MODELLING OF CARDIOVASCULAR SYSTEM REGULATION

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1 MODELLIG OF CARDIOVASCULAR SYSTEM REGULATIO Sefa Bor, Ivo Cap Deparme of Elecromagec ad Bomedcal Egeerg, Uvery of Zla Abrac The arcle deal wh prcple of cardovacular yem regulao. The wor decrbe phyologcal ba of eleced regulao mecham uch a auoregulao ad baroregulao. A overvew of cardovacular yem model preeed. The paper coa decrpo of lear regulao model ad alo model coag olear pheomeo. The arcle decrbe mplemeao of hear rae ad vacular reace regulao o exg model of areral yem baed o he prcple of elecromechacal aalogy. The mplemeed regulao model baed o barorecepor regulao applcao. There are aalyed characerc propere of barorecepor regulao uch a aurao, hyere or ep chage barorecepor mpule rae. Keyword Barorecepor, cardovacular yem, hear rae, modellg, regulao, vacular reace ITRODUCTIO The regulaory mecham play mpora role a operao of a whole lvg orgam. They fluece gle par allowg adapao o chage of exeral ad eral codo. Whe we focu o cardovacular yem regulao, we ca ar wh auoregulao ad fh wh barorecepor regulao. The boh ype of regulao cover hor me chage of gle parameer of vacular yem. Impac of humoral regulao exer laer f comparg o he prevouly ype. The modellg of bologcal fuco ca be ueful more cae. We wa o model cera phyologcal yem becaue we wa o prepare a model for educaoal purpoe or we wa o udy cera yem more deeply ad we wa o compare o a phycal equvale. By modellg of he cardovacular yem we ca oba valuable formao abou fuco ad abou chagg parameer. By comparg mulaed ad meaured gal poble o evaluae he fuco of really lvg yem. We ca oberve appearg of he fr model of cardovacular yem or model of h yem regulao a he begg of he 0 h ceury. The ereg wor ca be meoed whch deal wh law of areral flow ad areral brachg [], [] or he wor focued o cardovacular regulao [3], [4], [5], [6]. CARDIOVASCULAR REGULATIO There are may publcao purug o decrbe cardovacular regulao mecham, hough hey ca be addreed o dffere group of reader. Whe we eed o mpleme prcple ad depedece of cardovacular regulao o pecfc yem ug echcal ool, we eed approprae formao ource for prg or comparg of gve problem oluo whch dpoe of brgh ad relave mply explaao of gve pheomeo ad alo of approprae mahemacal apparau, e.g. [7]. I h arcle we fr of all focu o wo ype of cardovacular yem regulao uch a auoregulao whch volve vacular reace corol ad baroregulao adhere o hear rae corol ad alo o vacular reace regulao [7], [8] ad [9]. A. Auoregulao I local mecham whch modulae blood flow hrough ue baed o oxyge (meabolc) demad. Th mecham corol perpheral vacular reace by bochemcal chage correpodg o meabolc acvy of ue (e.g. mucle wor) whch cocerao of H +, CO, O or lacacd are moored [7], [8], ad [9]. 8

