# The Multiple Dimensions of Networks in Cancer: A Perspective

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Audi Konfuzius-Institut Ingolstadt Lab at the Technische Hochschule Ingolstadt, 85049 Ingolstadt, Germany

Department of Computer Science, University of Surrey, Guildford GU2 7XH, UK

IBM Research Europe, 8803 Rüschlikon, Switzerland

Author to whom correspondence should be addressed.

All authors contributed equally to this work. Order is alphabetic.

Academic Editors: Adam Glowacz and Alexander Shapovalov

Received: 18 July 2021 / Revised: 18 August 2021 / Accepted: 22 August 2021 / Published: 25 August 2021

(This article belongs to the Special Issue Networks in Cancer: From Symmetry Breaking to Targeted Therapy)

This perspective article gathers the latest developments in mathematical and computational oncology tools that exploit network approaches for the mathematical modelling, analysis, and simulation of cancer development and therapy design. It instigates the community to explore new paths and synergies under the umbrella of the Special Issue “Networks in Cancer: From Symmetry Breaking to Targeted Therapy”. The focus of the perspective is to demonstrate how networks can model the physics, analyse the interactions, and predict the evolution of the multiple processes behind tumour-host encounters across multiple scales. From agent-based modelling and mechano-biology to machine learning and predictive modelling, the perspective motivates a methodology well suited to mathematical and computational oncology and suggests approaches that mark a viable path towards adoption in the clinic.

Cancer is a highly complex condition that causes approximately 10 million annual deaths in the world, according to the International Agency for Research on Cancer of the World Health Organization [1]. While cancer comprises a group of heterogeneous diseases, genomic alterations result in uncontrolled cellular proliferation and progression to metastasis in most cases.

The last decades have shown that, in addition to genomic alterations, the microenvironment surrounding cancer cells also plays an essential role in shaping and enabling a whole range of underlying fundamental processes. Beyond gene expression, the microenvironment influences tumour initiation, progression, immune evasion, and treatment response. As cancer progresses from a small clone of abnormal cells to a clinically apparent disease, tumours alter the structure and function of surrounding tissues through biochemical and physical processes. Notably, these alterations comprise, among others, factors promoting vasculogenesis and angiogenesis. Overall, such processes shape the abnormal physiology of the tumour–host interaction and are often responsible for tumourigenesis and treatment resistance.

Network theory is helpful to model and investigate the abnormal molecular and cellular processes associated with cancer. Capable of capturing dynamics and interactions at very different spatial and temporal scales, e.g., gene regulatory networks, protein–protein interaction networks, cellular communication events, etc., network theory provides a very attractive modelling and computational approach for understanding cancer complexity. Interestingly, the tools and approaches used in the field of mathematical and computational oncology have very often been used in other disciplines, such as computational neuroscience, ecology or artificial intelligence, the overlapping point being the use of modelling techniques based on the analysis of complex networks.

The current review attempts to unify different perspectives regarding the use of network theory to advance cancer research. The manuscript is structured as follows. After a short preamble in Section 1, we unfold, in Section 2, four different perspectives on network model applications, spanning current network approaches to link cancer genomic dysregulation and cellular phenotypes using network mechanistic modelling and data-driven techniques, such as machine learning and deep learning (Section 2.1); the development of mechanistic and hybrid models to optimise cancer immunotherapies (Section 2.2); agent-based models of the physical tumour–host interactions (Section 2.3); and end-to-end systems that combine mechanistic models and machine learning for clinical decision support (Section 2.4). In Section 3, we discuss the bridging of the four perspectives and emphasise the need for and benefits that network theory and applications can bring to cancer research, both at the fundamental and translational level. We conclude with an outlook (Section 4) towards new research avenues that can stimulate the design and development of new theoretical and experimental approaches in the mathematical and computational oncology community.

Cancer is a genetic disease caused by the accumulation of somatic [2] and epigenetic [3] alterations throughout the lifespan of an organism. While most of these alterations are neutral and do not result in the loss of cellular fitness, a few mutations have a detrimental effect to the cell’s normal functioning. These driver mutations increase an individual’s susceptibility to developing cancer and, when combined with other driver mutations, might result in cancer onset and development. Despite some well-studied examples, e.g., hereditary breast and ovarian cancers driven by inherited mutations in the tumour suppressor genes BRCA1 or BRCA2, most cancers are driven by the combined effect of multiple mutations that jointly dysregulate important cancer genes, such as oncogenes or tumour suppressors. Supporting this view, most cancers have been observed to present 2–8 driver mutations [4,5].

An additional fundamental observation has been the high level of heterogeneity in the observed somatic mutational landscape, where patients with the same type of cancer only share a small fraction of mutations. Indeed, the analysis of the genomic information from 50 cancer types catalogued by The International Cancer Genome Consortium (ICGC [6]) showed that only a small set of well-studied driver genes are frequently mutated, i.e., at frequencies higher than 10%, while the bulk of the mutational burden happens on infrequently mutated genes [7,8]. This variability can be understood when somatic mutations are analysed in the context of cancer networks. It then becomes apparent that mutations preferentially target pathways essential to controlling cellular homeostasis, such as pathways associated with signal transduction, cell cycle and apoptosis, DNA repair, etc.

The application of network approaches to model such genomic dysregulation has led to a better understanding of the tumourigenic mechanisms that underlie cancer onset and development [9]. In a cancer network model, nodes represent relevant components, e.g., genes, proteins, or cells, and edges account for interactions between these components. Ideally, a faithful characterisation of the network dynamics would require knowledge of the causal structure of the network. However, causal inference requires specific data, either time-series experiments or carefully planned perturbation experiments, both of which are not frequently available. In most cases, the available data are snapshots, such as the gene transcripts expressed at a particular time-point. An example of such a snapshot could be, for instance, the transcriptomic profile of a cancer patient at the time of diagnosis.

At the omics level, many techniques have been developed to infer undirected networks of gene or protein associations, also known as interactomes [10], using cohorts of patients’ molecular profiles, typically transcriptomic or proteomic data. While each method has its own merits and weaknesses, it has been shown that no single method performs optimally across all data types and experimental conditions; however, consensus networks built by integrating the predictions across different methods achieve excellent overall performance and robustness [11]. User-friendly applications have been developed to infer such consensus networks [12], as shown in Figure 1.

Although in an interactome the causal dependencies between genes are not known, other important statistical and topological properties can be inferred and potentially correlated with important biological or clinical information. For instance, a human B cell interactome revealed a hierarchical, transcriptional control module, where MYB and FOXM1 acted as synergistic master regulators controlling the proliferation of B cells [22].

Additional network analysis can reveal important information by analysing node degree, betweenness centrality index, network density, the average number of neighbours, etc. [23]. Similarly, spectral graph analysis of multilayer networks can reveal similarities across cancer types [24]. To take advantage of the availability of prior knowledge regarding pathways, i.e., expert-based definitions of gene sets that share a common function, pathway-based approaches have been developed to statistically evaluate the possible over-representation or under-representation of mutated genes in those pathways [25,26]. More sophisticated approaches, which additionally take into consideration other factors, such as the magnitude of the genes’ expression changes in the diseased compared to the normal state, the alteration type and position in the given pathways, or the type of interactions, have also been proposed [27].

To enhance the possibility of modelling genetic dynamics across scales, networks can be further coupled with mathematical or computational modelling techniques to enable a detailed simulation of the network. For instance, ordinary differential equations (ODEs) [28,29] (Figure 2), stochastic models [30,31], or even hybrid models combining both continuum and stochastic descriptions [32,33] have been used to study the dynamical interplay of smaller subsets of important genes or cells, where the mechanistic interactions are defined according to prior knowledge.

The use of ODEs is sometimes limited by the need to infer kinetic parameters for which little information is typically available, as will be discussed in Section 2.4.2. To overcome this limitation, Boolean modelling approaches approximate genes as binary entities with only two possible states: on or off. This formalism results in discrete, deterministic, and parameter-free models that are easier to simulate and explain than ODE models, although the found dynamical properties might not always be consistent with the biological observations. Several extensions have been developed to render Boolean simulations more realistic, including iterative parameterisation, asynchronous updates, and the incorporation of continuous and stochastic elements [34,35,36]. In the context of cancer, Boolean modelling techniques have been applied to identify attractors corresponding to different cell phenotypes. Interestingly, mutations in key driver genes were shown to promote phenotypic transitions, which supported the discovery of routes of carcinogenesis [37]. The model was also used to evaluate the outcome of targeted cancer therapies. An interesting aspect of Boolean models is that they can be tailored to reproduce a particular biological sample such as a patient tumour, hence enabling the prediction of patients’ clinical data [38], and the prediction of personalised drug responses [39].

More sophisticated machine learning and deep learning techniques have been designed that exploit cancer networks. For instance, network smoothing techniques that diffuse information over a network have been proposed to identify disease-causing genes [40], to identify cancer sub-types by integrating somatic tumour genomes [41], to integrate biological prior knowledge into machine learning workflows [42], or to gain insight into the molecular mechanisms underlying a sample’s classification [43], just to name a few applications.

