et où lesontrlesreprésentent l'ationdes médiaments. La fontion
objetifestdénieàpartird'élémentspropresdumodèlenonontrléetd'unpriniped'entropie
relativegénéralisée,avel'idéedemaximiserl'eaitédesprotooles. Lesontraintesprennent
en ompte la toxiité desmédiaments. Nous présentonsdans e rapport lesaspets de mod-
élisation, ainsi que desrésultats théoriques et numériques sur le problèmede ontrle optimal
obtenu.
Mots-lés : Leuémies aigües myéloïdes, ontrle optimal, équations diérentielles à re-
tards,dynamique de populations, entropie relative généralisée, optimisation thérapeutique in
vitro
1 Introdution
Autemyeloidleukemias(AML)areanersofthemyeloidlineageofwhitebloodells. Thepro-
essofbloodprodution,alledhematopoiesis,takesplaeinthebonemarrow,withhematopoi-
eti stemells(HSC)at itsroot. HSC havetheabiltytoself-renew, i.e. todivide withoutdif-
ferentiating,andtodierentiatetowardsanylineageofblood ellsbydividingintoprogenitors.
These progenitorsare ommitted stem ellswhih follow apath of dienrentiation, produing
ells whih are moreand more engagedinto one lineageand lose progressivelytheir ability to
self-renew. One they are fully mature and funtional, ells of eah lineage are released into
the bloodstream. Thehematopoiesis onsists in theregulation of theself-renewaland thedif-
ferentiationofellpopulations[22℄. InAML, thedierentiationisblokedat someearlystage,
leadingto theaumulationofimmaturewhitebloodells,alledblasts,ofthemyeloidlineage.
Thisblokadebeingassoiatedwithaproliferationadvantage,theblastsquiklyrowdthebone
marrowandareeventuallyreleasedintothebloodstream.
Oneoftherstmathematialmodelonhematopoiesiswasproposedin 1978byMakeyand
foused on the HSC population dynamis [18℄. Makey onsidered two phases in his model,
a resting phase and a proliferating phase, and desribed the dynamis of the two HSC sub-
populationsbyasystemofdelaydierentialequations;theseequations anbejustiedbyage-
strutured partial dierential equations. To represent the blokade of the dierentiation in
AML, Adimy etal. onsidered thedynamis ofellpopulationsofseveral maturitystages and
developpedamulti-ompartmentalmodel,whereeahompartmentrepresentsamaturitystage
and is again divided in two phases [2℄. Özbay et al. proeeded with the stabiliy analysis of
this delay dierential system in [21℄, and Avila et al. rened the model in [3℄ by onsidering
more than two phases per ompartement and modeling the fast proliferation in AML. Stiehl
and Mariniakalso proposed a multi-ompartmental model onleukemias[25℄; theyonsidered
healthy and leukemi ellpopulations,but did notdistinguishresting and proliferating phases
andthusdidnotgetdelays.
Thetreatment for mostof thetypes of AML is ahallenge [24℄. Cliniians of the depart-
ment of hematology at Saint-Antoine hospital in Paris would be interested for some ases in
o-administratingtwodrugs: aytotoxi(Araytin),whihenhaneselldeath,andaytostati
(AC200), whih slowsdownproliferation. A rststepis to determinehowsuh aombination
should be sheduled in in vitro experiments. Tothat purpose, biologists of the same hospital
havesampledbloodfrompatientswithAML,sortedanerblasts,andarriedoutleukemiell
ultures. Thenumberof ells,theirstate in theellyle, and theirmaturitystage havethen
beendailymeasuredduring5days,withoutandwitheahofthetwodrugsatdierentonstant
onentrationsin theulture[4℄.
Inthispaper,weidealizetheseexperimentsandonsiderleukemiellultureswithvarying
onentrationof bothdrugs. Weare lookingfor in vitroprotoolsof drugsadministration,i.e.
shedules of the onentration of both drugs during the experiment, whih are as eient as
possiblewithoutbeingtootoxi. Toformulatethisissueasanoptimalontrolproblem,astate
equation,anobjetivefuntion,andonstraintshavetobeenset.
Thestate equation models theell population dynamis under the ationof the drugs; we
onsideranage-struturedmodelwithonematurityompartement,dividedinonerestingphase
and one proliferating phase. Adimy and Crauste used suh a model in [1℄ to represent the
dependeneofelldeathandproliferationongrowthfators. Here,theationoftheytotoxion
elldeathisage-dependent,andthedrugonentrationsarenotsolutionsofevolutionequations
but are ontrol variableswhih denean in vitro protool. Gabriel et al. identied theation
of a drug induingquiesene (erlotinib) with afration of quiesentells in [11℄. Theation
of the ytostati in our model is also represented by a fration of resting ells, whih is here
time-dependent,andnotbyavaryingveloityin theproliferatingphaseasHinowetal. in[15℄.
