SUPPLEMENTAL DIGITAL CONTENT. APPENDIX 1
(Methods Appendix)
Dynamic model of human in vivo cortisol secretion, binding and clearance:
We assume the system of three simultaneous, non-linear differential equations given below
govern the dynamic response of the human cortisol compartments to the endogenous secretion of
free cortisol Zc [1-3].
Supplemental Figure 1 (Supplemental Digital Content 2,
http://links.lww.com/CCM/B109) represents these three differential equations and illustrates the
concept that there are two mechanisms that simultaneously influence the compartmental
distribution and concentration of circulating cortisol. One mechanism is related to the binding
(and unbinding) of free cortisol in equations 2) and 3). The second mechanism is related to
cortisol secretion and free cortisol clearance and is represented in equation 1). Dynamic change
of total cortisol concentration over time, as well as reasonable estimates of on-rates (κ1) and offrates (κ-1) for cortisol binding to serum proteins, are needed to obtain unique estimates of cortisol
secretion rate (Zc) and free cortisol elimination rate constant (). Equilibrium between
compartments occurs rapidly (<1 sec), which is to be distinguished from the longer period of
time (>1 hr) required to achieve steady-state conditions following a significant change in ZC.
Steady state solution: For conditions where the rates of cortisol appearance and disappearance
are equal, setting the derivative in equations 1) – 3) equal to zero (i.e. steady state) and the
subsequent algebraic solution show that the steady state secretion rate satisfies the following
equation:
ZC (t) = α XF
(Eq4)
Estimation of baseline free-cortisol secretion (CSRbase): Assuming that steady state obtains at
baseline, Eq. 4 provides a method of estimating baseline free cortisol secretion rate:
CSRbase = α XF ,
4’)
where free cortisol concentration XF at baseline is either measured or computed using a cubic
equilibrium equation [4] and α = ln(2)/t½ is computed by fitting the dynamic model equations 1)3) to measured cortisol concentrations.
Least squares solution algorithm (pseudocode):
Initialization
Set albumin and CBG concentrations to measured values
Read experimental measurement pairs (time, total concentration)
Select secretion rate and half life starting values
Specify convergence tolerance and maximum iterations
Set Free, CBG-bound, and albumin-bound cortisol initial values to
(first measured total cortisol )/3
Repeat Levenberg Marquardt algorithm
Integrate 3 simultaneous equations for cortisol concentrations
Sum the three compartments to obtain computed values
Interpolate the computed values to the measurement times
Compute squared error of difference between measured and computed
concentrations
Compute new estimates of secretion rate and half life using gradient and damping
Until convergence fit criteria are met
References
1. Dorin RI, Qiao,Z, Qualls,CR et al. Estimation of maximal cortisol secretion rate in healthy
humans. J Clin Endocrinol Metab 2012;97:1285-93.
2. Keenan DM, Roelfsema,F, Veldhuis,JD. Endogenous ACTH concentration-dependent drive
of pulsatile cortisol secretion in the human. Am J Physiol Endocrinol Metab 2004;287:E652E661.
3. Keenan DM, Roelfsema,F, Carroll,BJ et al. Sex defines the age dependence of endogenous
ACTH-cortisol dose responsiveness. Am J Physiol Regul Integr Comp Physiol
2009;297:R515-R523.
4. Dorin RI, Pai,HK, Ho,JT et al. Validation of a simple method of estimating plasma free
cortisol: role of cortisol binding to albumin. Clin Biochem 2009;42:64-71.