Beyond gene-level descriptions, machine learning models have been used to enhance network models with insights from tumour growth data [44], describe the phenotypic stages of a tumour from histopathological [45] or from radiomic features of contrast-enhanced spectral mammography images [46], and even suggest personalised therapy sequences tailored to each patient [47], as further discussed in Section 2.4.2. More sophisticated approaches for data integration have been achieved by constructing networks of samples, i.e., networks of patients, for each available data type and then efficiently fusing these into one network that represents the full spectrum of underlying data [48].

Cancer onset and development are strongly influenced by the immune system, which might either promote or attenuate tumourigenesis and strongly influences therapeutic outcomes. While during the early phases of tumour development the immune system can identify and eliminate abnormal cells, chronic inflammation is associated with tumour development, progression, metastatic dissemination, and treatment resistance. Furthermore, chronic inflammation can be immunosuppressive and extinguish antitumour immune responses that emerge as a response to the expression of neoantigens, i.e., tumour-specific antigens expressed by cancer cells as a result of newly acquired mutations. The antagonism between inflammation and immunity can strongly affect the outcome of cancer treatment, and its implications are only now starting to be investigated [49].

In recent years, cancer immunotherapies focused on reactivating the immune system are emerging as powerful therapeutic options. Immunotherapies generally seek to redirect the immune system’s natural cytotoxic activity against malignant tissues, either by reactivating exhausted T cells or by blocking immune checkpoint regulators that prevent immune cells activation. One of the most promising approaches are chimeric antigen receptor (CAR) T cell therapies, where patients’ T cells are engineered to express a surface receptor designed to bind a cancer neoantigen, expanded ex vivo and re-infused into the patient [50]. CAR T cell therapies have resulted in durable remissions for some patients with certain types of cancers [51], although much work remains to be done to understand and overcome current challenges, such as gaining a better understanding of why some patients do not respond to the therapy, investigating and reducing severe associated toxicities, improving infiltration in solid tumours, extending the approach to a larger set of cancer types, etc. [52].

Many of these questions can be investigated with the help of network models. For instance, to accurately simulate the activation and therapeutic effect of CAR T cells, models need to jointly account for the binding of a cancer neoantigen by a CAR, and the subsequent activation of the T cell intracellular signalling pathways leading to the T cell antitumour function. Supporting these efforts, several network-based models have been developed to capture T cell signalling dynamics [53]. Some of them integrate different modelling approaches, such as logical, constraint-based, agent-based, and ODE models, to capture processes taking place at different spatial scales [54].

It is important, however, to keep in mind two considerations: First, although logical models and constraint-based models can provide good approximations of the qualitative behaviour of a biochemical system without the burden of parameter optimisation, obtaining a detailed description of a network’s dynamics, especially in the presence of nonlinear effects associated with irreversible transitions, bifurcations, or attractors, usually requires the use of a modelling formalism based on continuous differential equations [55,56]. Second, the activation mechanism of engineered T cells may differ substantially from T cells carrying native T cell receptors, and therefore models specifically designed to understand these differences are crucially needed.

In addition, the accurate modelling of complex immune responses is likely to require the development of multi-cellular mechanistic models that can capture the dynamical interplay between different types of immune cells. For instance, models to explain B cell development and its dependence on effective interactions with T cells and follicular dendritic cells have already been developed based on ODEs [28], stochastic hybrid models [33], and agent-based models [57], although their integration in the context of larger immune multi-cellular frameworks remains to be realised.

As the field moves forward and new and more informative datasets are being generated, the field of mathematical and computational oncology might also have to move to the development of hybrid models that combine traditional mathematical approaches, such as ODEs, partial differential equations and stochastic models, with machine learning and deep learning frameworks. In that sense, deep neural networks (DNNs) can be viewed as a coarse-grained iterative process, where the first layers learn low-level features that are combined in a nonlinear fashion into more abstract and complex features. The final prediction is often made by using simple classifiers that exploit the high-level features, thus combining all the information extracted from the sample in a highly nonlinear fashion. Indeed, DNNs have been shown to be powerful feature extractors that can effectively mimic the re-normalisation process [58], an approach that has been used in physics to accurately describe the macroscopic behaviour of a system without knowing the exact microscopic state of all its components. Similarly, by summarising low-level features into increasingly complex higher-level features that are predictive of macroscopic properties, DNNs can be seen as re-normalisation machines. Based on this idea, hybrid models can be developed to mechanistically integrate abstract DNN-generated features with observable variables that quantify the molecular and cellular responses after, for instance, T cell activation.

The advantage of hybrid models is that different modelling approaches can be used to model each system’s components based on its symmetries. For instance, DNNs are especially adept at modelling raw low-level data, and can therefore be used to model systems that are too complex or lack the necessary symmetries to be described with mathematical or statistical approaches, such as aggregates of cells or interactions between protein complexes. For instance, a multimodal deep learning model was used to successfully predict the binding affinity between a T cell receptor and an epitope [59], a system that would have been very difficult to model through mechanistic approaches. Other components, however, can be more efficiently described using mechanistic formalisms depending on informative biological observables that can be directly quantified. An example of the latter are, for instance, protein signalling cascades or gene regulatory networks. In both cases, the activity and expression level of the involved actors can be directly quantified, and a wealth of prior knowledge about molecular and cellular interactions can be used to further constrain the topology of the interaction networks. Communication between different compartments can be designed in a modular fashion, with communication channels that summarise the collective state of a compartment and send the information to the neighbouring compartment. Alternatively, new hybrid networks can be designed that merge both measured and DNN-derived variables and seamlessly simulate the ensemble together.

Computational modelling/simulation constitutes a fundamental tool in the scientific method. It complements the other existing approaches, namely, the mathematical/analytical approach and the experimental approach. The advent of high-performance computing resources has enabled a dramatic acceleration of computer simulations and revolutionised many fields, including biological, e.g., neuroscience, systems biology, oncology, etc., as well as non-biological fields, e.g., financial economics, mechanical engineering, etc. The computational approach offers a number of advantages, particularly in combination with the experimental approach, as it can reduce the number of experiments needed, serve as a test bed for competing hypotheses, and yield experimentally testable predictions. As such, it is an attractive tool for cancer research, particularly in modern times where large and complex datasets are collected.

In principle, computational models can be divided into mechanistic and non-mechanistic models. The former type is characterised by the presence of assumptions on the governing causal relationships, while the latter remains agnostic on these, but aims to describe and predict observable features. Both types have evident advantages and disadvantages, and should be chosen carefully depending on the available information and purpose of the study. It has been shown that computational models based on mechanistic as well as non-mechanistic interaction networks can provide valuable insights into dynamics of cancer. In many cases, the lack of (a priori) knowledge renders it difficult to initiate the modelling with assumptions on these dynamics. In such cases, it is appropriate to treat the system as a black or grey box. Machine learning-based models that are trained using large amounts of experimental data are attractive candidates for this approach [60], particularly due to the availability of high-performance computers that minimise the required simulation time.

Here, we describe examples of mechanistic and non-mechanistic models in computational oncology. Both types of computational models comprise network-based formalisation, including networks of interactions. Notably, mechanistic models can be differentiated into two types, namely, continuum models and discrete models [61].

Continuum models incorporate mathematical methods based on differential equation models, namely, ordinary differential equations (ODEs) and partial differential equations (PDEs), allowing to quantitatively capture aspects of cancer dynamics. In contrast to ODE-based continuum models, PDE models enable the modelling of cancer dynamics in space. Given the well-established importance of spatial aspects in the outcome of cancer treatments [62,63], the inclusion of these is a crucial feature for computational models of cancer progression. One of the earliest examples where spatial dependencies were considered is the study of modelling interactions at the tumour–host interface [64]. Since then, numerous studies have yielded important insights into cancer progression and the impact of therapy. However, these models have the shortcoming that they do not allow to incorporate genetic dynamics and behaviours of individual cells. However, these issues can be addressed with so-called discrete models. Given the focus of our current study on network-based and multi-scale models, we will elaborate on discrete models in the following sections.

With more detailed, high-throughput data it is becoming viable to make concrete assumptions on mechanistic interactions, across all spatial levels. Combined with the presence of dedicated software, e.g., PhysiCell [65] or BioDynaMo [66], these data enable the creation of computational models that have high explanatory power. Along those lines, agent-based models (ABMs) are arguably the most powerful type of discrete models for tissue dynamics [67]. Indeed, ABMs are becoming more and more prevalent in cancer research [68], as their particularly high computational requirements have become less problematic as they used to be [69]. These models allow to specify in a “bottom-up” approach not only intracellular, metabolic, and genetic dynamics, but also mechanical intercellular interactions in 3D space [70]. Therefore, they constitute a well-suited framework for integrating network-based dynamics. It is therefore not surprising that ABMs are used not only for cancer research, but also in many other disciplines such as developmental biology [71], synthetic biology [72], cell biology [73], or public health [74].

Generally, networks in ABMs can be formulated on three levels: the sub-agent level, the agent-level, and the extra-agent level (Figure 3). In the context of biological simulations, the sub-agent level could for instance denote the sub-cellular level of molecular interactions, gene regulatory networks or metabolic networks. Their dynamics can affect other cells but only indirectly by exerting impact via the host cell within which they reside.

