Mathematical modeling of tumor growth and tumor growth inhibition in oncology drug development

Abstract

Introduction:
Approaches aiming to model the time course of tumor growth and tumor growth inhibition following a therapeutic intervention have recently been proposed for supporting decision making in oncology drug development. When considered in a comprehensive model-based approach, tumor growth can be included in the cascade of quantitative and causally related markers that lead to the prediction of survival, the final clinical response.

Areas covered:
The authors examine articles dealing with the modeling of tumor growth and tumor growth inhibition in both preclinical and clinical settings. In addition, the authors review models describing how pharmacological markers can be used to predict tumor growth and models describing how tumor growth can be linked to survival endpoints.

Expert opinion:
Approaches and success stories of application of model-based drug development centered on tumor growth modeling are growing. It is also apparent that these approaches can answer practical questions on drug development more effectively than that in the past. For modeling purposes, some improvements are still needed related to study design and data quality. Further efforts are needed to encourage the mind shift from a simple description of data to the prediction of untested conditions that modeling approaches allow.

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Expert Opinion on Drug Discovery
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Mathematical modeling of efficacy and safety for
anticancer drugs clinical development
Silvia Maria Lavezzi, Elisa Borella, Letizia Carrara, Giuseppe De Nicolao,
Paolo Magni & Italo Poggesi
To cite this article: Silvia Maria Lavezzi, Elisa Borella, Letizia Carrara, Giuseppe De
Nicolao, Paolo Magni & Italo Poggesi (2018) Mathematical modeling of efficacy and safety for
anticancer drugs clinical development, Expert Opinion on Drug Discovery, 13:1, 5-21, DOI:
10.1080/17460441.2018.1388369
To link to this article: https://doi.org/10.1080/17460441.2018.1388369
Accepted author version posted online: 03
Oct 2017.
Published online: 12 Oct 2017.
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REVIEW
Mathematical modeling of efficacy and safety for anticancer drugs clinical
development
Silvia Maria Lavezzi
a
*, Elisa Borella
a
*, Letizia Carrara
a
, Giuseppe De Nicolao
a
, Paolo Magni
a
and Italo Poggesi
b
a
Dipartimento di Ingegneria Industriale e dell’Informazione, Università degli Studi di Pavia, Pavia, Italy;
b
Global Clinical Pharmacology, Janssen
Research and Development, Cologno Monzese, Italy
ABSTRACT
Introduction: Drug attrition in oncology clinical development is higher than in other therapeutic areas.
In this context, pharmacometric modeling represents a useful tool to explore drug efficacy in earlier
phases of clinical development, anticipating overall survival using quantitative model-based metrics.
Furthermore, modeling approaches can be used to characterize earlier the safety and tolerability profile
of drug candidates, and, thus, the risk-benefit ratio and the therapeutic index, supporting the design of
optimal treatment regimens and accelerating the whole process of clinical drug development.
Areas covered: Herein, the most relevant mathematical models used in clinical anticancer drug
development during the last decade are described. Less recent models were considered in the review
if they represent a standard for the analysis of certain types of efficacy or safety measures.
Expert opinion: Several mathematical models have been proposed to predict overall survival from earlier
endpoints and validate their surrogacy in demonstrating drug efficacy in place of overall survival. An
increasing number of mathematical models have also been developed to describe the safety findings.
Modeling has been extensively used in anticancer drug development to individualize dosing strategies
based on patient characteristics, and design optimal dosing regimens balancing efficacy and safety.
ARTICLE HISTORY
Received 17 August 2017
Accepted 2 October 2017
KEYWORDS
Adverse events; attrition
rate; clinical oncology;
efficacy; mathematical
modeling;
pharmacodynamics;
resistance; safety
1. Introduction
Clinical drug development implies the definition and optimiza-
tion, via dose and dose regimen selection, of the risk-to-benefit
ratio (RBR). When an acceptable RBR cannot be achieved, the
development of a new drug is stopped and the ensemble of
these occurrences is known as attrition. In the oncology field, the
attrition rate is higher compared to other therapeutic areas, and
it is mainly due to lack of efficacy and unacceptable safety [1–5].
In oncology, to evaluate the treatment efficacy of a new
drug and compare it with the available standard of care, the
overall survival (OS), measured from either the date of diag-
nosis or the start of treatment up to the time of death from
any cause, is generally assessed and is considered the golden
standard. However, during the development of new com-
pounds, exploratory markers (target engagement, pharmaco-
dynamic [PD], and disease biomarkers), demonstrating clinical
antitumor efficacy earlier than OS, are used to support the
decision-making process [6]. Also, surrogate endpoints were
proposed to substitute OS for regulatory approval of drugs.
Whilst pharmacological biomarkers validation may be limited
to the assessment of the bioanalytical performance and the
demonstration of the relationship with the mode of action of
the compound, surrogate endpoints should instead undergo a
more stringent validation. This requires the statistical demon-
stration that both the surrogate endpoint is predictive for the
disease outcome and the effect of therapeutic interventions
on the marker is correlated with the effect on the clinical
endpoint [7,8].
Commonly employed surrogates are progression-free survival
(PFS), that is, the time that passes from, for example, the first day
of treatment and the date on which disease progresses or the
patient dies from any cause, and time to tumor progression (TTP),
not counting deaths fromcauses other than disease progression.
During oncology clinical trials, not only information on
efficacy is collected, but also drug safety is investigated,
recording adverse events (AEs) associated to the administra-
tion of the drug under study. Anticancer therapies, in fact, are
often characterized by significant toxicities that are usually off-
target for cytotoxic agents, whilst more often on-target, and
thus related to the mechanism of action, for molecular tar-
geted therapies. Unfortunately, whilst biomarkers are effi-
ciently used to anticipate the clinical efficacy, they are rarely
used to anticipate potential downstream toxicity effects [9].
Mathematical modeling provides a means to reduce attri-
tion, gathering and leveraging the information gained
throughout all the investigational studies about the mechan-
ism of action and the safety and tolerability profile of a poten-
tial drug [5,10]. The use of mathematical models as potentially
valuable tools to improve drug development has been encour-
aged by the European Medicines Agency (EMA) [11] and by
the Food and Drug Administration (FDA) [12–14].
Pharmacometric models for clinical efficacy can be used to
confirm or drive new hypotheses on anticancer therapy response
CONTACT Italo Poggesi ipoggesi@its.jnj.com
*These authors contributed equally to this work.
EXPERT OPINION ON DRUG DISCOVERY, 2018
VOL. 13, NO. 1, 5–21
https://doi.org/10.1080/17460441.2018.1388369
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mechanisms, provide indication of antitumor efficacy, and corre-
late the surrogate endpoint to survival (hence providing also
surrogate validation) [15]. Finally, mathematical models for clin-
ical safety can be used to evaluate quantitatively the outcome of
dose-escalation approaches, find the optimal relationship
between exposure and safety, and, by considering variability
between patients, build individualized dosing strategies.
Analogously to our previous review [16], in which the most
recent and relevant models used in preclinical and transla-
tional anticancer drug discovery were surveyed, herein we
will present some recent modeling examples regarding the
study of clinical efficacy and safety of oncology drugs. We will
not focus on pharmacokinetic (PK) models, as classical meth-
ods are usually employed.
Given the high number of works published every year on
modeling in clinical oncology, we do not intend to provide a
comprehensive literature review. Works published in the last
10 years, presenting innovative methodological aspects or
offering a new context of application, were selected; less
recent models whose use is currently widespread in the mod-
eling community were also considered.
Models for efficacy will be presented following the concep-
tual order illustrated in Figure 1(a). Note that, since time-to-
event (TTE) modeling of OS belongs to the repertoire of widely
applied and well-established methods, the review of survival
models is beyond the scopes of the survey (for more informa-
tion, see for example [17]). Conversely, since the development
of resistance is a main cause for lack of efficacy, its modeling
will be reviewed. Models for safety will be presented following
the scheme of Figure 1(b). Since the mechanisms that lead
from PK to AEs are various and complex [10], and do not obey
to a generalized conceptual order, quantitative models for AEs
are hereafter classified according to the type of recorded data.
Readers are referred to Tables 1 and 2for a general over-
view of the models selected for efficacy and safety, respec-
tively; not all models reported in the summary tables are also
described in the text.
