Mathematical modeling for novel cancer drug discovery and development.

Review

Mathematical modeling for novel cancer drug discovery and development 1.

Introduction

2.

Current status of the field

3.

Mathematical modeling in cancer drug discovery and

Expert Opin. Drug Discov. Downloaded from informahealthcare.com by The University of Manchester on 10/19/14 For personal use only.

development 4.

Application of mathematical modeling in cancer diagnostics and treatment

5.

Conclusion

6.

Expert opinion

Ping Zhang & Vladimir Brusic† †

Dana-Farber Cancer Institute, Cancer Vaccine Center, Boston, MA, USA

Introduction: Mathematical modeling enables: the in silico classification of cancers, the prediction of disease outcomes, optimization of therapy, identification of promising drug targets and prediction of resistance to anticancer drugs. In silico pre-screened drug targets can be validated by a small number of carefully selected experiments. Areas covered: This review discusses the basics of mathematical modeling in cancer drug discovery and development. The topics include in silico discovery of novel molecular drug targets, optimization of immunotherapies, personalized medicine and guiding preclinical and clinical trials. Breast cancer has been used to demonstrate the applications of mathematical modeling in cancer diagnostics, the identification of high-risk population, cancer screening strategies, prediction of tumor growth and guiding cancer treatment. Expert opinion: Mathematical models are the key components of the toolkit used in the fight against cancer. The combinatorial complexity of new drugs discovery is enormous, making systematic drug discovery, by experimentation, alone difficult if not impossible. The biggest challenges include seamless integration of growing data, information and knowledge, and making them available for a multiplicity of analyses. Mathematical models are essential for bringing cancer drug discovery into the era of Omics, Big Data and personalized medicine. Keywords: cancer, computational models, drug discovery, mathematical modeling Expert Opin. Drug Discov. (2014) 9(10):1133-1150

1.

Introduction

Cancer is characterized by uncontrolled cell multiplication and invasion of tissues enabled through aberrant biological processes, the hallmarks of cancer, driven by the mutations or inherited genetic traits (Figure 1) [1,2]. Data from clinical observations and from animal experiments are used for development of mathematical models of cancer that are implemented as computational, or in silico, models [3-6]. Large quantities of data and related information have been generated using highthroughput methods (Omics) and from clinical information stored in the patient record data [7,8]. Mathematical modeling of cancer and its computational applications enable in silico observation of cancer growth, prognostic modeling and selection of therapy [3,9,10]. Oncogenesis is driven by mutations that disrupt signaling, metabolic and regulatory pathways resulting in heterogeneity within the same tumors, and between tumors that display the same phenotype [11,12]. Aberrations in glycosylation patterns and lipid metabolism contribute to oncogenesis [13,14]. Some tumors have small number of tumor-driving genetic mutations but show aberrant epigenetic modifications [15]. Mathematical modeling is now helping reshape our understanding of cancer, its origins, evolution and interactions within the organism [16]. These analyses help classify individual cancers, optimize therapies, identify promising drug 10.1517/17460441.2014.941351 © 2014 Informa UK, Ltd. ISSN 1746-0441, e-ISSN 1746-045X All rights reserved: reproduction in whole or in part not permitted

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P. Zhang & V. Brusic

Article highlights. .

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In silico modeling helps improve the efficiency of drug development and predictability of results, while at the same time it reduces the size and extent of preclinical and clinical trials. Mathematical modeling has been applied for improving and guiding the diagnosis, prognosis and treatment of cancer. Mathematical models, in particular the network models and multiscale models, help discover new drug targets, single or combinatorial, generally applicable or individualized. Observations of biological systems or clinical cases result in generation of new hypotheses followed by the initial experiments that produce data needed for building of mathematical models. These models are used to perform simulations that suggest further experiments. The mathematical model -- simulations -- further experiments loop produces new knowledge that, in turn refines previous or generates new hypotheses. Multiscale network models identify perturbed pathways, shed light on the effects of genetic mutations, integrate molecular networks to explain cellular behavior, integrate cellular patterns into processes at tissue/organ level and predict the effect of molecular targeting and drug effects. Key technological advances that drive new cancer drug development include Omics, Big Data, mathematical modeling and simulation, advanced flow cytometry, single cell analysis, microfluidics and nanotechnologies.

This box summarizes key points contained in the article.

targets and predict resistance to anticancer drugs [17-19]. Mathematical modeling has shown the potential to help discover new therapeutic targets and treatment strategies and is increasingly important for cancer drug discovery [6,20-23]. The main challenges for mathematical modeling are the need for integration of multiscale biological data and information with conceptual models of validated knowledge as well as the integration of computational models with laboratory experimentation [1,10,24]. This integration can be achieved through the knowledge-based systems (KBS) such as cancer research and drug discovery knowledgebase canSAR [25]. Mathematical models can be built to simulate the biological environment and activities based on the observations and hypothesis of biological systems or clinical cases from the initial experiments. The results of mathematical modeling depend on the quality of data used to develop these models. The growing quantities of data generated from various Omics technologies and clinical trials are not necessarily matched by the increase in the quality of data volumes. These data are subject to noise, various design and selection biases. Modeling techniques that deal with noisy data, filtering techniques and careful model development, which prevents data overfitting, need to be deployed. Statistical methods, filtering and noise reduction techniques, and experimental validation are needed 1134

for the design of robust and accurate models that are useful for drug discovery. Furthermore, these data are based on incomplete knowledge, and there is missing data. The mathematical model ! simulations ! further experiments ! refined mathematical model loop produces new knowledge that, in turn, helps refine previous or generates new hypotheses.

2.

