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Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students

Roles Conceptualization, Investigation, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliations Department of Biological Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada, Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada

Affiliation Department of Biology, Memorial University of Newfoundland, St John’s, Newfoundland, Canada

• Korryn Bodner, 
• Chris Brimacombe, 
• Emily S. Chenery, 
• Ariel Greiner, 
• Anne M. McLeod, 
• Stephanie R. Penk, 
• Juan S. Vargas Soto

Published: January 14, 2021

• https://doi.org/10.1371/journal.pcbi.1008539
• Reader Comments

Citation: Bodner K, Brimacombe C, Chenery ES, Greiner A, McLeod AM, Penk SR, et al. (2021) Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students. PLoS Comput Biol 17(1): e1008539. https://doi.org/10.1371/journal.pcbi.1008539

Editor: Scott Markel, Dassault Systemes BIOVIA, UNITED STATES

Copyright: © 2021 Bodner et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Biologists spend their time studying the natural world, seeking to understand its various patterns and the processes that give rise to them. One way of furthering our understanding of natural phenomena is through laboratory or field experiments, examining the effects of changing one, or several, variables on a measured response. Alternatively, one may conduct an observational study, collecting field data and comparing a measured response along natural gradients. A third and complementary way of understanding natural phenomena is through mathematical models. In the life sciences, more scientists are incorporating these quantitative methods into their research. Given the vast utility of mathematical models, ranging from providing qualitative predictions to helping disentangle multiple causation (see Hurford [ 1 ] for a more complete list), their increased adoption is unsurprising. However, getting started with mathematical models may be quite daunting for those with traditional biological training, as in addition to understanding new terminology (e.g., “Jacobian matrix,” “Markov chain”), one may also have to adopt a different way of thinking and master a new set of skills.

Here, we present 10 simple rules for tackling your first mathematical models. While many of these rules are applicable to basic scientific research, our discussion relates explicitly to the process of model-building within ecological and epidemiological contexts using dynamical models. However, many of the suggestions outlined below generalize beyond these disciplines and are applicable to nondynamic models such as statistical models and machine-learning algorithms. As graduate students ourselves, we have created rules we wish we had internalized before beginning our model-building journey—a guide by graduate students, for graduate students—and we hope they prove insightful for anyone seeking to begin their own adventures in mathematical modelling.

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Boxes represent susceptible, infected, and recovered compartments, and directed arrows represent the flow of individuals between these compartments with the rate of flow being controlled by the contact rate, c , the probability of infection, γ , and the recovery rate, θ .

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Rule 1: Know your question

“All models are wrong, some are useful” is a common aphorism, generally attributed to statistician George Box, but determining which models are useful is dependent upon the question being asked. The practice of clearly defining a research question is often drilled into aspiring researchers in the context of selecting an appropriate research design, interpreting statistical results, or when outlining a research paper. Similarly, the practice of defining a clear research question is important for mathematical models as their results are only as interesting as the questions that motivate them [ 5 ]. The question defines the model’s main purpose and, in all cases, should extend past the goal of merely building a model for a system (the question can even answer whether a model is even necessary). Ultimately, the model should provide an answer to the research question that has been proposed.

When the research question is used to inform the purpose of the model, it also informs the model’s structure. Given that models can be modified in countless ways, providing a purpose to the model can highlight why certain aspects of reality were included in the structure while others were ignored [ 6 ]. For example, when deciding whether we should adopt a more realistic model (i.e., add more complexity), we can ask whether we are trying to inform general theory or whether we are trying to model a response in a specific system. For example, perhaps we are trying to predict how fast an epidemic will grow based on different age-dependent mixing patterns. In this case, we may wish to adapt our basic SIR model to have age-structured compartments if we suspect this factor is important for the disease dynamics. However, if we are exploring a different question, such as how stochasticity influences general SIR dynamics, the age-structured approach would likely be unnecessary. We suggest that one of the first steps in any modelling journey is to choose the processes most relevant to your question (i.e., your hypothesis) and the direct and indirect causal relationships among them: Are the relationships linear, nonlinear, additive, or multiplicative? This challenge can be aided with a good literature review. Depending on your model purpose, you may also need to spend extra time getting to know your system and/or the data before progressing forward. Indeed, the more background knowledge acquired when forming your research question, the more informed your decision-making when selecting the structure, parameters, and data for your model.

Rule 2: Define multiple appropriate models

Natural phenomena are complicated to study and often impossible to model in their entirety. We are often unsure about the variables or processes required to fully answer our research question(s). For example, we may not know how the possibility of reinfection influences the dynamics of a disease system. In cases such as these, our advice is to produce and sketch out a set of candidate models that consider alternative terms/variables which may be relevant for the phenomena under investigation. As in Fig 2 , we construct 2 models, one that includes the ability for recovered individuals to become infected again, and one that does not. When creating multiple models, our general objective may be to explore how different processes, inputs, or drivers affect an outcome of interest or it may be to find a model or models that best explain a given set of data for an outcome of interest. In our example, if the objective is to determine whether reinfection plays an important role in explaining the patterns of a disease, we can test our SIR candidate models using incidence data to determine which model receives the most empirical support. Here we consider our candidate models to be alternative hypotheses, where the candidate model with the least support is discarded. While our perspective of models as hypotheses is a view shared by researchers such as Hilborn and Mangel [ 7 ], and Penk and colleagues [ 8 ], please note that others such as Oreskes and colleagues [ 9 ] believe that models are not subject to proof and hence disagree with this notion. We encourage modellers who are interested in this debate to read the provided citations.

(A) A susceptible/infected/recovered model where individuals remain immune (gold) and (B) a susceptible/infected/recovered model where individuals can become susceptible again (blue). Arrows indicate the direction of movement between compartments, c is the contact rate, γ is the infection rate given contact, and θ is the recovery rate. The text below each conceptual model are the hypotheses ( H1 and H2 ) that represent the differences between these 2 SIR models.

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Finally, we recognize that time and resource constraints may limit the ability to build multiple models simultaneously; however, even writing down alternative models on paper can be helpful as you can always revisit them if your primary model does not perform as expected. Of course, some candidate models may not be feasible or relevant for your system, but by engaging in the activity of creating multiple models, you will likely have a broader perspective of the potential factors and processes that fundamentally shape your system.

Rule 3: Determine the skills you will need (and how to get them)

Equipping yourself with the necessary analytical tools that form the basis of all quantitative techniques is essential. As Darwin said, those that have knowledge of mathematics seem to be endowed with an extra sense [ 10 ], and having a background in calculus, linear algebra, and statistics can go a long way. Thus, make it a habit to set time for yourself to learn these mathematical skills, and do not treat all your methods like a black box. For instance, if you plan to use ODEs, consider brushing up on your calculus, e.g., using Stewart [ 11 ]. If you are working with a system of ODEs, also read up on linear algebra, e.g., using Poole [ 12 ]. Some universities also offer specialized math biology courses that combine topics from different math courses to teach the essentials of mathematical modelling. Taking these courses can help save time, and if they are not available, their syllabi can help focus your studying. Also note that while narrowing down a useful skillset in the early stages of model-building will likely spare you from some future headaches, as you progress in your project, it is inevitable that new skills will be required. Therefore, we advise you to check in at different stages of your modelling journey to assess the skills that would be most relevant for your next steps and how best to acquire them. Hopefully, these decisions can also be made with the help of your supervisor and/or a modelling mentor. Building these extra skills can at first seem daunting but think of it as an investment that will pay dividends in improving your future modelling work.

When first attempting to tackle a specific problem, find relevant research that accomplishes the same tasks and determine if you understand the processes and techniques that are used in that study. If you do, then you can implement similar techniques and methods, and perhaps introduce new methods. If not, then determine which tools you need to add to your toolbox. For instance, if the problem involves a system of ODEs (e.g., SIR models, see above), can you use existing symbolic software (e.g., Maple, Matlab, Mathematica) to determine the systems dynamics via a general solution, or is the complexity too great that you will need to create simulations to infer the dynamics? Figuring out questions like these is key to understanding what skills you will need to work with the model you develop. While there is a time and a place for involving collaborators to help facilitate methods that are beyond your current reach, we strongly advocate that you approach any potential collaborator only after you have gained some knowledge of the methods first. Understanding the methodology, or at least its foundation, is not only crucial for making a fruitful collaboration, but also important for your development as a scientist.

Rule 4: Do not reinvent the wheel

While we encourage a thorough understanding of the methods researchers employ, we simultaneously discourage unnecessary effort redoing work that has already been done. Preventing duplication can be ensured by a thorough review of the literature (but note that reproducing original model results can advance your knowledge of how a model functions and lead to new insights in the system). Often, we are working from established theory that provides an existing framework that can be applied to different systems. Adapting these frameworks can help advance your own research while also saving precious time. When digging through articles, bear in mind that most modelling frameworks are not system-specific. Do not be discouraged if you cannot immediately find a model in your field, as the perfect model for your question may have been applied in a different system or be published only as a conceptual model. These models are still useful! Also, do not be shy about reaching out to authors of models that you think may be applicable to your system. Finally, remember that you can be critical of what you find, as some models can be deceptively simple or involve assumptions that you are not comfortable making. You should not reinvent the wheel, but you can always strive to build a better one.

Rule 5: Study and apply good coding practices

The modelling process will inevitably require some degree of programming, and this can quickly become a challenge for some biologists. However, learning to program in languages commonly adopted by the scientific community (e.g., R, Python) can increase the transparency, accessibility, and reproducibility of your models. Even if you only wish to adopt preprogrammed models, you will likely still need to create code of your own that reads in data, applies functions from a collection of packages to analyze the data, and creates some visual output. Programming can be highly rewarding—you are creating something after all—but it can also be one of the most frustrating parts of your research. What follows are 3 suggestions to avoid some of the frustration.

Organization is key, both in your workflow and your written code. Take advantage of existing software and tools that facilitate keeping things organized. For example, computational notebooks like Jupyter notebooks or R-Markdown documents allow you to combine text, commands, and outputs in an easily readable and shareable format. Version control software like Git makes it simple to both keep track of changes as well as to safely explore different model variants via branches without worrying that the original model has been altered. Additionally, integrating with hosting services such as Github allows you to keep your changes safely stored in the cloud. For more details on learning to program, creating reproducible research, programming with Jupyter notebooks, and using Git and Github, see the 10 simple rules by Carey and Papin [ 13 ], Sandve and colleagues [ 14 ], Rule and colleagues [ 15 ], and Perez-Riverol and colleagues [ 16 ], respectively.

Comment your code and comment it well (see Fig 3 ). These comments can be the pseudocode you have written on paper prior to coding. Assume that when you revisit your code weeks, months, or years later, you will have forgotten most of what you did and why you did it. Good commenting can also help others read and use your code, making it a critical part of increasing scientific transparency. It is always good practice to write your comments before you write the code, explaining what the code should do. When coding a function, include a description of its inputs and outputs. We also encourage you to publish your commented model code in repositories such that they are easily accessible to others—not only to get useful feedback for yourself but to provide the modelling foundation for others to build on.

Two functionally identical codes in R [ 17 ] can look very different without comments (left) and with descriptive comments (right). Writing detailed comments will help you and others understand, adapt, and use your code.

https://doi.org/10.1371/journal.pcbi.1008539.g003

When writing long code, test portions of it separately. If you are writing code that will require a lot of processing power or memory to run, use a simple example first, both to estimate how long the project will take, and to avoid waiting 12 hours to see if it works. Additionally, when writing code, try to avoid too many packages and “tricks” as it can make your code more difficult to understand. Do not be afraid of writing 2 separate functions if it will make your code more intuitive. As with writing, your skill as a writer is not dependent on your ability to use big words, but instead about making sure your reader understands what you are trying to communicate.

