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Short description:

PKPD model of tumor growth kinetics in xenograft mice after administration of anticancer agents given in combination

Format:	PharmML 0.8.x (0.8.1)
Related Publication:	A predictive pharmacokinetic-pharmacodynamic model of tumor growth kinetics in xenograft mice after administration of anticancer agents given in combination. Terranova N, Germani M, Del Bene F, Magni P Cancer chemotherapy and pharmacology, 8/2013, Volume 72, Issue 2, pages: 471-482 Affiliation: Dipartimento di Ingegneria Industriale e dell'Informazione, Università degli Studi di Pavia, Via Ferrata 3, Pavia, Italy. paolo.magni@unipv.it Abstract: PURPOSE: In clinical oncology, combination treatments are widely used and increasingly preferred over single drug administrations. A better characterization of the interaction between drug effects and the selection of synergistic combinations represent an open challenge in drug development process. To this aim, preclinical studies are routinely performed, even if they are only qualitatively analyzed due to the lack of generally applicable mathematical models. METHODS: This paper presents a new pharmacokinetic-pharmacodynamic model that, starting from the well-known single agent Simeoni TGI model, is able to describe tumor growth in xenograft mice after the co-administration of two anticancer agents. Due to the drug action, tumor cells are divided in two groups: damaged and not damaged ones. The damaging rate has two terms proportional to drug concentrations (as in the single drug administration model) and one interaction term proportional to their product. Six of the eight pharmacodynamic parameters assume the same value as in the corresponding single drug models. Only one parameter summarizes the interaction, and it can be used to compute two important indexes that are a clear way to score the synergistic/antagonistic interaction among drug effects. RESULTS: The model was successfully applied to four new compounds co-administered with four drugs already available on the market for the treatment of three different tumor cell lines. It also provided reliable predictions of different combination regimens in which the same drugs were administered at different doses/schedules. CONCLUSIONS: A good and quantitative measurement of the intensity and nature of interaction between drug effects, as well as the capability to correctly predict new combination arms, suggest the use of this generally applicable model for supporting the experiment optimal design and the prioritization of different therapies.
Contributors:	Paolo Magni

Context of model development:	Combination Therapy Dose Selection;
Model compliance with original publication:	Yes;
Model implementation requiring submitter’s additional knowledge:	No;
Modelling context description:	PURPOSE: In clinical oncology, combination treatments are widely used and increasingly preferred over single drug administrations. A better characterization of the interaction between drug effects and the selection of synergistic combinations represent an open challenge in drug development process. To this aim, preclinical studies are routinely performed, even if they are only qualitatively analyzed due to the lack of generally applicable mathematical models. METHODS: This paper presents a new pharmacokinetic-pharmacodynamic model that, starting from the well-known single agent Simeoni TGI model, is able to describe tumor growth in xenograft mice after the co-administration of two anticancer agents. Due to the drug action, tumor cells are divided in two groups: damaged and not damaged ones. The damaging rate has two terms proportional to drug concentrations (as in the single drug administration model) and one interaction term proportional to their product. Six of the eight pharmacodynamic parameters assume the same value as in the corresponding single drug models. Only one parameter summarizes the interaction, and it can be used to compute two important indexes that are a clear way to score the synergistic/antagonistic interaction among drug effects. RESULTS: The model was successfully applied to four new compounds co-administered with four drugs already available on the market for the treatment of three different tumor cell lines. It also provided reliable predictions of different combination regimens in which the same drugs were administered at different doses/schedules. CONCLUSIONS: A good and quantitative measurement of the intensity and nature of interaction between drug effects, as well as the capability to correctly predict new combination arms, suggest the use of this generally applicable model for supporting the experiment optimal design and the prioritization of different therapies.;
Modelling task in scope:	estimation;
Nature of research:	Preclinical development; In vivo;
Therapeutic/disease area:	Oncology;

Validation Status:	Annotations are correct.
Certification Comment:	This model is not certified.

Mandatory file
- Executable_Terranova_2013_oncology_TGI_combo.xml

Additional Files

Model owner: Paolo Magni
Submitted: Dec 10, 2015 11:21:34 PM
Last Modified: Oct 10, 2016 8:00:38 PM

Revisions

Version: 7
- Submitted on: Oct 10, 2016 8:00:38 PM
- Submitted by: Paolo Magni
- With comment: Edited model metadata online.
Version: 5
- Submitted on: May 24, 2016 11:31:41 PM
- Submitted by: Paolo Magni
- With comment: Updated model annotations.
Version: 4
- Submitted on: Dec 10, 2015 11:21:34 PM
- Submitted by: Paolo Magni
- With comment: Edited model metadata online.

