Mathematical Model of the Pharmacokinetic Behavior of Orally Administered Erythromycin to Healthy Adult Male Volunteers

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1 Sybiosis wwwsybiosisolieorg wwwsybiosisoliepublishigco Research Article SOJ Pharacy & Pharaceutical Scieces Ope Access atheatical odel of the Pharacokietic Behavior of Orally Adiistered Erythroyci to Healthy Adult ale Voluteers Received: Septeber 06, 205, Accepted: Noveber, 205, Published: Deceber 6, 205 *Correspodig author: Abstract Objectives: The ai objective of the curret study was to preset a further exaple which showed a victorious use of a superior atheatical odelig ethod based o the cocept of a dyaic syste i atheatical odelig i pharacokietics A pharacokietics to use of a alterative odelig ethod to those odelig ethods cooly used i pharacokietic studies The curret study is a escort piece of a correlated (Yakata et al 979) study i voluteers published i the Joural of Pharacokietics ad Biopharaceutics I the study cited at this poit, a ivestigatio of bioequivalece of erythroyci stearate tablets i a was described ethods: Data published i the study cited above ad a advaced odelig ethod were used (For the ethod used, please see eg the explaatory picture ad the full-text studies at: wwwuefsavsk/advacedht) Results: the easured data Based o the atheatical odels developed, ai pharacokietic variables of erythroyci were deteried Coclusio: The atheatical odelig ethod used i the it ca be used to develop atheatical odels ot oly i Keywords: Erythroyci; Oral adiistratio; atheatical odel Itroductio Erythroyci is a atibiotic useful for the treatet of a uber of bacterial ifectios This icludes respiratory tract ifectios, ski ifectios, chlaydia ifectios, ad syphilis It ay also be used durig pregacy to prevet Group B streptococcal ifectio i ewbors [] It is a effective ihibitor of CYP3A4 that patetly icreases circulatig levels of soe other HG-CoA reductase ihibitors [20] The ai objective of the curret study was to preset a further exaple which showed a victorious use of a advaced atheatical odelig ethod based o the cocept of a dyaic syste i atheatical odelig i pharacokietics [3-4] A additioal objective was to otivate researchers i pharacokietics to use of a alterative odelig ethod to those odelig ethods cooly used i pharacokietic studies Previous exaples presetig a advatageous use of the odelig ethod used i the curret study ca be foud i full-text articles available copletely free of cost o the authors web pages at: ad wwwuefsavsk/advacedht ethods The data published i the study [] were eployed For odelig purposes, a advaced atheatical odelig ethod odelig ethod based o the cocept of a dyaic syste was used; see eg the studies cited above The developet of a atheatical odel of each dyaic syste H was based o the followig siplifyig assuptios: a) iitial coditios of each dyaic syste H be zero; b) pharacokietic processes occurrig i the body oce oral erythroyci adiistratio; were liear ad tie-ivariat, c) cocetratios of erythroyci were the sae throughout all subsystes of the dyaic systes H (where subsystes were itegral parts of whole dyaic systes H); d) o barriers to the distributio ad/or eliiatio of erythroyci existed The odelig process used i the preset study ca be described as follows: ito the body, deoted was I the secod step of the ethod, the dyaic systes were used to atheatically represet dyaic relatios betwee erythroyci iputs ito the body ad erythroyci Sybiosis Group *Correspodig author eail: exfaadu@savbask

