An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool

Abstract

Cancer is a complex disease, responsible for a significant portion of global deaths. The increasing prioritisation of know-why over know-how approaches in biological research has favoured the rising use of both white- and black-box mathematical techniques for cancer modelling, seeking to better grasp the multi-scale mechanistic workings of its complex phenomena (such as tumour-immune interactions, drug resistance, tumour growth and diffusion, etc.). In light of this wide-ranging use of mathematics in cancer modelling, the unique memory and non-local properties of Fractional Calculus (FC) have been sought after in the last decade to replace ordinary differentiation in the hypothesising of FC’s superior modelling of complex oncological phenomena, which has been shown to possess an accumulated knowledge of its past states. As such, this review aims to present a thorough and structured survey about the main guiding trends and modelling categories in cancer research, emphasising in the field of oncology FC’s increasing employment in mathematical modelling as a whole. The most pivotal research questions, challenges and future perspectives are also outlined.

1. Introduction

Cancer is the quintessential complex disease, posing a major public health problem and being one of the leading worldwide causes of death [1]. The term itself is often used to designate more than 100 distinct diseases. Nonetheless, all relate to the unregulated growth and spread of abnormal cells in the body, resulting from a malfunction of the natural biological control mechanisms of cell division [2], in which the cells grow and multiply uncontrollably [3], despite restrictions of space or nutrients shared by other cells [4]. This phenomenon produces what is known as malignant/cancerous tumours, distinguished from benign tumours in that, unlike the latter, the former can propagate its malfunction from a primary tumour, leading it to invade and destroy nearby normal tissues (locally invasive cancer), or spread its cells to other tissues in the body, distant from the primary tumour (metastatic cancer) [4]. Some benign tumours may also develop to the state of cancer, as with some instances of adenoma and leiomyoma [5]. In any case, tumours are not to be viewed merely as a collection of mutated cells growing in isolation, since they interact with and modify their physical micro-environment [6], being better regarded instead as a part of a larger ecosystem in which they actively compete with and disrupt the attempts of a complex interacting cellular community to maintain homeostasis [7]. If untreated, cancer may inevitably lead to the patient’s death, thus requiring cautionary clinical response through surgery, chemotherapy, radiotherapy, immunotherapy or other [8].

According to the World Health Organisation (WHO), statistics reveal an estimate of 10 million cancer deaths in 2020 alone [9]. In the United States, it affects approximately 1 in 3 people, with around 442.4 new cases per 100,000 men and women every year [10], and a predicted 609,360 total deaths in 2022—corresponding to almost 1700 deaths per day—with the greatest predicted deaths being due to lung, prostate and colorectum cancer in men and lung, breast and colorectum cancer in women [11] (

Table 1. Ranking of the ten leading types of cancer for both men and women, pertaining to estimated new cases and deaths, as projected for 2022 in the United States (adapted from [11]).

Estimated New Cases
Males			Females
Prostate	268,490	27%	Breast	287,850	31%
Lung and bronchus	117,910	12%	Lung and bronchus	118,830	13%
Colon and rectum	80,690	8%	Colon and rectum	70,340	8%
Urinary bladder	61,700	6%	Uterine corpus	65,950	7%
Melanoma of the skin	57,180	6%	Melanoma of the skin	42,600	5%
Kidney and renal pelvis	50,290	5%	Non-Hodgkin lymphoma	36,350	4%
Non-Hodgkin lymphoma	44,120	4%	Thyroid	31,940	3%
Oral cavity and pharynx	38,700	4%	Pancreas	29,240	3%
Leukaemia	35,810	4%	Kidney and renal pelvis	28,710	3%
Pancreas	32,970	3%	Leukaemia	24,840	3%
All Sites	983,160	100%	All Sites	934,870	100%
Estimated Deaths
Males			Females
Lung and bronchus	68,820	21%	Lung and bronchus	61,360	21%
Prostate	34,500	11%	Breast	43,250	15%
Colon and rectum	28,400	9%	Colon and rectum	24,180	8%
Pancreas	25,970	8%	Pancreas	23,860	8%
Liver and intrahepatic bile duct	20,420	6%	Ovary	12,810	4%
Leukaemia	14,020	4%	Uterine corpus	12,550	4%
Esophagus	13,250	4%	Liver and intrahepatic bile duct	10,100	4%
Urinary bladder	12,120	4%	Leukaemia	9980	3%
Non-Hodgkin lymphoma	11,700	4%	Non-Hodgkin lymphoma	8550	3%
Brain and other nervous system	10,710	3%	Brain and other nervous system	7570	3%
All Sites	322,090	100%	All Sites	287,270	100%

).

Despite there still being no definite cause for cancer [4], there have been positive prospects of oncological understanding in light of the wide-ranging use of mathematics in cancer research, since it has become clear that many biological problems demand methods and techniques requiring not only traditionally applied mathematics, but also pure mathematics, statistics and computation [7]. Thus, with the rapid evolution of biological modelling, control and optimisation theory and its potential of application in the cutting edge of the medicinal field (e.g., in nanomedicine [12]), the role of mathematics in cancer research has continued to be increasingly prominent in whatever theoretical forms it manifests its importance [13], as presented through the field of Mathematical Oncology.

One sub-area of exponential incidence of mathematics in Mathematical Oncology has been in the modelling of tumour responses [14], since it poses a valuable and inexpensive tool for prediction of treatment outcomes and identification of potential therapeutic combinations with minimal adverse effects [15]. A large part of this modelling research has been obtained through the development of models comprising Ordinary Differential Equations (ODEs) [16], though the existent and severely studied ones for cancerous tumour growth lack in non-locality effect, affecting the system’s overall robustness [17]. In response to this major shortcoming, Fractional Calculus (FC) has come into the oncological fold through fractional order modelling (also referred to as non-integer or arbitrary order), i.e., through models employing Fractional Differential Equations (FDEs), since its mathematical nature may allow for a better description and understanding of oncological particularities [18]. Yet, in spite (or even because) of the rise of Mathematical Oncology’s relevance in cancer research, criticism has surged as to the responsibility of the scientific community to “critically evaluate and discuss if model predictions are an academic exercise or have true translational merit” [14].

In light of all this, and with the relatively recent implementation of FC in oncology, attempts have risen in prospects of conceiving mathematical models that may describe the real cancerous phenomena with greater fidelity in its utmost complexity, thus aiming for a better understanding of the system’s interactions and potentially the apt prediction of its future states.