Seealso[6℄aboutthemodelingoftheationofthedrugs.
The objetive funtion aims at minimizing the leukemi ell population at the end of the
experiment, in order to maximize the eieny of the orresponding protool. Its denition
atuallyrequiresalongtimeasymptotianalysisto avoidanhorizoneet. Thisanalysisrelies
onthespeializationofthegeneralrelativeentropyprinipleintroduedbyMiheletal. [20℄to
ourmodel. Variouskindsofobjetivefuntionsexistinthelitterature: nalormaximalnumber
oftumor ells[5℄, nal tumorvolume [16℄,performane index[17℄, oreigenvalue[7℄;the useof
anage-dependentweightgivenbyeigenelementsinthispaperseemstobenew.
The onstraintsome from biologialbounds ontheation ofthe drugsand from maximal
umulative doses that weimpose to limit the toxiity of the protools, asin [16℄; there is no
healthy populationin our model on whih weould set a toxiity threshold asin [5, 7℄. The
optimizationproblemthat wegetisequivalent,bythemethodof harateristis,toanoptimal
ontrol problem of delay dierential equations. Forsuh a problem, optimality onditions are
available in the form of Pontryagin's minimum priniple [14℄; it an also be redued to an
undelayedoptimalontrolproblem[12,13℄, andthensolvednumeriallybystandardsolvers.
Thepaperisorganizedasfollows. InSetion2,wemodelthepopulationdynamisunderthe
ationof the drugs. Setion 3ontains theanalysis of this model, inluding ageneral relative
entropy priniple and a longtime asymptoti analysis. The optimal ontrol problem is set in
Setion 4, and theoretial resultsand numerial optimal protools are presented in Setion 5.
Thepreise statement ofPontryagin'sminimumpriniplefor ourproblemhas beenpostponed
totheappendix, togetherwiththeparametersusedforthenumerialresolutions.
2 Modeling
Wepresenthere the dynamis of leukemi ell populations in ulture and under the ation of
thetwodrugs.
2.1 Cell populations
Weonsideraleukemiellpopulation,invitro,andwedistinguishtwosub-populations[1,18℄:
therestingells,whihareinative(
G
0 phase),andtheproliferatingells,whihareengagedin theiryle(G
1SG
2M
phase).Restingellsareintroduedintotheproliferatingphaseatarate
β
,independentlyofthetime spentintherestingphase. ConsideringthattheproliferationisunontrolledinaseofAML,wedonotrepresentanyfeedbakfrom aellpopulation[18,19℄oragrowthfator[1℄,and thus
β
isonstant in ourmodel.
Proliferating ells die by apoptosis at arate
γ
, and ifit doesnot die, aell divide duringmitosis,afteratime
2τ
spentinthephase,intwodaughterellswhihentertherestingphase.Weonsider thatthe duration of the proliferating phase
2τ
is the same for all ells; this isnottruebiologially[2℄but oneanthinkof
2τ
asanaverageduration[19℄.We struture the proliferating population by an age variable
a
whih represents the timespentin theproliferating phasebyaell. Wedenote by
R(t)
therestingpopulationat timet
,andby
p(t, a)
theproliferatingpopulationdensitywithagea
attimet
.2.2 Ation of the drugs
Thetwodrugsareaytotoxi(Araytin)andaytostati(AC220).
TheytotoxidamagestheDNAoftheellsduring the
S
sub-phaseoftheiryle;itresultsin anextradeathrate. Tosimplifyfurtheralulusandnumerialissues,weonsiderthat
u(t)
,thedeathratedue totheytotoxiat time
t
, aetsthe seond-halfoftheproliferatingphase, i.e. proliferatingellswithagea ∈ [τ, 2τ]
.Theytostatiinhibitsareeptor tyrosinekynase (Flt3)ofthe ellsin theresting phase;it
resultsinafration
k(t)
ofinhibitedellsamongtheresting ells,whihannomoreentertheproliferating phase. The global introdution rate to the proliferating phase at time
t
is then(1 − k(t))β
.τ τ
(1 − k(t))β
γ γ + u(t)
R(t) p(t, a)
proliferating phase resting phase
×2
Figure1: Themodel.
Wedenote by
v(t)
theinhibition ratedue to the ytostatiat timet
, and byα
the rateofnaturaldis-inhibition. Weonsider thatthedynamisof
k
isgivenbydk
dt (t) = v(t)(1 − k(t)) − αk(t).
