International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November ISSN

Transcription

1 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN mahemaial model of oxygen diffsion ino he blood seam G C E Mbah, SI Ionmbe, niak, S E Depamen of Mahemais,Univesiy of Nigeia, Nigeia gembh@yahooom godwinmbah@nnedng , BSTCT The fnion of he mammalian espiaoy sysem is he failiaion of he ansfe of gas/ exhange of gas beween he oganism s envionmen and is inenal anspo medim, he blood Evolionay poesses have opimied he anaomi se of he lng as a ee-like banhing newok of aays eminaing in hin walled elasi ds and alveoli, whee his gas exhange os Hee we develop a mahemaial model based on he esls and heoies fom aiflow and gas exhange beween he alveoli and blood apillay The model sdied he oxygen diffsion ino he blood, aimed a he diffsion aoss he membane The model is based on he mass balane in he sysem whih esls in he fis ode linea odinay diffeenial eqaion The model eqaion is solved expliily and he, ppe bond fo hikness of he membane is esablished Keywods :xygen,diffsion,espiaoy membane INTDUCTIN Beahing onsiss of wo phases, inspiaion and expiaion Ding inspiaion, he diaphagm and he ineosals msles ona Ding expiaion he diaphagm and msles elax When a beah is aken, ai passes in hogh he nosils, hogh he nasal passages, ino he phaynx, hogh he laynx, down he ahea, ino one of he main bonhi, hen ino smalle bonhial bles, hogh he smalle bonhioles, and ino a miosopi ai sa alled alveols I is hee ha exenal espiaion ends Hmans need a oninos spply of oxygen fo ellla espiaion, and hey ms ge id of exess abon dioxide, he poisonos wase pod Gas exhange sppos he ellla espiaion by onsanly spplying oxygen and emoving abon dioxide The oxygen we need is deived fom he Eah's amosphee, whee oxygen is % of he ai omposiion This oxygen in he ai is exhanged in he body by he espiaoy sfae In hmans he alveoli in he lngs seve as he sfae fo gas exhange, ino and o of he ilaing blood i flow os only when hee is a diffeene beween pesses in he alveoli and he apillaies aying he ilaing blood i will flow fom a egion of high pesse o one of low pesse, whee he bigge he diffeene, he fase he flow of he ai Diffsion is he movemen of a gas fom an aea of highe paial pesse onenaion o an aea of lowe paial pesse onenaion Diffsion helps o ansfe gases beween ( he lngs and he blood and ( he blood and he peipheal isses The aes of diffsion of a gas aoss he plmonay membanes depend on is solbiliy in wae Fo example, abon dioxide is imes moe solble in wae han oxygen and eadily osses he plmonay apillay membanes [0] In he lngs, oxygen leaves he aea of highe P, he alveoli, and enes he aea of lowe P, he aeial blood in he plmonay apillaies Conenly, abon dioxide leaves he aea of highe PC, he 03 hp://wwwijseog

2 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN aeial blood, and enes he aea of lowe PC, he alveoli The blood ens via he plmonay vein o he hea and hen moves ino he sysemi ilaion Gas exhange oing in he onsan-volme dead spae and vaiable-volme ollapsible and alveola ompamens was desibed by sing speies-onsevaion laws n he ai side of he exhange, i was assmed ha inspied ai was insananeosly wamed o body empeae and flly saaed wih wae vapo The gaseos mixe was pesmed o obey he ideal gas law n he blood side, i is of he same empeae wih he body and flly saaed wih wae vapo The gaseos mixe was pesmed o obey he ideal gas law n he blood side, he disee onsiens (plasma and eyhoyes wee lmped ogehe and assmed o saisially behave as a nifom, homogeneos phase[] Wihin a onol volme, he insananeos speifi eaions wee hen onsideed o be a eqilibim; elaionship beween speies onen and hei oesponding eqilibim paial pesses was onseqenly epesened by empiial dissoiaion ves [3],[5] and [] ne-dimensional axial onveion povided he sole means fo blk anspo of blood and movemen of oxygen along he plmonay ilaion; diffsion in he adial and axial dieions wee ignoed Two-phase flow eaed de o blood heeogeneiy was fhe disegaded Tanspo of gaseos speies aoss he alveola-apillay membane, assmed o be solely by diffsion, was haaeied by a lmped speies ling diffsing apaiy, whih aoned fo he oal diffsionesisive pah aken by speies I ( i, C, N as i diffsed aoss he alveola-apillay baie, was aken p by he blood and C was exeed, wheeas N (a elaively ine gas diffsed in eihe dieion, depending on he insananeos oveall venilaion-pefsion aio [9] The onibion of he physiologial shn [7] was negleed The model sed hee was diely adaped fom [], wih he povision ha alveola pesse was no held amosphei b ahe was allaed via he aay mehanis model In his pesen sdy we ae onened wih mahemaial model of gas exhange beween he alveoli and blood apillay fo nomal hman body FMULTIN F MDEL EQUTIN ssmpions of he model The faos ha deemine he ae of gas diffsion hogh he espiaoy membane inlde; The hikness of he membane The sfae aea of he membane 3 The diffsion oeffiien The paial pesse gadien of he gas aoss membane 5 The adia op (Blood 6 iflow ino he lngs is peiodi and is oninos 7 The espiaoy membane is ylindial in shape 03 hp://wwwijseog

