Gamma Dose

This occurs when the release of the drug into a specific compartment follows a Gamma curve. See:-

https://en.wikipedia.org/wiki/Gamma_distribution

Note

See the Gamma Dose with no elimination. for Tut Script used to generate results in this section.

The mathematical expression for an Gamma dose starting at time $t_S$ with Weibull parameters $\lambda$ and $\kappa$ and total dose amount $\text{AMT}$ in compartment Central is:-

$\text{d[CENTRAL]} = \text{d[CENTRAL]} + \text{R(t)}$

where:-

$R(t)= \Biggl \lbrace { \frac{ \beta^\alpha (t-t_S)^{\alpha-1} e^{-\beta (t-t_S)}}{\Gamma(\alpha)} ,\text{ if } { t \geq t_S } \atop 0.0, \text{ otherwise } }$

Where R(t) is the rate of drug absorption at time (t) and $\lambda$ and $\kappa$ are parameters of the Gamma. R(t) is the Gamma density function scaled by the AMT parameter. Since a density function has unit area under the curve, the AMT scaling ensures that the total dose administered is equal to AMT.

This can be implemented in PoPy using:

@gamma{amt: c[AMT], lag: m[LAG], alpha: m[ALPHA], beta: m[BETA]}

in the equation for the appropriate compartment in the DERIVATIVES section of a Fit Script or ref:tut_script. See @gamma.

Here the parameters map to the Gamma equation as follows:-

Parameter	Symbol
alpha	$\alpha$
beta	$\beta$

The lag is coded in the same way as for a bolus dose. A lag time merely delays the start time of the gamma dose.

An example of a gamma dose added to a one compartment model with no lag time is:

DERIVATIVES: |
    d[CENTRAL] = @gamma{amt: c[AMT], alpha: m[ALPHA], beta: m[BETA]} - m[KE]*s[CENTRAL]

Fig. 7 Cumulative amount following a Gamma dose with no elimination