(1)Theationratesdue to thedrugsare inreasingfuntions oftheir onentration inthe ell
ulture, thelatterbeinghosenduring in vitroexperiments. Thusweonsider thatweontrol
diretly theationrates
u
andv
.2.3 The age-strutured model
Thedynamisoftheellpopulationsisgivenbythefollowingpartiallyage-struturedsystem:
dR
dt (t) = −(1 − k(t))βR(t) + 2p(t, 2τ)
(2)∂p
∂t (t, a) + ∂p
∂a (t, a) = −(γ+χ
(τ,2τ)(a)u(t))p(t, a) 0 < a < 2τ
(3)p(t, 0) = (1 − k(t))βR(t)
(4)The equation (2) isa balane equationfor the resting phasebetween theoutwardand inward
ow;thetransportequation(3) desribestheevolutionof theageohortsofproliferating ells,
sine they are aging with veloity
1
; the boundary ondition(4) givesthe inward ow to theproliferatingphase.
2.4 A ontrolled version of Makey's model
Wedenoteby
P
andP
2thetotalproliferatingpopulationandsub-populationintheseond-half ofthephase,respetively:P (t) :=
Z
2τ 0p(t, a)da, P
2(t) :=
Z
2τ τp(t, a)da.
Formally,and this ouldbejustied withthe resultsofSetion 3, ifwe dierentiate
P, P
2 andusethemethodof harateristisasin[1℄, wederivefrom (1) -(4)thefollowingsystemof delay
dierentialequations:
dR
dt (t) = −(1 − k(t))βR(t) + 2(1 − k(t − 2τ))βR(t − 2τ)e
−(
γ2τ+Rtt−τu(s)ds)
(5)
dP
dt (t) = −(γP (t) + u(t)P
2(t)) + (1 − k(t))βR(t)
(6)− (1 − k(t − 2τ))βR(t − 2τ)e
−(
γ2τ+Rtt−τu(s)ds) dP
2dt (t) = −(γ + u(t))P
2(t) + (1 − k(t − τ))βR(t − τ)e
−γτ (7)− (1 − k(t − 2τ))βR(t − 2τ)e
−(
γ2τ+Rtt−τu(s)ds) dk
dt (t) = v(t)(1 − k(t)) − αk(t)
(8)Unsurprisingly,wegetaontrolledversionofMakey's1978model[18℄. Theoriginalmodelisa
systemoftwodierentialequationswithonedisretedelay,andanonlinearityin
β
;itisoneoftherstmathematial model ofthedynamis of hematopoietistemells(HSC), whih are at
therootof thehematopoiesis,theproessof bloodprodution. We havein (5)-(8) twoontrol
variables,
u
andv
,andtwoextrastatevariables,P
2 andk
,beauseoftheontrols.As we willexplain in Setion 4.1, the age-struturein theproliferating populationatually
matters,andthus itisofinterestto analysetheage-struturedmodel(2)-(4).
3 Analysis of the age-strutured model
3.1 Existene of solutions
Given
(β, γ) ∈ L
∞loc(0, ∞) × L
∞loc((0, ∞) × (0, 2τ))
and(R
0, p
0) ∈ R × L
∞(0, 2τ)
,weonsiderthesystem
dR
dt (t) = −β(t)R(t) + 2p(t, 2τ) 0 < t
(9)∂p
∂t (t, a) + ∂p
∂a (t, a) = −γ(t, a)p(t, a) 0 < t, 0 < a < 2τ
(10)p(t, 0) = β(t)R(t) 0 < t
(11)withtheinitialondition
R(0) = R
0, p(0, ·) = p
0.
(12)Wefollow[10℄forthenotionofsolution.
Denition3.1. Wesaythat(10)holdsalongtheharateristisa.e. ifandonlyifthereholds,
fora.a.
(t, a) ∈ (0, ∞) × (0, 2τ)
,p(s, a + s) = p(0, a) −
Z
s 0(γp)(θ, a + θ)dθ
fora.a.s ∈ (0, 2τ − a),
(13)p(t + s, s) = p(t, 0) − Z
s0
(γp)(t + θ, θ)dθ
fora.a.s ∈ (0, 2τ).
(14)Lemma3.2. If
p ∈ L
∞loc((0, ∞) × (0, 2τ))
issuhthat (10)holdsalong the harateristisa.e., thenp
is Lipshitz along the harateristis{t − a = c}
for a.a.c
,t 7→ R
2τ0
p(t, a)da
is loallyLipshitzandthere holdsa.e.
d dt
Z
2τ 0p(t, a)da = p(t, 0) − p(t, 2τ) − Z
2τ0
(γp)(t, a)da.