2. Efficacy
2.1. Biomarkers of pharmacological activity
Biomarkers are objective indicators of normal biological pro-
cesses, pathogenic processes, or pharmacological responses to
a therapeutic intervention [69]. The mechanism of action of
some anticancer treatments is based on the inhibition or
stimulation of specific molecules, which are supposed to trig-
ger other reactions, in turn influencing tumor progression.
Once demonstrated a correlation to clinical outcomes (e.g.
Figure 1. a) Pharmacokinetic (PK) exposure metrics can drive biomarkers of pharmacological activity that trigger a physiological response (e.g. they inhibit
proliferation, or they induce apoptosis); such a response should imply a reduction in tumor size (represented by different types of measures). The tumor mass should
in turn lower the production of possible tumor markers. All this chain of events will influence the survival of the cancer patient. Despite the whole chain could be
modeled, more frequently some steps are skipped, for example linking directly tumor size to overall survival. b) PK metrics can also be the drivers of various drug-
related adverse events (AEs); they can be described either via continuous, ordered categorical or time-to-event data. AEs can be distinguished in on-target and off-
target events. On-target AEs are related to the drug reaching the target via its mechanism of action: for instance, cardiovascular toxicities may result from the
intended inhibition of a signaling pathway. Off-target AEs are due to undesired interactions between the drug and unintended targets: for instance, cytopenias are
often associated with general cytotoxic activity of the compound.
Article highlights
●The reduction of attrition in drug development is of paramount
importance, in particular in the oncology field, where it is higher
than in the other therapeutic areas.
●Mathematical models, by gathering and quantitatively integrating
knowledge about the drug efficacy and safety profiles throughout
every development phase, can provide a fundamental tool to reduce
attrition and improve the decision-making process.
●Many relevant examples of modeling approaches applied to drug
development in clinical oncology can be found in the recent
literature.
●Approaches employed for the assessment of drug efficacy profile
consist in models describing and/or linking the dynamics of: biomar-
kers of pharmacological activity, tumor size, tumor markers, and OS
(or surrogate endpoints). Empirical and more mechanistic models
accounting for the development of resistance in cancer patients
have also been developed.
●A variety of models have been proposed for characterizing the drug
safety profile, according to the type of data recorded during AEs
monitoring: continuous, ordered categorical, and time-to-event.
This box summarizes key points contained in the article.
6S. M. LAVEZZI ET AL.
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Table 1. Efficacy models: details about tumor type, drug, measured variable, and pharmacometric models are reported for the cited works. As in many cases numerous observed variables are modeled, the listing order in the
column ‘Model type’follows the order of the column ‘Observed variables’, to indicate to which model each variable corresponds.
Tumor type Drug Drug description Endpoint Observed variable Model type
Driver of PD
response Driver(s) of survival Resistance Reference
GIST Sunitinib Targeted therapy (TKI) Biomarkers, OS VEGF, sVEGFR-2, sKIT,
sVEGFR-3, SLD, OS
IRM (type II, VEGF), IRMs
(type I, sVEGFR-2,
sVEGFR-3, sKIT), Claret TGI
model, TTE Weibull
distribution
AUC Baseline tumor size,
relative change of
sVEGFR-3 from
baseline at time t
Yes [18]
Advanced solid
tumors
Pembrolizumab mAb (immune
checkpoint
inhibitor)
Biomarker IL-2 Emax model C(t) - No [19]
PC Triptorelin Hormone therapy
(GnRH agonist)
Tumor marker TST Semi-mechanistic model Fraction of
activated
receptors
-No[20]
Advanced solid
tumors, GIST,
mRCC
Sunitinib Targeted therapy
(Tyrosine Kinase
Inhibitor, TKI)
Tumor size (1-D),
TTP, OS,
RECIST
SLD, TTP, OS, RECIST Claret TGI model, TTE
Weibull distribution, TTE
Weibull distribution,
logistic regression
C(t), AUCss, AUCss,
AUCss
AUCss Yes [21]
NSCLC Docetaxel Cytotoxic (taxane
antimicrotubule
agent)
ORR, OS ORR, OS Logistic regression (binomial
distribution), TTE Cox
Proportional Hazard
Actual cumulative
dose
Various demographic
covariates
No [22]
Colorectal cancer Capecitabine or Fluorouracil Cytotoxics
(antimetabolites)
Tumor size (1-D),
OS
SLD, OS Claret TGI model, TTE
Lognormal distribution
Daily dose Baseline and change
in tumor size at
week 7
Yes [23]
Advanced
epithelial OC
Carboplatin or
Gemcitabine + Carboplatin
Cytotoxic (alkylating
agent,
antimetabolite)
Tumor size (1-D),
OS
SLD, OS Claret TGI model, TTE
Weibull distribution
Per-cycle mean
concentration
TSR(t), new lesions at
time t
No [24]
Advanced Solid
Tumors
BYL719 Phosphoinositide 3-
kinase inhibitor
(PI3K)
Tumor size (1-D) SLD IRM (type I) C(t) - No [25]
Advanced/
metastatic RCC
Pazopanib Targeted therapy (TKI) Tumor size (1-D) SLD ODE system (including
tumor growth and
angiogenesis inhibition)
AUC - Yes [26]
LGG Procarbazine + 1-(2-
chloroethyl)-3-cyclohexyl-l-
nitrosourea + vincristine (PCV)
or temozolomide (TMZ) or
radiotherapy
Cytotoxics (alkylating
agent, alkylating
agent, plant
alkaloid, alkylating
agent)
Tumor size (1-D) Mean Tumor Diameter ODE system (considering
different cells
subpopulations)
C(t) - No [27]
LGG Temozolomide Cytotoxic (alkylating
agent)
Tumor size (1-D) Mean Tumor Diameter ODE system (considering
different cells
subpopulations)
C(t) - Yes [28]
NHL Inotuzumab ozogamicin ADC Tumor size, ORR SPD, ORR Modified Claret TGI model
(no resistance)
C(t) - No [29]
VS Bevacizumab or Bevacizumab +
Everolimus
Antiangiogenic mAb,
targeted therapy
(mTOR inhibitor)
Tumor size (3-D),
tumor markers
Volume (MRI), VEGF,
mTORC1
Mechanistic model C(t) - No [30]
NSCLC Erlotinib Targeted therapy
(protein-TKI)
Tumor size (3-D),
OS
FDG, FLT, OS Claret TGI model, TTE
exponential distribution
C(t) Baseline and change
in tumor size at
week 1
Yes [31]
GIST Sunitinib Targeted therapy (TKI) Tumor size (1-D,
3-D), tumor
marker, OS
SLD, SUVmax, VEGF,
sVEGFR-2, sVEGFR-3,
sKIT, OS
Claret TGI model, IRM (type
IV), TTE exponential
distribution
Daily AUC Relative change in
SUV
max
for the
lesion that
responded the best
at week 1
No [32]
(Continued )
EXPERT OPINION ON DRUG DISCOVERY 7
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Table 1. (Continued).
