Phoenix models the input function as a piecewise linear “precursor” function fp(t) convolved with an exponential “dispersion” function fd(t). The former provides substantial flexibility whereas the latter provides smoothness. The piecewise linear component is parameterized in terms of a sum of hat-type wavelet basis functions, hj(t):
where xj is the dose scaling factor within a particular observation interval and Tj are the wavelet support points. The hat-type wavelet representation enables discontinuous, finite duration drug releases to be considered together with other factors that result in discontinuous delivery, such as stomach emptying, absorption window, pH changes, etc. The dispersion function provides the smoothing of the input function that is expected from the stochastic transport principles governing the transport of the drug molecules from the site of release to subsequent entry to and mixing in the general systemic circulation.
The wavelet support points (Tj) are constrained to coincide with the observation times, with the exception of the very first support point that is used to define the lag-time, if any, for the input function. Furthermore, one support point is injected halfway between the lag-time and the first observation. This point is simply included to create enough capacity for drug input prior to the first sampling. Having just one support point prior to the first observation would limit this capacity. The extra support point is well suited to accommodate an initial “burst” release commonly encountered for many formulations.
The dispersion function fd(t) is defined as an exponential function:
where d is denoted the dispersion or smoothing parameter. The dispersion function is normalized to have a total integral (t=0 to ¥) equal one, which explains the scaling with the d parameter. The input function, f(t), is the convolution of the precursor function and the dispersion function:
The general convolution form of the input function above is consistent with stochastic as well as deterministic transport principles. The drug level profile, c(t), resulting from the above input function is, according to the linear disposition assumption, given by:
where “*” is used to denote the convolution operation.