Current status of the field

The ability to better understand biological processes in a system-wide fashion will have dramatic effects for better understanding of the disease and the improvement of healthcare [26]. Examples of interest include target identification and drug discovery in cancer [27], prediction of drug effects in humans for speeding up clinical trials [28] and greater understanding of the molecular and cellular interactions [29]. The fundamental methodology suitable for computational application of mathematical models in biology was described in 1950s [30]. Mathematical models are particularly useful when it is impossible or impractical to measure certain biological processes, when experimental solutions are either impossible or too complicated to perform, when it is impossible or prohibitively expensive to collect data experimentally and when it is dangerous or unethical to perform experiments, and complement or replace them with simulations when needed [9,26,31-37]. Mathematical models provide a formal coherent framework for interpreting biological data by helping [26]: . . . . .

inform experimental design evaluate, optimize and eliminate unnecessary experiments assess variables that are difficult to measure provide in silico predictions and enable simulations of systems and processes, and the analysis of their details.

Standard use of mathematical models involves the description of concepts (conceptual model), their translation into formal mathematical representations, standardization of data and operations, solving problem in silico, interpretation of results and translation of results into the biological domain (Figure 2). Hybrid models combine multiple mathematical representations, whereas multiscale models integrate various temporal, spatial and other dimensional scales. Several general methods can be used for the development of mathematical models [30]: . Make a conceptual model described from biological data

and formalize it (describe mathematically). . Start with a simple model then expand with the details

as they become available. Select a workable number of independent variables (feature selection and feature reduction) and define the level of abstraction achieved by the model.

Expert Opin. Drug Discov. (2014) 9(10)

Mathematical modeling for novel cancer drug discovery and development

Evasion of immune destruction

Metabolic aberrations Selfsufficiency in growth signaling


Resisting cell death

Cancer cells

Evasion of growth inhibition

Invasion and metastasis

Sustained angiogenesis Unlimited replicative potential Instability of genome

Inflammation

Figure 1. The hallmarks of cancer [2]. Self-sufficiency in growth signaling is necessary to sustain chronic proliferation. The evasion of growth inhibition signals circumvents biological programs that stop cell proliferation. Resisting cell death makes cancer cells avoid programmed cell death. Unlimited replicative potential enables cancer cells to form macroscopic tumors. Sustained angiogenesis provides sustenance by supplying nutrients and oxygen and removal of metabolic waste. Invasion and metastasis enables cancer cells to form local invasion and distant metastases. Additional hallmarks involve reprogramming of metabolic framework to support cell growth and division, instability of genome to support cancerous transformations, tumor promoting inflammation and evasion of immune destruction of cancer cells. These processes occur in multiple temporal and spatial scales and are enacted through network-type interactions [3].

. Models developed in other fields can be ported into a

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new domain; for example, agent-based models [9,38] describe cancer development mechanistically and better than earlier stochastic models [31]. When mathematical formalism is insufficiently rigorous it should be discarded and replaced with heuristics (learning from experience), such as from the trial-anderror approaches. Judicious use of robust statistical methods for missing data estimation, error reduction, bias reduction, and filtering to enhance data quality is highly desirable. Modeling approaches that refine model formalism may include Boolean modeling, multistep logic, fuzzy logic or ordinary differential equations based logic approaches [39]. The established models in a particular biological field must always be challenged and assessed against the new data. Multiscale models [1,40] that include more complex data, for example, are better approximation of tumor growth and progression than the earlier models [32]. We should continuously test the models using new data or by performing systematic validation experiments.

In particular, mathematical modeling of biological systems needs to consider the complexity and hierarchical nature of processes that generate biological data, fuzziness of biological data, biases and potential misconceptions in data, and the effects of noise and errors [41,42].

Mathematical modeling in cancer drug discovery and development

3.

Drug discovery and development starts with identification of targets, followed by in vitro, and then in vivo experimental screening to generate data about particular targets. Mathematical models use these data to help identify compounds that are potential drug targets in experimental screening systems, and through simulations can explore numerous possibilities including drug dosage, drug regimens, effect of drug combinations and toxicity. They help identify promising leads, help optimize therapies and support translation from animal systems into human studies [6].


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Conventional approach

Approach driven by mathematical modeling

Biological system

Biological system

Formal description

Computational analysis

Computational solution

Mathematical model

Biological interpretation

Conversion of results

Biological interpretation

Figure 2. The conventional and mathematical modeling-driven approaches to solving biological problems. The conventional approach includes three main steps: ‘biological system’ descriptions -- observations and measurements of the biological system, ‘computational analysis’ that is performed case-by-case in an ad hoc fashion and ‘biological interpretation’ that converts results into knowledge. The approach that deploys mathematical modeling involves three additional steps: ‘formal description’ -- a description of the biological problem, including conceptual model and formal representation, ‘mathematical model’ description in the form of mathematical model and implementation including standards for data and operations and ‘conversion of results’ that transforms mathematical results into the domain of biology to generate new knowledge.

Discovery of novel molecular targets for therapeutic intervention

3.1

Molecular screening of cancers provides another avenue for improvement of cancer diagnostics. A large-scale analysis of >400 cancers that included breast cancer identified >2500 mutations across >1500 genes [43]. Some of these alterations are common across all cancers, while some are less common. A study of a breast cancer (HER2+, HR+ , triple negative ER-, PR-, HER2/neu-) showed a high number of mutations in PIK3CA and TP53 genes. The analysis of protein neighbors of these genes using STRING database [44] and identification of Reactome pathways [45] from MsigDB [46] (Figure 3) has identified that mutations in P53 possibly affect the pathways involved in DNA repair, cell cycle, mitosis, meiosis, ERBB2 signaling, GAB1 signaling and PI3-AKT activation. The mutations in PIK3CA have possible consequences on immune response pathways, including CD28 activation [47] where B7-CD28 costimulation is needed to activate T cells. Possible aberrations in CD28 co-stimulation may lead to dampening of immune responses, thus enabling the immune escape. These perturbed pathways can help profile individual cancers and inform oncologists about promising drug targets. The knowledge in this field is constantly evolving, and current knowledge is being constantly updated. The discovery of new pathways and molecular interactions from the study of cancer samples increases our knowledge and helps personalization of treatment. Mathematical modeling alone can lead to wrong conclusions either because of peculiarities of 1136

