Invariantmanifoldsin amodelfor theGABA

receptor

Martin Caberlin∗ Nilima Nigam† SanjiveQazi‡

January8, 2003

Abstract

The reactionschemeof GABA interactingwith the ligand-gatedion-

channeldemonstratesnumericalstiffness. We comparedvariousexplicit

andimplicit numericalmethodsto solve for thereactionscheme,andfound

implicit methods(BackwardEuler, ode23s)performedordersof magnitude

betterthanexplicit methods(Forward Euler, ode23,RK4, ode45)in terms

of stepsizerequiredfor stability, numberof stepsand cputime. Interest-

ingly, we observed the existenceof low dimensionalinvariant manifolds

∗Dept.of MathematicsandStatistics,McGill. email:caberlin@math.mcgill.ca†Dept.of MathematicsandStatistics,McGill. email:nigam@math.mcgill.ca‡Parker-HughesCancerCenter. email:sqazi@ih.org

1

uponsolving thesystemof 8 ordinarydifferentialequationsfor theGABA

reactionscheme.Theemergenceof simpledirectrelationshipsbetweenthe

receptorstatesmayhavesomeimportantrolesin biologicalsignaltransmis-

sion.

1 Intr oduction

Receptor mediatedsignal transmissionin chemical synapsesis a fundamental

componentof neuronalcommunication.GABAA receptorsconductchlorideions

to generateinhibitory post-synapticcurrents.Networksof inhibitory neuronsare

thoughtto playanimportantrole in modulatingneocorticaloscillationsandhence

brain function (Connors& Amital, 1997;Castro-Alamancos& Connors,1996).

Receptor gatedion-channelssuch asthe GABAA receptorcanexist in multiple

conformationalstates:bound,unbound,open,closedanddesensitized. Sincethe

bindingof thetransmitterto thereceptoris thefirst constraintin thepassingof in-

formationfrom one neuronto thenext, we wereinterestedin therelationshipbe-

tweenthebindingeventandthesubsequenttransitionthroughthedifferentstates

of thereceptor. TheGABAA reactionschemewasmodeledvia basicbiochemical

processes.It is essentiallyan adaptationof the fundamentalMichaelis-Menten

mechanismof enzyme-substrateinteractions,andis characterizedby a systemof

2

ordinarydifferentialequations.

Our numericalinvestigationrevealedthatthemodelexhibits stiffness; a char-

acterizationof stiffnesswill be provided. In this particularsituation,this phe-

nomenonmeansthatthedynamicalsystemreducesfrom beingeight-dimensional

(correspondingto eightchemicalconcentrations)to seven,andthento five. This

dimensionreductionhasbiological implications,which we conjecture uponand

would like to investigatefurther.

Stiff dynamicalsystemsareexceedinglydifficult to integrateusingstandard

explicit methods, andrequirevery small time-stepsin orderto maintainnumeri-

cal stability. The useof adaptive algorithmsdoesnot significantlyalleviate this

problem;theintegratorselectsanunreasonablysmallallowablestep-size.Heuris-

tically, the time-stepis selectedto resolve the behavior of the fastesttransient

in the system.However, this transientmay not be the dominantmodeandmay

contribute negligibly to the over-all dynamics ofthe system.The net result is a

significantwasteof computationaleffort. A well-known strategy to numerically

integratestiff systemsis to usean implicit method. In Section2, we first provide

a working definitionof stiffness,andthendemonstratethroughsimpleexamples

thedifference betweenusingexplicit andimplicit methods.

Webeganour investigationby first usingstandard(explicit) algorithmsto nu-

3

merically simulatethe model. During the experimentswe encounteredstability

problemswhich necessitatedthe useof stiff solvers. The successof theselat-

ter algorithmsled us to characterizethe modelasstiff. Oncewe hadstableand

accuratealgorithmsto study the model,we observed the existenceof invariant

manifolds.

The organizationof this paperis asfollows. We begin with a brief descrip-

tion of themodelin question;for furtherdetails we refer the interestedreaderto

(Qazi & Trimmer, 1999). In Section2, we describestiffnessandpresentexam-

pleswhich demonstratethepower of implicit methods.In section3, we present

the resultsof our investigationsof the GABA model. We endthis paperwith a

tentativebiologicalexplanationfor theobservedphenomena,aswell assuggested

experimentsto verify ourfindings.

