Fluid Mechanics Governing Left Ventricle Outflow
The flow of blood in our veins and arteries can be understood by the principles of “Fluid Mechanics” of Physics. The three primary principles of fluid mechanics are the “continuity equation” governing the conservation of mass, momentum principle which governs the “conservation of momentum”, and the “energy equation”. However, before going into the detail of analyzing the blood flow, it is important to see what makes one a compressible and another incompressible fluid.
In Physics, gas is considered a fluid (we may discuss specifically this in another blog), and the equations to analyze the behavior of gases are identical to what are for water, a fluid. The difference is on account of the molecules of a gas, which have enough space between them, hence when pressure is applied in a container having a gas, they can be compressed to a smaller volume. Meaning, the density of the gas (or compressible fluids) can be changed under pressure. But the water which is an incompressible fluid has its molecules closely packed. Hence under pressure water in a container does not change its density. Having understood this difference we must not forget that 90% of blood plasma is water. In other words, blood as well falls into the “incompressible fluid” category.
Now let us recall a simple thing which one has often done while watering the plants and grass in our backyard. We use a flexible pipe, connect it to a water outlet/tap and start watering our plants. But some plants and grass are far away from where one is standing. So being lethargic one does not want to move, but uses the “continuity equation” without actually knowing, to let the water stream reach farther away by pressing the flexible pipe and reducing its cross-section. No sooner it is done, immediately the stream of water coming out of the pipe starts falling at a much farther point in comparison to when the pipe had not been pressed to reduce its cross-section. The equation which makes this happen is the “continuity equation” viz.,
Q = A’V’ = A” V”
Where Q is the quantity of water coming out at a steady flow at Velocity V’ across the pipe’s cross-section A’ initially. And when the cross-section of the pipe reduces on pressing it to A”, the velocity of flow of water coming out now increases to V”. This can be instantly noticed as the water now starts reaching at a farther point in the backyard!
Having experienced it in our daily life, now let us shift our focus to the Left Ventricle (LV), Aortic Valve (AoV), Ascending Aorta (Asc-Ao) for a moment. So, in a normal healthy human being, the LV, LVOT, AoV, etc., function in a normal manner as per the standards. It means the diameter of the Left Ventricle Outflow Tract (LVOT), the diameter of the AoV, and of the Asc-Ao are all within the normal limits. However, if there is any deposition across the LVOT, or on the Leaflets of the AoV, in Asc-Ao, leading to stenosis etc., then there will be an obstruction in the flow of the blood. So, in the above equation if A’ is the diameter of say the LVOT (which is normal) and blood is flowing with a velocity V’, but A” which is the diameter of the AoV or of the Asc-Ao, that gets reduced due to certain pathologies, then it will instantly change the value of V” to uphold the validity of “continuity equation” as blood is an incompressible fluid.
So, the continuity equation ensures that the amount (mass) of the blood being pumped out by the LV through LVOT across the AoV and in the Asc-Ao, is maintained by increasing the velocity of blood flow if there is any reduction in the diameter due to stenosis, etc. As any deficiency in maintaining the continuity equation would then result in the blood may otherwise start getting accumulated in LV! But this does not change across the systemic and pulmonic circulation of blood in the human body. It implies to stop this from happening the LV muscles start working harder to push out the blood for upholding the continuity equation.
Here for a moment recall what happens to our “muscles” when one goes to a gym and exercises. Our muscles become stronger, harder, and increase in size. The same thing then happens to the LV muscles and can lead to Hypertrophic or Restrictive cardiomyopathy. These are the pathologies that cardiologists then deal with to restore the heart’s functioning to normalcy as much as possible of a patient
Let us stop here and take up few other things which contribute to the blood flow and can be understood through fluid mechanics principles, before jumping to look at the two cardiomyopathies in another blog.
Cameron Ahmad
Code PHDM-5
April 18, 2021
Brampton ON
Canada