2 I the third step of the ethod, the trasfer fuctio, deoted (see Eq ) was derived for each active syste H by relatig C(s), oral adiistratio of erythroyci, deoted (the lower case letter eg the studies cited above ad the followig equatio: ( ) ( ) () ( ) Thereafter, the dyaic syste H of each voluteer was described with trasfer fuctio deoted see, eg the followig studies [3-4] ad refereces therei For odelig purpose, the software aed CTDB [8] ad trasit the fuctio odel ( ), described i the followig equatio were used: 0 ( ) ( ) [9], is a estiator of the odel paraeter kow as the gai of the dyaic syste a, a, b, b are the additioal odel paraeters, is the axiu degree of the oiator polyoial, ad is the axiu degree of the deoiator polyoial, where <, see eg the followig studies [3-4] I the fourth step of the ethod, the trasfer fuctio H( ) was coverted ito equivalet frequecy respose fuctio, deoted ( ) j [4] published earlier [3-4] was used to deterie a atheatical odel of the frequecy respose fuctio ( j ) ad poit estiates the paraeters of the odel frequecy respose fuctio ( j ) i the coplex doai for each voluteer The odel of the frequecy respose fuctio ( j ) used i the recet study is described by the followig equatio: ( ) ( j ) 0 j j ( j ) Aalogously as i Eq [2]: is the axiu degree of the uerator polyoial of the odel frequecy respose fuctio ( j ), is the axiu degree of the deoiator polyoial of the odel frequecy respose fuctio ( j ),, is the iveted uit, ad is the agular frequecy i respose fuctio ( ) j was advaced, usig the ote-carlo ad the Gauss-Newto ethod i the tie doai I the sixth step of the ethod, the Akaike iforatio was used to differetiate aog odels of frequecy respose fuctios ( j ) of dissiilar coplicatio ad to select the best odel of the frequecy respose fuctio ( j ) with the iiu value of the Akaike iforatio criterio [5] I the ( j ) were deteried (2) (3) After the iproveet of atheatical odels of the dyaic systes H, the followig priary pharacokietic variables were deteried: The tie occurrece of the axiu observed plasa cocetratio of erythroyci, deoted t the axiu observed plasa cocetratio of erythroyci, deoted C the eliiatio half-tie of erythroyci, deoted t /2 o -, ad total body clearace of erythroyci, deoted by CI The trasfer fuctio odel ( ) ad the frequecy respose fuctio odel ( ) j are ipleeted i the coputer progra CTDB [8] A saple versio of the coputer progra CTDB is available at: ht Trasfer fuctios ad frequecy respose fuctios are ot ukow i pharacokietics, see eg the followig studies [20,2] Results ( j ) selected with the Akaike iforatio criterio was described by Eq (4): 0 j ( j ) j cocetratio data i all voluteers ivestigated i the previous [] ad the curret study Estiates of the odel paraetersa 0, a, a 2, a 3 are listed i Table odel-based estiates of priary pharacokietic variables were listed i Table 2 Voluteer No was arbitrarily chose fro fourtee voluteers ivestigated i the previous study [] ad i the curret study, to illustrate the results obtaied Figure showed developed odel of the dyaic syste H Aalogous results also hold for all voluteers participatig i the previous [] ad the curret study Discussio The dyaic systes used i the curret study were atheatical objects, without ay physiological applicatio They were used to odel dyaic relatioships betwee Table : Paraeters of the third-order odel of the dyaic syste describig the pharacokietic behavior of orally adiistered erythroyci to voluteer No odel paraeters G (hl - ) 0007 Estiates of odel paraeters (95% CI) a 0 (-) to 02 a (i) 595 b (i) b 2 (i 2 ) b 3 (i 3 ) (4) Citatio: Healthy Adult ale Voluteers SOJ Phar Phar Sci,3() -5 Page 2 of 5

3 Table 2: Pharacokietic variables of orally adiistered erythroyci to voluteer No Pharacokietic variable The tie of the axiu observed cocetratio of erythroyci t ax (h) Estiates 32 * ad outputs of the dyaic systes H I geeral, a dyaic syste is odeled usig the trasfer fuctio odels ( ), as it was the case i the recet study, the the accuracy of the odel depeds i large part of the degrees of polyoials of the trasfer fuctio odels ( ) eg the followig studies [3-4] erythroyci C ax The plasa eliiatio half-tie of erythroyci t /2 (hod) ea absorptio tie of erythroyci (hr) e residece tie of erythroyci CI (li - ) fro plasa * stadard deviatio 293, betwee the agitudes of a output of the dyaic syste to a agitude of a iput to the dyaic syste i steady state Or i other words, the paraeter gai of a dyaic syste is a proportioal value that shows the relatioship betwee the agitudes of a output to a agitude of a iput of a dyaic syste i steady state The pharacokietic eaig of the paraeter gai depeds o the ature of the dyaic syste; see eg studies available at: The o-iterative ethod published i the study [4] ad used a optial structure of a odel frequecy respose This is a up the developet of frequecy respose odels The reaso for coversio of ( ) to ( ) j was as explaied i the Citatio: Figure : - voluteer s No dyaic syste which atheatically represeted the dyaic relatio betwee erythroyci iput to the body ad behavior of erythroyci i voluteer No Table 3: Observed ad odel predicted blood cocetratios of erythroyci i voluteer No Tie (h) Observed cocetratio of erythroyci (g/ l) odel predicted cocetratio of erythroyci (g/ l) , erythroyci iputs ito the body [5-7] ad behavior of erythroyci i the body The ethod used i the recet study has bee described i detail i the previous studies [3-3] As i previous studies, authored or co-authored by the author of the recet study, the developet of atheatical odels of the dyaic systes was based o the kow iputs Healthy Adult ale Voluteers SOJ Phar Phar Sci,3() -5 followig text The variable: s i ( ) is a real variable, what is suitable for odelig purposes The atheatical odels developed i the recet study erythroyci iput to the body ad behavior of erythroyci i the body i the voluteers ivestigated i the previous [] ad the curret study The curret study agai showed that atheatical ad coputatioal tools fro syste egieerig ca be successfully used i atheatical odelig i pharacokietics Frequecy respose fuctios are coplex fuctios, therefore odelig is perfored i the coplex doai I additio, the odelig ethods used to develop odel frequecy respose fuctios are coputatioally itesive, ad odelig require as a iiu partial kowledge of the theory of dyaic syste, ad a abstract way of thikig about the dyaic syste uder study The priciple differece betwee pharacokietic odelig ethods traditioally used i pharacokietic studies ad odelig ethods that use atheatical ad coputatioal tools fro the theory of dyaic systes, is as follows: the forer ethods are based o odelig plasa (or blood) cocetratioo odelig dyaic relatioships betwee a atheatically represeted drug iputs ito the body ad atheatically represet resultig plasa (or blood) cocetratio-tie eg the previous studies authored ad/or coauthored by the author of the curret study ad the explaatory exaple, available free of cost at the author s Web page: Page 3 of 5

