Question

V = (ΔPπr4)/8nL

indicate how application of the equation to pulmonary air flow differs from blood flow through arterioles

Answer 1

Basic of Poiseuille equation:

There are several important determinants of airway resistance including:

• The diameter of the airways
• Whether airflow is laminar or turbulent

Hagen–Poiseuille equation

In fluid dynamics, the Hagen–Poiseuille equation is a physical law that gives the pressure drop in a fluid flowing through a long cylindrical pipe. The assumptions of the equation are that the flow is laminar viscous and incompressible and the flow is through a constant circular cross-section that is substantially longer than its diameter. The equation is also known as the Hagen–Poiseuille law, Poiseuille law and Poiseuille equation.

The law of Poiseuille states that the flow of liquid depends on the following variables such as the length of the tube(L), radius (r), pressure gradient (∆P) and the viscosity of the fluid (η) in accordance with their relationship.

The entire relation or the Poiseuille’s Law formula is given by

V= ΔPπr4 / 8ηl

Wherein,

The Pressure Gradient (∆P) Shows the pressure differential between the two ends of the tube, defined by the fact that every fluid will always flow from the high pressure (P1) to the low-pressure area (P2) and the flow rate is calculated by the  ∆P = P1-P2.

The radius of the narrow tube:

The flow of liquid direct changes with the radius to the power four.

Viscosity (η):

The flow rate of the fluid is inversely proportional to the viscosity of the fluid.

Length of the arrow tube (L):

The flow rate of the fluid is inversely proportional to the length of the narrow tube.

Resistance(R):

The resistance is calculated by 8Ln / πr4 and hence the Poiseuille’s law is

V= (ΔP) R

Applications:

The circulatory system provides many examples of Poiseuille’s law in action—with blood flow regulated by changes in vessel size and blood pressure. Blood vessels are not rigid but elastic. Adjustments to blood flow are primarily made by varying the size of the vessels, since the resistance is so sensitive to the radius. During vigorous exercise, blood vessels are selectively dilated to important muscles and organs and blood pressure increases. This creates both greater overall blood flow and increased flow to specific areas. Conversely, decreases in vessel radii, perhaps from plaques in the arteries, can greatly reduce blood flow. If a vessel’s radius is reduced by only 5% (to 0.95 of its original value), the flow rate is reduced to about of its original value. A 19% decrease in flow is caused by a 5% decrease in radius. The body may compensate by increasing blood pressure by 19%, but this presents hazards to the heart and any vessel that has weakened walls. Another example comes from automobile engine oil. If you have a car with an oil pressure gauge, you may notice that oil pressure is high when the engine is cold. Motor oil has greater viscosity when cold than when warm, and so pressure must be greater to pump the same amount of cold oil.

Determinants of Airway Resistance

Certain equations can be used to determine airway resistance.

Ohm’s law usually refers to electrical circuits, in which current = voltage/resistance. However, it can be applied to describe the relationship between airflow, pressure gradient and resistance.

The equation is: Flow = Pressure gradient / Resistance

This demonstrates that as resistance increases, the pressure gradient must also increase to maintain the same airflow to the alveoli.

the respiratory system, the components of this equation are very hard to measure accurately, and the law only applies when there is laminar flow. However, it shows that the airway resistance is inversely proportional to the radius (to the power of 4). Hence a small change in diameter has a huge effect on the resistance of an airway e.g. halving the radius of an airway would cause a 16-fold increase in resistance.

Therefore, smaller airways such as bronchioles and alveolar ducts all individually have much higher flow resistance than larger airways like the trachea. However, the branching of the airways means that there are many more of the smaller airways in parallel, reducing the total resistance to air flow. So due to the vast number of bronchioles that are present within the lungs running in parallel, the highest total resistance is actually in the trachea and larger bronchi.

Nervous System Control

Airway diameter is usually determined by the autonomic nervous system. Sympathetic innervation causes relaxation of bronchial smooth muscle via beta-2 receptors, which increases diameter to allow more airflow. This is useful in situations such as exercise, as sympathetic nerve stimulation triggers airway muscle relaxation, increasing the diameter to allow more air into the lungs. This increases the rate of gas exchange at alveolar level compared to normal breathing.

Parasympathetic innervation works on muscarinic (M3) receptors to increase smooth muscle contraction and reduce diameter, as when resting it is not necessary to have an increased airflow to the lungs.

Inspiration vs Expiration

Resistance differs between inspiration and expiration due changes in the diameter of the airways. On inspiration, the positive pressure within the alveoli and small airways causes their diameter to increase, and therefore resistance to decrease. The opposite is true for expiration, as airways narrow due to the reduced pressure, thus increasing resistance.

In forced expiration, the thoracic cavity further reduces in size compared to quiet expiration, leading to a greater degree of compression of the lungs. This leads to the small airways becoming narrowed to the extent that the resistance is sufficient to trap some air in the alveoli which cannot be expelled – this volume of air is known as the residual volume..