specific cancer data sets or because of uncertainty due to incomplete knowledge. Therefore, the initial results should be validated either by using multiple modeling methods or by experimental validation. The tumor suppressor p53 is the most frequently mutated and inactivated protein in human cancer [43,48]. The p53 signaling pathway is an oscillatory system [49] and several mathematical models have been developed to explain the mechanism of p53 oscillations. Multiple models generated dynamics similar to that described from the observations of experimental data. An example model is the delay oscillator, which hypothesized a time delay between upregulation of p53 and the expression/ maturation of Mdm2 [50]. Some models simulated a network structure using differential equations to explain the undamped oscillations [50] and predict the changing level of p53 transcripts and the related protein [51-53]. These models help understand the dynamics of oncogenic processes, offer insights into diagnostics and identification of drug targets, as well as optimization and dosage of therapies. An example of a network model that can be used for the study of causal relationships between genes, proteins and cancer and an oscillatory network are shown in (Figure 4). Such oscillatory systems have been described by the systems of differential equations, such as those shown in Equation (2a -- j) [51]: (1) dP53T ’ = K s 53 − K d 53 × P53T − K 53 × P53 U dt



ATM MDM2

SP1

PIK3CA TP53 Functional neighbors: MDM2 CDKN1A ATM SIRT1 BRCA1 KAT2B RCHY1 USP7 SP1 EP300

USP7

BRCA1 RCHY1

CDKN1A TP53


KAT2B

EP300

PIK3CA

SIRT1

REACTOME gene set name P53_DEPENDENT_G1_DNA_DAMAGE_RESPONSE CELL_CYCLE_CHECKPOINTS CELL_CYCLE GAB1_SIGNALOSOME PI3K_AKT_ACTIVATION

p-value 6.97E-10 1.64E-08 2.78E-08 8.57E-08 8.57E-08

Figure 3. An illustration of the network model for the study of possible causality of genetic mutation, proteins and cancer. Ten functional protein neighbors of TP53 (encoded by p53) and PIK3CA have been identified using MSigDB database [46]. The analysis of REACTOME pathways indicates that mutations in Pik3ca and p53 genes are highly likely to affect molecular pathways involved in DNA repair, cell cycle, mitosis, meiosis, ERBB2 signaling, GAB1 signaling and PI3-AKT activation. The oscillatory behavior of TP53 and MDM2 has been described in the main text (Expressions 2a -- 2j). Similarly, interaction models of other protein neighbors could be developed.

+

Mdm2C

Mdm2PC

p53

+++

∑

p53T

++ +

Mdm2N

p53U

Repair +

DNADAM

DNA Damage

p53UU +

IRAD

Figure 4. Molecular interactions and wiring diagrams of the oscillatory network [51]. Three ubiquitination variants with zero (p53), one (p53U) and two (p53UU) ubiquitination moieties together form total p53 (p53T). Ubiquitinated variants degrade faster, contributing less to the p53T. Ubiquitination is mediated by the nuclear form of Mdm2 that increases with DNA damage due to ionizing radiation (IRAD). Phosphorylated form Mdm2P can enter Ccleus. p53T reduces DNA damage, increases cytoplasmic Mdm2 (Mdm2C) and blocks phosphorylation of Mdm2C. The system displays oscillatory behavior described by Expressions 2a -- 2j. MDM2 and TP53 are protein neighbors.

(2)

dP53U = K f × M N × P53 + K r dt

(

)

53U

53

(3)

dP53UU = K f × M N × P53U − K r dt

× K r + K f × M N − K d’ 53 × P53U


× P53UU − P53UU × K d’ 53 + K d 53

1137


(4)

dM N = Vr × K i × M PC − K 0 − dt

2

× (5)

m

K ×P dMC = K s 2 + sm2 53T m − K d 2 × MC + K dp dt J s + P53T × M PC −

K ph J ph + P53T

× MC


(6)

K ph dM PC = × MC − K dp × dt J ph + P53T

−

P

× M PC + K × M N − K d’ × M PC (7)

dDNAD = K DNA × I RAD − K DDNA dt DNAD × P53T × J DNA + DNAD (8)

DNAD Kd2 = Kd2 + × K d’ 2 J DDNA + DNAD P53 = P53 + P53 + P53UU

(9)

(10) MN MT = MC × + M PC Vr P53, P53U, P53UU, P53T are the quantities of p53 -- free, single, double ubiquitinated variants and total p53; MN, MC, MPC are the quantities of nuclear, cytoplasmic and phosphorylated Mmd2; K s 53 , K d 53 , K d’ 53 , K f , K r , K i , K 0 , K d’ 2 , K d’’ 2 , K s 2 , K s’ 2 , K s’’2 , K dp , K ph , K DNA , K DDNA are various pre-defined rate constants and m, Vr, Js, Jph, JDNA, JDDNA are dimensionless constants. For more details a reader should consult [51]. The compounds that reactivate or enhance the TP53 response represent promising drug candidates studied in clinical trials [54,55]. The anticancer strategies include targeting cancer with small molecules and selective inhibition of p53’s degradation pathways [56]. For example, RG7112 as small molecule that serves as p53-MDM2 inhibitor has been developed to restore P53 activity [57]. Discovery of new cancer drugs Statistical approaches can be used to identify significantly mutated cancer genes [58], but they may not capture rare but functionally relevant changes. Additional analysis and experimental validation are important for confirming the in silico identified drug targets. Drug lenalidomide that targets CD28 directly through tyrosine phosphorylation [59] has shown efficacy in multiple myeloma trials [5]. However, lenalidomide has been investigated by FDA for possible safety and toxicity problems [60]. Another drug, TGN1412 that targets CD28 as a superagonist caused multiple organ failures in 3.2

