Poiseuille's law and vascular resistance: why arterioles matter
A normal pump can still produce a high pressure. Mean arterial pressure is cardiac output times resistance, and Poiseuille's law shows why arteriolar radius controls that resistance.
Chapters (10)
Use MAP = CO × TPR and Poiseuille's relationship (resistance inversely proportional to radius⁴) to explain why arteriolar radius dominates vascular resistance, and add resistances correctly in series and in parallel.
The short version
- Flow equals the pressure drop divided by resistance, so mean arterial pressure (MAP) ≈ cardiac output × total peripheral resistance (TPR).
- MAP is not the simple average: it is the diastolic pressure plus one third of the pulse pressure, at a normal heart rate.
- Resistance rises with vessel length and blood viscosity, but falls with radius to the fourth power: halve the radius and resistance rises 16-fold.
- Arterioles are the main site of resistance, and radius is the factor that can change within seconds.
- Series resistances add, so the biggest term controls the total; parallel resistances add as reciprocals, so the total is lower than the lowest single bed.
A normal pump, a high pressure
A 50-year-old woman has a heart that pumps 6 litres every minute. Her blood pressure is 160/100. The pump is fine, so what is squeezing the system?
This lesson uses Poiseuille's law and the relationship between pressure, flow and resistance to answer that. The answer is vascular resistance, and in particular the radius of the arterioles.[1][2]
Flow, resistance and MAP = CO × TPR
Flow through any vascular bed equals the pressure drop across it divided by its resistance.[1]
ΔP = pressure drop across the bed; R = its resistance
Rearranged for the whole body, mean arterial pressure equals cardiac output times total peripheral resistance. Venous pressure is close to zero, so it is ignored.[1]
Venous pressure is close to zero, so it is left out.
Mean arterial pressure from the pulse pressure
Mean pressure is not the simple average of systolic and diastolic. It sits closer to the diastolic value: diastolic pressure plus one third of the pulse pressure, at a normal heart rate.[2]
valid at a normal heart rate
- Pulse pressure = 160 − 100 = 60
- One third of 60 = 20
- MAP = 100 + 20 = 120
- Resistance = MAP / CO = 120 / 6 = 20 mmHg per L/min
Pressure is flow times resistance. With flow normal, resistance must be the problem.[1] The lesson's numbers are illustrative, not patient data.
Poiseuille's law: resistance and the radius to the fourth power
Poiseuille worked out resistance for a long, straight tube. Resistance rises with the length of the vessel and with the viscosity of the blood. Double the length and resistance doubles; double the viscosity and resistance doubles.[1][2]
η = blood viscosity; L = vessel length; r = vessel radius
Radius is different, because resistance falls with the radius to the fourth power.[1][2]
| Change in radius | Change in resistance |
|---|---|
| Halved | Rises 16-fold |
| Doubled | Falls to 1/16 |
Length and viscosity change little from moment to moment. Radius can change within seconds, and it is raised to the fourth power.[2]
Arterioles: the resistance vessels
The arterioles have that control. They are narrow, wrapped in smooth muscle, and they are the main site of resistance in the circulation.[2]
When the smooth muscle contracts, the lumen narrows and the pressure upstream rises. A small narrowing, raised to the fourth power, is a big rise in resistance, and pressure follows.[1][2]
Back to the patient: her cardiac output is normal, but her mean pressure of 120 is above the normal range of about 70 to 110 mmHg, so her resistance is high, most likely from narrowed arterioles. Mean pressure equals output times resistance, and arteriolar radius sets resistance.[1][2]
Resistance in series and in parallel
In series, resistances add. The lesson uses illustrative numbers for the chain of vessels from artery to vein.[1]
| Vessel | Resistance |
|---|---|
| Small arteries | 20 |
| Arterioles | 50 |
| Capillaries | 20 |
| Venules | 6 |
| Veins | 4 |
| Doubled | New total | Change |
|---|---|---|
| Veins | 104 | +4% |
| Arterioles | 150 | +50% |
In a series circuit the biggest term controls the total, and the arterioles are the biggest term.[1][2]
The organs are different: they sit side by side, in parallel. For parallel beds, add the reciprocals.[1]
- 1/20 + 1/20 = 2/20 = 1/10
- R_total = 10
The parallel total is always lower than the lowest single bed. That is why opening one more organ bed drops total resistance, and why changing a few vessels among many in parallel barely moves the total.[1]
Practice question
Original · USMLE styleA 22-year-old woman jogs on a treadmill during an exercise physiology study. Metabolites released from her working thigh muscle dilate its arterioles so that their radius is 50% larger than at rest. The pressure difference across the muscle vascular bed, the length of the vessels and the viscosity of her blood are unchanged. Assume laminar flow, and that the arterioles supply nearly all of the resistance of the muscle bed.
