II.  BLOOD FLOW

Blood is an incompressible fluid for which the principle of conservation of mass implies the conservation of volume. FIG II.1 shows a flow tube with varying cross section where A is that area, v is the linear velocity of the fluid of density D, and v·)t = )s is the distance traveled by the fluid in time )t. Then, A·v·)t=)V is the volume crossing any cross section of the tube in time )t. Consequently,  DA₁v₁)t is the mass of fluid crossing surface A₁ in )t, and, since mass is conserved,

DA₁v₁)t = DA₂v₂)t,

A₁v₁ = A₂v₂.,  and    )V₁ = )V₂.

A.  GENERATION OF POTENTIAL ENERGY

FIG II.2 shows a pump that pumps a liquid from a tank into a cylinder pushing the piston against a force F. This device may be used as a model that represents some of the properties of the circulatory system. The pump is an external agent doing work against F, while F does the work w = - F· )l. The negative sign results from the opposite directions of the force and the displacement. This negative work is done on the cylinder-piston system, and represents the energy gained by that system as potential energy. So,  )U = - F· )l. and )U /)l = - F. But pressure is force per unit area, and   F = P·A, where A is the area of the piston. Then,

                )U = - P·A·)l = - P·)V,                     EQ. II.

which shows that the product P·V is work, has the units of joules and may represent the change in potential energy of a system. When the piston is at rest, the pressure exerted by the piston on the liquid is equal, but of opposite direction, to the pressure exerted by the liquid on the piston. By using the latter, the negative sign in the equation disappears. EQUATION II.1.,  then, becomes

)U = P · )V         EQUATION   II. 1'

FIG II.3 is similar to FIG II.2, except that the piston is not exposed to a constant force, but to a variable one proportional to the compression of the supporting coiled spring. The fluid compartment inside the cylinder represents the arterial compartment of the circulatory system and the spring represents the elastic components of the arterial walls. Under these conditions,

)U = )(PV) = P)V + V)P

 where P is the pressure exerted by the liquid on the piston (as well as on all the surfaces that bound it inside the cylinder). Then, using the notation of calculus,

.              
EQUATION  II.2

The force exerted by the piston on the fluid is    F = - k · l, where k is the elastic coefficient of the spring. Therefore, at equilibrium the pressure of the liquid is P = (k · l) / A from which   

 .

Substitution in EQ. II.2 yields 

                .

Integration from V₀ to V will yield an expression for the change in potential energy due to that change in volume.

Notice that, since   dP / dV = k / A² and compliance is defined by   C = dV /dP,   k / A² = 1 / C and     )U = )V² / C.

Instead of compliance, we could use distensibility, defined by  D = dV / VdP = A² / kV.   In that case,

)U = (kV / A²) · V  =   )V / D.

With these substitutions we eliminate from the equation all variables related to properties of components of the model and replace them with variables related to properties of the circulatory system itself, as distensibility and compliance, which are measurable properties of blood vessels.

Further development of the model will allow us to represent additional properties of the system. FIG.  4 shows that the blood reservoir that was previously open at the top is now closed and with a piston and spring that represents the elasticity of the walls of the veins. The difference in distensibility between arteries and veins is indicated by the tighter spring in the arterial compartment. The higher distensibility of the veins allows them to have a significantly larger volume at a much lower pressure. The connection between arteries and veins represents small arteries, arterioles and capillaries that constitute the resistive path in which most of the potential energy is dissipated by friction. The arrows labeled SYMP are used to indicate the reduction of the distensibility of both arteries and veins and the constriction of the diameter of small arteries and arterioles that result from increased sympathetic activity. The representation of the smaller vessels by a single narrow tube in the drawing may be misleading because it may suggest that the linear velocity of flow is high in these vessels when, in fact, it is quite slow. This region of the circulation consists of a very large number of thin tubes connected in parallel. Their collective cross section is much greater than, for example, the cross section of the aorta and, since the flow is the same in both, the linear velocity in the aorta must be much higher. Still, the collective resistance to flow of the small vessels is much higher, as will be seen later.unit mass and the gravitational field E as the force per unit mass. In a similar manner, we may define a potential and a field corresponding to the potential energy in a to a P-V system. Accordingly, we can write the following equations

The flow from arteries to veins is spontaneous and can be described by a flow equation similar to EQ. I.4.(see sections I.B.b and I.B.c):

Flow = - k (d A/dx),

in which - (d A/dx) is minus the gradient of a potential with respect to a spatial dimension along which the potential varies continuously.

In I.B.b. Potential and Field, the gravitational potential A was defined as the potential energy per kg.

  U_G = F · d   =    A_G · m   =   E_G · m · d     and    A_G = E_G · d        EQUATIONS   II.2

  U_P-V = P·V =  A_P-V · V =  E_P-V · V · d    and     A_P-V = P = E_P-V · d

where the units are

A_G :    Joules · Kg^-1   or    N · Kg^-1 · m                 E_G :     N · Kg^-1

A_P-V : Joules · m^-3     or    N · m^-3 · m                   E_P-V :  N · m^-3  or  N · m^-2 · m^-1

            or N · m^-2    or    Pa (Pascals)                               or Pa · m^-1

B.  THE CARDIAC PUMP

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