# Analytic solutions of ODEs

The compartmental models for PET data (assuming linear
system) can be described in terms of a set of ordinary differential
equations (ODEs).
The analytic solutions of ODEs can be derived using Laplace transformation, which utilize the
operation of convolution, ⊗.
Convolution integral of the input function with a response
function *h(t)* gives the concentration in the tissue compartment

The response function is zero when *t≤0*, and sum of exponentials when *t>0*;
the number of exponentials is the same as the number of tissue compartments.

As an example, for one-tissue compartmental model (1TCM) the solution is:

The measured PET data is discrete, with relatively long
sampling intervals. To achieve unbiased results, convolution operation in practise
(linear discrete time convolution method) requires short sample time intervals (*Δt*).
The time-activity curves (discretely sampled data), and the response function, need to be
interpolated to even sample times and sampling durations. The value of each sample in calculation
of the convolution integral should represent the mean value during the (short) *Δt*:
in practise this means that PET data should be interpolated to the midpoint of the even intervals,
and if the response function is an integrable function, then its definite integral during
*Δt*, divided by *Δt*, should be used. Resulting convolution integral
on the other hand should be scaled by *Δt*; thus *Δt* is cancelled out.
For example, in the case of the 1TCM, the definite integral of the response function between times
*t _{1}* and

*t*is:

_{2}When convolution integral is calculated at *N* time points of equal length
*Δt*, that is, at time points *½Δt*,
*Δt+½Δt*, *2*Δt+½Δt*, ...,
*(N-1)*Δt+½Δt*, the response function values (requiring no further
scaling) are calculated as

Convolution of input TAC with response function consisting of up to three exponential can be performed using convexpf, or specifically applying one-tissue compartment model using conv1tcm.

Alternative solutions that avoid the computationally slow convolution operation include the direct numerical approach (Graham, 1985) and 2nd order Adams-Moulton method based approach (Kuwabara et al., 1993). As an example, the solution of the one-tissue compartmental model ODE is:

Continuing with the example of the one-tissue compartmental model, utilized in the model for
[^{15}O]H_{2} PET studies, we simulate
renal cortex tissue curve with realistic input function
(arterial blood curve) and parameters *K _{1}*=3 mL*(min*mL)

^{-1}and

*k*=3.2 min

_{2}^{-1}. Arterial data is assumed to be measured from the dynamic image, and therefore have time frames of duration 4 s in the beginning of the scan, and the last time frame is 60 s. Linear discrete time convolution method requires even sample intervals, and to test how the selected interpolation intervals affect the accuracy, three different interpolation intervals are used, 0.1, 1.0, and 4.0 s. For comparison, tissue data is simulated also using the second-order Adams-Moulton method, without any interpolation. Simulations can be done using programs conv1tcm and sim_3tcm, and batch file is available in GitLab. The simulated TACs are shown in Figure 1a, with only the initial part of the data in Figure 1b.

**Figure 1a and 1b.** Renal cortex curve is simulated from arterial blood data
using the one-tissue compartmental model applying convolution method and Adams-Moulton method.
Convolution method is used with three different sample time intervals, 0.1, 1, and 4 seconds.
When observing the whole data range (**a**), all methods seem to lead to similar result.
When looking closer to just the initial phase of the simulated tissue curve (**b**),
it is easier to see that with convolution method with long interpolation intervals leads to biased
simulation, and with shorter sampling intervals the simulated tissue curve becomes closer to the
curve produced by Adams-Moulton used without any interpolation.

## See also:

- Compartmental model ODEs
- Fitting compartmental models
- Compartmental models
- PET data
- Input function
- Simulation of PET data
- Simulations in GitLab

## Literature

Graham MM. Model simplification: complexity versus reduction. *Circulation* 1985;
72(5 Pt 2):IV63-IV68. PMID: 4053329.

Kuwabara H, Cumming P, Reith J, Léger G, Diksic M, Evans AC, Gjedde A. Human striatal L-DOPA
decarboxylase activity estimated in vivo using 6-[^{18}F]fluoro-DOPA and positron emission
tomography: error analysis and application to normal subjects.
*J Cereb Blood Flow Metab.* 1993; 13: 43-56.
doi: 10.1038/jcbfm.1993.7.

Thompson WJ. *Computing for Scientists and Engineers - A Workbook of Analysis, Numerics, and
Applications*. Wiley, 1992. ISBN 0-471-54718-2.

Tags: Modeling, Compartmental model, Simulation, ODE, Convolution

Updated at: 2023-06-14

Created at: 2018-03-10

Written by: Vesa Oikonen