Proof. The rst assertion follows from (13) -(14). For the last two assertions, it is enough to
ompute
R
2τ0
p(t, a)da
usingthesamerelations.Denition3.3. Asolutionof (9)-(12)isany
(R, p) ∈ W
loc1,∞(0, ∞) × L
∞loc((0, ∞) × (0, 2τ))
suhthat(9)holdsa.e.,(10)holdsalongtheharateristisa.e.,(11)holdsa.e.,and(12)holds.
Lemma3.4. Given any
(β, γ)
and(R
0, p
0)
,thereexistsauniquesolution(R, p)
of (9)-(12). If(β, γ)
and(R
0, p
0)
arenon-negative, then(R, p)
isnon-negative. Moreover, deningΓ : (t, a) 7→
(R
t0
γ(s, a − t + s)ds
if0 < t < a < 2τ, R
a0
γ(t − a + s, s)ds
if0 < a < 2τ, a < t,
if
β
,Γ
,p
0 areloally Lipshitzandp
0(0) = β(0)R
0,thenp
isloally LipshitzandR ∈ W
loc2,∞.Proof. Fora.a.
c ∈ (−2τ, 0)
,p
isdeterminedon{t − a = c}
byp(t, a) = p
0(a − t)e
−R0tγ(s,a−t+s)ds.
Then(9)beomesalinearODEon
(0, 2τ)
, fromwhih wegetR
,and thenp
on{t − a = c}
fora.a.
c ∈ (0, 2τ)
,andsoon. Thesignof(R, p)
follows.Observethata.e. on
{t − a > 0}
,p(t, a) = β(t − a)R(t − a)e
−R0aγ(t−a+s,s)ds.
Theontinuityof
p
on{t − a = 0}
isequivalenttop
0(0) = β(0)R
0.3.2 General relative entropy
Weintroduethedualsystemassoiatedwith(9)-(11)
dΨ
dt (t) = β(t)Ψ(t) − β(t)φ(t, 0) 0 < t
(15)∂φ
∂t (t, a) + ∂φ
∂a (t, a) = γ(t, a)φ(t, a) 0 < t, 0 < a < 2τ
(16)φ(t, 2τ) = 2Ψ(t) 0 < t
(17)Solutionsof (15) -(17)aredenedasfortheprimal system,seeDenition 3.3.
Adapting[20℄ toourmodel,wegetageneralrelativeentropypriniple. Let
(β, γ)
bexed.Given a solution
(R, p)
and apositive solution( ˆ R, p) ˆ
of (9)-(11) , a positivesolution(Ψ, φ)
of(15)-(17) ,and
H ∈ L
∞loc( R )
,wedeneH
byH(t) := Ψ(t) ˆ R(t)H R(t)
R(t) ˆ
! +
Z
2τ 0φ(t, a)ˆ p(t, a)H
p(t, a) ˆ p(t, a)
da.
Theorem3.5 (GeneralRelativeEntropy). Let
H
beloally Lipshitzand dierentiable every- where. ThenH
isloally Lipshitzandthereholdsa.e.dH
dt (t) = φ(t, 2τ )ˆ p(t, 2τ )
H
p(t, 0) ˆ p(t, 0)
− H
p(t, 2τ) ˆ p(t, 2τ)
+H
′p(t, 0) ˆ p(t, 0)
p(t, 2τ ) ˆ
p(t, 2τ ) − p(t, 0) ˆ p(t, 0)
.
(18)Corollary3.6. Let
H
beonvex,possibly non dierentiable. ThenH
isnon-inreasing.Proof ofCorollary 3.6. Let
H
be onvex. ThenH
is loally Lipshitz and has left and rightderivativeseverywhere. Then, as in the proof of Theorem3.5,
H
is loally Lipshitzand (18)holdsa.e. ifwereplaethederivativeof
H
byitsrightderivativeandthederivativeofH
byitsleftorrightderivative,dependingonthesignoftherightderivativeof
p(t,0) ˆ
p(t,0). Observenowthat
thisright-handsideof (18)isnon-positiveif
H
isonvex.Proof ofTheorem3.5 . Observethat,alongtheharateristisa.e.,thereholds
∂
∂t + ∂
∂a
φ(t, a)ˆ p(t, a) = 0, ∂
∂t + ∂
∂a
p(t, a) ˆ
p(t, a) = 0,
andthen
∂
∂t + ∂
∂a
φ(t, a)ˆ p(t, a)H
p(t, a) ˆ p(t, a)
= 0.
