Hysteresis Analysis

I have been exploring some aspects of hysteresis analysis. A few preliminary results of some simulation studies are presented here for your comments/feedback. Later, I hope to have access to some "real" data to further explore some of these ideas.

In a pharmacodynamic (PD)-pharmacokinetic (PK) context, hysteresis is said to obtain when a plot of the drug effect, E, against the corresponding plasma drug concentration (PDC) is multivalued, i.e. a given PDC value corresponds to two or more E-values. Figure 1 shows a typical hysteresis plot (HP) plot of miotic effect versus PDC.

In this plot, in the region of maximum PDC, (at the right end), E, continues to to increase, despite the essentially stationary plasma values of about 17.5 units.

When effect, E, is observed to be a single valued function of plasma drug concentration, a frequently used "Effect" model is Hill's equation and its generalizations. We have:-

………………………………………………………..( 1 )

where E denotes the PD effect (on a zero to 1.00 scale), and C denotes the corresponding PDC. The parameter, C_50, represents that value of the PDC for which E = 0.5, while represents an arbitrary "shape" parameter controlling the curvature of the E-PDC curve.

In the context of PD-PK analysis, equation (1) may be simply derived using methods going back to Nernst, Langmuir, and others. Let C be the concentration of drug in some phase (such as the plasma) in contact with a surface containing N receptor sites, and let effect E be directly proportional to the fraction of sites which have formed drug-site complexes. Let denote the fraction (0 1.0) of the N sites complexed with drug. We have:

and

where and denote, respectively, the rates of condensation onto, and evaporation from, the active surface sites.

At equilibrium and solving for yields = E= C/(K+C). The shaping parameter, , is introduced to improve agreement between the observed data and the fitted function.

When such quasi equilibrium does not obtain the resulting HP will be a "loop", rather than a single valued function.

To examine the non-equilibrium case we let and denote, respectively, the PDC and PD effect at time, t, measured from time of administration (t=0) of a single dose of drug. Let and denote the corresponding derivatives with respect to time.

We have:-

…………………………(2a)

…………………………………………..(2b)

with

and

denote, respectively, the rate of elimination from, and absorption into the plasma phase, the rates of exchange between the plasma and "effect" phases, and the rate of destruction of the complexes,

as the differential equations connecting PDC and effect, E.

I put together several pieces of software to play with this. One package (not presented here) takes "real" data and estimates the rate constants (not a trivial task).on a subject - by - subject basis for input to subsequent statistical analysis. Another routine (in MS-DOS, and Windows 16 bit mode) simply takes a set of parameters and plots the solution curves for Equations (2). The idea was to get some idea of how the system looks.

Table 1 presents the 5 sets of parameter values chosen for exploration of the behavior of this system.

Table 1-Trial Rate Parameters used in Simulated Hysteresis.
Case No.	M*	k1	k2	k12	k21	kd	Comment
Case 1	25.00	0.100	0.200	0.100	0.050	0.020	Moderate Binding
Case 2	25.00	0.100	0.200	0.100	0.900	0.020	Loose Binding
Case 3	25.00	0.100	0.050	0.100	0.050	0.020	Retarded absorption, Moderate Binding
Case 4	25.00	0.100	0.050	0.100	0.900	0.020	Retarded absorption, Loose Binding
Case 5	25.00	0.100	0.050	0.900	0.050	0.020	Retarded absorption, Strong Binding

*

Figures 2 (a&b) through 6 show plots of the solution curves.

If and when I get hold of real data I will try out the parameter estimation process. Meanwhile your comments and criticisms would be appreciated. If you want a no warranty, "as-is", copy of the little routines I mentioned above, let me know by email. You are welcome to the source code as well.

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