Demystifying mathematical modeling in medical device development

24 Jun 2026 7min read

Mathematical modeling is an incredibly powerful tool that provides meaningful insight into complex processes and helps you to make decisions with confidence.

The simplest definition of mathematical modeling is the use of fundamental mathematical relationships and concepts to represent a system and predict how it might behave under different conditions. It is a white-box modeling technique, meaning the underlying physics and assumptions are explicitly understood and directly modeled. This allows engineers to predict answers to a range of questions early on in medical device development such as: how stiff does the spring need to be in an autoinjector? Is active cooling required for a MedTech device? How long does it take a syringe to deliver its dose?

Mathematical modeling can quickly de-risk decisions before committing to anything more costly. But how do you create an effective mathematical model to guide your medical device development?

How mathematical modellng supports medical device design

In the initial design phases of a device (and often beyond), mathematical modeling quickly builds understanding of a system, especially when requirements are evolving and details are not yet defined. By using informed assumptions, unnecessary complexity is removed, allowing engineers to focus on the core behavior and get a feel for how their system reacts to different conditions. Assumptions can reduce accuracy however when the model is built well, the results are often close enough to guide early design decisions.

This approach speeds up the conversion of initial client specifications into design parameters of the device. It translates how the drug needs to be delivered into concrete design specifications such as the size and performance of a spring or gas canister – a process that is much faster than guessing based upon past experience and refining by prototyping. The mathematical model can inform so many other sections of the design as well, leading to a faster and more efficient development cycle.

A key advantage of mathematical modeling is the speed of iteration. These models typically take a few seconds to compute, compared with minutes (or even hours) for a full computational simulation. This makes it easy to tweak the model and rerun it quickly, so you can see how key parameters interact and gain an empirical sense of how the system behaves. If the outcome looks very sensitive to a particular parameter, you can examine it quickly and make a judgement. For example, you might ask how the molding tolerance on a vent affects cooling performance in a MedTech device; this can be analyzed and the tolerance tightened if necessary.

How does mathematical modeling differ from finite element approaches?

Mathematical models break the system down into large blocks which can be modeled individually using physical relationships, working in broad strokes to define the system. Finite element approaches break every single block down into thousands of tiny blocks and then use even more simple relationships to model the system. The greater resolution gained by breaking the system down so much more is what allows for more complex behaviors to be modeled, however then required a greater computational power.

These two approaches serve a different purpose. High-fidelity simulations, such as Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD) usually require every detail to be properly defined to return thorough, accurate answers. However, that detail can also limit how quickly you can explore broader, early-stage questions. They are invaluable in later design stages when small details need attention, but they are often less suitable when major aspects are still being defined.

The right approach depends on the question you are trying to answer. There isn’t a “one size fits all” method. The approaches can coexist: a mathematical model might be the main way to understand the overall system while a particularly complex part is informed by targeted FEA simulations.

Ultimately that core understanding of the system is why a mathematical model is developed: what factors really drive performance, which parts are sensitive and where the design can be improved. This understanding supports confident decision-making and, in the end, helps create a better medical device for patients and users.

Understanding limitations of the mathematical model

Mathematical models are based on assumptions. Each one defines the conditions in which the model remains valid. The goal is to choose assumptions carefully and to be clear about where the model can (and cannot) be used.

For example, when developing a model to predict the delivery time of a drug in a syringe you may assume that the viscosity of the drug is constant. This is a valid assumption for syringe use in a clinical setting where the environment is stable. However, for emergency-use autoinjectors, the device needs to work across a much wider range of conditions where the viscosity may change dramatically with temperature, so this assumption needs to be re-evaluated.

Using a model outside its bounds is similar to extrapolating a statistical data set: the result might be accurate but you have no reliable way to know.

Developing an effective mathematical model for a medical device

Define the scope and objectives. What do you need to know and importantly, what do you not need to know at this stage? Keep this conservative at first so that you cover your key concerns but do not bloat the model with unnecessary complexity that can introduce errors.
Decompose the system into its constituent blocks and look at the relations between each of the blocks. A block is any smaller part of the overall system where its behavior can be directly and easily modeled. For a thermal model this would be each lumped thermal mass and the thermal transfer paths between them.
Keep a detailed record of assumptions. Assumptions underpin the entire model. While they should have already been considered in earlier steps, they now need to be properly recorded. This ensures that the bounds of the model are clear.
Identify the fundamental mathematical equations. These should cover changes in the internal properties of each block and the relationships between them. If needed, include methods for solving differential equations.
Build the model up with all of its necessary features. This includes the needed user inputs, fundamental physical properties, useful outputs and other features. This can be done in something like a spreadsheet or as a script or program.
Examine the information. Examine the outputs of the model and adjust the inputs to create a working system and gain understanding of the complex interactions. This is the invaluable information that will drive the early and even detailed design stages.
Refine the model. You can compare results with test data to check accuracy, or the model may reveal a fundamental flaw that means the operating principles need to change. It is also possible that the predicted operating point sits outside the model’s bounds. For any of these reasons, it may be worthwhile improving and refining the model, moving back to step one, re-evaluating key decisions and making the necessary changes.

Mathematical modeling is an invaluable tool in medical device design. It improves understanding of the system throughout the design process, from de-risking decisions in the early stages to informing the tolerances in later stages. All of this contributes to a better design with fewer risks while saving time and money during development.