ByLemma3.2, theseond termof
H
isloallyLipshitz,with derivativea.e.φ(t, 0)ˆ p(t, 0)H
p(t, 0) ˆ p(t, 0)
− φ(t, 2τ)ˆ p(t, 2τ)H
p(t, 2τ) ˆ p(t, 2τ)
.
Thersttermof
H
isobviouslyloallyLipshitz,anda.e.d
dt Ψ(t) ˆ R(t)
H R(t) R(t) ˆ
!
= − β(t)φ(t, 0) ˆ R(t) + Ψ(t)2ˆ p(t, 2τ)
H R(t) R(t) ˆ
!
= − φ(t, 0)ˆ p(t, 0) + φ(t, 2τ)ˆ p(t, 2τ) H
p(t, 0) ˆ p(t, 0)
,
and
Ψ(t) ˆ R(t) d
dt H R(t) R(t) ˆ
!
= Ψ(t) ˆ R(t)H
′R(t) R(t) ˆ
! 1 R(t) ˆ
dR
dt (t) − R(t) R(t) ˆ
d ˆ R dt (t)
!
= Ψ(t)H
′p(t, 0) ˆ
p(t, 0) 2p(t, 2τ) − p(t, 0) ˆ
p(t, 0) 2ˆ p(t, 2τ)
= φ(t, 2τ)ˆ p(t, 2τ)H
′p(t, 0) ˆ p(t, 0)
p(t, τ) ˆ
p(t, τ) − p(t, 0) ˆ p(t, 0)
.
3.3 Eigenelements
Let
β > 0
andγ ≥ 0
beonstant. Looking forpartiularsolutionsof (9)-(11)and (15)-(17)oftheform
R : t 7→ Re ¯
λtΨ : t 7→ Ψe ¯
−λtp : (t, a) 7→ p(a)e ¯
λtφ : (t, a) 7→ φ(a)e ¯
−λtwith
λ ∈ R
andp ¯
,φ ¯
dierentiable,wegetthefollowingeigenvalueproblem:(λ + β) ¯ R = 2¯ p(2τ) (λ + β) ¯ Ψ = β φ(0) ¯
(19)d¯ p
da (a) = −(λ + γ)¯ p(a) d ¯ φ
da (a) = (λ + γ) ¯ φ(a)
(20)¯
p(0) = β R ¯ φ(2τ) = 2 ¯ ¯ Ψ
(21)Equations(20)-(21)give
¯
p(a) = β Re ¯
−(λ+γ)a, φ(a) = 2 ¯ ¯ Ψe
(λ+γ)(a−2τ).
(22)If
R, ¯ Ψ ¯ 6= 0
,(19)isthenequivalenttoλ + β = 2βe
−(λ+γ)2τ.
(23)Theorem 3.7 (First eigenelements).There exists a unique solution
(λ, R, ¯ p, ¯ Ψ, ¯ φ) ¯
of (19)-(21)suhthat
R > ¯ 0, p > ¯ 0, R ¯ + Z
2τ0
¯
p(a)da = 1,
Ψ ¯ > 0, φ > ¯ 0, Ψ ¯ ¯ R + Z
2τ0
φ(a)¯ ¯ p(a)da = 1.
Proof. Itisenoughtoobservethat (23)hasauniquerealsolution.
3.4 Long time asymptoti
3.4.1 without theation ofthe drugs
We onsiderhere thatthere is noation ofthe drugsfor
t > 0
, i.e. thatβ > 0
andγ ≥ 0
areonstant. Thersteigenelements
(λ, R, ¯ p, ¯ Ψ, ¯ φ) ¯
aregivenbyTheorem3.7.Theorem3.8. Let
(R
0, p
0)
bean initial ondition,C > 0
besuhthat|R
0| ≤ C R, ¯ |p
0(·)| ≤ C p(·), ¯
and
(R, p)
bethe solutionof (9)-(12). Then,for allt > 0
,|R(t)| ≤ C Re ¯
λt, |p(t, ·)| ≤ C p(·)e ¯
λt,
(24)ΨR(t) + ¯ Z
2τ0
φ(a)p(t, a)da ¯
e
−λt= ¯ ΨR
0+ Z
2τ0
φ(a)p ¯
0(a)da := ρ,
(25)Ψ|R(t)| ¯ + Z
2τ0
φ(a)|p(t, a)|da ¯
e
−λt≤ Ψ|R ¯
0| + Z
2τ0
φ(a)|p ¯
0(a)|da,
(26)t
lim
→∞Ψ|R(t)e ¯
−λt− ρ R| ¯ + Z
2τ0