On the intra-agent level (Figure 3B), networks of interactions on the sub-cellular scale are modelled. This includes gene regulatory networks, metabolic networks, and protein–protein interactions (PPIs). On the agent level (Figure 3C), the networks of interactions usually reside within the extracellular neighbourhood. These can comprise mechanical interactions, for instance, when cells adhere to or push one another. Alternatively, these can be behavioural interactions, where cells detect nearby cells and interact via receptors and ligands. The crucial dynamics between cancer cells and immune cells described earlier can be attributed to this level. Finally, on the extra-agent level (Figure 3D), processes occur without requiring the active action of cellular components (agents). This can be for instance the reaction and diffusion of interacting substances in extracellular space, as well as oxygenation across space due to the effects of vasculature [76]. As drugs or radiation impact cancer cells via the extra-agent level, ABMs of therapeutic interventions, e.g., chemotherapy or radiotherapy, need to take into account the transmission on this particular level, as well as its impacts on the intra-agent level [77,78].

Notably, networks of interactions on these three levels differ not only in terms of their spatial scales, but also with regard to their temporal scale. Along those lines, certain physical, molecular interactions such as those in PPIs act on shorter timescales than, for instance, functional interactions governing gene regulation. The agent-based approach offers a powerful framework to integrate such heterogeneous spatial and temporal networks into coherent multi-scale models. To this end, ABMs require networks of interactions to be directed (uni- or bidirectional), because they are mechanistic and so need to have defined directions of action. A schematic visualisation of a cancer tissue ABM is shown in Figure 4.

ABMs have been successfully employed to elucidate various processes in cancer growth and interactions of cancer cells with immune cells. For instance, the authors of [79] employ agent-based modelling to model cancer growth and interactions between cancer cells and immune cells. Their model reproduces certain realistic spatial patterns, and could potentially be used to predict immune checkpoint blockade treatment outcome. Notably, their model reproduces the observed correlation between mutational burden and the response to immune checkpoint therapy [80] (Figure 5). In particular, it yields quantitative estimates on suitable antigen strengths that have a high impact on the number of cancer cells. Along those lines, we anticipate a stronger utilisation of agent-based modelling for pharmaceutical research in the future: it constitutes a cost-effective method for predicting the impact of different spatial, temporal, and dosage parameters in oncology.

One of the main challenges in the agent-based modelling of cancer is combining datasets to create an integrated and coherent picture. This is a complex task because individual experimental labs and research projects usually focus on highly specific questions. Overall, deeper and more far-reaching questions can be addressed with the availability of datasets spanning across different scales, and comprising information on cells, the immune system, the extracellular matrix, and other important factors.

The following dimension of networks in cancer examines the various physical interactions, including cancer kinetics, tumour-immune systems interactions, and treatment outcomes, through the lens of processes and dynamics at the system and clinical application levels [81]. Such processes are described by coupled interactions from a network perspective, combining the detailed modelling described in Section 2.1 and Section 2.2 and the computational framework described in Section 2.3 towards clinical use. This section unifies the insights and outcomes of using network analysis to study both bottom-up and top-bottom interactions in cancer. It further elaborates on a possible framework and research strategies in this area by emphasising and discussing the multiple facets of such a perspective across scales.

Fundamentally, the dynamics governing cancer development are informed by quantitative measurements that describe tumour growth, host cell encounters, and drug transport in order to predict a patient’s outcome [82]. For instance, spatio-temporal modelling of intracellular pathways associated with cancer can now, with the power of machine learning, be linked across scales to tissue level manifestations of malignant neoplastic processes [83]. At the same time, this approach can strengthen the links between cancer biology and its physics through data-driven learning systems, which hold the potential for the discovery of new drugs and treatment strategies [84].

This perspective advocates the combination of mechanistic modelling and machine learning that goes beyond measurement-informed bio-physical models and towards personalised disease evolution profiles learnt from clinical oncology data [85]. Tackling both bottom-up and top-down interactions, such a framework could offer a better understanding of the causes and consequences of (physical) interactions in cancer and their connections to the biological hallmarks of cancer broadly described in Section 2.1, Section 2.2 and Section 2.3.

Physical interactions of cancer cells with their environment, e.g., local tissue, immune cells, or drugs, determine the physical characteristics of tumours through distinct and interconnected mechanisms. For instance, cellular proliferation and its inherent abnormal growth patterns lead to increased solid stress [86]. Subsequently, cell contraction and cellular matrix deposition modify the architecture of the surrounding tissue, which can additionally react to drugs [87], modulating the stiffness [88], and interstitial fluid pressure [89]. However, such physical characteristics also interact among each other, initiating complex dynamics as shown in Section 2.3. Learning mechanistic interaction networks can capture such complex dynamics and can unfold a network-based paradigm for modelling, computation, and prediction. They can extract the interactions among the various entities, e.g., tumour, host cells, and cytostatic drugs, by learning the physics of their interactions for producing informed outcome predictions. A simple, biologically grounded, example is depicted in Figure 6.

This simple interaction network can then describe and predict specific phenomena and interactions among the tumour and the various arms of the immune system broadly described in Section 2.2. For example, starting from the interaction network in Figure 6, the underlying computation and learning functionality could, for instance, simultaneously capture the power-law tumour growth under immune escape [90] and the potentiation—inhibition model of Natural Killer Cells (NKs)–tumour interactions [91], while exhibiting the known overlapping Cytotoxic T Lymphocytes (CTLc)–Natural Killer Cells (NKc) regulation [92]. Such a basic instantiation is depicted in Figure 7.

As shown in Figure 7, the learning mechanistic interaction networks offer the means to learn the physical laws/relations governing the tumour–immune interactions from clinical data (see Figure 7 Decoded vs. Input relation) in order to make predictions on the effects of modifying the nonlinear pattern of interactions among the tumour and the immune system (after learning). Such networks are generic, data-driven, and not constrained to a certain cancer type or cell line. Their networked structure allows to easily interconnect multiple computational maps for more biological components [93] or a different granularity or scale [94] of representation of the underlying interaction physics.

In this section, we go through different studies that support the unified view of learning mechanistic interaction networks. The purpose of this section is to demonstrate that there is a large body of research describing learning mechanistic interaction networks, from processes and dynamics [45] at the system level to the clinical application level [47]. More precisely, we will focus on four problems where learning mechanistic interaction networks are applied, namely, extracting cancer kinetics, phenotypic staging, therapy sequencing, and therapy outcome prediction.

As we have seen up to now, learning mechanistic interaction networks allow for in silico modelling and analysis of cancer growth and therapies. Furthermore, they can (1) describe their effects on the physics of the tumour–host–drug interactions [84]; (2) understand the development and evolution of tumour–host–drug interactions over time [133]; and (3) extract the interaction patterns under possible perturbations [134], such as drug re-dosing, while characterising disease dynamics across scales [135]. Such an approach nicely complements the low-level genetic processes modelling and therapy design approaches described in Section 2.1 and Section 2.2 and the physics modelling approaches from Section 2.3 with a set of available and powerful tools offered by machine learning.

Conventional cancer treatments are often sub-optimal due to a limited understanding of the underlying molecular and cellular mechanisms driving the disease and insufficient accuracy when predicting how an individual patient is going to respond to an administered therapy. Yet, in the last decade advances in both fundamental and clinical oncology and data science have clearly illustrated the need for a computational oncology-based approach to investigating and treating cancer. This new approach sees cancer as a combination of mathematical, physical and biological problems, rather than a strictly biological one.

Tumour evolution and response to treatment depend on a plethora of factors, including physical constraints, nutrient dynamics, the tumour–host interaction, vasculature development, drug delivery processes, etc. These processes are in turn influenced by the genomic, phenotypic, and microenvironment changes fuelled by cancer cell proliferation, and ultimately determine the tumour’s response to treatment. Mathematical and computational models that describe the genetics of tumour initiation, the physics of a tumour’s growth patterns, and a tumour’s response to therapies, among others, are becoming a useful instrument for physicians in the oncology practice. Capturing the multi-factorial interactions among tumour and host, network theory models hold the promise of providing an unified framework to model, analyse, and predict a tumour’s evolution and response to therapy.

Each tumour is unique and should be treated as such. Although the pattern of interactions among tumour and host is similar, the peculiarities of the interactions at each level, i.e., cell, tissue, and whole body, are patient-specific. Network theory enables scientists and physicians alike to account for multiple scales of tumour evolution under treatment, from the cellular to the tissue scale, and ultimately, to predict the effect on the whole body. Our review integrates this view into a unified perspective that combines detailed molecular descriptions, as discussed in Section 2.1, cellular descriptions, such as agent-based modelling described in Section 2.3, and hybrid mechanistic and machine learning approaches, discussed in Section 2.2 and Section 2.4, respectively.

