A THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER
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1 A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail: valupadas@gmail.com ABSTRACT ahmaical modling of pharmacoinics is an imporan growing fild in drug dvlopmn. Pharmacoinics concrns wih h disribuion of drugs, chmicals or racrs by a fluid among h various comparmn of human body. In his wor w discuss h comparmn mahmaical modl of livr funcion basd on fundamnal biological pharmacological principls. Hr w prsn bhavior of hyroxin, iodin bil ovr a priod of im. Kywords: livr, mahmaical modl, diffrnial quaions, pharmacoinics, comparmn modl. INTRODUCTION Pharmacoinics concrns wih h disribuion of drugs, chmicals or racrs by a fluid among h various comparmn of human body []. Th comparmns could b ficiious spacs hrough which biomarials pass hrough various locaions (comparmns of h body. Th prsn invsigaion is on a hr comparmn modl rlad o h livr in human body [,, 4]. Whn a chmical, hyroxin is injcd ino h blood sram i is carrid o h livr. Th livr convrs hyroxin o iodin, which is absorbd ino h bil [5]. Howvr, nihr h convrsion nor h absorpion of which ino h bil, would occur insananously. Som of h hyroxin (unconvrd rnrs ino h blood sram gs rcirculad. Th ahmaical modl is composd of hr comparmns: Comparmn I, Comparmn II comparmn III which rprsn blood vssls, livr bil rspcivly as shown in h Figur-. Figur-. ahmaical modl for livr. Noaions adopd: x ( = h quaniy of hyroxin in h blood vssl a h insan im x ( = h quaniy of iodin in h livr x ( = h quaniy of iodin absorbd in o h bil = Th ra of convrsion of hyroxin ino iodin = Th ra of h quaniy of unabsorbd hyroxin sn ou for rcycling from Comparmn II o comparmn I = ra of absorpion of Iodin from comparmn II ino bil comparmn III x 0, x 0 x 0 ar iniial h valus of x, x, x rspcivly h ra consans x, x x ar all posiiv. I is assumd ha h ra x ( a which hyroxin is convrd o iodin as i ransfrrd from comparmn I o comparmn II is proporional o concnraion x ( of hyroxin in h comparmn I. ODEL BLOCK - DIAGRAS AND ODEL EQUATIONS dx x x, x(0 x0 ( 0
2 dx x x, x(0 x0 ( ( 0 4 (8 Evidnly is ral lss han. Hnc ar boh ral ngaiv roos. Th soluions of quaions (, (, ( can b wrin as dx, x (0 x 0 ( x Th quaions (, (, ( can b pu ino h of marix form d whr X AX (4 A 0 ( 0 x 0 X x 0 x L X X 0 b a rial soluion wih iniial condiions X 0 [x 0 x 0 x 0 ] T Th xponn λ saisfis h characrisic quaion of A: d AI 0 i, ; ( 0 (6 h roos of which ar λ=0, (7 whr = (5 x AB C (9 ( x ( A B( C( B( C( x( A E (0 ( whr A, B, C E ar arbirary consans. Using h iniial condiions, w g h valus of h consans A = 0 ( x0 ( x0 B ( x x 0 ( 0 C (4 E ( x0 x0 x0 (5 Subsiuing hs valus in (., (., (. w g afr som simplificaion x ( x ( 0 0 x x0 ( x0 x0 x0( sinh ( x0cosh + (6 0
3 x0( x0 x0x0( x( ( ( ( x0 x0 sinh ( x0cosh (7 x0 ( x0 x0 x0( x( ( ( ( x0 x0 x0 ( ( x0 ( x0 x0 sinh ( x0 x0cosh ( x0 x0 x0 (8 Th variaions of x (, x ( x ( Vrs ar illusrad for a slc rang of valus of,, (vid Figur- o Figur-5 for h iniial valus x0 50, x0 5 x0 65 of hyroxin, iodin bil, rspcivly. Figur-. Variaion of hyroxin, iodin bil for for h ransfr cofficins =0.85 =0.05 = Figur-. Variaion of hyroxin, iodin, bil for h ransfr cofficins =0.85 =0.056 = Figur-4. Variaion of hyroxin, iodin, bil for h ransfr cofficins =0.85, =0.05 =
4 Figur-5. Variaion of hyroxin, iodin, bil for h ransfr cofficins =0.85, =0.055 = CONCLUSIONS Thyroxin monoonically dcrass raching zro lvl. Iodin iniially riss o rach a maximum falls asympoically o approach o zro as is h cas wih hyroxin. Th bil incras monoonically approachs asympoic lvl. Figur-6. =0.85, =0.05, = Incipin variaions of x (, x ( x ( (i..; variaions for small rang im I is now ( cosh O( sinh O(! Nglcing rm of O (, w g xx0 ( 0 x x0 ( ( x0 0 x (9 Figur-7. =0.85, =0.05, = x x ( x x x ( x (0 x x0( x0 ( x0 ( x0 ( Th variaions of x (, x ( x ( Vrs small im ar illusrad for a slc rang of valus of,, (vid Figur-6 o Figur-9 for h iniial valus x0 50, x0 5 x0 65 of hyroxin, iodin bil, rspcivly. Figur-8. =0.85, =0.05, =
5 [] Jacquz John A. 97. Comparmn analysis in biology mdicin. Nw Yor:Elsvir Scinific publishing company. [4] J.. Wa Andrw Young. 96. An amp o simula h livr on a compur. Compur. Journal. 5: -7. [5] Goffry Gordon Sysm simulaion. Prnichal India. pp Figur-9. =0.85, =0.05, = Asympoic variaion of x (, x ( x ( (variaion for larg rang im In his cas dominas ovr (sinc >0 cosh sinh Hnc h asympoic xprssion for x (, x (, x ( ar ( K x ( x0 x0( K ( K x ( x0 x0 ( K ( x K ( x ( x x ( K ( x x x0 ( ( (4 CONCLUSIONS For larg rang im, ach of x (, x ( x ( dcras xponnially wih h Characrisic im. REFERENCES [] J.N. Kapur ahmaical modls in Biology dicin. Affiliad as-ws prss pv. Ld. pp [] Evr C.F.F. Ral Formulaion compuaion of comparmn modls. J. pharm. Sci. 9(:
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