Here, we present a comprehensive review on both the trends of mathematical modelling in oncology research and FC’s own increasing usage in mathematical modelling, with special emphasis in oncology. It is organised in two main sections, as follows: Section 2 provides a thorough contextualisation of mathematical modelling in oncology as a research procedure, exposing different modelling categories, techniques, considerations and methodological frameworks; Section 3 presents a brief history of FC as a theoretical field, emphasising its advantageous features as a modelling tool through general applications and, most crucially, through the study of its use in Mathematical Oncology, highlighting the ways in which Fractional Calculus has been shown to be a welcome fit for this still novel research branch. Section 4 closes the paper with concluding remarks.

2. Mathematical Modelling in Oncology

Mathematical models may describe a system through abstraction and mathematical formalism, possibly enabling extrapolation beyond cases originally analysed, quantitative predictions and/or inference of mechanisms, among other purposes [19]. In fields such as chemical, electrical and biotechnological engineering, mathematical modelling has already cemented its wide-ranging value in representing an essential step for problem solving and product development, by using the available information of a given system to obtain an integrated picture that permits its analysis and prospects of optimization [20]. When applied to biology, a general mathematical modelling approach may be defined as “the art of using mathematical tools and concepts, usually supported by computing power, to represent natural systems, properties and phenomena” [18], thus providing a rigorous framework enabling the comprehension of disease evolution and testing of biological hypotheses [21].

In this line of thought, for more than 15 years now mathematical models of cancer have significantly increased their impact and number in the research field [14]. In fact, its history goes back more than half a century, yet it had been largely ignored by most experimental biologists [22]—though many of its initial conceptions dating back as far as 1928 can be consulted in [23]. Nonetheless, as expressed, renewed interest has been shown for the last few years, which may be partly explained by the research community’s realization of their potential for a more mechanistic understanding of such complex systems [7]. Its driving principle in oncology is the use of mathematical approaches able to maintain deductive–reductionist model features without the mischaracterisation of eventual complexities of the biological system [18]. The mechanistic paradigm in the modelling world is easily understood as a welcome fit in an oncological context, given the accepted notion that cancer is a fundamentally physical process subject to laws of nature studied in chemistry and physics [13]. Thus, enabling the value of mathematical modelling in the extraction of fundamental behaviour from such multiple and complex components and their relations [22]. As put forward in [19], “the power of mathematical modelling lies in its ability to reveal previously unknown or counter-intuitive physical principles that might have been overlooked or missed by a qualitative approach to biology”. This is moreover prompted by the shortcomings of strictly molecular reductionist approaches—the dominant mode of current biology [22]—in their inability to model the pivotal interactions between individual components, which has had experimental biologists recently embrace a more systemic view of cancer that mathematical models can indeed cover [7].

The focus on separate cancerous molecular species through traditional in vitro and in vivo approaches has been partially unable to grant significant insights, by failing to provide a global view on the pathogenesis of this disease, for which in silico techniques—understood as studies performed and/or relying on computer simulations—have then been gradually sought after in oncology [20]. These in silico approaches—of which mathematical modelling is part of—are also crucial in the realization of the 3R’s (replacement, reduction and refinement) that will lead cancer research toward efficient and effective precision medicine [24], while also allowing for the reduction in animal experiments in clinical studies [25]—and even though validation of computational models still requires results stemming from in vitro and in vivo experiments, the shift still does reduce laboratory work [24]. This transition may also be given to research activities progressively prioritising know-why over know-how, thus contributing to the further conception of models suitably combining both data-oriented and phenomenological approaches [18], though the latter still remains an underexplored niche to a large extent [22]. Additionally, the financial angle also presents a major upshot when adhering to mathematical modelling methods, considering that it may help researchers answer questions of oncological nature by modelling therapeutic options and thus predicting treatment and toxicity outcomes prior to the onset of lengthy and expensive trials [15], since studies have shown the costly reality of drug developments [26].

In this context, Mathematical Oncology has gradually developed as an accepted area of cancer research [22] that broadens the development and potential application of models in comprehending manifold phenomena, of which tumour growth dynamics, personalized treatment and anti-cancer therapies are part of [18]. It is mostly characterized by two main ideas [21]: (1) that mathematics can be applied to improve biomedical knowledge of cancer; (2) that biology itself produces new mathematical challenges which compel the development and enhancement of mathematical tools. As of the last years, the authors of [7] specifically list some of mathematical oncology’s main research goals which it aims to achieve in the future (here adapted, with recent cases of positive progression), as its role becomes more and more prominent in oncology: (i) prediction and optimization of patient-specific treatment strategies (e.g., [27]); (ii) definition and facilitation of the immune system’s role in cancer (e.g., [28]); (iii) integration of molecular scale data into multi-scale models (e.g., [29]); (iv) understanding and minimisation of the emergence of drug/treatment resistance (e.g., [30]); and (v) and greater integration with systems biology (e.g., [15]). Naturally, this implies the pivotal demand of an inter- and multidisciplinary approach in the research field [22] (Figure 1), further evidenced by the many distinctions to be made of mathematical models according to their assumptions and purpose, as well as their invoking of concepts from different areas [18]. It has also been supported and further shown how cross-disciplinary collaboration can accelerate positive results in the further understanding and treatment of cancer [31].

Given the plethora of methods pertaining to the mathematical modelling approach of cancer, it can be difficult to decide which type of model best suits a particular problem, as well as to what level of detail [31]. In response to this, many categories have been traced over the years in order to characterise each model and set their purposes apart, usually falling in each of the following distinctions: (1) white-, grey- or black-box [20]; (2) continuous, discrete or hybrid [20]; (3) deterministic or stochastic [1]; (4) its placement in the spatio-temporal length scale, from the gene level to a population level [20]; (5) the type of cancer, (6) its features [20]; (7) treatments in question, if any, combined or not [15]; (8) the modelling techniques themselves [32], along with their assumptions [33], approach [30]; and even (9) their underlying philosophy [34]. As noted, most of these distinctions are pertinently reviewed in [20], which shall be referred to extensively in the following explanations of each modelling category.

2.1. White-, Grey- and Black-Box Modelling

In order for a mathematical model to be constructed, information pertaining to the system to be modelled is key. However, this same information may be available through qualitative knowledge or quantitative data. The nature of the information used thus gives rise to the labelling of mathematical models as either white-, grey- or black-box. As concisely expounded in [20] and hereinafter summarised, white-box models (also known as mechanistic, hypothesis-driven or physics-based models) base their conception on first principles of physics. The term directly refers to the “transparent walls” the concept assumes in the system’s presenting of its events and all dynamic stages as directly visible [35]. Given this, by evidently showing the mechanisms of the studied phenomena, most of their parameters usually have a direct significance physically or physiologically, as is the case for plenty of models of cancer using differential equations (e.g., [36]).