As we have seen in the four sections, network theory can provide tools to:

- enable the investigation of fundamental mathematical, physical and biological principles derived from experimental data;
- fuse data from different sources, such as genetics, imaging, pathology, and mammography, to capture patterns at multiple scales to characterise tumour evolution;
- predict a tumour’s evolution after a specific treatment in a personalised manner, such as immunotherapy or conventional drug administration; and
- alleviate over-treatment, where a patient receives treatments or invasive procedures that might not be necessary.

Our unified perspective introduces a clinically relevant approach that exploits quantitative measurements, including molecular, phenotypic, and diagnostic information, to generate robust predictions that are personalised and go beyond current subjective and heuristic assessments. We advocate for the combined use of mathematical modelling, network theory, and machine learning to optimise therapy design in clinical oncology. We suggest that, given the demonstrated potential such an approach provides, the next focus should be on adapting the models for clinical use. The framework we propose considers the development of practical mathematical and computational tools that can be used in the clinic to predict treatment outcome for each individual patient prior to administering a treatment.

This section is an attempt to unify the discussed perspectives and dimensions of networks in cancer. Across temporal and spatial scales, network theory models and tools capture the interactions among the multiple entities involved in cancer initiation, development, and therapy, as depicted in Figure 8.

Going beyond the cross-layer combination of methods and analysis in Figure 8, we can provide a more structured view on how the different methods and network systems contribute between and within the layers towards reaching the clinical goal. From the complexity point of view, ODEs can capture known gene dynamics and interactions at the cancer genetics level but with the risk of not capturing the interpatient variability. On the other side, consensus networks mine large datasets that support generalisation by capturing long-range mutations, but are complex and need very elaborate data processing pipelines. ABM networks offer typically a trade-off when it comes to complexity, exploiting both the biophysical knowledge and the available data to describe tumour dynamics. From the explainability point of view, ODE methods and ABM methods dominate use cases at both cancer genetics and cancer physics levels. Yet, their reductionist tendencies incline to offer machine learning methods the pole-position in cases where large heterogeneous data sources are available. Additionally, today’s advances in explainable and trustworthy machine learning will minimise this gap allowing such methods to be more openly adopted in the clinic. Independent of data quantity, complexity, and explainability features of the individual network-based methods, the current technological and data availability landscape motivates more research to be conducted towards hybrid systems, where physics-informed machine learning is a key tool in the design of clinical decision support systems. We anticipate that the unification and combination of mechanistic modelling and machine learning across scales will be of major significance for clinical decision-making.

Despite the great progress in both clinical oncology and data science, more research is needed to bring mathematical and machine learning models to patients in the clinic. The research and clinical oncology community is facing pressing scientific and practical problems that need to be overcome. For instance, we need a better understanding beyond mechanistic models of the emerging biological and physical phenomena that govern tumour-host interactions in the context of new therapeutic strategies, such as targeted drugs, immunotherapy, metabolic therapy, and nanomedicine. This is only possible when data-driven approaches, such as machine learning and deep learning, are combined with classical mechanistic approaches to leverage both large and small heterogeneous datasets to produce personalised patient recommendations.

Future multi-scale and multicellular models trained on new multi-omics high- throughput data will undoubtedly better recapitulate the molecular and cellular dynamics that drive cancer onset, progression, and response to therapy, as we have seen in Section 2.1. The combination of different modelling approaches, described in Section 2.3 and Section 2.4, including both mathematical and data-driven techniques, will be essential, as it is unlikely that mechanistic models, whether based on continuous, discrete or qualitative frameworks, will be expressive enough to accurately and faithfully reproduce the complex landscape of cancer phenotypes. For instance, as discussed in Section 2.2, the activation of cytotoxic T cells relies on carefully tuned molecular recognition events between T cell receptors and cancer epitopes, which are unlikely to be modelled precisely enough through mechanistic approaches.

The path to move forward might rely on the development of hybrid models that combine different modelling approaches, each one designed to model a specific data type or specific modelling problem. In this sense, they might support the development of multicellular and multi-scale frameworks to model complex cellular systems. Examples of such hybrid multicellular approaches are starting to become available. For instance, a multi-scale ABM has been developed to integrate the physical dimension and cell signalling [136], but we anticipate that newer and more powerful hybrid frameworks, which exploit a much more diverse range of modelling approaches and data types, will be become broadly available soon. Yet, coming closer to clinical decision-making, there is very encouraging work combining ODEs and machine learning towards a hybrid system capable of deciding the clinical intervention path: adjuvant vs. neoadjuvant chemotherapy. For instance, the work in [47] combined the nonlinear Gompertzian tumour growth model with neural networks to simultaneously learn the tumour growth curve and the pharmacokinetics of Paclitaxel to predict the therapy course of action for a breast cancer patient. This work emphasises the advantages of hybrid predictive systems combining machine learning and mechanistic modelling in oncology.

For the first time, we have the data, the tools, and the mindset to finally tackle cancer complexity. Exploiting the vast and heterogeneous data from in vitro and in vivo studies along with the multitude of models, we can now build learning systems capable of predicting patient-specific tumour development and, subsequently, patient-tailored treatments. For instance, modelling clonal evolution under antigen receptor CAR T therapies could decipher the activation mechanism of the engineered T cells that describe the dynamical interplay between tumour and immune cells for a certain patient. Subsequently, ABMs could simulate in detail the physics of tumour–host interactions after immune checkpoint therapy and predict the shrinkage or growth depending on the mutational burden of each patient. Equipped with such intimate insights into a patient’s tumour evolution, we can consolidate predictive models to simultaneously capture tumour dynamics, therapeutic drug response and kinetics, as well as the therapy outcome.

As one can already see, networks provide an excellent modelling, analysis, and processing substrate to investigate in depth a complex disease like cancer. By describing the relevant interactions between tumour and host before and after therapy and at different scales, we are able to guide the individualised assessment of each tumour and predict the optimal therapeutic intervention for each patient.

C.A., R.B. and M.R.M. designed the perspective structure. M.R.M. structured and wrote Section 2.1 and Section 2.2. R.B. structured and wrote Section 2.3. C.A. structured and wrote Section 2.4. C.A., R.B. and M.R.M. wrote and reviewed all other sections. All authors have read and agreed to the published version of the manuscript.

MRM’s work was supported by Horizon 2020 (H2020) programs 826121, 765158, 813545, and 955321. R.B. was supported by the Engineering and Physical Sciences Research Council of the United Kingdom (EP/S001433/1).

The authors declare no conflict of interest.

