Analysis of G-CSF Treatment of CN using Fast Fourier Transform

Transcription

1 Reseach Jounal of Recen Sciences ISSN 77-5 Res.J.Recen Sci. Analysis of G-CSF Teamen of CN using Fas Fouie Tansfom Absac Balamualihaan S. and Rajasekaan S. Deamen of Mahemaics, Si Ramanuja Engineeing College, Chennai-8, INDIA Pofesso and COE, Deamen of Mahemaics, BS Abdu Rahman Univesiy, Chennai-8, INDIA Available online a: (Received 9 h Febuay, evised h Febuay, acceed 8 h Febuay ) In his eseach ae we y o find ou and invesigae he Fas Fouie Tansfom model and G-CSF Teamen of CN (Cyclical Neuoenia), in deail. In his analysis of G-CSF eamen of Neuoenia, we ge daa fom CN. They ae gey collies. They ae usually used o build an exended model of i. I oduces he dynamics of ciculaing blood cells. They ae found fom he dogs wih and wihou daily G-CSF heay. I is a model which is vey useful fo collecion of laboaoy daa. This mahemaical model hels us o eoduce he lage vaiaion of daa oo. They occu fom one dog o anohe. I has long em effecs on he oscillaions when he fequency of dug delivey is made. This model is also useful o accoun fo he feaues of uneaed G-CSF. I is also useful fo eamen of dogs wih CN. Theefoe his model is consideed as an accomlished one. Thee is fiing aamees fo days and no fo dogs fo esimaion o evaluaion. I is also essenial and necessay o model he moe samles fo incease in Neuohil amlificaion. The oosed inevenions ae acical. I may educe he amoun of G-CSF. I equied oenial mainenance. Someimes, i may even imove he eamen effecs oo. This model gives us good esul in eamen. The changes would be acical and educe he isk side as well as he cos of eamen in G-CSF. Key Wods: FFT modeling, neuohil, ganulocye colony-simulaing faco (G-CSF). Inoducion All blood cells ae deived fom he hemaooieic sem cells (HSC), which ae undiffeeniaed cells having a high olifeaive oenial. These mulioen sem cells can olifeae and maue o fom all yes of blood cells. Poducion in hese cell lines is egulaed by a vaiey of cyokines, including eyhooiein (EPO), which mediaes he egulaion of eyhocye oducion, hombooiein (TPO), which egulaes oducion of laeles, as well as ganulocye colony-simulaing faco (G-CSF), which egulaes leukocye numbes. In a comehensive mahemaical model fo he egulaion of hemaooiesis was esened. This wok was moivaed by he exisence of seveal hemaological diseases ha dislay a highly dynamic naue chaaceized by oscillaions in one o moe of he ciculaing cell lines. Examles of hese ae cyclical neuoenia, eiodic chonic myelogenous leukemia, cyclical hombocyoenia and eiodic hemolyic anemia. In his chae, we concenae on cyclical neuoenia, a ae hemaological disode chaaceized by oscillaions in he ciculaing neuohil coun. These levels fall fom nomal o baely deecable levels wih a yical eiod of 9 o days in humans, even hough eiods u o days have been obseved. These oscillaions in he neuohil coun ae geneally accomanied by oscillaions wih simila eiod in he laeles, lymhocyes and eiculocyes. Cyclical neuoenia also occus in gey collies wih eiods on he ode of o 6 days. This animal model has ovided exensive exeimenal daa ha has eniched ou undesanding of cyclical neuoenia. Though he gene modified esonsible fo canine cyclical neuoenia has been idenified, he dynamic oigin of he cycling is only aially undesood. Because of is ineesing dynamical naue, many mahemaical models have been fomulaed o aem o answe his quesion. While many have modeled cyclical neuoenia as aising only fom desabilizaion of neuohil dynamics, he wok of FFT and sugges ha he oigin of cyclical neuoenia lies in a desabilizaion of he combined HSC and neuohil conol sysem. The hyohesis ha oscillaions oiginae in he sem cells is suoed by he obsevaion ha in cyclical neuoenia oscillaions ae