Name

Generated from MDL. MOG ID: terranova_2013

Independent Variables

Function Definitions

proportionalError : real (proportional : real, f : real) = proportional \cdot f

Parameter Model: $pm$

Random Variables

{eps_RES_W}_{vm_err.DV} ~ Normal2 (mean = 0, var = 1)

Population Parameters

LAMBDA0_POP

LAMBDA1_POP

K1a_POP

K2a_POP

K1b_POP

K2b_POP

W0_POP

GAMMA_POP

K10A_POP

K12A_POP

K21A_POP

V1A_POP

K10B_POP

K12B_POP

K21B_POP

V1B_POP

CV

Individual Parameters

LAMBDA0 = pm.LAMBDA0_POP

LAMBDA1 = pm.LAMBDA1_POP

K1a = pm.K1a_POP

K2a = pm.K2a_POP

K1b = pm.K1b_POP

K2b = pm.K2b_POP

W0 = pm.W0_POP

GAMMA = pm.GAMMA_POP

K10A = pm.K10A_POP

K12A = pm.K12A_POP

K21A = pm.K21A_POP

V1A = pm.V1A_POP

K10B = pm.K10B_POP

K12B = pm.K12B_POP

K21B = pm.K21B_POP

V1B = pm.V1B_POP

Structural Model: $sm$

Variables

PSI = 20

WTOT = sm.X00 + sm.X10 + sm.X20 + sm.X30 + sm.X01 + sm.X11 + sm.X21 + sm.X31 + sm.X02 + sm.X12 + sm.X22 + sm.X32 + sm.X03 + sm.X13 + sm.X23 + sm.X33

Ca = \frac{sm.Q1A}{pm.V1A}

\begin{matrix} \frac{ⅆ}{ⅆ T} Q1A = pm.K21A \cdot sm.Q2A - (pm.K10A + pm.K12A) \cdot sm.Q1A \\ Q1A (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} Q2A = pm.K12A \cdot sm.Q1A - pm.K21A \cdot sm.Q2A \\ Q2A (T = 0) = 0 \end{matrix}

Cb = \frac{sm.Q1B}{pm.V1B}

\begin{matrix} \frac{ⅆ}{ⅆ T} Q1B = pm.K21B \cdot sm.Q2B - (pm.K10B + pm.K12B) \cdot sm.Q1B \\ Q1B (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} Q2B = pm.K12B \cdot sm.Q1B - pm.K21B \cdot sm.Q2B \\ Q2B (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X00 = \frac{pm.LAMBDA0 \cdot sm.X00}{{(1 + {\frac{sm.WTOT \cdot pm.LAMBDA0}{pm.LAMBDA1}}^{sm.PSI})}^{\frac{1}{sm.PSI}}} - pm.K2a \cdot sm.Ca \cdot sm.X00 - pm.K2b \cdot sm.Cb \cdot sm.X00 - pm.GAMMA \cdot sm.Ca \cdot sm.Cb \cdot sm.X00 \\ X00 (T = 0) = pm.W0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X10 = pm.K2a \cdot sm.Ca \cdot sm.X00 - pm.K1a \cdot sm.X10 - pm.K2b \cdot sm.Cb \cdot sm.X10 \\ X10 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X20 = pm.K1a \cdot sm.X10 - pm.K1a \cdot sm.X20 - pm.K2b \cdot sm.Cb \cdot sm.X20 \\ X20 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X30 = pm.K1a \cdot sm.X20 - pm.K1a \cdot sm.X30 - pm.K2b \cdot sm.Cb \cdot sm.X30 \\ X30 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X01 = pm.K2b \cdot sm.Cb \cdot sm.X00 - pm.K2a \cdot sm.Ca \cdot sm.X01 - pm.K1b \cdot sm.X01 \\ X01 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X11 = pm.K2b \cdot sm.Cb \cdot sm.X10 + pm.K2a \cdot sm.Ca \cdot sm.X01 - pm.K1a \cdot sm.X11 - pm.K1b \cdot sm.X11 + pm.GAMMA \cdot sm.Ca \cdot sm.Cb \cdot sm.X00 \\ X11 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X21 = pm.K1a \cdot sm.X11 + pm.K2b \cdot sm.Cb \cdot sm.X20 - pm.K1a \cdot sm.X21 - pm.K1b \cdot sm.X21 \\ X21 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X31 = pm.K1a \cdot sm.X21 + pm.K2b \cdot sm.Cb \cdot sm.X30 - pm.K1a \cdot sm.X31 - pm.K1b \cdot sm.X31 \\ X31 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X02 = pm.K1b \cdot sm.X01 - pm.K2a \cdot sm.Ca \cdot sm.X02 - pm.K1b \cdot sm.X02 \\ X02 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X12 = pm.K1b \cdot sm.X11 + pm.K2a \cdot sm.Ca \cdot sm.X02 - pm.K1a \cdot sm.X12 - pm.K1b \cdot sm.X12 \\ X12 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X22 = pm.K1a \cdot sm.X12 + pm.K1b \cdot sm.X21 - pm.K1a \cdot sm.X22 - pm.K1b \cdot sm.X22 \\ X22 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X32 = pm.K1a \cdot sm.X22 + pm.K1b \cdot sm.X31 - pm.K1a \cdot sm.X32 - pm.K1b \cdot sm.X32 \\ X32 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X03 = pm.K1b \cdot sm.X02 - pm.K2a \cdot sm.Ca \cdot sm.X03 - pm.K1b \cdot sm.X03 \\ X03 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X13 = pm.K1b \cdot sm.X12 + pm.K2a \cdot sm.Ca \cdot sm.X03 - pm.K1a \cdot sm.X13 - pm.K1b \cdot sm.X13 \\ X13 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X23 = pm.K1a \cdot sm.X13 + pm.K1b \cdot sm.X22 - pm.K1a \cdot sm.X23 - pm.K1b \cdot sm.X23 \\ X23 (T = 0) = 0 \end{matrix}

