Models for the circulatory system INF5610 – p.1/36
Outline Overview of the circulatory system Important quantities Resistance and compliance vessels Models for the circulatory system Examples and extensions INF5610 – p.2/36
Important quantities Heart rate, measured in beats per minute. Cardiac output: The rate of blood flow through the circulatory system, measured in liters/minute. Stroke volume: the difference between the end-diastolic volume and the end-systolic volume, i.e. the volume of blood ejected from the heart during a heart beat, measured in liters. INF5610 – p.3/36
The cardiac ouput Q is given by Q = FV stroke Typical values: F = 80 beats/minute. V stroke = 70 cm 3 / beat = 0.070 liters/beat. Q = 5 . 6 liters/minute. INF5610 – p.4/36
Resistance and compliance vessels P ext = 0 Q 1 Q 2 V V = vessel volume, P ext = external pressure, P 1 = upstream pressure, P 2 = downstream pressure, Q 1 = inflow, Q 2 = outflow. INF5610 – p.5/36
Resistance vessels Assume that the vessel is rigid, so that V is constant. Then we have Q 1 = Q 2 = Q ∗ . The flow through the vessel will depend on the pressure drop through the vessel. The simplest assumption is that Q ∗ is a linear function of the pressure difference P 1 − P 2 : Q ∗ = P 1 − P 2 , R where R is the resistance of the vessel. INF5610 – p.6/36
Compliance vessels Assume that the resistance over the vessel is negligible. This gives P 1 = P 2 = P ∗ Assume further that the volume depends on the pressure P ∗ . We assume the simple linear relation V = V d + CP ∗ , where C is the compliance of the vessel and V d is the “dead volume”, the volume at P ∗ = 0 . INF5610 – p.7/36
All blood vessels can be viewed as either resistance vessels or compliance vessels. (This is a reasonable assumption, although all vessels have both compliance and resistance.) Large arteries and veins; negligible resistance, significant compliance. Arterioles and capillaries; negligible compliance, significant resistance. INF5610 – p.8/36
Heart Systemic veins Systemic arteries Body INF5610 – p.9/36
The heart as a compliance vessel The heart may be viewed as a pair of compliance vessels, where the compliance changes with time, V ( t ) = V d + C ( t ) P. The function V ( t ) should be specified so that it takes on a large value C diastole when the heart is relaxed, and a small value C systole when the heart contracts. INF5610 – p.10/36
Periodic functions for LV and RV compliance 0.015 0.01 0.005 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.04 0.03 0.02 0.01 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 INF5610 – p.11/36
Modeling the heart valves Characteristic properties of a heart valve: Low resistance for flow in the “forward” direction. High resistance for flow in the “backward” direction. INF5610 – p.12/36
The operation of the valve can be seen as a switching function that depends on the pressure difference across the valve. The switching function can be expressed as � 1 if P 1 > P 2 S = 0 if P 1 < P 2 INF5610 – p.13/36
The flow through the valve can be modeled as flow through a resistance vessel multiplied by the switching function. We have Q ∗ = ( P 1 − P 2 ) S , R where R will typically be very low for a healthy valve. Inserting for S , we have Q ∗ = max(( P 1 − P 2 ) /R, 0) . INF5610 – p.14/36
Dynamics of the arterial pulse For a compliance vessel that is not in steady state, we have dV dt = Q 1 − Q 2 . From the pressure-volume relation for a compliance vessel we get d ( CP ) = Q 1 − Q 2 . dt When C is constant (which it is for every vessel except for the heart muscle itself) we have C dP dt = Q 1 − Q 2 . INF5610 – p.15/36
The circulatory system can be viewed as a set of compliance vessels connected by valves and resistance vessels. For each compliance vessel we have d ( C i P i ) = Q in i − Q out , i dt while the flows in the resistance vessels follow the relation Q j = P in − P out . R j INF5610 – p.16/36
A simple model for the circulatory system Consider first a simple model consisting of three compliance vessels; the left ventricle, the systemic arteries, and the systemic veins. These are connected by two valves, and a resistance vessel describing the flow through the systemic tissues. For the left ventricle we have d ( C ( t ) P lv ) = Q in − Q out , dt with Q in and Q out given by Q in = max(( P sv − P lv ) /R mi , 0) , (1) Q out = max(( P lv − P sa ) /R ao , 0) . (2) INF5610 – p.17/36