3 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN Vaiables and paamees of he model P -alveola gas paial pesse (mm Hg P ν -venos blood gas paial pesse (mm Hg P -paial pesse of gas a any poin along he plmonay "apillay" (mm Hg P B -baomei pesse (mm Hg ( P av -aveage paial pesse diffeene beween gas in alveols and in plmonay apillay blood (mm Hg C -onenaion of blood gas a any poin along he plmonay "apillay" (dimensionless Q -adia op (lies/min φ -plmonay shn flow (lies/min Γ -faion of Q whih is Q8; ie, /Q (dimensionless α -solbiliy oeffiien (am- dq -ne ae of diffsion of oxygen aoss he plmonay membane (lies/min d dq -ne ae of diffsion of a gas ino o o of he small ylinde shaped elemen of Fig (lies/min d f -blood flow hogh a single plmonay "apillay" (lies/min -adis of a plmonay "apillay" (mees χ -lengh of a plmonay "apillay" (mees x -disane fom venos blood end of "apillay" o any poin along he "apillay" (mees n -nmbe of eqivalen plmonay "apillaies" (dimensionless h -effeive plmonay membane hikness (mees - oal plmonay apillay sfae aea available fo diffsion (mees K -onsan whih depends pon he physial se of he plmonay membane (kg-lies-am/min-mees M - Molela weigh (kilogams V -oxygen onsmpion (lies/min D C -diffsing apaiy of he hin ylinde (lies/min/mmhg D -oal diffsing apaiy (lies/min/mm Hg 03 hp://wwwijseog

4 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN β -abiay onsan whih vaies fom eo a he venos blood end of plmonay apillay o niy a he aeial blood end (dimensionless πdx - is he sfae aea of he ylinde α - is he solbiliy oeffiien expessed in mmhg P B 3 The flow diagam Fig Shows a shemai diagam of he plmonay sysem Blood enes he plmonay sysem a a ae Q and aies a gas of onenaion C The blood flow hen splis, he majo poion passing hogh he plmonay apillaies whee i exhanges gases wih he alveoli small pa of he blood flow, howeve, bypasses he apillaies and ens o he aeial blood, heeby asing he venos admixe effe (a The Plmonay Sysem (b Single Plmonay Capillay Figb above shows a shemai diagam of one of he hypoheial plmonay apillaies The ne ae of flow of a gas aoss he sfae aea of he hin ylinde an be expessed as; dq D P P d whee P has been ignoed, sine P << P The diffsing apaiy of he ylinde an be expessed as a ombinaion of faos as follows: ( ( D πdx α PB K h M ( Combining eqaions ( and (, we obain dq d πdx α PB K P h M ( P ( 3 03 hp://wwwijseog

