A heart cell’s behaviour is influenced by the different ion (e.g. sodium, potassium, etc.) currents flowing into and out of the cell across its membrane. As part of the drug cardiac safety assessment process, drugs are applied to specially-engineered cells that allow the drug’s effect on a particular ion current of interest to be measured. Most of the drugs considered reduce the flow of particular ions by physically blocking these particular ion channels. During ion channel screening, the reduction of a particular ion current, as a percentage, is recorded.

One of the simplest mathematical models describing ion channel blocking is a sigmoid (“S”-shaped) function called a Hill curve, which is defined by two parameters:

1. IC50 value — the concentration at which the ion current is 50% blocked, and
2. Hill coefficient — determines the steepness of the curve at the IC50 value.

The Hill curve is also a function of the drug concentration, which we will call x, and is defined as

which we call the dose-response model.

A dataset is of the form $\{ (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\},$ where xi is the ith experimentally-applied drug concentration, and yi is the ith experimentally-observed percentage block. Often, the same concentrations will be repeated, so there will be repeats of xi values with (probably) different corresponding yi values.

We want to fit Equation \eqref{dose-response-model} to this experimental dataset, and to see how much information is contained about the parameters IC50 and Hill.