2 The GABA reactionscheme

GABA, which is γ-aminobutyric acid, is an inhibitory neurotransmitter (Alberts

et al., 1994). The synapseis the site at which neuronalsignalsaretransmitted.

Thepredominantmechanismof signaltransmissionat thesynapseis via chemi-

cal exchangefrom thepresynapticcell to thepostsynapticcell (Qazi& Trimmer,

4

1999).Thenarrow gapbetweentheseis calledthesynapticcleft, into which neu-

rotransmitters,such asGABA, arereleased.Uponrelease,the neurotransmitteris

rapidly removedfrom the synapticcleft. In orderto understandthechemicalsig-

nal, the postsynapticcell mustbe equippedwith GABA binding receptors.One

suchexampleis the GABAA receptor. The GABA reactionschemeconsidered

heremodelsthe variousconcentrationsof GABA, boundandunboundreceptor,

anddesensitizedandopenstatesafterthesimulatedreleaseof aGABA pulse.

TheGABA reactionschemeis basedon theclassicalbimolecularreactionof

Michaelis-Menten,suitablyadaptedto accountfor reversiblereactionsandmulti-

ple receptorstates.TheMichaelis-MentenreactiondescribesasubstrateR, react-

ing with anenzymeN , which form acomplex RN , soonconvertedinto aproduct

P andtheenzyme(Murray, 1990).Schematically, thereactionlookslike

R + Nk−1

k1

RNk2

→ P + N, (1)

wherek1, k−1 andk2 are the rate constantsof the reaction. The Law of Mass

Action saysthat the reactionrate is proportional to the productof the concen-

tration of thereactants(we usethesamelettersto denoteboth thereactantsand

their respective concentrations,with theexception[RN ] = C). This leadsto the

5

Michaelis-Mentenequations,

R = −k1RN + k−1C, E = −k1ES + (k−1 + k2)C,

C = k1ES − (k−1 + k2)C, P = k2C.

The GABA schemeassumesall the reactionsare reversible(ie. all arrows in

the reactions(1) are doublepointed). One of the modificationsof Michaelis-

Mentenin theGABA schemestemsfromthefactthattheendproductP is abound

receptor. In this way, GABA ≡ N is not a catalyst- it is used upin thereaction.

So thesecondhalf of thereactionis not present.A furtheradaptationis that the

GABA schemeinvolvesprocessesin which the receptorR hasmultiple GABA

bindingsites.Theappropriatechangesbecomeapparentafter closeexamination

of theschematicdiagramsin Table1. TheGABA reactionvariablescanbefound

in Table2.

As Qazietal. explain,anadequatemodelof theGABA interactionwith there-

ceptormusthaveat leasttwobindingsites,twoopenstatesandtwodesensitization

states(Qazi& Trimmer, 1999; Joneset al., 1998;Shenet al., 2000;Mozrzymas

et al., 1999). Transitionsfrom the slow desensitizedDs to the fastdesensitized

stateDf have alsobeenincludedin subsequentmodels(Qazi& Trimmer, 1999;

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Table1: Schematicrepresentationof GABA reactions(Joneset al., 1998;Qazi& Trim-mer, 1999).

BindingReactions StateTransitions

R + Nkoff

2kon

B1 B1

a1

b1

O1

B1 + N2koff

kon

B2 B1

d1

r1

Ds

Ds + Nq

p

Df B2

a2

b2

O2

B2

d2

r2

Df

Ds

p -�q

Df

R2kon -�koff

B1

d1

6

r1

? kon -�2koff

B2

d2

6

r2

?

O1

a1

6

b1

?O2

a2

6

b2

?