5 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN oding o he law of mass balane (Fik s piniple, he ne ae of flow of gas aoss he ylinde ms be eqal o he blood flow f, imes he blood gas onenaion diffeene dc beween he ends of he ylinde Hene dq d fdc ( Seing he igh hand side of eqaion (3 eqal o he igh hand side of eqaion ( eaanging, and inegaing boh sides yields Cv π α P dc K P P fhm π α P dx K fhm X B B x 0 ( 5 Examining eqaion5, we see ha i onains he hypoheial qaniies, x and f We ms now eliminae hese This an be done as follows: The oal nmbe of Paallel "apillaies" n, an be expessed as n Qφ f ( 6 The oal apillay sfae aea available fo diffsion is ( πχ n ( 7 Combining eqaions 6 and 7, eliminaing n and solving fo χ yields f χ π φ ( Q ( 8 If we define an abiay onsan β, sh ha 0 β, we an wie he disane fom he venos blood end x along he apillay pah as 03 hp://wwwijseog

6 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN x βχ ( 9 Whee β is eo a he venos blood end and niy a he aeial blood end Combining eqaions8 and 9 yields f f x β β π Γ ( Q φ πq( ( 0 in whih Γ Q φ Sbsiing eqaion (0 ino eqaion 5 yields he geneal eqaion of plmonay Capillay gas exhange: Cv P Whee dc P α PB β K h M Q( Γ, x, f have sessflly been eliminaed, K h is omposed of hee membane faos, K, and h I shold be noed ha he nknown hypoheial qaniies ( Fo a paila gas, all onsans in he igh hand side of eqaion have vales whih ae well sbsaniaed, exep fo he membane onsan K h The vale of his fao ms, heefoe, be deemined This is done as follows: The oal ne ae of diffsion of oxygen dq d aoss he enie plmonay membane an be expessed as dq d K h α P ( M ( P,( B av P av In whih ( is he aveage pesse diffeene beween alveols and plmonay apillay blood 03 hp://wwwijseog

7 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN Solving eqaion fo K h yields K h α ( dq d ( M P ( P B av ( 3 In he seady sae, dq is eqal o oxygen onsmpion V and ( P av an be expessed as d V D whee D is he diffsing apaiy of oxygen Sbsiing hese elaionships ino eqaion 3 yields K h α ( M P B D ( ll of he ems on he igh hand side of eqaion have vales whih ae well sbsaniaed so ha K h an now be deemined Sbsiing fo; M 03kg, α 00am, P 760mmHg, D 00lies / min/ mmhg [] 0 0 B Yields a vale of 9 3 fo K h K is heefoe K 9 3h Vales of 6 h and m yield a nmeial vale fo K as appoximaely The eason fo sepaaing eqaions Sine he enie em K h is ha h an be mainained as a vaiable paamee in he final se of K h will appea in he final eqaion, aay is no impaied in he Slighes by sing appoximae vales of K and h ny eo in he esimaion of hese vales will be oeed by he deemined vale of K 03 hp://wwwijseog

8 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN Using eqaion, hanging he limis of he inegal of eqaion o paial pesses, and eaanging he igh hand side, he geneal eqaion of plmonay Capillay gas exhange in ems of paial pesse beomes p pv P df P ( P α P β K M hq B ( Γ, Whee F ( P denoes he fnion of paial pesse elaed o he onenaion of he espeive gas nde sdy nd i is given as, aoding o Heny's law in blood; ie C ( α PB P Hene (5 an be wien as, ( 5 p pv P dp P β K M ( Γ h Q, I shold be noed ha fo a given gas, all he faos in he igh hand side of eqaion 6 ae fixed, wih he exepion of he em hq ( 6 This means ha he apillay gadiens an be aleed by hanges in sfae aea, membane hikness, adia op, o any ombinaion of he hee 3 METHD F SLUTIN Eqaion 6 an be solved by die inegaion as, P [ ] log P P βk Pν M hq ( Γ 03 hp://wwwijseog