We get d ( C ( t ) P lv ) = max(( P sv − P lv ) /R mi , 0) dt − max(( P lv − P sa ) /R ao , 0) Note: this is an ODE with discontinuous right hand side (if-tests), which is not ideal from a numerical viewpoint. Implicit solvers must be used, and the order of convergence is limited to 1. INF5610 – p.18/36
Similar calculations for the two other compliance vessels gives the system d ( C ( t ) P lv ) = max(( P sv − P lv ) /R mi , 0) dt − max(( P lv − P sa ) /R ao , 0) , dP sa = max(( P lv − P sa ) /R ao , 0) − P sa − P sv C sa , dt R sys dP sv = P sa − P sv C sv − max(( P sv − P lv ) /R mi , 0) . dt R sys INF5610 – p.19/36
Heart Systemic veins Systemic arteries Body INF5610 – p.20/36
With a specification of the parameters R mi , R ao , R sys , C sa , C sv and the function C lv ( t ) , this is a system of ordinary differential equations that can be solved for the unknown pressures P lv , P sa , and P sv . When the pressures are determined they can be used to compute volumes and flows in the system. INF5610 – p.21/36
A more realistic model The model can easily be improved to a more realistic model describing six compliance vessels: The left ventricle, P lv , C lv ( t ) , the right ventricle, P rv , C rv ( t ) , the systemic arteries, P sa , C sa , the systemic veins, P sv , C sv , the pulmonary arteries, and P pv , C pv , the pulmonary veins, P pv , C pv . INF5610 – p.22/36
The flows are governed by two resistance vessels and four valves: Systemic circulation; R sys , pulmonary circulation; R pu , aortic valve (left ventricle to systemic arteries); R ao tricuspid valve (systemic veins to right ventricle); R tri , pulmonary valve (right ventricle to pulmonary arteries); R puv , mitral valve (pulmonary veins to left ventricle); R mi , INF5610 – p.23/36
This gives the ODE system d ( C lv ( t ) P lv ) = max(( P pv − P lv ) /R mi , 0) − max(( P lv − P sa ) /R ao , 0) dt dC sa P sa = max(( P lv − P sa ) /R ao , 0) − P sa − P sv , dt R sys dC sv P sv = P sa − P sv − max(( P sv − P rv ) /R tri , 0) , dt R sys d ( C rv ( t ) P rv ) = max(( P sv − P rv ) /R tri , 0) − max(( P rv − P pa ) /R puv , 0) , dt d ( C pa P pa ) = max(( P rv − P pa ) /R puv , 0) − P pa − P pv , dt R pu dC pv P pv = P pa − P pv − max(( P pv − P lv ) /R mi , 0) . dt R pu INF5610 – p.24/36
LV compliance, pressures and flows 0.015 0.01 0.005 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 150 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 80 60 40 20 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Top: C l v , Middle: P pv (blue), P lv (green), P sa (red), Bottom: Q mi , Q ao . INF5610 – p.25/36
RV compliance, pressures and flows 0.04 0.03 0.02 0.01 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 20 15 10 5 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 60 40 20 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Top: C r v , Middle: P sv (blue), P rv (green), P pa (red), Bottom: Q tri , Q puv . INF5610 – p.26/36
Systemic and pulmonary flows 8 7 6 5 4 3 2 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Q sys (blue) and Q pu (green). Note the higher maximum flow in the pulmonary system despite the lower pressure. This is caused by the low resistance in the pulmonaries. INF5610 – p.27/36
Pressure volume loops 120 100 80 60 40 20 0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 20 15 10 5 0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Top: left ventricle, bottom: right ventricle. INF5610 – p.28/36
Mitral valve stenosis R mi changes from 0.01 to 0.2. INF5610 – p.29/36
LV compliance, pressures and flows 0.015 0.01 0.005 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 150 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 80 60 40 20 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Reduced in-flow to the LV causes reduced filling and thereby reduced LV pressure and arterial pressure. INF5610 – p.30/36
RV compliance, pressures and flows 0.04 0.03 0.02 0.01 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 30 20 10 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 60 40 20 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 The RV pressure increases because blood is shifted from the systemic circulation to the pulmonary circulation. INF5610 – p.31/36
Systemic and pulmonary flows 8 7 6 5 4 3 2 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Blood is shifted from the systemic circulation to the pulmonary circulation. INF5610 – p.32/36
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