\begin{matrix} \frac{ⅆ}{ⅆ T} X33 = pm.K1a \cdot sm.X23 + pm.K1b \cdot sm.X32 - pm.K1a \cdot sm.X33 - pm.K1b \cdot sm.X33 \\ X33 (T = 0) = 0 \end{matrix}

Observation Model: $om1$

Continuous Observation

Y = sm.WTOT + proportionalError (proportional = pm.CV, f = sm.WTOT) + pm.eps_RES_W

External Dataset

OID	$nm_ds$
Tool Format	NONMEM

File Specification

Format	$csv$
Delimiter	comma
File Location	Simulated_terranova_2013_data.csv

Column Definitions

Column ID	Position	Column Type	Value Type
$ID$	$1$	$id$	$int$
$TIME$	$2$	$idv$	$real$
$DV$	$3$	$dv$	$real$
$AMT$	$4$	$dose$	$real$
$CMT$	$5$	$cmt$	$int$
$MDV$	$6$	$mdv$	$int$

Column Mappings

Column Ref	Modelling Mapping
$TIME$	$T$
$DV$	$om1.Y$
$AMT$	${\begin{cases} sm.Q1A & if & CMT = 1 \land AMT > 0 \\ sm.Q1B & if & CMT = 3 \land AMT > 0 \end{cases}$

Estimation Step

OID	$estimStep_1$
Dataset Reference	$nm_ds$

Parameters To Estimate

Parameter	Initial Value	Fixed?	Limits
pm.LAMBDA0_POP	$0.149$	true	$()$
pm.LAMBDA1_POP	$0.203$	true	$()$
pm.K1a_POP	$2.24$	true	$()$
pm.K2a_POP	$0.0512$	true	$()$
pm.K1b_POP	$1.68$	true	$()$
pm.K2b_POP	$0.0984$	true	$()$
pm.W0_POP	$0.0566$	true	$()$
pm.GAMMA_POP	$- 2$	false	$()$
pm.K10A_POP	$28.1$	true	$()$
pm.K12A_POP	$4.94$	true	$()$
pm.K21A_POP	$5.58$	true	$()$
pm.V1A_POP	$1.42$	true	$()$
pm.K10B_POP	$97.3$	true	$()$
pm.K12B_POP	$20.4$	true	$()$
pm.K21B_POP	$45.2$	true	$()$
pm.V1B_POP	$0.887$	true	$()$
pm.CV	$0.1$	false	$(0)$

Operations

Operation: $1$

Op Type

generic

Operation Properties

Name	Value
algo	$foce$

Step Dependencies

Step OID	Preceding Steps
$estimStep_1$

DDMODEL00000096: Terranova_2013_oncology_TGI_combo

Revisions

Name

Independent Variables

Function Definitions

Parameter Model: pm

Random Variables

Population Parameters

Individual Parameters

Structural Model: sm

Variables

Observation Model: om1

Continuous Observation

External Dataset

File Specification

Column Definitions

Column Mappings

Estimation Step

Parameters To Estimate

Operations

Operation: 1

Operation Properties

Step Dependencies

Parameter Model: $pm$

Structural Model: $sm$

Observation Model: $om1$

Operation: $1$