Cardiovascular Physics 101

Q: WHY IS POISEUILLE’S EQUATION OR THE OHM'S LAW IMPORTANT IN THE CARDIOVASCULAR SYSTEM?


A: Hemodynamics refers to the physical factors which govern the flow of blood. These physical factors are the same ones which dictate the flow of any fluid and are based upon the fundamental laws of physics. The laws of most relevance in the medical field are the Poiseuille’s equation and Ohm's Law.

Poiseuille’s equation

This relationship was first described in 19th century by the French physician Poiseuille and sheds light on the relationship between the primary determinants of resistance (R) to blood flow within a single vessel namely :-
  • Vessel radius (r)
  • Vessel length (L)
  • Viscosity of blood (n)

This tells us that resistance is directly proportional to both viscosity of blood and the length of the vessel and inversely proportional to the fourth power of the radius of the vessel.
We can also infer that radius is the key determinant of the resistance to blood flow not only because changes in radius get amplified to the power of four but also because radius (or diameter) is the only readily variable determinant amongst the three and hence the main mechanism for altering the resistance to blood flow is either increasing or decreasing the radius i.e. vasodilation or vasoconstriction of the blood vessels.
To elaborate further, a vessel having twice the length of another vessel (and each having the same radius) will have twice the resistance to flow. Similarly, if the viscosity of the blood increases 2-fold, the resistance to flow will increase 2-fold. In contrast, an increase in radius will reduce resistance. Furthermore, the change in radius alters resistance to the fourth power of the change in radius. For example, a 2-fold increase in radius decreases resistance by 16-fold! Therefore, vessel resistance is exquisitely sensitive to changes in radius.


Ohm's Law

This law was first described by German physicist Georg Ohm in 1827 and states that current (I) equals voltage difference (delta V) divided by resistance (R)  

Upon relating the Ohm's Law for fluid flow the Current (I) becomes the flow (F), the voltage difference (delta V) is the pressure difference (delta P) and the resistance is the resistance to flow (R)


For the flow of blood in a blood vessel, the  delta P is the pressure difference between any two points along a given length of the vessel. When describing the flow of blood for an organ, the pressure difference is generally expressed as the difference between the arterial pressure (PA) and venous pressure (PV). For example, the blood flow for the kidney is determined by the renal artery pressure, renal vein pressure, and renal vascular resistance.

The equation suggests that the blood flow is directly proportional to the pressure gradient and inversely proportional to the resistance to blood flow and altering resistance is the primary mechanism by which blood flow to organs is varied as control mechanisms within the body maintain arterial and venous blood pressures within a narrow range.


Relating the two equations together

Poiseuille’s equation :-



Ohm's Law :-


Substituting the above determinants of resistance (R) from the Poiseuille’s equation within the Ohm's Law.


The final relationship after combining both the Poiseuille’s equation and Ohm's Law helps us infer that flow (F) is directly proportional to the pressure gradient (delta P) and the fourth power of the radius (r) and inversely proportional to the Viscosity (n) of blood as well as the length of the blood vessel.






The relationship between flow and vessel radius to the fourth power (assuming constant delta P, L, Viscosity) is illustrated in the figure above. This figure shows how very small decreases in radius dramatically reduces flow.

By Dr Masroor S. Ahmed



References

  2. Determinants of resistance to blood flow (Poiseuille’s equation)









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