9 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN P P log βk P Pν M hq ( Γ P P exp βk P Pν M hq ( Γ P P ( P Pν exp βk M hq ( Γ Disssion of he model s blood flows along he plmonay apillaies, i eeives oxygen fom he alveoli and delives abon dioxide o hem Ths, onenaion gadiens fo hese wo gases develop along he plmonay apillaies, ineasing fo oxygen and deeasing fo abon dioxide Seveal faos, sh as adia op, plmonay membane aea and hikness of he membane, affe gas diffsion ino he blood seam and hene he paial pesse diffeenes beween alveola gas and end apillay blood gas Cabon dioxide, having a diffsion oeffiien abo 0 imes ha of oxygen, diffses mh moe apidly so ha eqilibim is hogh o be invaiably eahed, egadless of hanges in hese faos xygen eqilibim, howeve, is hogh o be limied eihe by blood flow o diffsion nde eain ondiions In he flow-limiing siaion, adia op is oo gea o allow eqilibim of alveola and blood oxygen paial pesses In he diffsion limiing siaion, he plmonay membane, pehaps owing o a disease sae sh as emphysema o plmonay ongesion, is pahologially aleed so ha diffsion os oo slowly o allow eqilibim o be eahed beween alveola gas and blood gas Boh ases (ha is, flow and diffsion limiaion may of ose o simlaneosly Tha is fom eqaion 6 paial pesse elaed o he onenaion of he espeive gas nde sdy I shold be noed ha fo a given gas, all he faos in he igh hand side of eqaion 6 ae fixed, wih he exepion of he em hq This means ha he apillay gadiens an be aleed by hanges in sfae aea, membane hikness, adia op, o any ombinaion The gaphs of pesse agains sfae aea shows ha he pesse gadien is ineasing as he sfae aea is ineasing nil he pesse gadien eahes maximm sh ha inease in sfae aea anno ase any fhe hange o he pesse gadien as shown in he fige below 03 hp://wwwijseog

10 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN Fige : Effe of sfae ea on gas diffsion ino he blood seam Fige 3 and below shows ha wiho hanging he vale of he pesse gadien, he diffsion ae hogh he espiaoy membane is deeasing signifianly as he hikness of he membane and Cadia op espeively ineases Fige : Effe of membane hikness on gas diffsion ino he blood seam Fige 53: Effe of adia op on gas diffsion ino he blood seam 03 hp://wwwijseog

11 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN Conlsion Fom o esl, ineasing he hikness of he espiaoy membane above of diffsion The inease in membane hikness an be ased by diseases deeases he ae The plmonay edema ased by faile of he lef side of he hea is he mos ommon ase of an inease in he hikness of he membane heeby ineasing venos pesse in he plmonay apillaies and esling in he amlaion of flid on he alveoli In a nomal hman, he oal sfae aea is appoximaely and a small deeases in he sfae aea advesely affes he espiaoy exhange of gases This is ased by disease ondiion sh as emphysema and lng ane efeene [] Bidani,, Flmmefel, (978nalysis of he effes of plsaile apillay blood flow and volme on gas exhange espi Physiol 35:7- [] Cole, B and Bishop J M (963 Jonal of ppl Physiol 8:03 [3] Doglas,,Jones, N L and eed,j W (988, Callaion of whole blood C onen,jpplphysiol65:73-77, [] Flmmefel, W and Candall, D (989 n analysis of exenal espiaion in manmahbios3:05-30 [5] Leoppky,J, Flehe, E, oah, C and Lf, U C (993elaionship beween whole blood base exess and C onen in vivoespiphysiol9:09-0 [6] Milhon, H T (J, Benon,, oss, and Gyom, (965 mahemaial model of he hman espiaoy onol sysem, Jonal Biophysial 5, 7-6, [7] Milhon, H T and Plley, P E(965 heoeial sdy of plmonay apillay gas exhange and venos admixebiophysj8: [8] Nnn J F (987 pplied espiaoy Physiology(3 d edbewohs [9] Nnn J F (993 pplied espiaoy Physiology( d edbewohs Heinemann [0] od S,Ten DS and Philip T( 008 naomy and Physiology MGaw-Hill Campanies in New Yok 03 hp://wwwijseog

