Sets where the pulse-width optimum falls. Fit it so the predicted optimum matches a protocol you know works.
Ω at the bulge, log scale. Dashed vertical stub on each curve marks the epidermal ceiling — fluence beyond it burns the epidermis before it kills the follicle.
| Wavelength | Local F | Bulge T | Ω | Epi T | F to kill | Epi ceiling | Verdict |
|---|
Margin is the gap between “F to kill” and “Epi ceiling”. Where the ceiling is the lower of the two, no fluence on that wavelength destroys the follicle at this depth without burning the epidermis — change the wavelength, the cooling, or the depth you are aiming at.
Ω against pulse width at the marker fluence. The peak is the confinement time of the follicular unit.
Ω against target depth at the marker fluence. Where each curve crosses Ω = 1 is the kill depth for this configuration.
Percentage of the whole follicle population affected per session, convolved over depth, calibre and pigment. Solid = destroyed (Ω ≥ 1). Dashed = driven to catagen without destruction (10⁻² ≤ Ω < 1).
Depth is the whole argument. The default target is the bulge at 1.3 mm, because that is where permanence is decided — the stem cell reservoir. The matrix sits deeper, around 3–4 mm for coarse terminal hair. Destroying the matrix while sparing the bulge gives loss that regrows, because the follicle rebuilds from above. A configuration is only permanent if it clears Ω = 1 at the bulge. Check both depths before judging any device: a configuration that kills at the matrix and stuns at the bulge produces months of clearance followed by full return, which is the pattern most often mistaken for treatment failure.
The pulse-width optimum is a fitted parameter, not a prediction. Its position is set almost entirely by the bulge offset — 4 ms at 15 µm, 7 ms at 25 µm, 13 ms at 40 µm, 22 ms at 60 µm. The model predicts the shape of the curve reliably: a broad maximum, a gentle approach from short pulses, and a collapse beyond it of three orders of magnitude by 30 ms past the peak. It does not independently predict where the maximum falls. Fit the offset against a clinical protocol known to work, then read everything else as conditional on that fit. The optimum shifts longer with coarser hair, because the bulge sits further from the shaft axis; that direction holds at any offset.
The population curve is capped by the anagen fraction. Telogen follicles have no pigmented shaft coupling the beam to the bulge, so no fluence reaches them. That ceiling — not the efficacy curve — is what limits clearance per session, and it is why the destruction curve saturates rather than climbing. The catagen curve rises first, peaks at low fluence, then falls as follicles that were being stunned start being killed instead. A device sitting on the rising edge of the catagen curve and the flat part of the destruction curve is doing almost nothing permanent while producing a great deal of visible shedding.
Ω is per follicle, not per population. It is the damage integral for one follicle of the stated calibre at the stated depth with the stated pigment. A real leg carries a spread of all three, so a device sitting at Ω = 1 on this chart clears the shallower, coarser, darker half of the population and leaves the rest. Population clearance per session is always lower than the curve suggests.
Bands. Ω ≥ 1 is follicle destruction. Ω between 10⁻² and 1 is sub-lethal injury sufficient to drive the follicle into dystrophic catagen — visible shedding, temporary reduction, and the regime in which repeated regular exposure entrains the growth cycle. Below 10⁻⁴ nothing measurable happens. The catagen threshold is the least certain number in this model; treat the stun band as indicative rather than settled.
What the model omits. Multiple hairs per follicular unit, pigment in the outer root sheath, pulse stacking and in-motion bulk heating, output droop over a treatment run, and any difference between the emitting aperture and the contact window. The first two make it pessimistic; the rest make manufacturer figures optimistic. Measure the aperture and the delivered energy before trusting any quoted fluence, including on this chart.
Parameters. Melanin absorption after Jacques (6.6×10¹¹·λ−3.33 cm⁻¹); reduced scattering 45.3·(λ/500)−1.292 cm⁻¹; f_mel 0.02–0.33 for types I–VI; Arrhenius A = 1.8×10⁵¹ s⁻¹, Ea = 327 kJ/mol (Jia et al.). Diffuse transport with a subsurface buildup factor of 2.0. Heat spread modelled as an energy-conserving radial Gaussian, w² = (d/2)² + 4ατ, with the bulge at 25 µm outside the shaft.