Summary
Poiseuille's equation, named after the French physicist Jean Louis Marie Poiseuille, describes the steady, laminar flow of a viscous fluid through a cylindrical pipe. It's a fundamental equation in fluid dynamics and plays a crucial role in understanding the behavior of fluids in various systems, such as blood flow in arteries or the flow of liquids in pipes.
The Poiseuille’s Equation takes into account factors such as blood viscosity, length and cross sectional area of a blood vessel and uses it to determine the resistance to the flow of blood.
R=8 n Lπr^4 (Assuming that the flow is laminar and the volume is constant)
- Where: n: Viscosity of the blood
L: Length of the blood vessel
r: Radius of the blood vessel
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FACTORS CONTRIBUTING TO THE POISeUILLE’S EQUATION
Blood viscosity
has a direct relation with resistance, according to the Poiseuille’s equation. This means that increasing the viscosity of blood will result in an increase in resistance. Blood viscosity undergoes changes with alterations in the hematocrit. Hematocrit is the percentage of total number of particulate matter (RBCs, WBCs, proteins etc) in a given volume of blood. Under normal conditions, the value of hematocrit is 45%. Changes in the hematocrit can occur with both, physiological and pathological conditions. Dehydration causes decrease in the volume of blood which results in a relative physiological increase in hematocrit. Similarly, secondary polycythemia also causes physiological increase in the value of hematocrit as a result of increased production of red blood cells by the bone marrow. Pathological increase in the hematocrit is primarily seen in malignancy of the bone marrow in which large numbers of undifferentiated cells are produced leading to an increase in the cell number. Multiple Myeloma is the neoplastic proliferation of plasma cells which produce excess immunoglobulins. This results in an increased protein content of the blood and a resultant increase in hematocrit.
On the other hand, anemia will cause a decrease in hematocrit and hence, a decrease in viscosity of blood. As mentioned earlier, a decrease in blood viscosity will result in a decrease in resistance to the flow of blood. Flow rate increases when there’s a decrease in resistance offered to blood flow. Increase in the flow rate can be problematic because it generates Eddy currents within the blood vessels. These Eddy currents cause pressure damage to the blood vessel and cause the blood flow to become turbulent. Turbulent blood flow within the heart results in murmurs. Turbulent blood flow within the blood vessels results in bruit. Most common sites of bruit are carotid artery and the aorta. Both murmur and bruit can be appreciated upon auscultation at the relevant sites.
The resistance to blood flow within the blood vessel is inversely proportional to the radius of the blood vessel raised to the fourth power. If the cross sectional area of the cylindrical blood vessel is decreased, there’s a resultant increase in resistance to blood flow in that vessel. So that basically means, if we take the radius and simply divide it by two (decreasing the radius of the vessel to half), the resistance would increase by a factor of 16. This would result in a considerable decrease in the flow rate of the blood. Application of this concept is seen in pathological cases, such as the narrowing of blood vessels in atherosclerosis and arteriosclerosis, resulting in increased resistance and reduced blood flow. The diameter of blood vessels in our body is physiologically controlled by our body’s autonomic nervous system (ANS). Arterioles are the vessels in which the autonomic effects are seen most prominently. For this reason they are also known as the resistance vessels or the functional sphincters of the cardiovascular system. The smooth muscles surrounding the arterioles contract or relax under the effect of the autonomic nervous system. When stimulated by the sympathetic nervous system (SANS), the alpha 1 receptors on the arteriolar smooth muscle get activated. These receptors are coupled intracellularly with Gq proteins which get activated and increase intracellular levels of second messengers, namely IP3 and DAG. The second messengers cause movement of calcium ions into the sarcoplasmic reticulum resulting in contraction of the smooth muscles. The same sympathetic innervation when stimulates the Beta 2 adrenergic receptors on the arteriolar smooth muscle of arterioles perfusing skeletal muscles, a Gi coupled response is initiated. This will cause a decrease in cAMP levels in smooth muscle cells of the arterioles. As a consequence, the smooth muscle will relax and the arteriole will dilate allowing more blood to perfuse the skeletal muscles. The distribution of extracellular receptors (beta 2, alpha 1, etc) on vascular smooth muscles depends on the demand for blood by different organs according to the situation. For example, during a fight and flight response, the blood vessels of the skeletal smooth muscles (innervated by beta 2 receptors) will dilate to provide more blood to the muscles. At the same time, the vascular smooth muscles of arterioles perfusing the visceras (GIT, Kidneys, etc) will contract, and the lumen will constrict in order for the blood to redirect to parts where it is needed more.
USE OF POISEUILLE’S EQUATION IN THE FLOW EQUATION
The value of the resistance, calculated using the POISEUILLE’S EQUATION, can be plugged into the flow equation (given below) to determine the flow rate:
- Flow = ΔP/R,
ΔP = MAP – Right Atrial Pressure
Therefore,
Flow = (MAP – RAP) /Resistance
ΔP: change in Pressure
R: Resistance
MAP: Mean Arterial Pressure
So, according to the equation, increase in resistance will cause a decrease in the flow of blood through a vessel. The resistance of a blood vessel is increased in conditions such as atherosclerosis and vasospasm due to increased sympathetic activity. Similarly, if the resistance is decreased, the flow of blood through a vessel will increase.