Tumor type Drug Drug description Endpoint Observed variable Model type
Driver of PD
response Driver(s) of survival Resistance Reference
GEP-NET Lanreotide
Autogel (LAN)
Somatostatin analog Tumor marker,
PFS
CgA, PFS Linear (Box-Cox scale) and
inhibitory Emax, TTE
Weibull distribution
C(t) CgA(t)/CgA0 (PFS) No [33]
SCLC Cisplatin or carboplatin +
etoposide
Cytotoxics (alkylating
agents,
topoisomerase II
inhibitor)
Tumor marker,
TTP
LDH, NSE, TTP Turnover models, TTE log-
logistic
AUC Change in disease
level at time t(TTP)
Yes [34]
mCRPC Not specified Chemotherapy and/or
hormonotherapy
Tumor markers PSA, CTC K-PD approach, latent
variable, IRM (type III),
cell life span
- - No [35]
mCRPC Docetaxel + prednisone Cytotoxic (taxane
antimicrotubule
agent),
glucocorticosteroid
Tumor marker,
OS
PSA, OS ODE system (distinction
among cells), TTE Weibull
distribution
Efficacy index Sensitive and resistant
cells at time t
Yes [36]
mCRPC Abiraterone acetate +
prednisone
Hormone therapy
(adrenal inhibitor),
glucocorticosteroid
Tumor marker,
OS
PSA, OS Claret TGI model, TTE Cox
Proportional Hazard
C(t) Model-based PSA
Doubling Time from
baseline
Yes [37]
CRPC Eribulin Cytotoxic (non-taxane
microtubule
inhibitor)
Tumor marker,
OS
PSA, OS Claret TGI model, TTE
Weibull distribution
AUC after each
dose
administration
PSA baseline and
growth rate
constant
Yes [38]
CRPC Eribulin Cytotoxic (non-taxane
microtubule
inhibitor)
Tumor marker,
OS, ECOG
PSA, OS, ECOG Claret TGI model, TTE
Weibull distribution,
Markov-transition
AUC PSA baseline and
growth rate
constant
Yes [39]
Relapsed/
refractory MM
Carfilzomib Targeted therapy
(proteasome
inhibitor)
Tumor marker,
OS
M-protein, OS Claret TGI model, TTE
lognormal distribution
Dose(t)ECTS at week 4 Yes [40]
OC Carboplatin + Doxorubicin (CD)
or Carboplatin + Paclitaxel
(CP)
Cytotoxics (alkylating
agent, anthracycline
antibiotic, taxane
antimicrotubule
agent)
Tumor size (1-D),
tumor marker,
PFS
SLD, CA-125, PFS K-PD approach, IRM (type I,
SLD), IRM (stimulation of
production, CA-125), TTE
log-logistic distribution
- Predicted change in
CA-125 from
baseline at 6 weeks
(PFS)
No [41]
Advanced
melanoma
with BRAF
mutations
Dabrafenib or Dabrafenib +
Trametinib
Targeted therapies
(BRAF kinase
inhibitor, MEK
inhibitor)
PFS PFS SDE system Drug
concentration in
the body
Disease progression
onset based on a
predefined
threshold of tumor
cell number
Yes [42]
PC: prostate cancer; NSCLC: non-small-cell lung cancer; GIST: gastrointestinal stromal tumors; mRCC: metastatic renal cell carcinoma; LGG: low grade glioma; RCC: renal cell carcinoma; OC: ovarian cancer; NHL: non-Hodgkin’s
lymphoma; VS: vestibular schwannomas; GEP-NET: nonfunctioning gastroenteropancreatic neuroendocrine tumor; SCLC: small cell lung cancer; mCRPC: metastatic castration-resistant prostate cancer; CRPC: castration-
resistant prostate cancer; MM: multiple myeloma; ADC: antibody-drug conjugate; TKI: tyrosine kinase inhibitor; mAb: monoclonal antibody; ORR: objective response rate; OS: overall survival; TTP: time to tumor progression;
RECIST: Response Evaluation Criteria in Solid Tumors; PFS: progression-free survival; ECOG: Eastern Cooperative Oncology Group status; TST: testosterone; SLD: sum of longest diameters; SPD: sum of the product of the
longest diameters; FDG: [18F-]-fluorodeoxyglucose; FLT: 3′-[18F]fluoro-3′-deoxy-L-thymidine; SUVmax: maximum standard uptake value; VEGF: vascular endothelial growth factor; sVEGFR-2: soluble vascular endothelial
growth factor receptor 2; sVEGFR-3: soluble vascular endothelial growth factor receptor 3; sKIT: soluble KIT; IL-2: interleukin-2; MRI: magnetic resonance imaging; mTORC1: mammalian target of rapamycin complex 1; CgA:
chromogranin A; LDH: lactate hydrogenase; NSE: neuron-specific enolase; PSA: prostate-specific antigen; CTC: circulating tumor cells; CA-125: cancer antigen 125; TGI: tumor growth inhibition; IRM: indirect response model;
ODE: ordinary differential equations; TTE: time-to-event; SDE: stochastic differential equations; AUC: area under concentration-time curve; AUCss: AUC at steady state; C(t): drug concentration over time; TSR: tumor size ratio;
ECTS: early change in tumor size.
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Table 2. Safety models: details about tumor type, drug, type of adverse event (AE), observed variable, and pharmacometric models are reported for the cited works.
Tumor type Drug Drug description AE Observed variable Model type
Driver of
PD
response Reference
- Docetaxel, Paclitaxel,
Etoposide, DMDC,
Irinotecan vinflunine
Cytotoxic Neutropenia Leukocytes and
neutrophils
counts
Myelosuppression model C(t) [43]
LC Combination of BI 2536
and Pemetrexed
Cytotoxic Neutropenia Neutrophils counts Myelosuppression model (Friberg model with additional ANC input
in Circ comp. for transient ANC flare-up following drug
administration)
C(t) [44]
BC Docetaxel Cytotoxic Neutropenia Neutrophils counts Myelosuppression model C(t) [45]
OC Trabectedin Cytotoxic Neutropenia Neutrophils counts Myelosuppression model (Friberg model with additional effect
compartment)
C
e
(t) [46]
OC, NSCLC, and
other solid
tumors
Paclitaxel Cytotoxic Neutropenia Neutrophils counts Myelosuppression model C(t) [47]
BC Eribulin mesilate Cytotoxic Neutropenia Neutrophils counts Myelosuppression model C(t) [48]
Lymphoma and
solid tumors
Abexinostat Cytotoxic Thrombocytopenia Platelets counts Myelosuppression model (Friberg model with additional feedback
on MTT and Imax disease progression model for baseline
reduction with time)
C(t) [49]
BC Docetaxel + epirubicin Cytotoxic Neutropenia ANC Myelosuppression model C(t) [50]
BC Docetaxel + epirubicin Cytotoxic Neutropenia ANC Myelosuppression model (modified version of Meille model [50]) C(t) [51]
Indolent NHL Inotuzumab ozogamicin ADC Thrombocytopenia Platelets counts Myelosuppression model (Friberg model with zero order
proliferation, no feedback mechanism, and drug-induced
inhibition on Circ)
C(t) [29]
Advanced solid
tumors
Diflomotecan Cytotoxic Neutropenia Neutrophils counts Myelosuppression model (Friberg model with incorporation of cell
cycle dynamics within the stem
cell compartment)
C(t) [52]
BC Trastuzumab Emtansine ADC Thrombocytopenia Platelets counts Myelosuppression model (Friberg model where Prol is divided in
two components, one of which sensitive to the drug with a
separate Slope parameter)
C(t) [53]
BC Docetaxel, Capecitabine,
FEC
Cytotoxic Neutropenia Neutrophils counts,
G-CSF
Myelosuppression model (Friberg model with GCSF-driven
feedback mechanisms on proliferation and MTT)
C(t) [54]
Non-myeloid
malignancies
Darbepoetin alfa Erythropoiesis-stimulating
agent
Anemia Hb IRM with stimulation of production rate C(t) [55]
Solid tumors Trabectedin DNA-interacting agent
blocking cell cycle
progression in G2/M phase
Hepatotoxicity ALT IRM with stimulation of the
production rate
C(t) [56]
BC Trastuzumab Emtansine ADC Thrombocytopenia
and hepatotoxicity
Platelets counts,
ALT,
AST
Myelosuppression model (Friberg) + modified Friberg model for
ALT and AST (zero order production, no feedback)
C(t) [57]
GIST Sunitinib Targeted therapy Hypertension dBP IRM with stimulation of the production rate C(t) [58]
Solid tumors Lenvatinib Targeted therapy Hypertension dBP, sBP IRM with stimulation of the production rate [59]
BC and GC Trastuzumab Humanized mAb Cardiotoxicity LVEF Indirect (delayed) effect compartment model with recovery C(t) [60]
- Moxifloxacine Antibiotic QT prolongation QTc Circadian rhythm model including power correction for RR C(t) [61]
Solid tumors Paclitaxel Cytotoxic CIPN CTC-AE Proportional odds model T
C>0.05µM
[62]
Solid tumors Capecitabine Cytotoxic HFS CTC-AE Proportional odds model with a Markov process Cumulative
dose
[63]
mCRC Capecitabine Cytotoxic HFS CTC-AE Proportional odds model with a Markov process Cumulative
dose
[64]
GIST Sunitinib Targeted therapy Fatigue, HFS CTC-AE Proportional odds model with a Markov process sVEGFR-3(t) [58]
Solid tumors Lenvatinib Targeted therapy Proteinuria CTC-AE Markov transition model AUC [59]
NSCLC Erlotinib Targeted therapy Rash, diarrhea CTC-AE Markov transition model C(t) [65]
(Continued )
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tumor size [TS], OS), the study of biomarkers dynamics can
provide a deeper understanding of drug efficacy.