Table2: GABA schemevariables

R ReceptorconcentrationN Ligand(GABA) concentrationB1 First boundstateconcentrationB2 SecondboundstateconcentrationDf FastdesensitizedstateconcentrationDs Slow desensitizedstateconcentrationO1 First openstateconcentrationO2 Secondopenstateconcentration

Joneset al., 1998;Mozrzymaset al., 1999). Thesearerepresentedin the seven

statemodelshown in Table1. Qazi et alia assertthat the seven statemodel is

theminimumnecessaryto predictGABA responses(Qazi& Trimmer, 1999). It

is a modelwhich hasbeenwell establishedin the literature(Qazi et al., 1998;

Jonesetal., 1998; Jones& Westbrook,1996;Shenetal., 2000;Mozrzymasetal.,

7

Table3: GABA reactionschemeequations

R = Ψ1(koff , 2kon, R(t)) (2)

N = Ψ1(koff , 2kon, R(t)) + Ψ2(2koff , kon, B1(t)) + pDs(t)N(t) (3)

B1 = −Ψ1(koff , 2kon, R(t)) + Ψ2(2koff , kon, B1(t)) − F1(t) − G1s(t) (4)

B2 = −Ψ2(2koff , kon, B1(t)) − G2f (t) − F2(t) (5)

Df = H(t) + G2f (t) (6)

Ds = −H(t) + G1s(t) (7)

O1 = F1(t) (8)

O2 = F2(t) (9)

Ψj(k1, k2, f(t)) := k1Bj(t) − k2f(t)N(t) (10)

Fj(t) := bjBj(t) − ajOj(t) (11)

Gj`(t) := djBj(t) − rjD`(t) (12)

H(t) := qDs(t)N(t) − pDf (t) (13)

1999).

TheGABA reactionschemeis modelledby thesystemof ODE (2) - (13) in

Table3, andthemodelparametersaregivenin Table4. At this stage,thereis no

a priori reasonto suspectthat thesystemis stiff, thoughit seemslikely thatdue

to the numberof processespresent,somemay occuron fastertime-scalesthan

others.Wediscussthenumericalsimulationsof thismodelin Section4.

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Table4: GABA reactionschemeparameters

a1 = 1100 sec−1 b1 = 200 sec−1

a2 = 142 sec−1 b2 = 2500sec−1

r1 = 0.2 sec−1 r2 = 25 sec−1

p = 2 sec−1 q = 0.01 sec−1

d1 = 13 sec−1 d2 = 1250 sec−1

kon = 5 · 106 M−1·sec−1 koff = 131 M−1·sec−1

3 Numerical stiffness

We shallbegin this sectionwith a sampledynamicalsystemwhich exhibits stiff-

ness,andseekto convincetheaudiencethatmerelyusinghigher-orderor adaptive

step-sizealgorithmsdoesnot alleviatethe problem,aslong asthesemethodsare

explicit. This motivates thediscussionon numericalstiffness,andwe describe

well-known fixes for the problem.Readerswho are familiar with the issuesof

stiffnessareencouragedto skip to Section4.

To begin with, we areinterestedin computingthesolutiony(x) to thesystem

of ODE

y′(x) = f(x, y(x)), (14)

y(a) = α, x ∈ [a, b].

We notethattheIVP maybea singleequation,or a system.In thelattercasethe

9

solutiony(x) is a vector. In Henrici’s notation(Haireret al., 1993),therecursion

usedto advancethesolutionof (14) from xn to xn+1 is

yn+1 = yn + hnΦ(xn, yn, hn), (15)

whereyn is thenumericalapproximationto theexactsolutiony(xn). Thefunction

Φ is calledtheincrementfunctionof themethod. For example,whenΦ(xn, yn) =

f(xn, y(xn)), thealgorithmobtainedis theusualForwardEuler.

3.1 Stiffness- what is it?

Webegin thissectionwith asimpleexampleto illustratethephenomenonof “stif f-

ness”.

Example1 Consider theIVP

y′ = −10000y − et + 10000, y(0) = 1, t ∈ [0, 5]. (16)

Theexactsolutionof thisproblemis givenby

y(t) = −1

10001et + 1 +

1

10001e−10000t.

10

We seefrom theexpressionabove that thelastexponentialterm 1

10001e−10000t

shouldbecomenegligible in comparisonto 1. Indeed,whent ≈ .072, this term

contributeslessthan10−15 to theoverall computation.

In orderto comparetherelativeefficiency of variousalgorithmsin theexample

above, we requiredthat therelative errorof thecomputedsolutionstayedwithin

10−6 during the interval [0, 5]. For eachalgorithm,we reducedthe step-sizeor

error toleranceuntil thedesiredaccuracy was achieved. We presentthestep-size

used,aswell asthenumberof time-stepstakento integrateover theinterval [0, 5].

