Resistance, Compliance and Local Blood Flow

In the skeletal muscle of an anesthetized cat, only about 15 percent of arterial pressure is spent crossing the capillary bed, although capillaries are its narrowest vessels. So where does the pressure go, and how does each tissue set its share of the cardiac output? Poiseuille’s law and a first resistance calculation are in blood and vessels.

Pressure falls most where the arterioles are

Flow through any segment equals the pressure difference across it divided by its resistance. For the whole systemic circuit, aorta to right atrium:

TPR = (MAP - Pra) / CO

Total peripheral resistance (TPR) lumps every systemic vessel into one resistance; MAP is mean arterial pressure and Pra right atrial pressure. Its unit, mmHg min per L, is the Wood unit; multiplying by 80 gives dyn s cm^-5, because 1 mmHg is 1,333 dyn per cm^2 and 1 L per min is 16.7 cm^3 per s. It is a steady-state number that reflects arteriolar tone.

Resistances in series add (R = R1 + R2 + R3). The same flow passes each, so a segment’s share of the pressure drop equals its share of the resistance.

Poiseuille’s law, R = 8 x viscosity x length / (pi x r^4), assumes steady laminar flow of a Newtonian fluid in a rigid tube, which blood in vessels only approximates: in rat mesentery, 10-micrometre microvessels at normal hematocrit had about four times the resistance of a glass tube of the same bore (Pries and colleagues 1994). By continuity, flow equals mean velocity times cross-sectional area, so blood moves slowest in the capillary bed, whose combined area dwarfs the aorta’s. Flow stays laminar, in smooth layers, while the Reynolds number (density x velocity x diameter / viscosity) is low; in a straight pipe it tends to turn turbulent above roughly 2,000. Fast flow in a wide vessel (the ascending aorta at peak ejection) or a jet past a narrowing favours turbulence, which costs extra pressure and makes noise.

Where does the drop happen? Fronek and Zweifach (1975) measured pressure inside single microvessels of the tenuissimus, a thin thigh muscle, in cats under pentobarbital anesthesia. Arterioles 35 to 20 micrometres across were the principal source of resistance setting the pressure downstream; the capillary bed proper took only about 15 percent of arterial pressure, and the smallest venules (8 to 15 micrometres) averaged about 24 mmHg. Capillaries are narrow but short, and very many lie in parallel. In the heart, too, small arteries and arterioles hold most resistance. The brain may differ: the authors of a 2021 mouse study state that its capillary bed has the highest resistance in the cerebral tree.

Beds in parallel share one pressure

The organs are fed side by side from the aorta and drain side by side into the venae cavae, so each sees nearly the same driving pressure, MAP - Pra. Resistances in parallel add as reciprocals:

1/R = 1/R1 + 1/R2 + 1/R3 …

Every added or widened parallel path lowers the total. So if a large muscle mass dilates and nothing constricts elsewhere, TPR falls and, unless cardiac output rises enough, so does MAP. When sympathetic restraint is lost, as in autonomic failure, pressure falls during large-muscle exercise. The reflexes that normally prevent this are in regulating arterial pressure.

Lumped, illustrative numbers, no reflexes.

Muscle arteriolar radius ×1.00 (resting = ×1.00) Cardiac output 6.0 L per min (worked problem: 6.0) Resistance of the other beds ×1.00 (×1.00 = 24; constriction raises it)

