Biology · Gas Exchange, Circulation & Water Balance
Hearts, vessels, pressure and circulatory regulation
A branch can receive more fluid even while the total circuit flow falls. To understand how, follow the pressure difference across that branch and the resistance of the whole network. A drawing's vessel width or arrow thickness is not enough unless its numerical meaning is specified.
Homeostasis, Nutrition & Biological Organisation introduces regulated internal conditions and physiological budgets. Cells, Membranes & Organisation provides further background on selective boundaries; Chemistry's Solutions, Concentration & Solubility distinguishes amount from concentration. The opening lessons define the gradient, pressure and transport quantities used here, so their symbols need not be known in advance.
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Relate pressure differences, resistance and total cross-sectional area to conserved circulation, while distinguishing simplified flow models from regulation in animals.
A heart creates a pressure difference over time
Contraction changes chamber pressure and can drive fluid toward a lower-pressure route. Relaxation permits filling under the appropriate pressure conditions. In a familiar mammalian heart, valves open and close in response to the pressure differences across them, helping restrict backflow. The valve directs passage; it does not supply the work that raises the fluid's pressure.
Pressure at one point differs from a pressure difference between two points. A vessel segment's steady flow depends on the difference across it and on resistance to movement. In the simple relation flow equals pressure difference divided by resistance, doubling the difference doubles flow only while resistance stays fixed. Real hearts and vessels also pulse, stretch and change their resistance.
Regulation connects these variables. Neural, hormonal and local tissue signals can alter heart rate, the amount ejected per beat and vessel tone. Heart rate multiplied by stroke volume gives a volume throughput for the specified pumping side and interval. Feedback and local responses can alter delivery among tissues rather than retaining the fixed pressures used in a classroom network.
Parallel paths share a pressure difference
The network has an inlet resistance, two parallel branches and an outlet resistance. The two branches connect the same inlet and outlet junctions, so they experience the same pressure difference. Their flows add at the junction; their resistances do not simply add as they would for serial segments. A larger resistance in one branch makes that route carry less at a given branch pressure difference.
The pump in this comparison always raises pressure by twelve units. Changing branch B also changes the total resistance and flow, which changes the pressure drops along the common inlet and outlet routes. Branch A can consequently experience a different pressure difference even though its own resistance has not changed.
This is a conservation model, not a prediction of a particular artery during exercise or disease. It omits gravity, fluid storage, elasticity and physiological feedback. Its useful lesson is to examine the connected circuit before concluding that one local change must leave all other branch flows unchanged.
A change in one branch redistributes a conserved flow
Read the diagram in words
Initially total resistance three supports flow four at the fixed pressure rise twelve. Each branch carries two across a pressure difference eight. Increasing B's resistance to twelve gives total resistance four and flow three. The new branch pressure difference nine gives A flow 2.25 and B flow 0.75, adding to three.
| Identifier | Structure or interpretation | Layout x | Layout y |
|---|---|---|---|
| low | Pump inlet | 0 | 0 |
| high | Pump outlet | 0 | 4 |
| arterial | Branch inlet junction | 2 | 4 |
| venous | Branch outlet junction | 6 | 0 |
| Fluid route | From identifier | To identifier | Relationship code | Relationship meaning |
|---|---|---|---|---|
| pump | low | high | pressure-raising-pump | Pressure-raising pump |
| inlet | high | arterial | resistive-fluid-route | Passive fluid route |
| bed-a | arterial | venous | resistive-fluid-route | Passive fluid route |
| bed-b | arterial | venous | resistive-fluid-route | Passive fluid route |
| outlet | venous | low | resistive-fluid-route | Passive fluid route |
| Fluid route | Order within the face or route | Layout x | Layout y |
|---|---|---|---|
| bed-a | 1 | 2 | 4 |
| bed-a | 2 | 4 | 4 |
| bed-a | 3 | 6 | 0 |
| bed-b | 1 | 2 | 4 |
| bed-b | 2 | 4 | 0 |
| bed-b | 3 | 6 | 0 |
| Assigned state | Network junction | Assigned steady pressure (Model fluid-pressure units) |
|---|---|---|
| Equal branch resistances | Pump inlet | 0 |
| Equal branch resistances | Pump outlet | 12 |
| Equal branch resistances | Branch inlet junction | 10 |
| Equal branch resistances | Branch outlet junction | 2 |
| Larger resistance in branch B | Pump inlet | 0 |
| Larger resistance in branch B | Pump outlet | 12 |
| Larger resistance in branch B | Branch inlet junction | 10.5 |
| Larger resistance in branch B | Branch outlet junction | 1.5 |
| Assigned state | Fluid route | Resistance (Pressure units per unit volume flow) | Volume flow (Fluid-volume units per interval) |
|---|---|---|---|
| Equal branch resistances | pump | Not assigned to this route | 4 |
| Equal branch resistances | inlet | 0.5 | 4 |
| Equal branch resistances | bed-a | 4 | 2 |
| Equal branch resistances | bed-b | 4 | 2 |
| Equal branch resistances | outlet | 0.5 | 4 |
| Larger resistance in branch B | pump | Not assigned to this route | 3 |
| Larger resistance in branch B | inlet | 0.5 | 3 |
| Larger resistance in branch B | bed-a | 4 | 2.25 |
| Larger resistance in branch B | bed-b | 12 | 0.75 |
| Larger resistance in branch B | outlet | 0.5 | 3 |
| Assigned state | Parallel-branch resistance (Pressure units per unit volume flow) | Total circuit resistance (Pressure units per unit volume flow) |
