# gamma_variate

Model number
0053

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.

References
``` Thompson HK, Starmer CF, Whalen RE, and McIntosh HD. Indicator transit time
considered as a gamma variate. Circ Res 14: 502-515, 1964.

Davenport R. The Derivation of the Gamma-Variate Relationship for
Tracer Dilution Curves. J Nucl-Med 24: 945-948, 1983.

Gergo Nemes factorial approximation, presented by www.luschny.de/
and found at www.luschny.de/math/factorial/approx/SimpleCases.html
```
Key terms
gamma variate distribution
probability density function
pdf
moments
Acknowledgements

Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.

Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.