Muscle bed resistance

Other beds resistance

Total peripheral resistance, TPR

Mean arterial pressure, MAP

Flow to the muscle bed

Flow to the other beds

In this sketch: steady state, lumped resistances, starting values from the lesson’s worked problem (adult human at rest), right atrial pressure 3 mmHg, and no reflexes, so nothing changes cardiac output or the other beds when pressure changes. Resistances are in mmHg min per L. The worked problem’s made-up round numbers: the muscle bed has arteries 4, arterioles 24, capillaries 6 and venules and veins 6 (total 40), all other beds together are 24, and cardiac output is 6.0 L per min. Poiseuille scaling (R proportional to 1/r⁴) is applied to the muscle arterioles only; the lesson says it holds only approximately in vivo. At the starting setting this sketch gives MAP 93.0 mmHg, where the lesson’s 93.3 comes from the one-third rule with the driving pressure rounded to 90. Made up for the sketch: the slider ranges (radius 0.7 to 1.6 times resting, cardiac output 4 to 15 L per min, other beds 1 to 2 times 24), and that “constricting the other beds” simply multiplies their 24. The lesson’s exercise step lowers the muscle bed to 8 by dilating arterioles and feeding arteries and recruiting capillaries; here only the arterioles change, so the muscle bed cannot go below about 20. Arteriole block height follows radius; all drawings are schematic, not to scale.

Rule to keep: parallel beds share one driving pressure, MAP − 3 mmHg, so every widened parallel path lowers TPR; within the muscle bed the segments are in series with the same flow, so each segment’s share of the pressure drop equals its share of the resistance.

Worked problem: one muscle bed inside the whole circulation

Made-up round numbers, adult human at rest, steady state: heart rate 75 per min, stroke volume 80 mL, pressure 120/80 mmHg, right atrial pressure 3 mmHg. Assume lumped resistances, Poiseuille scaling and the one-third rule for MAP at this heart rate.

1. Whole circuit. CO = 75 per min x 0.080 L = 6.0 L per min. Pulse pressure (PP) = 120 - 80 = 40 mmHg. MAP = 80 + 40/3 = 93.3 mmHg. TPR = (93.3 - 3) / 6.0 = 15.05, about 15 mmHg min per L or 1,200 dyn s cm^-5.
2. Series. Muscle bed: arteries 4, arterioles 24, capillaries 6, venules and veins 6 mmHg min per L, total 40. All other beds together: 24. Check: 1/40 + 1/24 = 0.0250 + 0.0417 = 0.0667, so R = 15.0. (The split suits the arithmetic; resting human muscle takes a smaller share of output.)
3. Drops. With 90 mmHg of driving pressure (rounded), muscle flow = 90 / 40 = 2.25 L per min and the rest 90 / 24 = 3.75; sum 6.0. In muscle: arteries 2.25 x 4 = 9 mmHg, arterioles 54 mmHg (60 percent), capillaries 13.5 mmHg (15 percent, close to the cat value), veins 13.5 mmHg. Total 90.
4. Radius. Halving arteriolar resistance (24 to 12) needs the radius up by the fourth root of 2, 1.19, about 19 percent. Halving the radius would raise resistance 16-fold.
5. Exercise. Dilation of arterioles and feeding arteries lowers the muscle bed to 8. Then 1/8 + 1/24 = 0.1667, so TPR = 6.0. At unchanged output, MAP = 6.0 x 6.0 + 3 = 39 mmHg; to hold 93 mmHg, output must reach 90 / 6.0 = 15 L per min, or other beds must constrict (exercise puts the two together).
6. Compliance. C = SV / PP = 80 mL / 40 mmHg = 2.0 mL per mmHg (next section), treating 120/80 as aortic pressure; a cuff reading on the arm usually has a wider pulse than the aorta. Halve it at the same stroke volume and MAP: PP = 80 mmHg, DBP = 93.3 - 80/3 = 66.7 mmHg, SBP = 66.7 + 80 = 146.7 mmHg. So 120/80 becomes about 147/67: systolic and pulse pressure up, diastolic down, mean unchanged. Real stiffening also changes how the pulse wave travels and reflects. Its main effect is a wider pulse; mean pressure depends on cardiac output and resistance.

Compliance turns a pulsing pump into steadier flow

Compliance is the change in volume per change in pressure, in mL per mmHg; its inverse, elastance, measures stiffness.

C = delta V / delta P

During ejection the arteries fill faster than they drain, so their walls stretch, then recoil in diastole and keep capillary flow going. Otto Frank’s two-element Windkessel (a compliance feeding a resistance), popularized in 1899, explains the smooth diastolic fall of aortic pressure but fits systole poorly; Frank’s other work is in circulation and the heart. In 31 adults at rest during cardiac catheterization, 7 of them controls and the rest patients with heart disease (Chemla and colleagues 1998), stroke volume divided by aortic pulse pressure gave a reliable estimate of total arterial compliance, a resting rule of thumb.