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clinical trials [61,62]. Target identification should always be combined with an adequate assessment of toxicity and safety. The standardized process of cancer drug development recommended by the NCI’s Developmental Therapeutics Program includes several steps: discovery and development, preclinical research, clinical research, FDA review and postmarket safety monitoring. The initial screening identifies targets from various sources such as the literature or the Natural Products Repository [63]. Once a compound is selected, in vitro screening is performed using three human tumor cell lines followed by in vitro screening using 60 human tumor cell lines. Successful candidates are then tested for activity against human tumor cells inserted or injected into mouse. The remaining candidates are then tested in animals for basic pharmacology and formulation, followed by toxicology studies in animals. If no problems are identified, the initial dose, route and schedule are determined for Phase I clinical trials in humans. In a groundbreaking anticancer drug study of breast cancer cell lines [64], linear model for microarray analysis has been used to help the selection of target proteins for anticancer drugs followed by the principal component analysis (PCA) and partial least-squares (PLS) regression that were employed to relate signaling data to cell phenotypes. The differences captured by PCA and PLS analyses confirmed that the signaling molecules measured from the lab can be used to define both the cell-type-specific targets and drug treatment protocols. The systems-level analysis of signaling networks, gene expression profiles and cell phenotypic responses was combined with mathematical modeling revealed that the drug combination, order and time are essential to the development of effective therapies against cancer [64]. This study focused on analysis of binary combinations of seven genotoxic drugs with eight targeted signaling inhibitors, revealing combinations and dosages that are better in inducing cell death given the expression of signaling targets. Optimization of immunotherapies Combination therapies offer a great promise in improving cancer treatment. Mathematical models help design those treatments rationally [65]. Mathematical models of the immune system/immune responses to cancer cells are based mainly on the differential equations [66-68] or cellular automata (agent-based systems) [9,69-72]. A cellular automaton comprises a grid of cells. Each cell may assume one among several possible states, usually binary (0 -- 1, on-off). The states of each cell depend on the initial state (t = 0) and on the states of the neighboring cells that interact. A new generation of states by advancing tn = tn-1 + 1 is created, and the states of the cells are changed according to a defined rule. The agent-based systems can encode the description of physical or biological entities. The agent-based systems that simulate the immune system/immune responses are based on the Celada-Seiden framework [73] that describe in detail the behavior of immune system and on the tumor growth 3.3




models [74,75]. Numerical simulations were performed in a study of several immunotherapeutic regimens: constant, periodic and impulsive immunotherapy against tumors [76,77]. The simulations suggested that the most important factor is the amount of drug, while the delivery regimen is important only for therapies of long durations. A striking application in this field is the combination of in silico modeling with in vivo efficacy study in mouse model of breast cancer [78]. A vaccine schedule was designed to reduce the number and duration of vaccinations without the reduction of efficacy. An agent-based model designed a protocol that was validated in a subsequent in vivo study showing that the intensity of initial vaccination is the key factor, while the immune aging is a key feature that needs to be considered in the later stages of vaccine protocols [69]. Integrated in silico models are necessary for capturing the complex dynamics across different levels of organization [4,79]. One such proposed system is ImmunoGrid framework that implements computational models of antigen processing and presentation with system-level models of the immune system for the study of immunity and vaccine development [80].

Enabling personalized medicine Pharmacogenomics studies how individuals respond to drugs. The correlates of drug responses may include gene expression levels, splice variation, single-nucleotide polymorphisms, copy number or epigenetic variation. Some are single-feature correlates, whereas most are multifeature correlates of drug responses [81]. Pharmacoproteomics studies individual responses to drugs based on protein expression profiles in individuals [82], whereas pharmacometabolomics studies the effects of individualized metabolic phenotypes to drug responses [83]. As the overall gene and protein expressions do not correlate well [84], these technologies are complementary and provide tools for the analysis of personalized molecular signatures that provide insights into regulatory processes, cross-talk between molecular pathways, non-linear reaction kinetics of molecular processes, cellular signaling, and interaction between cancer and normal tissue around and inside the cancer [24,85]. The multiscale network models are critical for complex analysis of biology and pharmacology of cancer [6,9,10,40,79]. Such models identify perturbed pathways, shed light on the effects of genetic mutations, integrate molecular networks to explain cellular behavior, integrate cellular patterns into processes at tissue/ organ level and predict the effect of molecular targeting and drug effects. The examples [86] include identification of dysregulated pathways and pathway interactions, for example, the activation of ERK pathway downstream of the ErbB; discovery that the death rate of individual cells vary because of protein concentration and not because of genetic or stochastic processes; the finding that targeting a single pathway is insufficient to stop tumor expansion and proliferation; and prediction of outcomes of molecular interventions such as blockade of ErbB1-3 products. 3.4

The disease progression and drug effects are often observed in longitudinal studies that measure outcome variables and the same set of covariates in repeated observations for each individual in the study cohort over a period of time. Linear models can be used for the analysis of the output variables that have Gaussian distribution [86], whereas log-linear, logistic regression, probit or quasi-likelihood models can be used for non-Gaussian output variables [87]. Omics technologies, on the other hand, measure hundreds of thousands if not millions of features (genes, proteins, metabolites, their expression and mutations) at multiple time points and from multiple tissues changing the landscape of cancer research [88] requiring advanced methods for complex systems analysis and advanced mathematical models [89]. These models must be integrated with experimentation [90,91] and refined cyclically with the new experimental data (Figure 5). Guiding preclinical and clinical trials The analysis of multiple reported studies by Laird [92] first determined that the rate of growth of solid tumors can be described by a Gompertz formula (see Section 4.4 for description). The cancer and leukemia group B (CALGB) clinical trial showed that patients benefited from the therapy regimen were predicted by mathematical modeling. The benefit included improved efficacy, lower toxicity and improved survival [93]. This study demonstrated the utility of statistical design of experiments and of mathematical modeling for optimization of drug regimens. Cancer drug dosage is an optimization problem -- the desired rate of killing cancer cells must be accompanied by low-to-moderate toxicity levels. The optimal administration of treatment includes initially high dose of chemotherapeutic drug, followed by moderate rate of further administration. An efficacy-toxicity model combined pharmacokinetic modeling of concentrations of drug in plasma and tumor site with pharmacodynamics modeling of anti-tumor toxicity and hematological toxicity [94]. A biomarker is a single feature or combination of several measurable biochemical, molecular or cellular features that are used to evaluate biological processes or responses and their outcomes [95]. Diagnostic biomarkers indicate the presence and characteristics of the disease, prognostic biomarkers indicate the likely progress of the disease and outcomes given a particular treatment (or lack of it). Some biomarkers can be used to estimate drug efficacy or toxicity, or its pharmacological activity [96]. The advantage of biomarkers is that they can be used to predict drug efficacy or toxicity faster than using conventional end points and are easier to measure. A well-defined biomarker in breast cancer is human epidermal growth factor receptor 2 (HER2), present in 25% of breast cancers. HER2+ cells can be targeted by a range of drugs including trastuzumab (Herceptin), pertuzumab (Perjeta) and lapatinib (Tykerb) [97]. Other breast cancer biomarkers include ER and PR. CD44+/ CD24-/low is a marker of oncogenicity -- human cells with this phenotype form tumor when injected in 3.5


