Open-Source Programming of Cardiovascular Pressure-Flow Dynamics Using SimPower Toolbox in Matlab and Simulink

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1 The Ope Pacig, Electrophysiology & Therapy Joural, 1, 3, Ope Access Ope-Source Programmig of Cardiovascular Pressure-Flow Dyamics Usig SimPower Toolbox i Matlab ad Simulik Ofer Barea * Departmet of Biomedical Egieerig Tel Aviv Uiversity Ramat Aviv Israel Abstract: A toolbox for simulatio of the cardiovascular system has bee costructed for use i Matlab Simulik (The MathWorks Ic.). The toolbox cotais basic elemets such as vessel segmets, cardiac chambers, valves ad shuts that ca be graphically arraged ad coected to directly simulate the circulatio with the desired level of detail. The toolbox is based o the SimPower toolbox to allow simple coectios betwee the differet elemets. Oe difficulty that remais is the eed to kow the values of the various parameters. The toolbox is a ope-source script that is available for dowload. It is expected that researchers that will use the toolbox will refie it, ehace its capabilities ad share their code with others. Keywords: Cardiovascular system, mathematical models. INTRODUCTION The cardiovascular system (CVS), the heart, the blood vessels ad the blood flowig i the system, received a cosiderable amout of attetio from scietists of various fields icludig physiologists, biologists, egieers ad physicists. Models arisig from may differet approaches have bee developed to elucidate mechaisms of the ormal ad pathological fuctio ad structure of the system. Models ad computer simulatios were also desiged to study specific effects ad iteractios of the CVS with physiological as well as ma-made systems. These models help to uderstad ad demostrate behavior of the system's various aspects as well as help i elimiatig or miimizig the eed for aimal experimets. Mathematical models may be subdivided ito two major categories: a) mathematical descriptios of observed pheomea or iteractios ad b) mathematical descriptios of physiological or physical mechaisms or structures. The first category icludes relatios such as polyomials or expoetial fuctios that describe the effect of oe variable o others based o averages of observed values. Coefficiets ad costats i the mathematical descriptio are ot ecessarily related to physical or physiological parameters. I secod category, the oe is described i this paper, coefficiets ad costats are related to physical or physiological parameters of the modeled system or orga. These models are based o physical/physiological behavior of the system. Pressure ad flow relatios i the cardiovascular system would ideally retai all the terms i the Navier-Stokes equatios ad a expressio describig elastic wall motio. Fiite-elemet methods are used to compute the flow ad *Address correspodece to this author at the Departmet of Biomedical Egieerig Tel Aviv Uiversity Ramat Aviv Israel; Tel: ; Fax: ; bareao@eg.tau.ac.il pressure with high accuracy.however, the mathematical ad computatioal complexity is very large ad ofte uecessary. O the other had, lumped parameter models ca describe pressure flow pheomea without providig such exact solutio but do provide reasoable pressure-flow relatios ad allow the study of a system behavior. The heart ca be represeted by fiite elemet models where the dyamics of each segmet of tissue is calculated [1, ]. I cotrast, whe exact regioal details are ot required, a lumped-parameters model ca represet the fuctio of the whole cotractig heart ad provide reasoable predictios of volume, pressure ad flow [3]. LUMPED PARAMETERS MODELS For simulatig blood vessels, a balace betwee accuracy of the fiite elemet solutios ad the lumped parameter models ca be achieved by assumig that the vessels ca be divided ito segmets, each with lumped values of hemodyamic parameters. Noordergraaf ad co-workers costructed a aalog model for the huma systemic arterial tree composed of 113 segmets, each about 5 cm log [4]. By varyig elemets of the model, they were able to study hemodyamic chages i the system ad to predict its respose to various iputs. A major cotributio to lumped parameter multi-elemet models was published i 1967 by Rideout ad Dick [5]. Followig a assumptio that the secod partial derivative of the velocity with respect to x (alog the vessel) is of egligible value, they arrived at the followig liearized equatios: P w w 1 w = + (1) x t r r r P u u 1 u u = + () r t r r r r X/1 1 Betham Ope

