# Input function time lag using transit-time model

Catenary chain of compartments with one-way flow (Figure 1) can be used to generate a distribution of time lags (“linear chain trick”).

If a perfect bolus (unit impulse) would be injected into compartment 1 at *t=0*,
the outflow from compartment *n* as a function of time can be exactly described with
probability density function of the Erlang distribution
(Equation 1).

, where *n≥1*, *k≥0*, and *t≥0*.
Assuming that compartment number *n* is known, the delayed response can be described
with just one parameter *k*.

With *n=1* Erlang distribution simplifies into exponential distribution.
With two compartments (*n=2*) the probability density function becomes similar to the gamma
variate ‑based surge function. Related functions have
been used to fit input TACs in DSC- or DCE-MRI and
contrast-enhanced CT, and, when extended with terms accounting for recirculation, also the PET
input TACs.
In the decay of radioactive isotope, the events occur independently with
certain average rate, and the waiting times between *n* events are Erlang distributed.
The related Poisson distribution describes the number of events in a given time.

If, instead of a perfect bolus, we have measured the input TAC to a system, and we wish to
model the delayed response in the system (vasculature, organs), we cannot use the
Erlang probability density function given above, but instead we
can convolve the input function with response function,
which in case of the *n*-compartmental transit-time model is

, where *β=1/k*. This approach has been used to estimate the
portal vein input to the liver.

## See also:

- Fitting input function
- Exponential functions as PET model input
- Gamma variate functions as PET model input
- Compartmental models for input function
- Input function
- Input for simulations
- Fitting compartmental models
- Fitting TTACs
- Plasma pharmacokinetics
- Area under curve (AUC)
- Integral Calculator
- Derivative Calculator

## References

DiStefano III J. *Dynamic Systems Biology Modeling and Simulation*. Academic Press, 2013.
ISBN: 9780124104112.

Jacquez JA. Density functions of residence times for deterministic and stochastic compartmental
systems. *Math Biosci.* 2002a; 180: 127-139.
doi: 10.1016/s0025-5564(02)00110-4.

Jacquez JA, Simon CP. Qualitative theory of compartmental systems with lags.
*Math Biosci.* 2002b; 180: 329-362.
doi: 10.1016/s0025-5564(02)00131-1.

MacDonald N: *Time Lags in Biological Models*. Springer, 1978.

Tags: Input function, Fitting, Transit-time model

Updated at: 2019-11-05

Created at: 2019-11-05

Written by: Vesa Oikonen