This approach stands in stark contrast to black-box modelling (resulting in models known as data-driven empirical), which is purely established on experimental data and hence does not usually seek the attribution of any clear biological significance to its parameters. Instead, black-box models—of which deep learning and neural networks are part of—merely determine abstract operational connections between the system’s defined inputs and outputs, “hiding” their internal logic [37] and tracing predictions without evident explanation, which has been increasingly demanded in the medical field, though there have been some positive prospects [38]. Reacting to these black-box tensions and, on the other extreme, criticism sprouting from white-box models relying on some form of experimental data for their theoretical constructions [20], grey-box modelling has appeared in attempts to merge both notions, thus marrying experimental, clinical data with theoretical structures of a hypothesis-driven nature through data fitting strategies.

2.2. The Spatial Scale

As has been noted before, cancer is in its biological nature a multi-scale process. However, decades of dedicated efforts have made possible advanced scale-specific models which are significant in their own right [39]. In this respect, a classification to be made to a potential mathematical model of cancer resides in the biological level it assumes and operates in, thus prioritising a position in a spatio-temporal length scale (Figure 2) ranging from the gene to the protein, cell, tissue, organ, patient and population levels [16]. Defining the spatio-temporal realm is pivotal for what is merely one of several modelling decisions, and noteworthy advancements have been registered on the basis of a tissue level [16], since it groups many different cells and their integral interactions based on sets of rules [20], being frequently linked with white- or grey-box approaches. Nevertheless, resources stemming from these various scales have inspired attempts at conceiving multi-scale models combining the different levels, though defining the links between them still poses a great barrier [39].

As a consequence of the modelling scale decision, another modelling differentiation can be traced, being the numerical type of the variables involved: continuous, discrete or hybrid. Continuous models, usually used in the tissue level [20], describe the tumour as a continuous medium by considering concentrations of cells (i.e., a population) being a choice apt for modelling larger-scale volumetric dynamics [39] and one supported to some degree by fundamental physical principles [46]. Models of this type usually make use of partial or integro-differential equations, with common variables such as nutrient, oxygen and growth factors being relatively simple to obtain when compared to the discrete case [39]. Despite their advantages in characterising global properties of tumour growth, this numerical type cannot be used to examine individual cell dynamics, for which discrete modelling is often used, being a particularly useful approach in the study of carcinogenesis [47], natural selection and cell–cell interaction mechanisms [39]. Its primary upshot—as implemented, e.g., through cellular automaton models [48]—resides in the possibility of translating detailed biological findings on a cell level into rules for the model, though the computational cost is heavily increased by the number of cells considered [39].

The limitations on both ends are alleviated with the use of hybrid modelling approaches, which conjugate both continuous and discrete notions into having specific cells of interest be discretely modelled and continuous methods for environmental variables [20] (e.g., [36]). However, on this final type, a distinction must be traced between the term “hybrid” (numerical type), when used to denote the presence of both numerical types in a model, and “hybrid” (method), when descriptive of a methodological conjugation of both continuous (e.g., ordinary or partial differential equations) and discrete approaches (e.g., agent-based modelling) in a single model [49].

2.3. Models of Ordinary and Partial Differential Equations

Population, ecological and kinetic models are common approaches when aiming to grasp phenomenological foundations of general avascular tumour growth, for which ODEs are often used [50]. Though elementary in nature when compared to Partial Differential Equations (PDEs), ODE-based models employ deterministic, time-dependent equations that enable analytical solutions and mathematical description of phenomena evolution [18], while allowing for the fine-tuning of its parameters when faced against clinical data in a grey-box procedure [51], favouring their flexibility [18]. Relevant hallmarks have been the earlier models thoroughly mentioned in [52] and some in [53], while recent developments on this end can be seen in [33,54], with there being a wide array of assumed tumour growth laws [51]. The different considerations in constructing an ODE-based model are well documented in [55]. On this latter note, this mathematical technique also has the primary modelling assumption of the tumour being viewed as a spatially uniform homogeneous cell population, thus neglecting its spatial effects [32] and posing a weakness in the capturing of the complexities of real-life carcinogenesis. Moreover, there still is no consensus as to choosing the most appropriate ODE-based model for a particular type of cancer [56]. In light of this, criticism such as in [57] has claimed that ODE models should be carefully employed and only used to describe general trends in tumour behaviour, instead of characterizing specific clinical cases. Additionally, other downsides as put forward in [32] are the inability to relate its system’s parameters to behaviour of individual cells as well as the focus on a limited number of species at a time. Other general limitations relate to the phenomenon of tumour shrinkage, in which deterministic differential equations fail to accurately describe the system’s dynamics when predicting response to treatment [32], though this may be mitigated by the switching of these models to their stochastic counterparts [20] (e.g., [58]).

It has been largely noted that a lot of cell functions depend on their microenviroment, as with cases of tumour cells proliferating in well-oxygenated regions [32], thus requiring a modelling approach that matches its natural heterogeneity. In this context, many of the downsides of ODE-based models pertaining to their oversimplistic spatial assumptions have been accounted for and assuaged with the employment of Partial Differential Equations (PDEs) instead [18]. As such, these PDE models take into consideration the effects of spatial gradients in the local microenvironment of the tumour as well as those provoked by tumour cell density [32], by not only having their equations be time dependent as also spatial dependent [18]. Moreover, these models have a propensity to integrate well-established conservation laws into their modelling assumptions. This has thus enabled the modelling of phenomena such as vascular growth processes through the transport equation [59], the tumour’s metastatic spreading adjoined to the possibility of drug resistance from treatment [60] or the evolution of cell population density across tissues by means of diffusion equations [61]. Noteworthy initial contributions on this end have been the Greenspan and proliferation-invasion models (both approached in [32]) with other relevant developments seen in [18,62]. For these reasons, PDE-based models have been considered to be a more comprehensive choice when studying tumour growth into surrounding tissue [18].

2.4. Other Modelling Techniques

Despite the listed advantages of differential equation-based models, certain important features of tumour biology still cannot be accounted for through their mathematical techniques, particularly given cancer’s aforementioned multi-scale nature [39]. As referred to, other modelling techniques relying on computational simulations such as Agent-based Models (ABM) may be of preference here, which admit discrete cell-scale spatial resolution and may even present a methodologically hybrid approach so as to consider the various biological scales and their interactions [32] (e.g., [22], for the modelling of brain cancer). In this sense, by being modelled on an individual-cell level, ABMs are able to represent the behaviour of discerning autonomous agents—encoded with a set of pre-specified rules [47]—while highlighting the whole system as the integration of the different actors involved [20]. Through their adaptability to a hybrid approach when bridging the different scales, these models reveal a pertinent and accessible cancer modelling tool to a biologist in terms of understanding and experimental validation [22]. The different scales to be considered and their overarching modelling implications have been well documented in [39] while several contributions can be consulted also in [47]. When modelling under an ABM approach, the behaviour of the agents varies according to the assumed framework, which can be: lattice-based, in which agents are confined to specific locations on a rigid grid (e.g., cellular automata [48] or Potts models); or off-latice, with agents moving freely in space (e.g., vertex-based models). Thorough details for both frameworks and given examples are well documented in [63].

