Gradients, pressure and flow
1Why this matters
Ms. Alvarez, 34, has lost a lot of blood in a car crash, and her blood pressure is falling. The paramedic passes over the small 22-gauge catheter and places a large 14-gauge one in her arm, hangs the fluid bag as high as it will go, and squeezes it in a pressure bag. The larger catheter is only a few times wider, yet it lets fluid run in many times faster. Every one of those choices comes from a single rule: flow equals the pressure gradient divided by resistance.
2What this builds on
3Quick check before you start
1. A solution has 10 mmol of glucose dissolved in 1 liter. What does its concentration describe?
- How much glucose is dissolved in each liter of the solution
- The total mass of the solution
- How fast the glucose moves through the solution
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Concentration is the amount of solute in a given volume of solution, here 10 mmol per liter. A difference in concentration between two places is the first kind of gradient on this page.
- Correct: How much glucose is dissolved in each liter of the solution:
- The total mass of the solution:
- How fast the glucose moves through the solution:
2. Which vessels carry blood away from the heart toward the tissues?
- Veins
- Arteries
- Capillaries
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Arteries carry blood away from the heart. Capillaries are the tiny vessels in the tissues, and veins carry blood back to the heart. Blood pressure is highest in the large arteries and lowest in the large veins.
- Veins:
- Correct: Arteries:
- Capillaries:
3. A ball held at the top of a staircase has stored energy because of its position. What kind of energy is that?
- Kinetic energy
- Chemical energy
- Potential energy
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Stored energy, whether from position or structure, is potential energy. When the ball rolls down, it becomes kinetic energy. A gradient is also a kind of potential energy.
- Kinetic energy:
- Chemical energy:
- Correct: Potential energy:
4How it works, step by step
- A tube carries fluid with a pressure gradient of 80 mm Hg and a resistance of 2 mm Hg for each mL/min, so flow is 80 ÷ 2 = 40 mL/min. The tube is squeezed until its radius is half of what it was.The radius to the fourth power falls to ½ × ½ × ½ × ½ = 1/16 of its old value.
- Resistance depends on 1 ÷ radius to the fourth power, and the radius to the fourth power is now 1/16 as large.Resistance rises 16-fold, from 2 to 32 mm Hg for each mL/min.
- Resistance is 16 times higher while the pressure gradient stays at 80 mm Hg.Flow falls to 80 ÷ 32 = 2.5 mL/min: one-sixteenth of the original, not one-half.
- To restore 40 mL/min through the narrowed tube, the pressure gradient would have to rise 16-fold.It would need 1,280 mm Hg, so changing the radius, not the pressure, is the powerful way to control flow.
5Core concepts
6A common mistake
The wrong idea: Fluid flows because the pressure is high: the higher the pressure, the more it flows.
What actually happens: Fluid flows only when there is a pressure difference between two points, and flow depends on that difference, not on how high the pressures are. A closed garden hose can be under high pressure with no flow at all. A tube with 100 mm Hg at one end and 90 at the other carries the same flow as the same tube with 20 at one end and 10 at the other, because both have a 10 mm Hg gradient.
7Check yourself
Anything you miss goes into your review queue.
1. The pressure at the start of a tube is 100 mm Hg and at the end is 20 mm Hg. The tube's resistance is 4 mm Hg for each mL/min. What is the flow?
- 25 mL/min
- 320 mL/min
- 20 mL/min
- 30 mL/min
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First find the pressure gradient: ΔP = 100 − 20 = 80 mm Hg. Then F = ΔP / R = 80 ÷ 4 = 20 mL/min.
- 25 mL/min: This divides the starting pressure alone by the resistance (100 ÷ 4). Flow depends on the pressure difference, 80 mm Hg, not the starting pressure.
- 320 mL/min: This multiplies the pressure gradient by the resistance (80 × 4). Resistance holds flow back, so you divide by it.
- Correct: 20 mL/min: Correct. ΔP = 80 mm Hg, and 80 ÷ 4 = 20 mL/min.
- 30 mL/min: This adds the two pressures (100 + 20 = 120) and divides by 4. The gradient is the difference between the pressures, not their sum.
2. The radius of a tube is cut to half its original value. The pressures at both ends stay the same, and so do the tube's length and the fluid. What happens to the flow?
- It falls to 1/2 of the original
- It falls to 1/4 of the original
- It is unchanged: pressure alone sets the flow
- It falls to 1/16 of the original
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Resistance depends on 1 ÷ radius to the fourth power. Halving the radius makes the radius to the fourth power (½)⁴ = 1/16 as large, so resistance becomes 16 times larger. With ΔP unchanged, F = ΔP / R becomes 1/16 of the original.
- It falls to 1/2 of the original: This treats flow as simply proportional to radius. Resistance depends on the fourth power of the radius, so the effect is far larger.
- It falls to 1/4 of the original: This squares the radius (½ × ½). Resistance depends on the radius multiplied by itself four times, not twice.
- It is unchanged: pressure alone sets the flow: Flow depends on both the pressure gradient and resistance. Narrowing the tube raises resistance, so flow falls even though the pressures are unchanged.
- Correct: It falls to 1/16 of the original: Correct. Half the radius gives 16 times the resistance, so one-sixteenth of the flow.
3. Two identical tubes carry the same fluid. Tube A has 150 mm Hg at its inlet and 140 mm Hg at its outlet. Tube B has 40 mm Hg at its inlet and 10 mm Hg at its outlet. Which statement is correct?
