
MBW:Describing the Pumping Heart as a Pressure SourceFrom MathBioContentsArticleThe following is paper from M.Danielsen and J.T.Ottesen discussing the pumping heart as a pressure source
Introduction200px The human heart has four chambers of two types: atria and ventricles. The blood from atria goes to ventricles which act as pumps to generate the flows.The ventricles does not put any resistance to the blood flow.That is, ventricles are characterized by pressure,which then determine the blood volume and consequently the compliance of the ventricles.The elastance of ventricle is reciprocal of compliance.The ventricular description based on time varying elastance function does not cover both isovolumic and ejecting heart beats. A new mathematical approach describing the pumping ventricle is presented which considers the heart as a pressure source depending on time,ventricular volume and ventricular outflow rather than driven by time varying elastance function. An animation of the heart to understand the procedures of the heart can be found here. ModelThe model simulates the pressure and the volume of the left ventricle of the heart. The approach consists of two parts. The first part concerns about the pressure in the left ventricle without a change in the volume. This simulates the “pressure source” caused by the blood inflow from the atrium. The second adds the effect of the blood ejection in the ventricle, i.e. the volume changes and allows to have a closer look the the pressure within the ventricle and the aorta during a heart beat. Part one – the isovolumic pressure source:The first part simulates the “isovolumetric pressure source” in the ventricle. It just simulates the increasing and decreasing of pressure when the aorta would be closed and the volume wouldn’t change. See picture(). 200px This ventricular pressure “pv” in the isovolumetric contracting ventricel is modeled by the following equation: 400px(1) Where f(t) is given by: 400px (2) and tb is calculated by: 400px (3) Parameters: a measure of ventricular elastance during relaxation b ventricular volume for zero diastolic pressure c,d volumedependent and volumeindependent component of developed pressure Tc,Tr characterize the relaxation(pressure decrease) and the contraction (pressure increase) processes. th peak of isovolumic pressure tb time when erlaxatoin process begins to evolve th time scale of the heart beat Parameter Value a 0.007 mmHg/ml^2 b 5 ml c 1.6 mmHg/ml d 1 mmHg Tc 0.15s Tr 0.175s tp 0.3s th 1s
Part two – blood ejection:The second part combines the isovolumetric model with a so called windkessel model. It did not only give the non ejecting properties of the isolated heart, but it also describes the ejecting ventricles.
The focus of this picture is on pv(Vv,t) what symbolizes the pressure in the ventricle. When the ventricular pressure (pv) raises beyond the aortic pressure (pa) –i.e. pv>pa – the aortic valve opens and blood ejects from the ventricular into the aorta. The volume of the left ventricle changes with respect to the pressure source from the atrium (ps) and the ejection of blood. Based on this model the ventricular pressure pv and the Volume Vv are given by the following differential equations during the heart beat: 300px (4) Parameters The parameters of the equation can also be found in the windkesselmodel. Where Ro and Rs symbolize the characteristic blood pressure impedance. Cs stands for the totatl arterial compliance and ps denotes the simulated pressoure source. Parametric values: Ro 0.08 mmHg/ml Rs 1 mmHg/ml Cs 2.75 ml/mmHg
Discussion of the resulting picturesThe result in the paper is summarized in FIG (X) . On the left hand the values for the isovolumetric pressure, the pressure in the ventricle and the aorta are summarized. On the right hand describes the outflow of blood from the ventricle into the aorta with respect to time. As mentioned before, “piso” simulates only the pressure source for the left ventricle without changes in volume, i.e. function pv(t,Vconst). So it gives the maximum value for the pressure in the ventricle at each time point. Adding the ejection effect we receive the curve of pv. At the point where the pressure in the ventricle (pv) increases beyond the pressure in the aorta (pa) the aortic valve opens and blood flows out of the left ventricle. At this point the isovolumic phase ends and the volume of the ventricle decreases. The quantitaty of the outflow can be seen in the right figure. The outflow starts when pv is bigger than pa and ends when pv falls below pa and the aortic valve closes again. 200px The result of our simulation doesn’t exactly reproduce the results in the paper.
Since a reasonable aortic pressure is around 80 mmHg, Ro and Vv’ is small we used 80 mmHg as initial value for ps. The volume has to be equal at the end and the beginning of the heart beat. And with this initial pressure the only initial volume that holds that constraint during the heart beat is around 100ml.
matlab codefunction returnvalue = heatmap() clear;clc; % INITIAL VALUES FOR DIFFERENTIAL EQUATIONS psIni=80; VIni=100; %CALL THE DIFFERENTIAL EQUATIONFUNCTION [t psV]= ode15s(@heartdiff01,[0:0.005:1],[psIni,VIni]);
volumeValues=psV(:,2); %pv  pressure in ventricle pvValue(1)=0;%initial for i=1:length(t) pvValue(i)=pv(t(i),volumeValues(i)); end; %Piso  pressure in isovolumetric contraction pisoValues = piso(t,max(volumeValues)); %Elastance E=pvValue'./volumeValues; %Q  the Outflow Q=q(t,volumeValues); %pa  pressure in the Aorta help=volumeValues; help(1)=[]; help(length(help)+1)=0; % to equal length delta=helpvolumeValues; delta(length(delta))=0; % last value is 0 Ro=0.08; % see table 2 pa=psV(:,1)Ro*delta;
subplot(3,1,1); plot(t,volumeValues); title('Volume'); % %elastance subplot(3,1,2); plot(t,E); title('Elastance = pv/(VvVo)'); %Q subplot(3,1,3); plot(t,Q); title('q'); axis([0 1 0 max(Q)]) figure; plot(t,psV(:,1),'r'); %ps hold on; plot(t,pisoValues,'b'); %piso plot(t,pvValue,'g'); %pv plot(t,pa,'y'); %pao legend ps piso pv pa;
function pvDot = pv(t,Vv) %Values from TABLE 2 in the paper a=0.007; b=5; c=1.6; d=1; %equation equates to EQUATION 3 pvDot=(a*((Vvb)^2))+(c*Vvd)*f(t); %equation 3 calls function f in file f.m function pisoDot=piso(t,Ve) piso(1)=0; for i=1:length(t) pisoDot(i)=pv(t(i),Ve); end; function qdot =q(t,V) t2=t; t2(1)=[];t(length(t))=[]; dt=t2t; V1=V;
V2=V;
V2(1)=[];
V1(length(V1))=[];
dV=V2V1;
result=dV./dt;
qdot=[0;result];