also esen in laeles and eiculocyes. Cyclical neuoenia in humans is ofen eaed using ganulocye colony simulaing faco (G-CSF), which is known o inefee wih aoosis. In he auhos showed ha, deending on he saing dae of he G-CSF eamen, he neuohil coun could eihe be sabilized o show lage amliude oscillaions. Thei model suggesed ha ohe G-CSF eamen schemes could be effecive while using less G-CSF. Howeve, his model included neihe eyhocye no laele dynamics even hough clinical daa indicaes oscillaions in hose cell lines in cyclical neuoenia aiens. Thus i is no known if he esuls would be consisen wih obseved laele and eiculocye daa. Second, he simulaions did no ake ino accoun he hamacokineics of G-CSF. In his chae, we esen a new model fo assessing he effecs of G- CSF eamen in cyclical neuoenia. To do his, we augmen he comehensive model of he hemaooieic sysem fom by Inenaional Science Congess Associaion

2 couling i wih a wo-comamen hamacokineic model ha accouns fo G-CSF kineics 5. Maeial and Mehods Peiodic Auo-Immune Hemolyic Anemia: Le (, α, β) be he oulaion of ecuso cells a ime, age α and ae of oducion β. le V(R) and U(R) be he velociies of ae of oducion and mauaion, which may deend on he homone (EPO) concenaion. If N (R) is he numbe of cells ecuied ino he eoduce ecuso oulaion, and hen he eny of new ecuso cells ino he sucued model will saisfy he bounday condiion, ( V( R) U( R)) (,, ) N( R) () and bounday condiion ae cells exchange ( V( R) U( R)) (,, ) Em(,, ) () Le he bih ae fo eoducing ecuso cells be β and α eesen he deah ae hough aoosis. Le be he densiy of he disibuion of mauiy levels of he cells when eleased ino he ciculaion blood, whee eesens he mean age of maue ecuso cells and D( ) d D( ) d The disaeaance ae funcion is given by: D( ) D( ) () D( x ) D( x ) Wih hese condiion he age sucued model fo he oulaion of ecuso cells wih >, < μ < μ and < μ < μ saisfies: V ( R) U ( R) ( V ( R) U ( R))[ (, R) (, R) D( ) ] () le m (, α, ) be he oulaion of maue non eoducing cells a and age α. Fom he disaeaance ae funcion, he bounday condiion fo cells eneing he maue oulaion is given by Em(,,) V ( R) D( ) (, ) d whee he mauiy U ( R) D( ) (, ) d, (5) levels μ and μ eesens he maximum age fo a cell eaching mauiy. We assumed ha desucion of RBC occus by acive emoval of he old cells 6. Fom a modeling oin of view, his esuls in a moving bounday condiion wih he age of he oldes RBC, ϴ() vaying in. The bounday condiion is hen given by ( E ) m(, ( )) F (6) whee F is he fixed RBC emoval ae. If γ(θ) is he deah ae of maue cells, hen he aial diffeenial equaion descibing m (,ϴ)is given by: m m E E ( ) m,,, (7) The oal oulaion of maue cells funcion is given by: () M ( ) m(, ) d. (8) The diffeenial equaion fo R (ed blood cells oducion) is hus: dr kr, (9) KM Whee k is he decay consan fo he homone and he ae of oducion β is given by a monoone deceasing Hill funcion. The given equaion is linea in R. dr kr () KM IF (inegaing faco) = = KT KT KT Re e C e C KM ( KM ) K R C () ( KM ) K Cyclical Thombocyoenia: Plaeles ae blood cells whose funcion is o ake a in he cloing ocess, and hombocyoenia denoes a educed laele (hombocye) coun. In cyclical hombocyoenia (CT), laele couns oscillae geneally fom vey low values ( 9 cells/l) o nomal (5 5 9 laeles/l) o above nomal levels ( 9 cells/l). These oscillaions have been obseved wih eiods vaying beween and days. In addiion, aiens may exhibi a vaiey of clinical symoms indicaive of imaied coagulaion such as uua, eechiae, eisaxis, gingival bleeding, menohagia, easy buising, ossibly emensually, and gasoinesinal bleeding. Thee ae wo oosed oigins of cyclical hombocyoenia. One is of auoimmune oigin and mos evalen in females. The ohe is of amegakayocyic oigin, moe common in males. Auoimmune