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Blood flow is the continuous circulation of blood in the cardiovascular system. This process ensures the transportation of nutrients, hormones, waste, O2 and CO2 to different body parts to maintain cell-level metabolism, the regulation of the pH, osmotic pressure and temperature for the whole body, and the protection from microbial and mechanical harms.

The science dedicated to describe the physics of blood flow is called hemodynamics. For basic understanding it is important to be familiar with anatomy of the cardiovascular system and hydrodynamics. However it is crucial to mention that blood is not a Newtonian fluid, and blood vessels are not rigid tubes, so classic hydrodynamics is not capable to explain hemodynamics.

Blood is not a fluid
Blood is a suspension. Blood plasma is the straw-colored/pale-yellow liquid component of blood that normally holds the blood cells in suspension. Blood is not a fluid, thus it cannot be a Non Newtonian fluid either. In a blood flow, the deformable red cells migrate in the direction of the highest velocity values by inertial effects. These deformable red cells are also able to modify the blood flow organisation. Blood is not homogeneous, however, the shear stress, rates and viscosity values are never homogeneous in a gradient flow. A viscometer is able to measure the viscosity of blood flow. Nevertheless, a viscometer, pipes and vessels all have different characteristics, which causes the results of viscosity diagrams to not be transposable to body circulation. This is due to the fact that the nature of blood flow depends on geometry, size and velocity. Furthermore, blood is a polyphasic material that can flow, harden, aggregate, and change organisation (Fåhræus–Lindqvist effect).

Blood and its composition
Blood is a viscous fluid composed of plasma and formed elements. The plasma contains 91.5% water, 7% proteins and 1.5% other solutes. The formed elements are platelets, white blood cells and red blood cells. The presence of these formed elements and their interaction with plasma molecules are the main reasons why blood differs so much from ideal Newtonian fluids.

The mechanics of blood circulation
Mechanics is the study of motion (or equilibrium) and the forces that causes it. The blood moves in the blood vessels, while the heart serves as the pump for the blood. The vessel walls of the heart are elastic and are moveable, therefore causing the blood and the wall to exert forces on each other, which in turn influence their respective motion. Therefore, to understand the mechanics of circulation of the heart, it will be worth the while to go through a review of basic mechanics of fluid, and elastic solids (momentum) and the nature of the forces exerted between two moving substances in contact.

The basics of motion
The study of motion is born from the argument that there is no change in motion without force. These beliefs were somewhat obscured until the seventeenth century when Isaac Newton formulated his three laws of motion.

Velocity
Another quantity of interest in describing the motion of a particle is its velocity. The velocity is the rate of change of the position of an object with time.
 * $$v = \frac{\Delta x}{\Delta t}$$

The velocity of blood flow is often expressed in cm/s, this value is inversely related to the total cross-sectional area of the blood vessels, and also differs per cross-sections, because in normal condition the blood flow has laminar characteristics. Due to this fact, the blood flow velocity is the fastest in the middle of the vessel and the slowest at the vessel wall. In most cases the mean velocity is in use. There are many ways to measure blood flow velocity, such as video capillary microscoping with frame to frame analysis or laser Doppler anemometry. Blood velocities in arteries are higher during systole than during diastole. One parameter to quantify this difference is pulsatility index (PI), which is equal to the difference between the peak systolic velocity and the minimum diastolic velocity divided by the mean velocity during the cardiac cycle. This value decreases with distance from the heart.


 * $$PI = \frac{v_{systole} - v_{diastole}}{v_{mean}}$$

Acceleration
The rate of change of position, as we saw, is the velocity. The rate of change of velocity is referred to as the acceleration. For a motion along a line, the acceleration is given as:
 * $$a = \frac{\Delta v}{\Delta t}$$

This is the same as the slope of the tangent of the graph v against t. The unit of acceleration is meter per seconds squared (m/s2).

Newton's Laws of Motion
Newton’s first law: every particle continues in a state of rest of uniform motion in a straight line unless acted on by some external force or forces. In other words the velocity remains constant (zero acceleration) if no force is acting on the body.

Newton’s second law: when a particle of mass m is acted on by a force so that it experiences an acceleration, a, the net force acting on it is equal to the mass multiplied by the acceleration. That is to say the net force is a vector which we can call F. The term net force means the sum of all forces acting on the particle, which may be exerted in different ways. The equation that is formed is F= ma and is an important equation in mechanics. It is called the equation of motion of a particle.

Newton’s third law: Newton’s third law has commonly been described as action and reaction. Basically the law is defined as to every action there is equal and opposite reaction. That is to say, if one body exerts a force, F, on another body, the second body must also exert an equal and opposite reaction on the formal body (-F). The negative on the force by no means suggests that force can be negative, the negative only means that the force is acting in the opposite direction, it has nothing to do with the magnitude it is simply the direction. As an illustration, if you press a stone to your finger, unknown to you, your finger exerts a force on the stone as well, which is equal in magnitude.