Common questions
What is the formula for mean arterial pressure?
MAP ≈ cardiac output × total peripheral resistance, with venous pressure taken as close to zero. From a cuff reading, MAP is the diastolic pressure plus one third of the pulse pressure at a normal heart rate.
What does Poiseuille's law say about vascular resistance?
Resistance rises with vessel length and blood viscosity, and falls with the fourth power of the radius. Halving the radius raises resistance 16-fold; doubling it lowers resistance to 1/16.
Why do arterioles control blood pressure?
They are narrow, wrapped in smooth muscle and the main site of resistance. Their radius can change within seconds, and because it is raised to the fourth power, a small change moves resistance a lot.
How do you add resistances in series and in parallel?
In series, resistances add. In parallel, add the reciprocals; the total is always lower than the lowest single bed.
Transcript
Read the full narration, by chapter695 words
0:00A normal pump, a high pressure
A 50-year-old woman has a heart that pumps six litres every minute. Her blood pressure is 160 over 100. Her pump is fine, so what is squeezing the system? Today: use mean arterial pressure equals cardiac output times resistance, plus Poiseuille's law, to see why arteriole radius controls resistance.
0:22Flow = pressure drop / resistance; MAP = CO x TPR
Start with the basic idea. Flow through any bed equals the pressure drop across it, divided by its resistance. Rearrange it for the whole body, and mean arterial pressure equals cardiac output times total peripheral resistance. Venous pressure is close to zero, so we ignore it.
0:42MAP from the pulse pressure (diastolic + 1/3 pulse pressure)
Mean pressure is not the simple average. It sits closer to the diastolic: diastolic plus one third of the pulse pressure, at a normal heart rate. For our patient, the pulse pressure is 60, one third is 20, so mean pressure is 120. One 120 divided by six litres per minute gives a resistance of 20. Pressure is flow times resistance, so with flow normal, resistance must be the problem. So what sets resistance?
1:12Poiseuille's law: radius to the fourth power
Poiseuille worked it out for a long, straight tube. Resistance rises with the length of the vessel, and with the viscosity of the blood. Double the length, double the resistance. Double the viscosity, double the resistance. But radius is different. Resistance falls with radius to the fourth power. Halve the radius, and resistance rises 16 times. Double the radius, and resistance falls to one sixteenth. Length and viscosity change slowly. Radius changes in seconds, and radius is raised to the fourth power.
1:49Arterioles, the resistance vessels
Which vessels have that control? The arterioles. They are narrow, wrapped in smooth muscle, and they are the main site of resistance in the circulation. When the muscle contracts, the lumen narrows and pressure upstream rises.
2:05Resistances in series
Now, how do resistances combine? In series, resistances add. Take illustrative numbers: small arteries 20, arterioles 50, capillaries 20, venules six, veins four. Total, 100. Double the veins and the total goes to 104. Only 4%. Double the arterioles and the total goes to 150. Fifty %. In a series circuit, the biggest term controls the total, and arterioles are the biggest term.
2:40Resistances in parallel
The organs are different. They sit side by side, in parallel. For parallel beds, add the reciprocals. Two beds, each with resistance 20, give one over 20 plus one over 20, which is one tenth. So the total is 10. Parallel total resistance is always lower than the lowest single bed. That is why opening one more organ bed drops total resistance, and why changing a few vessels barely moves the total.
3:09Back to our patient
Back to our patient. Her output is normal, so resistance is high, most likely from narrowed arterioles. A small narrowing, raised to the fourth power, is a big rise in resistance, and pressure follows. Mean pressure equals output times resistance, and arteriolar radius sets resistance.