Thealgorithmsusedwere:

1. Forward Euler: The simplestexplicit integratorpossible,low-order, and

fixedstep-length.

2. ODE23: MATLAB’ s adaptivestep-size,second-orderalgorithm

3. RK4: A fixedstep-length,explicit fourth-orderalgorithm

4. ODE45: MATLAB’ s adaptivestep-size,fourth-orderalgorithm

In Table 3.1, we compile someresultsusing variousstandardexplicit numeri-

cal algorithms. All explicit solvers, with or without variablestep-size,work in

the sameway. That is, the computationof the approximatesolutionat xn+1 de-

pends onthevaluesof thesolutionalreadycomputed.Eachexperimentwascon-

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Table5: Resultsof usingexplicit numericalmethodsonexample1.Thetruesolu-tion at t = 5 is y(5) = .9851601681...

algorithm no. of steps maxstep cputime Y(5)ForwardEuler 100000 5×10−5 339.8 .9851609

ode23 19902 3.803×10−4 67.37 .9851602RK4 50000 1×10−4 91.31 .9851616ode45 60273 1.088×10−4 109.41 .9851602

ductedin MATLAB, on a Linux workstationwith a 1.3GHzAthlon processor.

The cputimesreportedarein seconds.The algorithmswould blow up, yielding

NaN , if thestep-sizesweremuchlarger. We seethat thealgorithmsarechoos-

ing a step-sizesmall enoughto accuratelyresolve the fastesttransient- in other

words,the insignificantterm 1

10001e−10000t governsthechoiceof step-size!!This

is neithera problemof accuracy, nor of adaptivity. Thepoorperformanceof all

thesealgorithmsontheexamplepointsto adeeperissue.Merelyusinghigherand

higherorderalgorithms,evenif they areadaptive,doesnotsuffice; thedynamical

system16 exhibits behavior thatconfoundseachof thesemethods.In particular,

all of thesealgorithmshavestabilityproblems;if thestepsizeis notexcruciatingly

small,thecomputedsolutions“blow up”. Wedescribethis systemasstiff .

Heuristically, oneusefuldefinitionof stiff dynamicalsystemsis: A systemin

whichoneor severalpartsof thesolutionvaryrapidly (asanexponential,e−kx for

example,with k large),while otherpartsof thesolutionvary muchmoreslowly

12

(asan oscillator, or linearly) is characterizedasa stiff system.Thesearepreva-

lent in thestudyof dampedoscillators,chemicalreactionsandelectricalcircuits.

Essentially, the derivativeor partsthereofchangeveryquickly.

Theprecisedefinitionof numericalstiffnessis difficult to state;however, the

phenomenonis onethat is widely observedwhile numericallyapproximatingso-

lutionsof differential equations(asseenabove). For our purposes,we adoptthe

following definitionof stiffnessdueto Lambert(Lambert,1991):

Definition 1 If a numericalmethodwith a finite region of absolutestability, ap-

plied to a systemwith anyinitial conditions,is forcedto usein a certain interval

of integration a steplengthwhich is excessivelysmall in relation to thesmooth-

nessof theexactsolutionin that interval, then thesystemis saidto bestiff in that

interval.

This definition allows us to concedethat the stiffnessof a systemmay vary

overtheinterval of integration.Theterm‘excessively small’ requiresclarification.

In regionswherethefasttransientis still alive,weexpectthesteplengthto remain

small.However, in regionswherethefast(or slow) transientshavedied,wewould

expectthesteplengthto increaseto asizereasonableto the problem.

13

A “cure” for stiffnessis theuseof implicit algorithms.Thesearealgorithms

wheretheapproximationYn+1 to the truesolutiony at tn + h is givenby

Yn+1 = Yn + hΦ(Yn+1, t),

whereΦ is chosenaccordingto the systembeingsolved, the accuracy desired,

etc. Note that the updateYn+1 is now the solution of a (typically) non-linear

system.Thus,eachstepof ouralgorithmrequiresanonlinearsolve,anexpensive

proposition.However, sincethedomain ofstabilityof implicit algorithmsis much

larger thanthat of explicit algorithms,moststate-of-the-artstiff solversemploy

adaptive implicit algorithms.For example,thebuilt-in MATLAB solversode23s

andode23tb areimplicit Runge-Kuttamethodswith variablestepsizecontrol.