2 56The Ope Pacig, Electrophysiology & Therapy Joural, 1, Volume 3 (where P is pressure ad w ad v are velocities i the x ad r directios, respectively). With the equatio of cotiuity that expressed the icompressibility of blood (i cylidrical coordiates): u u w + + = (3) r r x Itegratio was performed alog the radius, pressure was assumed costat i the whole cross sectio area ad mea velocity was replaced by volumetric flow. Thus, eq. 1 was coverted to a discrete form show by eq. 4 dq P P +1 = L + RQ (4) dt Where P is the pressure at the 'th segmet, Q is volumetric flow, L represet iertia ad R viscosity. This model was further developed i two ways. Vessel compliace was added ad a o-parabolic velocity profile was cosidered to calculate R ad L i eq. 4. A capacitor represetig vessel compliace eglected before was added to the model as show i eq. 5: dp Q Q = C (5) +1 dt This satisfies the cotiuity equatio regardig logitudial ad radial flow compoets. Equatios 4 ad 5 describe the pressure-flow relatios i a vessel segmet. Elemets such as resistor ad capacitor were also used to represet viscoelastic behavior of the vessel wall such as Voigt ad Maxwell models. Barea et al. [6] described hybrid models that use also experimetal data for the vessel segmet pressure-volume relatios i the form of a polyomial alog with a series resistor ad iertace to obtai more accurate oliear effects at differet pressures. This model also eabled istataeous update of vessel radius with istataeous pressure ad volume chages. A model of the vetricle is expected to be idepedet of both preload ad afterload ad is supposed to predict the vetricular cotractio ad ejectio patter for various preload ad afterload coditios. The suga-sagawa widely accepted varyig-elastace model achieves most of these goals. The model is based o Otto Frak s work. He represeted the frog's cardiac fuctio i the pressurevolume plae more tha oe hudred years ago, layig the foudatio for today's most widely accepted "varyig elastace" vetricular model. Suga [3, 7, 8] further developed that model ad established the load idepedece of the ed systolic pressure-volume relatioship of the heart [9, 1]. This made the cocept very attractive for modelig sice cardiac fuctio ca be described idepedetly of the fillig pressure ad aortic load. Whe this vetricular model is coupled to a model of the circulatio, oe ca predict the behavior of the heart i respose to various perturbatios. The vetricular "elastace" was defied: P ( t) E( t) = (6) V ( t) V where P (t) ad V (t) are the left vetricular istataeous pressure ad volume respectively, ad V is the volume axis Ofer Barea itercept of the lie coectig the ed systolic pressurevolume poits for differetly loaded beats. The model was further supported by the demostratio of the straight regressio lie of isochroal pressure-volume ratios for differetly loaded beats at steady iotropic state [11]. Ed systolic pressure-volume relatioship was widely accepted as a idex of cardiac cotractility ad has bee amed ESPVR (ed systolic pressure volume relatioship). ESPVR is ofte used iterchageably with E max which is the maximum value of E(t) ot ecessarily at ed systole. I order to describe istataeous left vetricular pressure, equatio 6 may be rewritte as follows: P [ V ( t) ] ( t) = E( t) V (7) This relatioship approximates the cardiac fuctio for small stroke volumes, but does ot predict correct pressure flow relatioships for larger stroke volumes ad it is also somewhat affected by loadig coditios. I the late 197's several works demostrated the weak poit of this model. Suga ad Yamakoshi ad Suga ad colleagues preseted P-V trajectories where for the same ed systolic volume, the ed systolic pressure of a isovolumic beat was higher tha the ed systolic pressure of a ejectig beat [1, 13]. This pressure deficit was observed by others ad there were several attempts to improve the model. Campbell ad his colleagues added a series resistor to the electrical aalog model of the heart, which was based o the varyig elastace model [14, 15]. The role of the resistor was to accout for flow related pressure drop. Shroff et al. demostrated a close fit of pressure ad flow patters geerated by a model where the series resistor is proportioal to istataeous left vetricular pressure ad experimetal data [16, 17]. THE TOOLBOX To model the hemodyamic aspects of the cardiovascular system, we eed a few elemets. We eed a model of a cotractig chamber, a model of a valve, ad a model of a blood vessel. All models should be able to represet oliear aspects of the physical properties of the orgas as well as pathologies. Also choices must be made sice several differet models are available. The heart model was based o the varyig elastace. Pressure-volume relatios are used