Lastly, other significant modelling techniques of black-box nature used over the last years have been neural networks [38], neuro-fuzzy systems [64] and Petri net models [65]. Several of all the aforementioned modelling techniques—from ODE-based to Petri nets—are often too complex and thus unlikely to be amenable to standard mathematical analysis, hence relying on computational solutions, whether it be numerical or simulation based [22]—though limitations stemming from computing power have been well documented [39]. Given this, the authors of [15] provides a thorough list of existing software for implementing many of the mentioned mathematical models, detailing their relevant packages and designed purposes.

2.5. General Modelling Assumptions

Past the modelling techniques themselves, another referred distinction regarding the model employed pertains to the cancer features it tackles in their agents’ individual study and overarching interactions [20]. For this purpose, many hallmarks of cancer cells have been collected in sources such as [66], detailing characteristics of normal cells that enable them to become malignant during the multi-step development of tumours, whatever their origin and phenotype [20], as well as other processes/phenomena, such as the activation of invasion and metastasis or resistance of cell death. Thus, mathematical models in oncology may also be described by the cancer trait they aim to question and analyse, with relevant curated examples being the cases of tumour growth [51], cancer metabolism [67], blood and lymphatic vasculature [59] and tumour immunity [16], as well as its microenvironment/heterogeneous nature [68] or its referred invasion and metastasizing [69]—most features already best associated to each previously described modelling techniques. Accordingly, to approach such cancer features also implies the commitment to certain biological assumptions (e.g., tumour growth laws or the involvement and interaction of certain species in the system [70]) yet given that this trait is common across all models and widely varied [33], it is not here considered as a category in itself.

2.6. Modelling Treatment Intervention

A great part of Mathematical Oncology also has its focus on modelling phenomena stemming from treatment-related effects, such as optimal regimens [71] and dosages in drug delivery [72] and their impacts on tumour and healthy cells in general [18]. Reasonably then, the treatment in question also plays a crucial role in choosing which mathematical model to conceive, of which there is an existing large variety [8]. In fact, it is through the knowledge of the treatment in question that mathematical models may be developed to support drug decision making, dosages and regimens, e.g., by characterising and predicting the relationships between drug exposure/pharmokinetics (PK), drug effects/pharmacodynamics (PD) and disease progression [33], as enabled by some PK/PD models [73]. In light of the treatment under consideration, the authors of [21] select many noteworthy publications pertaining to the modelling of certain breast cancer features that advances interventions in treatments such as chemotherapy, immunotherapy, radiotherapy or surgery. Other treatments have also been of interest for many mathematical modelling contributions, as is the case for virotherapy, hormone therapy and hyperthermia (some examples in [74]), all requiring the consideration of specific features of the cancer type in question, as well as biological assumptions [75].

Moreover, the adherence to novel therapies stemming from the combination of more than one treatment and the rise of personalised patient-specific strategies [76] has further required the advancement of modelling approaches, as thoroughly documented in [15], with instances of mathematical models of cancer to be treated with chemoimmunotherapy, chemovirotherapy, chemoradiotherapy, radioimmunotherapy, etc. For most of the treatments mentioned, whether mono-therapeutic or combined, one of the biggest challenges facing cancer research today is to account for and understand the causes underlying drug resistance [77], it being one of the major reasons for oncology patients experiencing treatment failure [33]. To advance research in the understanding of this, the authors of [33] have exhaustively covered the prospects of various assumptions and techniques when tackling clinical data and considering the possibility of drug resistance in mathematical modelling of cancer.

2.7. General Modelling Limitations

Besides the referred specific limitations residing in most categories for mathematical models of cancer, there are also general shortcomings when working with these in silico approaches. As thoroughly listed in [20] and here summarized, most challenges Mathematical Oncology still faces are, namely: (1) the necessity of simplifying assumptions, thus unable to grasp all the necessary intricacies of the real biological system, with the counter-risk of overparameterization in large, complex models; (2) the critical importance of parameter estimation, usually proceeded through data fitting techniques—since it is sometimes impossible to measure such values in subsequent in vitro or in vivo experiments; (3) the severe computational limitations that forces a trade-off between accuracy and time efficiency, only leading to the (at best) mean behaviour of the species modelled; (4) the dire need for the aforementioned multidisciplinary effort to develop pipelines for data sharing between research communities, given that experimental inaccuracies are directly transmitted in model outcomes; (5) the incorporation of the development of biomarkers capable of predicting degrees of response in various cancer patients [30]; and (6) the ethical and philosophical implications when recurring to simulation-based predictions for decision making [78].

Noticeably, limitations such as these have posed great barriers in enabling the use of most models in a clinical context [20], though contributions have surged in efforts to mitigate these shortcomings. Guidelines and best practices for development of mathematical models have been approached in [55] while a code of professional ethics for simulationists has been proposed in [78], as well as steps toward a verification procedure in this context, for a correct potential implementation of in silico techniques in clinical trials—similarly approached in the concept of virtual clinical trials in [76]. However, the most affecting drawback preventing most mathematical models’ application has pertained to the scarcity of the data itself [7], with cases of models recurring to parameterisation from multiple sources mixing cancer types, experimental conditions or spatio-temporal scales, sometimes leading to interesting yet biologically and clinically unrealistic dynamics [14]. Therefore, despite the rapid evolution of this field prompted by the increasing data availability of bio-informatics [18], there still has been a lack of proper theoretical models that manage to understand, organise and apply real clinical data effectively, though the principles of Mathematical Oncology do recognisably represent a necessary next step beyond verbal reasoning and linear intuition in research [79]. Sources such as [7] centre the problem on there still not being the “right” data that allows for a better definition of the cancer system in connecting its multi-scale processes, since most of the available data is often scale-specific and scarce in spatial information, being averaged and homogeneous. This has prompted the development of specific databases such as for the employment of combined treatment, gathered in [15].

2.8. Mathematical Oncology or Oncological Mathematics?

Unresolved tensions stemming from the aforementioned inabilities of mathematical modelling of cancer have prompted what is known as the Oncological Mathematics/ Mathematical Oncology split, as put forward by [14]. It evokes the necessity for the mathematical oncology profession to deploy a standard to only predict novel treatments if the model in question has been properly trained and validated for such—echoing the criticisms posed in [78]—thus clearly identifying a model’s purpose and its predictions as either “academic” (Oncological Mathematics) or “translational” (Mathematical Oncology), i.e., apt for potential medical use and providing pertinent clinical insight [14]. Naturally, this results in a new category with which to distinguish models.