4 The coputatioal ad odelig ethods that use coputatioal ad odelig tools fro the theory of dyaic systes that ca be exaple for adjustet of drug adiistratio aied at achievig ad the aitaiig required eg the followig ad cost-effective idividualizatio of drug dosig eg through coputer-cotrolled ifusio pups This is iportat exaple for adiistratio of clottig factors to heophilia patiets, as 4 5 Pharacokietics fro a dyaical systes poit of view Cli Pharacokiet 989;7: ;22(4) deteriatio of the aalytical for of the pharacokietic syste weightig fuctio Coput ethods Progras Bioed The advatages of the odel ad odelig ethod used i the recet study are evidet here: The odels developed ad used overcoe oe of the typical liitatios of copartetal odels: For the developet ad use of the odels cosidered here, a assuptio of well-ixed spaces i the body is ot ecessary The basic structures of the odels developed are used largely applicable to atheatical odelig differet of tie-varyig cotiuous drug ifusio i idividual patiets: A 444 odelig Basic Cli Pharacol Toxicol pharacokietic couity, is a advatage of the odels developed usig coputatioal tools fro the theory of dyaic systes is that the odels cosidered here ephasize dyaic relatioships betwee drug iputs ito the body ad behavior of a drug i a hua ad/or a aial body The ethod used i the curret study ca be easily geeralized Therefore, there is o ot ukow i pharacokietics; see eg the followig studies [20-22] Coclusio The odels developed ad used i the curret study are successfully described the pharacokietic behavior of oral adiistratio to healthy ale adult voluteers The odelig it ca be applied to a broad rage of dyaic systes i the study repeatedly preseted a attept to visualize the successful use of atheatical ad coputatioal tools fro the theory of dyaic systes i pharacokietic odelig For the previous attepts with the use of the odelig ethod used i the curret study please visit ht The curret study repeatedly showed that a itegratio of pharacokietic ad bioegieerig approaches is a good ad this is that such itegratio cobies atheatical rigor with biological isight Refereces Y et al Sigle-dose fastig bioequivalece assesset of erythroyci Diabetes Res Cli Pract 2007;77(3): ties Ge Physiol Biophys 204;33():75-80 doi: 0449/ Orally Adiistered Predisoloe i Healthy Voluteers J Pharaceu Pharacol 204;2(2):-5 2 successful use of a advaced atheatical odelig ethod based o the theory of dyaic systes i pharacokietics It J Res Sci 3 idicates oliearity of oradrealie effect o rat real artery Physiol Res 2008;57(5): ties Ge Physiol Biophys 204;33():75 80 doi: 0449/ 5 Weiss, Pag, KS Dyaics of drug distributio The role of the secod ad third curve oets J Pharacokiet Biophar 992;20(3): Verotta, D Cocepts, properties, ad applicatios of liear systes to describe distributio, idetify iputs, ad cotrol edogeous substaces ad drugs i biological systes Crit Rev Bioed Eg 7 deliver cisplati (IV) ad paclitaxel i aoicelles to iprove effect of erythroyci o the pharacokietics of rosuvastati Eur J 9 009/ TAC jsp?tp=&aruber=00705 Citatio: Healthy Adult ale Voluteers SOJ Phar Phar Sci,3() -5 Page 4 of 5

5 20 ad sie Nu Algorith 2005;40: DOI 0007/s Siegel RA Pharacokietic trasfer fuctios ad geeralized 22 Segre G The sojour tie ad its prospective use i pharacology J 23 pharacokietic odels J Pharacokiet Pharacody 24 Rescigo A Copartetal aalysis ad its aifold applicatios to Citatio: Healthy Adult ale Voluteers SOJ Phar Phar Sci,3() -5 Page 5 of 5