Blood Pressure And Blood Flow Variation During Postural .

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1Blood Pressure and Blood Flow Variation duringPostural Change from Sitting to Standing: ModelDevelopment and ValidationMette S. OlufsenDepartment of Mathematics &Center for Resarch in Scientific ComputationNorth Carolina State UniversityRaleigh, NC 27695email: msolufse@math.ncsu.eduPhone: (919) 515 2678, Fax: (919) 513 7336Johnny T. OttesenDepartment of Mathematics and PhysicsRoskilde University, Roskilde, DenmarkHien T. Tran and Laura M. EllweinDepartment of Mathematics &Center for Resarch in Scientific ComputationNorth Carolina State University, Raleigh, NC 27695Lewis A. LipsitzHebrew SeniorLife, Research and Training Institute,Division of Gerontology, Beth Israel Deaconess Medical Center,and Harvard Medical School, Boston, MA 02215Vera NovakDivision of Gerontology Beth Beth Israel Deaconess Medical Centerand Harvard Medical School, Boston, MA 02215Abstract— Short term cardiovascular responses to postural change from sitting to standing involve complexinteractions between the autonomic nervous system thatregulates blood pressure, and cerebral autoregulation thatmaintains cerebral perfusion. We present a mathematicalmodel that can predict dynamic changes observed inbeat-to-beat arterial blood pressure and middle cerebralartery blood flow velocity during postural change fromsitting to standing. Our cardiovascular model utilizes 11compartments to describe blood pressure, blood flow,compliance and resistance in the heart and systemiccirculation. To include dynamics due to the pulsatile natureof blood pressure and blood flow, resistances in the largesystemic arteries are modeled using nonlinear functionsof pressure. A physiologically based sub-model is usedto describe effects of gravity on venous blood poolingduring postural change. Two types of control mechanismsare included: (i) Autonomic regulation mediated by sympathetic and parasympathetic responses that affect heartrate, cardiac contractility, resistance, and compliance. (ii)Autoregulation mediated by responses to local changesin myogenic tone, metabolic demand, and concentrationof carbon dioxide (CO2 ) that affect cerebrovascular resistance. Finally, we formulate an inverse least squaresproblem for parameter estimation and to demonstrate thatour mathematical model is in agreement with physiologicaldata obtained from a young subject during postural changefrom sitting to standing.Keywords: Cardiovascular system; Mathematicalmodeling; Cerebral blood flow; Gravitational effect;Autonomic regulation; Cerebral Autoregulation

