Stephanie Parragh et al. / IFAC-PapersOnLine 48-1 (2015) 017–022

8th Vienna International Conference on Mathematical Modelling 8th Vienna International Conference on Mathematical Modelling 8thFebruary Vienna 18 International - 20, 2015. Vienna Conference University on Mathematical of Technology, Modelling Vienna, FebruaryAustria 18 - 20, 2015. Vienna UniversityAvailable of Technology, online at Vienna, www.sciencedirect.com AustriaFebruary 18 - 20, 2015. Vienna University of Technology, Vienna, Austria ScienceDirect

IFAC-PapersOnLine 48-1 (2015) 017–022 Influence of an Asymptotic Pressure Level Influence of an Asymptotic Pressure Level Influenceon the Windkessel of an Asymptotic Models of Pressure the Arterial Level on the Windkessel Models of the Arterial on the WindkesselSystem Models of the Arterial System , Stephanie Parragh ∗,∗∗ Bernhard Hametner ∗∗ Stephanie Parragh ∗,∗∗ Bernhard Hametner ∗∗ StephanieSiegfried Parragh Wassertheurer∗ ∗∗ Bernhard Hametner∗∗ ∗∗ Siegfried Wassertheurer ∗∗ Siegfried Wassertheurer ∗∗ Institute for Analysis and Scientific Computing, Vienna University of ∗ Institute for Analysis and Scientific Computing, Vienna University of ∗ Institute for AnalysisTechnology, and Scientific Vienna, Computing, Austria Vienna University of ∗ Institute for AnalysisTechnology, and Scientific Vienna, Computing, Austria Vienna University of (e-mail: [email protected]).Technology, Vienna, Austria Health(e-mail: & Environment [email protected]). Department, AIT Austrian Institute of ∗∗ Health(e-mail: & Environment [email protected]). Department, AIT Austrian Institute of ∗∗ Health & EnvironmentTechnology Department, GmbH, Vienna, AIT Austria Austrian Institute of ∗∗ Health & EnvironmentTechnology Department, GmbH, Vienna, AIT Austria Austrian Institute of (e-mail: Siegfried.Wassertheurer,Technology GmbH, Bernhard.Hametner Vienna, Austria @ait.ac.at). (e-mail: {Siegfried.Wassertheurer, Bernhard.Hametner}@ait.ac.at). (e-mail: {Siegfried.Wassertheurer, Bernhard.Hametner}@ait.ac.at). { } Abstract: Windkessel models are lumped-parameter models of the arterial system that describe Abstract: Windkessel models are lumped-parameter models of the arterial system that describe Abstract:the dynamicWindkessel relation between models are blood lumped-parameter flow and pressure models in the of the aorta. arterial Despite system their that simplicity, describe thethey dynamic are used relation in many between applications blood flowincluding and pressure methods in for the the aorta. non-invasive Despite theirstratification simplicity, of theythe dynamic are used relation in many between applications blood flowincluding and pressure methods in for the the aorta. non-invasive Despite theirstratification simplicity, of theycardiovascular are used inrisk. many However, applications even though including they methods have been for studiedthe non-invasive extensively, stratification there is still of cardiovasculardisunity regarding risk. the However, question even if pressure though they should have be been modelled studied as resulting extensively, solely there from is still the disunitycardiovascular regarding risk. the However, question even if pressure though they should have be been modelled studied as resulting extensively, solely there from is still the disunityejection of regarding the heart the or if question an asymptotic if pressure pressure should level be should modelled be included as resulting which solely is independent from the ejectionfrom cardiac of the beating. heart or The if an aim asymptotic of this work pressure is to mathematically level should be analyseincluded the which influence is independent of such a fromejection cardiac of the beating. heart or The if an aim asymptotic of this work pressure is to mathematically level should be analyseincluded the which influence is independent of such a frompressure cardiac level beating.P on the The model aim ofbehaviour this work of is the to four mathematically most widely analyseused Windkessel the influence models of such (two-, a pressure level P∞ on the model behaviour of the four most widely used Windkessel models (two-, pressurethree- and level four-elementP∞ on the model Windkessel, behaviour the oflatter the four in a most series widely as well used as Windkessela parallel configuration). models (two-, Therefore,three- and the four-element model∞ equations Windkessel, are introduced the latter and in a Fourier series as analysis well as is a performed parallel configuration). to clarify the Therefore,three- and the four-element model equations Windkessel, are introduced the latter and in a Fourier series as analysis well as is a performed parallel