Hypotheses · perforators, up close
Can a knot hold?
If a knot is a perforator held shut, something has to hold it, and it has to let go within seconds. A first model, built only from published measurements, finds that the vessel can hold itself.
One perforator, running
A perforator's small artery, 0.24 mm across. Blood pushes its wall out; the wall pulls in with its own elastic tension and with the tone of its muscle. Over a band of tone both answers balance: an open vessel snaps shut when tone passes the fold, and a shut one stays shut until tone falls well below it. Every number comes from a measurement listed at the foot of the page, and the few that are guesses say so.
state open
- tone
- 0.19
- flow
- 1.00× rest
- debt
- 0.00
- time
- 0:00
These are the model's inputs, not instructions. It illustrates a hypothesis and measures nothing.
Four checks
What could hold a knot and let it go within seconds, checked with published numbers.
Can the vessel hold itself?
Yes: a small artery is a switch.
With measured wall strength (3.4 mN/mm; Zhang 2007), the stretch at which its muscle is strongest (El Assar 2022), how its force falls as the muscle shortens (Mulvany 1979), the thickness of healthy human small arteries' walls (Schiffrin 1995), the pressure inside it, and the timing of breath and reactive hyperaemia, the band runs from 0.62 to 0.93 of maximal tone: a shut vessel stays shut while its tone stays above 3.4 times rest (0.19). It is the instability Burton 1951 described, which clinicians now read as a closing threshold (Miller 2026); nerves hold some arterioles shut in living tissue (Bohlen 1979). A switch in 52% of 49,152 plausible parameter sets.
Can hyaluronan hold it?
No: a liquid resists motion, not position.
At rest its drag is zero. At its most viscous (healthy synovial fluid, up to 175 Pa·s; Nicholls 2018), in a collar 20 µm thin around a vessel snapping open in a second, the drag is 654 Pa: 12 times weaker than the pressure holding the vessel open. Fascia holds 6–90 µg of hyaluronan per gram (Fede 2018), 38–567 times less than joint fluid (Nakayama 2000), and without protein hyaluronan does not gel (Krause 2001). If a collar holds, it is something stiffer.
Can cooling let it go within a breath?
No: heat is too slow.
When the vessel opens, the patch it feeds warms with a time constant of at least 44 s even at the body's maximal skin blood flow (6–8 L/min; Charkoudian 2003), and about 6 minutes at a tenth of it, while losing heat to the room at 7.9 W/m²·K (de Dear 1997). Letting go within a breath needs something faster than heat.
Does a latch remember?
No: it economises.
“The rate-limiting step for relaxation appears to be removal of [Ca2+] and not latch-bridge detachment.”
Smooth muscle can hold force with less calcium and ATP, the latch (Murphy 2005), but in arteries relaxation follows the fall of calcium (Rembold 1991). A latch lets a held vessel stay held cheaply while the drive lasts; the memory is the switch. Its rate constants (Hai 1988) are not yet in an open paper; with them, the latch will run here too.
What it predicts
- Blood returns where a knot lets go. Within seconds of release, flow over the spot rises from near nothing to 1.07 of resting flow when stress eases, or 0.33 after a press while drive stays raised. Laser speckle imaging over the spot would see it; a theory with nothing held in the tissue predicts no local change. It rises above rest only as tone falls.
- Pressure, then release, lets go faster than calm alone: 0.6 s after the pressure lifts against 7 s after stress eases, with a burst from the sensory nerves of 0.15 against 0.19. The pressure runs up a debt, and returning blood repays it through the nerves.
- Slowing near the edge. As stress carries a vessel toward its fold, it answers small changes more and more slowly; flow over the spot should flicker more slowly just before a knot forms.
- Through drive, breath lets go of knots only if it is uneven. Tone rises in about a second and fades over 12 s, so an even swing with each breath holds a knot slightly tighter. The out-breath has to lower the skin's sympathetic drive more than the in-breath raises it; laser Doppler over hairy skin can measure whether it does. Breath may also act through movement: rhythmic stretch cuts smooth muscle's force at once (Ljung 1975), and squeezed arteries widen within seconds of release, more after several squeezes than after one long one (Clifford 2006). The model now carries that route, fitted to those squeezes; it acts where the breath moves the tissue, and adds up over breaths (what a breath would have to do, theory by theory, is in the field guide).
- What to measure next: how much force a small artery's muscle keeps when the vessel is nearly shut, and how thick its wall is, in skin perforators themselves (the model takes them from rat arteries and healthy human walls). More than anything else, they decide whether a knot can hold at rest.
