Why red and near-infrared

Shine a torch through your fingers in a dark room and the light that emerges is red. Blue and green light are absorbed within a millimetre or so by haemoglobin in the blood and by melanin in the skin; red and near-infrared light, from about 650 nm to about 1,350 nm, are absorbed much less, because haemoglobin and melanin absorb weakly there and water does not yet absorb strongly. This range is called the optical window of tissue. Beyond about 1,350 nm, water absorption rises steeply and closes the window again (Jacques 2013).

Inside the window, what limits penetration is mostly not absorption but scattering. Tissue is full of membranes, fibres and organelles with slightly different refractive indices, and a photon in the dermis changes direction many times per millimetre. Most of the light that enters the skin is scattered back out or wanders sideways; only a share goes deep.

Absorption, scattering and the effective attenuation coefficient

Tissue optics describes this with a few numbers, each measured per centimetre of path: the absorption coefficient $\mu_a$; the scattering coefficient $\mu_s$; and the anisotropy $g$, the average cosine of the scattering angle, which is about 0.9 in tissue, meaning that most scattering is in a forward direction. The combination that matters for deep light is the reduced scattering coefficient, $\mu_s' = \mu_s(1-g)$. Far from the source, where light has been scattered many times, the diffusion approximation gives an exponential fall of fluence with depth,

$$\phi(z) \propto e^{-\mu_{\text{eff}}\,z}, \qquad \mu_{\text{eff}} = \sqrt{3\mu_a(\mu_a + \mu_s')},$$

so the “1/e depth” is $1/\mu_{\text{eff}}$ (Jacques 2013).

import math

# Reduced scattering, power law fitted by Jacques (2013): µs' = a (λ/500 nm)^−b, cm⁻¹
SCAT = {"skin": (46.0, 1.421), "brain": (24.2, 1.611), "bone": (22.9, 0.716)}
def mus_r(tissue, nm):
    a, b = SCAT[tissue]
    return a * (nm / 500) ** -b

# Absorption of the dermis: blood (150 g/L haemoglobin) plus water.
EPS = {660: (319.6, 3226.6), 850: (1058.0, 691.3)}   # HbO2, Hb, cm⁻¹/M
WATER = {660: 0.0036, 850: 0.043}                     # cm⁻¹
def mua_dermis(nm, blood=0.02, sat=0.75, water=0.65):
    oxy, deoxy = EPS[nm]
    c = 150 / 64500                                   # mol/L of haemoglobin
    mua_blood = 2.303 * c * (sat * oxy + (1 - sat) * deoxy)
    return blood * mua_blood + water * WATER[nm]

def mu_eff(mua, musr):
    return math.sqrt(3 * mua * (mua + musr))

for nm in (660, 850):
    ma, ms = mua_dermis(nm), mus_r("skin", nm)
    me = mu_eff(ma, ms)
    print(f"{nm} nm dermis: µa = {ma:.3f}, µs' = {ms:4.1f}, µeff = {me:.2f} cm⁻¹, "
          f"1/e depth ≈ {10 / me:.1f} mm")

# The epidermis adds melanin absorption, µa = 519 (λ/500)^−3.5 cm⁻¹ × volume fraction,
# over about 0.1 mm.
for nm in (660, 850):
    mel = 519 * (nm / 500) ** -3.5
    light, dark = (math.exp(-mel * f * 0.01) for f in (0.03, 0.30))
    print(f"{nm} nm epidermis passes ≈ {light:.0%} (light skin), {dark:.0%} (dark skin)")
# 660 nm dermis: µa = 0.114, µs' = 31.0, µeff = 3.27 cm⁻¹, 1/e depth ≈ 3.1 mm
# 850 nm dermis: µa = 0.131, µs' = 21.6, µeff = 2.93 cm⁻¹, 1/e depth ≈ 3.4 mm
# 660 nm epidermis passes ≈ 94% (light skin), 55% (dark skin)
# 850 nm epidermis passes ≈ 98% (light skin), 78% (dark skin)

Two points stand out. In the dermis itself the difference between 660 nm and 850 nm is modest: both have a 1/e depth of about 3 mm, with near-infrared a little deeper, because scattering falls with wavelength. The larger difference is in the epidermis: melanin absorbs much more at 660 nm, so darkly pigmented skin passes roughly half of the red light but three quarters of the near-infrared. In both cases, by 1 cm depth the fluence has fallen to a few per cent of its value near the surface.

Through the skull

Claims that light shone on the head reaches the brain need numbers. The light must pass through the scalp, about 5 mm of tissue rich in blood; the skull, several millimetres of bone; and a thin layer of cerebrospinal fluid.

import math

# A crude layered head: each layer attenuates fluence by exp(−µeff · thickness).
# µeff in cm⁻¹: scalp from the dermis model above; skull chosen inside the
# published range 2.2–5.1 cm⁻¹; CSF from cadaver measurements (Pitzschke 2015).
LAYERS = [("scalp", 0.5, {660: 3.3, 850: 2.9}),
          ("skull", 0.7, {660: 3.8, 850: 3.1}),
          ("CSF",   0.2, {660: 2.3, 850: 1.0})]
H_scalp = 0.050 * 20 * 60          # J/cm²: 20 minutes at 50 mW/cm² on the scalp

for nm in (660, 850):
    frac = math.exp(-sum(mu[nm] * d for _, d, mu in LAYERS))
    print(f"{nm} nm: ≈ {frac:.1%} reaches the cortical surface; "
          f"scalp {H_scalp:.0f} J/cm² → cortex ≈ {H_scalp * frac:.1f} J/cm²")
print("(order of magnitude only: hair, beam size and back-scatter change it several-fold)")
# 660 nm: ≈ 0.8% reaches the cortical surface; scalp 60 J/cm² → cortex ≈ 0.5 J/cm²
# 850 nm: ≈ 2.2% reaches the cortical surface; scalp 60 J/cm² → cortex ≈ 1.3 J/cm²
# (order of magnitude only: hair, beam size and back-scatter change it several-fold)

On this rough estimate, of the order of one to a few per cent of the light reaching the scalp arrives at the surface of the brain, more at near-infrared than red wavelengths, and correspondingly less at any depth within the cortex. Hair, which absorbs and scatters strongly, can reduce it much further. A dose that is large at the scalp therefore becomes a dose of the order of a joule per square centimetre at the cortex, which, as Chapter 5 shows, is not negligible on the scale of laboratory studies. Whether such doses do anything useful in the living human brain is a separate question that only clinical trials can answer.