cyclical hombocyoenia is chaaceized by a shoened laele lifesan a he ime of deceasing laele couns. This is consisen wih nomal o high levels of bone maow megakayocyes and wih an inceased desucion ae of ciculaing laeles. Auoimmune CT has also been osulaed o be a ae fom of idioahic (immune) hombocyoenic uua (ITP). Cyclical Neuoenia: We esened a wo vaiable delay diffeenial equaion (DDE) sysem ha has negaive feedback loos in boh he eiheal loo and he sem cell loo and illusaes he fou comamens of he model: he hemaooieic sem cell (HSC) comamen (S), he neuohil comamen (N), he eyhocye comamen (R) and he laeles comamen (P). The HSCs ae assumed o be selfenewing, and hus cells in he esing (S ) hase can eihe ene he olifeaive hase a ae K (S) o diffeeniae ino neuohils (N) a ae F(N). As he neuohil ecusos diffeeniae, hei numbes ae amlified by a faco A, which accouns fo boh successive divisions and cell loss due o Inenaional Science Congess Associaion 5

3 aoosis. I is also assumed ha aoosis occus duing he olifeaive hase a ae γ and ha maue neuohils die a ae α. As can be seen in figue, he sysem is conolled by fou negaive feedback loos. The fis one egulaes he ae K (S) of eeny of HSCs o he olifeaive cycle, and i oeaes wih a delay τ ha accouns fo he ime equied o oduce wo daughe cells fom one mohe cell. dn N AF( N) S () dr R AF( R) S () dp P AF( P) S () The equaions fo he wo vaiables N, R, P and S can be deived fom a ime age mauaion fomulaion, o wien diecly fom consuling figue. Fo he comamen N, he loss is he efflux o deah N and he oducion of maue neuohils is equal o he influx F (N) S fom he HSC comamen imes he amlificaion A. Since one needs o ake ino accoun he ansi ime τ, he oal oducion of maue neuohils ae AF(N ( - τ ))S( - τ), AF(R ( - τ ))S( - τ) and AF(P ( - τ ))S( - τ) o equivalenly AF(N )S, AF(R)S and AF(P)S (ecall ha N = N ( - τ ), R = R ( - τ ) and P = P ( - τ )). Fo he second vaiable, he loss fom he comamen S is he flux eeneing he olifeaive hase, K (S)S, lus he efflux going ino diffeeniaion, F(N )S. The oducion of S is equal o he flux of cells eeneing and suviving he olifeaive hase, given by K (S)S, imes he cell division faco. The dynamics of S is hen descibed by ds F( N) S K( S) S K( S) Se (5) ds F( R) S K( S) S K( S) Se (6) ds F( P) S K( S) S K( S) Se (7) The feedback funcions F and K ae monoone deceasing Hill funcions. F conols he numbe of comamens while K(S) egulaes he level of HSCs. n ( ) f (8) n n N F N ( ) f (9) R F R s F( P) f () P n s ( ) f () n s S K S I develoed a comehensive model ha conains fou comamens: he HSC (Q), he neuohils (N), he eyhocyes (R) and he laeles (P). This model combines a numbe of comamenal models. The sem cell and neuohil dynamics ae based on he model and he eyhocye and laele comamen ae simlified models. The ciculaing cells ae couled o each ohe via hei common oigin in sem cell comamen. Regulaoy negaive feedback loos deemine how much diffeeniaion fom he sem cells each cell line will undego. Since i akes seveal days o oduce a maue cell fom a newly diffeeniaed cell, ime delays aea in he equaions. The model consiss of a se of fou couled delay diffeenial equaions. dq Q ( K N KR KP ) Q e Q () dn dr dp N AK( N) Q () R A{ K( R) Q e K( R ) Q} () P A{ K( P) Q e K( P ) Q} Peiodic Chonic Myelogenous Leukemia: Leukemia is a cance of he blood o bone maow chaaceized by an abnomal olifeaion of blood cells, usually leucocyes. Chonic myelogenous leukemia (CML) is disinguished fom ohe leukemias by he esence of a geneic abnomaliy in blood cells, called he Philadelhia chomosome, which is a anslocaion beween chomosomes 9 and ha leads o he fomaion of he BcAbl fusion oein. This oein is hough o be esonsible fo he dysfuncional egulaion of myelocye gowh and ohe feaues of CML. DDE