12 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN mahemaial model on he ai inflow ino he Lng G C E Mbah, SI Ionmbe, niak, S E Depamen of Mahemais,Univesiy of Nigeia, Nigeia gembh@yahooom godwinmbah@nnedng , BSTCT Evolionay poesses have opimied he anaomi se of he lng as a ee-like banhing newok of aays eminaing in hin walled elasi ds and alveoli, whee his gas exhange os Boh dissipaive and elasi popeies of he espiaoy sysem onibe o is niqe sai and dynami mehanial behavio Hee we develop a mahemaial model based on he esls and heoies fom aiflow and gas exhange beween he alveoli and blood apillay The oniniy and Navie- Sokes eqaions in he ylindial oodinae ae sed o desibe he flow sysem The flow is assmed o be lamina and nidieional Conseqenly he oniniy eqaion shows ha he axial veloiy is independen of he axial oodinaes The model eqaion is solved expliily and he esl show ha, he flow of ai hogh he espiaoy aay is peiodi Keywods: iflow, espiaoy aay INTDUCTIN The ai we beahe passes hogh he ahea whih divides ino wo main banhes, he lef and igh bonhi, eah of whih again divides ino ds, eah of whih one again divides ino wo This poess onines fo0- geneaions, as shown in fig Eah of he eminal d has abo 300 alveola sas o alveoli a he end The diamee of eah alveola sas is beween 75 and 300 mions,( m o 0 mm Fige ays geneaions Pahways Fo he pposes of gas anspo, he lng is divided anaomially ino he ondive and espiaoy ones [8] Beginning wih he ahea, he onding aays fom a seies of banhing ondis, inlding he main, loba, and segmenal bonhi, down o he eminal bonhioles The onding aays lead inspied ai o he espiaoy one, onsising of espiaoy bonhioles ha onain a few bdding alveoli, and finally o he alveoli-lined aini (Fig lveoli ae he eminal 03 hp://wwwijseog

13 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN gas-exhanging nis of he lng The hman lng onains abo 300 million alveoli, whih amons o a sfae aea beween 50 and 00 m n ains efes o he anaomial ni of he poion of lng disal o a eminal bonhiole Ding inspiaion, a pesse gadien beween he aay opening and he alveoli esls in he onveive movemen of ai fom he amosphee o he eminal nis of gas exhange Fo sponaneos beahing, his gadien is geneaed by he onaion of he diaphagm and ineosal msles, asing he hes wall o expand and deease he inapleal pesse beween iself and he lng Unde ondiions of aifiial espiaion, a mehanial venilao povides posiive pesse a he aay opening elaive o he alveoli In eihe siaion, he lng will expand and ai will move onveively hogh he bonhial ee While onveion dominaes he avel of inspied gas fom he ahea hogho he ondive one, pon eahing he espiaoy one, is veloiy beomes negligible and diffsion beomes he pimay means of gas anspo [8] This deease in veloiy is de o he apid inease in oal aay oss-seional aea ove he sho disane fom he ahea o he espiaoy one he alveoli, inspied oxygen will diffse down is paial pesse gadien ino he bloodseam Simlaneosly, abon dioxide will follow is diffsion gadien fom he plmonay apillaies ino he alveoli o be expied The mos impoan espiaoy msle is he diaphagm The movemen of diaphagm ases pesse diffeenes and hanges he volme of he lngs Ding inspiaion, onsiion of he diaphagm msle fibes esls in a downwad movemen of diaphagm and he deeased pesse ases ai o flow in he lngs Nomal expiaion eqies no msle aivaion When he diaphagm elaxes, elasi foes of he lngs and aay walls ase elasi eoil pesse and he ai flows o of he lngs aoding o he pesse diffeene In aays he flow an be lamina, blen, o a ombinaion of he wo (ansiional When aiflows a highe veloiies espeially hogh an aay wih iegla walls, he flow is blen and is fond mainly in he lages aays, like he ahea When he veloiy is low and goes hogh naow bes, flow is Lamina Lamina flow is he mos effiien way fo oxygen anspo o he disal pas of he bonhial ee The oal effeive oss seional aea of he aays ineases wih eah division of he aays fom he ahea o bonhi, so ha he flow ae of inspied gas falls s he flow ae dops, he flow beomes ahe Lamina han blen The ansiion beween onveion and diffsion depends on he Mophomei popeies of he aays so ha a smalles bonhiole level (geneaion 5-8 he gas anspo hanges fom onveion o diffsion and gas exhanges in he alveoli [8] [3] deives he eqaions fo model idenifiaion in espiaoy mehanis nde ondiions of mehanial venilaion [7] developed physial based model of hman espiaion He modeled he slow deep beahing by nnel diode osillao [9] demonsaes ha pediive neal newok may pove o be valable as a ool in aomai mehanial venilaion sppo In his sdy a mehod siable fo an adapive oninos onol of nonlinea plmonay mehanis has been poposed[] model of espiaoy mehanis was fi o he daa, and he paamees of he model ae hen aken o be meases of impoan physiologial qaniies [6] developed a wo-omponen simlaion model fo espiaoy mehanis ompehensive ndesanding of espiaoy mehanis is pivoal fo he aae diagnosis and eamen of lng disease, fo adeqae aifiial o assised venilaion, and fo he analysis of envionmenal effes, like in diving o nde hypebai ondiions on lng mehanis and fnion[] sessed ha gaviy is impoan in he deposiion of 05 and miomee pailes in he hman ains[] inopoaed gaviaional sedimenaion wih wall movemens, employing a 3-D hemispheial alveols model eenl, [5] simlaed ajeoies of 5 miomee pailes in 3-D alveolaed ds epesening geneaions 8 wih diffeen gaviy angles They onlded ha he oal deposiion an be a fnion of he gaviy angle and he aio of he eminal seling veloiy o mean lmen flow veloiy In his sdy we ae onened wih he aiflow hogh aays model inegaing aiflow hogh he ai way is pesened MTHEMTICL MDEL We model flow hogh he espiaoy aay as a flid defined as any maeial whih defoms oninosly nde shea sess The ai is a Newonian flid and he dynami visosiy is independen of he ae of shea Theefoe, he aiflow in he espiaoy sysem an be eaed as an inompessible, visos flow ie eah gas has onsan volme ove ime, homogeneos, densiy of he flid says onsan in spae visos, ae of defomaion is same nde he same sess fo all flid The Veloiy and he Pesse of flid flow being a fnion of ime and spae iflow allaion is based on he famos Navie s-sokes eqaion We assme ha he espiaoy ai way is ylindial in shape Hene we onside flow hogh a ila be sbje o a sep hange in pesse Theefoe, he Navie- Sokes eqaion in he ylindial fom is given as; 03 hp://wwwijseog