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Observation

Hypothesis

RelativeRisk = AM × NB AFlbNR

Initial experiments

Mathematical model


New knowledge Further experiments

Simulations

Figure 5. Observations of biological systems or clinical cases result in generation of new hypotheses. Initial experiments are then performed to produce data needed for building of mathematical models. These models are used to perform simulations that rapidly and effectively search many possibilities that suggest further experiments. The new data are then used for cyclical refinement of mathematical models and design of further experiments. The mathematical model -- simulations -- further experiments loop produces new knowledge that, in turn, refines previous or generates new hypotheses.

immunocompromised mice [98]. The panel of five biomarkers ER, PR, HER2, EGFR and cytokeratin 5/6 can predict survival in basal-like breast cancer better than triple-negative phenotype (described in Section 3.1) [99]. DNA microarray studies are used to identify gene expression signature that is predictive of various outcomes, for example, of forming distant metastases [100]. Cellular markers are used to predict survival in patients -- a multi-centre study determined that the number of circulating tumor cells is an independent predictor of progression-free survival and overall survival [101]. 4. Application of mathematical modeling in cancer diagnostics and treatment

The US FDA Critical Path initiative identified in silico modeling (application of mathematics, statistics and computational analysis) of biological information as a key challenge area for improving the development of new or improved drugs [102]. To illustrate how mathematical modeling can add a mechanistic insight to intuition and improve understanding the processes and mechanisms from cancer initiation to metastasis and how it can be used to assist in diagnosis, prognosis and the treatment of cancer, we use breast cancer examples to demonstrate the utility of mathematical modeling. Identification of populations at high risk of breast cancer

4.1

The Gail model calculates 5-year risk of breast cancer using several risk factors (Equation 3a and 3b). 1140

(11)

AH

(12) FiveYearRisk RelativeRisk BAR where AM, NB, AFBNR, AH and BAR are predetermined factors for specific groups: AM is based on the age of first menstruation, NB is based on number of biopsies before last consultation, AFDNR is based on the age of first live birth corrected for number of relatives with breast cancer and AH is based on atypical hyperplasia found in biopsies, while BAR is the baseline risk based on age and race [103]. A two-stage model based on Armitage-Doll multi-stage theory [104], of response of breast tissue to hormonal changes in females defined well the age-specific incidence of breast cancer in multiple populations and predicted the protective effects of pregnancy [105]. Similar models use the status of key genes (e.g., BRCA1, BRCA2, EGFR, ERbb2), plus lifestyle factors, mammographic density measures and levels of hormones to provide predictions of breast cancer risk [106]. Table 1 shows some developed risk assessment tools for breast cancer and the risk factors included in the models. Coupling of diagnostic screening with computational predictive models promises long-term and sustainable benefits. Risk factors for breast cancer are numerous (Table 2). The comparison of model features from Table 1 that are in routine use and risk factors from Table 2 that are predictive of risk for breast cancer indicate that prognostic models for breast cancer risk assessment can and should be improved. Mathematical models identify high-risk individuals who should have regular and frequent screening for breast cancer. Models for guiding cancer screening strategies Screening for breast cancer using mammography is useful, but recently its value has been scrutinized. For every 2000 screened women, one will be correctly diagnosed with breast cancer and will benefit from subsequent therapy. However, for each true positive case, 10 healthy women will also be diagnosed (false positives) and will undergo unnecessary treatment [107]. This problem increases the number of unnecessary biopsies and has negative implications for drug discovery. Computeraided diagnosis systems (CADS) deploy advanced computational and mathematical modeling techniques to improve diagnostic accuracy of digital mammograms [108]. These models and features are used by CADS for improved diagnostics using mammograms. Precise prediction of breast abnormality uses classification methods such as statistical regressions, classification trees, support vector machines and neural networks [109]. The connection of classification models in CADS and clinical diagnostics is achieved through identification of areas within breast that show contrast enhancement, detection and analysis of calcifications, detection and analysis of masses and tumors, analysis of bilateral asymmetry and detection of architectural distortion within the tissue. A neural-genetic algorithm [110] was reported to improve the classification of breast cancer from benign abnormalities 4.2




Table 1. A selection of mathematical models for the assessment of risk of development of breast cancer is described. Tool

Description

Ref.

Gail

Features include: race, reproductive history, number of previous breast biopsies and number of first-degree relatives with breast cancer. Identifies risk of developing breast cancer Features include: number of relatives with breast cancer, who are these relatives, and their ages at diagnosis, including paternal history. Identifies risk of developing early-onset breast cancer. Limited to Caucasians only Features include: personal history of breast cancer, history among first- and second-degree relatives, and Ashkenazi Jewish ancestry. Identifies probability of mutation of BRCA1 or BRCA2 genes Features include: family history, ER status, triple-negative status (ER, PR and HER2), expression of basal markers CK5, CK6 and CK14. Identifies probability of mutation of the BRCA1 or BRCA2 genes and the risk of developing breast cancer Features include: family history, reproductive history, Ashkenazi Jewish ancestry and personal factors such as height and weight

[149]

Claus

BRCAPRO*

BODAICEA*

Tyrer-Cuzick models

[150]

[151]

[152]

[153]

*The carriers of BRCA1 mutations are at highest risk of breast cancer, followed by the carriers of BRCA2 mutations. ER: Estrogen receptor; PR: Progesterone receptor.