How sure
Every uncertain parameter sampled across its range, 49,152 sets in a Sobol design. A single deep breath rarely shuts a relaxed vessel; stress has to carry tone most of the way to the fold first.
What decides whether a knot can hold at rest (total Sobol index):
Against measurements
| model | measured | used as | |
|---|---|---|---|
| lowest flow after a deep breath | 0.35 of rest | 0.29–0.41 (Lau 1995) | fitted |
| when it is lowest | 5.4 s | 4.6–5.2 s (Mayrovitz 2026) | test: 0.2 s late |
| recovered | 34 s | about 34 s, one recording (Mayrovitz 2026) | fitted |
| flush after a 5 min occlusion | +159% | +187 ± 61% (Alexandrou 2021) | sets resting tone; falls a little short |
| its peak | 11.1 s | 11.1 ± 5.1 s (Alexandrou 2021) | fitted |
Every number
Each value with its source and the line it comes from, checked word for word against the paper.
measured in the source inferred from it fitted to a measured target guessed and marked so
The vessel
| parameter | value | range | source | |
|---|---|---|---|---|
| vessel radius r₁₀₀ | 120 µm | 60–250 µm | measured | Zhang 2007 · abstract“adjacent small arteries ( approximately 240 microm)” Half the lumen diameter of the small arteries whose wall tension was measured. Perforators below 0.5 mm are small arteries in Mulvany's sense (mulvany1990). |
| maximal active wall tension Tmax | 3.4 N/m | 2.6–4.2 | measured | Zhang 2007 · abstract“the maximal active tension of arteries (3.4 +/- 0.42 mN/mm)” Maximal active wall tension (1 mN/mm = 1 N/m); the range is ± 2 SE. Rat mesenteric small arteries. |
| stretch of greatest force xopt | 0.9 r₁₀₀ | 0.85–0.95 | measured | El Assar 2022 · full text, section 2.2 (Methods)“The arteries were then set to the 90% of the determined internal circumference under a transmural pressure of 100 mmHg (L 100 ), at which point the force development was close to maximal.” Human small arteries develop their greatest active force near 0.9 of the relaxed circumference at 100 mmHg; the same sentence defines r100. |
| pressure inside the vessel P | 60 mmHg | 40–90 | inferred | Fronek 1975 · abstract“The microvascular pressure in arterioles of 70 mum in diameter or larger was proportional to the systemic arterial pressure (PA).” Pressure inside the perforator. Small arteries lie upstream of the main resistance (the 20–35 µm arterioles), so their pressure is a large fraction of arterial pressure; results are shown across the whole range. |
| stiffness of the relaxed wall β | 7 | 4–10 | guessed | Stiffness of the relaxed wall (passive tension rises as exp(β·(x − 1))). To be sourced from pressure-myograph curves. |
| width of the length–tension curve w | 0.37 | 0.3–0.45 | inferred | Mulvany 1979 · abstract“Below L0 (the length at which maximum tension, delta T0, was developed) the active tension fell with decreasing length along a line which extrapolated to 0.38 L0” Width of the active length–tension curve, as a fraction of the optimal muscle length. The Gaussian that best fits the measured line (zero at 0.38 of the optimal circumference) has width 0.37. Rat mesenteric arteries of 100–200 µm; taken as the length of the muscle cells, which followed the vessel's circumference in the same study. |
| thickness of the wall aw | 0.177 r₁₀₀² | 0.13–0.3 | inferred | Schiffrin 1995 · abstract“at which time it was not different from that of arteries from normotensive subjects (5.2 +/- 0.2%)” Cross-sectional area of the media, in units of r100². Human gluteal subcutaneous small arteries: media thickness 5.2% of the lumen diameter in normotensive people (7.5–8% in untreated hypertension). Measured at 0.9 of the relaxed circumference at 100 mmHg, which gives an area of 0.177 r100² (hypertension: 0.26–0.28). Sets how short the muscle becomes when the lumen closes: thicker walls keep it longer, and stronger. |
| shut radius xc | 0.08 r₁₀₀ | 0.04–0.15 | guessed | Lumen radius at which the vessel is shut to flow; flow scales as radius⁴, so flow here is under 0.1% of rest. |
| speed of the wall τx | 0.3 s | 0.1–1 | guessed | How fast the wall moves to its mechanical balance. |
| latency of the gasp reflex tlat | 3.7 s | 3.6–3.8 | measured | Mayrovitz 2026 · full text, Review“The median latency, measured from the start of the IG to the onset of the vasoconstriction, was reported as 3.6-3.8 seconds with no difference between genders, and the time to the minimum LDF value was 4.6-5.2 seconds, depending on the hand measured and gender” Delay between a deep inspiratory gasp and the constriction of skin vessels. |