14 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN hp://wwwijseog ( p F ( p F (3 p F Wih he oniniy eqaion; 0( Sine he aiflow hogh he espiaoy aay is nseady and flid flow is eihe Lamina, Navie- Sokes eqaion edes o he fom in axial dieion in one dieion only, ha is 0 and 0,he oniniy eqaion beomes, o(5 Fom eqaion (3 he seond and he hid ems on he LHS ae also eos The foh em is also eo by (5 n he igh hand side he hid, foh and fifh ems ae eo bease; ( f nly, and hee is no body foe Hene o model now beomes: (6 p whee he ae of flow in axial dieion, p is he pesse gadien, is he flid densiy is he flid visosiy, is he adis of he apeed be of he espiaoy aay as in fig Wih he bonday ondiions as showed below; Fige : Tapeed be ( Tahea ( α an L I α o fo o ( α

15 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN SLUTIN PCEDUE We shall assme ha he pesse gadien asing he moion is hamoni and is given by ; p k os( (7 p ( ke (8 aibing physial signifiane only o he eal pa Sine he moion is hamoni de o pesse gadien whih is ime dependen and he veloiy vaies wih espe o whee is apeing, veloiy has he fom; (, ( e ( (9 Then o obain he axial fom of he veloiy, we seek o find ( by Sbsiing eqaion (8 and (9 ino eqaion (6, we obain d ( d( ( k d d d ( d( ( k (0 d d d ( d( ( 0( d d Eqaion (0 is a non-homogeneos eqaion whih an be solved as follow; Homogeneos pa, Eqaion ( beas esemblane o he Bessel diffeenial eqaion and hs we appoximae is solion by he Bessel fnion of eoeh ode, wih he finie solion as: X J ( Fo he non-homogeneos pa, he paila inegal is given as; ( C p sh ha; d p( d p( 0 d d Sbsiing hese in eqaion (0, we have k C and so k C (3 03 hp://wwwijseog

16 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN hp://wwwijseog Theefoe, he geneal solion of eqaion (0 is; k J p ( ppling he bonday ondiion (BC ha a We obain 0 k J k J J k Theefoe; ( ( k J kj Theefoe, ( e k J kj (5 ( J J e w ik Sine ( x x x J o, 0