Mean arterial pressure is the time average of the pressure curve, the area under it divided by the cycle length, not the average of systolic and diastolic. At resting heart rates diastole fills roughly two thirds of the cycle, which gives the standard estimate, attributed to Gauer (1960):

MAP ≈ DBP + PP/3

Invasive studies suggest one third underestimates the mean, perhaps nearer 0.4, and the error grows with heart rate because diastole shortens more than systole; a heart-rate-corrected formula, checked against aortic catheter recordings at paced rates in 12 patients (Razminia and colleagues 2004), adds a heart-rate term.

Stiffer arteries widen the pulse. In Framingham longitudinal data on 2,036 untreated adults, after age 50 to 60 diastolic pressure declined, pulse pressure rose steeply and mean pressure levelled off, consistent, the authors judged, with large-artery stiffening.

Veins are the compliant side. In humans they hold about 70 percent of the blood and are roughly 24 times more compliant than arteries in the standard textbook estimate. Unstressed volume fills a vessel without stretching its wall; stressed volume stretches it and makes pressure. About 75 percent of venous volume is normally unstressed, so veins act as a reservoir, and venoconstriction shifts unstressed into stressed volume without adding blood (cardiac output and venous return).

Tissues set their own flow

Autoregulation holds a bed’s flow relatively constant over a range of perfusion pressures while its metabolic demand is unchanged. Its lower limit is the pressure at which the resistance vessels are maximally dilated; below it, flow follows pressure. One mechanism is the myogenic response, the acute reaction of a small artery to a change in transmural pressure, told with Bayliss’s 1902 experiment in smooth muscle. The kidney adds tubuloglomerular feedback (glomerular filtration).

Reactive hyperemia is the surge of flow when a limb’s arterial inflow is released after a brief occlusion. Folkow proposed in 1949 that removing pressure-induced stretch of small vessels is a major factor in its first stage, and accumulated metabolites probably add to it. In the human forearm, nitric oxide plays a minimal to modest role at the peak and prostaglandins matter for peak flow (Tagawa and colleagues 1994; Engelke and colleagues 1996), so reactive hyperemia is not a test of endothelial nitric oxide. Flow-mediated dilation of a conduit artery is a separate measurement.

The endothelium sets basal tone. How its relaxing factor was found (Furchgott and Zawadzki 1980) and identified as nitric oxide (1987) is in senses and cGMP. In healthy volunteers, Vallance, Collier and Moncada (1989) infused L-NMMA, a nitric oxide synthase (NOS) inhibitor, into the brachial artery: basal forearm flow halved, and dilation to acetylcholine was blunted but not dilation to glyceryl trinitrate, a nitric oxide donor. Resting arterioles are dilated continuously. Mice lacking endothelial NOS (Huang and colleagues 1995) lost acetylcholine relaxation of aortic rings and were hypertensive, though non-endothelial NOS isoforms may help maintain pressure, and the rise need not be purely vascular.

Metabolism sets flow most clearly in the heart, where flow tracks oxygen use. In 1963 Robert Berne asked whether oxygen-poor heart muscle releases a dilator in amounts that could explain this. He used isolated perfused cat hearts and open-chest dogs. If such a dilator exists, hypoxia should lower coronary resistance and dilator-like material should appear in the coronary venous effluent in sufficient quantity. Hypoxia did lower coronary vascular resistance, and the myocardium released inosine and hypoxanthine, breakdown products of adenosine: by his estimate, 3 to 27 times more than would double coronary flow if infused as adenosine. He proposed that adenosine, formed as ATP breaks down when oxygen supply falls short of use, diffuses to the arterioles and dilates them. That rules in a sufficient amount, if the purines leave the cell as adenosine, an assumption Berne stated; he did not measure adenosine, and release does not prove necessity. Later receptor blockade in dogs, reviewed by Goodwill and colleagues (2017), found adenosine a potent dilator whose role is not evident unless oxygen delivery is compromised.