- Global Cancer Observatory. Available online: https://gco.iarc.fr/ (accessed on 21 August 2021).
- Alexandrov, L.B.; Nik-Zainal, S.; Wedge, D.C.; Aparicio, S.A.J.R.; Behjati, S.; Biankin, A.V.; Bignell, G.R.; Bolli, N.; Borg, A.; Børresen-Dale, A.L.; et al. Signatures of mutational processes in human cancer. Nature
**2013**, 500, 415–421. [Google Scholar] [CrossRef] [PubMed] - Esteller, M. Cancer epigenomics: DNA methylomes and histone-modification maps. Nat. Rev. Genet.
**2007**, 8, 286–298. [Google Scholar] [CrossRef] [PubMed] - Vogelstein, B.; Papadopoulos, N.; Velculescu, V.E.; Zhou, S.; Diaz, L.A.; Kinzler, K.W. Cancer Genome Landscapes. Science
**2013**, 339, 1546–1558. [Google Scholar] [CrossRef] [PubMed] - Bauer, R.; Kaiser, M.; Stoll, E. A computational model incorporating neural stem cell dynamics reproduces glioma incidence across the lifespan in the human population. PLoS ONE
**2014**, 9, e111219. [Google Scholar] [CrossRef] - ICGC Data Portal. Available online: https://dcc.icgc.org/ (accessed on 21 August 2021).
- Lawrence, M.S.; Stojanov, P.; Mermel, C.H.; Garraway, L.A.; Golub, T.R.; Meyerson, M.; Gabriel, S.B.; Lander, E.S.; Getz, G. Discovery and saturation analysis of cancer genes across 21 tumor types. Nature
**2014**, 505, 495–501. [Google Scholar] [CrossRef] - Garraway, L.A.; Lander, E.S. Lessons from the cancer genome. Cell
**2013**, 153, 17–37. [Google Scholar] [CrossRef] - Creixell, P.; Reimand, J.; Haider, S.; Wu, G.; Shibata, T.; Vazquez, M.; Mustonen, V.; Gonzalez-Perez, A.; Pearson, J.; Sander, C.; et al. Pathway and Network Analysis of Cancer Genomes. Nat. Methods
**2015**, 12, 615–621. [Google Scholar] [CrossRef] - Vidal, M.; Cusick, M.E.; Barabási, A.L. Interactome Networks and Human Disease. Cell
**2011**, 144, 986–998. [Google Scholar] [CrossRef] - Marbach, D.; Costello, J.C.; Küffner, R.; Vega, N.M.; Prill, R.J.; Camacho, D.M.; Allison, K.R.; Kellis, M.; Collins, J.J.; Stolovitzky, G. Wisdom of crowds for robust gene network inference. Nat. Methods
**2012**, 9, 796–804. [Google Scholar] [CrossRef] - Manica, M.; Bunne, C.; Mathis, R.; Cadow, J.; Ahsen, M.E.; Stolovitzky, G.A.; Martínez, M.R. COSIFER: A Python package for the consensus inference of molecular interaction networks. Bioinformatics
**2020**, 37, 2070–2072. [Google Scholar] [CrossRef] - Butte, A.J.; Kohane, I.S. Unsupervised knowledge discovery in medical databases using relevance networks. Proc. AMIA Symp.
**1999**, 711––715. [Google Scholar] - Margolin, A.A.; Nemenman, I.; Basso, K.; Wiggins, C.; Stolovitzky, G.; Dalla Favera, R.; Califano, A. ARACNE: An algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context. BMC Bioinform.
**2006**, 7, 1–15. [Google Scholar] [CrossRef] - Faith, J.J.; Hayete, B.; Thaden, J.T.; Mogno, I.; Wierzbowski, J.; Cottarel, G.; Kasif, S.; Collins, J.J.; Gardner, T.S. Large-scale mapping and validation of Escherichia coli transcriptional regulation from a compendium of expression profiles. PLoS Biol.
**2007**, 5, e8. [Google Scholar] [CrossRef] - Meyer, P.E.; Kontos, K.; Lafitte, F.; Bontempi, G. Information-theoretic inference of large transcriptional regulatory networks. EURASIP J. Bioinform. Syst. Biol.
**2007**, 2007, 1–9. [Google Scholar] [CrossRef] - Friedman, J.; Hastie, T.; Tibshirani, R. Sparse inverse covariance estimation with the graphical lasso. Biostatistics
**2008**, 9, 432–441. [Google Scholar] [CrossRef] - Haury, A.C.; Mordelet, F.; Vera-Licona, P.; Vert, J.P. TIGRESS: Trustful inference of gene regulation using stability selection. BMC Syst. Biol.
**2012**, 6, 1–17. [Google Scholar] [CrossRef] - Zhang, Y.; Song, M. Deciphering Interactions in Causal Networks without Parametric Assumptions. arXiv
**2013**, arXiv:1311.2707. [Google Scholar] - Petralia, F.; Song, W.M.; Tu, Z.; Wang, P. New Method for Joint Network Analysis Reveals Common and Different Coexpression Patterns among Genes and Proteins in Breast Cancer. J. Proteome Res.
**2016**, 15, 743–754. [Google Scholar] [CrossRef] - Huynh-Thu, V.A.; Irrthum, A.; Wehenkel, L.; Geurts, P. Inferring Regulatory Networks from Expression Data Using Tree-Based Methods. PLoS ONE
**2010**, 5, e12776. [Google Scholar] [CrossRef] - Lefebvre, C.; Rajbhandari, P.; Alvarez, M.J.; Bandaru, P.; Lim, W.K.; Sato, M.; Wang, K.; Sumazin, P.; Kustagi, M.; Bisikirska, B.C.; et al. A human B-cell interactome identifies MYB and FOXM1 as master regulators of proliferation in germinal centers. Mol. Syst. Biol.
**2010**, 6, 377. [Google Scholar] [CrossRef] - Li, Z.; Ivanov, A.A.; Su, R.; Gonzalez-Pecchi, V.; Qi, Q.; Liu, S.; Webber, P.; McMillan, E.; Rusnak, L.; Pham, C.; et al. The OncoPPi network of cancer-focused protein–protein interactions to inform biological insights and therapeutic strategies. Nat. Commun.
**2017**, 8, 14356. [Google Scholar] [CrossRef] - Rai, A.; Pradhan, P.; Nagraj, J.; Lohitesh, K.; Chowdhury, R.; Jalan, S. Understanding cancer complexome using networks, spectral graph theory and multilayer framework. Sci. Rep.
**2017**, 7, 41676. [Google Scholar] [CrossRef] - Khatri, P.; Drăghici, S. Ontological analysis of gene expression data: Current tools, limitations, and open problems. Bioinformatics
**2005**, 21, 3587–3595. [Google Scholar] [CrossRef] - Subramanian, A.; Tamayo, P.; Mootha, V.K.; Mukherjee, S.; Ebert, B.L.; Gillette, M.A.; Paulovich, A.; Pomeroy, S.L.; Golub, T.R.; Lander, E.S.; et al. Gene set enrichment analysis: A knowledge-based approach for interpreting genome-wide expression profiles. Proc. Natl. Acad. Sci. USA
**2005**, 102, 15545–15550. [Google Scholar] [CrossRef] - Draghici, S.; Khatri, P.; Tarca, A.L.; Amin, K.; Done, A.; Voichita, C.; Georgescu, C.; Romero, R. A systems biology approach for pathway level analysis. Genome Res.
**2007**, 17, 1537–1545. [Google Scholar] [CrossRef] - Martinez, M.R.; Corradin, A.; Klein, U.; Alvarez, M.J.; Toffolo, G.M.; di Camillo, B.; Califano, A.; Stolovitzky, G.A. Quantitative modeling of the terminal differentiation of B cells and mechanisms of lymphomagenesis. Proc. Natl. Acad. Sci. USA
**2012**, 109, 2672–2677. [Google Scholar] [CrossRef] - Korkut, A.; Wang, W.; Demir, E.; Aksoy, B.A.; Jing, X.; Molinelli, E.J.; Babur, O.; Bemis, D.L.; Onur Sumer, S.; Solit, D.B.; et al. Perturbation biology nominates upstream–downstream drug combinations in RAF inhibitor resistant melanoma cells. eLife
**2015**, 4, e04640. [Google Scholar] [CrossRef] - Arkin, A.; Ross, J.; McAdams, H.H. Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Phage Lambda-Infected Escherichia coli Cells. Genetics
**1998**, 149, 1633–1648. [Google Scholar] [CrossRef] - Rodríguez Martínez, M.; Soriano, J.; Tlusty, T.; Pilpel, Y.; Furman, I. Messenger RNA fluctuations and regulatory RNAs shape the dynamics of a negative feedback loop. Phys. Rev. E
**2010**, 81, 031924. [Google Scholar] [CrossRef] - Thomas, M.J.; Klein, U.; Lygeros, J.; Rodríguez Martínez, M. A Probabilistic Model of the Germinal Center Reaction. Front. Immunol.
**2019**, 10, 689. [Google Scholar] [CrossRef] - Pélissier, A.; Akrout, Y.; Jahn, K.; Kuipers, J.; Klein, U.; Beerenwinkel, N.; Rodríguez Martínez, M. Computational Model Reveals a Stochastic Mechanism behind Germinal Center Clonal Bursts. Cells
**2020**, 9, 1448. [Google Scholar] [CrossRef] [PubMed] - Chaves, M.; Albert, R.; Sontag, E.D. Robustness and fragility of Boolean models for genetic regulatory networks. J. Theor. Biol.
**2005**, 235, 431–449. [Google Scholar] [CrossRef] [PubMed] - Shmulevich, I.; Dougherty, E.R.; Kim, S.; Zhang, W. Probabilistic Boolean networks: A rule-based uncertainty model for gene regulatory networks. Bioinformatics
**2002**, 18, 261–274. [Google Scholar] [CrossRef] [PubMed] - Chaves, M.; Sontag, E.D.; Albert, R. Methods of robustness analysis for Boolean models of gene control networks. IEE Proc. Syst. Biol.
