Sets where the pulse-width optimum falls. Fit it so the predicted optimum matches a protocol you know works.
Solid = dermal papilla, heated through the pigmented matrix cup. Dashed = bulge, heated through the shaft. Vertical stub marks the epidermal ceiling. A permanent result needs both curves above Ω = 1.
| Wavelength | Bulge T | Ω bulge | DP T | Ω DP | Epi T | F permanent | Epi ceiling | Outcome |
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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.
Solid = DP, dashed = bulge. There is no interior optimum — efficacy is flat at short pulses and falls beyond a knee. Triangles mark each compartment's knee, the point at which Ω has halved. That boundary, not a peak, is what limits usable pulse width.
Ω at the DP against bulb depth, at the marker fluence. Where it crosses Ω = 1 is the deepest papilla this configuration reaches.
Percentage of the whole population per session. Solid = permanent (both compartments destroyed). Dashed = reversible loss (one compartment only). Dotted = catagen without destruction. Clearest with one or two wavelengths selected.
Two targets, two escape routes. The dermal papilla is the signalling centre that instructs the follicle, and it is the better optical target: during anagen the matrix that envelops it is loaded with melanosomes and runs roughly three times the shaft diameter, so it absorbs far more energy per unit length than the shaft does. But the papilla can be reconstituted by mesenchymal cells of the dermal sheath. The bulge is the epithelial stem cell reservoir and cannot be replaced, but it couples only to the thin shaft. Destroy one and the follicle recovers; destroy both and it cannot. That is why this model computes them separately and why the verdict column requires both.
Pulse width has a knee, not an optimum. Ω is computed by numerical integration of dT(r,t) = E′(t)/(ρc·πw²(t))·exp(−r²/w²(t)), with w²(t) = r₀² + 4αt and E′ accruing through the pulse — uniform steps through the pulse, log-spaced steps through the decay tail. At fixed absorbed energy a shorter pulse gives a higher peak temperature, and Arrhenius is superexponential, so efficacy is flat at short pulse widths and falls beyond a knee rather than peaking. The DP knee always sits beyond the bulge knee, because heat must cross the matrix cup to reach the papilla. What bounds short pulses is outside this model: shaft vaporisation and mechanical rupture rather than conduction, and epidermal peak temperature.
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. A telogen follicle keeps its club hair, but the proximal end of a club hair is depigmented — matrix melanocytes shut down before the shaft finishes forming, which is why a plucked telogen hair has a white bulb. There is no absorber where it matters, and the papilla has retracted upward away from any heated tissue. 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.
Outcomes are classified follicle by follicle, then aggregated. The two compartments are strongly correlated — the same beam, depth, calibre and pigment drive both — so the chance of a permanent result is not the product of two independent probabilities. Each sampled follicle is classified on its own pair of Ω values before the weights are summed. Ω itself is per follicle: 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. Perfusion, any latent heat sink at 100 °C, and the possibility that papilla fibroblasts and bulge keratinocytes have different Arrhenius constants — one pair is used for both, which is a real simplification. Also omitted: 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.
IPL bands are not reduced to a representative wavelength: the band is sliced at 25 nm and each slice propagated with its own μ_eff, melanin absorption and epidermal transmission, then summed. Emission and action centroids differ substantially — 650–1200 nm emits at a centroid of 870 nm but its absorption-weighted centroid is 782 nm. The lamp spectrum is an analytic approximation, output is truncated at 1200 nm (so water-absorbed IR that heats epidermis without reaching follicles is omitted), and multi-pulse trains are not modelled.
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.