Basic ideas In fluid mechanics
The force experienced by fluids can be long range and short range. The long range force includes gravitational and electromagnetic forces. The electromagnetic force on an element depends on quantities such as its electric charge, however, the gravitational force depends only on its mass. We will consider from this point the gravitational force alone. If we have a fluid with element p, which occupies the point x at a certain time, t, and has a volume, v, and if the fluid in the neighbourhood of x at that time has a density, ϼ, then the gravitational force on the element is given as:
 * ϼ v g

Stress
When we apply force to a material it begins to deform or move. The force that is needed to deform a material (let a fluid flow) increases with the size of the surface of the material A. The magnitude of this force, F, is proportional to the area, A, of the portion of the surface. Therefore the quantity (F/A) that is the force per unit area is called the stress. The shear stress that is associated with blood flow through an artery is around 8-12 dynes/cm^2
 * $$\sigma = \frac{F}{A}$$.



Hydrostatic pressure
In a stationary fluid, the only stress present is the pressure. If we assume a body force that is vertical, we can then assume that the horizontal component of the pressure force must balance out. Pressure varies with height,so, if we increase or decrease height, the pressure changes. If A is the horizontal cross-sectional area of the element and if $$p_{1}$$ and $$p_{2}$$ are the pressure on the surface, where $$p_{1}$$ is the pressure on the bottom and $$p_{2}$$ is the pressure on the top, then the net upward force on the element can be given as ($$p_{1}$$ – $$p_{2}$$ )A This must be equal to the weight which acts downwards and is equal to the density of the fluid times its volume ( AZ’) where Z’ is the depth of the element times g.

($$p_{1}$$ – $$p_{2}$$ )A = gϼAZ’

($$p_{1}$$ – $$p_{2}$$ ) = gϼZ’

If we assume that the pressure is atmospheric and Z=0 (that is to say Z is negative in the fluid). p is given as: $$p_{1}$$ = $$p_{2}$$ - gϼZ

The above equation is called the hydrostatic pressure formula.

Viscosity
The magnitude of the viscous stresses depends upon the rate of deformation. For example, when a body is moved rapidly through a fluid it causes more rapid deformation of fluid element than one moving slowly. Viscosity is giving as:
 * $$\tau=\mu \frac{\partial u}{\partial z}$$

where mu is a constant called the coefficient of viscosity

Conservation of mass
Before discussing the circulation of blood there are two principles that must be addressed. The first is the conservation of mass, which states that mass can neither be created nor destroyed. The implication of this is that the mass of the fluid flowing into a closed system must be equivalent to the mass of the fluid flowing out. In vivo, however, is not strictly the case because the density of blood can be variable (so that a greater mass occupies the same volume), or the volume of the system of blood vessels increases through the expansion of flexible blood vessel walls. The second principle is that blood is incompressible. Due to these two principles, it can be roughly assumed that the blood flowing into a tube (circulatory system) has the same properties (mass, volume, etc.) as the blood that flows out. Therefore, based on the principles of fluid dynamics, in a tube of cross sectional area, $$A$$, and an average fluid velocity, $$u$$, the volume flow rate is given by the equation:
 * $$ Q = u  \cdot A$$

This is related to the mass flow rate as
 * $$ M = p  \cdot Q$$

where


 * $$M$$ is the mass flow rate,
 * $$p$$ is the density,
 * $$Q$$ is the volumetric flow rate.

Bernoulli’s equation
The second general principle involves the conservation of energy. Recall the laws of thermodynamics, it states that the rate of change of the total energy of a system is equal to the rate at which applied forces do work on it. When there are no applied forces, the total energy is constant i.e. energy is conserved. It is now possible to introduce the idea of a streamline. Consider a steady flow in which its boundary consists of a streamline. If we assume that the fluid is constrained to remain on the streamline and if we assume the two ends of a finite length of the tube are at different levels and the fluid is flowing up from an initial velocity $$ u_1 $$ and a cross sectional area $$ A_1 $$ to a level $$ Z_2 $$ with velocity $$ u^2 $$ and area $$ A_2 $$ therefore we can have the Bernoulli’s equation given as $$ p_1 +0.5 \rho (u_1 )^2 + \rho g Z_1 = p_2 +0.5 \rho (u_2)^2 + \rho g Z_2 $$

where


 * $$p$$ is the pressure,
 * $$u$$ is the velocity,
 * $$Z$$ is the height.

The mechanics of blood
The flow measurement, as well as the mechanics of blood itself is a very broad topic, we will only give a brief overview of what is contained therein. The mechanical property of blood which is of interest to us is the viscosity.