|---|---|---|
| Equal branch resistances | 2 | 3 |
| Larger resistance in branch B | 3 | 4 |
| Identifier | Structure or interpretation |
|---|---|
| pressureUnit | Model fluid-pressure units |
| flowUnit | Fluid-volume units per interval |
| resistanceUnit | Pressure units per unit volume flow |
| A | Exchange branch A |
| B | Exchange branch B |
| pump | Pump adds 12 pressure units |
| passive | Passive flow down a pressure difference |
| nodeHeader | Network junction |
| pressureHeader | Assigned steady pressure |
| edgeHeader | Fluid route |
| resistanceHeader | Resistance |
| flowHeader | Volume flow |
| balance | Total flow equals the sum of both branch flows |
An invented incompressible steady network with zero fluid storage, linear positive vessel resistance and a pressure-raising pump fixed at 12 pressure units. No pulsation, compliance, gravity, viscosity change or feedback. Layout lengths do not define resistance or anatomy.
Mean speed depends on the combined available area
For incompressible steady flow, volume throughput equals mean speed times the cross-sectional area through which that flow passes. A bed of many parallel small vessels can have a large combined area and therefore a low mean speed. The small radius of an individual vessel alone does not establish the speed through the whole bed.
The area panels hold total flow at four. One circle of radius one has area π; four of radius one-half together also have area π. Both arrangements have mean speed 4 ÷ π. Sixty-four circles of radius one-quarter have combined area 4π and mean speed 1 ÷ π. Each panel's constituent routes share its flow equally by assumption.
A separate ideal tube comparison uses Poiseuille's relation: resistance is proportional to length and viscosity and inversely proportional to radius to the fourth power. With pressure difference, length and viscosity fixed, doubling radius gives sixteen times the flow. Area increases fourfold, so mean speed increases fourfold. This is a different condition from the fixed-flow area panels, and a rigid laminar-tube model does not represent every feature of real blood flow.
Total cross-section and single-vessel radius have different effects
Read the diagram in words
Combined areas are π, π and 4π for the one-route, four-route and sixty-four-route panels. At total flow four, mean speeds are 4/π, 4/π and 1/π. The separate fixed-pressure tube comparison doubles radius, quadruples area, divides resistance by sixteen and multiplies flow by sixteen, giving four times mean speed.
| Identifier | Constructed arrangement | Number of parallel routes | Radius of each route (Model length units) | Area coefficient multiplying π (Square model units) | Volume flow (Fluid-volume units per interval) | Flow through each equal-sharing route (Fluid-volume units per interval) | Speed coefficient dividing by π (Model length units per interval) |
|---|---|---|---|---|---|---|---|
| one | One route | 1 | 1 | 1 | 4 | 4 | 4 |
| four | Four narrower parallel routes | 4 | 0.5 | 1 | 4 | 1 | 4 |
| many | Sixty-four small parallel routes | 64 | 0.25 | 4 | 4 | 0.0625 | 1 |
| Selected comparison | Circular route | Centre x | Centre y | Radius |
|---|---|---|---|---|
| One route | v1 | 1.5 | 1.5 | 1 |
| Four narrower parallel routes | v1 | 0.5 | 0.5 | 0.5 |
| Four narrower parallel routes | v2 | 1.5 | 0.5 | 0.5 |
| Four narrower parallel routes | v3 | 0.5 | 1.5 | 0.5 |
| Four narrower parallel routes | v4 | 1.5 | 1.5 | 0.5 |
| Sixty-four small parallel routes | v1 | 0.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v2 | 1.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v3 | 2.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v4 | 3.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v5 | 4.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v6 | 5.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v7 | 6.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v8 | 7.5 | 0.5 | 0.25 |
| Sixty-four small parallel routes | v9 | 0.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v10 | 1.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v11 | 2.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v12 | 3.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v13 | 4.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v14 | 5.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v15 | 6.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v16 | 7.5 | 1.5 | 0.25 |
| Sixty-four small parallel routes | v17 | 0.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v18 | 1.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v19 | 2.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v20 | 3.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v21 | 4.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v22 | 5.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v23 | 6.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v24 | 7.5 | 2.5 | 0.25 |
| Sixty-four small parallel routes | v25 | 0.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v26 | 1.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v27 | 2.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v28 | 3.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v29 | 4.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v30 | 5.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v31 | 6.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v32 | 7.5 | 3.5 | 0.25 |
| Sixty-four small parallel routes | v33 | 0.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v34 | 1.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v35 | 2.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v36 | 3.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v37 | 4.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v38 | 5.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v39 | 6.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v40 | 7.5 | 4.5 | 0.25 |