to describe both the cotractig phase (eq 8) ad the diastolic phase (eq 11). Systolic cotractio is based o a ormalized elasticity fuctio E N P t) = E EN ( t / t ) ( V ( t) V ) (8) ( max max Where P is the chamber's pressure, E max is the maximal chamber pressure-volume relatio, V(t) is istataeous chamber volume ad V is a costat. t max is the time whe the elastace fuctio assumes its maximum value ormally close to ed systole ad may be adjusted to correct for chages i systolic duratio at differet heart rates. Left vetricular E N (t ) (t =t/t max ) fuctio is described i Eq. 9 while Eq 1 describes the RV, RA ad LA E N fuctio. Note, oly the positive values are used. N 3 E ( t ) = 1.841t +.685t t (9)

3 Ope-Source Programmig of Cardiovascular Pressure-Flow Dyamics The Ope Pacig, Electrophysiology & Therapy Joural, 1, Volume 3 57 N 3 E ( t ) = 3.374t t t (1) The diastolic passive stage is related to the blood fillig stage of the chamber is characterized by a simplified pressure-volume relatio: ( KV ( t)) P ( t) = e 1 (11) The implemetatio uder the SimPower toolbox is show i Fig. (1). Curret is aalogous to blood flow, voltage is aalogous to pressure, ad charge to blood volume. This special implemetatio allows the use of Simulik where blocks are simply attached to other blocks, thereby trasferrig pressure ad flow data. I a cardiac chamber iput flow is measured ad itegrated to obtai istataeous value of chamber blood volume. Usig eq. 8 with a polyomial (eqs. 9, 1 or 11 depedig o the case), we obtai the chamber pressure. The charge output idicates blood volume. The user may cotrol all cardiac chamber parameters. Reduced cotractility ca simulate global ischemia, icreased diastolic compliace ca simulate diastolic dysfuctio, ad icreased V shifts the PV loop to the right to simulate local ischemia. The valve is described by a versatile model that accouts for steosis ad isufficiecy as well as for leakage ad reverse flow. It also icludes the dyamic pressure-volume relatios of the leaflets represeted by a vsicoelastic model (resistor ad capacitor). The diodes represet the ideal valve but the software allows the user to alter forward resistace, backward resistace ad opeig pressure as show i the right pael of Fig. (). Blood vessels may be modeled usig the basic segmet model show i the left pael of Fig. (3). This model ca be used either as a lumped parameter represetatio of the whole vascular load (widkessel model) or as a very short segmet i a cascaded structure (show i the right pael of Fig. 3). For example the aorta ca be modeled as a multi-segmet vessel with differet parameter values at each segmet if iformatio alog the aorta is required ad the user wishes to emphasize the effects of delay ad reflected waves. To assist i buildig the model Y type vessel coectios are also provided. The same model ca be used for ay vessel sice the parameter values are preset by the user. I the case of a vessel that is exposed to exteral pressure, like a coroary artery, a exteral cotrolled pressure source ca be added. The SimPower implemetatio is show i Fig. (4). It icludes preset resistor, iductor (for itertace) ad a oliear capacitor that represets the oliear compliace usig a polyomial. The toolbox was writte usig Matlab versio 7 with Simulik ad the SimPower toolbox. This cofiguratio allows the user to simply coect ad elemet to the other ad ru the program. Several examples were previously publishedi. As a ope source code, the whole toolbox ca be dowloaded at the address CVT.ZIP. The package icludes help files ad istallatio istructios. Due to chages i ewer versios of the Matlab applicatio, the toolbox is ot compatible. It is recommeded to use it uder versio 7. Fig. (1). Implemetatio of a cardiac chamber. Fig. (). Left pael: Valve model implemetatio that icludes viscoelstic elemets ad forward ad reverse valves with variable resistaces. Right pael: diode characteristics ad parameters where voltage is aalogous to pressure ad curret to flow.

4 58The Ope Pacig, Electrophysiology & Therapy Joural, 1, Volume 3 AN EXAMPLE A example is show i Fig. (5) to demostrate the toolbox. A varyig elastace vetricle model is coupled to a three elemet widkessel model via a valve. The vetricle is Ofer Barea filled by a reservoir of fixed pressure via a valve. I additio, there are pressure ad flow moitors that preset waveforms. Fig. (6) shows sample waveforms for this model. More elaborate models are preseted i Sheffer et al. [18]. Fig. (3). Electrical aalog models of flow i blood vessels. Fig. (4). Implemetatio of a vessel segmet. Fig. (5). A example of model costructio usig the toolbox.