Viscosity of plasma
Normal plasma behaves like a Newtonian fluid at rates of shear stress. Typical values for the viscosity of normal human plasma at 37 °C is 1.2 N·s/m2. The viscosity of normal plasma varies with temperature in the same way that its solvent water does. A 5 °C increase of temperature in the physiological range reduces plasma viscosity by about 10%.

Osmotic pressure of plasma
The osmotic pressure of a solution is determined by the number of particles present by the temperature. For example a 1 molar solution of a substance contains $6.022$ molecules per liter of that substance and at 0 °C has an osmotic pressure of 2.27 MPa. The osmotic pressure of the plasma affects the mechanics of the circulation in several ways. An alteration of the osmotic pressure difference across the membrane of a blood cell will cause a shift of water and a change of the cell volume. The change both in shape and flexibility will affect the mechanical properties of whole blood. A change, therefore, in plasma osmotic pressure will alter the hematocrit, which is the volume concentration of red cells in the whole blood, by redistributing water between the intravascular and extravascular spaces. This in turn will affect the mechanics of the whole blood.

The red cells
The red cell is a highly flexible bi-concave disc. The red cell membrane has a Young's modulus in the region of 106 Pa. The deformation in the red cells is induced by the shear stress. When a suspension is sheared, the red cells are seen to deform and spin, because of the velocity gradient However, the rate of deformation and spin depends on the shear-rate and the concentration. This topic can influence the mechanics of the circulation and may complicate the measurement of the blood viscosity. In a steady-state flow of a viscous fluid through a rigid spherical body that is immersed in the fluid, we assume that the inertia is negligible. In such a flow, it is believed that the downward gravitational force of the particle is balanced by the viscous drag force. From this force balance the speed of fall can be shown to be given by Stokes' law:
 * $$U_s = \frac{2}{9}\frac{\left(\rho_p - \rho_f\right)}{\mu} g\, a^2$$

Where

From the above equation we can see that the sedimentation velocity of the particle depends on the square of the radius. If the particle is released from rest in the fluid, its sedimentation velocity $$U_s$$ increases until it attains the steady value called terminal velocity ($$U$$) as shown above.
 * $$a$$ is the particle radius,
 * ρ_p and ρ_f are the respective particle and fluid densities,
 * μ is the fluid viscosity,
 * $$g$$ is the gravitational acceleration.

History of the use of blood
We have looked at blood flow, and blood composition. Before we look at the main issue, hemodilution, let us take a brief history into the use of blood. The therapeutic use of blood is not a modern phenomenon. Egyptian writings that date back to at least 2000 years, suggest oral ingestion of blood as a ‘sovereign remedy’ for leprosy. Experiments with the first intravenous blood transfusions began at the start of the 16th century. In the last 50 years, the field of transfusion medicine has progressed remarkably, bringing with it an increase in the use of blood and blood product. However, the therapeutic use of blood comes with significant risks. As a result, many persons are searching for alternatives to transfusion of whole blood. Today, bloodless medicine and surgery (BMS) programs have been developed, not only for religious beliefs, but they are also sought after by patients who fear the risks of blood transfusion and desire the best medical care.

Hemodilution
Hemodilution is the dilution of the concentration of red blood cells and plasma constituents by partially substituting the blood with colloids or crystalloids. It is a strategy which is used to avoid exposing patients to the hazards of homologous blood transfusions.

Hemodilution can be normovolemia which, implies the dilution of normal blood constituents by the use of expanders. During acute normovolemic hemodilution (ANH), blood lost during surgery contains proportionally fewer red blood cells per millimetre, thus minimizing intraoperative loss of the whole blood. Therefore, blood lost by the patient during surgery is not actually lost by the patient, since this volume is purified and redirected into the patient.

There is, however, the hypervolemic hemodilution (HVH). Here, instead of simultaneously exchanging the patient’s blood as in ANH, hypervolemic technique is carried out by the use of acute preoperative volume expansion without any blood removal. In choosing a fluid, however, we must be sure that when mixed, the remaining blood behaves in the microcirculation as the original blood fluid, thereby retaining all its properties of viscosity.

In presenting what volume of ANH should be applied, one study suggests a mathematical model of ANH, which calculates the maximum possible RCM savings using ANH, given the patients weight $$H_i$$ and $$H_m$$. Attached to this document is a glossary of the term used.

