Biomechanics/Hemodynamics

Introduction

• There are 2 types of flow :

a) Pulsatile flow

Represented by cycles.

Each cycle consists of 2 parts:

1. A working part (where V of blood flowing increases to max).

2. A resting part (where V of blood flowing decreases to 0.

E.g. if the cycle occurs in 1 sec, 100 mL pass in 0.5 sec and 0 mL pass in the next 0.5 sec.

b) Continuous flow

Represented by a constant line.

The line can be considered as cycles, but with the time of resting part = 0 (always working).

E.g. if the cycle occurs in 1 sec, 50 mL pass in 0.5 sec and 50 mL pass in the next 0.5 sec.

• Note: In lungs, the force that was acting on the walls was IPP, from outside the lung. In arteries, the force acting on the walls is the blood itself from inside the artery.

Cardiac work

Assuming a:

- Volume rate = V / t = 100 mL / sec.

- Pressure rate = P / t = 100 mm Hg / sec.

And noting that W = PV, 3 cases are considered:

1) Continuous flow with rigid arteries

Constant continuous output at 100mL / sec.

Constant pressure = 100 mm Hg

Thus, W = 10000 (mmHg)(mL/s) = 1.3 J

2) Pulsatile flow with rigid arteries

P is 2 times the 1st case since radius is constant and all flow occurs in half the time.

Thus, P = 200 mmHg.

Thus, W = 20000 (mmHg)(mL/s) = 2.6 J

3) Pulsatile flow with very compliant arteries

P in the vessel remains constant no matter how much fluid has been added:

When blood is added, the walls of the vessels balloon outwards, carrying a surplus of blood. This keeps P low.

When heart stops adding blood, as soon as some blood is drained, some blood in the balloon moves into the normal vessel. The walls move inwards by recoiling forces, maintaining P constant.

Thus P remains 100 mmHg.

Thus, W = 10000 (mmHg)(mL/s) = 1.3 J

Note

Case 2 and 3 are ideal cases. In real organisms, arteries have certain compliance so that P↑ slightly as blood is added and P↓ as blood is drained.

The more compliant the artery, the less work a pulsatile heart needs to do and the lower the P needed to pump the blood.

Arterial pulse

• Rate and intensity of pulse can be used in diagnosis.

• In this case, the artery is pressed from all directions by a band around the arm.

• When P done on artery = blood pressure, a resonance sound is heard.

• Normal count rate = 72 / min. Normal blood P = 80 (diastolic) and 120 (systolic).

Arteries functions

1) The heart pumps blood in arteries as a pulsatile flow. Moving from great arteries to capillaries, the compliance of blood vessels converts the pulsatile flow into a continuous one by the time it reaches the cells.

2) Allow heart to rest during half the cycle (they do the other half of the work).

3) Reduce work done by heart (if they weren’t compliant, W would have been doubled).

4) Decrease blood pressure to a normal value.

5) Allow us to measure rate and intensity of pulse for diagnosis.

Effect of cholesterol deposits on blood pressure and flow

High blood pressure (hypertension) is generally caused by factors such as the renin-angiotensin system, salt and water retention etc.. Contrary to popular belief hypertension is usually a cause of cholesterol deposits rather than being caused by these. Hypertension damages the artery walls and allows atheromatous plaques to form. Once cholesterol has been deposited in an artery wall perfusion is reduced for a given level of blood pressure:

1) From continuity and Bernoulli’s equations

From the continuity eqn, A1v1 = A2v2.

Due to cholesterol deposits, A1 > A2 ----> v1 < v2.

From Bernoulli’s equation, ½ ρv2　　+ ρgh + P = constant.

Thus, since v1↓ ----> P1↑

2) From Poiseuille’s law

C = ∆V / ∆P = (∆V/t) / (∆P/t) = ∆Q / (∆P/t) ----> ∆P / t = ∆Q / C

Thus, as cholesterol deposits on walls, it makes them rigid (decreases their compliance).

Thus, as C↓ (∆P/t) ↑ i.e. P1↑

Conclusion from both views

Blood pressure would need to increase to ensure a given level of tissue perfusion downstream of a plaque. This does not always occur with localised plaques and patients can report symptoms of underperfusion. In the heart underperfusion can lead to angina and sudden heart attacks, especially if a plaque becomes detached (e.g.: coronary thrombosis). Atheromatous plaques are usually fairly localised in the initial stages of the disease. In places such as the kidneys, where regulatory mechanisms exist to ensure adequate blood flow, the existence of atheromatous plaques can cause the further elevation of blood pressure. In the case of the kidneys the renin-angiotensin-aldosterone system depends upon pressure sensors that may be downstream from plaque, these will then signal incorrectly that the systemic blood pressure is low and cause water retention and other reflex events to raise blood pressure over the whole body.

• Arteriosclerosis = general term for build up of cholesterol and fibrous tissue in artery walls. Calcification or even ossification can occur within old sclerosed artery walls.Arteriosclerosis decreases the diameter and compliance of arteries, which increases the work done by the heart.

• Atherosclerosis = a form of arteriosclerosis where deposits of cholesterol and fibrous tissue occur on the walls of large arteries.

Effect of exercises on blood pressure

• R = ∆P / ∆Q ---> ∆P = (Qi – Qo) R

Where Qi = cardiac output (artery inflow), Qo = artery outflow

• During exercises, the cardiac output must increase to supply muscles.