significantly reducing the false positive rate (FPR) from mammogram screening. This algorithm can also be applied with other imaging technologies such as MRI and ultrasound for prediction of individuals with genetic or hereditary predispositions [111]. Because it is often difficult to interpret images, computational classification algorithms and mathematical models deployed by these algorithms, both the FPR and false negative rate (FNR) in diagnosis of breast cancer are improving [112]. For example, as a rough estimate, the FPR of screening mammograms may be as high as 60% and FNR as high as 20%. The use of advanced algorithms may reduce both these numbers to ~ 10% [113]. Models for prediction of tumor growth Interactions between the oncogenic cells and the microenvironment are complex and depend on many factors. Multiscale 4.3

simulation tools in combination with high-fidelity ex vivo experimental models can help unravel this complexity [40]. Tumor growth has been simulated in both 2D and 3D multiscale models based on differential equations (continuous models) or cellular automata (discrete models) or combinations thereof [9,114]. In continuous models, cancer-related variables such as cell population, nutrient and drug concentrations and concentration of other factors are modeled using a set of ordinary or partial differential equations to simulate tumor progression. In the cellular model, cancer cells are simulated individually, and cell behaviors are governed by a set of deterministic or probabilistic rules. Cellular models are well suited for computer simulations of biological systems. The general framework for capturing dynamic behavior in the models can be precisely tuned to mimic the behavior of the real system [115]. Molecular and cellular interactions can be simulated at the first two levels. Multiscale models require fine-grained training and validation data that are not always possible with current in vitro and in vivo approaches [40]. New technologies of tissue engineering help bridge this gap and enable modeling the reaction-diffusion processes of solid tumors [9,116]. Ductal carcinoma in situ (DCIS) is noninvasive breast cancer that exhibits several distinct morphologies, representing an intermediate step between normal breast tissue and invasive cancer [117,118]. Computational models have been developed to help investigate both 2D and 3D characteristics of DCIS development [117,119-123]. The simulations resembled the patterns of DCIS development [117]. Another mathematical model of DCIS explained how the tumor growth deforms the duct wall, and the dependence of the tumor progression along the duct upon the stiffness of the wall [119]. Simulations of DCIS have shown how its treatment with tamoxifen, protease production and the tissue environment affects the tumor growth [120]. Three mathematical models were developed to predict morphological stability of the tumor and evaluate the consistency between the predictions and experimental data from in vitro studies [121]. These models simulated tumors as various types of fluid with partial differential equations based on constitutive laws that provide formulas describing deformation and stress of the tissue, including Darcy’s law, Stokes flow and the combined Darcy-Stokes law [121]. The extraction of the model parameters from a limited set of data was feasible and sufficient to create a self-consistent modeling framework that can be extended to the multiscale study of cancer. A mathematical analysis using a free boundary problem was performed to diagnose the growth of DCIS and demonstrated how scientists could develop mathematical methods to support the diagnosis and prognosis of DCIS [122]. Models for guiding cancer treatment Mathematical models can capture the features describing molecular and cellular processes involved in cell metabolism, pharmacokinetics of drugs and toxicity. For example, mathematical modeling provided important insights into chronic 4.4


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Table 2. Risk factors for breast cancer according to the American Cancer Society [154]. NPL stands for non-proliferative lesion and PLA for proliferative lesion with atypia. Risk factor

Description

Quantification

Gender

Breast cancer is common in females not in males Risk of invasive breast cancer increases with age Some breast cancers result directly from defects inherited from a parent Mutation BRCA1 and BRCA2 genes

100-fold higher risk in women

Age Genetic factors


Primary genetic factors Secondary genetic factors Family history Personal history

1 in 8 if < 45 years of age, 2 of 3 if older than 55 5 -- 10% of all breast cancers

Mutations of ATM, TP53, CHEK2, PTEN, CDH1 and STK11 genes increase risk Having first-degree relatives with breast cancer increases risk Having cancer in one breast increases risk

Race/ethnicity

Some races have higher risk of developing breast cancer

Density of breast tissue

Dense breasts have more glandular and fibrous tissue and less fat NPL, PLA-, or PLA+

Benign breast conditions Lobular carcinoma in situ Menstrual periods Chest radiation Diethylstilbestrol exposure

Hormone therapy after menopause Breastfeeding Alcohol consumption Obesity Physical activity

One relative doubles, two relatives triple the risk three- to fourfold increase in risk of new cancer in other breast Caucasians and African-Americans have higher risk than Asians, Hispanics and Native Americans Denser breasts are associated with higher risk NPL: no risk, PLA-: slight risk, PLA+: higher risk 10-fold increased chance of breast cancer

LCIS cells grow in the milk-producing glands lobules Age of first menstruation and of menopause Radiation therapy at young age DES was used for lowering incidence of miscarriages in 1940s -- 1960s

Early menstruations and late menopause increase risk of breast cancer Significantly increased risk Slightly increased risk

Having children Age when having first child Birth control

BRCA1 -- lifetime risk of 80% BRCA2 -- lifetime risk of 45% NA

Having children reduces risk of breast cancer Having first child at age > 30 slightly increases the risk Slightly higher risk, but no effect if drug was used 10 (oral) or 5 (DMPA) years ago Increased risk for CHT), no risk for ET Slightly reduces risk Increases risk, particularly in heavy drinking Increased risk 2 h walking per week reduces risk by 18%

Oral contraceptives and depotmedroxyprogesterone acetate CHT or ET

CHT: Combined hormone therapy; ET: Estrogen therapy; NPL: Non-proliferative lesions; PLA-: Proliferative lesions without atypia; PLA+: Proliferative lesions with atypia.

myeloid leukemia responsiveness and resistance to treatment with imatinib, an ABL tyrosine kinase inhibitor and the dynamics of relapse. Molecular responses to imatinib were modeled by a set of differential equations (Equation 4a -- c) [124], suggesting that imatinib depletes differentiated leukemic cells but does not deplete leukemic stem cells. (13) x0 = ⎡⎣ l ( x0 ) − d 0 ⎤⎦ x0 x1 = kx x0 − d1 x1

x2. = l x1 − d 2 x2 , x3. = m x2 − d 3 x3 y0 = ⎡⎣ R y (1 − PR ) − d 0 ⎤⎦ y0 y1 = k y y0 − d1 y1

