To relate measured changes in fundus reflectance to rhodopsin, we now present a formal description of rhodopsin absorption, bleaching, and regeneration in situ. During the presentation of a step of light of wavelength
λ whose duration
t is sufficiently short that regeneration can be neglected, the rate of change of the fraction
p of rhodopsin in a rods obeys
34 where
αλγ is the photosensitivity, with
αλ (unit: μm
2 molecule
−1) being the extinction coefficient of rhodopsin in situ for axially propagating light and
γ the quantum efficiency of isomerization/bleaching,
Iabs (unit: photons μm
−2 s
−1) is the rate of photon capture by rhodopsin per unit area of the retina occupied by rods,
Iλ,cornea (unit: photons μm
−2 s
−1) is the retinal irradiance in the absence of losses in the ocular media,
τλ is the transmissivity of the ocular media and retina up to the depth where light is guided into the rod,
fwg is a dimensionless factor (≤1) by which the rod waveguide condenses or funnels light into the outer segment,
DRho,λ is the optical density of the pigment in the dark adapted eye for axially propagating light, and parameters indexed by
λ are wavelength-dependent. Solution of Equation 1 yields a relation in which
p(
t) is expressible as a function of the bleaching energy
Iλ,corneat, and “bleach versus energy” (
p versus
Iλ,corneat) curves can be generated readily from it with numerical methods. In the special case of low pigment density (
DRho,λ ≤ 0.1), the solution of Equation 1 is an exponential decay:
The product
represents the time-integrated photon irradiance incident on the base of the outer segment (after adjustment for media losses
τλ and for waveguide condensation by the waveguide factor
fwg), multiplied by the photosensitivity
αλγ of rhodopsin in the disc membranes. The argument of the exponential in Equation 2 is dimensionless, and can be reparametrized in other useful ways. Thus, when the time-integrated irradiance is converted to illuminance (unit: scot Td-s), Equation 2 can be rewritten as:
where
, and
sλ a scaling constant proportional to
αλ.