# 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").

Ordinary differential equations (ODEs) for this
*n*-compartmental model are:

, where *u* is the input as a function of time,
*q _{n}* is the concentration in compartment

*n*and

*q̇*is its derivative. The

_{n}*q*could then be used as the time-lagged input to the central compartment of a PK model. Laplace transform gives (DiStefano, 2013)

_{n}(t)and thus the transfer function is

The temporal response in the *n* th compartment to input *u(t)* is the inverse
Laplace transform of *Q _{n}* (DiStefano, 2013).
If a perfect bolus (unit impulse) would be injected into compartment 1 at

*t=0*,

*U(s) = Dose × 1*, the temporal response function is

and the time-lagged input function to the central compartment (blood) of the PK model is then

, which with *Dose = 1* is the probability density function (PDF) of the
Erlang distribution:

, 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*.
The overall mean transit time (OMTT) for the whole chain of compartments is
*OMTT = n/k * (DiStefano, 2013).
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.

The cumulative distribution function (CDF) of the Erlang distribution (integral of PDF from 0 to t) is

, that is, with *n=1* , *g(t)=1-e ^{-kt}* ;
with

*n=2*,

*g(t)=1-e*; with

^{-kt}-kxe^{-kt}*n=3*,

*g(t)=1-e*; and so on.

^{-kt}-kxe^{-kt}-(kx)^{2}e^{-kt}/2If, 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 the
response function. This approach has been used to estimate the
portal vein input to the liver, in which in case the
2-compartmental transit-time model was used (*n=2*), with transfer function

, where *β=1/k*.

The ODEs of the *n*-compartmental transit-time model for
discrete-time data could alternatively be solved using the
second-order Adams-Moulton method

, where *C _{0}(t)* is the discrete input function,

*Δt*is the sample time difference, and the output of this system is

*k × ∫*. The integrals can be calculated stepwise using equation

_{0}^{T}C_{n}(t)## 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.
doi: 10.1007/978-3-642-93107-9.

Tags: Input function, Fitting, Transit-time model

Updated at: 2020-01-29

Created at: 2019-11-05

Written by: Vesa Oikonen