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Delay differential equations based models in NONMEM

Abstract

Delay differential equations (DDEs) are commonly used in pharmacometric models to describe delays present in pharmacokinetic and pharmacodynamic data analysis. Several DDE solvers have been implemented in NONMEM 7.5 for the first time. Two of them are based on algorithms already applied elsewhere, while others are extensions of existing ordinary differential equations (ODEs) solvers. The purpose of this tutorial is to introduce basic concepts underlying DDE based models and to show how they can be developed using NONMEM. The examples include previously published DDE models such as logistic growth, tumor growth inhibition, indirect response with precursor pool, rheumatoid arthritis, and erythropoiesis-stimulating agents. We evaluated the accuracy of NONMEM DDE solvers, their ability to handle stiff problems, and their performance in parameter estimation using both first-order conditional estimation (FOCE) and the expectation–maximization (EM) method. NONMEM control streams and excerpts from datasets are provided for all discussed examples. All DDE solvers provide accurate and precise solutions with the number of significant digits controlled by the error tolerance parameters. For estimation of population parameters, the EM method is more stable than FOCE regardless of the DDE solver.

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Acknowledgements

This work was supported by Research Grants Council Early Career Scheme Project 24103120 from University Grants Committee, Hong Kong SAR, China.

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Correspondence to Wojciech Krzyzanski.

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Appendices

Appendix 1

Solution to the linear DDE model Eq. (4)

An explicit solution to Eq. (4) can be obtained by the method of steps [8] and is provided as in a form of the following recursive formulas. Let \( i = 0, 1, \ldots \) be a non-negative integer and \(i\tau \le t\le \left(i+1\right)\tau \). The solution is

$$A\left(t\right)={a}_{i0}+{a}_{i1}\left(t-i\tau \right)+\cdots +{a}_{ii+1}{(t-i\tau )}^{i+1}$$
(52)

where

$${a}_{00}={A}_{0}\quad {\rm{and}}\; {a}_{01}=-k{A}_{0}$$
(53)

and for \(i\ge 1\)

$${a}_{i0}={a}_{i-10}+{a}_{i-10}\tau +\cdots +{a}_{i-1i}{\tau }^{i}$$
(54)
$$ a_{{ij}} = - \frac{k}{j}a_{{i - 1j - 1}} \quad {\text{for}}\;j = 0,1, \ldots ,i + 1. $$
(55)

Appendix 2

R code to calculate the exact solution to the delayed linear model Eq. (4) based on method of steps

figurea

Appendix 3

NONMEM code for solving Stiff DDE model Eqs. (5)–(6) and an excerpt from the data file

figureb

First 14 lines of data fileNM3.csv

C ID AMT TIME DV EVID CMT
1 . 0 1 0 1
1 . 0 1 0 2
1 . 0.1 1 0 1
1 . 0.1 1 0 2
1 . 0.2 1 0 1
1 . 0.2 1 0 2
1 . 0.3 1 0 1
1 . 0.3 1 0 2
1 . 0.4 1 0 1
1 . 0.4 1 0 2
1 1 0.5 . 1 1
1 . 0.5 1 0 1
1 . 0.5 1 0 2
1 . 0.6 1 0 1

Appendix 4

NONMEM code for solving delayed logistic growth model shown in Eq. (8) and example of data file

figurec

The first 10 rows of data file NM4.csv

C.ID AMT TIME DV EVID CMT TAU
1 . 0 1 0 1 2
1 . 1 1 0 1 2
1 . 2 1 0 1 2
1 . 3 1 0 1 2
1 . 4 1 0 1 2
1 . 5 1 0 1 2
1 . 6 1 0 1 2
1 . 7 1 0 1 2
1 . 8 1 0 1 2
1 . 9 1 0 1 2
1 . 10 1 0 1 2

Appendix 5

MATLAB script for solving delayed logistic growth model Eq. (8) using dde23

figured

Appendix 6

NONMEM code for solving lifespan tumor growth inhibition model shown in Eqs. (9)–(14) and data file

figuree

The first 12 rows of data file NM6.csv

CID TIME AMT RATE CMT EVID MDV DV
0 0 0 0 1 0 1 0
0 9 0 0 3 0 0 0.222772
0 12 0 0 3 0 0 0.482673
0 14 0 0 3 0 0 0.831683
0 16 0 0 3 0 0 1.061881
0 19 0 0 3 0 0 1.492574
0 21 0 0 3 0 0 1.737624
0 23 0 0 3 0 0 2.287129
100 0 0 0 1 0 1 0
100 9 0 0 3 0 0 0.230198
100 12 100 0 2 1 1 0
100 12 0 0 3 0 0 0.475248
100 13 100 0 2 1 1 0
100 14 100 0 2 1 1 0
100 14 0 0 3 0 0 0.660891
100 15 100 0 2 1 1 0
100 16 100 0 2 1 1 0
100 16 0 0 3 0 0 0.831683
100 19 0 0 3 0 0 1.002475
100 21 0 0 3 0 0 1.232673
100 23 0 0 3 0 0 1.477723