Rule 6: Sweat the “right” small stuff

By “sweat the ‘right’ small stuff,” we mean considering the details and assumptions that can potentially make or break a mathematical model. A good start would be to ensure your model follows the rules of mass and energy conservation. In a closed system, mass and energy cannot be created nor destroyed, and thus, the left side of the mathematical equation must equal the right under all circumstances. For example, in Eq 2 , if the number of susceptible individuals decreases due to infection, we must include a negative term in this equation (− cγIS ) to indicate that loss and its conjugate (+ cγIS ) to the infected individuals equation, Eq 3 , to represent that gain. Similarly, units of all terms must also be balanced on both sides of the equation. For example, if we wish to add or subtract 2 values, we must ensure their units are equivalent (e.g., cannot add day −1 and year −1 ). Simple oversights in units can lead to major setbacks and create bizarre dynamics, so it is worth taking the time to ensure the units match up.

Modellers should also consider the fundamental boundary conditions of each parameter to determine if there are some values that are illogical. Logical constraints and boundaries can be developed for each parameter using prior knowledge and assumptions (e.g., Huntley [ 18 ]). For example, when considering an SIR model, there are 2 parameters that comprise the transmission rate—the contact rate, c , and the probability of infection given contact, γ . Using our intuition, we can establish some basic rules: (1) the contact rate cannot be negative; (2) the number of susceptible, infected, and recovered individuals cannot be below 0; and (3) the probability of infection given contact must fall between 0 and 1. Keeping these in mind as you test your model’s dynamics can alert you to problems in your model’s structure. Finally, simulating your model is an excellent method to obtain more reasonable bounds for inputs and parameters and ensure behavior is as expected. See Otto and Day [ 5 ] for more information on the “basic ingredients” of model-building.

Rule 7: Simulate, simulate, simulate

Even though there is a lot to be learned from analyzing simple models and their general solutions, modelling a complex world sometimes requires complex equations. Unfortunately, the cost of this complexity is often the loss of general solutions [ 19 ]. Instead, many biologists must calculate a numerical solution, an approximate solution, and simulate the dynamics of these models [ 20 ]. Simulations allow us to explore model behavior, given different structures, initial conditions, and parameters ( Fig 4 ). Importantly, they allow us to understand the dynamics of complex systems that may otherwise not be ethical, feasible, or economically viable to explore in natural systems [ 21 ].

Gold lines represent the SIR structure ( Fig 2A ) where lifelong immunity of individuals is inferred after infection, and blue lines represent an SIRS structure ( Fig 2B ) where immunity is lost over time. The solid lines represent model dynamics assuming a recovery rate ( θ ) of 0.05, while dotted lines represent dynamics assuming a recovery rate of 0.1. All model runs assume a transmission rate, cγ , of 0.2 and an immunity loss rate, ψ , of 0.01. By using simulations, we can explore how different processes and rates change the system’s dynamics and furthermore determine at what point in time these differences are detectable. SIR, Susceptible-Infected-Recovered; SIRS, Susceptible-Infected-Recovered-Susceptible.

https://doi.org/10.1371/journal.pcbi.1008539.g004

One common method of exploring the dynamics of complex systems is through sensitivity analysis (SA). We can use this simulation-based technique to ascertain how changes in parameters and initial conditions will influence the behavior of a system. For example, if simulated model outputs remain relatively similar despite large changes in a parameter value, we can expect the natural system represented by that model to be robust to similar perturbations. If instead, simulations are very sensitive to parameter values, we can expect the natural system to be sensitive to its variation. Here in Fig 4 , we can see that both SIR models are very sensitive to the recovery rate parameter ( θ ) suggesting that the natural system would also be sensitive to individuals’ recovery rates. We can therefore use SA to help inform which parameters are most important and to determine which are distinguishable (i.e., identifiable). Additionally, if observed system data are available, we can use SA to help us establish what are the reasonable boundaries for our initial conditions and parameters. When adopting SA, we can either vary parameters or initial conditions one at a time (local sensitivity) or preferably, vary multiple of them in tandem (global sensitivity). We recognize this topic may be overwhelming to those new to modelling so we recommend reading Marino and colleagues [ 22 ] and Saltelli and colleagues [ 23 ] for details on implementing different SA methods.

Simulations are also a useful tool for testing how accurately different model fitting approaches (e.g., Maximum Likelihood Estimation versus Bayesian Estimation) can recover parameters. Given that we know the parameter values for simulated model outputs (i.e., simulated data), we can properly evaluate the fitting procedures of methods when used on that simulated data. If your fitting approach cannot even recover simulated data with known parameters, it is highly unlikely your procedure will be successful given real, noisy data. If a procedure performs well under these conditions, try refitting your model to simulated data that more closely resembles your own dataset (i.e., imperfect data). If you know that there was limited sampling and/or imprecise tools used to collect your data, consider adding noise, reducing sample sizes, and adding temporal and spatial gaps to see if the fitting procedure continues to return reasonably correct estimates. Remember, even if your fitting procedures continue to perform well given these additional complexities, issues may still arise when fitting to empirical data. Models are approximations and consequently their simulations are imperfect representations of your measured outcome of interest. However, by evaluating procedures on perfectly known imperfect data, we are one step closer to having a fitting procedure that works for us even when it seems like our data are against us.

Rule 8: Expect model fitting to be a lengthy, arduous but creative task

Model fitting requires an understanding of both the assumptions and limitations of your model, as well as the specifics of the data to be used in the fitting. The latter can be challenging, particularly if you did not collect the data yourself, as there may be additional uncertainties regarding the sampling procedure, or the variables being measured. For example, the incidence data commonly adopted to fit SIR models often contain biases related to underreporting, selective reporting, and reporting delays [ 24 ]. Taking the time to understand the nuances of the data is critical to prevent mismatches between the model and the data. In a bad case, a mismatch may lead to a poor-fitting model. In the worst case, a model may appear well-fit, but will lead to incorrect inferences and predictions.

Model fitting, like all aspects of modelling, is easier with the appropriate set of skills (see Rule 2). In particular, being proficient at constructing and analyzing mathematical models does not mean you are prepared to fit them. Fitting models typically requires additional in-depth statistical knowledge related to the characteristics of probability distributions, deriving statistical moments, and selecting appropriate optimization procedures. Luckily, a substantial portion of this knowledge can be gleaned from textbooks and methods-based research articles. These resources can range from covering basic model fitting, such as determining an appropriate distribution for your data and constructing a likelihood for that distribution (e.g., Hilborn and Mangel [ 7 ]), to more advanced topics, such as accounting for uncertainties in parameters, inputs, and structures during model fitting (e.g., Dietze [ 25 ]). We find these sources among others (e.g., Hobbs and Hooten [ 26 ] for Bayesian methods; e.g., Adams and colleagues [ 27 ] for fitting noisy and sparse datasets; e.g., Sirén and colleagues [ 28 ] for fitting individual-based models; and Williams and Kendall [ 29 ] for multiobject optimization—to name a few) are not only useful when starting to fit your first models, but are also useful when switching from one technique or model to another.

After you have learned about your data and brushed up on your statistical knowledge, you may still run into issues when model fitting. If you are like us, you will have incomplete data, small sample sizes, and strange data idiosyncrasies that do not seem to be replicated anywhere else. At this point, we suggest you be explorative in the resources you use and accept that you may have to combine multiple techniques and/or data sources before it is feasible to achieve an adequate model fit (see Rosenbaum and colleagues [ 30 ] for parameter estimation with multiple datasets). Evaluating the strength of different techniques can be aided by using simulated data to test these techniques, while SA can be used to identify insensitive parameters which can often be ignored in the fitting process (see Rule 7).

Model accuracy is an important metric but “good” models are also precise (i.e., reliable). During model fitting, to make models more reliable, the uncertainties in their inputs, drivers, parameters, and structures, arising due to natural variability (i.e., aleatory uncertainty) or imperfect knowledge (i.e., epistemic uncertainty), should be identified, accounted for, and reduced where feasible [ 31 ]. Accounting for uncertainty may entail measurements of uncertainties being propagated through a model (a simple example being a confidence interval), while reducing uncertainty may require building new models or acquiring additional data that minimize the prioritized uncertainties (see Dietze [ 25 ] and Tsigkinopoulou and colleagues [ 32 ] for a more thorough review on the topic). Just remember that although the steps outlined in this rule may take a while to complete, when you do achieve a well-fitted reliable model, it is truly something to be celebrated.

Rule 9: Give yourself time (and then add more)

Experienced modellers know that it often takes considerable time to build a model and that even more time may be required when fitting to real data. However, the pervasive caricature of modelling as being “a few lines of code here and there” or “a couple of equations” can lead graduate students to hold unrealistic expectations of how long finishing a model may take (or when to consider a model “finished”). Given the multiple considerations that go into selecting and implementing models (see previous rules), it should be unsurprising that the modelling process may take weeks, months, or even years. Remembering that a published model is the final product of long and hard work may help reduce some of your time-based anxieties. In reality, the finished product is just the tip of the iceberg and often unseen is the set of failed or alternative models providing its foundation. Note that taking time early on to establish what is “good enough” given your objective, and to instill good modelling practices, such as developing multiple models, simulating your models, and creating well-documented code, can save you considerable time and stress.

Rule 10: Care about the process, not just the endpoint

As a graduate student, hours of labor coupled with relative inexperience may lead to an unwillingness to change to a new model later down the line. But being married to one model can restrict its efficacy, or worse, lead to incorrect conclusions. Early planning may mitigate some modelling problems, but many issues will only become apparent as time goes on. For example, perhaps model parameters cannot be estimated as you previously thought, or assumptions made during model formulation have since proven false. Modelling is a dynamic process, and some steps will need to be revisited many times as you correct, refine, and improve your model. It is also important to bear in mind that the process of model-building is worth the effort. The process of translating biological dynamics into mathematical equations typically forces us to question our assumptions, while a misspecified model often leads to novel insights. While we may wish we had the option to skip to a final finished product, in the words of Drake, “sometimes it’s the journey that teaches you a lot about your destination”.

There is no such thing as a failed model. With every new error message or wonky output, we learn something useful about modelling (mostly begrudgingly) and, if we are lucky, perhaps also about the study system. It is easy to cave in to the ever-present pressure to perform, but as graduate students, we are still learning. Luckily, you are likely surrounded by other graduate students, often facing similar challenges who can be an invaluable resource for learning and support. Finally, remember that it does not matter if this was your first or your 100th mathematical model, challenges will always present themselves. However, with practice and determination, you will become more skilled at overcoming them, allowing you to grow and take on even greater challenges.

Acknowledgments

We thank Marie-Josée Fortin, Martin Krkošek, Péter K. Molnár, Shawn Leroux, Carina Rauen Firkowski, Cole Brookson, Gracie F.Z. Wild, Cedric B. Hunter, and Philip E. Bourne for their helpful input on the manuscript.

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Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic

1 Laboratory of Systems Tumor Immunology, Comprehensive Cancer Center Erlangen and Deutsches Zentrum Immuntherapie (DZI), Department of Dermatology, FAU Erlangen-Nürnberg, Universitätsklinikum Erlangen, 91054 Erlangen, Germany; [email protected] (C.L.); [email protected] (X.L.); [email protected] (M.E.)

Christopher Lischer

Momchil nenov.

2 Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 4, 1113 Sofia, Bulgaria; [email protected] (M.N.); [email protected] (S.N.)