**2006**, 153, 154–167. [Google Scholar] [CrossRef] - Fumiã, H.F.; Martins, M.L. Boolean Network Model for Cancer Pathways: Predicting Carcinogenesis and Targeted Therapy Outcomes. PLoS ONE
**2013**, 8, e69008. [Google Scholar] [CrossRef] - Béal, J.; Montagud, A.; Traynard, P.; Barillot, E.; Calzone, L. Personalization of Logical Models With Multi-Omics Data Allows Clinical Stratification of Patients. Front. Physiol.
**2019**, 9, 1965. [Google Scholar] [CrossRef] - Eduati, F.; Jaaks, P.; Wappler, J.; Cramer, T.; Merten, C.A.; Garnett, M.J.; Saez-Rodriguez, J. Patient-specific logic models of signaling pathways from screenings on cancer biopsies to prioritize personalized combination therapies. Mol. Syst. Biol.
**2020**, 16, e8664. [Google Scholar] [CrossRef] - Vanunu, O.; Magger, O.; Ruppin, E.; Shlomi, T.; Sharan, R. Associating Genes and Protein Complexes with Disease via Network Propagation. PLoS Comput. Biol.
**2010**, 6, 9. [Google Scholar] [CrossRef] - Hofree, M.; Shen, J.P.; Carter, H.; Gross, A.; Ideker, T. Network-based stratification of tumor mutations. Nat. Methods
**2013**, 10, 1108–1115. [Google Scholar] [CrossRef] - Oskooei, A.; Manica, M.; Mathis, R.; Martínez, M.R. Network-based Biased Tree Ensembles (NetBiTE) for Drug Sensitivity Prediction and Drug Sensitivity Biomarker Identification in Cancer. Sci. Rep.
**2019**, 9, 15918. [Google Scholar] [CrossRef] - Manica, M.; Cadow, J.; Mathis, R.; Rodríguez Martínez, M. PIMKL: Pathway-Induced Multiple Kernel Learning. Npj Syst. Biol. Appl.
**2019**, 5, 1–8. [Google Scholar] [CrossRef] - Axenie, C.; Kurz, D. GLUECK: Growth pattern learning for unsupervised extraction of cancer kinetics. In Machine Learning and Knowledge Discovery in Databases, Proceedings of the ECML2020—Applied Data Science and Demo Track, Ghent, Belgium, 14–18 September 2020; Dong, Y., Ifrim, G., Mladenić, D., Saunders, C., Van Hoecke, S., Eds.; Springer International Publishing: Cham, Switzerland, 2021; pp. 171–186. [Google Scholar]
- Axenie, C.; Kurz, D. Tumor Characterization using Unsupervised Learning of Mathematical Relations within Breast Cancer Data. In Proceedings of the International Conference on Artificial Neural Networks 2020, Bratislava, Slovakia, 15–18 September 2020; pp. 838–849. [Google Scholar]
- Massafra, R.; Bove, S.; Lorusso, V.; Biafora, A.; Comes, M.C.; Didonna, V.; Diotaiuti, S.; Fanizzi, A.; Nardone, A.; Nolasco, A.; et al. Radiomic Feature Reduction Approach to Predict Breast Cancer by Contrast-Enhanced Spectral Mammography Images. Diagnostics
**2021**, 11, 684. [Google Scholar] [CrossRef] - Axenie, C.; Kurz, D. CHIMERA: Combining Mechanistic Models and Machine Learning for Personalized Chemotherapy and Surgery Sequencing in Breast Cancer. In Proceedings of the International Symposium on Mathematical and Computational Oncology 2020, San Diego, CA, USA, 8–10 October 2020; pp. 13–24. [Google Scholar]
- Wang, B.; Mezlini, A.M.; Demir, F.; Fiume, M.; Tu, Z.; Brudno, M.; Haibe-Kains, B.; Goldenberg, A. Similarity network fusion for aggregating data types on a genomic scale. Nat. Methods
**2014**, 11, 333–337. [Google Scholar] [CrossRef] - Shalapour, S.; Karin, M. Immunity, inflammation, and cancer: An eternal fight between good and evil. J. Clin. Investig.
**2015**, 125, 3347–3355. [Google Scholar] [CrossRef] - Jackson, H.J.; Rafiq, S.; Brentjens, R.J. Driving CAR T-cells forward. Nat. Rev. Clin. Oncol.
**2016**, 13, 370–383. [Google Scholar] [CrossRef] - Park, J.H.; Rivière, I.; Gonen, M.; Wang, X.; Sénéchal, B.; Curran, K.J.; Sauter, C.; Wang, Y.; Santomasso, B.; Mead, E.; et al. Long-Term Follow-up of CD19 CAR Therapy in Acute Lymphoblastic Leukemia. N. Engl. J. Med.
**2018**, 378, 449–459. [Google Scholar] [CrossRef] - Rafiq, S.; Hackett, C.S.; Brentjens, R.J. Engineering strategies to overcome the current roadblocks in CAR T cell therapy. Nat. Rev. Clin. Oncol.
**2020**, 17, 147–167. [Google Scholar] [CrossRef] - Konstorum, A.; Vella, A.T.; Adler, A.J.; Laubenbacher, R.C. A mathematical model of combined CD8 T cell costimulation by 4-1BB (CD137) and OX40 (CD134) receptors. Sci. Rep.
**2019**, 9, 10862. [Google Scholar] [CrossRef] - Wertheim, K.Y.; Puniya, B.L.; La Fleur, A.; Shah, A.R.; Barberis, M.; Helikar, T. Multi-Approach and Multi-Scale Model of CD4+ T Cells Predicts Switch-Like and Oscillatory Emergent Behaviors in Inflammatory Response to Infection. bioRxiv
**2020**. [Google Scholar] [CrossRef] - Bouchnita, A.; Bocharov, G.; Meyerhans, A.; Volpert, V. Hybrid approach to model the spatial regulation of T cell responses. BMC Immunol.
**2017**, 18. [Google Scholar] [CrossRef] - Mayer, A.; Zhang, Y.; Perelson, A.S.; Wingreen, N.S. Regulation of T cell expansion by antigen presentation dynamics. Proc. Natl. Acad. Sci. USA
**2019**, 116, 5914–5919. [Google Scholar] [CrossRef] - Meyer-Hermann, M.; Mohr, E.; Pelletier, N.; Zhang, Y.; Victora, G.D.; Toellner, K.M. A Theory of Germinal Center B Cell Selection, Division, and Exit. Cell Rep.
**2012**, 2, 162–174. [Google Scholar] [CrossRef] - Mehta, P.; Schwab, D.J. An exact mapping between the variational renormalization group and deep learning. arXiv
**2014**, arXiv:1410.3831. [Google Scholar] - Weber, A.; Born, J.; Rodríguez Martínez, M. TITAN: T Cell Receptor Specificity Prediction with Bimodal Attention Networks. arXiv
**2021**, arXiv:2105.03323. [Google Scholar] - Dlamini, Z.; Francies, F.Z.; Hull, R.; Marima, R. Artificial intelligence (AI) and big data in cancer and precision oncology. Comput. Struct. Biotechnol. J.
**2020**, 18, 2300–2311. [Google Scholar] [CrossRef] - Enderling, H.; AJ Chaplain, M. Mathematical modeling of tumor growth and treatment. Curr. Pharm. Des.
**2014**, 20, 4934–4940. [Google Scholar] [CrossRef] - Heindl, A.; Sestak, I.; Naidoo, K.; Cuzick, J.; Dowsett, M.; Yuan, Y. Relevance of spatial heterogeneity of immune infiltration for predicting risk of recurrence after endocrine therapy of ER+ breast cancer. JNCI J. Natl. Cancer Inst.
**2018**, 110, 166–175. [Google Scholar] [CrossRef] - Gallaher, J.A.; Enriquez-Navas, P.M.; Luddy, K.A.; Gatenby, R.A.; Anderson, A.R. Spatial heterogeneity and evolutionary dynamics modulate time to recurrence in continuous and adaptive cancer therapies. Cancer Res.
**2018**, 78, 2127–2139. [Google Scholar] [CrossRef] - Gatenby, R.A.; Gawlinski, E.T. A reaction-diffusion model of cancer invasion. Cancer Res.
**1996**, 56, 5745–5753. [Google Scholar] - Ghaffarizadeh, A.; Heiland, R.; Friedman, S.H.; Mumenthaler, S.M.; Macklin, P. PhysiCell: An open source physics-based cell simulator for 3-D multicellular systems. PLoS Comput. Biol.
**2018**, 14, e1005991. [Google Scholar] [CrossRef] - Breitwieser, L.; Hesam, A.; de Montigny, J.; Vavourakis, V.; Iosif, A.; Jennings, J.; Kaiser, M.; Manca, M.; Di Meglio, A.; Al-Ars, Z.; et al. BioDynaMo: A general platform for scalable agent-based simulation. bioRxiv
**2021**, 2020-06. [Google Scholar] [CrossRef] - Kaul, H.; Ventikos, Y. Investigating biocomplexity through the agent-based paradigm. Brief. Bioinform.
**2015**, 16, 137–152. [Google Scholar] [CrossRef] [PubMed] - Metzcar, J.; Wang, Y.; Heiland, R.; Macklin, P. A review of cell-based computational modeling in cancer biology. JCO Clin. Cancer Inform.
**2019**, 2, 1–13. [Google Scholar] [CrossRef] [PubMed] - Gonzalez-de Aledo, P.; Vladimirov, A.; Manca, M.; Baugh, J.; Asai, R.; Kaiser, M.; Bauer, R. An optimization approach for agent-based computational models of biological development. Adv. Eng. Softw.
