<Dd> M (D) = tanh − 1 ⁡ (D − 4.9 3) d M d D d D d t + 2.3026 tanh − 1 ⁡ (D − 4.9 3) = 5.2 − 0.45 ln ⁡ (Φ (t − τ) 4.8118 × 10 − 10) (\ displaystyle (\ begin (aligned) M (D) () & = \ tanh ^ (- 1) \ left ((\ frac (D - 4.9) (3)) \ right) \ \ (\ frac (\ mathrm (d) M) (\ mathrm (d) D)) (\ frac (\ mathrm (d) D) (\ mathrm (d) t)) + 2.3026 \ tanh ^ (- 1) \ left ((\ frac (D - 4.9) (3)) \ right) & = 5.2 - 0.45 \ ln \ left ((\ frac (\ Phi (t - \ tau)) (4.8118 \ times 10 ^ (- 10))) \ right) \ end (aligned))) </Dd> <P> where D (\ displaystyle D) is the pupil diameter measured in millimeters and Φ (t − τ) (\ displaystyle \ Phi (t - \ tau)) is the luminous intensity reaching the retina in a time t (\ displaystyle t), which can be described as Φ = I A (\ displaystyle \ Phi = IA): luminance reaching the eye in lumens / mm times the pupil area in mm . τ (\ displaystyle \ tau) is the pupillary latency, a time delay between the instant in which the light pulse reaches the retina and the beginning of iridal reaction due nerve transmission, neuro - muscular excitation and activation delays . d M (\ displaystyle \ mathrm (d) M), d D (\ displaystyle \ mathrm (d) D) and d t (\ displaystyle \ mathrm (d) t) are the derivatives for the M (\ displaystyle M) function, pupil diameter D (\ displaystyle D) and time t (\ displaystyle t). </P> <P> Since the pupil constriction velocity is approximately 3 times faster than (re) dilation velocity, different step sizes in the numerical solver simulation must be used: </P> <Dl> <Dd> d t c = T c − T p S d t d = T c − T p 3 S (\ displaystyle (\ begin (aligned) \ mathrm (d) t_ (c) & = (\ frac (T_ (c) - T_ (p)) (S)) \ \ \ mathrm (d) t_ (d) & = (\ frac (T_ (c) - T_ (p)) (3S)) \ end (aligned))) </Dd> </Dl>

What is the pathway of light into the eye