**INTRODUCTION**

The desire to quantitatively understand both the microscopic and macroscopic behaviour of blood when it flows through the vessels has supplied the motivation for appropriate investigations. Young summarized the essence of this desire as understanding the physical events that take place in normal animal and thereby make contribution to physiology. A secondary aim is to make contribution to medicine by analyzing particular abnormal or diseased states in the hope of improving diagnosis or treatment (Pedley, 1980) .

Within the blood vessels are red blood cells, which contain a protein called haemoglobin whose function is to carry oxygen from the lungs to all parts of the body. Most people have normal haemoglobin or HbA; their blood cells are doughnut shaped and flexible, allowing them to pass easily through even the smallest vessels. Those with sickle cell anaemia have a different type of haemoglobin, HbS. Their blood cells are sickle shaped and rigid and therefore cannot pass through small blood vessels. This results in blockage and a lack of oxygen reaching parts of the body causing pain or damage to vital organs.

There are three primary factors that determine resistance to blood flow within a single vessel: diameter R (radius), length of vessel L and viscosity of blood η (Klahunde, 2005). The relationship is given by:

The relationship between flow F, the Pressure gradient ΔP and Resistance R is given by the well-known Poiseuille equation as

This implies that flow is inversely proportional to blood viscosity. The viscosity (thickness) of blood or any liquid, for that matter, is controlled by both the chemical components and temperature (Bedham, 2004). Temperature induced changes are attributable to the way molecules in the liquid react to a gain or loss of thermal energy. Just as the Poiseuille equation suggests viscous liquids resist internal flow; viscous blood would resist flow through blood vessels and would consequently not be as effective at distributing heat to either areas of the body that are in need of it or away from places that have excess of it.

It is well known that the viscosity of a sickle cell patient is greater than that of a normal patient. Mazzoni *et al*. (2002) found that as blood viscosity decreases, cardiac output increases. This implies also that increase in blood viscosity decreases cardiac output. Chien’s measurement (Chien, 1970) of effective viscosity as a function of particle shape in dilute suspension found that thin discs have higher viscosity. Chien’s results also suggest that continuous surface deformations in response to flow conditions may further reduce their contribution to blood viscosity. This implies that in a sickle cell patient where the cell are sickle and resist deformation the viscosity will be higher than normal. In this study we investigate the effect of increase viscosity on small pressure disturbance and the arterial compliance. We also determine the effect of this increase on the time of shock formation i.e., the onset of sickle cell anaemia patient crisis.

**MATHEMATICAL FORMULATION**

Due to the large wavelength of the arterial pulse waves compared to the vessel diameters, one-dimensional models permits an efficient simulation of the wave propagation in the large arteries which allows us to study the effects of local changes on the global system. The governing equations of the one dimensional model are obtained by applying conservation of mass and momentum to a one dimensional impermeable tubular control volume of inviscid, incompressible and Newtonian fluid and by considering a tube law that relates changes in pressure to changes in cross-sectional area. The continuity equation is one of the equations governing the fluid dynamics of the blood and states the conservation of matter i.e., what goes into a section must either come out or be stored. Under the assumption of one dimensionality the continuity equation is

where A denotes the cross sectional area of the artery, u the axial blood velocity (average over the cross-section), x the axial coordinate and t is the time. The first term is the rate of storage of blood within the arterial segment; the second is the difference between the blood flowing into and out of the segment in the axial direction.

The second fluid dynamics relation is the momentum equation, which balances inertia, friction and pressure forces acting on the blood. In one dimension it is given as

where P is the pressure difference across the tube and p is the fluid density. The first term is the acceleration of the blood; the second is the convective acceleration due to axial rate of change of velocity, the third the pressure gradient and the fourth term represents the friction at walls of the arteries. It is assumed that the lumen area is related instantaneously to the pressure (Vander Werff, 1974) and there is no phase lag; thus the one dimension state equation is given as

Although the shear stress in pulsating flows is not directly related to the instantaneous mean velocity, it is assumed to be given by the laminar Poiseuille relation

where μ is the blood viscosity.

Equation 4 then becomes

**METHOD OF SOLUTION**

If we assume that the wave amplitude is smaller than the wavelength, then the cross-sectional area can be assumed to be constant along the elastic vessel.

and Eq. 3 reduces to

A special example of a pressure area relationship which has been empirically formulated to model the pulmonary artery is

where R_{0} is the vessel radius at zero pressure and α is the compliance constant of the vessel wall.

We can thus use Eq. 7, 9 and 10
to determine the speed of propagation of a small disturbance in an elastic vessel.
This may allow us quantify the compliance and the effect of increase in viscosity
on it. If we substitute Eq. 10 into 9 we
have

If the initial fluid velocity is zero, then again because the wave amplitude is small the convective acceleration term in Eq. 7 can be neglected. Thus we have

Differentiating Eq. 11 with respect to t and Eq.
12 with respect to x, subtracting the resulting equation and neglecting
the second order terms we obtain

Equation 12 is the wave equation where the quantity c is
the wave speed and is given as

We can the determine the vessel compliance α as

Therefore we can determine the vessel compliance if we know the other measurable parameters.

Recall Eq. 3, 5 and 7

Let us introduce the wave speed in the form

Note that this form is conforms with Eq. 14

Using (5) and (16) Eq. 3 becomes

if we consider a uniform arterial segment as in Eq. 8, we have

Now let us consider the flow as a perturbation of the undisturbed state u = 0, P = P_{0}= constant, A = A_{0} = constant (Hoogstraten and Smit, 1978; Akinrelere and Ayeni, 1983; Ayeni and Akinrelere, 1984). We also assume that

for x≥0 i.e., the tube wall react instantaneously on pressure change in the blood. Thus if we use the subscript 0 to denote the initial value then we have

Let

Let also the wave expansion for P, u, T be of the form.

Thus P_{1}, u_{1} are a measure of the jump in the normal derivatives
of P, u, respectively. We now transform Eq. (7) and (18)
to ordinary differential equations by noting that

This leads us to have

Using Eq. (24) to eliminate u_{1} from Eq.
(25) and (26) we have

If we let

then equation can be written in the form

We consider first the case when μ = 0 i.e., b = 0 and we have

This gives us

P_{1} becomes infinite when

We now consider when the viscosity μ ≠ 0 by solving the equation

This gives us

We plot below the graph of P_{1}(t) against t for various values of
μ

P_{1}(t) and invariably v_{1}(t) become infinite at time

This is the time when shock occurs.

**DISCUSSION**

The circulation time of blood is given by the ratio of volume to flow. A minimum
volume will result in a reduced transformation time of information through the
circulation delivered by hormones, oxygen carbon dioxide and other blood solute.
Chien (1970) has shown that continuous surface deformation in response to flow
conditions reduces their contribution to blood viscosity. For a sickle cell
anaemia patient the sickle nature which causes rigidity prevents this thus making
the blood density higher than would have been reduced. From Eq.
(14) we see that the wave speed is inversely proportional to the to the
density of the fluid. Therefore as the density and invariably the viscosity
increases, the wave-speed decreases. The wall compliance is also inversely proportional
to the density and as it increases the compliance decreases and the vessel response
is decreased. Therefore there is a blockage and a lack of oxygen reaching parts
of the body causing pain or damage to vital organs.

The occurrence of shock is synonymous with the breakdown of flow or what is
termed crisis in the patient. Equation (35) clearly shows
when this is likely to occur and we can also see that as μ increases t
decreases i.e., the existence of shock becomes prominent as the viscosity increases.