Mathematic Models of HPA Axis

Overview of HPA axis edit

Hypothalamus - Activated by stress to release CRH , negative feedback from Cortisol
Pituitary gland - Activated by CRH to release ACTH , negative feedback from Cortisol
Adrenal glands - Activated by ACTH to release Cortisol

This model assumes that all three chemicals undergo degradation, each at their own constant rates.

{\frac {\mathrm {d} \mathbf {C} }{\mathrm {d} t}}=-K_{cd}

{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}=-K_{ad}

{\frac {\mathrm {d} \mathbf {O} }{\mathrm {d} t}}=-K_{od}

In this model, the hypothalamus is stimulated by two sources-- circadian rhythm input (K_c) and stress input (F):

Input =

K_{c}+F

In this model, Cortisol decreases hypothalamic sensitivity to input:

\left(1-{\frac {O}{K_{il}}}\right)

That is-- When O=0, sensitivity would be at 100%. As O increases, sensitivity decreases.

Cortisol production is stimulated by ACTH. Additionally, cortisol undergoes linear degradation.

Both effects are captured in the following equation:

{\frac {\mathrm {d} \mathbf {O} }{\mathrm {d} t}}=K_{O}A-K_{od}O

{\frac {\mathrm {d} \mathbf {C} }{\mathrm {d} t}}=(K_{c}+F)*\left(1-{\frac {O}{K_{i1}}}\right)-K_{cd}C

{\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}=K_{a}C*\left(1-{\frac {O}{K_{i2}}}\right)-K_{ad}A