3:30Practice question
Your turn. A 22-year-old woman runs on a treadmill. Metabolites from her working thigh muscle widen its arterioles, so their radius is 50% larger than at rest. The pressure drop across the muscle bed, the vessel length and the blood viscosity do not change. Assume laminar flow, and that the arterioles supply nearly all of the bed's resistance. By roughly how much does blood flow through the muscle bed increase? A, 1.5 fold. B, 2.25 fold. C, 3.4 fold. D, five fold. E, 16 fold. Pause here and decide. A is out: that treats flow as linear in radius, but the radius carries a power of four. B is out: that squares the radius, which is the wrong power. C is out: that cubes it, still the wrong power. E is out: 16 is for a doubling or halving of the radius, but here it rose by only half. So the answer is D: 1.5 to the fourth power is about five, so flow rises about five fold.
4:45Recap and next lesson
Three things to keep. One: mean pressure equals cardiac output times resistance, and mean pressure is diastolic plus a third of the pulse pressure. Two: resistance goes as one over radius to the fourth power, so arterioles dominate. Three: series resistances add, parallel ones add as reciprocals. Next, we put this beside the heart: venous return and cardiac function curves.
Sources
Numbers in the text, e.g. [1], point here- Klabunde RE. CV Physiology: Mean Arterial Pressure (BP006); Arterial Blood Pressure basics (BP001); Hemodynamics: Vascular Resistance, Poiseuille's equation (H003); Resistance in series (H004); Resistance in parallel (H005). cvphysiology.com/Blood%20Pressure/BP006 · cvphysiology.com/Blood%20Pressure/BP001 · cvphysiology.com/Hemodynamics/H003 · cvphysiology.com/Hemodynamics/H004 · cvphysiology.com/Hemodynamics/H005
- OpenStax. Anatomy and Physiology 2e, 20.1 Structure and Function of Blood Vessels; 20.2 Blood Flow, Blood Pressure, and Resistance. openstax.org/books/anatomy-and-physiology-2e/pages/20-1-structure-and-function-of-blood-vessels · openstax.org/books/anatomy-and-physiology-2e/pages/20-2-blood-flow-blood-pressure-and-resistance
The practice question is original and its patient is synthetic. Every fact is checked against the sources above in independent accuracy reviews. Always follow your local guidelines and your own laboratory’s ranges.
Image credits. 3D heart: "Anatomical Human Heart" by Nexal 3D (BlenderKit). The tube, arteriole cross-section, resistor circuits, charts, formulas and all handwriting are drawn in code (original). Previous: EP 09 · HFrEF vs HFpEF: Two Failing Loops · Next: EP 11 · Venous Return and Cardiac Function Curves
Corrections. None so far. Spotted an error? Email info@medservatory.com with the lesson code MS-S1-CARD-016.
For exam study only, not medical advice. Read the medical disclaimer.
In this series
Playlist on YouTube- EP 01Pressure-volume loop explained: four phases, four valve events5:00
- EP 02Preload on the pressure-volume loop: EDV, stroke volume and the EDPVR5:04
- EP 03Afterload on the pressure-volume loop4:49
- EP 04Contractility on the pressure-volume loop: the ESPVR explained5:17
- EP 05Wiggers diagram explained: the cardiac cycle step by step5:30
- EP 06The Frank-Starling curve: preload, inotropy and heart failure4:44
- EP 07Aortic stenosis vs mitral regurgitation on the pressure-volume loop5:41
- EP 08Fick principle: how to calculate cardiac output5:07
- EP 09HFrEF vs HFpEF on the pressure-volume loop5:33
- EP 10Poiseuille's law and vascular resistance: why arterioles matter5:15
- EP 11Venous return and cardiac function curves: where they cross5:11
- EP 12The baroreceptor reflex: from stretch to vagal and sympathetic output5:17
- EP 13Myocardial oxygen supply and demand: why tachycardia causes ischaemia4:50
- EP 14Unstable angina vs NSTEMI vs STEMI: from plaque to ACS4:50
- EP 15Antianginal drugs: nitrates, beta blockers, calcium channel blockers4:56
- EP 16Complications of myocardial infarction, by timeline4:59