For instance, the vascular endothelial growth factor (VEGF)
is the target of numerous antiangiogenic therapies, as it pro-
motes the formation of new blood vessels. In [18] and [32], the
time courses of VEGF, soluble fragments of KIT (sKIT), and
soluble VEGF receptors were evaluated as potential predictors
of clinical outcome, in sunitinib-treated patients with gastro-
intestinal stromal tumors (GISTs). In particular, in [18] sKIT and
soluble VEGF receptor 3 (sVEGFR-3) were identified as predic-
tors of TS, which in turn drove OS together with sVEGFR-3.
For targeted therapies, measures of target engagement can
be used as biomarkers of pharmacological activity. In [19],
measurements of interleukin-2 (IL-2) stimulation ratio in blood
(obtained from IL-2 concentrations) were employed. A model
approach based on PK and IL-2 PD data was used to support
the assessment of dose regimens for an immune checkpoint
inhibitor, pembrolizumab, to reliably determine target engage-
ment in patients with advanced solid tumors, hence character-
izing successfully the benefits of the immunotherapy drug.
As it is believed that prostate cancer development and
progression is driven by androgens, it can be treated with
hormone therapies, aimed to maintain serum testosterone
(TST) levels consistently below the castration limit (0.5 ng/ml).
This can be achieved by administration of triptorelin (TRP), a
gonadotropin-releasing hormone (GnRH) agonist. TRP initially
stimulates TST production; however, after continuous exposure,
GnRH receptors synthesis undergoes downregulation, even-
tually decreasing TST production. The mechanism-based popu-
lation model for TST dynamics developed in [20] by Romero
et al. reflects the mechanism of action of TRP (see Figure 2).
Noticeably, the model was used to compute PD descriptors (e.g.
castration time), and a PK descriptor, CTRP min. This represents
the minimal drug concentration maintaining TST under the
castration limit, and it can be used for designing optimal dosing
schedule for achieving minimal flare-up in TST levels, and rapid
and long-term castration.
2.2. Tumor size
2.2.1. Categorical assessment
In clinical trials, TS measurements and additional evaluations are
combined, and the response is categorized. According to the
Response Evaluation Criteria in Solid Tumors (RECIST), the sum
of the longest linear dimensions of the measurable target lesions
(SLD), together with the evaluation of nonmeasurable lesions and
lymph nodesdimensions, contributes to the assessment of objec-
tive response (OR) [70,71]. Analogously, in case of malignant
lymphomas, the sum of the products ofthe two largest diameters
of the target lesions (SPD) [72], is used together with liver, spleen,
and bone marrow assessments.
OR is divided in four classes: complete response (CR, i.e.
disappearance of the target measurable lesions), partial
response (PR, a predefined % decrease of the target measurable
lesions), progressive disease (increase of the target measurable
lesions), and stable disease (declared when none of the pre-
vious definitions do apply). OR rate (also called overall response
rate, ORR), usually defined as the proportion of overall
responses (i.e. PR and CR), is sometimes used as surrogate
Table 2. (Continued).
Tumor type Drug Drug description AE Observed variable Model type
Driver of
PD
response Reference
BC Pertuzumab,
trastuzumab and
docetaxel
Humanized mAb, cytotoxic Various Grade of AE severity
and Time to AE
Kaplan–Meier, multi-state models, competing risk models - [66]
Advanced
melanoma,
NSCLC, RCC,
mCRPC, CRC
Nivolumab Humanized mAb AE > Grade 3 and AEs
leading to
discontinuation
Time to AE Kaplan–Meier Dose [67]
Metastatic MTC Cabozantinib Targeted therapy Dose modification Time to first
occurrence of a
dose modification
Kaplan–Meier AUCss [68]
BC: breast cancer; OC: ovarian cancer; NSCLC: non-small-cell lung cancer; LC: lung cancer; NHL: non-Hodgkin’s lymphoma; GC: gastric cancer; mCRC: metastatic colorectal cancer; GIST: gastroIntestinal stromal tumors; mRCC:
metastatic renal cell carcinoma; CRC: colorectal cancer; MTC: medullary thyroid cancer; DMDC: dimethyl dicarbonate; FEC: fluorouracil (5-FU), epirubicin and cyclophosphamide; ADC: antibody drug conjugate; mAb:
monoclonal antibody; CIPN: chemotherapy-induced polyneuropathy; HFS: hand-and-foot syndrome; ANC: absolute neutrophils count; MTT: mean transient time; G-CSF: granulocyte-colony stimulating factor; ALT: alanine
transaminase; AST: aspartate transaminase; Hb: hemoglobin; QTc: corrected QT interval; LVEF: left ventricular ejection fraction; dBP: diastolic blood pressure; sBP: systolic blood pressure; CTC-AE: NCI common toxicity criteria
for adverse events classification; IRM: indirect response model; C(t): drug concentration over time; C
e
(t): drug concentration over time in the effect compartment; T
C>0.05µM
: time above a drug plasma concentration of 0.05 µM;
sVEGFR-3: soluble vascular endothelial growth factor receptors; AUC: area under concentration-time curve; AUCss: AUC at steady state.
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endpoint for accelerated or regular approval [73]. Logistic
regression was employed in [21]and[22] for modeling the
relationship between metrics of drug exposure and response.
In particular, in [22] ORR was considered as the efficacy end-
point (together with OS) in non-small-cell lung cancer (NSCLC)
patients treated with docetaxel. The number of patients with PR
or CR (Nresp) is described by a binomial distribution with para-
meters p;NðÞ,whereNis the sample size and pis the prob-
ability of response occurrence, which is in turn modeled as an
inverse logit. Hence, the model is represented by Equation 1:
Nresp,Binðp;NÞ(1)
p¼invlogit intercept þfD;covðÞðÞ;
where fD;covðÞis a linear combination of docetaxel actual
cumulative dose (D) and possible covariates covðÞ. As ORR
was found to correlate with D, this model could represent a
tool to further explore the drug therapeutic window.
2.2.2. Unidimensional assessment
Although the categorization of tumor response offers a simple
criterion for standardization, it omits the time course of drug
potency and effect. The longitudinal PK-PD modeling frame-
work applied to TS measurements can improve understanding
of drug action.
Claret et al. developed a Tumor Growth Inhibition (TGI)
model based on SLD data obtained from metastatic colorectal
cancer (mCRC) patients treated with capecitabine or fluorour-
acil [23]. A single differential equation describes the longitu-
dinal TS dynamics (Equation 2):
dTS tðÞ
dt ¼KLTS tðÞKDtðÞExposure tðÞTS tðÞ
KDtðÞ¼KD0eλt;(2)
TS t ¼0ðÞ¼TS0;KDt¼0ðÞ¼KD0
Natural tumor growth is described via a first order process with
rate constant KL, whilst tumor shrinkage (KDtðÞ) due to drug
exposure (Exposure tðÞ) is decreased in time by the development
of resistance, modeled through an exponential with parameter λ.
An OS hazard model was also proposed: observed baseline and
model-predicted relative change in TS from baseline at week 7
were used as predictors. Overall, the TGI + OS model was able to
predict survival in a phase III trial on the basis of phase II and
historical data, hence representing a good tool for supporting
end-of-phase II decisions.
To describe SLD dynamics in platinum-sensitive metastatic
ovarian cancer patients treated with carboplatin or carboplatin
+gemcitabine, in [24] Zecchin et al. modified the Claret TGI
model, by including the additive effect of the two drugs and
by removing the resistance term (not supported by the data).
The exposure metrics chosen to drive TGI was the mean
concentration reached in each cycle of treatment.
Furthermore, a TTE model was built for describing the appear-
ance of new lesions, with descriptors given by: (i) presence of
liver metastasis at baseline, and (ii) predicted TS Ratio at time t
(TSR tðÞ). The outcome of this TTE model, together with base-
line SLD, Eastern Cooperative Oncology Group (ECOG) perfor-
mance status, and TSR tðÞ, were found to be significant
predictors for OS. Hence, the whole model-predicted time
profile of TS was related to OS, in contrast with [23], where
the summary TS metric, that is, model-predicted TS change at
week 7 versus baseline, was exploited. Noticeably, according
to [24], this is the first work in which a TTE model for new
lesions appearance including TS time course is considered.
Furthermore, the importance of monitoring also the birth of
new lesions and, possibly, the status of non-target lesions
during the treatment was reinforced by the strong correlation
found between the onset of new lesions and OS [24].
Models for SLD distinct from [23] were also proposed [25,26].
Furthermore, different TS linear assessments were considered for
modeling tumor growth progressionand inhibition: for example,
mean tumor diameters (the geometric mean of the three largest
perpendicular diameters of the lesion) in [27,28]. In lymphoma
indications, SPD is instead used, as in [29].