To maintain the normovolemia, the withdrawal of autologous blood must be simultaneously replaced by a suitable hemodilute. Ideally, this is achieved by isovolemia exchange transfusion of a plasma substitute with a colloid osmotic pressure (OP). A colloid is a fluid containing particles that are large enough to exert an oncotic pressure across the micro vascular membrane. When debating the use of colloid or crystalloid, it is imperative to think about all the components of the starling equation:
 * $$\ Q = K ( [P_c - P_i]S - [P_c - P_i] )$$

To identify the minimum safe hematocrit desirable for a given patient, the following equation is useful:
 * $$\ BL_s = EBV \ln \frac{H_i}{H_m} $$

Where $$EBV$$ is the estimated blood volume (70 mL/kg) was used in this model and Hi (initial hematocrit) is the patient’s initial hematocrit. From the equation above, it is clear that the volume of blood removed during the ANH to the $$H_m$$ is the same as the $$BL_s$$. The collection of blood needed to be removed is usually based on the weight, not the volume. The number of units that needed to be removed to hemodilute to the maximum safe hematocrite (ANH) can be found by:
 * $$ANH = \frac {BL_s}{450}$$

This is based on the assumption that each unit removed by hemodilution has a volume of 450 mL (the actual volume of a unit will vary somewhat since completion of collection is dependent on weight and not volume). The model assumes the hemodilute value to the Hm prior to surgery, therefore, the re-transfusion of blood obtained by hemodilution must begin when SBL begins. The RCM available for retransfusion after ANH (RCMm) can be calculated from the patient's $$H_i$$ and the final hematocrit after hemodilution($$H_m$$):
 * $$ RCM = EVB \times (H_i - H_m) $$

The maximum SBL that is possible when ANH is used without falling below Hm(BLH) is found by assuming that all the blood removed during ANH is returned to the patient at a rate sufficient to maintaining the hematocrit at the minimum safe level:
 * $$ BL_H = \frac {RCM_H} {H_m}$$

If ANH is used as long as SBL does not exceed $$BL_H$$, there will not be any need for blood transfusion. We can conclude from the foregoing, that $$BL_H$$ should therefore not exceed $$BL_s$$. The difference between the $$BL_H$$ and the $$BL_s$$ is the incremental surgical blood loss ($$BL_i$$) possible when using ANH.
 * $$\ {BL_i} = {BL_H} - {BL_s} $$

When expressed in terms of the RCM
 * $$ {RCM_i} = {BL_i} \times {H_m} $$

Where $$RCM_i$$ is the red cell mass that would have to be administered using homologous blood to maintain the $$H_m$$ if ANH is not used, and blood loss equals $$BL_H$$.

The model used assumes ANH used for a 70 kg patient with an estimated blood volume of 70 mL/kg (4900 mL). A range of $$H_i$$ and $$H_m$$ was evaluated to understand conditions where hemodilution is necessary to benefit the patient.

Result
The result of the model calculations are presented in a table given in the appendix for a range of $$H_i$$ from 0.30 to 0.50 with ANH performed to minimum hematocrits from 0.30 to 0.15. Given a $$H_i$$ of 0.40, if the $$H_m$$ is assumed to be 0.25, then from the equation above, RCM count is still high and ANH is not necessary. This is true if $$BL_s$$ does not exceed 2303 mL, since the hemotocrit will not fall below $$H_m$$. Though 5 units of blood must be removed during hemodilution. Under these conditions, to achieve the maximum benefit from the technique, and if ANH is used, no homologous blood will be required to maintain the $$H_m$$ if blood loss does not exceed 2940 mL. In such a case, ANH can save a maximum of 1.1 packed red blood cell unit equivalents. Thus, homologous blood transfusion will be necessary to maintain $$H_m$$ even if ANH is used. This model can be used to identify when ANH may be used for a given patient and the degree of ANH necessary to maximize that benefit.

For example, if $$H_i$$ is 0.30 or less, it is not possible to save a red cell mass equivalent to 2 units of homologous PRBC even if the patient is hemodiluted to an $$H_m$$ of 0.15. That is because, from the RCM equation, the patients RCM falls short. If $$H_i$$ is 0.40, one must remove at least 7.5 units of blood during ANH, resulting in an $$H_m$$ of 0.20 to save 2 units equivalence. Clearly, the greater the $$H_i$$ and the greater the amount of units removed during hemodilution, the more effective ANH is for preventing homologous blood transfusion. The model here is designed to allow doctors determine where ANH may be beneficial for a patient based on their knowledge of the $$H_i$$, the potential for SBL, and estimate of the $$H_m$$. Though the model used a 70 kg patient, the result can be applied to any patient. To apply these results to any body weight, any of the values $$BL_s$$, $$BL_H$$ and $$ANH_U$$ or $$PRBC$$ given in the table need to be multiplied by the factor we will call $$T$$:
 * $$ T = \frac {70 \times \text{patient's weight in kg}} {4900} $$

The model we have considered above is designed to predict the maximum RCM that can be saved ANH. In summary, the efficacy of ANH has been described mathematically by means of measurements of surgical blood loss and blood volume flow measurement. This form of analysis permits accurate estimation of the potential efficiency of the techniques, and it shows the application of measurement in the medical field.

Table
Table model calculations for hypothetical 70 kg patient.

Glossary of terms
glossary of terms used