Basic Circulation Model

In the circulation, the heart (H) is the active element, pumping blood through the vascular system, the passive element. This vascular system has both a transporting and a delivering function. The arterial side (a) is to transport blood from the heart to the tissues, the venous side (v) from tissues to heart, and in between the microcirculation. The heart has two compartmental sides, two pumps, where the left supplies the body and the right the lungs. As a basis for understanding how the circulation works, mostly only one heart half is modelled, the left one, as shown in the figure. The relevant variables are pressures P, volume V and blood flow Q for each compartment, and compliance C ⁽¹⁾ except the microcirculation where it is the flow resistance TPR  ⁽²⁾ (Total Periferal Resistance). Changes in P or V are indicated with a Δ, ΔP and ΔV respectively

The model starts with the functioning of the pump, the heart chamber (H). There are two valves. The outlet valve opens when P_H > P_a, the inlet valve opens when P_v > P_H. The heart chamber switches between two states, in the figure to the right shown as red and blue curved lines:
  – contracted or systolic state, red curve;
  – relaxed or diastolic state, blue curve.
The black curve shows what the heart does.
In point A, the heart reaches the relaxed state and the inlet valve opens, allowing blood to fill the heart.
In point B, contraction starts and pressure builds up, causing the inlet valve to close.
In point C, pressure reaches the point where the outlet valve opens. Now, a further pressure build-up is opposed by pressure loss because blood flows out of the heart into the arterial system, which results in the curved line; heart volume decreases while pressure goes up and down.
In point D, the heart muscle relaxes causing the outlet valve to close.
The black cycle is called the heart's Pressure-Volume Loop. Together with the heart frequency f_H it gives a representation of heart volumes and heart flows as a function of time. The volume pumped per beat is the Stroke Volume (SV).

This heart pump has to be coupled to the circulation. Here, the following laws of physics apply:
- Arterial compliance: ΔP_a = C_aΔV_a. If C_a does not vary too much, P_a = P_a0 + C_aV_a
- Arterial volume change = in − out: dV_a/dt = Q_H,out − Q where Q_H,out is the blood coming from the heart and Q (without index) is the 'systemic' blood flow, through the entire peripheral resistance.
- Peripheral Resistance: P_a − P_v = Q×TPR
- Venous volume change = in − out: dV_v/dt = Q − Q_H,in where Q_H,in is the blood flowing into the heart.
- Venous compliance: ΔP_v = C_vΔV_v. Again, if C_v does not vary too much, P_v = P_v0 + C_vV_v

Some remarks:
- The heart atria here are considered to belong to the venous side.
- The treatment above only leads to a set of equations; to solve for blood and heart pressures, a computer program is needed.
- The heart is within the thoracic cavity which 'pulls' at the heart with an intrathoracic pressure P_th, but most of the vasculature is not. This is neglected above but can be accounted for by incorporating P_th as indicated in the figure.
- In a diseased heart, there may be not enough time to reach the systolic curve in point D. Then, contraction duration also will be relevant.

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