**2018**, 121, 262–275. [Google Scholar] [CrossRef] - Macnamara, C.K. Biomechanical modelling of cancer: Agent-based force-based models of solid tumours within the context of the tumour microenvironment. Comput. Syst. Oncol.
**2021**, 1, e1018. [Google Scholar] - Bauer, R.; Clowry, G.; Kaiser, M. Creative destruction: A basic computational model of cortical layer formation. Cereb Cortex
**2021**, 31, 3237–3253. [Google Scholar] [CrossRef] [PubMed] - Gorochowski, T.E. Agent-based modelling in synthetic biology. Essays Biochem.
**2016**, 60, 325–336. [Google Scholar] [PubMed] - Mogilner, A.; Manhart, A. Agent-based modeling: Case study in cleavage furrow models. Mol. Biol. Cell
**2016**, 27, 3379–3384. [Google Scholar] [CrossRef] - Tracy, M.; Cerdá, M.; Keyes, K.M. Agent-based modeling in public health: Current applications and future directions. Annu. Rev. Public Health
**2018**, 39, 77–94. [Google Scholar] [CrossRef] - Bauer, R.; Zubler, F.; Hauri, A.; Muir, D.R.; Douglas, R.J. Developmental origin of patchy axonal connectivity in the neocortex: A computational model. Cereb. Cortex
**2014**, 24, 487–500. [Google Scholar] [CrossRef] - De Montigny, J.; Iosif, A.; Breitwieser, L.; Manca, M.; Bauer, R.; Vavourakis, V. An in silico hybrid continuum-/agent-based procedure to modelling cancer development: Interrogating the interplay amongst glioma invasion, vascularity and necrosis. Methods
**2021**, 185, 94–104. [Google Scholar] [CrossRef] - Jalalimanesh, A.; Haghighi, H.S.; Ahmadi, A.; Soltani, M. Simulation-based optimization of radiotherapy: Agent-based modeling and reinforcement learning. Math. Comput. Simul.
**2017**, 133, 235–248. [Google Scholar] [CrossRef] - Hadjicharalambous, M.; Wijeratne, P.A.; Vavourakis, V. From tumour perfusion to drug delivery and clinical translation of in silico cancer models. Methods
**2021**, 185, 82–93. [Google Scholar] [CrossRef] - Gong, C.; Milberg, O.; Wang, B.; Vicini, P.; Narwal, R.; Roskos, L.; Popel, A.S. A computational multiscale agent-based model for simulating spatio-temporal tumour immune response to PD1 and PDL1 inhibition. J. R. Soc. Interface
**2017**, 14, 20170320. [Google Scholar] [CrossRef] - Le, D.T.; Uram, J.N.; Wang, H.; Bartlett, B.R.; Kemberling, H.; Eyring, A.D.; Skora, A.D.; Luber, B.S.; Azad, N.S.; Laheru, D.; et al. PD-1 blockade in tumors with mismatch-repair deficiency. N. Engl. J. Med.
**2015**, 372, 2509–2520. [Google Scholar] [CrossRef] - Cristini, V.; Koay, E.; Wang, Z. An Introduction to Physical Oncology: How Mechanistic Mathematical Modeling Can Improve Cancer Therapy Outcomes; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Werner, B.; Scott, J.G.; Sottoriva, A.; Anderson, A.R.; Traulsen, A.; Altrock, P.M. The cancer stem cell fraction in hierarchically organized tumors can be estimated using mathematical modeling and patient-specific treatment trajectories. Cancer Res.
**2016**, 76, 1705–1713. [Google Scholar] [CrossRef] - Chamseddine, I.M.; Rejniak, K.A. Hybrid modeling frameworks of tumor development and treatment. Wiley Interdiscip. Rev. Syst. Biol. Med.
**2020**, 12, e1461. [Google Scholar] [CrossRef] - Nia, H.T.; Munn, L.L.; Jain, R.K. Physical traits of cancer. Science
**2020**, 370. [Google Scholar] [CrossRef] - Kondylakis, H.; Axenie, C.; Bastola, D.K.; Katehakis, D.G.; Kouroubali, A.; Kurz, D.; Larburu, N.; Macía, I.; Maguire, R.; Maramis, C.; et al. Status and recommendations of technological and data-driven innovations in cancer care: Focus group study. J. Med. Internet Res.
**2020**, 22, e22034. [Google Scholar] [CrossRef] - Nia, H.T.; Liu, H.; Seano, G.; Datta, M.; Jones, D.; Rahbari, N.; Incio, J.; Chauhan, V.P.; Jung, K.; Martin, J.D.; et al. Solid stress and elastic energy as measures of tumour mechanopathology. Nat. Biomed. Eng.
**2016**, 1, 1–11. [Google Scholar] [CrossRef] - Griffon-Etienne, G.; Boucher, Y.; Brekken, C.; Suit, H.D.; Jain, R.K. Taxane-induced apoptosis decompresses blood vessels and lowers interstitial fluid pressure in solid tumors: Clinical implications. Cancer Res.
**1999**, 59, 3776–3782. [Google Scholar] [PubMed] - Rouvière, O.; Melodelima, C.; Dinh, A.H.; Bratan, F.; Pagnoux, G.; Sanzalone, T.; Crouzet, S.; Colombel, M.; Mège-Lechevallier, F.; Souchon, R. Stiffness of benign and malignant prostate tissue measured by shear-wave elastography: A preliminary study. Eur. Radiol.
**2017**, 27, 1858–1866. [Google Scholar] [CrossRef] [PubMed] - Nathanson, S.D.; Nelson, L. Interstitial fluid pressure in breast cancer, benign breast conditions, and breast parenchyma. Ann. Surg. Oncol.
**1994**, 1, 333–338. [Google Scholar] [CrossRef] [PubMed] - Benzekry, S.; Lamont, C.; Beheshti, A.; Tracz, A.; Ebos, J.M.L.; Hlatky, L.; Hahnfeldt, P. Classical Mathematical Models for Description and Prediction of Experimental Tumor Growth. PLoS Comput. Biol.
**2014**, 10, 1–19. [Google Scholar] [CrossRef] - Ben-Shmuel, A.; Biber, G.; Barda-Saad, M. Unleashing Natural Killer Cells in the Tumor Microenvironment–The Next Generation of Immunotherapy? Front. Immunol.
**2020**, 11, 275. [Google Scholar] [CrossRef] - Uzhachenko, R.V.; Shanker, A. CD8+ T lymphocyte and NK cell network: Circuitry in the cytotoxic domain of immunity. Front. Immunol.
**2019**, 10, 1906. [Google Scholar] [CrossRef] - Markowetz, F.; Troyanskaya, O.G. Computational identification of cellular networks and pathways. Mol. BioSyst.
**2007**, 3, 478–482. [Google Scholar] [CrossRef] - Cornish, A.J.; Markowetz, F. SANTA: Quantifying the functional content of molecular networks. PLoS Comput. Biol.
**2014**, 10, e1003808. [Google Scholar] [CrossRef] - Haeno, H.; Gonen, M.; Davis, M.B.; Herman, J.M.; Iacobuzio-Donahue, C.A.; Michor, F. Computational modeling of pancreatic cancer reveals kinetics of metastasis suggesting optimum treatment strategies. Cell
**2012**, 148, 362–375. [Google Scholar] [CrossRef] - Benzekry, S. Artificial intelligence and mechanistic modeling for clinical decision making in oncology. Clin. Pharmacol. Ther.
**2020**, 108, 471–486. [Google Scholar] [CrossRef] - Vaghi, C.; Rodallec, A.; Fanciullino, R.; Ciccolini, J.; Mochel, J.P.; Mastri, M.; Poignard, C.; Ebos, J.M.L.; Benzekry, S. Population modeling of tumor growth curves and the reduced Gompertz model improve prediction of the age of experimental tumors. PLoS Comput. Biol.
**2020**, 16, 1–24. [Google Scholar] [CrossRef] - Volk, L.D.; Flister, M.J.; Chihade, D.; Desai, N.; Trieu, V.; Ran, S. Synergy of nab-paclitaxel and bevacizumab in eradicating large orthotopic breast tumors and preexisting metastases. Neoplasia
**2011**, 13, 327–338. [Google Scholar] [CrossRef] - Benzekry, S.; Lamont, C.; Weremowicz, J.; Beheshti, A.; Hlatky, L.; Hahnfeldt, P. Tumor growth kinetics of subcutaneously implanted Lewis Lung carcinoma cells. PLoS Comput. Biol.
**2019**. [Google Scholar] [CrossRef] - Simpson-Herren, L.; Lloyd, H.H. Kinetic parameters and growth curves for experimental tumor systems. Cancer Chemother. Rep.
**1970**, 54, 143–174. [Google Scholar] - Tan, G.; Kasuya, H.; Sahin, T.T.; Yamamura, K.; Wu, Z.; Koide, Y.; Hotta, Y.; Shikano, T.; Yamada, S.; Kanzaki, A.; et al. Combination therapy of oncolytic herpes simplex virus HF10 and bevacizumab against experimental model of human breast carcinoma xenograft. Int. J. Cancer
**2015**, 136, 1718–1730. [Google Scholar] [CrossRef] - Edgerton, M.E.; Chuang, Y.L.; Macklin, P.; Yang, W.; Bearer, E.L.; Cristini, V. A novel, patient-specific mathematical pathology approach for assessment of surgical volume: Application to ductal carcinoma in situ of the breast. Anal. Cell. Pathol.
**2011**, 34, 247–263. [Google Scholar] [CrossRef] - Burstein, H.J.; Polyak, K.; Wong, J.S.; Lester, S.C.; Kaelin, C.M. Ductal carcinoma in situ of the breast. N. Engl. J. Med.
