Principles of Diffusion and Flow
Materials are carried around the body by a combination of bulk flow and diffusion. Bulk flow simply means transport with the carrying medium (blood, air). Passive diffusion refers to movement down a concentration gradient, and accounts for transport across small distances, e.g. within the cytosol and across membranes. The rate of diffusion in a solution is described by Fick’s law:
Js = −DA(∆C/∆x) (11.1)
where Js is the amount of substance transferred per unit time, ΔC is the difference in concentration, Δx is the diffusion distance and A is the surface area over which diffusion occurs. The negative sign reflects movement down the concentration gradient. D is the diffusion coefficient, a measure of how easy it is for the substance to diffuse. D is related to temperature, solvent viscosity and the size of the molecule, and is normally inversely proportional to the cube root of the molecular weight. Diffusion across a membrane is affected by the permeability of the membrane. The permeability (p) is related to the membrane thickness and composition, and the diffusion coefficient of the substance. Fick’s equation can be rewritten as:
Js = −pA∆C (11.2)
where A is the membrane area and ΔC is the concentration difference across the membrane. The rate of diffusion across a capillary wall is therefore related to the concentration difference across the wall and the permeability of the wall to that substance (Fig. 11a).
Flow through a tube
Flow through a tube is dependent on the pressure difference across the ends of the tube (P1 − P2) and the resistance to flow provided by the tube (R):
Flow = (P1 − P2 )/R (11.3)
This is Darcy’s law (analogous to Ohm’s law in electronics; Fig. 11b). Resistance is due to frictional forces, and is determined by the diameter of the tube and the viscosity of the fluid:
R = (8VL)/(πr4 ) (11.4)
This is Poiseuille’s law, where V is the viscosity, L is the length of the tube and r is the radius of the tube. Combining eqns. (11.3) and (11.4) shows an important principle, namely that flow ∝ (radius)4:
Flow = [(P1 − P2 )πr4 ]/(8VL) (11.5)
Therefore, small changes in radius have a large effect on flow (Fig. 11c). Thus, the constriction of an artery by 20% will decrease the flow by ∼60%.
Viscosity. Treacle flows more slowly than water because it has a higher viscosity. Plasma has a similar viscosity to water, but blood contains cells (mostly erythrocytes) which effectively increase the viscosity by three- to four-fold. Changes in cell number, e.g. polycythaemia (increased erythrocytes), therefore affect the blood flow.
Laminar and turbulent flow. Frictional forces at the sides of a tube cause drag on the fluid touching them. This creates a velocity gradient (Fig. 11d) in which the flow is greatest at the centre. This is termed laminar flow, and describes the flow in the majority of cardiovascular and respiratory systems at rest. A consequence of the velocity gradient is that blood cells tend to move away from the sides of the vessel and accumulate towards the centre (axial streaming; Fig.11e); they also tend to align themselves to the flow. In small vessels, this effectively reduces the blood viscosity and minimizes the resistance (the Fåhraeus–Lindqvist effect).
At high velocities, especially in large arteries and airways, and at the edges or branches where the velocity increases sharply, flow may become turbulent, and laminar flow is disrupted (Fig. 11f). This significantly increases the resistance. The narrowing of airways and large arteries (or valve orifices), which increases the fluid velocity, can therefore cause turbulence, which is heard as lung sounds (e.g. wheezing in asthma) and cardiac murmurs (Chapter 18).
Turbulent flow is also responsible for the sounds heard when measuring blood pressure using a sphygmomanometer and stethoscope (Korotkoff sounds). A rubber cuff round the arm is inflated to a pressure well above predicted arterial pressure and then slowly deflated. When the pressure in the cuff approaches systolic pressure, the blood is able to force its way through the constricted artery in the arm for part of the pulse. The high velocity of the blood through the narrowed artery causes turbulence and therefore a sound; the first appearance of this is taken as systolic pressure. As the pressure in the cuff falls further and so below diastolic pressure, flow is continuous because the pressure is greater than that in the cuff throughout the pulse. As a result the sound fades and disappears, and the cuff pressure at this point is taken as diastolic pressure.
Resistances in parallel and in series. The cardiovascular and respiratory systems contain a mixture of series (e.g. arteries ⇒ arteri- oles ⇒ capillaries ⇒ venules ⇒ veins) and parallel (e.g. lots of capillaries) components (Fig. 11g). Flow through a series of tubes is restricted by the resistance of each tube in turn, and the total resistance is the sum of the resistances:
RT = R1 + R2 + R3 +… (11.6)
In a parallel circuit, the addition of extra paths reduces the total resistance, and so:
RT = 1/(1/Ra + 1/Rb …) (11.7)
Although the resistance of individual capillaries or terminal bronchioles is high (small radius, Poiseuille’s law), the huge number of them in parallel means that their contribution to the total resistance of the cardiovascular and respiratory systems is comparatively small.
Wall tension and pressure in spherical or cylindrical containers
Pressure across the wall of a flexible tube (transmural pressure) tends to extend it, and increases wall tension. This can be described by Laplace’s law: Pt = (Tw)/r (11.8) where Pt is the transmural pressure, T is the wall tension, w is the wall thickness and r is the radius (Fig. 11h). Thus, a small bubble with the same wall tension as a larger bubble will contain a greater pressure, and will collapse into the larger bubble if they are joined. In the lung, small alveoli would collapse into larger ones were it not for surfactant which reduces the surface tension more strongly as the size of the alveolus decreases (Chapter 26). Laplace’s law also means that a large dilated heart (e.g. heart failure) has to develop more wall tension (contractile force) in order to obtain the same ventricular pressure, making it less efficient.