Models: Delay-diffeenial equaions (DDEs) ae a lage and imoan class of dynamical sysems. They ofen aise in biological sysems whee ime lags naually occu. In aicula, in hemaology seveal ocesses ae conolled hough feedback loos and hese feedbacks ae geneally oeaive only afe a ceain ime, hus inoducing a delay in he sysem feedback. The geneal fom of a DDE fo x() R n is f (, x( ), x ) (5) whee x is he delayed vaiable (x( oeao in R R n C. )) and f is a funcional DDE wih Consan Delays: Delay diffeenial equaions wih consan delays ake he fom f ( x( ), x( ), x( ),..., x( n)), whee he quaniies i, i =,,..n ae osiive consans. Fo simliciy, conside he DDE wih a single consan delay: f ( x( ), x( )) (6) To obain a soluion of equaion (6) fo >, one needs o secify a hisoy funcion on [, ]. Indeed, ecall ha fo an odinay diffeenial equaion (ODE) sysem wih n vaiables, one would only need o secify he iniial values x() fo each of he n sae vaiables. In ode o solve a DDE, one needs o secify no only he value a =, bu also all he as values of x() ove he ineval [, ]. Fo examle, le X() eesen he ciculaing cell oulaion of a ceain ye of blood cell, assume ha is he andom ae of loss of cells in he ciculaion and F is he flux of cells fom he evious Inenaional Science Congess Associaion 6

4 comamen. Then, he dynamics of he numbe of ciculaing cells will have he geneic fom X F( X ( )) DDE wih Disibued Delays: A disibuion of delays is hen be moe aoiae and he DDE becomes an inegodiffeenial equaion of he fom f ( x( ), x( ) G( ) d ) (7) The densiy G(u) of he disibuion funcion is efeed o as he memoy funcion o he kenel and is nomalized, i.e. G( u) du This ye of model can also be ineeed as allowing fo a sochasic elemen in he duaion of he delay. Also, we will see ha fo some densiies G(u), equaion (7) can be equivalenly viewed as a sysem of odinay diffeenial equaions 7. ODE Models: Delay diffeenial equaions naually aise in modeling biological sysems. Conside he following DDE sysem wih a disibued delay: f ( x( ), x( ) G( ) d ), (8) wih he secial choice of he densiy of he gamma disibuion fo he memoy funcion a u au G( u) Ga ( u) e,! whee a is a osiive numbe and is a osiive inege o zeo. Noe ha he funcion G(u) has a maximum a u = /a and ha, as a and incease, keeing /a fixed, he kenel aoaches a dela funcion and he disibued delay aoaches he discee ime delay wih = /a. Moeove, i is clea ha he following hee oeies ae saisfied: lim G ( u), G (), G () a u a a a The cenal idea of he mehod is o elace he disibued delay by an exension of he se of vaiables. Define + new vaiable as j j ( ) a ( ),,,..., x x G d j x x ( ) G( ) d, Then, using he oeies of G one can show ha hese new vaiables saisfy a sequence of linea ODEs Solving he following sysem is hus equivalen o solving he DDE oblem (8), given ha he new vaiables ae given aoiae iniial values: j f ( x, x ) a( x x ), j,,..., j j a( x x ) The linea chain ick could be useful fo numeical comuaions since i educes he oblem o an ODE sysem, fo which seveal numeical mehods ae available. Howeve, his mehod canno be used fo all sos of delays 8. Age-Sucued Models: Le x(, a) be he cell densiy a ime and age a in a geneic comamen. We assume ha cells disaea (die) a a ae (). We also assume ha he cells in he comamen age wih a velociy V () and ha a cell enes a comamen a age a = and exis his comamen a age a =. Theefoe, he equaion saisfied by x(, a) is an ime-age equaion (advecion, o eacion-convecion, equaion): x x V ( ) ( ) x,, a [, ], a The igh hand side in his equaion eesens he ae a which cells in he age ineval a o a + a disaea a ime. To eesen he manne in which new cells ene he comamen, we define he bounday condiion (B.C.) x(, ) = H(). Finally, o fully eesen he oblem, we secify he iniial condiion (I.C.) x(, a) = (a). Using he mehod of chaaceisics, we obain he following delay diffeenial equaion: V ( )[ H ( ) H( T ) T ex( ( w) dw)] ( ) X ( ), (9) whee X() is he oal numbe of cells X ( ) x(, a) da and T saisfies V ( w) dw. Noe ha if is a consan, equaion (9) T educes o T V ( )[ H( ) H( T ) e ] ( ) X ( ), In addiion, if he aging velociy is consan (V () = V ), we have ha T saisfies Vdw =VT T which imlies ha T = /V. Hence, if and V ae consan, we obain a delay diffeenial equaion wih consan delay: V H H V e X T / V ( )[ ( ) ( / ) ] ( ) ( ), Model Comosiion: The model has seveal negaive feedback funcions ha egulae sem cell olifeaion and diffeeniaion ino he hee ciculaing cell yes. In his model, as in he one esened, hese ae given by: s ( Q) k s s Q Inenaional Science Congess Associaion 7

5 n N ( ) n n N k N f k k( P) KP k kr( R) m KR We mus also secify an inu funcion I() ha eesens he subcuaneous G-CSF injecions. We assume ha his inu is bief in duaion, and ha he oal amoun of G-CSF added coesonds o he desied dosage, namely: I() dosage Noe ha if is small, a Gaussian-like inu aoximaes a Diac -funcion, and we can wie / ae a Theefoe o simulae eiodic injecions, we le T (( mod T) ) / I( ) H( d) ae whee H () denoes he Heaviside se funcion H( ), H( ), The day on which eamen is iniiaed is denoed by d, and he Heaviside funcion simly uns he injecions on. The em mod T ensues eiodiciy, and we equie ha T so ha he aoximaion o he inegal emains valid. Finally, we ensue ha holds by choosing he aamee a a = dosage 9. I emains only o descibe how he G-CSF acs on he hemaological oion of he model. Because we believe fom evious modeling effos ha A N, S, and ae he aamees ha need o change unde G-CSF, we model G-CSF injecions as causing flucuaions in hese hee aamees: u A A ( H( d)) H( d)(( m ( G G)) A ) N N A N u ( H( d)) H( d)(( m ( G G)) ) s s g s ( H( d)) H( d)(( m ( G G)) ) u The suescis and u esecively indicae values coesonding o values ha, in he model wihou he dynamics of G-CSF, mach eaed and uneaed daa esecively. The aamees m A, m g, and m ae sloes ha secify how much A N, S, and change in esonse o a given change in G-CSF concenaion, G. is he aveage G-CSF concenaion fo each daa se. These wee comued using he G-CSF model alone, and using he aveage neuohil levels in each daa se. Resuls and Discussion In each case, we found ha he neuohil amlificaion inceases subsanially unde G-CSF eamen, as does he ae of sem cell aoosis, and he diffeeniaion ino he neuohil line. We heefoe edic simila changes fo he emaining dogs hee is some edundancy in he model, in ha inceasing he neuohil amlificaion and he diffeeniaion ino he neuohil line fom he sem cells has simila effecs. This is no unexeced, since he imay effec of boh changes is o aise neuohil levels. Figue. shows he fi of he uneaed and eaed daa fo dogs, 8 and 7. Noe ha he analysis maches he daa easonably well fo he neuohils as well as fo he eyhocyes and laeles. This confims ha he new model, wih he G-CSF couled o he cell oulaion dynamics, is caable of eoducing he daa. The leas squaes diffeences beween he simulaions and he daa wee no significanly less han he values eoed. These simulaions and daa ae fo daily eamen. Figue shows he daa and analysis fo he ohe fou dogs (dogs,, 7 and 8), again wih daily eamen. Recall ha hese wee he esimaed, no fied, values fo he eaed aamees and noe he qualiy of he fis. Thus, we ae able o mach obseved daa wihou auomaed aamee fiing based simly on an examinaion of he eaed daa and he aamee changes fo dogs, 8 and 7. Fo each dog, we efomed simulaions comaing daily eamen, eamen evey ohe day, and evey hee days. We find ha aiculaly fo dogs,, 8 and 7, changing he eiod of he eamen can significanly affec he naue of he oscillaions. I shows he esuls of eaing