2 B. Baroregulao Barorecepor regulao global feedbac mecham of ervou yem corol whch eep he hear rae, vacular reace ad veou preure order o areral preure rema o demaded level. Barorecepor dvde o low- ad hgh-preure recepor placed aral hear wall ad aorc arch or carod [7], [8], ad [9]. I h wor we ued for our mplemeao ad aaly he hgh-preure barorecepor. REGULATIO MODELS There are may auhor who deal wh he modellg of cardovacular yem (CVS) ad regulao. I eeded good owledge of huma phyology for creao of CVS mahemacal model. May wor ry o exed exgmodel of CVS by mplemeg of regulaory mecham ad by ha way ca be creaed complex model of CVS whch hould repoe o eral ad exeral evrome chage. C. Lear model We ca expre oxyge coumpo ad blood flow by he ex equao [7]: M Q([ O ] a [O ] v ), () P a P v R Q, () where [O ] a ad [O ] v are areral ad veou paral preure of oxyge blood, M meabolc acvy (oxyge coumpo per u me), P a ad P v are areral ad veou preure, Q blood flow hrough veel egme wh reace R. Veel reace ca be expreed a follow: R R0 A O, (3) v where A > 0. Parameer A repree reace chage evy o oxyge cocerao. If A = 0, he veel reace uregulaed [7]. By he mlar way we ca wre equao for ympahec acvy whch relaed o barorecepor regulao: where S S S ad P a P a, (4) P a are ormal value of ympahec acvy ad mea areral preure, expree evy of ympahec acvy o chage of mea areral preure [7]. Prevou equao ad her combao offer model of regulao mecham. Ther advaage relave mplcy bu o he oher had her dadvaage ha hey dd o repree real fuco of regulao mecham ad ca be ued oly he lmed rage of pu parameer. D. Overvew of olear baroregulao model The decrpo of relao blood preure o barorecepor acvy ca be foud wor [4]. The auhor pree dffereal equao whch expree barorecepor acvy : dp p p, (5) 0 where p areral preure, ad are coa parameer, p 0 hrehold value of areral preure a whch barorecepor acvy chage ad characerze hrehold value of barorecepor acvy [4]. Aoher approach ca be foud [5]: 3 p ae ae a3e, (6) m where barorecepor acvy chage whch repod p. Parameer a, a a a 3 are coeffce repreeg adapao peed, m value of aurao acvy of barorecepor ad 3 are me coa whch characerze ucceve decreae of barorecepor acvy. Equao 6 ca be deoed a ufed baroreflex model. A aemp o creae oe complex model of cardovacular yem ad regulao he aemp of Guyo e al. [3]. I h wor roduced chemac dagram cog of gle bloc whch repree mahemacal operao bewee phyologcal varable [0]. The wor [0] pred by Guyo model ad mpleme h deg SIMULIK mulao evrome. CARDIOVASCULAR SYSTEM MODEL BASED O ELECTROMECHAICAL AALOGY A huma cardovacular yem ad propere repree a dyamc yem whch ca be raformed ug approprae elecromechacal aaloge o yem cog of dcree elecrcal eleme uch a reor, ducor or capacor. A blood preure equvale o elecrcal volage ad a blood flow ca be modeled a elecrcal curre. The propere of he vacular yem are expreed by gle elecrcal eleme. Magec feld eergy ercoeced wh ec eergy of blood flow by ducvy whch repree blood dey. O he oher had elecrc feld eergy ercoeced wh elac deformao of ube by capacy whch repree complace of blood veel ad reve compoe correpod o blood vcoy. The he whole areral ree ca be bul by coeco of veel egme cog of gle eleme (ee Fg. ) [], [], [3], [4], [5]. 9

3 where a ad are equal o coeffce equao (6). Expreo dp c repree me chage of mea areral preure. Fg. : The veel egme model IMPLEMETATIO OF BARORECEPTOR REGULATIO MECHAISMS Barorecepor are coeced o hor me regulao loop of everal CVS par uch a vacular reace ad hear rae. Thee wo parameer are cluded equao for calculao of mea areral preure p c (aumpo ha mea areral preure aorc arch equal o mea areral preure carod): pc Ra SV H (7) Parameer SV repree hear roe volume, H hear rae ad R a perpheral vacular reace. For modellg he barorecepor regulao we ca ue modfed form of equao (6) [6]: E. Sympahec ad paraympahec acvy Hear rae, coracly ad vacular reace creae by creag of ympahec acvy. O he oher had paraympahec acvy decreae he hear rae ad ha mall fluece o coracly ad vacular reace chage [9]. Relao of ympahec ad paraympahec acvy o preure carod ca be characerzed by gmodal fuco ad expreed by followg equao [5], [6]:, () p, () p where p c mea preure value carod defed a average value of gve preure durg oe hear cycle. Parameer repree lope of gle curve ad, p are auraed value of ympahec ad paraympahec acvy. dp m (8) d m c 3 ae ae a3e The he barorecepor acvy ca be dvded o four par [6]: 3, (9) where repree baal mpule frequecy of 3 parameer are relaed o mpule frequecy evy o chage of areral preure. The oluo ca be expreed by he ex dffereal equao: dpc a m m,,, 3, (0) Fg. 3: Depedecy of ympahec (blue curve) ad paraympahec acvy (red curve) o mea areral preure carod. F. Effere repoe Effere repoe of correpodg orga o ympahec ad paraympahec mulao ca be expreed a her lear combao [6]: p, (3) where coeffce,, a characerze regh of ympahec or paraympahec mulao. Idex E H,. decrbe approprae orga repoe: R p 0

