ABC of Vascular Disease
Advanced Arterial Haemodynamics
In order to understand fully the way in which blood flows in normal and diseased arteries it is necessary to represent the relationships between blood flow (Q) and blood pressure (P) on a more formal way.
Flow in a smooth tube
For a normal artery, which is a smooth tube, the relationship between flow and pressure is given by Poiseuille's Law
dP = k1Q
where dP is the difference in pressure between the ends of the tube (mmHg)
Q is the steady or average flow in the tube (ml/min)
k1 is a constant that is related to the dimensions of the tube and the viscosity of the fluid.
Flow in a partially occluded tube
In an artery that has a narrowing this relationship between pressure and flow changes because the flow is disrupted and becomes disordered and even turbulent. The relationship between flow and pressure is
dP = k1Q + k2 Q2
where k2 is a different constant that is related to the geometry of the stenosis and the density of the fluid. This equation is called non-linear because of the addition of the second term which is related to the square of the flow [1,2,3].
Flow in a network of normal arteries
The lower limb arterial tree consists of a branching network of major and collateral arteries that ultimately supply blood to the major muscle groups in the buttock, thigh and calf; as well as the foot, skin, bones and other structures. The majority of blood flow at rest is to the foot and skin; but during exercise the blood flow to the muscles increases dramatically; up to 20 times the resting flow!
The control of blood flow the tissues consists of two components
 | The heart and its associated reflex systems work to maintain a constant
average pressure in the aortic arch. This is achieved by adjusting
the rate and force of contraction of the heart through intrinsic and
extrinsic mechanisms. |
| The tissues autoregulate their own blood flow according to their immediate metabolic (and other) needs by adjusting the calibre of the small arteries: vasodilation to increase blood flow and vasoconstriction to reduce blood flow. |
The term "ischaemia" means inadequate blood flow and if a tissue cannot autoregulate its blood supply to provided enough blood it becomes ischaemic and the cardinal symptom of this is pain.
During exercise the muscles require more blood. The muscles vasodilate which increases the blood flow and would tend to reduce the blood pressure were it not for the fact that the heart compensates by increasing its rate of contraction. There is usually very little change in the arterial blood pressure despite an increase in blood flow of several fold.
To simulate this complex behaviour in a network of arteries requires a mathematical model which is held and manipulated using a computer program. This type of computer simulation is widely used in engineering design and allows the user to explore the complex relationships between the factors, interactively but without getting bogged down in the complex mathematics.
References
- Dodds SR, Bourne NK, Chant ADB. The effect of flow on the resistance of
modelled femoral artery stenoses. Br J Surg 1996; 83: 957-961.
- Dodds SR. The haemodynamics of asymmetric stenoses. Eur J Vasc Endovasc
Surg 2002; 24: 332 - 337.
- Dodds SR, Phillips PS. The haemodynamics of multiple sequential stenoses and the criteria for a critical stenosis. Eur J Vasc Endovasc Surg 2003; 26: 348-353.
© S.R.Dodds 2003