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Mathematical modelling of tumor

growth inhibition for the development

of anticancer drugs

Giuseppe De Nicolao

Department of Computer Science

and Systems Theory

University of Pavia

Italy

Summary

• Mathematical models in drug development

• Tumor growth inhibition (TGI) studies

• The pharmacokinetic-pharmacodynamic model

• Results: discovery candidates

• Benefits & Further developments

• Oncology drugs and extrapolation to patients

Mathematical models indrug development

Mathematical models in drug development

• Drug development

• It takes 12 – 15 years

• Average cost: 900 ML Dollar

• Only 3 drugs out of 10 pay back the costs

• Attrition (rate of):– Fraction of discarded candidates

• Facts:– Only 11% of Phase 1 candidates is eventually

registered: Attrition = 89%

– Phase II anticancer drugs: Attrition > 70%

– Pase III: Attrition = 45% (anticancer: 59%)

– Main causes of attrition in clinics:• 30% lack of efficacy

• 30% excessive toxicity


• Fact:– Approval of NCE (New Chemical Entities) is at

a historical minimum

• Moral:– pharma companies can maintain their growth

rate only by reducing attrition

• What to do?– predict efficacy and toxicity in discovery

and/or preclinical studies

– Improve predictivity of animal models (also bymeans of mathematical modeling and simulation)


• Computational methods for reducingattrition

• Systems biology

• ADME / ADMET property prediction

• PK / PD modelling


• Computational methods for reducingattrition

PK/PD modelling

Systems Biology

ADMET prediction


• Systems biology– Cellular models, mathematical models of

diseases, virtual patient

– Methodologies: system theory, dynamical models

– Firms: Entelos (1996), Beyond Genomics(2000), Bioseek (2000), Gene NetworkInternational (2001), etc ...

– Investments: (source: Nature Biotechnology, October 2004)

• Entelos: 45ML $

• Beyond Genomics: 26ML $

• Bioseek: 8.4ML $

• Gene Network International: 6.5 ML $

• …


• ADME / ADMET property prediction– Prediction of ADMET properties (Absorbtion,

Distribution, Metabolism, Excretion andToxicity) from molecular features.

– Methodologies: multivariate statistics, neuralnetworks, machine learning, data mining.

– Firms: Umetrics, Tripos, Spotfire, MolecularDiscovery, Entelos, SimulationsPlus,Shrodinger, Biorad, Inpharmatica, Accelryce,Compudrug, Leadscope, Lhasa, Lion Bioscience,Logichem, MDL Information Systems,Multicase, etc ….


• PK / PD modelling– Prediction of pharmacokinetic (PK) and

pharmacodynamic (PD) properties from invivo/in vitro experiment

– Firms: Bayer Technology Services, Pharsight,Globomax, Medeval, Cyprotex, Optimata,Simcyp, etc ...

– Investments: (source: Nature Biotechnology, October 2004)

– Optimata (1999): 2ML $

– …


• In silico technologies will enable drugmanufacturers to accelerate the selectionprocess, reduce the cost of preclinical andclinical studies and increase their overallchances of success. We estimate that theycould collectively save at least $200 ML andtwo to three years per drugPwC Report “Pharma 2005 - Silicon Rally: The race to e-R&D”

• by 2006 ~ 10% of pharmaceutical R&Dexpenditure will be on computer simulation andmodelling, a figure set to rise to 20% by 2016“A brave new world of drug development”, Curr. Drug Disc 2002


Tumor growth inhibition (TGI)studies

TGI studies: the experimental setting

Experiments on animals

•Human tumor cells inoculated into athymic mice

•Tumor dimensions measured by caliper

•Aim: assess drug effectiveness

Control

Treated

0

1

2

3

4

5

6

7

8

9

7 14 21 28 35

Time (day)

60mg/kg bid x 4 days

60mg/kg qd x 11 days

60mg/kg tid x 1 day

control

0

1

2

3

4

5

6

7

8

9

7 14 21 28 35

Time (day)

0

1

2

3

4

5

6

7

8

9

7 14 21 28 35

Time (day)


60mg/kg qd x 11 days

60mg/kg tid x 1 day

control

qd

tid


60mg/kg x 11 days

60mg/kg x 1 day

control

qd

tid


60mg/kg x 11 days

60mg/kg x 1 day

control


60mg/kg x 11 days

60mg/kg x 1 day

control

TW in control group = 8.9 g

TW in treated group = 4.8 g

Efficacy reported as:

(8.9-4.8)/8.9=46%

Uncertainty on the definition ofthe optimal time at which theefficacy should be evaluated.