dog 8 evey ohe day, ahe han evey day. We have also exloed he effecs of changing he ime a which he eamen is iniiaed. In mos cases, his did no significanly change he long-em behavio. Howeve, fo dog 7 he amliude of he oscillaions was significanly educed when he eamen was iniiaed in he lae half of he cycle. Moe secifically, measued fom day, we find ha smalle oscillaions occu if eamen is iniiaed on day 8 o afewads, o on days o 5 (see figue ). When eamen was iniiaed on ohe days, lage oscillaions in he model esuled. We wee awae fom ou evious sudy of simila models ha hee is he ossibiliy ha wo o moe qualiaively diffeen saes can be locally sable, and we have also found evidence fo his in he esen model. Namely, changing he eamen onse ime fom day o day 8 fo dog 7 caused he simulaion o sabilize o wo vey diffeen yes of behavio. I should also be noed ha inceasing he G-CSF dosage in he model someimes heled o sabilize oscillaions (dog 7), bu in seveal cases (dogs, 8 and ) a dosage incease fom 5 μg/kg o a dosage in he ange 5-5 μg/kg caused some simulaions o fail. In ha analysis, he diffeeniaion ae ou of he sem cells was so high, and he aoosis ae in he sem cells was so high, ha he sem cell oulaion was no longe able o mainain iself. Fo he ohe dogs, hee was always a dosage ha was sufficienly high o eminae he FFT analyze, bu i was someimes a faco of highe han he acual dosage given. Inenaional Science Congess Associaion 8

6 Neuohil (Dog 7) Neuohil (Dog 7) Neuohil(Dog 8) Neuohil (Dog 8) Neuohil(Dog ) Neuohil (Dog ) Figue Coninuous neuohil daa and analyze fo dogs, 8, 7,,, 7 and 8. The lef figue shows uneaed daa (ed) and fi (nomal aea). The igh figue shows eaed daa (geen) and analyze fo dogs unde daily G- CSF eamen. Noe ha he model accouns fo he diffeen scaling in neuohil couns. The calculaions wee obained using aamees esuling fom he FFT analysis mehod. Neuohil unis ae 8 cells-kg uneaed (dog ) eaed (dog ) ime (days) ime (days) uneaed (dog 8) eaed (dog 8) ime (days) ime (days) uneaed (dog 7) eaed (dog 7) ime (days) ime (days) Inenaional Science Congess Associaion 9

7 Neuohil (Dog 8) Neuohil (Dog 8) Neuohil (Dog 7) Neuohil (Dog 7) Neuohil (Dog ) Neuohil (Dog ) Neuohil (Dog ) Neuohil (Dog ) uneaed (dog ) eaed (dog ) ime (days) ime (days) uneaed (dog ) eaed (dog ) ime (days) ime (days) uneaed (dog 7) eaed (dog 7) ime (days) ime (days) uneaed (dog 8) eaed (dog 8) ime (days) ime (days) We have develoed a model of he hemaooieic sysem ha includes a hamacokineic model of G-CSF dynamics in issue and in ciculaion. The model is able o accoun fo he feaues of uneaed, and G-CSF-eaed, daa fo dogs wih cyclical neuoenia. This is accomlished, saing wih aamee fiing done, by fiing aamees fo dogs and heeby esimaing, no fiing, aamees fo ohe dogs. One of he mos iniguing obsevaions esuling fom he aamee fiing in his sudy is ha o fi obseved daa fo cyclical neuoenic dogs and human aiens duing G-CSF eamen i was necessay o assume ha hee was an incease in he ae of aoosis in he sem cell comamen duing G-CSF eamen, a he same ime as he moe execed incease in neuohil amlificaion. Inenaional Science Congess Associaion

8 The sudy we eo hee abou eamen schedules indicaes ha changing he eiod of he eamen fom daily o evey ohe day, and hen o evey hid day, almos always significanly ales he naue of he oscillaions. Since G-CSF is cosly and may have undesiable side effecs, i may be woh exloing his oion fuhe in humans. Fuhemoe, we found in one case (Dog 7) ha changing he ime of onse of eamen o he lae half of he cycle (as measued by seing day o be he day when he neuohil level is minimal) esuls in much smalle amliude oscillaions in he eaed FFT analysis. In he model, boh of hese inevenions had moe significan effecs on he oscillaions han did