configuration). to clarify the Therefore,impact of P the modelon the equations other parameters. are introduced Then, and a typical Fourier aortic analysis flow is wave performed is used to as clarify input the to impact of P∞ on the other parameters. Then, a typical aortic flow wave is used as input to impactthe models of P and∞ on simulation the other experiments parameters. with Then, varying a typical values aortic of P floware wave performed. is used asTheoretical input to the models and∞ simulation experiments with varying values of P∞ are performed. Theoretical considerationsthe models and as simulation well as numerical experiments results with show varying that, values in all four of P∞ models,are performed. including TheoreticalP mainly considerations as well as numerical results show that, in all four∞ models, including P∞ mainly affectsconsiderations the diastolic as well part as numerical of the modelled results show pressure that, wave in all and four could models, potentially including improveP∞ mainly the fittingaffects performance the diastolic during part of diastolic the modelled decay. pressureHowever, wave further and research could potentially is needed to improve clarify∞ the fittingaffects performance the diastolic during part of diastolic the modelled decay. pressureHowever, wave further and research could potentially is needed to improve clarify the fittingphysiological performance interpretation during diastolicof P as decay. well as However, its appropriate further size. research is needed to clarify the physiological interpretation of P∞ as well as its appropriate size. physiological interpretation of P∞ as well as its appropriate size. © 2015, IFAC (International Federation∞ of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Modelling, Physiological∞ models, Fourier transforms, Parameters, Differential Keywords:equations, WindkesselModelling, Physiological models, Arterial models, system Fourier transforms, Parameters, Differential equations,Keywords: WindkesselModelling, Physiological models, Arterial models, system Fourier transforms, Parameters, Differential equations, Windkessel models, Arterial system 1. INTRODUCTION element Windkessel as well as different configurations of 1. INTRODUCTION elementfour-element Windkessel Windkessel as well models as different (Burattini configurations and Gnudi, of 1. INTRODUCTION four-elementelement Windkessel Windkessel as well models as different (Burattini configurations and Gnudi, of The mechanisms shaping pressure and flow in the arterial four-element1982; Segers et Windkessel al., 2008). models (Burattini and Gnudi, The mechanisms shaping pressure and flow in the arterial 1982; Segers et al., 2008). Thesystem mechanisms have been shaping studied pressure since the and first flow measurements in the arterial 1982;Because Segers of their et al., simple 2008). yet (to a certain degree) accurate system have been studied since the first measurements Because of their simple yet (to a certain degree) accurate systemwere performed have been almost studied three since centuries the first ago. measurements The german Becausedescription of their of the simple arterial yet system, (to a certain these degree) models accurate are still were performed almost three centuries ago. The german description of the arterial system, these models are still werephysiologist performed and almost anatomist three Ernst centuries Heinrich ago. The Weber german was descriptionused in many of theapplications. arterial system, Especially, these since models all parame- are still physiologist and anatomist Ernst Heinrich Weber was used in many applications. Especially, since all parame- physiologistone of the first and to anatomist propose an Ernst explanation Heinrich based Weber on was the usedters have in many a physiological applications. meaning, Especially, methods since to all determine parame- one of the first to propose an explanation based on the ters have a physiological meaning, methods to determine oneWindkessel of the first effect to (Hildebrandt propose an explanationand Weber, 1831):based onduring the terscharacteristics have a physiological of the arterial meaning, system methods of a specific to determine person Windkessel effect (Hildebrandt and Weber, 1831): during characteristics of the arterial system of a specific person Windkesselsystole, when effect the (Hildebrandt heart is ejecting, and theWeber, walls 1831): of the during large characteristicsare often based of on the Windkessel arterial system models of(Stergiopulos a specific person et al., systole, when the heart is ejecting, the walls of the large are often based on Windkessel models (Stergiopulos