| rise of tone τup | 1.2 s | 0.8–1.6 | inferred | Mayrovitz 2026 · full text, Review“The median latency, measured from the start of the IG to the onset of the vasoconstriction, was reported as 3.6-3.8 seconds with no difference between genders, and the time to the minimum LDF value was 4.6-5.2 seconds, depending on the hand measured and gender” Tone rises fast; flow reaches its minimum 1.0–1.4 s after the constriction begins. |
| fade of nerve-driven tone τdown | 12.3 s | 3–20 | fitted | Mayrovitz 2026 · full text, Inspiratory gasp reflex“The overall duration of the reflex to recovery in this image is approximately 34 seconds, with the interval of near-zero LDF at approximately 10 seconds” How slowly nerve-driven tone fades once the drive stops (transmitter clearance and muscle relaxation). Fitted to one subject's recovery; to be refined with more recordings. Fitted: a gasp's constriction recovers (flow back to 95%) about 34 s after the gasp. |
| the gasp's burst tburst | 1 s | 0.6–1.6 | inferred | Mayrovitz 2026 · full text, Review“The median latency, measured from the start of the IG to the onset of the vasoconstriction, was reported as 3.6-3.8 seconds with no difference between genders, and the time to the minimum LDF value was 4.6-5.2 seconds, depending on the hand measured and gender” A gasp's burst of sympathetic firing is brief: flow is lowest 1.0–1.4 s after the constriction begins. |
| fall in flow after a gasp Δqgasp | 0.65 | 0.594–0.714 | measured | Lau 1995 · abstract“the index of vasoconstrictor response (per cent change) induced by inspiratory gasp was higher in young males than in young females (71.4% versus 59.4%; P < 0.05)” Fall in fingertip skin blood flow after a deep gasp; the model sets the gasp's drive so an open vessel loses this share of its flow. Hairy skin, where most knots are, may respond less. |
| size of reactive hyperaemia Δqflush | 1.87 | 1.26–2.48 | measured | Alexandrou 2021 · abstract“controls: 187 ± 61%, P = 0.001” Rise of forearm skin perfusion from baseline to the peak of reactive hyperaemia. Sets resting tone, since a fully relaxed vessel must carry 2.87 times resting flow. |
| timing of reactive hyperaemia tflush | 11.1 s | 6–16.2 | measured | Alexandrou 2021 · abstract“controls: 11.11 ± 5.1 s” Time from release of an occlusion to peak perfusion (± 1 SD). The model's flush is checked against it. |
| build-up of oxygen debt τdebt | 40 s | 20–120 | guessed | How fast an unsupplied patch runs up an oxygen debt that its sensory nerves sense. |
| speed of the sensory nerves τn | 4.77 s | 1–10 | fitted | Alexandrou 2021 · abstract“controls: 11.11 ± 5.1 s” How fast the sensory nerves release their dilator when blood returns to a patch in debt (sensory nerves and BKCa channels carry the skin's reactive hyperaemia, lorenzo2007). The dilator relaxes the muscle directly. Fitted so the flush peaks when the measured one does. Fitted: reactive hyperaemia peaks at porh_peak_time after a 5 min occlusion. |
| share supplied by neighbours c | 0.7 | 0.3–0.95 | guessed | Saint-Cyr 2009 · abstract“Each perforasome is linked with adjacent perforasomes by means of two main mechanisms that include both direct and indirect linking vessels.” Share of a shut perforator's territory that neighbours keep supplied through linking vessels. The linking is measured; the share is a guess. |
| myogenic relaxation under pressure kmy | 0.6 | 0.3–0.9 | guessed | Mulvany 1990 · abstract“investigation of the vasoconstrictor response to raised intravascular pressure (the Bayliss response)” How much tone a small artery gives up when its transmural pressure is taken away (the Bayliss response run backwards). The response is established; its size here is a guess. |
| timing of myogenic relaxation τmy | 20 s | 10–60 | guessed | How long the myogenic relaxation takes under sustained pressure. |
| breath's swing in tone δbreath | 0.03 | 0–0.08 | guessed | How much ordinary slow breathing swings tone (a fraction of maximal activation), rising on the in-breath and falling on the out-breath. |