Appendix 7

NONMEM code for simulation of RA model Eqs. (15)–(21)

figuref
figureg

The first 10 rows of data file NM7.csv

CID TIME AMT RATE CMT EVID MDV DV DOSE
1 0 0 0 1 0 0 1 0
1 0 0 0 2 0 0 1 0
1 0 0 0 3 0 0 1 0
1 0 0 0 4 0 0 1 0
1 1 0 0 1 0 0 1 0
1 1 0 0 2 0 0 1 0
1 1 0 0 3 0 0 1 0
1 1 0 0 4 0 0 1 0
1 3 0 0 1 0 0 1 0
1 3 0 0 2 0 0 1 0

Appendix 8

NONMEM code for simulation of data for PDLIDR model Eqs. (22)–(27) using the methods of steps. Note that MOS does not require a DDE solver. The generation of this code was facilitated by the ddexpand utility using its MOS expansion process.

figureh

The first 10 rows of data file NM8.csv

C ID AMT TIME DV EVID CMT DOSE
1 0.1 0 . 1 1 0.1
1 0.1 0 . 1 4 0.1
1 0 0 1.867322 0 3 0.1
1 0 4 2.005362 0 3 0.1
1 0 8 2.285038 0 3 0.1
1 0 12 2.37155 0 3 0.1
1 0 16 2.545543 0 3 0.1
1 0 20 2.596434 0 3 0.1
1 0 24 2.3411 0 3 0.1
1 0 28 2.276433 0 3 0.1

Appendix 9

NONMEM code for PDLIDR model Eqs. (22)–(27) using DDE solver in ADVAN13 with FOCEI

figurei

The first 10 rows of NM9.csv

ID AMT TIME DV EVID CMT DOSE
1 0.1 0 0 1 1 0.1
1 0 0 2.5516 0 3 0.1
1 0 4 2.5194 0 3 0.1
1 0 8 2.6586 0 3 0.1
1 0 12 2.8158 0 3 0.1
1 0 16 2.7879 0 3 0.1
1 0 20 2.9741 0 3 0.1
1 0 24 2.8118 0 3 0.1
1 0 28 2.8435 0 3 0.1
1 0 32 2.7563 0 3 0.1

Appendix 10

NONMEM code for ESA population model Eqs. (28)–(51), Importance Sampling

figurej
figurek
figurel

The first 10 rows of the data NM10.csv

ID TIME AMT RATE CP CMT BSL DOSE FLAG ET1 ET2 ET3 ET4
1008 0 2.04 32.64 0 1 8 0.03 0 0.353 0.241 0.253 0.191
1008 0 0 0 0.8 5 8 0.03 0 0.353 0.241 0.253 0.191
1008 0 0 0 4.75 6 8 0.03 0 0.353 0.241 0.253 0.191
1008 0 0 0 14.17 6 8 0.03 1 0.353 0.241 0.253 0.191
1008 1 0 0 0.8 5 8 0.03 0 0.353 0.241 0.253 0.191
1008 1 0 0 4.82 6 8 0.03 0 0.353 0.241 0.253 0.191
1008 1 0 0 14.17 6 8 0.03 1 0.353 0.241 0.253 0.191
1008 2 0 0 1 5 8 0.03 0 0.353 0.241 0.253 0.191
1008 2 0 0 5.06 6 8 0.03 0 0.353 0.241 0.253 0.191
1008 2 0 0 14.98 6 8 0.03 1 0.353 0.241 0.253 0.191

Appendix 11

NONMEM code for ESA population model Eqs. (28)–(51), MCMC Bayes

figurem
figuren
figureo
figurep

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Yan, X., Bauer, R., Koch, G. et al. Delay differential equations based models in NONMEM. J Pharmacokinet Pharmacodyn (2021). https://doi.org/10.1007/s10928-021-09770-z

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Keywords

  • Delay differential equations
  • NONMEM
  • DDE solver
  • Stiff differential equations