The characteristics pertaining to the former category of academic models—as opposed to the already extensively mentioned nature of the translational ones—are then noted by their employment of cancer biology to motivate the development of “interesting” mathematics [14], thus not positioned to engage in the answering of certain oncological questions, namely the speculation of optimal therapy, as such misapplication could ultimately hurt patients if treated with ill-informed protocols [78]. Clear suggestions towards the creation of a biologically sound academic model can be found in [31], listing the following stages of development: (1) formulation of a biological hypothesis; (2) stating of modelling assumptions, as well as identification of key physical variables; (3) conception of “word” equations for all involved variables and identification of of the physical processes regulating their dynamics, followed by (4) conversion of said equations into mathematical language, specifying all functional forms; (5) solving the resultant mathematical model through appropriate numerical and analytical techniques; and (6) checking the consistency of the results with the originally formulated biological hypothesis.

Lastly, regarding the conception of translational models, three distinct modelling approaches when dealing with data are depicted in [30]: (1) top-down data-driven modelling, built predominantly upon observed clinical data and leaning towards a more empirical description of the system (e.g., PK/PD models [73]), though they may neglect important tumour-immune interactions; (2) bottom-up modelling, being of a more white-box nature by invoking clinical information as input data, fitting a mechanistic structure; and (3) middle-out modelling, which combines both previous approaches into one that provides a quantitative platform for model development in clinical scenarios [30]. Noticeably, the mentioned approaches parallel the black-, white- and grey-box natures, respectively.

Furthermore, the authors of [14] extensively list the crucial stages that have to be followed and verified for a mathematical model to make reliable and testable predictions of novel treatment strategies (here adapted): (1) identification of a putative biomarker, which may range from the number of cancer cells in a petri dish to tumour volume derived from medical imaging; (2) development of a mechanistic model by means of one of the modelling techniques already referred, taking into consideration the spatio-temporal nature of the available data, as well as balancing the model’s complexity with the degrees of freedom—also employing Occam’s razor concept if possible [55]; (3) calibration of the model through existing data, derived as less as possible from empirical wisdom or the literature but from realistic conditions, using machine learning and statistical tools to derive values if need be; (4) validation of the model with untrained data, evaluated via methods such as $R^2$ analyses, for instance; (5) evaluation of the predictive performance for a known treatment, since a model’s ability to fit data does not imply its predictive power [57], being the calculation of Positive Predictive Value (PPV) and Negative Predictive Value (NPV) strongly advised—while yet there being no conventional notion of acceptable cut-offs for predictive performance; and (6) simulation and prediction of untested treatment alternatives, i.e., therapy dosages and schedules that can be derived from the model, posterior to calibration and validation, for which it was trained to predict—and being carefully considerate of determinants such as clinical feasibility, drug half-life or toxicity when evaluating optimal regimes and dosages, requiring a rigorous control approach.

3. Fractional Calculus in Mathematical Modelling

Fractional Calculus may be considered both an old and yet essentially novel topic [80]. Its early conception can be said to derive from a question of extension of meaning [81] and dates back more than 300 years now, first registered in a 1695 letter directed to Gottfried Wilhelm Leibniz by Guillaume de l’Hôpital [81]. Up until then, Calculus had made clear the iterative process of both differentiation and anti-differentiation (i.e., integration) of a given real-valued function y of a real variable t [82], being informed by n, the order of such process, but was that all there could be of it? Could the order only be understood as an integer? With this in mind, in the letter l’Hôpital questions the possibility and consequent interpretation of a fractional order derivative [81], i.e., having n be a fraction in the computation of $\frac{d^n}{d{t}^n}y\left(t\right)$—this latter notation for the derivative operator D having been invented by Leibniz himself. At the time, the question was promptly and prophetically answered by the philosopher as being “an apparent paradox from which, one day, useful consequences will be drawn” [81].

Later on, this mathematical conundrum did not escape Euler’s attention, who also questioned the subject’s nature, as well as that of other mathematicians, such as Lagrange and Laplace [81]. Years of speculations also prompted the reformulation of the initial question to one in which the derivative’s order n (now taken as $\alpha$) could generally be taken as any value, thus embracing the possibilities of it being irrational or even complex [82]. Despite a history of subsequent hypothesising, fractional derivatives remained for the longest time an abstract albeit interesting mathematical concept [83].

3.1. Early Development

The true beginning of FC as a mathematical field in itself could be said to have only begun in 1832 with Liouville’s contributions, as well as posterior 19th and 20th century developments [84], having its foundations built through several contributions from other mathematicians such as Abel, Riemann, Grünwald and Letnikov [85], some of which are thoroughly documented in [81]. However, given that the new formulations of the notion allowed the order to go beyond mere fractional values, the term “fractional calculus” really is a misnomer, being “differentiation of arbitrary order” a more apt designation of its mathematical theoretical nature [81], while other terms like “Non-Integer Calculus”—liable to criticism, since it implies the ruling out of integer orders—and “Generalized Calculus” never got any real hold [82]. In this sense, fractional integrals are understood to be a particular case, one of differentiation of negative order ($\alpha <0$), which is sensible due to the blurry distinction of both cases in such field [82]. For this reason, FC is strictly related to the theory of pseudo-differential operators [80].

Under these considerations though, agreement over the true qualifications of a fractional derivative has been hard to meet [86], while a number of competing definitions have been postulated, noticeably the Riemann–Liouville (late 1840s) and Caputo (1967), widely adopted in many fields [87], followed since by a plethora of adaptations and original contributions—though the Grünwald–Letnikov definition (late 1860s) has been shown to possess great mathematical coherence in its generalisation of integer order derivatives [87]. Indeed, there exists a desire to explore and create new definitions and models, possibly motivated by the pure mathematician’s desire to generalise [88]. Given that each possess its own particularities and presuppositions, debates have questioned the criteria that may secure a strict notion of the fractional derivative concept, which established definitions do comply to it [86] and what classifications may be drawn to distinguish the various approaches [88]. Analysis of these considerations are well put forward in [89], while also presenting the many definitions.

Moreover, definitions and their implications expanded again once the possibility of a variable arbitrary order came into the fold, leading to Variable Order Differential Equations (VODEs) and the systems they may constitute, well documented in [90]. Additionally, the progress of FC as a whole up until 2017 may be consulted in the survey [91]. Naturally, all these kaleidoscopic offerings have enabled many differing approaches to and from FC as a field, and since the early 1990s more practically oriented scientists and engineers have been working with such various forms in hopes of converging them with physically meaningful formalisms and applications [83].

