(Biomedical) Fluid mechanics: flow, resistance, and pressure

[DNA888]

Hello, I'm a physician with special interest in cardiovascular physiology. I'm trying to comprehend the basic physics of the cardiovascular system. Here's my thought process -

Simply stated - the heart muscle contracts, thereby causing a reduction in the cardiac chamber volume and release of blood out of the chanber in the process. Consider an ideal situation where cardiac contraction produces X units of force, which then pumps 70 mL of blood out into the arteries. If the heart beats 70 times a minute, this would add up to around 5L/min - aka the cardiac output (Q). Total resistance of the arterial network against which the blood is pumped (R) = 20 mm Hg/L/min. When blood is pumped against this resistance, it produces an arterial pressure (P) of 100 mm Hg.

The relationship between cardiac output (flow), mean arterial pressure (P), and total peripheral resistance (R) has been described as -

Q = P/R (some sources call it Ohm's law of fluid flow). I understand it is actually "delta P" that drives blood flow, but pressure on the venous end is considered negligible and not considered here. This formula is intuitive in the sense that if R is increased (referred to as increase in afterload in physiology texts), Q should decrease as the heart is pumping blood against higher resistance. Also, if P increases, the "delta P" increases, i.e. the pressure gradient between the arterial side and venous side increases, and blood flow should increase.

However, my biggest conundrum is, that P itself is dependent on R (as widely noted in literature). When R increases, blood is pumped against a larger resistance and that causes a larger outward push on the walls of the arteries - thereby increasing the P. Hence, this relation can be written as P = Q.R. Based on my current understanding, R is the only exclusive physical property here that is not influenced by other properties of the equation.

Quoting a thought experiment from a physiology textbook -

Experiment 1 - Imagine an instantaneous increase in total periperhal resistance (R) from 20 to 40. They cite "P = Q.R" and say P would jump from 100 to 200 mm Hg. Now Q = P/R --> both P and R doubled so Q would remain constant (5 L/min).

Experiment 2 - The alternate thought process in my head is - R doubles to 40 --> causing reduction in cardiac output to 2.5 from 5 L/min (because Q = P/R). Now, P = Q.R --> Q halved but R doubled so P would remain same (100).

I would guess the reality lies somewhere in between these two extreme outcomes (but perhaps more inclined towards the outcomes of experiment 1). Based purely on clinical experience, we often see a significant increase in P while increasing R but it has been experimentally established that increasing R does cause SOME (if not a huge) reduction in Q. I'm sure there has to be a more precise physical/mathematical explanation for this. I feel like I'm missing something obvious here. Any direction would be greaatly appreciated. Sorry for the long thread.

 

[Chestermiller]

Your analysis of this is great. The missing ingredient that "closes the loop" on this is the pumping characteristic of the heart. It isn't clear how the heart pumping rate responds to the pressure change (as you more or less concluded). The pulse rate is also part of this. If it were a typical centrifugal pump, for example, the flow rate Q would depend on the pump rotational speed N (rpm) and the pressure buildup that the pump provides. Increasing the rotational speed would increase the pump volumetric output Q; increasing the pressure buildup provided by the pump would decrease the volumetric output (because of back-flow).

 

[BillTre]

At what point in the vascular circuit is most of the peripheral resistance thought to reside? Capillary beds? Arterioles? Arteries?Perhaps it has to do with a change in the length of time of the pressure pulse that drives flow, or the blood volume stored in the arteriole compliance being driven out later.
Compliance would be like a capacitor in an electrical circuit, letting extra flow come in and storing it (electrons), along with voltage (driving force equivalent to pressure) which could then be released later.

The heart is a pulseatile pump which drives pressure pulses through the arteriole tree. These pressure pulses will drive the flow most at their highest pressures. The heart driven pressures will get smeared out over time as the pulse goes through greater numbers of smaller and smaller arteries.

The pressure pulse would expand the arteries (more flow into the artery that out) based on their compliance (expandability under pressure).

Because arteries are muscular, they will:

1. Eventually retain/return some of the pressure when the heart driven pulse starts to diminish. This prolonged pressure should push more flow across a resistance (capillary beds?) since it is available for a larger percentage of the heart beat cycle.
   
2. Have more flow out of the expanded artery as it deflates to "normal" after the initial pressure wave of the heart beat passes.

In this way, either increased total pressure over time (like an integral of pressure across a heartbeat) could drive more flow than might otherwise be expected, or delayed flow from the compliant arteries could explain the greater flow than expected at the higher pressures when the arteries get inflated.

 

[Andy Resnick]

Experiment 1 - Imagine an instantaneous increase in total periperhal resistance (R) from 20 to 40. They cite "P = Q.R" and say P would jump from 100 to 200 mm Hg. Now Q = P/R --> both P and R doubled so Q would remain constant (5 L/min).

Experiment 2 - The alternate thought process in my head is - R doubles to 40 --> causing reduction in cardiac output to 2.5 from 5 L/min (because Q = P/R). Now, P = Q.R --> Q halved but R doubled so P would remain same (100).

This is a little out of my field, but Silbernagl and Despopoulos "Color Atlas of Physiology" has a lot of good information on this.

The cardiac output Q is a combination of pulse rate and stroke volume (SV): if those do not change, Q must be constant. That's an alternate explanation for Experiment 1. However, and you would understand this better than I, the work diagram for the heart can display pathological changes that impact the SV.

Side note: 50% of vascular resistance comes from the lesser arteries and arterioles, 25% from capillaries. Silbernagl's book has a great diagram on page 191- can't find an online version.

 

[finnk]

Hello and welcome to PF. It's been years since i took physiology, but i remember vividly that we can't measure resistance directly. It is calculated based on measurement of pressure and flow. I agree that increasing resistance would mean increasing pressure, decreasing flow, or both. Although I'm not sure if it is possible to decrease flow without modifying the pressure.