2.2.3. Volumetric assessment
Uni- or bidimensional measurements can adequately summarize
TS for cancers characterized by regularly shaped lesions. In other
Figure 2. Scheme of the PD model for testosterone (TST) proposed in [20]. The fraction of gonadotropin-releasing hormone (GnRH) receptors activated by triptorelin
(FRAC) stimulates TST production. After continuous exposure to the drug, total GnRH receptors (RT) undergo down-regulation. CTRP represents triptorelin (TRP) serum
concentration, KDis TRP receptor equilibrium dissociation constant, and AGN is the ratio between endogenous agonist concentration and its equilibrium dissociation
constant. Synthesis and degradation constants for TST and RTare, respectively, kS T and kD T, and kS R and kD R. A zero-order TST production term, independent
from both luteinizing hormone (LH) and follicle-stimulating hormone (FSH), is also included (kin). A feedback mechanism regulates decrease and recovery of RT. The
dashed arrows represent the stimulation induced by activated receptors. Adapted from [20], with permission of ASPET.
EXPERT OPINION ON DRUG DISCOVERY 11
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cases, volumetric assessment may provide a more complete
picture, as it is also more sensitive to changes in tumor burden,
allowing early detection of response or progression [74].
Volumetric assessments were recommended for assessing
response of neurofibromatosis type 2 (NF2) related vestibular
schwannoma (VS) to drug treatment [75]. In [30], Ouerdani
et al. developed a mechanism-based model for VS based on
tumor volume measurements for describing the PD of beva-
cizumab and everolimus, which target VEGF and mammalian
target or rapamycin complex 1 (mTORC1), respectively. A
schematic description of the model is reported in Figure 3.
Both PK models for bevacizumab [77] and everolimus [78], and
a quasi-steady-state target-mediated drug disposition (qss-
TMDD) model for the effect of bevacizumab on VEGF [79]
were obtained from the literature. The overall model was not
over-parametrized. Furthermore, as it is mechanism-based, it is
able to carry relevant semi-physiological information. Once
further validated, the model will be able to help clinicians in
predicting the treatment efficacy and hence in personalizing
treatment.
2.2.4. Functional assessment
As an alternative or an adjunct to anatomic assessment of
tumor burden, also functional assessment may be considered,
after appropriate clinical validation.
In particular, the use of [18F-]-fluorodeoxyglucose positron
emission tomography (FDG-PET) imaging, already recom-
mended in 2009 [71], has been proposed to assess early response
for cytostatic drugs [80,81]. With respect to standard TS measure-
ments, functional assessment provides a complete picture of the
tumor burden and its metabolic activity in the patient’sbody.
Models of longitudinal measurements of FDG uptake, assessed
as the maximum standardized uptake value (SUV
max
) of a region
of interest, were developed in [31], for erlotinib treatment in
NSCLC, and in [32], for sunitinib treatment in GISTs.
In particular, in [32] SUV
max
was modeled by means of an
indirect response model (IRM) with stimulation of elimination
(type IV [82]), estimating also interlesion variability. In this
work, also SLD measurements were modeled, via the Claret
TGI model; interestingly, drug effects on individual lesion
SUV
max
and SLD were found to be highly correlated. OS was
not driven by SUV
max
or SLD dynamics; a time-invariant cov-
ariate summing up the drug-effects on SUV
max
was included
instead.
Even if SUV
max
is usually employed for the interpretation of
FDG-PET imaging, its clinical relevance remains controversial,
hence alternatives may be considered. In [83] a model-derived
parameter, described as the time required to reach 80% of the
amount of metabolized [18F-]-FDG, was proposed as a better
predictor of patients response.
2.3. Tumor markers
Tumor markers are substances that are produced by cancer or
normal cells in the presence of cancer. Their dynamics may be
linked to OS, directly or indirectly (e.g. via common surro-
gates), providing an early insight for antitumor efficacy: mod-
els describing these links are hence of great importance.
In [33], a quantitative relationship between PFS and
Chromogranin A (CgA) was established for patients with gas-
troenteropancreatic neuroendocrine tumors (GEP-NETs) trea-
ted with Lanreotide Autogel (LAN). Similarly, in [34] Buil-Bruna
et al. explored and linked to TTP the dynamics of the tumor
markers lactate hydrogenase (LDH) and neuron specific
Figure 3. Schematic representation of the mechanistic model proposed in [30]. Tumor natural proliferation is divided in two phases: an initial exponential growth,
followed by a linear one (as first proposed for xenograft models in [76]), whilst hypoxia-induced apoptosis is inhibited by unbound vascular endothelial growth
factor (VEGF). An inhibitory I
MAX
model describes everolimus effect on mammalian target or rapamycin complex 1 (mTORC1), whilst its effect on VEGF is exerted via
an mTORC1-related upregulation of VEGF production (zero-order process). The upregulation of proliferation signal pathways promoting tumor development is
triggered by the continuous inhibition of the mTORC1 pathway. Hence, a chain of transit compartments representing the different proteins involved in the
upregulation of tumor proliferation has been added to the model. Furthermore, a term accounting for the delayed upregulation triggered by these signal pathways
is included in the equation describing tumor size dynamics. Adapted from [30], with permission of Springer.
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enolase (NSE), using data from small cell lung cancer (SCLC)
patients, treated with chemotherapy and radiotherapy. Half of
the patients also received granulocyte colony-stimulating fac-
tor (G-CSF). LDH and NSE dynamics were described as in
Equation 3:
dD tðÞ
dt ¼λ1βRTðÞαCT tðÞDtðÞexp γò
t
0
CT uðÞdu
!
dLDH tðÞ
dt ¼KINLDH 1þ#GCSFðÞþKDLDH DtðÞKOUTLDH LDH tðÞ
(3)
dNSE tðÞ
dt ¼KINNSE þKDNSE DtðÞKOUTNSE NSE tðÞ
Dt¼0ðÞ¼1;
LDH t ¼0ðÞ¼
KINLDH þKDLDH Dt¼0ðÞλKDLDH
KOUTLDH

KOUTLDH
;
NSE t ¼0ðÞ¼
KINNSE þKDNSE Dt¼0ðÞλKDNSE
KOUTNSE

KOUTNSE
The latent variable DtðÞis defined as ‘disease level’and repre-
sents the unobserved TS (arbitrarily initialized to 1). It grows
with a zero-order process (with rate constant λ), fractionally
decreased with magnitude βby radiotherapy (RT), which is
treated as a categorical variable (RT ¼1 if the patient is trea-
ted with irradiation, RT ¼0 otherwise). Chemotherapy,
expressed as level of exposure CT tðÞ, acts on disease level
depletion as a second-order process with proportionality con-
stant α; the exponential term accounts for the development of
resistance, related to cumulative chemotherapy exposure
(ò
t
0
CT uðÞdu), appropriately scaled via γ. In the turnover models,
LDH and NSE production is a zero-order process with rate
constants KINLDH and KINNSE , respectively, and it is stimulated
by the disease, via the first-order rate constants KDLDH and
KDNSE . LDH and NSE are degraded with first-order rate con-
stants KOUTLDH and KOUTNSE ; G-CSF is included as a covariate on
LDH natural synthesis via #. This model separates SCLC-speci-
fic (λ,KDLDH ,KDNSE ) and treatment-specific parameters (α,γ,β),
hence it may be adapted to other cancer types or treatments.
Changes in disease level were found to drive TTP; correlation
between predicted disease level and observed TS was also
demonstrated. Moreover, it was shown that model-predicted
LDH and NSE are able to discriminate observed tumor pro-
gression: this is of particular interest, as these markers are not
routinely used as clinical outcome predictors for SCLC.
To investigate circulating tumor cells (CTC) as a potential
tumor marker for metastatic castration-resistant prostate can-
cer (mCRPC) a similar model was developed, including a latent
variable, representing the unobserved tumor burden, that
drives increase in prostate-specific antigen (PSA) and CTC
count [35]. Possible links with survival measures were pro-
posed by the authors of [35] as development of the work.
Even if contradictory results have been reported regarding
its surrogacy [84], PSA is routinely used as intermediate marker
for mCRPC [85]. Several mathematical models describing PSA
dynamics in response to anticancer therapy and correlating it
to OS have been developed. In [36], a semi-mechanistic model
was proposed for describing PSA, treatment-sensitive (StðÞ)
and resistant cells (RtðÞ) dynamics. The two unobserved vari-
ables, StðÞand RtðÞ, anticipated OS. In [37], Xu et al. applied
the Claret TGI model to PSA measurements, obtained from
mCRPC patients treated with abiraterone acetate (AA).