For example:

the qd schedule is less activeup to 21 days in comparison tothe bid schedule but then itappears the most efficaciousone.

Efficacy is then reported as percentage of decrease of the average tumorweight of treated animals in comparison to the average of the control group.

TGI studies: the experimental setting

The pharmacokinetic-pharmacodynamic model

The PK-PD approach

1

2

3

4

5

6

7

8

9

0

7 14 21 28 35

Time (day)

PHA-680632 (exp552): observed tumor weights

0

5000

10000

15000

20000

25000

30000

35000

8 9 10 11 12 13 14 15 16 17 18 19 20

Time (day)

60mg/kg TID x 1day

0

5000

10000

15000

20000

25000

30000

35000

8 9 10 11 12 13 14 15 16 17 18 19 20

Time (day)

60mg/kg QD x 11 days

The ideal situation: having a PK/PD model, linking the

dosing regimen to the tumor growth dynamics.In this way it

is possible to predict the response of tumor growth

dynamics at different regimens

Building the PK-PD model

Strategy:

• Model of tumor growth in control animals

(unperturbed growth)

• Model of tumor growth in treated animals,

including the effect of the anticancer agent

(perturbed growth)

The PK-PD model: unperturbed growth

0

1

2

3

4

5

6

7

8

7 14 21 28

Time (day)

A2780, 466

A2780, 557

A2780, 731

A2780, 746

A2780, CRI007

HTC116, 594

HTC116, 598

0

1

2

3

4

5

6

7

8

7 14 21 28

Time (day)

0

1

2

3

4

5

6

7

8

7 14 21 28

Time (day)

A2780, 466

A2780, 557

A2780, 731

A2780, 746

A2780, CRI007

HTC116, 594

HTC116, 598

Tumor growth has two phases: exponential and linear

!!

"

"

"1

1

0

0

)(1

)(

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%

&&

'

(

))*

+,,-

./+

/=

•

tW

tWW

0)0( LW =

!

W

•

=GF(W (t)) =

"0W (t) W (t) #W *

"1

W (t) >W*

$

% &

' &

time

W(t)

exp

linear

L0

!

W*

="1

"0



Results: individual fittings of control animals

The PK-PD model: perturbed growth

Modeling the drug – tumor cell interaction

The tumor growth in treated animals follows the same law of the control animalsminus a loss due to the effect of the drug, according to the following scheme:

( ) ( )tZ)t(cKGFtZ 121 !!"=•

PK

1Z

!!

"

"

"1

1

0

0

)(1

)(

##

$

%

&&

'

(

))*

+,,-

./+

/=

•

tW

tWW

0)0( LW =

cycling cells

k2

The PK-PD model: perturbed growth

Modeling the damage and the delay of the cell death.

Since the death of tumor cells is not immediate with respect to the drugtreatment, a delay in the time of death has to be introduced. A transitcompartment model is used for describing this feature.

Mortality chain

1K

1K

1K

1K

2Z

3Z

nZ

1+nZdamaged cells death

The PK-PD model: the general scheme

01 )0( LZ =

( ) )()()( 21122 tZKtZtcKtZ !"!!=•

( ) )()( 31213 tZKtZKtZ !"!=•

( ) )()( 41314 tZKtZKtZ !"!=•

0)0(2=Z

0)0(3=Z

0)0(4=Z

)()(

)(1

)(121

1

0

101 tZtcK

tW

tZZ !!"

##

$

%

&&

'

(

))*

+,,-

.!+

!=

•

!!

"

"

"

01 )0( LZ =

( ) )()()( 21122 tZKtZtcKtZ !"!!=•

( ) )()( 31213 tZKtZKtZ !"!=•

( ) )()( 41314 tZKtZKtZ !"!=•

0)0(2=Z

0)0(3=Z

0)0(4=Z

)()(

)(1

)(121

1

0

101 tZtcK

tW

tZZ !!"

##

$

%

&&

'

(

))*

+,,-

.!+

!=

•

!!