et al., systole,elastic arteries when the such heart as is the ejecting, aorta the expand walls andof the act large as are1999a; often Liu based et al., on 1986) Windkessel and they models have (Stergiopulos therefore become et al., elastic arteries such as the aorta expand and act as 1999a; Liu et al., 1986) and they have therefore become elastica reservoir arteries storing such blood. as the When aorta the expand pressure and decreases act as 1999a;an important Liu et al., tool 1986) for the and non-invasive they have therefore stratification become of a reservoir storing blood. When the pressure decreases an important tool for the non-invasive stratification of aduring reservoir diastole, storing the blood.walls contract When theagain pressure thereby decreases enabling ancardiovascular important risktool (Wassertheurer for the non-invasive et al., 2008;stratification Hametner of duringa steady diastole, blood flow the duringwalls contract the entire again cardiac thereby cycle. enabling cardiovascular risk (Wassertheurer et al., 2008; Hametner aduring steady diastole, blood flow the duringwalls contract the entire again cardiac thereby cycle. enabling cardiovascularet al., 2012). risk (Wassertheurer et al., 2008; Hametner a steady blood flow during the entire cardiac cycle. et al., 2012). In 1899, this concept was formalized in a mathematical etHowever, al., 2012). even though the Windkessel models have been In 1899, this concept was formalized in a mathematical However, even though the Windkessel models have been Inmodel 1899, by this Otto concept Frank was (Westerhof formalized et in al., a 2009). mathematical His so However,investigated even extensively, though the there Windkessel is still disunity models have regarding been model by Otto Frank (Westerhof et al., 2009). His so investigated extensively, there is still disunity regarding modelcalled two-element by Otto Frank Windkessel (Westerhof consisted et al., of 2009). the arterial His so investigatedthe question extensively, if an asymptotic there is pressure still disunity level should regarding be called two-element Windkessel consisted of the arterial the question if an asymptotic pressure level should be calledcompliance, two-element describing Windkessel the distensibility consisted of of the the large arterial ar- theincluded, question i.e. a if pressure an asymptotic that is not pressure caused level by the should ejecting be compliance, describing the distensibility of the large ar- included, i.e. a pressure that is not caused by the ejecting compliance,teries, and the describing peripheral the resistance, distensibility representing of thelarge the total ar- included,heart itself i.e. but a pressure is sustained that by is not the caused vascular by system. the ejecting The teries, and the peripheral resistance, representing the total heart itself but is sustained by the vascular system. The teries,resistance and of the the peripheral arterial system resistance, mostly representing caused by the the small total heartaim of itself this butwork is is sustained to investigate by the the vascular influence system. of such The a resistance of the arterial system mostly caused by the small aim of this work is to investigate the influence of such a resistancearteries and of arteriolesthe arterial (Kokalari system mostly et al., caused2013). The by the number small aimpressure of this level work on the is to behaviour investigate of the the most influence commonly of such used a arteries and arterioles (Kokalari et al., 2013). The number pressure level on the behaviour of the most commonly used arteriesof parameters and arterioles in Frank’s (Kokalari Windkessel et al., was2013). later The extended number pressureWindkessel level models. on the behaviour of the most commonly used of parameters in Frank’s Windkessel was later extended Windkessel models. ofto parametersimprove the in accuracy Frank’s of Windkessel the modelled was pressure later extended waves, Windkessel models. toresulting, improve among the accuracy many more, of the in modelled the very pressure popular waves, three- resulting,to improve among the accuracy many more, of the in modelled the very pressure popular waves, three- resulting, among many more, in the very popular three- Copyright2405-8963 ©© 2015,2015, IFAC IFAC (International Federation of Automatic Control)17 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2015, IFAC 17 CopyrightPeer review © under 2015, responsibility IFAC of International Federation of Automatic17 Control. 10.1016/j.ifacol.2015.05.125 MATHMOD 2015 18February 18 - 20, 2015. Vienna, Austria Stephanie Parragh et al. / IFAC-PapersOnLine 48-1 (2015) 017–022

