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 ⇒ arterioles ⇒ 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.
