Asymmetrical (five-parameter) logistic dose-response curves

The standard log(dose) vs. response curve is defined by the bottom, top, EC50, and slope. In this curve, the top and bottom parts are mirror images of each other -- the curve is symmetrical.

Some log(dose) vs. response curves are not symmetrical. This can be modeled by including a fifth parameter that describes the asymmetry of the curve. The standard curve is sometimes called a four-parameter logistic model, so the asymmetrical curve is called a five parameter logistical model. 

Of course, an equation should not be referred to by its number of parameters. Some authors assume that any nonspecific signal is already subtracted off, so present the equations in a form where Bottom is defined to be zero and doesn't appear in the equation. Then the symmetrical variable slope equation has three (not four) parameters, and the asymmetrical form has four (not five) parameters.

Asymmetrical dose-response curves can be described by several equations. Prism uses the RIchards version (from Giraldo et. al.), which  is built-in to the 'Dose-response -- Special' group of equations in Prism 5 and later. This is also called the generalized Hill equation.

LogXb = LogEC50 + (1/HillSlope)*Log((2^(1/S))-1)
Numerator = Top - Bottom
Denominator = (1+10^((LogXb-X)*HillSlope))^S
Y = Bottom + (Numerator/Denominator)

This equation assumes that X has been entered as (or transformed to) the logarithm of concentration, and that Y is the response in any convenient units. 

S is the asymmetry parameter. If s=1.0, then this is the same as the four parameter equation. When s is not 1.0, the curve will be asymmetrical. S must be greater than zero, but can be less than, or greater than, 1.0. 

Top and Bottom are the Y values at the top and bottom plateaus of the curve. If you have normalized the data, you may want to constrain these values to 100 and 0. 

The equation above fits the logEC50, which is the X value when Y is half-way between the Top and Bottom plateaus. It is in the same units as the X values -- the logarithm of concentration.  Note that the logEC50 is not the same as the inflection point (which can be higher or lower than the logEC50 depending on the value of S).

If your goal is to obtain meaningful best-fit parameters, then you'll need lots of high quality data. It is very hard to fit both slope and asymmetry with tight confidence intervals. If your goal is just to interpolate unknowns from a standard curve, the width of the confidence intervals of the parameters doesn't really matter. What you want is a curve that follows the data, and in some cases an asymmetrical five parameter model does so better than a four parameter model. 

 Links:

• Other formulations of asymmetrical dose-response curves have been developed. For example, Ricketts and Head developed a model for use in baroreflex studies. 
• Bindslev has written a lengthy on-line text, Drug-Acceptor Interactions.  Chapter 10,  Hill in Hell discusses many models of dose-response curves, including asymmetrical ones. 
• Liao and Liu have done simulations that show the advantage of fitting the EC50 rather Xb. 
• Gottschalk and Dunn review the properties of the 5Pl. 'Edited Oct 2014, to remove the implication that logXb is the inflection point.
• Cumberland and colleagues compare the 4PL and 5PL (and discuss various ways to write the equation).