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").

**Figure 1.** One-way catenary compartmental model.

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

$\begin{align*} \dot q_1 &= u - k \: q_1 \\ \dot q_2 &= k \: (q_1 - q_2) \\ \dot q_3 &= k \: (q_2 - q_3) \\ &\vdots \\ \dot q_n &= k \: (q_{n-1} - q_n) \\ \end{align*}$

, where u is the input as a function of time, q_n is the concentration in compartment n and q̇_n is its derivative. The q_n(t) could then be used as the time-lagged input to the central compartment of a PK model. Laplace transform gives (DiStefano, 2013)

$\begin{align*} Q_1 &= \frac{U}{s+k} \\ Q_2 &= \frac{k \: U}{(s+k)^2} \\ Q_3 &= \frac{k^2 \: U}{(s+k)^3} \\ &\vdots \\ Q_n &= \frac{k^{n-1} \: U}{(s+k)^n} \\ \end{align*}$

and thus the transfer function is

$H_n = \frac{Q_n}{U} = \frac{k^{n-1}}{(s + k)^{n}}$

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

$q_n (t) = \frac{ \text{Dose} \: (k \: t)^{n-1} \: e^{-k \: t} }{ (n - 1)! }$

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

$\dot C_B(t) = k \: q_n (t) = \frac{ \text{Dose} \: k^n \: t^{n-1} \: e^{-k \: t} }{ (n - 1)! }$

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

$f_{n,k}(t) = \frac{ k^n \: t^{n-1} \: e^{-k \: t} }{ (n - 1)! }$

, 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. 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

$g_{n,k}(t) = 1 - \sum_{i=0}^{n-1} \frac{(k \: t)^{i} \: e^{-k \: t}}{i!}$

, that is, with n=1 , g(t)=1-e^-kt ; with n=2 , g(t)=1-e^-kt-kxe^-kt ; with n=3 , g(t)=1-e^-kt-kxe^-kt-(kx)²e^-kt/2 ; and so on.

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 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

$h(t) = \frac{\beta^{n-1}}{(\beta + t)^{n}}$

, 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

$\begin{align*} C_{1}(T) &= \frac{1}{1 + \frac{\Delta t}{2} k} \left[ \int_{0}^{T} C_{0}(t)dt - k \: \int_{0}^{T - \Delta t} C_{1}(t)dt - \frac{\Delta t}{2} k \: C_{1}(T - \Delta t) \right] \\ C_{2}(T) &= \frac{k}{1 + \frac{\Delta t}{2} k} \left[ \int_{0}^{T} C_{1}(t)dt - \int_{0}^{T - \Delta t} C_{2}(t)dt - \frac{\Delta t}{2} C_{2}(T - \Delta t) \right] \\ C_{3}(T) &= \frac{k}{1 + \frac{\Delta t}{2} k} \left[ \int_{0}^{T} C_{2}(t)dt - \int_{0}^{T - \Delta t} C_{3}(t)dt - \frac{\Delta t}{2} C_{3}(T - \Delta t) \right] \\ &\vdots \\ C_{n}(T) &= \frac{k}{1 + \frac{\Delta t}{2} k} \left[ \int_{0}^{T} C_{n-1}(t)dt - \int_{0}^{T - \Delta t} C_{n}(t)dt - \frac{\Delta t}{2} C_{n}(T - \Delta t) \right] \end{align*}$

, where C₀(t) is the discrete input function, Δt is the sample time difference, and the output of this system is k × ∫₀^TC_n(t). The integrals can be calculated stepwise using equation

$\int\limits_{0}^{T} C_{n}(t)dt = \int\limits_{0}^{T-\Delta t} C_{n}(t)dt + \ \frac{\Delta t}{2} \: \left[ C_{n}(T - \Delta t) + C_{n}(T) \right]$

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

Updated at: 2020-01-29
Created at: 2019-11-05
Written by: Vesa Oikonen

Input function time lag using transit-time model

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References