3.2. The Memory Effect

While entirely comprehensible in its extension of mathematical meaning, the theory underlying FC could be said to evidence a logical formalism of sorts, i.e., a formal language of pure syntax and no semantics. This is since that, contrary to integer order derivatives and integrals, FC does not carry in itself a clear physical and/or geometrical interpretation [92]. As put forward in [93], a possible reading of fractional derivatives both geometrically and physically could be that of “shadows on the walls” and “shadows of the past,’, respectively—an interpretation directly related to fractional derivatives’ main property: its memory effect [74]. This feature, also commonly referred to as non-locality, derives from a given fractional derivative operator’s $D^{\alpha }$ ability to take into consideration a given function’s past values (if the independent variable is temporal, thus $y\left(t\right)$) or long-range interactions (if the independent variable(s) is/are spatial), since the fractional derivative computation of a specific point depends on an average over an interval containing such value [94], granting thus a history or accumulation of information which integer order differentiation, using a local operator, does not supply. This same property could then be exploited into the forms of short- and long-term memories [83].

Taking this property into account, a possible physical interpretation was provided by [95], defining the fractional order $\alpha$ as the index of memory for the physical process, for which then $\alpha =0$ means that nothing is memorized, while $\alpha =1$ that nothing is forgotten. It is then supposed that many physical processes are best modelled by means of a fractional order operating between these extremes [74]. As Westerlund boldly and optimistically claimed in 1990, this reveals a powerful advantage when employing fractional order differential equations (FDEs), when compared to their integer counterpart, i.e., ODE: “all systems need a fractional time derivative in the equations that describe them. (…) It is necessary to include this record of events in order to predict the future (…) [as such behaviour is impossible to predict] with less than an infinite number of initial conditions, which is the same as saying that the whole function describing the past must be known” [96].

Nonetheless, as expressed in [94], this mathematical property mainly sets out three fundamental questions. (1) Are models composed of space and/or time fractional derivatives consistent with the governing and fundamental laws and symmetries of nature? (2) Even if so, how is the fractional order $\alpha$ to be experimentally observed? (3) How may a fractional derivative emerge from models not containing them? Additionally, [97] identified yet more open problems stemming from FC’s mathematical analysis, numerical processing, and application in physics, specifically, mentioning issues such as the computational cost related to the non-locality property and its numerical implications. On the theoretical end, monograph [98] offers a selection of problems related to positive one- and two-dimensional fractional linear systems, mapping out its fundamentals notions. These are some of the questions Fractional Calculus’s theory and main property opened up, having prompted researchers to further investigate its possible applications in real-life phenomena.

3.3. General Applications

Integer-valued operators, as noted, are local and isotropic in both space and time, having dominated science and engineering for centuries [99]. However, with the increasing need to model complex phenomena with underlying properties as spatial heterogeneity or the effects of memory, new quantitative ways of thinking have steadily brought FC to the front end of dynamics’ modelling research, as it enables a framework for such thinking [99]. Indeed, the interest for FC’s many possible applications has been such to have prompted a number of cyclical conferences, adding to its already rich literature, with some resulting materials among them present in [100,101,102]. Ever since the establishment of FC as a respected branch of mathematics, its real-world applications have been plenty and wide-ranging, being used as a powerful tool in several areas of science and engineering for the last decades now [85], such as, e.g.: (1) signal and image processing, for instance in edge detection [103]; (2) the assessment of cryptocurrencies’ price dynamics in economy [104]; (3) the understanding of the epidemic spread of diseases [105]; (4) the flow of highway traffic and its control [106]; (5) determining the properties of viscoelasticity [92]; (6) the modelling betterment of thermoacoustic engines [107]; (7) the understanding of long-term memory and multi-scale phenomena in materials [108]; (8) the description of complex shapes of microbial survival and growth curves in food science [92]; (9) the modelling and analysis of industrial processes [109]; and (10) the study of pulse propagation in electromagnetism [110]. Other examples may be consulted in [94]. In several of these areas, it is the hereditary property of the fractional order models that has granted a significant advantage in comparison to other integer order models, as stated years before in [111]. Other noteworthy examples of applications across many fields are thoroughly exposed in [112], also highlighting its increasing use as a control tool, for instance in the improvement of a precision positioning stage, betterment of active damping controllers or solution of problems pertaining to the asymptotic stability of inverted pendulum systems (respective references in [112]).

On the realm of models making use of variable order fractional differential equations, their importance has been remarkably acknowledged as a precise and alternative approach in the description of real-life phenomena [113], with some notable applications in Duffing systems, time-dependent mechanical property evolution in materials with strain softening behaviour, transient dispersion in heterogeneous media, digital cryptography, with respective references in the survey [113] and other examples in [114].

3.4. Modelling Biological Phenomena

Nature has often revealed itself to follow simple rules that still lead to the emergence of complex phenomena [115], and of the many different applications pertaining to FC, the propensity to model biological phenomena has been one of the most notable. In this respect, the authors of [115] compose a thorough review of FC’s many different applications in biology, stating that evidence suggests that fractional order dynamic behaviour may be possibly linked to self-similar/fractal structures or fractal kinetics, e.g., anomalous diffusion, present in both physical and chemical cases [116].

Diffusion in general is of fundamental importance in several disciplines in the description of properties like growth phenomena [117]. In the anomalous case, these diffusive processes are unlike normal ones, i.e., informed by Brownian motion of particles (for which $\alpha =1$)—in the sense that they are instead governed by a probability density function (PDF) in space that is not of the Gaussian type, hence its variance not being proportional to the first power of time t [118]. Anomalous diffusion may be split in either sub-diffusion ($\alpha <1$) or super-diffusion ($\alpha >1$). In light of this, the adoption of FC has been advantageous for its straightforward way of including external force terms, calculating boundary value problems, having easy adaptability when incorporating standard techniques for solving PDEs and showing a proximity to the analogous standard equations of diffusion processes [119]. As such, FC has been employed in the modelling of heat transfer, water transfer through porous materials and gas exchange (see references in [115]), but one of its most pertinent contributions has been in the description of anomalous diffusive processes occurring in cancer-related phenomena [18], e.g., in characterising such behaviour in biological tissue [115]. Moreover, cancer dynamics rarely meet ideal conditions such as symmetry or isotropic or periodic paths of movement, for which then simple diffusion does not suffice to approximate the modelling of its complexities [68], thus revealing one of the many ways in which FC may be a powerful tool in mathematical oncology.

