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.
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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.
Updated at: 2019-11-05
Created at: 2019-11-05
Written by: Vesa Oikonen