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

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## Abstract

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

**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.

**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.

**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.

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

_{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”?

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

## 4. A Primer on Mechanistic Modelling of Biochemical Systems

**modelling workflow**. In a nutshell, the workflow includes the following sequential operations (Figure 3):

**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

#### 5.1. Mathematical Models Cannot Reproduce the Complexity of Biology

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

#### 5.3. You Should Employ Data in Your Mathematical Model

#### 5.4. Your Predictions Are Not Experimentally Validated

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

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

#### 6.1. Know Your Problem

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

#### 6.3. Build on Preceding Efforts

#### 6.4. The Size Does Not Always Matter

#### 6.5. Set Your Results in Stone

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

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**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].

**Figure 2.**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).

**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.

**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.

Mathematical models cannot reproduce the complexity of biology. Cells and biochemical systems are governed by the same laws as any other physicochemical system, most of which are successfully simulated with mathematical models (see, for example, computer model-based weather and climate prediction). |

Your model is not physiological. The real system is more complex than your mathematical model. A model must include the elements, interactions, and processes necessary to investigate the hypothesis in question (nothing else), and this is equally valid for mathematical and lab bench models. |

You should employ data in your mathematical model. A well-formulated mathematical model is based on quantitative data taken from databases and published reports or produced for calibration of the model. Also, one can use qualitative model analysis to formulate hypotheses and propose experiments; here, the experimental data comes after the model. |

Your predictions are not experimentally validated. Model derivation and simulation can be temporally detached from experimentation: a researcher can investigate a hypothesis with mathematical modelling and can leave the experimental validation for another team after publication of the model-based predictions. |

I do not see the clinical relevance of your predictions. There are mathematical models that are of mandatory use in biomedicine (see pharmocokinetics models for drug approval). However, mathematical modelling is in most cases theoretical research. Similar to any other basic research approach, it has an unpredictable long-term potential for enhancing clinical practice. |

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**MDPI and ACS Style**

Vera, J.; Lischer, C.; Nenov, M.; Nikolov, S.; Lai, X.; Eberhardt, M.
Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic. *Int. J. Mol. Sci.* **2021**, *22*, 547.
https://doi.org/10.3390/ijms22020547

**AMA Style**

Vera J, Lischer C, Nenov M, Nikolov S, Lai X, Eberhardt M.
Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic. *International Journal of Molecular Sciences*. 2021; 22(2):547.
https://doi.org/10.3390/ijms22020547

**Chicago/Turabian Style**

Vera, Julio, Christopher Lischer, Momchil Nenov, Svetoslav Nikolov, Xin Lai, and Martin Eberhardt.
2021. "Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic" *International Journal of Molecular Sciences* 22, no. 2: 547.
https://doi.org/10.3390/ijms22020547