Svetoslav Nikolov

Martin eberhardt, associated data.

In most disciplines of natural sciences and engineering, mathematical and computational modelling are mainstay methods which are usefulness beyond doubt. These disciplines would not have reached today’s level of sophistication without an intensive use of mathematical and computational models together with quantitative data. This approach has not been followed in much of molecular biology and biomedicine, however, where qualitative descriptions are accepted as a satisfactory replacement for mathematical rigor and the use of computational models is seen by many as a fringe practice rather than as a powerful scientific method. This position disregards mathematical thinking as having contributed key discoveries in biology for more than a century, e.g., in the connection between genes, inheritance, and evolution or in the mechanisms of enzymatic catalysis. Here, we discuss the role of computational modelling in the arsenal of modern scientific methods in biomedicine. We list frequent misconceptions about mathematical modelling found among biomedical experimentalists and suggest some good practices that can help bridge the cognitive gap between modelers and experimental researchers in biomedicine. This manuscript was written with two readers in mind. Firstly, it is intended for mathematical modelers with a background in physics, mathematics, or engineering who want to jump into biomedicine. We provide them with ideas to motivate the use of mathematical modelling when discussing with experimental partners. Secondly, this is a text for biomedical researchers intrigued with utilizing mathematical modelling to investigate the pathophysiology of human diseases to improve their diagnostics and treatment.

1. Science: A World of Systems (and Models)

A good portion of science relies on the cascades of models that represent our reality at different spatial and temporal scales. A model is a simplified and often abstract representation of a complex natural system. Models are used in science to help understand, hypothesize about, or simulate the behavior of the natural systems they represent. Since a model is an abstraction of the natural system, it often includes only those elements, interactions, and processes of the system that are required to investigate the hypotheses in question. By exploiting the innate ability of the human brain to work with abstractions, models make highly complex phenomena accessible to study. In this sense, they can be used as a tool to develop hypotheses and to conceive and execute experiments to test those hypotheses. Models also provide an overview of the current knowledge on the natural system in question, facilitating the exchange of up-to-date information between researchers working on the same topic. Moreover, models can be used to simulate aspects of the natural system. A simulation is a rule-based recapitulation of the natural system’s behavior under relevant conditions using the model.

Since the features of a model are tightly linked to the purpose we want to give to it, there are multiple types of models. Models that are common to any branch of science are semantic models ; these consist in verbalization of the natural system’s features and the hypothesis using natural language. This is the type of model that underlies the results and discussion sections in most scientific papers.

More important for our discussion are the “lab bench” models and the “mathematical” models ( Figure 1 ). A lab bench model is a simplified or analogous version of a natural system employed in experiments under controlled conditions. In molecular and cell biology, lab bench models are usually units of life that can be conveniently propagated and studied in experiments to understand a given biological phenomenon. The consensus is that discoveries made via experimentation with the lab bench model provide insights into the behavior of similar phenomena in other organisms, especially humans. This is the case for cell lines, organoids, and mouse or rat strains with given genotypes or phenotypes, which have been used consistently in biomedicine as models for many human diseases.

Science employs different types of models to represent natural systems. Let us suppose that we are interested in investigating the properties and potential vulnerabilities of a melanoma metastasis (here, the “ natural system ”, visualized with MELC microscopy as in Ostalecki et al. 2017 [ 1 ]). One can represent the natural system with a semantic model , that is, the verbalization in natural language of the key compounds and processes as well as the hypotheses about a melanoma micrometastasis. Under some simplifying assumptions, a melanoma metastasis can be studied with lab bench models . For example, if one is interested in the interplay between cancer and immune cells, it is possible to co-culture tumor cells with relevant types of immune cells in vitro, like in Vescovi et al. 2019 [ 2 ]. In most cases, an alternative option is mathematical models , that is, sets of parametric equations that encode the key properties of metastasis and the hypotheses. The mathematical model is the basis for computational simulations to design experiments or to formulate or explore hypotheses like in Santos et al. 2016 [ 3 ].

A mathematical model is a set of parametric equations or other mathematical entities that encode the basic properties of the investigated natural system and that can be used to perform computational simulations. The same way that there are lab bench models with different features and purpose, there are different classes of mathematical models. One can classify them based on their treatment of the system’s dynamics as static models , which describe the system’s state at a point in time; comparative static models , which compare the properties of the system at different points in time; and dynamic models , which follow changes in the system over time. One can also classify models based on the mathematical apparatus they employ or the knowledge they exploit. There are mathematical models grounded in statistics that are used to process, analyze, and impute quantitative data generated in lab bench experiments or obtained from patient samples. All hypothesis tests and estimators of statistical correlation or inference between biological data sets are essentially (bio)statistical models.

However, there are also mathematical models that encode a mechanistic description of the natural system. This means the model equations and variables account for the interactions between the system’s key biological components and can be used for computational simulation of molecular and cellular processes. There are many subclasses of these models. A significant number of them reside in biophysics, developed to allow physically accurate simulations of atomic movement and molecular interactions in biomolecules including protein–protein interactions, DNA folding, or biomembrane dynamics. Other mechanistic models describe biochemical processes that shape cell phenotypes and cell-to-cell interactions. These are inspired by the mathematical models built in chemistry to understand and predict the kinetics of chemical reactions. In any case, just as one cannot elucidate all the mysteries of modern biomedicine with a single experimental technique, say confocal microscopy, a single subclass of mathematical model is not useful for every purpose. Every problem or hypothesis requires a carefully selected mathematical modelling approach.

2. The Scientific Method and the Role of Mathematical Modelling in It

Contemporary science consists in application of the scientific method and careful examination of the results one obtains with it. In the classical view, the scientific method is composed of three steps: observation of a natural phenomenon, elaboration of a hypothesis based on the observation, and design of an adequate experiment to test the hypothesis. If this is the way one would conceive scientific work nowadays, we would be strictly following the approaches employed by Galileo Galilei, Johannes Kepler, and Isaac Newton at the inception of modern science [ 4 ].

However, if you are a 21st century scientist, you are probably implementing a subtle variation of this method, which we will call the Einstein-grade scientific method. Here, the work is not performed in long-term isolation of a scientific ivory tower. Rather, the scientist continuously interacts with a community of peers. We retain the three basic steps mentioned above, but after the experimental test, one (more or less) immediately communicates the results in a booklet-length scientific publication. Early communication allows for your peers to try and reproduce your results. When experimental evidence accumulates and the underlying hypotheses are accepted, they are integrated in the scientific corpus of the field, that is, the set of theories, data, and hypotheses commonly accepted by the majority of researchers. Furthermore, new observations and experiments are continuously cross-checked against the scientific corpus in a way such that the method has an actual cyclic structure.

In the Einstein-grade scientific method, mathematical and computational modelling plays a pivotal role in many different manners ( Figure 2 ). Firstly, today’s “observations” in the majority of fields of natural sciences, including biomedicine, come as quantitative data that have to be processed, assessed, and analyzed with statistical models. In biomedicine, this is especially true in the case of sequencing data for the detection of common patterns in the sequence or the expression of genes in large cohorts of patients, a task impossible without sophisticated statistical and bioinformatics methods [ 5 ]. The features of these special mathematical models have a significant impact on the conclusions one can derive from sequencing data.

Sketch of the Einstein-grade scientific method (grey boxes) and the place that mathematical modelling occupies in it (blue boxes): the photograph of Albert Einstein is modified from the photo “Albert Einstein colorized” by Michael W. Gorth as stored in Wikimedia (CC-BY-SA-4.0, accession date: 10.08.2020; https://commons.wikimedia.org/wiki/File:Albert_Einstein_colourised_portrait.jpg ).

Secondly, computational models can be utilized to derive, ponder on, or substantiate hypotheses. Mathematical modelling has been employed to design validation experiments or to evaluate whether the results of the experiment agree with the expectations derived from the hypothesis and the scientific corpus . Finally, models can be used to achieve a formalized and unambiguous description of a field’s accepted knowledge in which theories, data, and hypotheses are organized and interconnected through equations. A remarkable example of this power is contained in a recent study by Meyer-Hermann and colleagues, in which a comprehensive mathematical model was used to summarize and elaborate around the contemporary knowledge on B-cell differentiation and maturation in germinal centers [ 6 ]. In one sense, Meyer-Hermann’s model is conceptually close to the equations that make up large physical theories, such as electromagnetism. In Supplementary Table S1 , we include and comment on select publications in which the different manners to deploy models are illustrated for biomedicine.

There are a few more issues to mention and discuss about the role of mathematical models in the scientific method. The formal language of a mathematical model enhances precision and clarity compared with natural-language descriptions in the semantic models usually preferred by biomedical experimentalists. Everybody with adequate training can develop an identical understanding of a statement formulated by an equation, for example, v = v max * S /( K M + S ). However, statements made in natural languages can be vague and can provoke misunderstandings. For example, is there a difference between “transcription factor X activates the expression of gene Y” and “transcription factor X promotes the expression of gene Y”?

The Einstein-grade method constitutes a collective endeavor. This means that it is common to find several researchers from different research institutions and locations and even with backgrounds complementing each other completing one cycle of the scientific method. For example, a German theoretical biologist, a sort of mathematical modeler, can publish a hypothesis based on his modelling efforts which will years later pique the interest of an Indonesian molecular biologist to perform experimental validation. Such occurrences are rather common in physics or chemistry, with a world record in some predictions made by Albert Einstein hundreds of years ago that have been experimentally validated only recently (which is why we use the term Einstein-grade here). However, they can also happen in biomedicine: Hodgkin and Huxley established in the first half of the 20th century a long-lasting scientific collaboration to elucidate the biophysical and biochemical mechanisms behind the initiation and propagation of action potential in nerve cell axons, a key phenomenon in understanding the ability of the nervous system to process and store information. The remarkable culmination of this collaboration was the amalgamation of their discoveries into a mathematical model which formalizes, explains, and quantifies the electrical excitability in nerve cells (see their seminal paper in Alan Hodgkin and Andrew Huxley 1952 [ 7 ] and a historical analysis in Schwiening 2012 [ 8 ]). Interestingly, some of the assumptions and simulations in the Hodgkin–Huxley model led Hodgkin’s team to predict already in 1955 that potassium ion channels can be occupied by multiple ions simultaneously. This prediction was confirmed using X-ray crystallography only in 1998, more than 40 years later, in a paradigmatic instance of the Einstein-grade scientific method in neurobiology [ 9 ].

Lastly, the scientific method is nowadays not monolithic and there are multiple variations of the general workflow commented on here [ 10 ]. This plurality of scientific methods may correspond to the different phases of discovery in diverse disciplines, the nature of the problems tackled, or even the styles of thinking belonging to different scientific communities. This notion applies also to mathematical modelling in biology and biomedicine [ 11 ].

3. The Love-and-Hate Relationship of Biology and Mathematics

The story of mathematics and modelling in biology is rather long, which might come as a surprise to many researchers. Biostatistical modelling and analysis have always been a key method in fields like evolutionary biology. In the beginning of the 20th century and right after the rediscovery of the Mendelian inheritance theory, there were many fundamental contributions to biology from statisticians that tried to bridge the gap between Mendelian genetics and Darwin’s theory of evolution. For example, the journal Biometrika [ 12 ] was established as early as 1901 by several founding fathers of modern statistics like Francis Galton and Karl Pearson with the intention of promoting biometrics, that is, the application of statistics to the analysis of biological data. Nowadays, it is difficult to dive into advanced concepts of evolutionary biology and population genetics without an understanding of mathematical modelling [ 13 ]. We also point out the seminal works of K.L. von Bertalanffy, P.A. Weiss, or M.D. Mesarovic in the application of the general systems theory to organisms dating back to the first part of 20th century (see Drack and Wolkenhauer [ 14 ] for detailed discussion).