4 E clude alo hear rae H ad perpheral reace R p. Dyamc repoe modeled by ordary dffereal equao of fr order: dx dx H H Rp Rp x H x Rp, (4), (5) The equao (4) repree chrooropc effec whch dyamcal hear rae chage ad he equao (5) characerze dyamc chage of vacular reace [6], [8] ad [9]. G. umercal mplemeao Soluo of he equao (6) ad modfcao (0) are ba for our umercal mplemeao of regulao mecham o exg model of CVS baed o elecromechacal aalogy. We ued MATLAB programmg evrome ad SIMULIK par for performg of mulao. We dvded barorecepor regulao o 4 fucoal u:. Barorecepor. CS (Ceral erve Syem) 3. Hear rae ad vacular reace 4. Corolled gal ource of blood preure The Fg. 3 characerze fucoal coeco of gle regulao u. relaed o adapao of orgam o preure chage by ug he barorecepor regulao. H. Threhold ad aurao A hrehold value of preure ex whe he baal barorecepor acvy creae. By ucceve creag of barorecepor acvy aurao occur ad mpule frequecy of barorecepor eep uchaged. We ge he ame reul a [5] (ee Fg. 3). I. Hyere Barorecepor repoe o mea areral preure chage ca be expreed by gmodal fuco (fuco wh hape of leer S ee Fg. 3) [5]. Hyere ca appear whe he areral preure lowly creaed ad he lowly decreaed o arg value (ee Fg. 4). Fg. 4:Hyere curve barorecepor acvy chage (he blue curve repree barorecepor repoe o blood preure creag ad he red curve correpodg o decreag). J. Sep chage Durg hor-me blood preure ep chage barorecepor acvy alo creae correpodg o blood preure. A log-me blood preure chage barorecepor adap ad her mpule frequecy decreae oward ba hrehold value bu a eepg creaed blood preure [5]. The mulao oupu you ca ee he Fg 5. Fg. 3: Fucoal coeco of regulao u AALYSIS OF PROPERTIES OF BARORECEPTOR REGULATIO Whe he areral preure carod or aorc arch chage, olear pheomea appear ad hey are Fg. 5: Sep chage of barorecepor acvy durg blood preure chage from 00 mmhg o 50 mmhg ad vce vera.

5 COCLUSIO The barorecepor regulao corol he mpora parameer of cardovacular yem. Wh he wor hear rae ad vacular reace regulao were decrbed ad mplemeed o he exg model of cardovacular yem baed o elecromechacal aalogy. The barorecepor regulao o lear ad creaed model wa baed o dffereal equao by whch aaly mulao evrome everal olear pheomea appear uch a aurao, hyere or ep chage barorecepor mpule rae. Smulao reul are comparable o daa led wor [5] ad [6]. Our fuure wor wll be focued o exedg of regulao model, for example by cludg of furher regulao par of cardovacular yem uch a veel complace, hear mucle coracly or chage of meabolc eed. REFERECES [] MURRAY, Cecl D. The phyologcal prcple of mmum wor: I. The vacular yem ad he co of blood volume. Proceedg of he aoal Academy of Scece of he Ued Sae of Amerca, 96,.3: 07. [] MURRAY, Cecl D. A relaohp bewee crcumferece ad wegh ree ad bearg o brachg agle. The Joural of Geeral Phyology, 97, 0.5: [3] GUYTO, Arhur C.; COLEMA, Thoma G.; GRAGER, Harr J. Crculao: overall regulao. Aual revew of phyology, 97, 34.: [4] WARER, HOMER R. The frequecy-depede aure of blood preure regulao by he carod u uded wh a elecrc aalog. Crculao Reearch, 958, 6.: [5] TAHER, Moa F., e al. Barorecepor repoe derved from a fudameal cocep. Aal of bomedcal egeerg, 988, 6.5: [6] OTTESE, Johy T.; OLUFSE, Mee S.; LARSE, Jeper K. Appled mahemacal model huma phyology. Phladelpha: Socey for Idural ad Appled Mahemac, 004. [7] KEEER, Jame P.; SEYD, Jame. Mahemacal phyology. Sprger, 998. [8] JAVORKA, K., e al. Leára fyzológa. Ovea, Mar, 00. [9] HALL, Joh E. Guyo ad Hall Texboo of Medcal Phyology: Ehaced E-boo. Sauder, 00. [0] KOFRÁ dagram brough o lfe-from graphc char o mulao model for eachg phyology. Techcal Compug Prague, 007, - 3. [] ermal egme effec o blood preure ad blood flow propagao. I: Proc. of 8h eraoal coferece KRALIKY p [] Blood Flow Veel. Advace Elecrcal ad Elecroc Egeerg, 0, 7.-: [3] refleco veel. Ieraoal Joural of Appled Elecromagec ad Mechac, 0, 39.: [4] GOMBARSKA, D.; Compuer mulao of huma cardovacular yem. Aca Techca. 0, Vol. 57, o. 4, p [5] GOMBARSKA, D.; Modellg of areral bfurcao by mea of elecromechacal model wh drbued parameer. BIOMEDICAL TECHOLOGY 0 AD BEYOD,. Sefa Bor Deparme of Elecromagec ad Bomedcal Egeerg Uvery of Zla Uverza, 00 6 Zla Slova Republc efa.bor@fel.uza. Ivo Cap Deparme of Elecromagec ad Bomedcal Egeerg Uvery of Zla Uverza, 00 6 Zla Slova Republc vo.cap@fel.uza.

3.10 Convected Coordinates

3.10 Convected Coordinates Seco.0.0 Coveced Coordaes Some of he mpora resuls from secos.-.9 are ow re-epressed erms of coveced coordaes. As before, ay relaos epressed symbolc form hold also he coveced coordae sysem..0. The Sress