Numerous model-predicted summary PSA endpoints were
derived and tested as OS predictors. PSA doubling time from
baseline (PSADT) was included in the model as the most
significant descriptor; therefore, the whole PSA dynamics,
and not only a summary index referred to a particular week,
was considered for the computation of the predictor.
Interestingly, in this work it is suggested that the contradictory
results about PSA surrogacy may be due to the fact that
previous evaluations were based mainly on data from studies
for cytotoxic agents: AA is a direct inhibitor of androgen
biosynthesis, so it may have a more PSA-dependent mechan-
ism. In [38] and [39], the Claret TGI model was again employed
for PSA; its baseline and growth rate were selected as survival
predictors. Furthermore, in [39] a comprehensive framework is
reported, including PK-PD models for safety and drug expo-
sure effect on PSA. In turn, PSA influence on survival, ECOG
performance score, and dropout was modeled [39].
Similar models, linking tumor markers to survival, have
been developed for other types of cancer and markers [40,41].
2.4. Resistance
An important clinical problem in cancer research is the devel-
opment of resistance to therapy. To describe resistance, a term
representing an exponentially declining drug effect is usually
employed and included in tumor progression models
[23,28,34]. A variety of more complicated approaches, involving
evolutionary theory, stochastic processes, and differential equa-
tions, have been proposed [86]; yet these models are difficult to
connect with markers, tumor progression, and survival models,
and they are often not validated on clinical data.
In [42], an effort is made in this direction by Sun et al.:
drug-sensitive cancer cells are supposed to secrete into the
tumor microenvironment various soluble factors, which pro-
mote growth, dissemination, and metastases of drug-resistant
cancer cells and support the survival of drug-sensitive cells. A
model on cellular scale, expressed in terms of stochastic alge-
braic and differential equations, was proposed for describing
these dynamics. The model was validated on clinical data of
metastatic melanoma patients with BRAF V600 mutations trea-
ted with the BRAF inhibitor dabrafenib and the MEK inhibitor
trametinib [87], by comparing the distribution of observed and
predicted PFS, which appeared in good agreement.
3. Safety
3.1. Continuous AEs
3.1.1. Hematological toxicities
Hematological toxicities represent one of the most common
AEs associated with anticancer drugs. They consist in a reduc-
tion of the proliferation of blood cells in the bone marrow,
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resulting in a decreased number of circulating leukocytes, red
blood cells, and/or platelets. Several semi-mechanistic models
describing the effect of chemotherapy on neutrophil kinetics
have been published over the last decades [43,88–91]. They
generally comprehend a proliferation compartment containing
cells with self-renewal capacity, a compartment representing
circulating cells, and sometimes one or more transit-compart-
ments in between to account for the maturation process [92].
The golden standard for describing the time course of myelo-
suppression is considered the model developed by Friberg et al.
(Figure 4) from leukocytes and neutrophil counts of docetaxel-,
etoposide-, and paclitaxel-treated patients [43].
The main advantages of this model are the small number
of parameters to be estimated and the discrimination
between the system-related parameters ðCirc0,γ,Ktr Þ, and
the drug-related parameter (Slope), making it applicable to
sparse datasets and different drugs. The model has shown
consistency of system-related parameters across several stu-
dies involving either humans or different animal species
treated with cytotoxic or targeted therapies. It has been
successfully used to predict the neutropenic effect following
drug combinations [44], to identify covariate effects [45,93],
and to scale myelosuppression from animal data to
humans [94].
Starting from Friberg’s work, Luu et al. [29] recently pub-
lished a quantitative method to optimize inotuzumab ozoga-
micin dosing regimen balancing safety (represented by
incidence of AEs) and efficacy (represented by ORR), evaluated
via simulated longitudinal platelet count and TS data, respec-
tively. From ORR and incidence of AEs, the clinical utility index
(CUI) of each simulated trial was then derived. This is a general
approach that can be applied in the first phase of develop-
ment of other anticancer candidates to find the optimal
dosage to be administered in subsequent studies.
Modified versions of the Friberg model have been used to
accommodate specific behaviors observed in experimental
data [44,52] or to describe thrombocytopenia driven by cyto-
toxic drug and targeted therapies [53].
In the neutropenia model described in [52] by Mangas-
Sanjuan et al., the stem cell compartment was divided into
two components, the proliferative cells Prol and the quiescent
cells Qc(nonproliferating, non-maturing), considering that
only cells in the proliferative state are sensitive to drug effect.
The quiescent cells were modeled with two compartments
(Qc1and Qc2) to account for cell cycle dynamics (Figure 4).
These modifications were introduced to overcome some limits
of the Friberg model which, in this case, was not able to
describe the neutropenic effect of diflomotecan with the
Figure 4. Schematic representation of the model for myelosuppression presented by Friberg et al. and some of its variants. The model proposed by Friberg was
originally characterized by a stem cell compartment containing cells with self-renewal capacity, a blood compartment representing circulating cells, three transit-
compartments in between to account for the maturation process, and a feedback mechanism on the proliferation rate driven by circulating cells [43]. Bender et al.
distinguished between two types of proliferating cells (depletable and non-depletable) [53], whilst Mangas-Sanjuan et al. accounted for cell cycle dynamics in the
stem cell compartment [52]. Quartino et al. substituted the feedback mechanism proposed by Friberg with two granulocyte colony-stimulating factor (G-CSF)-based
feedback mechanisms on the proliferation rate and on the intercompartmental transit rate, respectively [54]. Adapted from [43] from the American Society of Clinical
Oncology, from [53] with permission of Springer, from [52] with permission of ASPET, and from [54] with permission of Springer.
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same set of drug parameters across different routes and/or
dosing schedules. Another modification was proposed in [53]
by Bender et al. to describe the effects of trastuzumab-emtan-
sine (TDM-1) on platelets and to capture the fact that the nadir
seemed to move down slowly over cycles of TDM-1 for some
patients. The proliferation compartment of the Friberg model
Prol was divided in two pools: one of cells sensitive to drug
(BASE2), and another one of nonsensitive cells (BASE1). Two
drug effects were modeled: Edrug1on the entire proliferative
cell compartment, and Edrug2only on the depletable pool
compartment BASE2(Figure 4).
Additional modifications were applied to the Friberg model
to increase the level of mechanistic description. In [54],
Quartino et al. considered the interplay between endogenous
G-CSF concentration (modeled via a one compartment turn-
over model) and neutrophils. A G-CSF-driven feedback
mechanism controlling the proliferation rate (FBkprol) and
another one reducing the intercompartmental transit rate
(FBktr) replaced the empirical Friberg feedback mechanism
(Figure 4).
Since one of the most critical points in anticancer drug
development is to find an optimal balance between efficacy
and toxicity, PK-PD models for myelosuppression could be
used to create a tool for neutrophil- or platelet-guided dose
optimization (especially for cytotoxic drugs), as in [29,46–49],
which could be employed also for combination therapies, as in
[50,51]. Another hematological AE due to anticancer treatment
consists in hemoglobin (Hb) reductions (anemia). In [55], an
IRM with stimulation of the production (type III [82]) was
proposed to describe Hb production after administration of
an erythropoiesis-stimulating agent to patients with che-
motherapy-induced anemia. This model was able to success-
fully predict Hb response in a test dataset.
3.1.2. Hepatotoxicity
Hepatotoxicity typically consists in transient, reversible eleva-
tions of serum transaminases. In particular, in [56] Fetterly
et al. described the time course of alanine aminotransferase
(ALT), following trabectedin administration. ALT dynamics was
modeled through an adaptive IRM, consisting in two compart-
ments: the hepatocyte compartment (ALTh), where ALT pro-
duction is affected by a feedback mechanism (FB);and a
plasma compartment (ALTp) where ALT release is stimulated
by trabectedin concentration (Figure 5). This model was used
in the clinic to explore different dosing regimens and identify
the most safe and efficacious one based on liver enzymes
measurements.
In [57], ALT and aspartate transaminase (AST) responses to
T-DM1 were described by a modified version of the Friberg
model, characterized by a zero-order production rate for the
proliferative compartment and no feedback mechanism.