| widening after one 1 s squeeze Δd₁ | 0.16 | 0.14–0.18 | measured | Clifford 2006 · abstract“the peak increases in diameter were, respectively, (mean +/-s.e.m.) 16 +/- 2, 14 +/- 2 and 27 +/- 3%” Rat soleus feed arteries at 68 mmHg, squeezed shut by 600 mmHg for 1 s; the rise in diameter after release (± s.e.m.). |
| widening after one 5 s squeeze Δd₅ₛ | 0.14 | 0.12–0.16 | measured | Clifford 2006 · abstract“the peak increases in diameter were, respectively, (mean +/-s.e.m.) 16 +/- 2, 14 +/- 2 and 27 +/- 3%” The same squeeze held five times longer: no larger. The wall answers the change, not how long it lasts. |
| widening after five 1 s squeezes Δd₅ₓ | 0.27 | 0.24–0.3 | measured | Clifford 2006 · abstract“the peak increases in diameter were, respectively, (mean +/-s.e.m.) 16 +/- 2, 14 +/- 2 and 27 +/- 3%” Five 1 s squeezes, 1 s apart: the loosening adds up, though less than five times. |
| peak after one 1 s squeeze t₁ | 4.1 s | 3.8–4.4 | measured | Clifford 2006 · abstract“times from release of pressure to peak diameter of 4.1 +/- 0.3, 4.6 +/- 0.7 and 2.8 +/- 0.4 s” From release to the widest diameter (± s.e.m.). |
| peak after one 5 s squeeze t₅ₛ | 4.6 s | 3.9–5.3 | measured | Clifford 2006 · abstract“times from release of pressure to peak diameter of 4.1 +/- 0.3, 4.6 +/- 0.7 and 2.8 +/- 0.4 s” |
| peak after five 1 s squeezes t₅ₓ | 2.8 s | 2.4–3.2 | measured | Clifford 2006 · abstract“times from release of pressure to peak diameter of 4.1 +/- 0.3, 4.6 +/- 0.7 and 2.8 +/- 0.4 s” |
| the wall's memory of its shape τmv | 0.2 s | 0.05–1 | guessed | How long the wall takes to register a change of shape. The squeeze measurements favour a short memory (the fit worsens above about 1 s); while it is short, what a movement does depends on how much the tissue moves, not how fast. |
| loosening per change of shape kmv | 1.95 1/s | 0.1–20 | fitted | Clifford 2006 · abstract“The magnitude of dilatation was not affected by increasing the duration of compression but was enhanced by increasing the number of compressions” How much each change of shape (a squeeze, its release, a movement) loosens the wall's force, saturating. Measured in rat muscle feed arteries and applied to a skin perforator: inferred across vessels. Rhythmic stretch cuts vascular smooth muscle's force directly (ljung1975). Fitted: the widening after one squeeze, one long squeeze and five squeezes, and when each peaks. |
| how long a loosening lasts τw | 9.72 s | 1–60 | fitted | Clifford 2006 · abstract“times from release of pressure to peak diameter of 4.1 +/- 0.3, 4.6 +/- 0.7 and 2.8 +/- 0.4 s” How long the loosening from a change of shape lasts before the wall's force returns. Fitted: the widening after one squeeze, one long squeeze and five squeezes, and when each peaks. |
| how fast a loosening shows τz | 2.07 s | 0.2–15 | fitted | Clifford 2006 · abstract“times from release of pressure to peak diameter of 4.1 +/- 0.3, 4.6 +/- 0.7 and 2.8 +/- 0.4 s” The delay between a change of shape and the wall's loosening: why the widening peaks seconds after release. Fitted: the widening after one squeeze, one long squeeze and five squeezes, and when each peaks. |
| the breath's movement at a knot δmove | 0.1 | 0–0.5 | guessed | How much each breath deforms the tissue at a knot, as a fraction of a squeeze that shuts the vessel. Unmeasured: ultrasound speckle tracking over a knot during the breath could measure it. The trials sweep it. |
The checks
| quantity | value | source | |
|---|---|---|---|
| hyaluronan over trapezius and deltoid | 6 µg/g | measured | Fede 2018 · abstract“6 μg g-1 in the fascia overlying the trapezius and deltoid muscles” Hyaluronan in the fascia over trapezius and deltoid, where knots are commonly reported. |
| hyaluronan in the retinacula | 90 µg/g | measured | Fede 2018 · abstract“In the fascia that surrounds joints, the hyaluronan increases to 90 μg g-1” The richest fascia measured, the retinacula of the ankle. |
| viscosity of synovial fluid | 175 Pa·s | measured | Nicholls 2018 · full text, Discussion“The observed zero shear rate viscosity for synovial fluid from a healthy joint ranges from 1 to 175 Pa s” Healthy synovial fluid holds milligrams of hyaluronan per millilitre, tens to hundreds of times the fascia's; its viscosity is an upper bound for any collar. |