2I. I NTRODUCTIONOrthostatic intolerance disorders, which are commonin every age, are difficult to diagnose and treat. Typically, these disorders whose clinical manifestations include dizziness, syncope, orthostatic hypotension, falls,and cognitive decline are a result of several biologicalmechanisms. To develop better strategies to treat anddiagnose orthostatic intolerance, it is important to understand the underlying mechanisms leading to thesedisorders. One of the main mechanisms involved is theshort term cardiovascular regulation of blood flow tothe brain, which include both autonomic regulation andcerebral autoregulation. The overall goal of this workis to develop a mathematical model that can predictdynamics in observed cerebral blood flow and peripheralblood pressure data, and to propose mechanisms that canexplain the interaction between autonomic regulation andcerebral autoregulation. To this end we have developed amathematical model that can predict these two regulatorymechanisms. To validate the model we compare modelpredictions with arterial finger blood pressure paf , andmiddle cerebral artery blood flow velocity vacp measurements from a young subject.Upon standing from a chair, blood is pooled in thelower extremities due to gravitational forces. As a result,venous return is reduced, which lead to a decreasein cardiac stroke volume, a decline in arterial bloodpressure, and an immediate decrease of blood flow tothe brain. The reduction in arterial blood pressure unloads the baroreceptors located in the carotid and aorticwalls, which leads to parasympathetic withdrawal andsympathetic activation through baroreflex-mediated autonomic regulation. Parasympathetic withdrawal inducesfast (within 1-2 cardiac cycles) increases in heart rate,while sympathetic activation yields a slower (within 6-8cardiac cycles) increase in vascular resistance, vasculartone, cardiac contractility, and a further increase in heartrate [4], [7], [37]. Simultaneously, cerebral autoregulation, mediated by changes in CO2 , myogenic tone, andmetabolic demand leads to vasodilation of the cerebralarterioles [2], [18], [34], [38].Our mathematical model includes two sub-models: (i)a cardiovascular model that can predict blood pressureand blood flow velocity during sitting (for t 60 sec.),and (ii) a control model that can predict autonomic andcerebral regulatory mechanisms during postural changefrom sitting to standing. Both sub-models are basedon the same closed-loop compartmental model with11 compartments that represent the heart and systemiccirculation. Our previous work [27], [29] also usedcompartmental models to describe the dynamics of thecardiovascular system. The model in [27] used an openloop model (the 3-element windkessel model) to analyzedynamics of cardiovascular control. This model usedarterial blood pressure measured in the finger as an inputto predict model parameters that describe dynamics ofcerebral vascular regulation for young people. Theseparameters were obtained by minimizing the error between computed and measured middle cerebral arterialblood flow velocity. Consequently, no equations wereused to describe possible mechanisms of the underlyingregulation. To further advance this study, we recentlydeveloped a 7 compartmental closed-loop model, thatcan predict the dynamics observed in the data. Thismodel did not rely on an external input, but includeda model that describe the pumping of the left ventricle.In addition, the 7 compartmental model included simpleequations that describe the short term regulation. Thismodel was able to accurately predict dynamics of bothcerebral blood flow velocity and arterial blood pressureduring sitting (for t 60 sec.) and standing (for t 80sec), as well as the mean values during the transition (for60 t 80 sec.), but it was not able to predict detaileddynamics during the transition region. Furthermore, wewere not able to obtain adequate filling of the left ventricle. To obtain a more accurate model we developed the11 compartmental model described in this work, whichovercome limitations of the 7 compartmental model asdescribed below.To obtain adequate filling of the left ventricle weadded a compartment that represents the left atrium. Inthe 7 compartmental model we used the blood pressurein the upper body to validate the model against data,which are measured in the finger. The pulse pressure(systolic minus diastolic pressure) in the upper body istoo wide and very sensitive to parameter changes. Hence,to improve our model we included a compartment thatrepresents the arteries in the finger. In addition, we addedtwo small compartments that represent the aorta andvena cava. These compartments were primarily addedto improve model simulation stability. In summary, theimprovements discussed above have led to the addition of4 additional compartments. Furthermore, the 7 compartmental model was not able to predict pulse pressure regulation immediately following standing. To compensatefor this we have modeled resistance of the large arteriesusing nonlinear functions of pressure. Finally, to obtainaccurate widening of the blood flow velocity, a featurethat our 7 compartmental model was not able to predict,we devised an empirical model of autoregulation, and aphysiological model that can predict pooling of blood inlower extremities, due to effects of gravity.A large body of work that describe cardiovascular