Design problem

Berne’s result leaves open whether adenosine is necessary. Using an isolated perfused heart or an anesthetized dog, design an experiment to test whether adenosine is needed to match coronary flow to oxygen demand.

One way to do it

Raise demand at normal oxygen supply (pacing or a positive inotrope), and separately lower the supply. Measure coronary flow and myocardial oxygen consumption (flow times the arteriovenous oxygen content difference) and plot one against the other. Repeat under adenosine receptor blockade or with an enzyme that removes adenosine, and prove the block works by showing that the response to infused adenosine is lost. Also measure coronary venous oxygen, because blockade can lower flow and oxygen consumption together and hide an effect on their slope. If adenosine is the coupler, venous oxygen should fall as demand rises. If it is one of several redundant signals, little changes at normal supply, but blockade worsens the oxygen balance when supply is compromised, as the dog studies found. Measure adenosine in interstitial fluid if possible; during the heart’s own adenosine production, venous levels underestimate it, but during adenosine infusion they overestimate it. A negative result in one preparation does not exclude a role elsewhere, and several mediators may each be small, as in exercising muscle.

Four circulations, four sets of rules

Heart. At rest the left ventricle extracts 60 to 80 percent of delivered oxygen, so more oxygen use is met mainly by more flow. Systole compresses vessels inside the muscle, so left ventricular inflow rises in diastole.

Add a fixed narrowing upstream of a bed. It is a resistance in series that climbs steeply as its radius falls; blood speeds through it, and past a tight narrowing the flow is disturbed and Poiseuille fails. Distal pressure falls, the downstream arterioles autoregulate by dilating, and resting flow stays near normal. The cost is reserve: once those vessels reach the lower limit, flow follows pressure and cannot rise with demand, and the inner layers of the left ventricle, fed mainly in diastole, fare worst. Net: stenosis resistance up, distal pressure and arteriolar resistance down, resting flow about unchanged, maximal flow down.

Brain. Lassen’s 1959 curve, often taught as constant flow from 60 to 150 mmHg, was extrapolated from a cross-sectional comparison of 376 people in 7 publications; a 2021 review argues that flow is held relatively stable, not constant, and that the range is no true plateau. Cerebral vessels are also very sensitive to arterial PCO2: a rise dilates them, a fall constricts them.

The modern experiment asks which segment controls brain flow. Hall and colleagues (2014), in rodent brain slices and in vivo, reported that capillaries dilated before arterioles during sensory input and estimated that they produced 84 percent of the flow increase. Hill and colleagues (2015) tested this in mouse cortex in vivo, imaging with a two-photon microscope while light activated channelrhodopsin in mural cells. If capillary pericytes contract, light should narrow the capillaries they cover. Vessels under 10 micrometres covered by pericytes but not by smooth-muscle-actin-expressing cells did not constrict, and the authors concluded that arteriolar smooth muscle regulates cerebral flow. Attwell and colleagues replied that the conflict depends on how a pericyte is defined. Hartmann and colleagues (2021) then reported that stimulated pericytes constricted capillaries slowly and that ablating one pericyte doubled blood-cell flux in capillaries that lost its contact, so pericytes contribute to basal resistance and slow modulation. Almost all of this work is in rodent brain, mostly mouse cortex. How much fast functional hyperemia depends on capillary dilation is unsettled, although capillaries can sense neural activity and signal arterioles upstream to dilate.

Skeletal muscle. In human studies no single dilator explains much more than 20 to 30 percent of exercise hyperemia, and working muscle blunts sympathetic constriction (functional sympatholysis, a contested term). Quadriceps perfusion in knee-extensor exercise reached about 2.5 L per kg per min (Andersen and Saltin 1985); exercise follows the redistribution.

Skin. Skin flow follows body temperature more than skin metabolism. In human whole-body heating it can reach 6 to 8 L per min, 80 to 90 percent of the dilation coming from a sympathetic active vasodilator system (skin and human heat).

Keep one contrast straight: hypoxia dilates systemic arterioles but constricts pulmonary arterioles (ventilation and perfusion).