| hyaluronan in synovial fluid | 3.4 mg/ml | measured | Nakayama 2000 · abstract“The mean HA concentration was 3.4+/-0.6 mg/ml.” Hyaluronan in normal knee joint fluid, the source of the viscosity bound (eta_max); roughly 40–570 times the fascia's. |
| thickness of the collar | 50 µm | guessed | Thickness of the hyaluronan-rich layer around the vessel where it passes through the fascia. A thinner collar shears faster, so the low end is the collar's best case. |
| radiative heat loss | 4.5 W/m²·K | measured | de Dear 1997 · abstract“The radiative heat transfer coefficient measured for the whole-body was 4.5 W/m2 per K for both the seated and standing cases” Heat lost from the skin by radiation, per degree. |
| convective heat loss | 3.4 W/m²·K | measured | de Dear 1997 · abstract“the whole-body natural convection coefficient for the manikin fell within the mid-range of previously published values at 3.4 and 3.3 W/m2 per K when standing and seated respectively” Heat lost from the skin by convection in still air, per degree. |
| maximal skin blood flow | 7 L/min | measured | Charkoudian 2003 · abstract“skin blood flow can reach 6 to 8 L/min during hyperthermia” Whole-body skin blood flow at its maximum; the fastest a patch can be warmed from within. |
| skin area | 1.8 m² | guessed | Skin area of an adult, to turn whole-body flow into flow per square metre. |
| heat capacity of tissue | 3.7 MJ/m³·K | guessed | Volumetric heat capacity of skin and subcutis, which are mostly water (water is 4.2e6). |
| heat capacity of blood | 3.8 MJ/m³·K | guessed | Volumetric heat capacity of blood. |
| depth of the warmed layer | 3 mm | guessed | Depth of skin and superficial fat whose temperature a perforator's flow sets. |
And in the sources' own words
- hyaluronan does not gel
“There is no evidence of intermolecule hydrogen bonding causing gel formation in NaHA solutions without protein present.”
Krause 2001 · abstract - the latch, in the model
“Comparison of the rate constants indicates that latch-bridge detachment is the rate-limiting step.”
Hai 1988 · abstract - the latch, in tissue
“The rate-limiting step for relaxation appears to be removal of [Ca2+] and not latch-bridge detachment.”
Rembold 1991 · abstract - the latch's economy
“In sustained contractions, Ca2+, cross-bridge phosphorylation, and ATP consumption rates fall, a phenomenon termed "latch".”
Murphy 2005 · abstract - age and adrenergic drive
“The inhibition of STIM/Orai with YM-58483 (20 μM) reversed adrenergic hypercontractility in MA from older subjects”
El Assar 2022 · abstract - arterioles held shut by nerves
“Following denervation, a 22.2 +/- 6.3% (SE) and 61.8 +/- 12 increase in the number of third-order arterioles open to flow occurred in WKY and SHR.”
Bohlen 1979 · abstract - the closing threshold
“When pressure in an arteriole falls below its critical closing pressure (CCP), the smooth muscle generates enough wall tension to snap the vessel shut.”
Miller 2026 · full text, section 5 - staying shut
“it falls abruptly towards zero and remains minimal until pressure rises above the CCP threshold”
Miller 2026 · full text, section 5 - the threshold in skin
“Results demonstrate that a relatively high intraarterial pressure must be applied to this skin flap before blood flow will commence.”
Cutting 1982 · abstract
How it was made
The model is written once, in Python, and the code this page runs is generated from it; a test holds the two to the same trajectories within a millionth. Parameters come from the library, each with the line it comes from, checked word for word against the paper. Fitted values are fitted automatically to their measured targets; guesses are marked, and the sampling above spans their ranges.
Limits: 10 parameters are guesses. The reflex timings are from the skin of the hands, while most knots are reported over hairy skin, where the gasp reflex may be weaker. The movement route is calibrated on squeezed muscle arteries of the rat, not on skin. The bench is one vessel alone; the tree of a parent and its children is on the research page, and every theory, this one among them, runs through the same scenes in the field guide. The simulation shows what a mechanism can do with measured numbers; it does not show that knots are perforators.
run 2026-09-28 · inputs 85d8aaae2e58 · numpy 2.4.6, scipy 1.17.1, sympy 1.14.0, SALib 1.6.0
the findings notethe modelthe parametersthe planthe perforator hypothesis