In parallel to the search for the link between genetics and evolution theory, a new branch of science originating in physical chemistry was created by researchers interested in the dynamics of chemical reactions in living cells. This field became known as biochemistry, and already in 1913, biochemists made use of mathematical modelling to understand the mechanisms behind enzymatic catalysis. The Michaelis–Menten equation, taught in biochemistry courses throughout the world, is the first-ever mathematical model describing the dynamics of a biochemical reaction (see an updated translation of the original paper in Michaelis et al. 2011 [ 15 ]). We find in this field scientists like Jacques Monod and his team, who used mathematical modelling to understand sophisticated features of enzyme activity, such as allosteric regulation (see the seminal contribution in Monod et al. 1965 [ 16 ] and a 50-year retrospective analysis in Changeaux 2012 [ 17 ]). Fascinated by the elegance of these results, for several decades, a myriad of biochemists devoted their efforts to enzymology, that is, elucidation of the mechanisms of reaction, quantification, and modelling of the enzyme catalysis.

If mathematical modelling is necessary to immerse oneself in the connection between genes, phenotypes, and evolution as well as to understand the inner workings of catalytic proteins, how is it that certain modern biomedical researchers are reluctant to incorporate mathematical and theoretical approaches into their work? In the 1970s and 1980s, new experimental techniques were invented allowing the targeted mutation of selected genes [ 18 ]. These techniques became so fundamental in the hunt for the link between genes, proteins, and cellular functions that they led to the inception of modern molecular biology. An experimental approach based on targeted mutation followed by assessment of their effects has dominated the field since then. It is an approach that relies on advanced experimental skills, trial and error, intuition, and small-scale studies, and perhaps, that was what taught several generations of biomedical researchers that mathematics and formalized systems were somehow unnecessary in biology.

However, nowadays, two developments are turning this trend. On the one hand, newly discovered techniques for producing quantitative high-throughput data on whole classes of biomolecules (the omics revolution) required statistical methodologies for processing and analyzing these massive amounts of data. Additionally, mathematical and computational methods are indispensable in finding insights and connections between genes in these data [ 19 ]. This is the rationale behind medical genomics, the field that scans quantitative high-throughput data to find the genetic causation of diseases. On the other hand, there is mounting evidence that proteins and genes in cells do not work in isolation but rather organize into tightly interconnected networks which are often disturbed in pathological conditions [ 20 ]. These networks contain feedback loops, feedforward loops, or network hubs and gene regulatory circuits that can induce nonlinear behavior like homeostasis, self-sustained oscillations, or biostability [ 21 ]. There features, rather than anecdotic, are intrinsic and necessary to many vital cellular processes, e.g., the cell cycle [ 22 ].

The situation in 2020 is peculiar. We are rapidly conjuring biomedicine that necessarily relies more and more on quantitative high-throughput data, advanced statistics, bioinformatics, and computational modelling [ 23 ]. However, a significant fraction of its practitioners have insufficient mathematical and computational skills, probably ones worse than the generation that was initiated into the quantification of enzyme catalysis through kinetic equations and has recently retired. In line with this and to substantiate further discussion, we will now introduce the basics of mathematical modelling of biochemical networks.

4. A Primer on Mechanistic Modelling of Biochemical Systems

There are mathematical and computational models that encode a mechanistic description of the natural system in which equations and variables account for the interactions between the key biological components of the system, for example, signaling proteins and transcription factors. Among them, there is a family of models that are conceived to integrate the topology of biochemical networks and the kinetics of their molecular interactions with their ability to control cell phenotypes. These models are inspired by the models built in chemistry to understand and predict the kinetics of chemical reactions. These mathematical and computational models are formulated, characterized, and utilized following a well-established procedure common to several branches of physics, chemistry, and engineering, which we name here as the modelling workflow . In a nutshell, the workflow includes the following sequential operations ( Figure 3 ):

The modelling workflow in biomedicine: during model derivation, the biological knowledge and hypotheses about the studied system are encoded in a mathematical model. In model calibration, quantitative experimental data are added to characterize the mathematical model and to give values to the model parameters. In model validation, the ability of the model to make predictions is assessed by judging the agreement between new quantitative data and equivalent simulations of the calibrated model. In model analysis, a validated model is used to investigate the system using computer simulations or other tools like stability analysis.

Model derivation: Biomedical information from scientific literature is surveyed to select relevant biomolecules and interactions for the investigated hypothesis. With this information, a graphical depiction of the network of interacting molecules or cells is sketched. Under some formal or heuristic rules, a mathematical model is derived from the network graph. The mathematical model consists of mathematical equations (i.e., ordinary, partial differential, or integrodifferential equations) or other computational entities (i.e., Boolean-logic networks or Petri nets).

Model calibration: To ensure that the model mimics the behavior of the natural system in a given biological scenario, one has to attribute values to free parameters in the model. In some cases, it is possible to discern these from published quantitative data. More often, however, one has to design and perform biological experiments that produce adequate quantitative data. Later, the mathematical model and the quantitative data are integrated using a computational process which assigns optimal values to the model parameters while minimizing the mismatch between experimental observations and corresponding model simulations.

Model validation: The ability of the model to predict the system’s behavior is judged based on the alignment between quantitative data from a different experiment not used for calibration and the corresponding simulations of the calibrated model. A mismatch between data and simulation leads to reformulation of the hypothesis or the model’s structure, which is reflected in a modification of its mathematical equations and a re-iteration of the entire procedure.

Model analysis: A validated model can be used to design and perform predictive simulations, that is, simulations of the system’s behavior under new biological scenarios. This type of simulations has been successfully deployed to detect potential drug targets or to identify biomarkers for diagnosis in cancer and other multifactorial diseases (see Table S1 for selected examples). Furthermore, tools like stability analysis and bifurcation analysis can uncover nonlinear properties of the investigated network, delineating regions in the system’s phase space with distinctive stability or critical values of the model parameters provoking qualitative changes in the system’s behavior. Despite all the power that mathematical models bring to the table, however, it goes without saying that any prediction will require further experimental validation with lab bench models.

5. Frequent Unfounded Criticisms to Mathematical Modelling in Biomedicine

Now, we will list and discuss misconceptions about mathematical modelling in biomedicine which one can hear rather often when talking to experimental researchers in seminars and conferences (see also Table 1 for a summary).

Top misunderstandings on modelling and how to fight them.

5.1. Mathematical Models Cannot Reproduce the Complexity of Biology

When applied to biology in contrast to other natural sciences like physics or chemistry, this statement is a rather elaborate instance of magical thinking. Cells and physicochemical systems are governed by the same thermodynamic laws as any other natural system, laws that have been formulated in mathematical terms. The complexity of stellar systems’ dynamics is at least comparable with that of physiological systems, yet mathematical modelling is the standard tool to postulate hypotheses, to design experiments, and to formulate theories in astrophysics. The dynamics of the Earth’s atmosphere and climate are governed by the same laws of chemical kinetics and reaction-diffusion that apply to biochemical reactions. However, sophisticated mathematical modelling is behind the daily weather forecast or the recommendations of the Intergovernmental Panel on Climate Change. Besides nineteenth-century holistic thinking, there is not a single solid argument to support the notion that mathematical models cannot reproduce the complexity of biochemical and physiological systems.

5.2. Your Model Is Not Physiological. The Real System Is More Complex Than Your Mathematical Model

It may look like a softer version of the previous statement, but here, the emphasis is different. The idea is that modelling may be a valid option, but the current model does not contain sufficient detail to meaningfully represent the physiological context. When inspected more closely, this statement does not actually criticize mathematical modelling in particular but rather the fundamental act of utilizing any sort of model in biology. As we said above, any model, whether mathematical or experimental, is an abstraction that, rather than contemplating every detail of the system, includes the elements, interactions, and processes necessary to investigate the natural system under the hypothesis in question. This implies an intentional attempt at simplification by the researcher that is a shared feature of mathematical models and cell lines, organoids, or mouse models. This criticism can be countered by stating that the model must be as complex as necessary to capture the hypothesis, a notion equally valid for mathematical and lab bench models.

5.3. You Should Employ Data in Your Mathematical Model

A mathematical model in the sense discussed here always relies on quantitative data. As indicated above, model calibration is only possible with experimental data. Thus, a well-formulated mathematical model is based on quantitative data, which makes the above objection moot. Moreover, the interplay between model and quantitative data can come in different flavors. While usually one both calibrates and validates the model with data, one can also forego calibration in favor of analytical tools like stability analysis and can derive qualitative predictions about the system’s dynamics which are accessible to further experimental validation. Bar-or et al. 2000 [ 24 ] is a classic example of the ability of mathematical models to make qualitative predictions about the regulation of gene circuits from data. In the paper, the authors collected and synthesized all the information available at the time about the interplay between the TF p53 and its transcriptional target and repressor Mdm2 to hypothesize a) that they form a negative feedback loop gene circuit and b) that, under DNA damage, the system displayed oscillations in the expression of its components. They derived a qualitative mathematical model based on these hypotheses and found out that the model simulations predicted actual oscillations in p53 and Mdm2 levels in some experimental scenarios. They further validated this model-based prediction utilizing in vitro experiments. Interestingly, this “design principle” associated to oscillating gene circuits with TFs and their targets and repressors has been found in other master regulator TFs also by integrating quantitative data and mathematical modelling (see the case of NFkB in inflammation in Nelson et al. 2004 [ 25 ]).

5.4. Your Predictions Are Not Experimentally Validated

As explained above, once you perform an Einstein-grade version of the scientific method, it is not mandatory that a single paper conveys all the steps of the method. One team of researchers can formulate and investigate a hypothesis with mathematical modelling and simulations, and another team can follow up with validation experiments once technology and effort allow it (see the case of the neuronal action potential above). This is not to say that modelers are generally exempt from the need to validate, though. They should attempt to engage with experimental collaborators to see their own work come to fruition. An important aspect here is to provide ways of facilitating communication between modelers and experimentalists. In this sense, scientific papers on modelling and simulation should be written in a manner that allows the design of experiments to validate their hypotheses and predictions. Ultimately, this requires using common scientific language understandable for both mathematical modelers and experimental researchers.

5.5. I Do Not See the Clinical Relevance of Your Predictions

Very little of the scientific research in biomedicine delivers immediate clinical relevance, be it results obtained via mathematical modelling or through experimentation. However, at the same time, all basic research has an unpredictable long-term potential for enhancing clinical practice. To illustrate its potential, let us analyze a series of results obtained in the context of miRNA regulation and cancer. MicroRNAs exert posttranscriptional repression of selected gene targets [ 26 ] and play a pivotal role in some cell phenotypes subverted in cancer [ 27 ]. Lai et al. [ 28 ] utilized mathematical modelling to investigate the overarching hypothesis that different miRNAs can cooperate in the repression of some of their targets, a prediction that they experimentally validated for miR-572′s and miR-93′s joint repression of CDKNA1, a key cell cycle protein deregulated in cancer. In a continuation of this work, Schmitz et al. [ 29 ] used hybridization and molecular dynamics simulations of the binding of two cooperative miRNAs on their target mRNA to illustrate the general biophysical feasibility of this mechanism and to elucidate how it works at the molecular level. Moreover, they performed a human genome-wide exploration to systematically look for this type of joint miRNA regulation. Based on these results and further computational simulations, they hypothesized that miRNA cooperativity and its modelling can predict drug targets in cancer. They tested the hypothesis in a case study on cooperativity between miR-205-5p and miR-342-3p and its capacity to repress E2F1-mediated chemoresistance in cancer [ 30 ]. Their model-based prediction was confirmed experimentally. Finally, Lai et al. [ 31 ] extended this approach to the whole genome and systematically identified pairs of miRNAs that cooperatively target upregulated genes in metastatic melanoma. In summary, this series of interconnected papers illustrates how mathematical modelling can lead the way from hypothesis formulation and basic research to identifying potential clinical applications.