2. METHODS

2.1 Two-element Windkessel

Windkessel models describe the dynamic relation between pressure and flow in the arterial system assuming that the system is in steady state, i.e. that both pressure and flow are periodic functions. They are so called lumped- parameter models, which means that the whole arterial system is modelled as one compartment and space is therefore not considered. The ejection pattern of the left ventricle represents the input, the resulting pressure wave the output of the models. Otto Frank originally considered two parameters to de- scribe the mechanisms in the arterial system: the arte- rial compliance Ca and the peripheral resistance Rp. The mathematical formulation of his model is based on conser- Fig. 1. Electrical analogue representation of the 2-element vation of mass, i.e. the change of blood volume V contained (a), 3-element (b) and the two configurations of the in the compartment must be equal to the amount of blood 4-element (serial c, parallel d) Windkessel models. flowing in from the heart q minus the amount of blood in Pressure corresponds to voltage, blood flow to current leaving the system to the periphery qout, see (1). flow. Characteristic impedance Zc and peripheral re- sistance Rp are represented by resistors, total arterial dV = q q (1) compliance Ca by a capacitor, inertance L by an dt in − out inductor and asymptotic pressure P by a pressure source. ∞ The parameter Ca specifies the elasticity of the arteries and is defined as the change of volume V arising from a change in pressure p : An electrical analogue representation of this model is wk shown in fig. 1a. In order to solve the ODE (5) for a dV C = (2) given flow profile in the time domain, an initial condition a dp wk pwk(0) = p(0) P has to be specified. − ∞ Rp describes the resistance the blood has to overcome in order to flow through the periphery, i.e. to leave the 2.2 Three- and four-element Windkessel compartment. According to Ohm’s law it relates pressure pwk to outflow qout: p = R q (3) Comparison of the aortic pressure p derived with the WK2 wk p out to measurements showed that the modelled pressure rise These equations can now be merged into one by first using in the beginning of systole was too slow. Therefore the the chain rule of differentiation and (2) WK2 was extended by a third element, the characteristic impedance Zc. Zc takes into account the compliance and inertance in the ascending aorta (Westerhof et al., 2009) dV dV dpwk dpwk = = Ca (4) and acts as an extra resistance on qin (see fig. 1b). dt dpwk dt dt and then substituting (4) and (3) in (1) which results in Model 2. 3-element Windkessel (WK3) the following linear ordinary differential equation (ODE) For Rp, Ca> 0 and Zc,P 0 the aortic pressure p(t) ∞ ≥ for the pressure pwk. corresponding to the flow qin(t) is modelled as

dpwk 1 1 (t)+ pwk(t)= qin(t) (5) p(t)=Zcqin(t)+pwk(t)+P (7) dt RpCa Ca ∞ with pwk(t) satisfying the ODE given in (5). pwk represents the pressure resulting from the ejection of blood into the arterial system. The aortic pressure p itself Efforts to further improve the model resulted in the inclu- is then given by pwk plus an asymptotic pressure level P ∞ sion of a fourth element, the hydraulic inductance L, which which is maintained by the vascular system (at least for represents the total arterial inertance (Stergiopulos et al., some time) even without activity of the heart. 1999b). L specifies the augmentation of pressure caused by a change in flow or, in other words, by an acceleration Model 1. 2-element Windkessel (WK2) or deceleration of blood. Two different configurations of For Rp, Ca > 0 and P 0, the aortic pressure p(t) this model exist, once Z and L are, electrically spoken, ∞ ≥ c corresponding to the flow qin(t) is modelled as connected in series (WK4s) and once in parallel (WK4p), see fig. 1c, d. In the first case, the total pressure p increases linearly with changes in qin, whereas the second case is a p(t)=pwk(t)+P (6) ∞ little more complex and will be discussed in the paragraphs with pwk(t) satisfying the ODE given in (5). following the formulation of the WK4s.

18 Download English Version: https://daneshyari.com/en/article/713543

Download Persian Version:

https://daneshyari.com/article/713543

Daneshyari.com