Table 1.
Steady-state values for concentrations of CRH, ACTH, free GR and circulating cortisol.
Table 2.
Parameter settings for the differential equation model of the HPA axis proposed by Gupta et al. [10].
Figure 1.
Steady states of the HPA axis system.
Steady-state concentration of CRH (x1), ACTH (x2), GR (x3) and cortisol (x4) as a function of the external stressor f for the model expressed as system H. The system naturally accommodates 3 stable steady states at rest f = 0 and over a broad range of increasing values for f.
Figure 2.
Migration of cortisol concentration from one stable point to another.
Concentration of circulating cortisol plotted as a function of the external stressor f. A first idealized trajectory (red - -) describes the displacement of the system from rest to a peak cortisol concentration followd by an eventual lapse into a chronic hypocortisolic state. A second idealized trajectory (green - -) illustrates the effects of treatment. Here removal of cortisol can be thought of as a negative stress f. An increase in ACTH concentration of ∼30% above baseline serves as a signal that the treatment may be discontinued.
Figure 3.
Idealized corrective control action.
Concentrations of CRH (x1), ACTH (x2), GR (x3) and cortisol (x4) as a function of time in response to an ideal externally applied perturbation in cortisol u(t). The negative supplement in cortisol signifies a pharmaceutical removal or inactivation of circulating cortisol. ACTH concentration serves to monitor the progress of the treatment which is discontinued when ACTH increases by ∼30% over baseline.
Figure 4.
A suboptimal but clinically realistic control treatment.
Concentrations of CRH (x1), ACTH (x2), GR (x3) and cortisol (x4) as a function of time in response to a suboptimal but more realistic externally applied perturbation in cortisol u(t). Once again the negative supplement in cortisol signifies a pharmaceutical removal or inactivation of circulating cortisol. Note a less severe reduction in cortisol is applied over a longer period. The corresponding ACTH response is slower but the threshold concentration for cessation of treatment remains the same.
Figure 5.
Balancing intensity and duration of treatment.
Diagram of the minimal perturbation in normalized circulating cortisol u(t) as a function of duration of treatment. In one extreme instance the perturbation in cortisol would be so small that no treatment would be effective regardless of how much treatment prolonged. Conversely an excessively short treatment would also be ineffective regardless of the intensity of cortisol reduction.