6. Rules to Build Mathematical Models That Can Be Understood by Experimentalists

To conclude, we elaborate on a few recommendations for biomedical modelers for when they conceive and implement their mathematical models, which hopefully will help bridge the cognitive chasm between them and experimentalists.

6.1. Know Your Problem

A good modeler should become an expert in the biomedical system that they plan to model. This is the best warranty that the structure and the hypotheses behind the mathematical model make sense and are consistent with the current biomedical knowledge. In addition, becoming an expert on the topic will help in choosing the right model assumptions, data, and hypotheses to be tested. Complementary to this, the best models emerge from constant interaction between competent biomodelers and experimental researchers. Remember, though, that collaboration is productive when communication is fluent, and this is only possible when a common language is spoken. In the present, this common language is the one that modelers need to learn when diving into the biology of the system they want to model. We refer again to Meyer-Hermann et al. 2012 [ 6 ] and their ability as modelers to acquire a deep understanding of B-cell biology and how they translated it into their mathematical model. However, we postulate that, in the long-term, biomedical researchers need in turn to rediscover the more precise language that math offers, which will also help them to quickly grasp advances in their own field of interest. There is an in-between methodology that could rekindle the growth of mathematical skills in experimentalists: network biology. In silico reconstruction, visualization, and modelling of intracellular biochemical networks provide a framework for connecting genes and molecules quantitatively to phenotypes and hence understanding the function and dynamics of cellular systems [ 32 ]. The network biology approach relies on mathematical concepts from graph theory, statistics, and mathematical modelling but is yet intuitive enough to allow a fluent discussion between wet- and dry-lab biologists. This has been, for example, advantageously used by yeast biologists to connect their experimentally detected biological interactions with their effect in cell phenotypes.

6.2. Select the Right Type of Mathematical Model, and Select It Early

The features of the mathematical model largely depend on the aim of the study; the scale and structural complexity of the investigated system; and the quantity, quality, and nature of the available experimental data. A model to investigate the nonlinearity associated with a feedback loop circuit has completely different requirements than a model of quantitative drug dosage in humans. This affects in particular the selection of the mathematical framework in which the model is derived and simulated. There is no single best modelling framework for every biomedical system or purpose, and therefore, the choice of model often relies on a trade-off between several requirements. We want to mention here (a) the computational demand and scalability, (b) the nature and necessary amount of calibration data, and (c) the way time and space are handled in the simulations. Sometimes, standard modelling frameworks are not suited for the problem in question and hybrid computational models of different types need to be considered (Chiam et al. 2006 illustrate how this type of hybridization can be done in the context of signaling pathways [ 33 ]; a discussion of this issue in the context of bacterial infection can be found in Cantone et al. 2017 [ 34 ]). When trying to facilitate the communication between modelers and experimentalists, one interesting approach is rule-based modelling. Compared to more math-heavy methods, rule-based modelling allows compact representations of reaction networks with a language-oriented structure; this makes them similar to semantic models and hence closer to the way of thinking of experimentalists (as a case study, see the epidermal growth factor receptor signaling network built in [ 35 ]).

6.3. Build on Preceding Efforts

To start the derivation of a mathematical model totally from scratch makes sense only when there is no alternative. A prudent modeler should reuse, extend, and adapt preexisting models when possible. In some cases, this will not be possible because the available models are based on different experimental conditions, formulated for a different biological scenario, or derived using an unsuitable modelling framework. In these cases, even if the model is found to be partially incompatible, its assessment will help judge the validity and portability of its assumptions and hypotheses in the context of one’s own modelling effort. If the problem lies in the modelling framework, it is sometimes worth translating the model into one’s chosen framework. In the ideal case, the model authors should have uploaded a fully annotated version of their model to a public repository (e.g., Biomodels [ 36 ]), which facilitates the work of incorporating the model into one’s own. Some tools even allow the semiautomatic translation of models from one formalism to another (see for example OdiFy [ 37 ]). As an example of this idea of building on preceding modelling efforts, in Csikász-Nagy 2009, one can find a comprehensive overview of the cascade of increasingly detailed mathematical models constructed since 1991 to understand the regulation of the cell cycle and how mathematical models are based on or have benefitted from the results obtained with previously developed models [ 38 ].

6.4. The Size Does Not Always Matter

There is nowadays a tendency to moon-shoot everything in biomedicine. This has translated also to biomodelling, and some researchers think that the quality of a mathematical model is measured in terms of the number of model variables as well as the required computational power and the complexity of the simulation algorithms. However, quality in modelling is primarily achieved through the biological precision of the assumptions encoded in the model equations. Thus, models can look simple and be small in terms of their number of equations but can actually possess the right features for the purpose of the specific modelling effort. This is somewhat similar to a BIC ballpoint pen. This is the simplest and cheapest ballpoint pen one can buy, but it actually displays a number of easy-to-overlook sophisticated features conceived to optimize its design in economic, ergonomic, and safety terms. One can also formulate BIC-like mathematical models, in which one gives priority to the description of the biological context and hypothesis and its planned utilization rather than to unnecessary size, complexity, or levels of detail ( Figure 4 ).

BIC pen-like mathematical models: prioritizing the purpose of the model instead of its detailedness . BIC pens look like the simplest and cheapest ballpoint pen one can buy, but their apparent simplicity conceals features conceived to optimize them in economic, ergonomic, and safety terms. In Santos and coworkers 2016 [ 3 ], a similar strategy was followed to build a mathematical model accounting for anticancer dendritic cell (DC) vaccination composed of only two ordinary differential equations, far simpler than other published models [ 41 ]. ( A ) Simplicity in design: BIC pens have a characteristically simple hexagonal structure; this apparently naïve choice significantly reduces the material consumption of the pen and minimizes the required space for storage. An important aspect to consider in DC vaccine modelling is the bioavailability of the cells after their injection. There are much elaborated models describing this process [ 42 ], but for our purpose, it was sufficient to model DC bioavailability with a cyclic piecewise linear function that mimics the known overall behavior of injected DCs. ( B ) Mathematics behind design principles: compared to standard circular pens, BIC pens hardly roll on the surface of a table. This feature was explicitly desired when drafting their design. In Santos et al., we wanted a simple enough model that was still able to mimic the interaction between the tumor and both innate and adaptive immunity; to this end, the model contained two nonlinear kinetic rates in a single equation, which are still able to mimic the basics of the interplay between the tumor and the two branches of immunity. ( C ) Ability to solve problems: in the end, simplicity has to be reconciled with effectiveness. The design of a BIC pen, for example, integrates more characteristics like minimizing the risk of suffocation when swallowing the cap. The predictions made in Santos et al. (2016) in terms of which phenotypic features sensitize the tumor to the therapy were aligned with patient data from clinical trials; furthermore, the model predicted alternative phenotypes that promote therapy resistance. The figures about DC vaccine modelling are adapted from Santos et al. 2016 under the conditions of an open access publication (CC BY 4.0). The figures about the BIC pen were inspired by the content of the webpage www.bicworld.com . Fight by doing: A route map to good mathematical modelling in biomedicine.

6.5. Set Your Results in Stone

One should prepare one’s own models in such a way that they can be understood and reused by other researchers, thereby closing the loop inherent to the scientific method. This is a frequently disregarded aspect in many fields in mathematical biomedicine, in which the number of modeler teams can be counted on two hands, and consequently, any model should have a high potential for reuse, adaptation, or integration. A model that is difficult to reproduce or understand by other modelers is guaranteed to collect virtual dust on its publication shelf. Thus, it is advantageous to implement equations and simulations in standard formats [ 39 ]. Further, whenever possible, the models and their supporting experimental data should be uploaded to repositories [ 36 ]. In closing, we emphasize the importance of the careful manual curation and annotation of mathematical models. In line with this, whenever possible, one should give preference to frameworks that allow for fast, simple, and standardized dissemination and model exchange [ 40 ].

7. The Best of Both Worlds—A Final Note on Mathematics, Models, Big Data, and Experimental Biology

The approach based on semantic and lab bench models has been very successful in past decades in describing many fundamental regulatory pathways in molecular biology. However, due to the massive amount of data produced nowadays, we are quickly reaching its limits. This is especially true (and dramatic) in the description of dynamic systems like heterogeneous cell populations, genomic regulation, or multifactorial diseases like cancer, for which this approach is clearly inadequate. There is no semantic solution to properly characterize multi-variable states, events, or diseases.

There is a clear understanding in the wet-lab biology community that advanced high throughput data analysis approaches are an urgent necessity for them, but in the opinion of many wet-lab biologists, data analysis and modelling are often conceived as a mere press of the button on computers that leads to publication-ready figures and plots within minutes. This has led to an entire industry producing “clickable” data analysis and modelling tools that make little demands on the knowledge of the operator. We think that this can lead to many problems, like reproducibility issues or the inability to critically judge the results produced by those tools, especially since most of them use proprietary algorithms. Additionally, the ability to generate an output from a purchased software or analysis suite might not automatically mean that this output is correct or sensible. Hence, an understanding of the underlying data and modelling theory is important and is often neglected in some wet-lab settings.

In the future, we as a community of biomedical researchers should strive to recognize that we need both sides of the coin, giving them equal weight in considerations like funding or time investment. Biology should orient itself towards the other two major natural science disciplines, physics and chemistry, and try to give their students a well-rounded education in mathematical understanding, data analysis, and modelling as well as in computer programming. Until that time, we need to make sure that we find ways to communicate between the two areas in a way that furthers productive collaboration.

Acknowledgments

Some of the misconceptions about modelling come from actual discussions with experimental and clinical researchers at the Department of Dermatology of the FAU Erlangen-Nürnberg, to whom we are grateful for their honest though at times scathing disagreement. Some parts of the text are based on a presentation given by J.V. at the Heidelberger Institut für Theoretische Studien (HITS), and others are from exchanges of ideas within the laboratory of Systems Tumor Immunology. J.V. also thanks Olaf Wolkenhauer (University of Rostock) for 15 years of continuous discussion on the foundations of systems biology.

Supplementary Materials

The following are available online at https://www.mdpi.com/1422-0067/22/2/547/s1 , Table S1: Real cases of mathematical modeling in biomedicine with additional bibliography and further reading.

Author Contributions

Initial draft: J.V.; Figures: J.V.; Final draft: J.V., M.E., C.L., S.N., M.N.; revision of the draft: J.V., X.L. All authors have read and agreed to the published version of the manuscript.

J.V.’s work in mathematical modelling applied to biomedicine has been generously funded over the years by the German Ministry of Education and Research (BMBF) through the initiatives FORSYS [CALSYS-FORSYS 0315264], e:Bio [e:Bio-miRSys 0316175A, e:Bio-MelEVIR 031L0073A], and e:Med [e:Med-CAPSyS 01ZX1304F, e:Med-MelAutim 01ZX1905A] and targeted initiatives in artificial intelligence [KI-VesD, 031L0244A] as well as by the German Research Foundation (DFG) [Ve642/1-1 in SPP1757] and the Bavarian Government [Gaminfection-UKER]. We also acknowledge support by Deutsche Forschungsgemeinschaft and Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding program Open Access Publishing.

