Mathematic Models of HPA Axis

Mathematical models of Hypothalamic–pituitary–adrenal axis

Anatomy

 * 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

Chemistry

 * CRH
 * ACTH
 * Cortisol

Receptors

 * Glucocorticoid Receptor
 * MR

GR Expression model (Gupta et. al)
(using equations and notation from http://www.ploscompbiol.org/article/info:doi%2F10.1371%2Fjournal.pcbi.1000273 )


 * F - input to system, stress, stimulates hypothalamus
 * C - CRH, released by Hypothalamus, stimulates pituitary
 * A - ACTH, released by pituitary, stimulates adrenals
 * O - Cortisol, output of the system, inhibits production of both CRH and ACTH

Constant Degradation Rate
This model assumes that all three chemicals undergo degradation, each at their own constant rates.
 * $$\frac{\mathrm{d}\mathbf{C}}{\mathrm{d}t} = -K_{cd}$$
 * $$\frac{\mathrm{d}\mathbf{A}}{\mathrm{d}t} = -K_{ad}$$
 * $$\frac{\mathrm{d}\mathbf{O}}{\mathrm{d}t} = -K_{od}$$

Inputs to Hypothalamus
In this model, the hypothalamus is stimulated by two sources-- circadian rhythm input (K_c) and stress input (F):
 * Input = $$K_c + F$$

Cortisol inhibition of Hypothalamus sensitivity
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.

CRH equation

 * Stress stimulation
 * Circadian stimulation
 * Linear inhibition from Cortisol
 * Linear Degradation

Cortisol Equation (Adrenals equation)
Cortisol production is stimulated by ACTH. Additionally, cortisol undergoes linear degradation.
 * ACTH stimulation    $$K_OA$$
 * Linear Degradation  $$-K_{od}$$

Both effects are captured in the following equation:
 * $$\frac{\mathrm{d}\mathbf{O}}{\mathrm{d}t} =K_OA - K_{od}O$$

CRH Equation (Hypothalamus Equation)

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

ACTH Equation (Pituitary Equation)

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