(14)

y2. = l y y1 − d 2 y2 , y3. = m y y2 − d 3 y3 (15) 1142

= ( R z − d 0 ) z0 + R y y u

= kz z − d1z

z2. = l z z1 − d 2 z2 , z3. = mz z2 − d 3 z3 where x0, x1, x2, x3 are respective abundances of healthy hematopoietic stem cells, progenitors, differentiated cells and terminally differentiated cells; y0, y1, y2, y3 are corresponding leukemic cell abundances of cells without imatinib resistance mutations; z0, z1, z2, z3 are corresponding leukemic cell abundances of cells with resistance mutations; and d0, d1, d2, d3 are corresponding death rates. The homeostasis of healthy hematopoietic stem cells is represented by a declining function l; Ry is the rate of division of leukemic stem cells; Rz is the rate of division of resistant leukemic stem cells; u is the rate of production of resistant leukemic stem cells from leukemic stem cells; rates of production of progenitors,




differentiated and terminally differentiated cells are given by constants k, l, m with indices that distinguish rates for healthy (x), leukemic (y) and resistant leukemic (z) cells. The Norton-Simon hypothesis [125] proposed that the reduction of tumor volume due to chemotherapy follows the growth formula proposed by Gompertz in 1825 [126], and it was applied in several case studies a century later [127]. The Gompertz curve is a sigmoidal function (Equation 5a). The growth of cancer can be described remarkably well by the Gomperz equation, both in human cancers and animal models (Equation 5b and c) [128]: (16) ct

y (t ) = ae be where a is the upper asymptote, b and c are negative numbers and e = 2.71828. (17) N (t ) = N (0) × e k 1−e

− bt

(18) ⎛ N max ⎞ k = ln ⎜ ⎝ N (0) ⎟⎠ N is the size of the tumor, N(max) is limiting size of the tumor and b is a constant. This hypothesis has been applied in a 2 2 factorial design of clinical trial experiments [129] that tested if there would be the difference in outcomes when patients with breast cancer were given drugs sequentially or in combination, and when the interval between drug administrations was either 2 or 3 weeks. This study experimentally demonstrated that dosedense treatment should provide significant improvements, which is consistent the hypothesis of Gompertzian growth. It represents an outstanding result based on hypotheses generated using mathematical modeling. The discovery of new cancer drugs requires a combined approach that takes molecular targets and correlates their profiles to tumor growth. Mathematical models provide a convenient bridge between wet lab experiments (cell lines or animal models) and clinical trials in human subjects [6]. 5.

Conclusion

There is an urgent need for advances in analytical tools that can adequately describe the underlying processes in sufficient detail and capture the complexity of cancer and carcinogenesis. In particular, there is a need for sophisticated mathematical models that can capture the essential features that can be used to model oncogenesis, tumor responses to drug and drug toxicity. New cancer drug discovery has already benefited from such models -- for example, models that describe and potentially simulate cancer growth, drug responses and metastatic dissemination of cancer cells have already been developed. In silico simulations that correspond to the actual biological processes and accurately predict various measurable features help reduce the experimental burden and enable rapid, economical and targeted analyses.

A recent report [130] proposed a workflow for integrated discovery, testing and clinical trial design of optimal cancer therapies. In this workflow, mathematical modeling and analysis are involved in acquiring study data, definition of possible drug discovery and validation, as well as the design of clinical trials. In addition to examples already discussed in this review, there are several additional success stories. Drug developments that have benefited from mathematical modeling include dose optimization of trastazumab for gastric cancer where pharmacokinetic model combined with statistical modeling indicated the minimal doses of the drug needed for the patient benefit [131]. Molecular modeling and simulations of randomly generated mutations was used for the in silico design of peptide (RASPADREV) that was validated for anticancer activity against HER-2-positive cancers, stability and solubility [132]. Mathematical modeling was used to identify the most effective therapeutic regimens using bevacizumab in mesenchymal chondrosarcoma [133]. Successful deployment of mathematical models in cancer drug discovery requires standardized workflows (Figure 6) that integrate data from multiple sources. Data can be accessed directly from repositories or indirectly from literature mining, database mining and simulations. Combining heterogeneous data have been used to improve the accuracy of protein and gene functional annotation [134,135]. To deal with the emerging Big Data, we need to dynamically integrate standardized data into knowledge bases and also make selected data sources accessible through integration with the analytical tools for cancer drug discovery. In the initial step, data must be subject to quality control, error elimination and filtering. This prevents fitting the noise to the model and ensures that integrity of data is maintained. The data is then analyzed and refined for different modeling purposes (e.g,. summarization, exploratory analysis or discovery). The refined data are subsequently added to the data sources. Multilevel mathematical models are also integrated with the knowledgebase to perform simulations and predictions and suggest further experiments [70,78]. They can be integrated and combined for applications in drug discovery. Through the experimental design and experiments, new data will be created and will be fed back to enrich the data sources. The specific purposes of knowledge discovery using combination of simulations/ predictions and selective experimental validation can be the discovery of drug targets, drug effects, drug toxicity, analysis of mechanism of action, quality control, data refinement or various combinations of these individual goals. 6.

