Mathematic Models of HPA Axis

Mathematical models of w:Hypothalamic–pituitary–adrenal axis

Overview of HPA 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 $K_{O}A$
• Linear Degradation $-K_{od}$

Both effects are captured in the following equation:

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

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)

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

References

http://www.ploscompbiol.org/article/info:doi%2F10.1371%2Fjournal.pcbi.1000273 http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1804264/