3.1.3. Cardiovascular toxicities
Hypertension, left ventricular dysfunction, and QT prolonga-
tion have emerged as potential cardiovascular AEs for both
cytotoxic and targeted therapies, especially for drugs targeting
the VEGF receptor. PD models have been developed for hyper-
tension, describing the change in blood pressure (BP) over
time induced by chemotherapeutic agents [21,58,59]. In parti-
cular, two separate IRMs with stimulation of production were
used to describe diastolic and systolic BP data from patients
treated with lenvatinib [59]. A similar IRM was used to model
hypertension following sunitinib treatment [58]. Then, the
predicted increase in BP, together with baseline TS and sever-
ity of myelosuppression, were found to be OS predictors. The
IRM + OS model allows to monitor in advance AEs and clinical
response, and, consequently, define the best dosing regimen.
In [60] cardiotoxicity data, expressed as left ventricular ejec-
tion fraction (LVEF), were used to develop a PK–PD model to
investigate the relationship between trastuzumab exposure
and LVEF decline. The observed cardiac damage is modeled
via an effect compartment (CEFÞlinked to the central PK com-
partment, and related to the observed LVEF via an Emax
equation (Equation 4):
Figure 5. a) Pharmacodynamic (PD) indirect response model for alanine aminotransferase (ALT). Adapted from [56], with permission of Springer. b) Simulated ALT
dynamics for different doses and inter-dose intervals.
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LVEF ¼LVEF01CEF
EC50 þCEF

;(4)
where LVEF0is the baseline LVEF value prior to treatment and
EC50 accounts for the drug effect. The recovery from trastuzu-
mab-related cardiac damage is modeled as an elimination
from the effect compartment via a first-order process. This
model can be used to find the optimal cardiac monitoring
and treatment strategy for trastuzumab cardiac toxicity.
Several models for QT prolongation have been published
for antiarrhythmic drugs [95], which could be applied also for
anticancer treatments. Recently, in [61], a PK–PD model for
translational purpose was developed, to investigate the rela-
tionship between the ether-a-gogo related gene (hERG) inhi-
bition (obtained in vitro) and the QT prolongation in dogs, and
between the QT prolongation in dogs and humans. This
model is a sum of three components: individual heart rate
correction, circadian rhythm, and drug effect.
3.2. Ordered categorical AE data
Some toxicities (e.g. fatigue, hand-and-foot syndrome [HFS],
diarrhea) cannot be quantified through continuous measure-
ments but are instead represented by ordered categories
defined by scoring systems. Grades for AEs usually go from 1
(mild or asymptotic symptoms) to 5 (death). To model ordered
categorical scores and to predict the probability of having a
certain grade of toxicity, two types of approaches can be used:
logistic regression analysis (binary logistic regressions and
proportional odds models) and Markov transition modeling
(discrete and continuous). A proportional odds model was
proposed in [62] to develop a dosing algorithm to target a
certain paclitaxel exposure, avoiding at the same time the
occurrence of chemotherapy-induced polyneuropathy (CIPN).
A limitation of the logistic regression analysis approach is due
to the fact that the probability of a certain grade of toxicity
does not depend on the previous observed grade. This issue is
overcome by implementing hybrid proportional odds models
integrating a Markov process. These models have been used
to predict HFS [63] and to develop an individual adaptation
dosing strategy limiting toxicity events [64] in patients receiv-
ing capecitabine, and to predict both fatigue and HFS in
patients receiving sunitinib [58].
Alternatively, continuous time Markov models keep into
account that probabilities of experiencing each grade of AE
are in fact time-varying. To apply this approach, the time
course of AEs scores must be recorded on a suitable timescale.
A continuous time Markov transition model was used to
describe proteinuria in patients treated with lenvatinib [59],
and rash and diarrhea following erlotinib administration [65].
3.3. Time-to-event AE data
AEs are sometimes reported as TTE data; application of TTE
statistical methods, although usual for efficacy endpoints (e.g.
OS) [17], is not as frequent for safety data. Indeed, the basic
approach of calculating the incidence proportions (IP) in dif-
ferent treatment groups is often preferred. However, AEs are
not necessarily acute and, in some cases, they might appear
with a delay because dependent on a continued drug expo-
sure. Whilst IP computation is appropriate for those AEs occur-
ring immediately after drug administration, it is not the best
approach in case of long-term AEs. In [66], Proctor et al.
compared the IP approach to several TTE methods for the
analysis of safety data [96]. In particular, the following meth-
ods were considered: incidence rate (IR), exposure-adjusted
incidence rate (EAIR), competing risk models and multistate
models [97,98]. Competing risk models and multistate models
are essentially survival methods which allow a further investi-
gation of the chronological sequence of AE and other events.
In case of different exposure times in various treatment
groups, IR is preferred over IP, but it requires the assumption
of time-constant hazard; this has been criticized because it is
rarely applicable to real data.
If the chronological sequence of events must be taken into
account, the Kaplan–Meier estimator could be used instead. In
[67], Kaplan–Meier curves were used to compare time to
serious AEs across different dose levels to investigate if there
was a correlation between incidence of AEs and nivolumab
administered dose. Analogously, the relationship between car-
bozantinib exposure and time to first occurrence of dose
modification after AE was evaluated by a Kaplan–Meier analy-
sis stratified according to the area under the concentration
versus time curve at steady state (AUCss), using a Cox propor-
tional hazards methodology [68]. In this way, starting from
carbozantinib-predicted individual patient exposures, patients
eligible for dose modifications are identified, and the subse-
quent potential impact of these modifications on PFS can be
described.
In case of competing events, that is, serious AEs and death,
only the probability of the composite event can be computed,
and not the probability of experiencing each single event. To
overcome this issue, competing risk models should be con-
sidered to determine the effect of a treatment on the hazard
of both the events of interest.
In case of recurrent events, probability estimates may still
be derived, but are typically complex: a possible solution
could be the use of an extended version of the multistate
model [97].
4. Conclusions
Some examples of modeling approaches, employed to
describe and/or predict different clinical endpoints and var-
ious types of AE data, have been reviewed.
Mathematical models can play a fundamental role in clin-
ical oncology inasmuch they have the potential to provide an
early prediction of the efficacy outcome (usually in terms of
OS), confirming or disproving treatment efficacy. Indeed, they
can support the validation of new surrogate endpoints (e.g.
biomarkers or tumor markers), or the establishment of differ-
ent types of TS assessment (e.g. volumetric and functional),
which can be linked to OS. As efficacy can be limited by the
onset of resistance to treatment, modeling can also play a
crucial role in understanding this phenomenon, hence provid-
ing an unbiased prediction of the clinical outcome.
AEs are important to be monitored for assessing the
patient’s safety and, together with efficacy, defining the RBR
16 S. M. LAVEZZI ET AL.
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of a new drug. Modeling approaches have shown several
advantages compared to the conventional analysis of AE inci-
dences in different dosing groups. In particular, mathematical
models have proven helpful in supporting the selection of
optimal dosage regimen. Furthermore, they are able to take
full advantage of more-informative longitudinal data.
Despite numerous works in which mathematical models
have been successfully applied in clinical oncology, some
gaps are still present: in this regard, some recommendations
will be presented in the next section.
5. Expert opinion
Pharmaceutical research and development must face increas-
ing costs and a growing difficulty in finding novel therapeutic
targets [99]. The need for an improvement in the whole drug
development process has been recognized by the FDA. In its
Critical Path Initiative [12], resorting to mathematical modeling
was proposed as valid tool to reduce attrition and improve
productivity. Furthermore, since 2009 the FDA
Pharmacometrics Division promotes the use of mathematical
modeling in drug approval process [100].
The role of modeling has been steadily expanding in the
last decade; however, it cannot be definitely concluded that
this reflected in a decrease of attrition. Indeed, the percentage
of oncology drugs achieving the market among those that
entered clinical development between 2006 and 2015 is
5.1% [4], very close to the 5% reported in [101] for the period
1991–2000. Nevertheless, the increasing impact of modeling
approaches on decision-making processes is well recognized
[102–105] and, in this respect, several case studies of the use
of mathematical models for efficacy and safety endpoints have
been presented in this review.
For efficacy assessment, OS represents the main clinical
endpoint: nevertheless, it requires a long time for evaluation.