3control modeling, e.g., [10], [11], [12], [30], [44] isbased on predictions of mean values for arterial bloodpressure and cerebral blood flow velocity. Consequently,these models can not predict the pulsatile dynamicsof the cardiovascular system. These models use optimal control to minimize the deviation between someobserved quantity (e.g., arterial blood pressure) and agiven set-point. While this strategy can provide goodparameter estimates, optimal control models do not describe the underlying physiological mechanisms. Othermodeling strategies have been proposed in the work byMelchior [19], [20] and Heldt [8], who devise pulsatilemodels that include pulsatility, autonomic regulation, andeffects of gravity. The latter was done by changing thereference pressure outside the compartments. However,these models do not include effects of autoregulation.One way to model the effect of autoregulation is to letthe cerebrovascular resistance be a function of time assuggested by Ursino [39]. However, this work does notinclude the effects of autonomic regulation.A second group of models has described parts ofthe control system without validation against experimental data, e.g., [5], [19], [20], [21], [31], [32], [35],[40], [41], [42], [43]. These models used a closed-loopcompartmental description of the cardiovascular systemcombined with physiological descriptions of the control.While these models can provide qualitative analysis ofthe system they cannot be used for quantitative comparisons with data. Furthermore, most of the models in thesecond group describe the effects of autonomic regulation without including the effects of cerebral autoregulation. In contrast, our model includes both autonomicand cerebrovascular regulations and provides quantitativecomparisons with physiological data.II. M ODELING B LOOD P RESSUREV ELOCITYANDB LOOD F LOWA. A Compartmental Model for the Cardiovascular SystemOur cardiovascular model is based on a closed-loopmodel with 11 compartments. This model is designedto predict blood pressure and volumetric blood in theleft atrium, the left ventricle, aorta, vena cava, arteriesand veins in the upper body, the lower body, and thehead, as well as arteries in the finger, see Fig. 1. Eachcompartment represents all vessels in areas of similarpressure. Hence, in its simplest form the systemic circuitcould consist of one arterial compartment (high pressure)and one venous compartment (low pressure). In ourmodel, we include 5 arterial compartments and 4 venouscompartments.The 11 compartments depicted in Fig. 1 are chosen toensure that the level of detail in the model is adequateto describe the complex dynamics observed in the dataand at the same time not too complex to be solvedcomputationally. Four compartments that represent theupper body and the legs are included to model venouspooling of blood and sympathetic contraction of thevascular bed. Two compartments that represent the brainare included to model effects of cerebral autoregulationand to enable model validation against cerebral arterialblood flow velocity measurements. One compartmentthat represents the finger is included to enable modelvalidation against arterial blood pressure measured in thefinger. To determine cardiac output and venous return,two compartments are included to represent the aortaand vena cava. Finally, to obtain a closed-loop model itis necessary to include a source (the heart) that pumpsblood through the system. Consequently, two compartments are included to represent the left atrium and theleft ventricle. Our previous work [29] only included theleft ventricle, but without an atrium it is not possible toobtain adequate filling of the heart.The major system not included in our model is thepulmonary circulation. Addition of compartments thatrepresent the pulmonary circulation would require moreparameters, which would increase the computationalcomplexity. Instead, the pulmonary circulation is represented as a resistance between vena cava and the leftatrium.To study dynamics of postural change from sittingto standing it is not important to know how blood isdistributed among various inner organs. Hence, the upperbody is simply represented by an arterial and a venouscompartment. Each compartment is represented by acompliance element (inverse elasticity) and is separatedby resistance to flow. The design of the systemic circulation with arteries and veins separated by capillariesprovides some resistance and inertia to the volumetricflow rate. In our model we include effects of resistancebetween compartments but neglect effects due to inertia.The major resistance to flow is located in peripheralregions between compartments that represent arteriesand veins. Compartments that represent large conduitvessels are also separated by resistances that representthe overall resistance of the compartment. Resistancesbetween conduit vessels are very small compared withthe peripheral resistances.The description of blood pressure and volumetric flowrate in a system comprised of compliant compartments(capacitors) and resistors are equivalent to that of anelectrical circuit, see Fig. 1, where blood pressure p[mmHg] plays the role of voltage and volumetric flow

4qacpVvc Cer VeinsCvcpvcRvcqacqvAtriumqmvplapvRvRacCla (t)Left ventricleqavVaRmvqaf pClv (t)qafSys AortaplvFinger ArtpaRavRafCaqauqvuRvuCacVlvVlaCvpacRacpqvcVv Vena cavaVacCer ArtpafVafCafRauRaf pqaupVvu Veins (up)CvupvuRvlRauppauqalqvlVvl Veins (legs)Sys Art (up)VauCauRalqalpSys Art (legs) ValRalpCalpvlpalCvlFig. 1. Compartmental model of the systemic circulation. The model contains 11 compartments. 5 compartments represent systemic arteries(the brain, upper body, lower body, aorta and the finger), 4 compartments represent the systemic veins (the brain, upper body, lower body,and vena cava), and 2 compartments represent the left atrium and left ventricle. Since the pulmonary system is not included, the systemicveins are directly attached to the left ventricle. Each compartment includes a capacitor to represent the compliant volume of the arteriesor veins. All compartments are separated by resistors representing resistance of the vessels. Furthermore, the compartment representing theleft ventricle has two valves (the aortic valve and the mitral valve). Following terminology from electrical circuit theory, the flow betweencompartments is equivalent to electrical current and the pressure inside each compartment is analogous to voltage. Resistors R [mmHgsec/cm3 ] are marked with zigzag lines, capacitors C [cm3 /mmHg] are marked with dashed parallel lines inside the compartments, whilethe aortic and mitral valves are marked with short lines inside the compartment that represents the left ventricle. The abbreviations are laleft atrium, lv left ventricle, av aortic valve, mv mitral valve, a aorta, au arteries in the upper body, al arteries in the lower body, aupperipheral arteries in the upper body, alp peripheral arteries in the lower body, ac cerebral arteries (in the brain), acp peripheral cerebralarteries, af finger arteries, af p peripheral finger arteries, vl veins in the lower body, vu veins in the trunk and upper body, v vena cava,and vc cerebral veins.rate q [cm3 /sec] plays the role of current. To compareour model with data we assume that the diameter ofthe middle cerebral artery remains constant, such thatblood flow velocity can be obtained by scaling volumetric blood flow by a constant factor that representsthe area of the vessel. Recent measurements of middlecerebral artery diameter by magnetic resonance imaging(MRI) combined with transcranial Doppler assessment ofcerebral blood flow velocity have demonstrated that themiddle cerebral artery diameter does not change despitelarge changes in cerebral blood flow velocity elicited bystimuli such as lower body negative pressure and CO2changes [36].To predict blood pressure and blood flow in andbetween the compartments we base our model on volumeconservation laws [41]. Blood pressure and volumetricblood flow can be found by computing the volume andchange of volume for each compartment. The equationsthat represent the arterial and venous compartments aresimilar. For each of these compartments the stressed volume V Cp [cm3 ] (volume pumped out during one cardiac cycle), where C [cm3 /mmHg] is the compliance andp [mmHg] is the blood pressure. The cardiac output fromthe heart is given by CO HVstroke [cm3 /sec], whereH [beats/sec] is the heart rate and Vstroke [cm3 /beat] isthe stroke volume. For each compartment, the net changeof volume is given bydVpin pout qin qout ,q ,(1)dtRwhere q [cm3 /sec] is determined analogously to Kirchhoff’s current law and R is the resistance to flow. Severalcompartments have more than one inflow or outflow.For example, the compartment that represent the aortahas 3 outflows (qout qaf qau qac ) while thecompartment that represent vena cava has 3 inflows(qin qaf p qvu qvc ), see Fig. 1.To model the left ventricle as a pump, the positionof the mitral and the aortic valves must be included.