17 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN hp://wwwijseog J Theefoe onsideing only he fis wo ems we have; ( e w ik ( e w ik ( e w ik ening o he eal noaion, [ ] ( e k [ ] ( 6 Cosw k DISCUSIN F THE MDEL Ding inspiaion, a pesse gadien beween he aay opening and he alveoli esls in he onveive movemen of ai fom he amosphee o he eminal nis of gas exhange Fo sponaneos beahing, his gadien is geneaed by he onaion of he diaphagm and ineosals msles, asing he hes wall o expand and deease he inapleal pesse beween iself and he lng Unde ondiions of aifiial espiaion, a mehanial venilao povides posiive pesse a he aay opening elaive o he alveoli In eihe siaion, he lng will expand and ai will move onveively hogh he bonhial ee While onveion dominaes he avel of inspied gas fom he ahea hogho he ondive one, pon eahing he espiaoy one, is veloiy beomes negligible and diffsion beomes he pimay means of gas anspo This deease in veloiy is de o he apid inease in oal aay oss-seional aea ove he sho disane fom he ahea o he espiaoy one he alveoli, inspied oxygen will diffse down is paial pesse gadien ino he bloodseam Simlaneosly, abon dioxide will follow is diffsion gadien fom he plmonay apillaies ino he alveoli o be expied Ding exhalaion, elaxaion of he diaphagm and ineosals msles allows he elasi lng o en passively o is peinspiaoy volme Fnional esidal apaiy (FC efes o he volme a whih he inwad elasi eoil of he lng is exaly balaned by he owad elasi foe of he hes wall The many banhing aay segmens povide visos pahways fo aiflow, and he apid, exponenial expansion in effeive aay esls in eded gas veloiies and hanging flow egimes The dissipaive and elasi popeies of he

18 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN espiaoy sysem onibe o is niqe peiodi and dynami mehanial behavio,ha is a a fixed disane veloiy ime gaph shows a peiodi gaph whih epesens inhalaion and exhalaion poess in beahing mehanisms as shown in fige 6 below 5 veloiy-ime gaph 05 veloiy, m/s ime, s Fige 3: Peiodi and dynami flow of ai hogh he espiaoy aay The gaph below shows ha, he ae of aiflow ino he lngs ineases linealy as yo go down hogh he espiaoy aay Fige : Peiodi and dynami flow of ai hogh he espiaoy aay 5 CNCLUSIN The flow of ai hogh he espiaoy aay is peiodi a a fixed disane fo one beah, in and o oninosly Ths a wavefom is geneaed in he fom of osine fnion as we beah The ae of flow is also linea a a given ime efeene [] Baes, J H T and Goldbeg, P (999 Fiing non linea ime domain models of espiaoy mehanis o pesse flow daa fom an inbaed paien, Seving Hmaniy dvaning Tehnology Medial eseah of Canada [] Daqenne, C, and Pisk, G K (003, Effe of gaviaional sedimenaion on simlaed aeosol dispesion in he hman ains, Jonal of eosol Siene, 3, 05 8, 03 hp://wwwijseog

19 Inenaional Jonal of Sienifi & Engineeing eseah, Volme, Isse, Novembe ISSN [3] Gane, S, Molle, K Kisian, ead L D and Gmann,j (007 Eqaion disovey fo model idenifiaion in espiaoy mehanis nde ondiion of mehanial venilaion, ICML07 US, Jne [] Habe, S Yihak, D, and Tsda, (003 Gaviaional deposiion in a hyhmially expanding and onaing alveols Jonal of pplied Physiology, 95, , [5] Haingon, L, Pisk, G K, and Daqenne, C (006 Impoane of he bifaion one and banh oienaion in simlaed aeosol deposiion in he alveola one of he hman lng, Jonal of eosol Siene, 37(, 37 6 [6] Keble, K Meens, M and xel, (007 wo omponen simlaion model o eah espiaoy mehanis, dvane Physiology Edaion 3,8-, [7] Sepehis, S (007 Physial model of hman espiaion, Yong eseahes lb, Islami ad nivesiy of Shia -7 [8] Wes J B (005 espiaoy physiology-the Essenials (7 h Ed Lippino Williams and Wilkins [9] Zh, H Gmann, J Molle, K (007 Conol of espiaoy mehanis wih aifiial neal newoks, IEEE, hp://wwwijseog