Conflicts of Interest

The authors declare no conflict of interest.

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• Review Article
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• Published: 12 April 2021

Regenerative medicine meets mathematical modelling: developing symbiotic relationships

• S. L. Waters 1 ,
• L. J. Schumacher   ORCID: orcid.org/0000-0003-0797-3406 2 &
• A. J. El Haj   ORCID: orcid.org/0000-0003-3544-5678 3  

npj Regenerative Medicine volume  6 , Article number:  24 ( 2021 ) Cite this article

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• Stem-cell therapies
• Translational research

Successful progression from bench to bedside for regenerative medicine products is challenging and requires a multidisciplinary approach. What has not yet been fully recognised is the potential for quantitative data analysis and mathematical modelling approaches to support this process. In this review, we highlight the wealth of opportunities for embedding mathematical and computational approaches within all stages of the regenerative medicine pipeline. We explore how exploiting quantitative mathematical and computational approaches, alongside state-of-the-art regenerative medicine research, can lead to therapies that potentially can be more rapidly translated into the clinic.

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Introduction and vision.

The concept of using regenerative medicine approaches to repair and regenerate tissue damaged through disease or trauma has been maturing over the past few decades. Translation of regenerative therapies to the patient – bench-to-bedside – is one of the global multidisciplinary challenges of our time, offering a vision of new therapies with the power to address major unmet healthcare needs 1 .

Regenerative therapies can utilise progenitor or stem cells which are delivered to a repair site or area of degeneration to restore tissue structure and function. Regenerative therapies may also include a molecule or biomaterial based approach which promotes endogenous recruitment and tissue repair. Key examples of the challenges which regenerative therapies are facing include: choice of the best cell type from multiple sources, both autologous and allogeneic, or adult and embryonic 2 ; new ways for manufacturing therapeutic doses of donor stem cells which are characterised as Advanced Therapy Manufacturing Platforms (ATMPs); new enabling technologies using optical, sensing and mechanical tools for routine use to support scaled up cell production 3 ; novel biomaterials providing structural tissue mimics and instructive cues based on topography and protein chemistry; 3D tissue models grown in bioreactors or growth chambers presenting new ways for testing potential therapeutic strategies before implantation; optimising clinical choice and patient stratification using cell-based assays which aim to improve efficacy, and long term outcomes in patients.; and finally, patient monitoring at cell resolution using MRI, PET and multi-modal approaches which support efforts to move to clinical first in man and 1st stage trials 4 .

These broad challenges which aim to advance a new type of medicine have relied on multiple disciplines covering cell biology to material chemistry, enabling physics and clinical medicine. It is now being recognised that additional maths-based approaches may speed the advance of these therapies and enable them to reach the clinic faster. One example of such an approach is in silico modelling. This disruptive perspective can bring mathematicians into the pathway at many stages of the translation; mechanistic modelling studies enable acceleration of translational research by optimisation of protocols, new algorithms and statistics help to define our quantitative metrics and new data science and AI innovations expand our use of patient derived databases to optimise therapies.

A key challenge in integrating mathematical approaches into the regenerative medicine pathway is to identify where mathematical modelling can make the most disruptive impact. Mathematical modelling approaches can be much faster and cheaper than performing numerous time consuming and expensive laboratory experiments 5 . Embedding mathematics within regenerative medicine enables researchers to go beyond the usual trial-and-error approach, be guided in their experimental design, and therefore accelerate advances in regenerative medicine 6 . Mathematical models provide mechanistic insight into complex biological systems exhibiting richly non-linear behaviour, and predictions from mathematical models can be used to optimise protocols both for the manufacture of regenerative medicine products as well as for treatment strategies, e.g. the delivery of cell therapies. Mathematical models that predict the dynamic behaviour of the regenerative product, e.g. tissue growth during in vitro culture, can potentially be used as online monitoring tools to ensure the reproducibility and safety of manufactured products, addressing challenges in product regulation. Finally, bespoke patient models may be built in an individualised medicine approach and these models can be used to predict the efficacy of regenerative medicine strategies 7 .

Mathematical models have traditionally been used to provide mechanistic insight into the many interactions between the biological components of a system, for example enabling quantitative assessment of the cellular microenvironment that can then be manipulated to guide cell behaviour during development or growth. By systematically varying the parameters of the models, or by the addition of new components, we can perturb the model leading to new predictions and insights that can be used to overcome bottlenecks. For example, understanding gained from in vitro systems can be translated to the in vivo scenario through the inclusion of an immune component in the in silico models 8 . Models can also be used to “bridge the gap” between sub-disciplines by integrating multiple quantitative data sources such as imaging and molecular or biomechanical data, for example 9 .

A brief introduction to common modelling approaches

Multicellular, multiscale biological systems can often be too complex to understand by interrogating experimental and clinical data with verbal thinking and linear reasoning alone, thus the addition of theoretical or in silico models, expressed in the precise and powerful language of mathematics, can provide new and deeper insights. The key steps in the development of mathematical models are model construction, calibration, prediction, and refinement. We discuss the choices to be made in model construction at greater length below. Briefly, theoretical models may be phenomenological or mechanistic and describe biological processes at different scales: on the whole patient, organ, tissue scale, single-cell-level, and even the molecular level (Fig. 1 ).

Diagram which highlights the many opportunities for utilising mathematical and computational approaches within regenerative medicine. This figure was created for the authors by the University of Edinburgh’s graphic design service team.

A key aspect in the development of biologically realistic, predictive mathematical models, is interfacing mathematical models with experimental data. The calibration of models through comparison of model outputs with experimental data poses additional formidable challenges because the available data are usually complex, high-dimensional, noisy, and often incompletely observed. Comparing models with data is vital for parameter inference, which is the inverse problem of determining which parameter values are most likely to produce the observed data. Another reason to compare models with data is for the purpose of model selection, i.e. determining the level of model complexity required to interrogate a given set of experimental data, or deciding which model out of several competing hypotheses is more likely to be true. Once calibrated, the theoretical models are validated via detailed comparison of mathematical model predictions with experimental data. The use of predictions, whether on existing and withheld data, or predictions that are to be tested by newly generated data, are a key aspect of any mathematical modelling process. Any discrepancies between model predictions and experimental data can then lead to further model refinement. An iterative cycle of predict-test-refine is fundamental to the development of all models (See Fig. 2 ).

An illustrative diagram showing the quantitative regenerative medicine pipeline with stages of the modelling process and types of modelling involved.

The choice of modelling approach is guided by the biological question being asked, and the nature of the quantitative experimental data (see Table 2 ). Here we give a brief and broad categorisation of commonly used types of mathematical modelling in biology and medicine.

Mechanistic models: Mechanistic models represent all the components of a hypothesis (cell-cell interactions, role of biomolecules on cell behaviour, etc.) mathematically 10 . Mechanistic model development is often guided by analysis of experimental data, allowing hypotheses to be made for the causal mechanisms underlying a biological system 5 . For example, in describing the growth of a mechanosensitive tissue such as bone in a bioreactor, causal mechanisms include the response of the mechanosensitive cells to the applied mechanical load (fluid shear stress, hydrostatic pressure, substrate deformation etc). A key step is to identify the dependent variables of the system e.g. cell number, fluid velocity, substrate density, and their dependence on the independent variables of the system e.g. space and time.

Mechanistic models can be multi-scale, incorporating processes on a range of spatial and temporal scales. The development of coarse-graining methods for models that contain disparate space and time scales is crucial to enable rigorous mathematical analysis, and for general classification of models according to their predicted emergent behaviours 11 , 12 . Efficient and accurate computational methods for simulation of multi-scale mathematical models 13 are necessary to enable full investigation of potential model behaviours, parameter sensitivity analysis, and data-driven model calibration.

The representation of a biological mechanism need not be reductionist and molecular, but can be phenomenological 14 . Phenomenological models aim to reproduce the experimental observations without the terms in the model equations necessarily corresponding to cellular or molecular processes directly. An example is a model of a homoeostatic epithelium in which cell division always co-occurs with another cell dying or migrating away 15 , so that the overall cell density stays constant (which is the important phenomenology to capture). Phenomenological models, despite what the name suggests, can still provide mechanistic understanding of a system’s function, for example what kind of regulatory interactions are important, even if the precise nature of the interactions and their molecular mediators remains obscure. In such cases, there may be many underlying molecular mechanisms that give rise to the same phenomenological models. Strictly speaking, all models are phenomenological at some level, as they simplify the underlying chemistry and physics considerably. The model development process for phenomenological models is largely the same as for other mechanistic models described above.

Mechanistic modelling approaches include continuum 16 /discrete 17 , hybrid 18 , and deterministic/stochastic 19 , 20 .

Discrete models treat cells as distinct entities and consider the behaviour of one or more individual cells, accounting for their interactions both with each other and with the surrounding microenvironment. Discrete cell models provide a natural framework for incorporating available quantitative experimental data at the cellular or subcellular scale. Often, discrete models are also stochastic models , meaning that the outcome is to some degree random, and only one of many possible realisations. Average model behaviours or a full distribution of predictions can be obtained by repeating stochastic model simulations many times 17 .

Continuum models average the cell behaviour over a number of cells, for example describing the behaviour of the cell population in terms of its density that depends continuously on space and time. Continuum models are also used to describe the surrounding mechanical and chemical environment, with variables such as fluid velocity and pressure, solid deformation and solute concentration again depending continuously on space and time 10 . Hybrid discrete-continuum models refer to the integration of discrete cell-based models with continuum models for the surrounding cellular microenvironment, or the use of discrete (low cell numbers) and continuum (high cell numbers) models in different regions of the spatial domain as appropriate 18 . In a deterministic mathematical model, the spatiotemporal evolution of the dependent variables is completely determined by the model parameters and initial conditions - such a model will therefore always produce the same output for a given initial state. A natural formulation of continuum models is in the form of differential equations .

Statistical models are another class of models which focus on prediction over mechanistic insight. Examples of statistical models are general linear models, logistic regression, and machine-learning techniques such as artificial neural networks 21 , 22 . Statistical models aim to fit or learn the relationship of input variables, such as experimental parameters or biological variables, to output variables, such as experimental measurements. Through this, statistical models can be used to predict how the distribution of e.g. experimental measurements should change under a change in the input variables. In the development of statistical models, data are typically divided into training, validation, and test data. Training data are used to train the model (i.e. fit its parameters), validation data are used to prevent overfitting, and test data are used to assess the model’s performance at prediction 23 . Unlike phenomenological models, in which individual model components implicitly represent biological processes, no such interpretability is offered a priori by statistical models. Interpretability is however possible by inspecting the model after training it on data, although the degree of interpretability varies depending on the statistical method used 24 .

Mathematical models in regenerative medicine research

Mathematical approaches have traditionally focused on the discovery science end of the spectrum of regenerative medicine research. This has stemmed from a strong research base in mathematical medicine and biology where there are existing successful interactions with biologists and medics. Major questions in developmental and stem cell biology have been investigated using experimental and theoretical approaches 25 , 26 , 27 , 28 . Another area that has received a lot of attention is modelling of tissue growth within bioreactors, as this draws on a long tradition of continuum mechanics and its applications to medicine and biology.