Expert opinion

Key technological advances that will be driving new cancer drug development include Omics [81-83], Big Data [7,8] and simulation and modeling [1,3,5,6,20-23,136]. Simulation and modeling will increasingly include the genomics [137], proteomics [138] and metabolomics [139] data. Additional


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DATA SOURCES

Integrated knowledge-based system


Clinical Omics Experimentation Databases Literature Simulations Other

New data: Drug targets, Drug effects, Properties, Combinations

Data analysis/ refinement

Mathematical models (various levels of complexity)

Experiments

Simulations/ predictions

Experimental design

Figure 6. An integrated workflow for the study of cancer drugs is shown. This analysis workflow starts from large data sources that can be utilized using literature and database mining. These data must be carefully pre-processed and refined for future use, in particular it is important that these data are relevant, filtered of most of the errors, duplicated or other artifacts. The mathematical model/simulation/experimentation loop can be as shown in Figure 3. The new data is then analyzed and refined and added to the data sources.

transformative technologies include single cell analysis, microfluidics, nanotechnologies and a variety of instruments. These technologies help identify biomarkers for early diagnosis, accurate disease classification, assessment of progression and relapse of disease, and profiling of drug responses [140]. Cancer displays chaotic behavior, as defined by the chaos theory [141-143]. The interactions between cell populations -- cancer, immune system cells and host cells -- can be modeled using chaotic attractors. The advantage of the chaotic attractor models is that they capture spatiotemporal data, can embed network models and describe cancer as a complex adaptive system. Drug effects can easily be incorporated in these models, and model parameter can easily be estimated from experimental data. Main approaches in the field Early mathematical models describing cancer were formalizations that described some observable aspects of cancer. These models include Gompertz formula, differential equations and cellular automata and, more recently, hybrid multiscale models. All these models have been validated using experimental or clinical data. We expect that the hybrid multiscale models will continue to grow in importance and they will be complemented by other emerging modeling methods. A new class of 6.1

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mathematical models based on utilization of chaos theory and strange attractors [141-143] promise to capture the complexity of cancer better than the previous models. A chaotic attractor represents a formula that defines the parameter space of a dynamic system. The system behavior is observed in discrete steps whereby the current state of the system defines parameters for the next step. Chaotic attractors show display stable dynamics of a quasi-cyclical behavior and occasional phase transitions of the system. Potential for mathematical modeling in cancer drug discovery

6.2

Drug discovery is a highly combinatorial problem that involves high-throughput screening and combinatorial optimization. Combinatorial chemistry uses chemical synthesis methods where tens of thousands compounds are made as mixtures, individual compounds or computationally generated structures. In combination with Omics, the combinatorial complexity of new drugs discovery makes a systematic experimental discovery very difficult, if not impossible. Simulations are used to identify drug candidates and explore various possibilities such as dosage, regimens, drug combinations and toxicity. An example of application is




CALGB 9741 clinical trial [93] described in Section 3.5. In combination with Omics, mathematical models can rapidly identify disrupted molecular pathways, identify potential drug targets and propose testing strategies.. Omics helps profile patient samples and identify suitable drug targets. On a larger scale, computational analysis using network models can identify new targets by studying molecular neighborhood of the key disrupted processes and identification of key regulatory genes. These applications are still in early development and over time it will become increasingly important. Enabling technologies and the knowledge base Traditional approaches that deploy combinatorial chemistry, pharmacokinetics and pharmacodynamics are increasingly combined with the high-throughput methods, advanced instrumentation, computational methods, mathematical models and simulations. Genomics, proteomics and metabolomics perform rapid and comprehensive screening of cancer profiles. Advanced instrumentation such as flow cytometry or microfluidics enable precise separation and quantification of cell types, and enable their capture. The knowledge base is growing, many molecular databases are catalogued at the Molecular Biology Database Collection that is updated annually [144]. A large selection of computational resources, including literature databases, molecular databases, various Omics databases, taxonomies and computational analytical tools, among others is available at the National Center for Biotechnology Information website [145]. Information about clinical trials is available at . Potential drug compounds are available from the Natural Products Repository [63]. The emerging main problem is the disparity between the rapid growth of relevant data and slower growth of our capabilities to manage and analyze that data. A KBS canSAR [25] integrates biological, pharmacological and chemical data with protein network and structural biology data. Such resources assist with the prioritization of drug targets, preliminary assessments, identification of potential risks, generation of driving hypotheses and the design of experiments. The biggest challenges will be the seamless integration of ever growing data, information, and knowledge and making them available for multiplicity of analyses. This process will be supported by emerging big data analytics techniques and methods that involve analysis of data sets that are so large

that traditional statistical and date management methods cannot be applied due to quantity, complexity and scalability of these data sets. In our opinion, new methods for data preprocessing, reduction and filtering will emerge. These methods will take huge data sets, filter errors and reduce data to workable size. The integrity of the resulting data will be validated by judiciously selected small number of experiments. The analytical environment will also see the integration of data and knowledge into knowledge-based systems that will incorporate mathematical models for simulation and analysis of the growing quantities of data.

6.3

Challenges and prospects and key development areas

6.4

We are in the era of Big Data -- data sets are excessively large and complex making its processing difficult and traditional data management and processing applications insufficient. Large amounts of data are generated by the next-generation sequencing technologies [146] or ChIP-Seq for the analysis of protein--DNA interactions [147]. The analysis of such massively parallel data can be done using cloud computing [148]. There will be ever-increasing needs for robust statistical methods for error and bias reduction, powerful data filtering techniques and estimation of missing data to preserve the integrity of data used for mathematical modeling. New cancer drug development will increasingly shift to development of personalized medicine, and we will see tighter integration of diagnostics and personalized drug discovery. Analysis of hundreds of thousands patient profiles using multiple Omics technologies (Genomics, Proteomics, Glycomics, Lipidomics and Metabolomics) will provide a comprehensive and holistic understanding of cancer environment and facilitate the discovery of new drug targets and drug development.

Declaration of interest P Zhang is an employee of the Commonwealth Scientific and Industrial Research Organisation (CSIRO) Australia. The authors have no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.


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Affiliation Ping Zhang1 PhD & Vladimir Brusic†2,3 PhD MBA † Author for correspondence 1 Research Scientist, CSIRO Computational Informatics, Marsfield, NSW, Australia 2 Director of Bioinformatics, Dana-Farber Cancer Institute, Cancer Vaccine Center, Boston, MA, USA Tel: +1 617 910 8058; E-mail: [email protected], E-mail: [email protected] 3 Professor, Boston University, Metropolitan College, Department of Computer Science, Boston, MA, USA

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