**2004**, 350, 1430–1441. [Google Scholar] [CrossRef] - Franks, S.; Byrne, H.; King, J.; Underwood, J.; Lewis, C. Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol.
**2003**, 47, 424–452. [Google Scholar] [CrossRef] - Franks, S.; Byrne, H.; Underwood, J.; Lewis, C. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast. J. Theor. Biol.
**2005**, 232, 523–543. [Google Scholar] [CrossRef] - Smith, J.; Martin, L. Do cells cycle? Proc. Natl. Acad. Sci. USA
**1973**, 70, 1263–1267. [Google Scholar] [CrossRef] - Marx, J. How cells cycle toward cancer. Science
**1994**, 263, 319–322. [Google Scholar] [CrossRef] - Cristini, V.; Lowengrub, J. Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Macklin, P.; Edgerton, M.E.; Thompson, A.M.; Cristini, V. Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): From microscopic measurements to macroscopic predictions of clinical progression. J. Theor. Biol.
**2012**, 301, 122–140. [Google Scholar] [CrossRef] - Benzekry, S.; Lamont, C.; Barbolosi, D.; Hlatky, L.; Hahnfeldt, P. Mathematical Modeling of Tumor–Tumor Distant Interactions Supports a Systemic Control of Tumor Growth. Cancer Res.
**2017**, 77, 5183–5193. [Google Scholar] [CrossRef] - Axenie, C.; Kurz, D. PRINCESS: Prediction of Individual Breast Cancer Evolution to Surgical Size. In Proceedings of the 2020 IEEE 33rd International Symposium on Computer-Based Medical Systems (CBMS), Rochester, MN, USA, 28–30 July 2020; pp. 457–462. [Google Scholar]
- Coffey, J.C.; Wang, J.; Smith, M.; Bouchier-Hayes, D.; Cotter, T.; Redmond, H. Excisional surgery for cancer cure: Therapy at a cost. Lancet Oncol.
**2003**, 4, 760–768. [Google Scholar] [CrossRef] - Pusztai, L.; Foldi, J.; Dhawan, A.; DiGiovanna, M.P.; Mamounas, E.P. Changing frameworks in treatment sequencing of triple-negative and HER2-positive, early-stage breast cancers. Lancet Oncol.
**2019**, 20, e390–e396. [Google Scholar] [CrossRef] - Afghahi, A.; Timms, K.M.; Vinayak, S.; Jensen, K.S.; Kurian, A.W.; Carlson, R.W.; Chang, P.-J.; Schackmann, E.; Hartmann, A.-R.; Ford, J.M.; et al. Tumor BRCA1 reversion mutation arising during neoadjuvant platinum-based chemotherapy in triple-negative breast cancer is associated with therapy resistance. Clin. Cancer Res.
**2017**, 23, 3365–3370. [Google Scholar] [CrossRef] - Killelea, B.K.; Yang, V.Q.; Mougalian, S.; Horowitz, N.R.; Pusztai, L.; Chagpar, A.B.; Lannin, D.R. Neoadjuvant chemotherapy for breast cancer increases the rate of breast conservation: Results from the National Cancer Database. J. Am. Coll. Surg.
**2015**, 220, 1063–1069. [Google Scholar] [CrossRef] - Reid-Lawrence, S.; Tan, A.R.; Mayer, I.A. Optimizing Adjuvant and Neoadjuvant Chemotherapy for Triple-Negative Breast Cancer. In Triple-Negative Breast Cancer; Springer: Cham, Switzerland, 2018; pp. 83–94. [Google Scholar]
- Loibl, S.; Treue, D.; Budczies, J.; Weber, K.; Stenzinger, A.; Schmitt, W.D.; Weichert, W.; Jank, P.; Furlanetto, J.; Klauschen, F.; et al. Mutational diversity and therapy response in breast Cancer: A sequencing analysis in the Neoadjuvant GeparSepto trial. Clin. Cancer Res.
**2019**, 25, 3986–3995. [Google Scholar] [CrossRef] - Fisher, B.; Gunduz, N.; Saffer, E.A. Influence of the Interval between Primary Tumor Removal and Chemotherapy on Kinetics and Growth of Metastases. Cancer Res.
**1983**, 43, 1488–1492. [Google Scholar] - Pauli, C.; Hopkins, B.D.; Prandi, D.; Shaw, R.; Fedrizzi, T.; Sboner, A.; Sailer, V.; Augello, M.; Puca, L.; Rosati, R.; et al. Personalized in vitro and in vivo cancer models to guide precision medicine. Cancer Discov.
**2017**, 7, 462–477. [Google Scholar] [CrossRef] - Wu, Y.; Deng, Z.; Wang, H.; Ma, W.; Zhou, C.; Zhang, S. Repeated cycles of 5-fluorouracil chemotherapy impaired anti-tumor functions of cytotoxic T cells in a CT26 tumor-bearing mouse model. BMC Immunol.
**2016**, 17, 29. [Google Scholar] [CrossRef] [PubMed] - Kessler, D.A.; Austin, R.H.; Levine, H. Resistance to chemotherapy: Patient variability and cellular heterogeneity. Cancer Res.
**2014**, 74, 4663–4670. [Google Scholar] [CrossRef] [PubMed] - Navin, N.E. Tumor evolution in response to chemotherapy: Phenotype versus genotype. Cell Rep.
**2014**, 6, 417–419. [Google Scholar] [CrossRef] [PubMed] - Henke, E.; Nandigama, R.; Ergün, S. Extracellular matrix in the tumor microenvironment and its impact on cancer therapy. Front. Mol. Biosci.
**2020**, 6, 160. [Google Scholar] [CrossRef] [PubMed] - Kurz, D.; Axenie, C. PERFECTO: Prediction of Extended Response and Growth Functions for Estimating Chemotherapy Outcomes in Breast Cancer. In Proceedings of the 2020 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), Seoul, Korea, 16–19 December 2020; pp. 609–614. [Google Scholar]
- She, Y.; Jin, Z.; Wu, J.; Deng, J.; Zhang, L.; Su, H.; Jiang, G.; Liu, H.; Xie, D.; Cao, N.; et al. Development and validation of a deep learning model for non–small cell lung cancer survival. JAMA Netw. Open
**2020**, 3, e205842. [Google Scholar] [CrossRef] [PubMed] - Benzekry, S.; Pasquier, E.; Barbolosi, D.; Lacarelle, B.; Barlési, F.; André, N.; Ciccolini, J. Metronomic reloaded: Theoretical models bringing chemotherapy into the era of precision medicine. Semin. Cancer Biol.
**2015**, 35, 53–61. [Google Scholar] [CrossRef] - Amoroso, N.; Pomarico, D.; Fanizzi, A.; Didonna, V.; Giotta, F.; La Forgia, D.L.; Latorre, A.; Monaco, A.; Pantaleo, E.; Petruzzellis, N.; et al. A Roadmap towards Breast Cancer Therapies Supported by Explainable Artificial Intelligence. Appl. Sci.
**2021**, 11, 4881. [Google Scholar] [CrossRef] - Simone, G.; Bianchi, M.; Giannarelli, D.; Daneshmand, S.; Papalia, R.; Ferriero, M.; Guaglianone, S.; Sentinelli, S.; Colombo, R.; Montorsi, F.; et al. Development and external validation of nomograms predicting disease-free and cancer-specific survival after radical cystectomy. World J. Urol.
**2015**, 33, 1419–1428. [Google Scholar] [CrossRef] - He, Y.; Liu, H.; Wang, S.; Zhang, J. A nomogram for predicting cancer-specific survival in patients with osteosarcoma as secondary malignancy. Sci. Rep.
**2020**, 10, 1–10. [Google Scholar] [CrossRef] - Zlotnik, A.; Abraira, V. A general-purpose nomogram generator for predictive logistic regression models. Stata J.
**2015**, 15, 537–546. [Google Scholar] [CrossRef] - Jalali, A.; Alvarez-Iglesias, A.; Roshan, D.; Newell, J. Visualising statistical models using dynamic nomograms. PLoS ONE
**2019**, 14, e0225253. [Google Scholar] [CrossRef] - Zhong, B.Y.; Ni, C.F.; Ji, J.S.; Yin, G.W.; Chen, L.; Zhu, H.D.; Guo, J.H.; He, S.C.; Deng, G.; Zhang, Q.; et al. Nomogram and artificial neural network for prognostic performance on the albumin-bilirubin grade for hepatocellular carcinoma undergoing transarterial chemoembolization. J. Vasc. Interv. Radiol.
**2019**, 30, 330–338. [Google Scholar] [CrossRef] - Wang, X.; Yuan, K.; Hellmayr, C.; Liu, W.; Markowetz, F. Reconstructing evolving signalling networks by hidden Markov nested effects models. Ann. Appl. Stat.
**2014**, 8, 448–480. [Google Scholar] [CrossRef] - Achim, T.; Florian, M. Structure Learning in Nested Effects Models. Stat. Appl. Genet. Mol. Biol.
**2008**, 7, 1–28. [Google Scholar] - Castro, M.A.; Wang, X.; Fletcher, M.N.; Meyer, K.B.; Markowetz, F. RedeR: R/Bioconductor package for representing modular structures, nested networks and multiple levels of hierarchical associations. Genome Biol.
**2012**, 13, 1–11. [Google Scholar] [CrossRef] - Letort, G.; Montagud, A.; Stoll, G.; Heiland, R.; Barillot, E.; Macklin, P.; Zinovyev, A.; Calzone, L. PhysiBoSS: A multi-scale agent-based modelling framework integrating physical dimension and cell signalling. Bioinformatics
**2019**, 35, 1188–1196. [Google Scholar] [CrossRef]

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