5 Ope-Source Programmig of Cardiovascular Pressure-Flow Dyamics The Ope Pacig, Electrophysiology & Therapy Joural, 1, Volume 3 59 Fig. (6). Vetricular blood volume, pressure ad flow computed by the toolbox for the model show i Fig. (5). DISCUSSION The toolbox offers easy costructio of cardiovascular models ad rus well o Matlab versio 7. The use of the SimPower toolbox allows easy coectios betwee elemets of the system oce the aalog circuit has bee determied icludig oliearities represeted by polyomials. However, oe difficulty, as i ay model, is the selectio of parameters values. I this versio the parameters must be properly set. Thus, to use the toolbox, prior kowledge regardig the physical aspects of the system is required. I future versios of the toolbox we ited to chage the type of parameters. Istead of settig a resistace value or a iertace value i a vessel model it could be more user friedly to set its diameter, legth, vessel wall elastic factor ad blood viscosity or hematocrit. I additio, users ca develop models of various orgas ad share them with other users. REFERENCES [1] Huter PJ, Pulla AJ, Smaill BH. Modellig Total Heart Fuctio. Au Rev Biomed Eg 3; 5: [] Guccioe JM, Costa KD, McCulloch AD. Fiite elemet stress aalysis of left vetricular mechaics i the beatig dog heart. J Biomech 1995; 8: [3] Suga H. Left vetricular pressure volume ratio i systole as a idex iotropism. Jp Heart J 1971; 1: 153. [4] Noordergraaf A. Aalog of the Arterial Bed. I: Attiger EO, Ed. Iteratioal Symposium o Pulsatile Blood Flow, New York: McGraw Hill [5] Rideout VC, Dick DE. Differece-differetial equatios for fluid flow i distesible tubes. IEEE Tras Biomed Eg 1967; 14: 171. [6] Barea O, Moore TW, Jaro D. Cardiac eergy cosideratios durig itra-aortic balloo pumpig. IEEE Tras Biomed Eg 199; 37(): [7] Suga H. Time course of left vetricular pressure-volume relatioship uder various ed-diastolic volume. Jap Heart J 1969; 1: 59. [8] Suga H. Time course of left vetricular pressure-volume relatioship uder various extets of aortic occlusio. Jap Heart J 197; 11: 373. [9] Suga H, Sagawa K, Shoukas A. A Load idepedece of the istataeous pressure-volume ratio of the caie left vetricle ad effects of epiephrie ad heart rate o the ratio. Circ Res 1973; 3: 314. [1] Suga H, Sagawa K. Istataeous pressure-volume relatioship ad their ratio i the excised, supported caie left vetricle. Circ Res 1974; 35: 117. [11] Sagawa K, Suga H, Shoukas A, Bakalar KM. Ed systolic pressure/volume ratio: a ew idex of vetricular cotractility. Am J Cardiol 1977; 4: [1] Suga H, Yamakoshi K. Effects of stroke volume ad velocity of ejectio o ed-systolic pressure of caie left vetricle. Circ Res 1977; 4: [13] Suga H, Kitabatake A, Sagawa K. Ed systolic pressure determies stroke volume from fixed ed diastolic volume i the isolated caie left vetricle, uder a costat cotractile state. Circ Res 1979; 44: 38. [14] Campbell KB, Rigo JA. Use of elastace-resistace models to characterize atural ejectig left vetricle. Proceedig of 6th Iteratioal Coferece ad Workshop, Cardiovascular Dyamics Society, Philadelphia [15] Campbell KB, Rigo JA, Kowle GG, Kirkpatrick RD, Schmidt SL. Validatio of optioal elastace-resistace left vetricle pump model. Am J Physiol 1986; 51: H38. [16] Shroff SG, Jaicki JS, Weber KT. Left vetricular systolic dyamics i terms of its chamber mechaical properties. Am J Physiol 1983; 45: H11. [17] Shroff SG, Jaicki JS, Weber KT. Evidece ad quatitatio of left vetricular systolic resistace. Am J Physiol 1985; 49: H358. [18] Sheffer L, Satamore WP, Barea O. Cardiovascular Simulatio Toolbox. Cardiovasc Eg 7; 7: Received: February 17, 1 Revised: April 4, 1 Accepted: April 4, 1 Ofer Barea; Licesee Betham Ope. This is a ope access article licesed uder the terms of the Creative Commos Attributio No-Commercial Licese ( by-c/3./) which permits urestricted, o-commercial use, distributio ad reproductio i ay medium, provided the work is properly cited.