| Sixty-four small parallel routes | v41 | 0.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v42 | 1.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v43 | 2.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v44 | 3.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v45 | 4.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v46 | 5.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v47 | 6.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v48 | 7.5 | 5.5 | 0.25 |
| Sixty-four small parallel routes | v49 | 0.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v50 | 1.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v51 | 2.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v52 | 3.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v53 | 4.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v54 | 5.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v55 | 6.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v56 | 7.5 | 6.5 | 0.25 |
| Sixty-four small parallel routes | v57 | 0.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v58 | 1.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v59 | 2.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v60 | 3.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v61 | 4.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v62 | 5.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v63 | 6.5 | 7.5 | 0.25 |
| Sixty-four small parallel routes | v64 | 7.5 | 7.5 | 0.25 |
| Tube radius (Model length units) | Area relative to the first tube | Resistance relative to the first tube | Flow relative to the first tube | Mean speed relative to the first tube |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 0.0625 | 16 | 4 |
Area coefficients multiply π; speed coefficients divide by π
Area panels retain total flow 4
Radius comparison retains the same pressure difference, length and viscosity
| Identifier | Structure or interpretation |
|---|---|
| lengthUnit | Model length units |
| areaUnit | Square model units |
| flowUnit | Fluid-volume units per interval |
| speedUnit | Model length units per interval |
| area | Sum of all circular cross-sections |
| speed | Mean flow speed: volume flow divided by total area |
| routeHeader | Constructed arrangement |
| countHeader | Number of parallel routes |
| radiusHeader | Radius of each route |
| totalAreaHeader | Combined area |
| speedHeader | Mean speed |
| piConvention | Area coefficients multiply π; speed coefficients divide by π |
| singleTube | Separate comparison: one rigid tube |
| fixedFlow | Area panels retain total flow 4 |
| fixedPressure | Radius comparison retains the same pressure difference, length and viscosity |
Constructed circular vessel cross-sections with equal assigned total flow 4 in each area panel. Parallel vessels within a panel share that flow equally. The separate radius comparison is steady laminar flow in rigid cylindrical tubes; its fixed-pressure condition is distinct from the fixed-total-flow panels.
Worked example
Constriction in B lowers total flow but raises flow in A
Initially, each parallel resistance is four. Their combined resistance is two. Adding inlet and outlet resistance one-half each gives total three, so a twelve-unit pressure rise supports flow four. The common inlet and outlet each lose two pressure units. Branch pressure difference is eight, giving flow two in each branch.
After B's resistance becomes twelve, the parallel combination is three and total resistance four. Total flow is now three. Common-route drops are each 1.5, leaving branch inlet pressure 10.5 and outlet pressure 1.5: a difference of nine.
Branch A then carries 9 ÷ 4 = 2.25, while B carries 9 ÷ 12 = 0.75. Together they carry three, matching the pump and both common routes. Flow in A has increased from two although total flow fell from four to three. The result follows from the fixed-pressure network, with no extra fluid created or feedback assumed.
Check your understanding
If both junction pressures in a branch rise by five units while their difference and resistance remain unchanged, what happens to its simple steady flow?
Show reasoning
Its flow stays unchanged in this model because the driving pressure difference stays the same. Absolute pressure could matter to a real vessel's shape or surrounding conditions, but those effects are excluded from the fixed-resistance comparison.
Why can many narrower vessels together give slower mean flow than one wider vessel?
Show reasoning
Their combined cross-sectional area can be larger. At the same total volume flow, speed is flow divided by that combined area. The model's sixty-four small routes give four times the area and one-quarter the mean speed of its one-route panel.
Does doubling one rigid tube's radius always give sixteen times the flow in a living animal?
Show reasoning
No. The sixteenfold result requires the specified laminar model with fixed pressure difference, length and viscosity. Living vessels and connected circuits can change pressure, tone, shape and throughput; the area panels instead hold total flow fixed.
What to carry forward
Compare connected pressure differences and total area under stated conditions before interpreting a vessel's flow.
References
Resources supporting this lesson’s explanations and examples.
- Biology 2e: gas exchange, circulation and excretion (2018; selected sections in chapters 39–41)Mary Ann Clark, Matthew Douglas and Jung ChoiTextbook sectionsExchange surfaces and respiratory systemsGas gradients across respiratory surfacesDissolved and carrier-associated gas transportOpen and closed circulationSelected respiratory pigmentsMammalian hearts, valves and vesselsFlow, pressure and vascular regulationOsmotic and ion balanceSelected nephron routesDiverse excretory structuresNitrogen-containing waste formsWater permeability and hormonal regulation
- College Physics 2e: diffusion, osmosis and ideal tube flow (Second edition; selected sections in chapter 12)Paul Peter Urone and Roger HinrichsTextbook sectionsViscosity and Poiseuille's relationDiffusion and osmosis