Basic regeneration biology

Models can be used to distinguish which cellular processes are important to the overall regenerative process. For example, models incorporating both cell proliferation and migration can be used to explore the contribution of each process to experimentally observed regeneration. The balance of quiescence vs proliferation has been investigated in several studies. For example, the balance of quiescence and proliferation in neural stem cells has been modelled by a compartment-based differential equation approach (a continuum model) to investigate the change in regenerative capacity due to increased quiescence with age 29 . By modelling a simplified signalling network and using single-cell RNAseq data 29 the authors were able to identify a potential niche signal that maintains quiescence, Wnt Antagonist sFRP5. Another study investigating the balance of quiescent and proliferative cells in regeneration in liver biliary epithelial cells found little interconversion (on shorter time-scales) based on dual labelling experiments, and used a discrete, stochastic model of symmetric and asymmetric cell divisions to explain distribution of clone sizes 30 . At the larger scale of cell population dynamics, axolotl spinal cord regeneration has been modelled with compartment-based differential equations to identify that acceleration of the cell cycle is a more important part of the regenerative response than cell influx and stem cell activation 31 .

Mathematical modelling and data analysis approaches can be used to identify similarities between developmental and regenerative processes, i.e. can “developmental processes be reinstated and adapted or are there entirely new regenerative processes to be discovered?” 32 . A recent single-cell scale analysis 33 investigated to what extent cells in axolotl limb regeneration are de-differentiating into multipotent states, and how similar these states are to their developmentally observed analogues. Another recent example using this approach of comparing cell states in single-cell sequencing data identified a “regeneration-organising cell” in Xenopus tails 34 . Another question underpinning regeneration and growth of tissues is why does regeneration occur in some animals but not others 35 ? One approach may be to compare regeneration and wound healing, and what factors affect successful healing vs scarring. Similarities in gene expression between regeneration and wound healing have been identified 36 , however the complexity of the involvement of the immune system has not been mapped. Modelling could provide a means to address these additional components before carrying out a large number of complex co-culture approaches, thus guiding experiments towards the inclusion of essential components. One study 37 used coupled differential equations to model cytokine signalling in microglia, and explained the pro- and anti-inflammatory effects of cytokine perturbations through differential kinetics in parallel negative feedback loops. This has implications for treatment e.g. of neuroinflammation/neurodegenerative associated conditions through application of cytokines.

Another example which demonstrates the utility of mathematics in defining the role of cell interactions for successful regeneration is in hair regeneration. Spatial simulations using both continuum and discrete models have shown how a collective cell behaviour akin to bacterial quorum sensing causes hair follicle regeneration in mice to occur only when the injury is large enough 38 . Other studies from the same group of authors 39 , 40 use spatial discrete and stochastic modelling to show how the coupling, i.e. the strength of communication, between hair follicles determines the pattern in which hairs regenerate, e.g. in spreading waves, and why regeneration may stop in human scalps where stem cell activities may be more independent and less coupled. Further work 41 has investigated the morphogenesis of skin layers and hair follicles in vitro from dissociated mouse epidermal and dermal cells, and thus identified crucial physical and molecular events in the process. This led to a partial rescue of hair forming ability in these reconstituted skin samples when formed from adult cells, through the timed application of growth factors, Wnts, and MMPs 41 .

Bioreactors

Mathematical modelling of tissue maintenance and growth within in vitro bioreactors is motivated by the desire to understand and control how the imposed experimental environment and operating conditions influences the time-dependent and spatial distribution of cells, nutrients, fluid flow and substrate deformation within the bioreactor. In vitro engineering of 3D tissues is characterised by a source of cells (autologous, allogeneic, xenogenic) which are seeded on a substrate or biomaterial scaffold which can be used to provide chemical, topographical and mechanical cues 42 . Scaffolds can be extremely varied materials – synthetic e.g. or natural (decellularised ECM) – and cell-seeded scaffolds are cultured within bioreactors. Significant tissue-engineering studies have progressed the field in bone tissue engineering 43 . Examples of bioreactors include perfusion, compression, hollow fibre, hydrostatic etc 44 , 45 .

Mathematical models of bioreactors range from details of the fluid-tissue interaction at the pore scale within a cell-seeded scaffold 46 to models of growing tissue constructs 47 , 48 . We do not present a comprehensive review of bioreactor modelling studies here, but instead highlight how mathematical modelling techniques have been applied to these systems. Recent work has shown that in addition to material scaffold properties such as surface roughness, elasticity and substrate chemistry, the macroscopic geometry of the substrate controls cell growth kinetics 49 . By using rapid prototyping to build artificial macro-pores of different controlled geometries, Rumpler et al. demonstrated that cells locally respond to high curvature through enhanced tissue growth 50 . Additionally, mechanosensitive cells respond to fluid shear stress, which is itself a function of the pore geometry. Sanaei et al. 46 developed a continuum mathematical model for the fluid flow through an individual scaffold pore, coupled to the growth of cells on the pore walls, to determine how the interplay between substrate geometry and fluid shear stress enhances tissue growth. In a complementary approach, Guyot et al. 51 developed a 3D computational model using the level set method to capture the growth, again depending on curvature and fluid shear stress, at the scaffold level in a perfusion bioreactor. These models offer simple frameworks for testing the behaviour of different scaffold pore geometries, and facilitates the prediction of operating regimes (inlet fluid flux etc) in which the tissue growth may be enhanced.

While computational approaches can be employed to scale-up mechanistic insights from the pore to the tissue scale, an alternative approach is to use mathematical homogenisation techniques to derive effective macroscale equations (construct level) that explicitly incorporate details of the structure and dynamics of the pore scale detail. Such coarse-graining approaches rely on a disparity in length scales e.g. between the pore scale and scaffold scale. A recent experimental approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor 52 . To understand how the applied load is distributed throughout the construct, Chen et al. 53 employed mathematical homogenisation theory to derive the effective macroscale equations. The resulting model captured the orthotropic nature of the composite material, and can be exploited to determine how local mechanical environment experienced by cells embedded within the construct 53 depends on the composite material properties (e.g. fibre dimension and properties). In a complementary approach, Castilho et al. 54 employed a finite element (FE) model to explore the reinforcement mechanisms of fibre-hydrogel constructs.

While the studies of Chen et al. 53 and Castilho et al. 54 , focused on the material properties of the scaffold, techniques of mathematical homogenisation can also be employed to derive systems of homogenised partial differential equations describing tissue growth within biomaterial scaffolds 11 , 55 , 56 . Alternative routes to describing an evolving biological tissue, in which the volume fraction of the constituents/phases – cells, ECM, interstitial fluid etc - change over time utilise multiphase mixture theory, based on the principles of mass and momentum conservation with specified constitutive laws describing the interactions between the phases 57 . Such a multiphase framework has been employed in a multiscale setting to describe the properties of a tissue growing on a rigid porous scaffold: again, mathematical homogenisation techniques can used to derive effective macroscale equations that describe the effective properties of the construct, and retain explicit dependence on both the microscale scaffold structure and the multiphase tissue dynamics 58 . When considering cell-seeded construct growth within bioreactors, these bioactive multiphase models must be coupled to surrounding single phase fluid through specification of the appropriate boundary conditions 59 .

Mathematical models and computational approaches describing bioreactor processes enable identification of optimal process conditions leading to robust and economically viable products 60 . Taking a mechanistic model for the growth of neotissue in a perfusion bioreactor, Mehrian et al. 61 applied model reduction techniques to extract a set of ordinary differential equations from the original set of partial differential equations. The simpler reduced system enabled rapid simulation, allowing the application of rigorous optimisation techniques. Bayesian optimisation was applied to find the medium refreshment regime in terms of frequency and percentage of medium replaced that would maximise neotissue growth kinetics during 21 days of culture. The simulation results indicated that maximum neotissue growth will occur for a high frequency and medium replacement percentage, supporting existing reports in the literature 61 .

Clinical translation

Mathematical models can also be used to ask “what if…?” questions (hypothesis testing), allowing us, for example, to generate experimentally testable predictions for the way cells or engineered tissues behave after implantation. A recent theoretical study using continuum models 62 of homoeostatic hematopoeisis put forward a novel interaction between hematopoeitic stem cells (HSCs) and niche cells, namely that niche cells could be quiescence-inducing, while the HSC in turn promote the survival of the niche cells. This mechanism would have the advantage that a large excess of niche cells can compensate large fluctuations in HSC number, unlike proliferation-inducing niche interactions. The differential equation model based on this premise was able to explain why there is a delay in HSC recovery after near-complete ablation, but not after irradiation (which kills a smaller fraction of cells). Such insights stemming from the basic regenerative biology can be exploited to make sense of the dynamics of recovery after cell transplantation, and how the ratio of niche to stem cells affects the performance of cell therapy or tissue engineering approaches.

Mathematical models can also be used for clinical optimisation of a regenerative therapy e.g. to optimise RM treatment strategies by understanding the trade-offs involved. One such trade-off is between quick repair and risk of fibrosis in ischaemia, which has been investigated using a combination of mouse experiments in a kidney injury model and a differential equation model of cell-cell communications 63 . In the model, Wnt overexpression would decrease the risk of death but increase fibrosis, while Wnt down-regulation would decrease fibrosis but increase risk of death. This led to an optimal treatment prediction of sequentially applying Wnt agonist and antagonist which ultimately could lead to increased survival and decrease fibrosis risk.

Mathematical models can assist at the end of the translational pathway, for example we can use models to gain a deeper understanding of the efficacy of treatments. In liver regeneration, mesenchymal stem/stromal cells are directed to sites of injury by SDF-1, which has potential for cell-based therapies. A differential equation model has recapitulated the in vivo response to treatment of liver injury for different SDF-1 concentrations and doses of transplanted cells 64 , including the beneficial effect of hypoxia-preconditioning to increase the CXCR4 receptor concentration.

Mathematical modelling can also give confidence to enable new protocols for RM to reach the clinic. Using clinical retrospective data, modelling can predict the importance of contributions of aspects of the protocol to the eventual outcome of the treatment. One example is theoretical modelling work for autologous Chondrocyte implantation (ACI) which is an effective treatment for cartilage defects 65 , 66 . From clinical and animal studies it was unclear whether the type or number of implanted cells is important. To determine the effect of the number and type of implanted cells on cartilage repair, Campbell et al. 65 , 66 formulated a reaction-diffusion model for repair after implanting chondrocytes or mesenchymal stem cells (MSCs). The model captured cell migration, proliferation and differentiation, nutrient diffusion and depletion, and cartilage matrix synthesis and degradation at the defect site, both spatially and temporally. They identified that the number of implanted cells had only a marginal effect on the defect fill time or the maturation time, and that the implantation of MSCs vs chondrocytes did not affect maturation time but did affect the nature of the maturation. Chondrocyte implantation gave the most mature cartilage towards the bottom of the defect, but MSC implantation gave the most mature cartilage towards the surface of the defect. The small effect of cell number in this study may explain why both clinical and animal studies have been inconclusive in defining dosing of cells. This result gave the clinical team’s MHRA-licensed cell manufacturing facility confidence to implement wide cell release criteria with respect to cell numbers. The small maturation difference between chondrocytes and bone marrow derived MSCs agrees with experimental studies 65 , 66 . Chen et al. 67 also considered a reaction-diffusion model for identifying optimal strategies for chondrogenesis in tissue engineering applications. Experimentally, a hydrogel is seeded with a layer of MSCs lying below a layer of chondrocytes, and the MSCs are stimulated from above with exogenous TGF-beta, and then cultured in vitro. Through mathematical modelling, Chen et al. 67 identified how the initial concentration of TGF-beta, the initial densities of the MSCs and chondrocytes, and the relative depths of the two layers influence the long time composition of the tissue construct 61 , 67 .