Therefore, it is important to identify earlier endpoints, demon-
strative of drug efficacy: to this end, a number of surrogate
endpoints have become of interest. For instance, markers
(such as PSA, CTC, LDH, NSE), easily measured, for example,
via blood or urinary samples, were linked directly to OS
[18,36–39,41], or to other efficacy endpoints (e.g. TS)
[18,30,33,34,40]. Modeling can support surrogacy validation
of new markers, as shown in [34] for LDH and NSE in SCLC,
and in [35] and [37] for CTC count and PSA, respectively, in
mCRPC. Furthermore, mathematical models might help disco-
vering markers for defining patients inclusion/exclusion cri-
teria. It has been demonstrated that the use of such marker-
based criteria can raise the success rates at each phase of
development [4]. Tumor burden is commonly assessed for
monitoring antitumor drug activity and can be used to obtain
an earlier prediction in terms of survival. Different kinds of
measurements have been proposed: unidimensional, bidimen-
sional, volumetric, and functional. To leverage all the available
information, and possibly to validate new assessment meth-
ods, integration of different evaluations of tumor burden
within a modeling framework is needed [32]. Furthermore,
additional clinical observations, such as the appearance of
new lesions [24], may provide a more complete picture of
the patient tumor load and may be used as OS predictors.
Predictors for OS (or for surrogates, e.g. PFS or TTP) can be
derived from biomarkers, TS, and tumor markers models. In
many cases, the computed descriptor is a single value sum-
marizing the tumor change and drug effect, not exploiting
information derived from longitudinal modeling [23,32,40,41]
as instead recommended in [15]. Other works, conversely,
succeeded in identifying dynamic descriptors
[24,33,34,36,37], which might be helpful in surrogacy valida-
tion of new markers. In the reviewed literature, the Claret
model [23], despite being empirical, is currently the most
widespread clinical TGI model [18,21,23,24,29,31,32,37,38,40]
and has become a de facto standard. It is also noticeable that
a number of alternative models, in some cases semi-mechan-
istic or mechanistic, have been proposed, for markers [20,36],
TS [27,28], or both [30].
As for the occurrence of resistance to treatment, simple
modeling approaches have been suggested, such as an expo-
nentially decreasing drug effect over time [23,28] or over
exposure [34]. More mechanistically grounded models have
been proposed, but not fully validated or integrated in classi-
cal PK–PD models [86]. Our suggestion is to reach a compro-
mise, that is, to search for more mechanistic models of
resistance, allowing their pragmatic validation with clinical
data and their integration with marker, TGI, and survival mod-
els. In this way, resistance modeling in the clinical setting can
become useful, if not crucial, for predicting drug efficacy and
OS in cancer patients. A step forward in this direction has been
made in [36].
Different types of AEs can be observed in anticancer ther-
apy and, consequently, several types of toxicity data can be
collected. Herein, the following model structures for three
types of safety outcomes have been discussed: (i) semi-
mechanistic or classical PD models for continuous measures,
(ii) TTE models for single or multiple events, and (iii) propor-
tional odds models (with or without transition probability for
Markovian dependency) for graded scores. Mathematical mod-
els are preferable to simple IP for toxicity assessment, as they
provide AE dynamics (e.g. the severity, the onset, and the
offset in case of reversible events).
Myelosuppression is one of the most frequent toxicities
encountered during chemotherapy. In several works, semi-
mechanistic or mechanistic PD models were developed to
describe and predict the full-time course of myelosuppression
in patients treated with either cytotoxic, targeted, or combina-
tion therapies. They can provide a guidance for dose and sche-
dule finding, and anticipate potential safety issues [29,46–49].
The good performances of these models can be explained by
the incorporation of prior information on the mechanism under-
lying myelosuppression. This should be taken as a paradigm also
for other toxicities models, so far lacking mechanistic content.
Another peculiarity of these models is the integration of patient
covariates, supporting dose individualization strategies [45,93].
At present, the most widely used myelosuppression model in
both preclinical and clinical studies is the one developed by
Friberg et al. [43], whose applicability was demonstrated also
for interspecies scaling [94]. Further developments of the origi-
nal model have been proposed to predict thrombocytopenia
[53], to model rescue therapy [54], or to account for stem cell
cycle dynamics [52].
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Together with (semi-)mechanistic models, also classical
direct and indirect response PD models are employed to
describe continuous AE data. They can be used to describe
AE time course when data do not support the implementation
of more mechanistic models, or information on the pathophy-
siology is not available. These empirical models have been
used to describe the time course of ALT [56], cardiovascular
AEs [21,58–60], and anemia [55]. Examples of mathematical
models for QT prolongation for anticancer drugs have also
been found [61].
When continuous data are not available or their collection
not feasible, logistic regression or Markov models can be used
to assess the frequency and severity of multiple occurring AEs
after drug administration [58,59,63,65]. Nevertheless, longitu-
dinal measures should be collected whenever possible, as they
contain more information than graded scores.
Despite the availability of statistical tools and models for
the analysis of TTE data, the common practice in safety assess-
ment is to report just observed frequencies and test statistics.
However, the use of TTE models has been demonstrated to
provide additional information (e.g. AE temporal dynamics)
overlooked in the IP approach [66].
The future will see an ever increasing use of clinical trial
simulations (CTS) in prospective analysis plans. For instance,
models for efficacy and safety can be coupled to individualize
dosing strategy, reaching a certain biomarker or drug concen-
tration level, and preventing at the same time the occurrence of
serious AEs, as demonstrated in [29,46–51,62,64]. The assess-
ment of the effective and safe dose becomes particularly chal-
lenging when drugs are given in combination. In fact, new
anticancer compounds are often combined either with other
treatments (e.g. chemotherapy and radiotherapy or surgery), or
other drugs, which may be of the same class (e.g. poly che-
motherapy) or of other pharmacological classes (e.g. chemother-
apy and immunotherapy). CTS may be used to find the best
schedule and doses, enhancing the combined effect and redu-
cing undesirable interactions [44,50,51], without need to test
experimentally all the possible dosing regimens of the com-
bined drugs. Of particular interest is the MODEL-1 trial, which
was the first-ever clinical trial where the dosing regimen of
docetaxel + epirubicin was entirely driven by a mathematical
model [51]. As recommended by the FDA in its guidance on end
of phase IIA meetings [106], CTS should also be systematically
carried out at the end of phase II before entering phase III,
where the highest attrition rates are reported [4]. The aim is to
inform end-of-phase II (e.g. go/no go) decisions (EOP2D) and
support the design (e.g. sample size, study duration) of the
phase III trial, based on the findings collected during the first
development phases. At the same time, different scenarios can
be explored to evaluate the robustness of the critical model
assumptions. Very few publications about phase III projections
based on phase II data exist in the literature (see e.g. [107,108]).
It is often the case that the choice of phase III study sample size
relies on optimistic predictions of the expected effect size [109].
Barriers for using CTS to support EOP2D have been reviewed in
[110]; this is a potential area where substantial improvements
can be achieved using PK–PD models predicted outcomes. In
general, prospectively exploring different study designs via CTS
may be used at all stages of drug development, for example, to
reduce trial costs and minimize the exposure of patients to
inactive or unsafe treatments [105].
Mathematical models might also be used retrospectively, in
order to routinely analyze failures, so as to identify both the
specific causes that led to attrition and the stages of develop-
ment that are more at risk [111]. In this way, modeling ana-
lyses could be rationally planned, to avoid the repetition of
the same errors and misconceptions.
In the near future, mathematical models should embrace the
complexity of the interactions between tumor, patient and drug,
and become an instrument for optimizing the tradeoff between
efficacy and safety [44]. Among the more mechanistic systems
models that have the potential to increase predictive capabilities,
one may mention network-based systems pharmacology models.
Mathematical descriptions of signaling pathways and/or chemical
reactions at the microscopic (e.g. cellular) scale may be consid-
ered for inclusion in whole body physiologically based PK mod-
els. Examples of cellular-scale models can be found for instance in
[112], where the development of malignancy in ductal carcinoma
in situ was explored, and in [113], where an ODE cellular model
was used to explore the hypothesis that extracellular pH normal-
ization can reduce tumor’s invasion. Systems pharmacology
approaches can help elucidating mechanisms that drive drug
efficacy, as well as identifying off-target toxicities (hence explain-
ing undesired effects) [114], and, in the end, are a valuable
resource to face the challenges connected to targeted drug
development and personalized medicine.
Funding
This manuscript has not been funded.
Declaration of interest
I Poggesi is an employee and holds stock with Janssen Pharmaceuticals
shares. The authors have no other relevant affiliations or financial involve-
ment with any organization or entity with a financial interest in or finan-
cial conflict with the subject matter or materials discussed in the
manuscript apart from those disclosed. Peer reviewers on this manuscript
have no relevant financial or other relationships to disclose.
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