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.

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

Appendix 3

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

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

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

Appendix 6

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

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)

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.

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

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

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