- Tube A carries more flow: its pressures are much higher
- Both carry the same flow: they are identical tubes
- Tube B carries three times the flow of tube A
- Neither carries flow: the outlets are not at 0 mm Hg
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Flow depends on the pressure difference. Tube A has ΔP = 150 − 140 = 10 mm Hg; tube B has ΔP = 40 − 10 = 30 mm Hg. With the same resistance, tube B carries three times the flow.
- Tube A carries more flow: its pressures are much higher: High pressures with a small difference drive little flow. Tube A's gradient is only 10 mm Hg.
- Both carry the same flow: they are identical tubes: Identical tubes have the same resistance, but their pressure gradients differ (10 versus 30 mm Hg), so their flows differ.
- Correct: Tube B carries three times the flow of tube A: Correct. 30 mm Hg ÷ 10 mm Hg = 3, and with equal resistance, flow is proportional to ΔP.
- Neither carries flow: the outlets are not at 0 mm Hg: Fluid flows whenever one end is at a higher pressure than the other. The outlet does not need to be at 0.
4. A paramedic needs to give fluid as fast as possible to a patient who is bleeding heavily. Starting from one IV setup, which single change would raise flow the most?
- Doubling the pressure squeezing the fluid bag, using a pressure bag
- Using a catheter of the same radius but half the length
- Switching to a catheter with twice the internal radius and the same length
- Warming the fluid enough to lower its viscosity by one-quarter
Show the answer
Flow depends on the radius to the fourth power. Doubling the radius cuts resistance to 1/16, so flow rises about 16-fold. Doubling the pressure or halving the length each only doubles flow, and a quarter less viscosity raises flow by about one-third.
- Doubling the pressure squeezing the fluid bag, using a pressure bag: Flow is proportional to ΔP, so doubling the pressure doubles flow: a real gain, but far smaller than the radius change.
- Using a catheter of the same radius but half the length: Resistance is proportional to length, so half the length halves resistance and doubles flow.
- Correct: Switching to a catheter with twice the internal radius and the same length: Correct. Twice the radius gives one-sixteenth the resistance and about 16 times the flow.
- Warming the fluid enough to lower its viscosity by one-quarter: Resistance is proportional to viscosity. Cutting viscosity to three-quarters raises flow by a factor of 4/3, about 33%.
5. A clamp narrows a tube carrying fluid. The pressures at the inlet and outlet are held constant. Predict each change.
| Variable | Change |
|---|---|
| Resistance of the tube | — |
| Pressure gradient between inlet and outlet | — |
| Flow through the tube | — |
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Narrowing a tube raises its resistance steeply. With the pressure gradient fixed, flow falls.
- Resistance of the tube: up. A smaller radius means more friction between the fluid and the wall for the fluid passing through, and resistance depends on 1 ÷ radius to the fourth power.
- Pressure gradient between inlet and outlet: no change. The inlet and outlet pressures are held constant, so their difference does not change.
- Flow through the tube: down. F = ΔP / R. The same ΔP divided by a larger R gives less flow.
6. A 28-year-old man loses about 1.5 liters of blood from a deep leg wound. His blood pressure falls from 124/78 to 92/60 mm Hg. Which explanation fits?
- Less blood in the same stretchy vessels stretches their walls less
- His remaining blood has thickened and now lowers the pressure
- The air pressure around him has fallen during the bleeding
- His blood vessels have lengthened and now lower the pressure
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Blood vessels form a closed, stretchy system. Blood pressure depends partly on how much blood fills it. With a lower blood volume, the walls are stretched less and the pressure falls.
- Correct: Less blood in the same stretchy vessels stretches their walls less: Correct. A smaller blood volume in the same vessels means less stretch and a lower pressure.
- His remaining blood has thickened and now lowers the pressure: Losing blood, cells and fluid together, does not thicken what remains. Higher viscosity would raise resistance, not explain a pressure drop.
- The air pressure around him has fallen during the bleeding: The surrounding air pressure is not changed by bleeding. Blood pressure readings are already relative to it.
- His blood vessels have lengthened and now lower the pressure: The length of the vessels does not change with blood loss.
7. A patient's blood has an unusually high concentration of cells. If the pressure gradient driving blood through a vessel stays the same, what happens to flow through that vessel?
- It rises: more cells carry more blood through the vessel
- It stays the same: the pressure gradient has not changed
- It falls: thicker blood raises the resistance
- It falls: the thick blood narrows the vessel
Show the answer
Most of blood's viscosity comes from its cells. More cells make blood more viscous, which raises resistance. With the same ΔP, F = ΔP / R falls.
- It rises: more cells carry more blood through the vessel: Flow is the volume of blood moving per minute. More cells make blood thicker and harder to push, so flow falls.
- It stays the same: the pressure gradient has not changed: The same ΔP drives less flow when resistance is higher. Flow depends on both.
- Correct: It falls: thicker blood raises the resistance: Correct. Higher viscosity means higher resistance and less flow at the same pressure gradient.
- It falls: the thick blood narrows the vessel: Thick blood does not narrow the vessel. The rise in resistance comes from viscosity.
8Summary
A gradient is a difference in a quantity, such as concentration or temperature, between two places; things move down gradients without an energy input. Pressure is force per unit area, measured in mm Hg and given relative to atmospheric pressure (760 mm Hg at sea level). Bulk flow moves a whole fluid from higher to lower pressure, and flow equals the pressure gradient divided by resistance (F = ΔP/R). Resistance rises with length and viscosity and falls steeply as radius grows: it depends on the fourth power of the radius, so halving a tube's radius makes resistance 16 times larger and flow 16 times smaller. Hydrostatic pressure is the pressure a fluid exerts, and it rises with depth. Blood pressure is the pressure of blood against vessel walls, and it falls when blood volume (about 5 liters) falls.