3.5. Toward a Fractional Mathematical Oncology

Indeed, FDE-based models have been under significant interest in Mathematical Oncology for the past few years [18]. As such, “Fractional Mathematical Oncology” as a branch is becoming more and more a possible reality [56], whether given the ability to model said anomalous diffusive processes (e.g., in the tumour microenvironment or in metastasis [68]) or other complex phenomena such as numerical solution and control of invasion systems [120], description of multi-stage tumour characteristics [121], response to treatments [74], exploration of ideal combined chemo- and immunotherapy through optimal control [71], modelling of tumourous bone remodelling [122] and potential general tumour growth [56]. Other noteworthy efforts in this domain are excellently reviewed in [18,123], with the latter listing the fit FC grants in this modelling world by straightforwardly being able to deal with memory effects, heterogeneous scales and dormancy periods related to the tumour’s onset and development, thus potentially contributing to decision-making regarding tumour evolution, early diagnosis and personalized treatment therapies (including that of combined treatment). Additionally, recent advances suggest that ODE-based models for tumour growth may be improved by adopting FC’s tools, and further prospects of modelling improvement have been shown with variable order fractional equations-based models [121], identifying the arbitrary order $\alpha$ as an index of tumour memory.

However, as noted, these are still early albeit promising prospects of application of Fractional Calculus in oncology, and current literature is still somewhat scarce in the systematisation and surveying of its many possible employments. In spite of this,

Table 2. Compilation of Fractional Calculus studies in cancer modelling.

Year	Title	Type of Cancer	Treatment	Fractional Operator	Ref.
2014	A fractional diffusion equation model for cancer tumour	Not specified	Not specified	Caputo	[125]
2014	Numerical simulation of fractional bioheat equation in hyperthermia treatment	Not specified	Hyperthermia	Caputo	[126]
2016	Dynamics of tumour-immune system with fractional-order	Not specified	Immunotherapy	Caputo	[127]
2016	Modelling doxorubicin effect in various cancer therapies by means of fractional calculus	Not specified	Chemotherapy	Grünwald–Letnikov	[128]
2017	Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy	Not specified	Hyperthermia	Caputo	[129]
2018	New observations on optimal cancer treatments for a fractional tumour growth model with and without singular kernel	Obesity-associated cancer	Chemotherapy, Immunotherapy and Combined	Caputo and Caputo–Fabrizio	[71]
2018	Variable order fractional derivatives and bone remodelling in the presence of metastases	Not specified	Chemotherapy	Grünwald–Letnikov	[130]
2019	A new fractional model for the cancer treatment by radiotherapy using the Hadamard fractional derivative	Not specified	Radiotherapy	Hadamard	[131]
2019	Numerical solutions for time-fractional cancer invasion system with nonlocal diffusion	Not specified	—	Caputo	[120]
2019	Stability analysis and numerical simulations via fractional calculus for tumour dormancy models	Not specified	—	Caputo	[132]
2019	A study on discrete and discrete fractional pharmacokinetics pharmacodynamics models of tumour growth and anti-cancer effects	Colon carcinoma	Not specified	Rieman–Liouville	[133]
2020	On multistep tumour growth models of fractional variable-order	Breast cancer	—	Caputo	[121]
2020	Can fractional calculus help improve tumour growth models?	Breast cancer	—	Caputo	[56]
2020	Mathematical modelling of cancer and hepatitis co-dynamics with non-local and non-singular kernel	Co-infection of cancer & hepatitis	—	ABC	[134]
2020	Mathematical modelling of radiotherapy cancer treatment using Caputo fractional derivative	Uterine cervical cancer	Radiotherapy	Caputo	[135]
2021	Memory effects on the proliferative function in the cycle-specific of chemotherapy	Breast & ovarian cancer	Chemotherapy	Caputo	[136]
2021	A mathematical model and numerical solution for brain tumour derived using fractional operator	Glioblastoma	Chemotherapy and Surgery	Caputo	[137]
2021	A fractional modelling of tumour-immune system interaction related to lung cancer with real data	Lung cancer	—	Caputo	[138]
2021	Comparison of fractional-order and integer-order cancer tumour growth models: an inverse approach	Prostate cancer	Chemotherapy	Caputo	[139]
2022	Modelling and analysis of breast cancer with adverse reactions of chemotherapy treatment through fractional derivative	Breast cancer	Chemotherapy	Caputo–Fabrizio	[124]
2022	The dynamics of a fractional-order mathematical model of cancer tumour disease	Not specified	Chemotherapy	Caputo	[123]
2022	Modelling the dynamics of tumour-immune cells interactions via fractional calculus	Not specified	Immunotherapy	Caputo	[140]

highlights and gathers 22 significant progresses registered in the last 9 years of literature in cancer research, as modelled through FC. The selection criteria was proceeded so as to provide a sufficiently wide-ranging representation of different fractional operators’ applications, across a variety of treatments and cancer types. Special emphasis is given to the Caputo definition, since it is the one most often resorted to, notably due to its suitability in dealing with initial-valued problems in natural phenomena modelling [18,121], and with variants such as Caputo–Fabrizio revealing to be a revolutionary fractional operator, given its nonsingular kernel, piquing the interest of scholars [124]. Examples of other lesser used operators in oncological modelling such as Grünwald–Letnikov and Hadamard are also present. While the following is not a systematic review, certain keywords resorted to in the search were (“non-integer” OR “fractional” OR “variable order” OR “Caputo” OR “Grünwald-Letnikov” OR “Riemann-Liouville” OR “Hadamard” OR “ABC”) AND (“mathematical oncology” OR ((“cancer” OR “tumour”) AND (“mathematical modelling” OR “differential equation(s)”))). Searches were furthermore complemented by citation searching. Finally, some entries are adapted from the overview conducted in [74].

From the wide array of contributions compiled in Table 2, common features, methodologies and conclusions may be traced. Regarding cancer dynamics, beyond the cancer types themselves, articles such as [127,132,138,140] rely on the modelling of tumour-immune interactions through FDEs. They thus consider the mutual effect of a given tumour’s cells with the immune system, represented by its effector cells (e.g., cytotoxic T-cells, natural killer cells and possibly assuming interleukin-2 concentration [127,140]—or macrophages and host cells [138]), and in cases adapt prey-predator dynamics [132]. However, most mathematical models of cancer in Table 2 either approach the tumour alone through, e.g., modelling its volumetric growth [56,121], or consider merely tumour-immune interactions with the possibility of treatment intervention, with few others considering populations beyond (e.g., fat cells [71]). Moreover, the majority of the models presented operate in the spatial scale of the tissue level. Other contributions concerning the modelling of specific treatment drug effects feature PK/PD models instead [128,130,133].