Thesavingsin termsof computationaltime is well worth theeffort of solvingthe

nonlinearproblem.To drive this messagehome,let us returnto theexamplewe

beganthissectionwith, anduseacoupleof implicit solvers:

1. Backward Euler : the implicit analogof Forward Euler, this methodre-

quiresafixedstep-size.

2. MATLAB’ s ODE23s: this methodusesan adaptive step-size,and is a

second-orderalgorithm.

14

Table6: Someimplicit algorithmsusedto computeexample1. Comparethesewith similar explicit algorithmsin Table3.1.

algorithm no. of steps maxstep cputime Y(5)BackwardEuler 2500 .002 .11 .9851898

ode23s 179 0.1094 0.61 0.9851601

Table3.1 summarizestheperformanceof theseimplicit algorithms.Despitethe

nonlinearsolve, thesealgorithmsoutperformthosein Table3.1 by a coupleof

orders ofmagnitudebothin termsof numberof time-steps,andcputimetaken.

4 The GABA reactionscheme

Thesenumericalexperimentswereconductedafterwehadcomputationallystud-

iedacompletelyunrelatedsystem,theSantillan-Mackey modelof genetranscrip-

tion (Caberlin,2002;Caberlinetal., 2002).Thelatterwasastiff-delaydifferential

systemof 4 reactants,andwe foundinvariantmanifoldspresentin theunderlying

dynamics.To our surprise,our numericalinvestigationsof the GABA reactions

led usto observe similar invariants.In locatinginvariantmanifoldsin theGABA

system,werelieduponobservationsfrom (Caberlin,2002;Caberlinetal., 2002).

Wetypically rantheGABA modelwith theinitial conditionsin Table7. When

describingexperiments,weshallspecifytheinitial GABA concentration,N0.

15

Table7: Initial conditionsfor theGABA scheme

R0 = 1 × 10−6 MN0 ∈ (0.5, 4096) × 10−6 MAll otherinitial variablessetto zero.

4.1 Stiffnessin the GABA reactionscheme

In an attemptto reproduceFigure 4 of (Qazi & Trimmer, 1999), we first ran

ode45 on the GABA system(2) - (13). We set the initial GABA concentra-

tion to N0 = 5 × 10−7 M and ran theexperimentfor t between0 and1 second.

We foundode45 to be unstable.It startsthe integrationwith a steplengthof

2.5× 10−4s. Neithersettingtheinitial stepsizeto 1.0× 10−4s,nor themaximum

stepsizeto1.0×10−3smakesthesolutionstable.Thiscanbeseenin Figure1 with

theaforementionedmaximumstepsize.With amaximumstepsizeof 5 × 10−4s,

ode45 integratesthe systemwell in 2000 steps. With smallerstepsizes,the

plots aregraphicallyindistinguishable atthis resolution. Note the saturationof

plusesfor theexplicit integratorsin Figure2, evenaftertheinitial transientphase.

This is typical of solving stiff systemsusingexplicit methods. With ode23s,

we find goodconvergencewith themaximumsteplengthsetto 0.01s.This step

lengthis used byode23s throughouttheinterval, completingtheintegrationin

100steps,a twenty-foldimprovementoverode45.

16

Figure1: Instabilityof ode45 runningGABA model.

Invariantmanifoldswerediscoveredin theGABA model.Sincethereareeight

dependentvariables, there are 28 possibletwo dimensionalphasespaceplots

and 56possiblethreedimensionalphasespaceplots. With this alone, it would

be quite tediousto identify invariant manifolds. We noticedthat the time plots

for thevariableswhich becamelinearly relatedon the invariantmanifoldsof the

Santillan-Mackey systemcloselyresembledeachotherqualitatively. Using this

asa startingpoint,we plottedthevariableswhosetime plots lookedqualitatively

similar in theGABA scheme.To our surprise,we found twoinvariantmanifolds

- again describedby linear relationshipsbetweenthe variables. In Figure3 we

seethevariablesB1 andO1 pair up. Therelationshipis givenapproximatelyby

O1 ≈ 0.182B1. Thesolutionreachesthis manifoldon theorder of10−2 seconds.