"

"

"

PK

2Z 4Z3Z1K1K 1K)(2 tcK !

damaged cells

cell

death1Z

)4()3()2()1()( ZZZZtW +++=

cycling cells

In control animals:

)1()( ZtW =

In treated animals:

PK

2Z 4Z3Z1K1K 1K)(2 tcK !

damaged cells

cell

death1Z

)4()3()2()1()( ZZZZtW +++=

cycling cells

In control animals:

)1()( ZtW =

In treated animals:1Z1Z

)4()3()2()1()( ZZZZtW +++=

cycling cells

In control animals:

)1()( ZtW =

In treated animals:

This can be written as a

system of differential

equations:

The complete PK/PD model assumes that all the cells (cycling plus the

damaged cells) contribute to the weight of the tumor and that all the

cells entered in the “motality chain” go irreversibly to death.

The PK-PD model: the drug-specific parameters

Dependence of the tumor

growth kinetics on K2: it is

a multiplicative factor

representing the drug

potency.

0

1

2

3

0 10 20 30 40

time [days]

Control

K2 = 0.081

K2 = 0.162

K2 = 0.243

K2 = 0.324

The PK-PD model: the drug-specific parameters

K1

(days-1)

Mean

(days)

95%

(days)

1.56 1.9 4

1.04 2.9 6

0.52 5.8 12

0.26 11.5 24

High K1 values give sharp

and immediate response.

Low K1 values give delayed

and smoothed response.Dependence of the

distribution of probability of

death on K1.

The PK-PD model: secondary parameters

0

0.5

1

1.5

2

0 7 14 21 28 35

Time (day)

control

60mg/day x 10 days

120mg/day x 10 days

180mg/day x 10 days

PHA-680632 exp552: simulated tumor weights

C(t) < CT

C(t) / CT

Treatment

start Treatment end

Derived parameters: the threshold concentration (CT)According to the model, it is possible to define a threshold

concentration: CT = 00/k2

such that, if CSS / CT ! complete tumor regression.

0

1

2

3

4

5

6

7

8

9

0 7 14 21 28 35

Time (day)

Observed 60mg/kg bid x 4 days

Predicted 60mg/kg bid x 4 days

Observed 60mg/kg qd x 11 days

Predicted 60mg/kg qd x 11 days

Observed 60mg/kg tid x 1 day

Predicted 60mg/kg tid x 1 day

Predicted control

Observed control

PHA-680632 (exp552): observed and predicted tumor weights

The PK-PD model: simultaneous fittingsSince in the model, the perturbed growth collapses into the

unperturbed one in the absence of treatment, average data can be

used for simultaneous modeling of control and treated groups, thus:

– Making an efficient use of all the information

– Giving robustness to the estimates

The PK-PD model: simultaneous fittings

Meaning of a ‘good’ simultaneous fitting1Predictive power of the model

Results: discovery candidates

The PK-PD model: discovery candidates

0

2

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12

14

16

18

0 5 10 15 20 25 30 35

Time (day)

obs 45 mg/kg (tid x 1d) q5d x 2

pred 45 mg/kg (tid x 1d) q5d x 2

obs 45 mg/kg bidx4d+stopx7d+bidx2d

pred 45 mg/kg bidx4d+stopx7d+bidx2d

obs control

pred control


0

2

4

6

8

10

12

0 7 14 21 28 35 42

Time (day)

obs 45 mg/kg qdx11d

pred 45 mg/kg qdx11d

obs control

pred control


For the development of

the project it was of

interest testing the

efficacy of a long-term

infusion.

Another compound was tested

after i.v. bolus with different

doses and schedules.

Example 1: Project ARExploring the response of the tumor at different doses

and schedules, from IV bolus to infusion.Exp. 1: 45mg/kg qdx11dExp. 2:

45mg/kg (tidx1d) q5dx2

45mg/kg bidx4d+stopx7d+bidx2d

0

1

2

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Time (days)

1

10

100

1000

10000

0 1 2 3 4 5 6 7 8 9

Time (days)

Prediction

?

IV infusion

Based on the derived parameter CT and the PK parameters of the

compound, the dose to be given by infusion for five days able to

give a significant tumor regression was calculated and the

corresponding tumor growth was predicted.


0

1

2

3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Time (days)

1

10

100

1000

10000

0 1 2 3 4 5 6 7 8 9

Time (days)


Based on the derived parameter CT and the PK parameters of the

compound, the dose to be given by infusion for five days able to

give a significant tumor regression was calculated and the

corresponding tumor growth was predicted.

Prediction

IV infusion

Conclusion: the compound maintains

the efficacy also after infusion,

excellent predictability of the response

by the model


Example 2: Project AUComparing the antitumor activity of two compounds;

due to their different toxicological profiles different

doses were administered.

