Sato–Tate conjecture

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The science dedicated to describe the physics of blood flow is called hemodynamics. For the 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 and its composition.

Blood is 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. 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 movable, 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.

Velocity

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 characteristic. 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 like videocapillary 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.

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

Relation between blood flow velocity and total cross-section area in human

Type of blood vessels	Total cross-section area	Blood velocity in cm/s
Aorta	3–5 cm2	40 cm/s
Capillaries	4500–6000 cm2	0.03 cm/s
Vena cavae inferior and superior	14 cm2	15 cm/s

Stress

When force is applied to a material it starts to deform or move. As the force that is needed to deform a material (e.g. to make 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

$\sigma ={\frac {F}{A}}$.

Laminar shear of fluid between two plates. $v=u,\tau =\sigma$. Friction between the fluid and the moving boundaries causes the fluid to shear (flow). The force required for this action per unit area is the stress. The relation between the stress (force) and the shear rate (flow velocity) determines the viscosity.

Viscosity of plasma

Normal plasma behaves like a Newtonian fluid at rates of shear. 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 just the same way as does that of its solvent water, a 5 °C increase of temperature in the physiological range reduces plasma viscosity by about 10%.

Osmotic pressure of plasma

The osmotic pressure of solution is determined by the number of particles present by the temperature. For example a 1 molar solution of a substance contains Template:Val molecules per liter of that substance and has at 0 °C an osmotic pressure of Template:Convert. 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 that 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.

Red blood cells

The red blood cell is highly flexible and biconcave in shape. 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, but 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. It is true that in a steady state flow of a viscous fluid through a rigid spherical body immersed in the fluid where we assume the inertia is negligible in such a flow it is believed the downward gravitational force of the particle is balanced by the viscous drag force. From this force balance the speed of fall this 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 a is the particle radius, ρ_p, ρ_f are the respectively particle and fluid density μ is the fluid viscosity, g is the gravitational acceleration. 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.

We have looked at blood flow, 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 dates back 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, and 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 its significant risks which are enormous. 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 a group of people because of their religious believes but they are also sought after by patient 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 and it is a strategy to avoid exposure of patients to the hazards of homologous blood transfusions.

Hemodilution can be normovolemia which, as we said, implies the dilution of normal blood constituents by the use of expanders. During acute normovolemic hemodilution (ANH) blood subsequently 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, for 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, 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. Not to worry. 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 Hm is the same as the BLs. 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 as we mentioned is dependent on weight and not volume). The model assumes that 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 Hi 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 BLs therefore 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 BLH.

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 Hi and Hm 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 Hi from 0.30 to 0.50 with ANH performed to minimum hematocrits from 0.30 to 0.15. Given a Hi 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, if BLs 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 Hm 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 equivalent, in such a case homologous blood transfusion will be necessary to maintain Hm 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 Hi is 0.30 or less it is not possible to save a red cell mass equivalent to 2 unit of homologous PRBC even if the patient is hemodiluted to an H_m of 0.15. That is because from the RCM equation the patient RCM falls short from the equation giving above. If Hi is 0.40 one must remove at least 7.5 units of blood during ANH, resulting in an Hm of 0.20 to save 2 units equivalence. Clearly the greater the Hi 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 Hm. Though the model used a 70 kg patient, the result can be applied to any patient. To apply these result to any body weight any of the values BLs, BLH and ANHH or PRBC given in the table need to be multiplied by the factor we will call T

$T={\frac {70\times {\text{patient's weight in kg}}}{4900}}$

Basically 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 goes to show the application of measurement in the medical field.

Glossary of terms

ANH	Acute Normovolemic Hemodilution
ANHu	Number of Units During ANH
BLH	Maximum Blood Loss Possible When ANH Is Used Before Homologous Blood Transfusion Is Needed
BLI	Incremental Blood Loss Possible with ANH.(BLH – BLs)
BLs	Maximum blood loss without ANH before homologous blood transfusion is required
EBV	Estimated Blood Volume(70 mL/kg)
Hct	Haematocrit Always Expressed Here As A Fraction
Hi	Initial Haematocrit
Hm	Minimum Safe Haematocrit
PRBC	Packed Red Blood Cell Equivalent Saved by ANH
RCM	Red cell mass.
RCMH	Cell Mass Available For Transfusion after ANH
RCMI	Red Cell Mass Saved by ANH
SBL	Surgical Blood Loss

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.