changing he G- CSF dosage. Indeed, inceasing he dosage was no seen o be a viable oion in ou analysis, as i fequenly led o he eminaion of he FFT ahe han o he sabilizaion of oscillaions. The obseved daa ae highly vaiable fom one dog o anohe, bu he simulaions can be individualized o accoun fo his. This esens he ossibiliy of using eal ime daa fo a given dog o individualize model analysis and make edicions abou he effecs of diffeen eamen schedules. Ealie modeling wok also suggesed ha significanly diffeen behavio would esul fom diffeen G-CSF eamen schedules. Ou model subsaniaes his, and quanifies he effecs using ealisic G- CSF dynamics and yielding analysis ha ae diecly comaable o obseved daa. Ou cenal esul is ha in he model, changing he ime of eamen iniiaion and o he eiod of eamen may esul in equally good, o bee, longem oucomes and may equie less G-CSF. These changes would be acical o imlemen and, if less G-CSF wee equied, would educe he isk of side effecs as well as he cos of eamen. Conclusion The model could be used o sudy diffeen mechanisms of ganulooiesis and of G-CSF adminisaion. In aicula, we assumed G-CSF was acing on boh he aoosis ae and he amlificaion faco in he olifeaive hase of neuohil ecusos. Since boh effecs ae modeled seaaely, we could sudy in moe deail he ecise effecs of each faco. As menioned above, clinical daa fo alenaive eamen schedules wih G-CSF ae needed o validae ou esuls and make fuhe model imovemens. If such daa wee available, he individualized aoach esened in he ae could be ineesing o imlemen since i fis he model o daa befoe and duing eamen fo a given subjec. FFT simulaions fo his subjec could hen edic he ossible oucomes of diffeen eamen schemes. In conclusion, hemaooiesis and G-CSF effecs ae no ye fully undesood and seveal asecs ae sill being sudied. Clinical findings and fuue sudies will ovide new insighs and hel o bee undesand he sysem. The models esened hee would hen have o be modified accodingly, as his is a of he evoluionay ocess of mahemaical modeling. Refeences. Adimy M. and Cause F., Global sabiliy of a aial diffeenial equaion wih disibued delay due o cellula elicaion, Nonlin. Analysis., 5, 69 9 (). Balamualihaan S. and Rajasekaan S., A Mahemaical Age-sucued Model on Aiha Using Delay Paial Diffeenial Equaions. Ausalian Jounal of Basic and Alied Sciences, 5(), 9-5 (). Balamualihaan S. and Rajasekaan S., A Mahemaical Age-sucued Model on CN Using Delay Paial Diffeenial Equaions, Poceedings of he Inenaional Confeence on Emeging Tends in Mahemaics and Comue Alicaions, MEPCO Schlenk Engineeing College, Sivakasi, India, 8-87 (). B elai J., Mackey M.C., and Mahaffy J.M., Agesucued and wo-delay models fo eyhooiesis, Mah. Biosci., 8, 7 6 (995) 5. Benad S., Belai J. and Mackey M., Oscillaions in cyclical neuoenia: New evidence based on mahemaical modeling, J. Theo. Biol.,, 8 98 () 6. Beue A., Glass L., Mackey M.C. and Ticombe M.S. Nonlinea Dynamics in Physiology and Medicine, Singe-Velag () 7. Colijn C., Fowle A.C. and Mackey M.C., High fequency sikes in long eiod blood cell oscillaions, J. Mah. Biol., 5, (6) 8. Dale D.C. and Hammond W.P., Cyclic neuoenia: A clinical eview, Blood Rev.,, (988), 9. Foley C. and Mackey M., Dynamic hemaological disease, A eview, J. Mah. Biol., In ess (8). Foley C., Benad S. and Mackey M., Cos-effecive G- CSF heay saegies fo cyclical neuoenia: Mahemaical modelling based hyoheses, J. Theo. Biol., 8, (6). Hauie C., Dale D.C., Rudnicki R., and Mackey M.C. Modeling comlex neuohil dynamics in he gey collie, J. Theo. Biol.,, 5 59 (). Mahaffy J., Belai J. and Mackey M., Hemaooieic model wih moving bounday condiion and sae deenden delay: Alicaions in eyhooiesis, J. Theo. Bio., 9, 5 6 (998) Inenaional Science Congess Associaion