The variablesB2, O2 andDf alsobecomedependent, asseenin Figure4. The

17

Figure2: GABA system,left: ode45, MaxStep= 5 × 10−4s,2000steps.right: ode23s, varyingMaxStep.

relationshiphereis givenapproximatelybyDf ≈ 48.4B2 ≈ 2.73O2. Thesolution

reachesthismanifoldontheorder oftenthsof seconds(around0.3s- 0.5s).In all,

the systemgoesfrom eight dimensionsto seven on the order of1/100seconds,

andthendown to dimensionfive, tenthsof secondslater. An exhaustive search

in the remainingvariablesrevealsthat thesearetheonly invariantmanifoldsthe

GABA reactionsystem.

As for theSantillan-Mackey model,theseinvariantmanifoldobservationspro-

videakey experimentaltestof themodel.

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Figure3: left: InvariantB1-O1 manifold. right: Rotationin theB1O1-plane.

4.2 Roleof desensitizationin generatinginhibitory curr ents.

Therateatwhich thereceptorentersthedesensitization statewill affect theshape

of inhibitory currents,andthis may be a meansby which endogenousenzymes

regulate receptoractivation (Jones& Westbrook,1997; Bianchi et al., 2001).

Desensitizationtendsto prolong the inhibitory currentand keepsthe transmit-

19

Figure4: top: InvariantB2-O2-Df manifold.bottoml: Rotationin Df O2-plane.bottomr: Rotationin B2O2-plane.

ter in the boundstateof the receptor. This would be expectedto affect output

of neuronalcircuits. GABA inhibitory currentsareknown to beimportantin the

generationof synchronizedoscillationsin thecortex, thalamusandhippocampus

(Connors& Amital, 1997; Staak& Pape,2001). Low frequency rhythmsin the

neocortexrangefrom 5-10 Hz (0.1-0.2s) coincidingwith the appearanceof the

20

lower dimensionalmanifoldsin the GABA reactionscheme.Thelinearcoupling

of the bound,desensitizedand openstatesof the receptorbetween0.1 - 0.5 s

occursataphysiologicallyrelevanttimescaleto affect timing of neuralcircuit ac-

tivity. This suggeststhatsimplerelationshipsmayexist betweenthestatesof the

receptorat low frequenciesof receptoractivation. Thenumberof channelopen-

ingswill bedirectly proportionalto theamount oftransmittertraversingthrough

the boundand desensitizationstates. So, if the transmitteris rapidly removed

someof it will still be boundto the receptorflipping throughbound,openand

desensitizedstates,andthe amount bufferedwill be directly proportionalto the

receptorin the openstatejust prior to the free GABA removal. Therefore,the

systemwill havesomememoryof theactivity at thepoint freeGABA is removed.

Theexistenceof sucha simple linearbuffer mayhave importantimplicationsfor

the transmissionof informationthroughtheGABA receptor. Sucha buffer may

aid in theroutingof signalsfrom one neuronto another. To investigatethis pos-

sibility we arenow building a modelin which pulsesof GABA aredeliveredto

thereceptor. We will monitor theamount ofGABA bufferedandto seeif this is

relatedto theaccuratetransferof frequency informationthroughthereceptor.

It is experimentallypossibleto deliver high concentrationpulsesof GABA

ontoneuronsandmeasuretheinhibitory potentialsthataregenerated.A rangeof

21

frequenciescould be deliveredto the cell with the GABA receptorswhile mea-

suringthefidelity of theresponseusinginformationtheoreticapproaches.These

responsescanbe comparedto the effect of GABA in the presenceof drugsthat

changethekinetic parametersin theGABA model(9, 12, 16). Thequantitative

relationshipbetweentherelative statesof thereceptorandfrequency transfercan

thenbeusedto assesthe roleof desensitization statein information processing.

5 Acknowledgements

Martin Caberlinand Nilima Nigam gratefully acknowledgethe supportof this

work by theNaturalSciencesandEngineeringCouncilof Canada.MC wassup-

portedthroughaPGS-Agrant,andNN wasfundedby anNSERCAppliedMath-

ematicsgrant. Sanjive Qazigratefullyacknowledgesthe institutionalsupportof

theParkerHughes CancerCenter.

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