A probability density function described by a gamma-variate distribution.

Description

A probability density function (PDF) described by a gamma-variate
distribution.  In place of the gamma function, this model 
utilizes a factorial approximation formula developed by Gergo Nemes. The gamma
variate function is extensively used to describe indicator dilution curves.

The moments of the PDF are:
The zeroth moment, M0, is the Area under the PDF curve and is often set to 1.
The first moment, M1, is the Mean of the PDF which allows us to calculate the 
    Mean transit time (MTT): MTT = M1/M0
The second moment, M2, is the Variance (SD squared)
The third moment, M3, is used to calculate the Skewness (asymmetry of the curve,
    if Skewness is zero then curve is symmetric):
    Skewness = M3/(M2^3/2)

For the gamma variate PDF, we can calculate alpha, beta, and ta from the MTT, skewn
and RD (All dependent on the moments calculated above):
    1) alpha = 4/(skewn^2) -1
    2) beta = MTT*RD*skewn/2, (RD = SD/Mean of the PDF = sqrt(M2)/M1)
    3) ta = MTT-beta*(alpha +1)


GENERAL RESULTS:
The model presented here reproduces the JSim function generator curve
within a range of parameter values.  However, the model returns NaN
in the initial portion of the curve where the function generator returns
0. A fit is maintained for t.min and t.max values that do not cut off portions 
of the curve that are significantly greater than zero.  The curves match well for
skew values <= 1.4 , but error increases as skew increases towards 2.

Equations

The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.