PK samples from Group 1


Experiment A Dose Volume Route Scheduling Tot mice TGI

drug X mg/Kg ml/Kg %

Control animals 10 iv 1<5 bid 8

Group 1 40 10 iv 1<5 bid 8 33

Group 2 60 10 iv 1<5 bid 8 44

Experiment B Dose Volume Route Scheduling Tot mice TGI

drug Y mg/Kg ml/Kg %

Control animals 10 iv 1<5 bid 8

Group 1 15 10 iv 1<5 bid 8 31

The PK-PD model: discovery candidatesUsing the PK parameters previously obtained, the plasma concentrations of the drug X

predicted after daily IV bolus at the doses of 40 and 60 mg/kg bid were derived and

included in the PK/PD model. TheTGI model was then applied to the tumor growth

curves observed in the efficacy experiment.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 50 100 150 200 250 300 350 400

Time (hr)

F1 Observed

F1 Predicted

F2 Observed

F2 Predicted

F3 Observed

F3 Predicted

parameter value CV%

K1 1/h 0.26199 123.6

K2 1/µM/h 0.00001 19.8!0 1/h 0.01358 9.8!1 g/h 0.00732 16.1

L0 g 0.01422 27.8

Ct µM 3.32


0

2

4

6

8

10

12

14

0 100 200 300 400 500 600 700 800

h

F1 Observed

F1 Predicted

F2 Observed

F2 Predicted

Conclusion:

Drug Y is much more potent

in comparison to drug X

(one order of magnitude,

0.23 vs 3.3)

Using the PK parameters previously obtained, the plasma concentrations of the drug Y

predicted after daily IV bolus at the doses of 15 mg/kg bid were derived and included in

the PK/PD model. TheTGI model was then applied to the tumor growth curves observed

in the efficacy experiment.

parameter value CV%

K1 1/h 0.0317 106.6

K2 1/µM/h 0.0673 23.8!0 1/h 0.0156 8.2!1 g/h 0.0236 15.7

L0 g 0.0141 28.7

Ct µM 0.2320

The PK-PD model: discovery candidatesExample 3: Project C2

Comparing two compounds, drug X given IV and drug Y

given orally.


PK samples from Group 1,3

Experiment A Dose Volume Route Scheduling Tot mice TGI

drug X mg/Kg ml/Kg %

Control animals 10 iv 1<10 daily 8

Group 1 30 10 iv 1<10 daily 8 48

Experiment B Dose Volume Route Scheduling Tot mice TGI

drug Y mg/Kg ml/Kg %

Control animals 10 oral 1<10 bid 8

Group 1 20 10 oral 1<10 bid 8 49

Group 2 30 10 oral 1<10 bid 8 59

Group 3 40 10 oral 1<10 bid 8 77


0

1

2

3

4

5

6

7

0 5 10 15 20 25 30

days

F1 Observed

F1 Predicted

F2 Observed

F2 Predicted

Observed and predicted tumor growth curves after obtained in A2780 tumor bearing

female mice after IV bolus administration of drug X given at the dose of 30 mg/kg/day for

10 days.

L0 g 0.02!0 1/day 0.31!1 g/day 0.37

K1 1/day 0.96

K2 1/µM/day 0.09

Ct µM 3.32

The PK-PD model: discovery candidatesPK/PD analysis of drug Y given orally at the doses of 20, 30, 40 twice a day for 10

consecutive days.

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600 700 800

time (hr)

F1 Observed

F1 Predicted

F2 Observed

F2 Predicted

F3 Observed

F3 Predicted

F4 Observed

F4 Predicted

0

500

1000

1500

2000

2500

0 50 100 150 200 250

time_(hr)

Observed

Predicted

dose_(mg/kg)=40, mouse=233

Conclusion:

drug Y (the oral compound)

has a potency similar or

even higher compared to

drug X

L0 (g) 0.005!0 (1/day) 0.445!1 (g/day) 0.605

K1 (1/day) 0.394

K2 (1/µM/day) 0.190

CT (µM) 2.34

Benefits & furtherdevelopments

• Simple model with few parameters.

• Identifiable and physiologically relevant model parameters.

• Estimates of drug potency can be obtained independently

from dose levels and schedules (ranking of compounds).

• Applicable to different cell lines.

• Prediction of tumor growth kinetics at different schedules.

• Savings in animals, time and resourses.

The PK-PD model: achievements

• Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I,Rocchetti M.

Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograftmodels after administrations of anticancer agents. Cancer Research. 2004 64: 1094-1101.

• Rocchetti M, Poggesi I, Germani M, Fiorentini F, Pellizzoni C, Zugnoni P, Pesenti E, SimeoniM, De Nicolao G.

A PK-PD model for predicting tumor growth inhibition in mice: a useful tool in oncology drugdevelopment. Basic & Clinical Pharmacology and Toxicolog. 2005, 96: 265-268.

• Magni P, Simeoni M, Poggesi I, Rocchetti M, De Nicolao G.

A mathematical model to study the effects of drugs administration on tumor growth dynamics.Mathematical Bioscience. 2006, 200: 125,151.

The PK-PD model: references

Oncology drugs andextrapolation to patients

The typical question made at the end

of the presentation:

The PK-PD model: what is going on….

How does this translate to humans?

Facing the dilemma of oncology drug failures in the clinic.

Making predictions from animals to humans

About 10% of Investigational New Drug

(IND) applications for new molecular

entities submitted to the US Food and Drug

Administration (FDA) progress beyond the

investigational phase 1. The success rate is even

lower in oncology (~5%)2. The problematic

issues that underlie the low rate of approval

of new oncological drugs include the lack of

preclinical systems (both in vitro assays and

in vivo animal models) that can accurately

predict the efficacy and toxicity of new

agents 3,4,5,

Kola I, Landis J. Can the pharmaceutical industry reduce attrition rates?

Nature Rev Drug Discov. 3, 711-715 (2004).

The PK-PD model: from animals to humans

Relationship of doses of anticancer drugs in humans

(cumulative doses given in 3-week cycles, midpoint of range)

vs. CTxCLh

r=0.939

0.01 0.1 1 10 100

1

10

100

1000

10000

g

f

c

a b

paclitaxelirinotecand

5-fluorouracil e

100100

2D

ose (

mg/m

, 3-w

eek)

2D

ose (

mg/m

, 3-w

eek)

Dose (

mg/m

, 3-w

eek)

0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

22CTxCLh (mg/m /h)CTxCLh (mg/m /h)

0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e


0

1

2

3

4

5

6

0 7 14 21 28 35 42

Time (day)

0.001

0.01

0.1

1

10

100

1000

10000

0 6 12 18 24 30 36 42 48

Time (hr)

0

1

2

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4

5

6

0 7 14 21 28 35 42

Time (day)

0.001

0.01

0.1

1

10

100

1000

10000

0 6 12 18 24 30 36 42 48

Time (hr)

Retrospective PK-PD analysis of drug D, whose development was

interrupted due to absence of clinical benefit at the dose adopted in the

phase II clinical trials of 12 mg/m2 q3wk.

CT =7.42 ng/mL

Observed and model-fitted tumor growth curves obtained in nude mice given i.v. either

the vehicle (!) or drug D (2 mg/kg " or 2.5 mg/kg #, given q4dx3 from Day 11).


0.01 0.1 1 10 100

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

100100

2D

ose (

mg/m

, 3-w

eek)

2D

ose (

mg/m

, 3-w

eek)

Dose (

mg/m

, 3-w

eek)

0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e


0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

0.01 0.1 1 10 100

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

0.01 0.1 1 10 100

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

100100

2D

ose (

mg/m

, 3-w

eek)

2D

ose (

mg/m

, 3-w

eek)

Dose (

mg/m

, 3-w

eek)

0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e


0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

2D

ose (

mg/m

, 3-w

eek)

2D

ose (

mg/m

, 3-w

eek)

Dose (

mg/m

, 3-w

eek)

2D

ose (

mg/m

, 3-w

eek)

Dose (

mg/m

, 3-w

eek)

2D

ose (

mg/m

, 3-w

eek)

Dose (

mg/m

, 3-w

eek)

0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

22CTxCLh (mg/m /h)CTxCLh (mg/m /h)22CTxCLh (mg/m /h)CTxCLh (mg/m /h)CTxCLh (mg/m /h)CTxCLh (mg/m /h)

0.01 0.1 1 10

1

10

100

1000

10000

g

f

c

a b


5-fluorouracil e

CTxCLh=0.5 mg/m2/h

Actual dose

from phase I

trial

Acknowledgments

• Maurizio Rocchetti (Nerviano Medical Sciences)

• Monica Simeoni and Italo Poggesi (currently

GlaxoSmithKline)

• Paolo Magni (University of Pavia)

• …

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