• The resistance of the vascular beds (R) is reduced by local and central reflexes to allow adequate perfusion of the muscles. These are much larger than any changes in resistance due to variations in arterial compliance.

• Note: ∆P / t = (Qi – Qo) / C. Thus, a severely atherosclerotic person doing exercises (low C) experience higher ∆P than normal person.

Effect of blood vessel diameter on viscosity

• Viscosity (η) = a frictional constant associated with the flow of a given liquid in a given system.

• $R=8\eta L/\pi r^{4}$ The resistance of a tube of diameter "r" decreases with the fourth power of the radius. This means that 10,000 1mm pipes have the same resistance to flow as one 10 mm pipe. The capillary beds contain many more vessels than the arteries or veins but less than might be expected for the observed flow rates. The circulation avoids excessive resistance in the capillary beds as a result of a curious property of blood known as the Fahraeus and Lindqvist effect: blood is a non Newtonian fluid because it contains large biconcave RBCs.

1) When blood flows in arteries of large r

RBCs have great random motion: some move horizontally, others vertically, and others with an angle. Thus, internal friction is great, which increases viscosity.

Thus, η↑.

2) When blood flows in capillaries of small r

RBCs have no random motion: each RBC must move singly, one after the other. Thus, internal friction is minor, which decreases viscosity.

Thus, η↓.

Thus, R doesn’t increase and capillaries don’t get too much P, so they have a lower resistance to flow.

Peripheral resistance

• Peripheral resistance = resistance of the blood vessels between the aorta and the right atrium.

• The greater the peripheral resistance, the greater the difficulty that blood has to leave the arteries and hence the greater the cardiac work for a given cardiac output.

• Peripheral resistance can be changed by the action of the autonomic nervous system, hormonal factors etc.

• Effect of peripheral resistance: increasing peripheral resistance at a given level of cardiac output will increase blood pressure in the arteries.

Blood pressure in different parts of the body

• ½ ρv2　　1 + ρgh1 + P1 = ρv2　　2 + ρgh2 + P2

• If the person is sleeping, h1 = h2. Assuming A1 = A2 ---> v1 = v2 ---> P1 = P2. Thus, all body receives the same P = 80/120 mmHg

• If the person is standing upright, P at the level of the heart (measuring at the arm) is 80/120 mmHg.

• If we measure at a lower level, blood pressure increases by ρgh (because blood loses some P.E), where h is the perpendicular distance between heart and the given point.

• If we measure at a higher level, blood pressure decreases by ρgh.

• Example: in the previous figure, a person has his legs up (C) and his head down (A).

PA = PB + ρgh.

PB = PC + ρgh’.

If θ and d are given, h can be calculated by h = d sin θ.

Previous: The Biomechanics Of The Lung

Venous return, venous compliance and blood pressure

The rigid pipe model of haemodynamics is applicable when considering any instantaneous flow of blood from the arteries to the veins. The instantaneous blood flow is equal to:

$F={\frac {P_{a}-P_{vb}}{R-R_{v}}}$

where: $F$ is the instantaneous blood flow $P_{a}$ is instantaneous arterial pressure $P_{vb}$ is instantaneous venous vascular bed pressure $R$ is total peripheral resistance $R_{v}$ is venous resistance

Over any period of time, especially periods greater than the interval between heartbeats, hemodynamics is controlled by the factors that control $F$, $P_{a}$, $P_{vb}$, $R$, $R_{v}$ and right atrial pressure. The cardiac output is equal to the summation of the instantaneous blood flow over a second.

The most important feature of the operation of the heart from the viewpoint of haemodynamics is that it nearly always pumps any blood that enters through the right atrium through the lungs and out of the left ventricle. This simplifies the analysis because the cardiac output is always equal to the venous return.

The venous return depends upon the resistance to flow and pressure difference between the venous vascular beds, including the splanchnic vena cava, and the right atrium:

$F_{venousreturn}={\frac {P_{vb}-P_{ra}}{R_{v}}}$

Physically the circulation is like a pump connected to a slightly elastic pipe (arteries) that joins to a fluid resistance (vascular bed) that in turn links to a set of rather floppy, elastic bags (veins, liver etc.). The veins contain much of the blood so even a small slackening of the tone in the venous walls can lead to pooling of the blood in the veins outside the thoracic cavity. When this occurs, such as when standing suddenly on a hot day, the patient may faint (syncope). In practice the circulation is intermediate between an open and a closed fluid system, with many features of the open system.

In the short term venous return, and hence cardiac output, is maintained by a continuous modulation of venous tone. Blood pressure is maintained by adjusting both the venous return and the total peripheral resistance. The total peripheral resistance is regulated largely through contracting the walls of arterioles.

In the longer term the volume of blood is probably the most important regulator of haemodynamic variables because this sets the pressures in the great veins and hence venous return. Pressures in the circulatory system can directly influence blood volume. Low arterial pressure causes the release of renin in the kidneys and the renin-angiotensin-aldosterone system causes an increase in sodium retention and hence blood volume. Atrial naturetic peptide is a hormone released as a result of stretching the right atrium and this causes a decrease in blood volume through sodium excretion. The control systems for blood pressure and cardiac output are complex and finely tuned so factors such as baroreflexes, lymphatic return, heart failure, elevated diuresis etc. can entirely mask blood volume changes over time.