5During diastole the mitral valve is open while the aorticvalve is closed allowing blood to enter the left ventricle.Then, isometric contraction begins, increasing the ventricular pressure. Once, the ventricular pressure exceedsthe aortic pressure, the aortic valve opens, propellingthe pulse wave through the vascular system. Note, forhealthy young people, both valves cannot be open simultaneously. To incorporate the state of the valves wehave modeled the resistances (Rav and Rmv , see Fig. 1)as follows Rv min Rv e 10(pin pout ) , 5000 ,where v mv, av . This equation result in a largeresistance (and no flow) while the valve is closed anda small resistance (and normal flow) while the valveis open. The minimum value is introduced to avoidnumerical problems due to large numbers.A system of differential equations is obtained by differentiating the volume equation V Cp and inserting(1), i.e.,dpdCdV C p qin qout .dtdtdt(2)The circuit in Fig. 1 gives rise to a total of 9 differentialequations in dp/dt, one for each of the arterial andvenous compartments. For the two compartments thatrepresent the atrium and the ventricle, differential equations are kept as dV /dt. For these two compartments,blood pressure is computed explicitly as a function ofvolume, see next section. A complete list of equationscan be found in the Appendix.B. Ventricular and Atrial ContractionAtrial and ventricular contraction leads to an increasein blood pressure from low values observed in the venoussystem to high values observed in the arterial system.Our model is based on the work by Ottesen [6], [33].This model predicts atrial (pla ) and ventricular (plv )pressure as a function volume and cardiac activation ofthe formp a(V (t) b)2 (c(t)V (t) d)g(t),p pla , plv .(3)The parameter a [mmHg/cm3 ] is related to elastanceduring relaxation and b [cm3 ] represents volume at zerodiastolic pressure, c(t) [mmHg/cm3 ] represents contractility, and d [mmHg] is related to the volume dependentand volume independent components of the developedpressure. The activation function g(t), which is definedover the length of one cardiac cycle, is described by apolynomial of degree (n, m): g(t) f (t)/f (t p ) withf (t̃) pp t̃n (β t̃)mnn mm0him nβm n0 t̃ β(4)β t̃ T.T [sec] is the duration of the cardiac cycle, t̃ mod(t, T ) [sec], β(H) [sec] denotes the onset of relaxation, H 1/T [1/sec] denotes the heart rate, n and mcharacterize the contraction and relaxation phases, andpp is the peak value of the activation. The ability to varyheart rate is included in the isovolumic pressure equation(3) by scaling time and peak values of the activationfunction f . The time for peak value of the contraction,tp [sec], is scaled by introducing a sigmoidal function,that depend on the heart rate H , of the formθν(tM tm ),(5)H ν θνwhere θ represents the median and ν represents the steepness, and tm [sec] and tM [sec] denote the minimumand maxim

Abstract—Short term cardiovascular responses to pos-tural change from sitting to standing involve complex interactions between the autonomic nervous system that regulates blood pressure, and cerebral autoregulation that maintains cerebral perfusion. We present a mathematical model that can predict dynamic changes observed in

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