These examples above demonstrate how mathematics has been used to model regenerative biology at multiple stages of the translational pathway but there are clearly further areas where the input of theoretical and computational approaches would benefit the speed of progress towards the clinic. Table 1 identifies stages in the process where help is needed to define the appropriate clinical regenerative protocol more rapidly and reproducibly. This table is intended to establish the potential for the mathematical community to contribute at each step of the translational process. Each part of the table underpins some basic biological and/or engineering question where mathematical modelling could potentially add value.

For many biomedical and clinical researchers, the concept of how to approach the relationship with mathematicians can be daunting. This review is a first step to try and provide a reference which illustrates previous work and signposts where to go for future studies. To help researchers to identify possible areas for collaboration, Table 2 identifies areas which can be modelled and specific approaches which may be or have been used in the literature. What is needed are interactive workshops, training pathways and defining some common languages to support this interaction.

In this review, we have highlighted the enormous potential for embedding mathematical and computational approaches within the regenerative medicine pipeline. To successfully achieve this however requires a number of challenges to be overcome. For example, theoretical model development often lags behind experimental approaches in the earlier stages of the research, prohibiting their early use as predictive tools to guide and inform experimental design. Another challenge arises when model parameterisation is hindered by a lack of experimental data (or the right kind of experimental data). Addressing issues of structural and practical identifiability of mathematical models is key, and, in simple terms, means to check to what extent (groupings of) parameters can be determined by statistical fitting of observable data in principle or in practice. Issues of non-identifiability can then drive further model reduction and/or additional experiments.

To overcome these potential bottlenecks it is essential to have mechanisms in place to allow integrated mathematical and experimental research programmes to be designed and implemented, including interactive workshops, combined and reciprocal training pathways for wet and dry scientists, and funding schemes to engage interdisciplinary teams of mathematicians, regenerative medicine scientists, and clinicians (see also 68 ).

In conclusion, the opportunity to engage mathematics within a growing regenerative medicine community has the potential to enable more rapid translation of cell-based approaches to the clinic. In contrast to laboratory experiments which are often time consuming and expensive, mathematical modelling approaches are much faster and cheaper. Embedding mathematics within regenerative medicine enables researchers to go beyond the usual trial-and-error approach, be guided in their experimental design, and therefore accelerate advances in regenerative medicine. In silico approaches can provide added value in understanding complex regeneration events in tissues in vivo and in growth environments in vitro. This review highlights the wealth of opportunities for collaboration between mathematicians and regenerative medicine scientists, and to identify where modelling approaches can contribute to the many stages of the regenerative medicine pipeline to address key challenges in translation.

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Acknowledgements

The authors gratefully acknowledge funding in the form of an EPSRC Healthcare Technologies Discipline Hopping Award (S.L.W.: EP/R013128/1, A.J.E.H.: EP/R013209/1); EPSRC Healthcare Technologies Awards (A.J.E.H.: EP/P031137/1, S.L.W.: EP/P031218/1 & EP/S003509/1); MRC (S.L.W. & A.J.E.H.: MR/T015489/1, AJEH: MR/R015635/1). L.J.S. was supported by a Chancellor’s Fellowship from the University of Edinburgh. A.J.E.H. is supported by ERC Advanced grant DYNACEUTICS No. 789119.

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Waters, S.L., Schumacher, L.J. & El Haj, A.J. Regenerative medicine meets mathematical modelling: developing symbiotic relationships. npj Regen Med 6 , 24 (2021). https://doi.org/10.1038/s41536-021-00134-2

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• Vol 7, Issue 15, 2023

Research on Integrating Mathematical Modeling Thinking into Large, Medium and Small School Teaching

DOI: 10.23977/aetp.2023.071507 | Downloads: 43 | Views: 490

Zhiqiang Hu 1 , Xiaodong Zhao 1 , Zhongjin Guo 1 , Xiaoqian Li 1 , Shan Jiang 1

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1 School of Mathematics and Statistics, Taishan University, Tai'an, Shandong, 271000, China

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This study explores how mathematical modeling thinking can be integrated into large, medium, and small school teaching to enhance students' mathematical abilities and interdisciplinary thinking. Using various research methods, including literature review, surveys, teaching experiments, statistical analysis, and expert interviews, we aim to establish a localized model for integrating mathematical modeling, thinking into large, medium, and small school teaching to optimize the quality of mathematics education. Research both in China and internationally has shown that mathematical modeling thinking has garnered significant attention in the field of education and holds promise as an effective approach to improving students' mathematical thinking skills and overall quality of education. The results of this research are expected to offer new insights and methods for mathematical education within the context of large, medium, and small school integration and provide scientifically sound assessment standards for education. This study is not only academically significant but also expected to support educational reform and practice in schools and educational institutions, contributing to the development of innovative and practical talents.

CITE THIS PAPER

Zhiqiang Hu, Xiaodong Zhao, Zhongjin Guo, Xiaoqian Li, Shan Jiang, Research on Integrating Mathematical Modeling Thinking into Large, Medium and Small School Teaching. Advances in Educational Technology and Psychology (2023) Vol. 7: 63-68. DOI: http://dx.doi.org/10.23977/aetp.2023.071507.

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Introduction to Mathematical Modelling

What is a Mathematical Model? #

A mathematical model is a mathematical representation of a system used to make predictions and provide insight about a real-world scenario, and mathematical modelling is the process of constructing, simulating and evaluating mathematical models.

Why do we construct mathematical models? It can often be costly (or impossible!) to conduct experiments to study a real-world problem and so a mathematical model is a way to describe the behaviour of a system and predict outcomes using mathematical equations and computer simulations .

Check out the following resources to get started with mathematical modelling:

Chapter 1: What is Mathematical Modelling? in Principles of Mathematical Modeling

What is Math Modeling?

Wikipedia: Mathematical Model

Outline of the Modelling Process #

Mathematical modelling involves observing some real-world phenomenon and formulating a mathematical representation of the system. But how do we even know where to start? Or how to find a solution? The modelling process is a systematic approach:

Clearly state the problem

Identify variables and parameters

Make assumptions and identify constraints

Build solutions

Analyze and assess

Report the results

Models can have a wide range of complexity ! More complex does not necessarily mean better and we can sometimes work with more simplistic models to achieve good results. In many instances, we often start with a simple model and then build-up the complexity by iterating through the steps in modelling process until the model accurately describes the real-world application.

Check out Math Modeling: Getting Started and Getting Solutions to read more about the modelling process.

Types of Models #

There are many different types of mathematical models! In this course we focus on the following:

Deterministic models predict future based on current information and do not include randomness. These kinds of models often take the from of systems of differential equations which describe the evolution of a system over time.

Stochastic models include randomness and are based on probability distributions and stochastic processes .

Data-driven models look for patterns in observed data to predict the output of a system. These kinds of models often take the form of functions with parameters computed to fit observed data.

Generative AI/ML Models for Math, Algorithms, and Signal Processing (MASP) RFI

The Intelligence Advanced Research Projects Activity (IARPA) seeks information regarding innovative approaches to generative artificial intelligence (AI) or machine learning (ML) models to achieve a revolutionary leap in applications of science and engineering by generating smaller evolutionary products of math, algorithms, or signal processing (MASP). While significant progress has been made for generators of text, image, and audio (TIA) modalities, AI/ML generators for more complex sciences, including the MASP modalities, have not received the same attention. It is important to note this RFI is  not  for AI/ML solutions that  perform  such calculations of math, algorithms, or signal processing; rather, this RFI is looking for AI/ML solutions that  create  math, algorithms, or signal processing products themselves at the output of the generator. The envisioned models and systems could enable exponential advances in scientific and engineering fields as the AI/ML generates many small evolutionary products, unfettered from delays in human creativity, to quickly accumulate into generational discoveries.

This RFI seeks understanding of innovative systems consisting of MASP input and output modalities for generative AI/ML frameworks. These systems, when fully realized, should have the opportunity to create marginal advances to MASP problems by generating novel MASP products. These marginal improvements, fed back into future machine iterations, in a positive feedback fashion, may provide generational improvements in some science and engineering fields.

For more information and submission instructions, please visit SAM.gov .

Published Date: March 5, 2024

Response Date: March 29, 2024, 5pm EST

New model allows a computer to understand human emotions

Researchers at the University of Jyväskylä, Finland, have developed a model that enables computers to interpret and understand human emotions, utilizing principles of mathematical psychology. This advancement could significantly improve the interface between humans and smart technologies, including artificial intelligence systems, making them more intuitive and responsive to user feelings.

According to Jussi Jokinen , Associate Professor of Cognitive Science, the model could be used by a computer in the future to predict, for example, when a user will become annoyed or anxious. In such situations, the computer could, for example, give the user additional instructions or redirect the interaction.

In everyday interactions with computers, users commonly experience emotions such as joy, irritation, and boredom. Despite the growing prevalence of artificial intelligence, current technologies often fail to acknowledge these user emotions.

The model developed in Jyväskylä can currently predict if the user has feelings of happiness, boredom, irritation, rage, despair and anxiety.

"Humans naturally interpret and react to each other's emotions, a capability that machines fundamentally lack," Jokinen explains. "This discrepancy can make interactions with computers frustrating, especially if the machine remains oblivious to the user's emotional state."

The research project led by Jokinen uses mathematical psychology to find solutions to the problem of misalignment between intelligent computer systems and their users.

"Our model can be integrated into AI systems, granting them the ability to psychologically understand emotions and thus better relate to their users." Jokinen says.

Research is based on emotional theory -- the next step is to influence the user's emotions

The research is anchored in a theory postulating that emotions are generated when human cognition evaluates events from various perspectives.

Jokinen elaborates: "Consider a computer error during a critical task. This event is assessed by the user's cognition as being counterproductive. An inexperienced user might react with anxiety and fear due to uncertainty on how to resolve the error, whereas an experienced user might feel irritation, annoyed at having to waste time resolving the issue. Our model predicts the user's emotional response by simulating this cognitive evaluation process."

The next phase of this project will explore potential applications of this emotional understanding.

"With our model, a computer could preemptively predict user distress and attempt to mitigate negative emotions," Jokinen suggests.

"This proactive approach could be utilized in various settings, from office environments to social media platforms, improving user experience by sensitively managing emotional dynamics."

The implications of such technology are profound, offering a glimpse into a future where computers are not merely tools, but empathetic partners in user interaction.

• Relationships
• Brain-Computer Interfaces
• Computer Modeling
• Mathematical Modeling
• Mathematics
• Computer Science
• Computer simulation
• User interface design
• Computer software
• Grid computing
• Mathematical model
• Application software
• 3D computer graphics
• Speech recognition

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Theoretical understanding of the relevant problem structure and consistent mathematical modeling are necessary keys to formulating operations research models to be used for optimization of decisions in real applications. The numbers of alternative models, methods and applications of operations research are very large. This paper presents fundamental and general decision and information structures, theories and examples that can be expanded and modified in several directions. The discussed methods and examples are motivated from the points of view of empirical relevance and computability.

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My thanks go to Professor Hadi Nasseri for kind, rational and clever suggestions.

Recommender: 2016 International workshop on Mathematics and Decision Science, Dr. Hadi Nasseri of University of Mazandaran in Iran.

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Lohmander, P. (2018). Applications and Mathematical Modeling in Operations Research. In: Cao, BY. (eds) Fuzzy Information and Engineering and Decision. IWDS 2016. Advances in Intelligent Systems and Computing, vol 646. Springer, Cham. https://doi.org/10.1007/978-3-319-66514-6_5

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DOI : https://doi.org/10.1007/978-3-319-66514-6_5

Published : 19 September 2017

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