As for the mathematical techniques employed, most selected contributions conceive FDE-based models of cancer dynamics [56,127,128,132,134,135,136,139,140], with the fractional operator most resorted to being Caputo’s, as noted in Table 2. On the other hand, the use of Fractional Calculus when modelling anomalous diffusive processes occurring in oncological phenomena is conducted through PDE-based models [120,123,125,126,129,137]. Furthermore, the possibility of a shifting arbitrary order (whether time-variable or not) is made possible with VODE-based cancer models, as expressed in contributions [121,130]. In spite of these differences, the majority of the aforementioned mathematical models assume time-fractional differential equations [56,120,121,124,125,126,127,128,129], thus endowing the cancer dynamics with memory effects as granted by a positive, fractional $\alpha$, and hence not challenging the spatially isotropic nature of these models, when diffusion through PDEs is concerned. Additionally, regarding parameterisation, seldom articles openly state data fitting procedures (e.g., least squares curve fitting method (LSCFM) [138]) using experimentally obtained clinical data [56,121,133,135,138], standing out as the ones with the methodological prospects to be classified as translational, Mathematical Oncology-pertinent models toward support in clinical decision making and prediction. At last, in spite of their academic nature, certain mathematical models conduct an optimisation of either the fractional order $\alpha$ that best renders treatment outcomes [126] or of the treatments themselves: chemotherapy [136] or mixed chemotherapy and immunotherapy doses [71].

Despite all contributions from Table 2 depicting the application of Fractional Calculus in the mathematical modelling of cancer, the studies on the obtained models vary widely, ranging from theoretically-veered mathematical investigations to the aforementioned conception of translational data-fitted models. On the former, the existence and uniqueness (E&U) of solutions is established through fixed point theory [124,131,134,138,140], with [120] adapting the Faedo-Galerkin approximation method for this effect. Equilibria and stability of the fractional differential cancer models is moreover proceeded (e.g., [134]), with cases of stability of finite difference scheme [71,126], local stability through Matignon’s theorem [127,132,138] (considering Routh-Hurwitz conditions) or Ulam-Hyers stability [140]. Furthermore, numerical solutions are computed through approximations enabled by methods such as: implicit finite difference method [126]; Euler approximation [127,128]; Legendre wavelet Galerkin method [129]; nonstandard finite difference (NSFD) method [132]; Adams-Moulton rule [134,138]; Bernoulli polynomials [137]; Adams-Bashforth [124] and Adams-Bashforth-Moulton predictor-corrector [138]; q-homotopy analysis method (q-HAM) [125]; Adomian decomposition method [139]; or reduced differential transform method (RDTM) [123].

Lastly, ultimately hypothesising the role of Fractional Calculus as a modelling tool in this research branch, several contributions in Table 2 claim that its usage provides interesting and new dynamics’ description into the fold. Indeed, the employment of FDEs has been shown to enrich the dynamics of tumour-immune interactions while increasing complexity of observed behaviours [127], and also its order plays an essential role on the stability of a system’s equilibria [132]. The order’s association with treatment procedures such as fractioned dose of radiotherapy in [135] yielded a model that can simulate radiotherapy process and predict results of other radiation protocols. In research resorting to experimental data, the study of a VODE-based model in [121] pointed to the superiority of the obtained model in describing such data, in this sense enabling new perspectives in tumour growth modelling; moreover, the application of FC adjoined with real non-small cell lung cancer data in [138] led to the stating that with the aid of this modelling tool, cancer’s different mode of progression when it starts to appear in the body can be better explained. In the context of PDEs, positive prospects have been registered in [123,129], in which the time-fractional derivative has been recognised as pivotal in the former to the success of modelling hyperthermia treatment for metastatic cancerous cells, making its its prediction precise, while the latter highlights its importance in achieving a better understanding of chemotherapeutic effects. Furthermore, contributions such as [128] claim that fractional modelling of PK/PD models provides insight into certain drug dynamics which would not be captured by classical techniques of ordinary differentiation.

As for the memory effect/anisotropic potential natural to fractional differentiation, the authors of [127] theorised that the fractional order could play the role of the cancer system’s memory. Indeed, as concluded in [56], the knowledge that tumours are constituted by cells which accumulate an array of mutations along their evolution, it is sound that fractional models’ account of non-local/past events is oncologically pertinent. FC’s application through FDEs to model the time-evolution of proliferating and quiescent cell masses in breast and ovarian cancer in [136] emphasises this advantage of the memory effect, in which it is claimed that FDE models are appropriate to the real-life problems since it gives important information on the proliferative function, with the model granting some qualitative insight on how to better the implementation of cycle-specific chemotherapy. Still along this notion, the VODE cancer study of [121] posited that a time dependent $\alpha \left(t\right)$ could be interpreted as a variable memory index of cancer, a key translational reading of its data-based results. Despite the novelty of these findings, they are still early albeit promising prospects of application of Fractional Calculus in oncology, and the current literature is still somewhat scarce in the systematisation and surveying of its many possible employments. Moreover, given the insufficient existent data, most attempts at cancer modelling thus still merely enable an academic, theoretical outlook on the matter.

4. Concluding Remarks

With the rise of mathematical modelling techniques into the fold of cancer research, it becomes ever more urgent to grasp the multiplicity of the models employed by discerning thorough categories pinpointing their modelling assumptions and approach, cancer features tackled, data availability, spatial scale and involvement of treatments. In a field initially prompted by the two main goals of applying mathematics in improving oncological knowledge and simultaneously advance the very mathematical tools it employs, it has become evident that the past years of mathematical oncology research have engendered at times confusing purposes for the cancer models developed, with some contributions adhering to literature parameters, conflicting experimental data and theoretical assumptions that disable their mathematical models as sound predictive treatment tools for clinical decision support and optimisation. In this context, the necessity to devise pertinent frameworks and distinctions between “academic” (Oncological Mathematics) and “translational” (Mathematical Oncology) models is revised and highlighted in this article.

In spite (or even because) of this general gap between the academically and the translationally relevant in the modelling sphere, new mathematical techniques have been sought after with the hopes to enhance pre-existing mathematical models and modelling frameworks, with one of the most novel tools being those offered by Fractional Calculus. Indeed, FC’s main purpose—to generalise integer order differentiation and integration to a fractional, irrational or even complex arbitrary order $\alpha$—has already shown positive modelling results in areas like signal and image processing, cryptocurrencies, epidemic spread of diseases, highway traffic flow and viscoelasticity. However, it is in the modelling of biological phenomena in which FC has made a bigger impact, since its non-locality/memory effect considers the weight of a function’s past values in the computation of its fractional derivative, thus aligned with real-life biological events. Given this, the employment of FC in the modelling of cancer (e.g., replacing ordinary differential equations or refining partial differential equations) has been a reasonable endeavour. To illustrate this current research landscape of a prospective Fractional Mathematical Oncology, several contributions were selected and reviewed, in which it was concluded that while there is a lack of models conceived with/from clinical data, the qualitative modelling of phenomena such as tumour growth and diffusion has revealed valuable insights in understanding the complexities of this disease.