DRUG DEVICES – Mathematical Modeling for Faster Autoinjector Design


There is an increasing demand for advanced injection devices that bring  benefits around self-administration and ease of use and work with new biological drugs, which are often required to be delivered in larger volumes and/or at higher viscosities than those for conventional drugs. In order to reduce development and manufacturing costs, there is also a desire for platform devices, which can meet a broader range of drug and user requirements through simple adaptation of a core design. Unfortunately, a number of devices currently in the market do not meet these requirements and furthermore, there is often a lack of a detailed understanding of how the key design parameters affect the overall device performance. In some cases, this has led to product recalls and in other cases, resulted in challenges in adapting the designs to meet new needs. Hence, there is a commercial need in the injectables industry for better understanding of how design fundamentals affect the performance of autoinjector devices.

Cambridge Consultants believe a fast and effective approach to a better injection device design is a novel combination of mathematical modeling on a desktop PC, and complementary experimental data. The use of mathematical modeling gives insight into the physical behavior of the device, and allows rapid prediction of the effect of different parameters during the design process. The more established experimental approach provides confirmatory data to support the modeling. Cambridge Consultants has deployed this methodology in a number of development projects and achieved benefits in terms of reduced development time, more robust and adaptable designs, and reduced product cost.

Mathematical modeling allows for quick simulations of device performance. The effect of important design parameters, such as injection time, injection force, and shear stress on the drug, can be predicted in real-time. This allows the engineer to investigate the design space and quickly optimize a suitable set of components for a new injection device. For example, the mathematical model can guide the designer in choosing the correct driving spring, needle gauge, and plunger in order to meet the device performance specification. In this paper we present a mathematical model of device performance applied to a typical autoinjector. This has the generic features of an autoinjector, but is not tied to any specific commercial device, and thus, can be tailored to investigate a wide range of interesting design features. The approach and the type of physics used inside the model are described. How experimental data can be used to calibrate and verify the model is also shown.

As a case study, we use the mathematical model to predict the effect needle tolerances have on injection times within an autoinjector. This is an essential piece of performance data and is something that would take weeks and an expensive budget to determine experimentally, but that can be done in a few hours with a mathematical model.


The generic autoinjector shown in Figure 1 and considered for the model has the following components, which are commonly found in most commercial devices:

“¢ A syringe containing the drug
“¢ A needle
“¢ A compliant plunger to seal the drug into the syringe
“¢ A drive spring as the energy source that powers the plunger
“¢ A liquid drug
“¢ An air bubble between the drug and the plunger

The trigger that activates the autoinjector by releasing the spring force on to the plunger has not been considered. The physics of each component can be considered in isolation using ordinary differential equations that vary with time.

The needle is modeled as a component that creates viscous pressure losses via the Hagen Poiseuille equation and inertial pressure losses at the entry and exit. The drug is modelled as a Newtonian fluid. The plunger is modelled as a rubber component with linear compliance, subject to a maximum compression limit. The syringe exerts a frictional force on the plunger. For the purpose of the model, a 1 ml BD Hypak syringe was used as a representative commercially available syringe. The air bubble is modelled as a compliance, with an internal pressure related to the amount of compression via Boyle’s Law. The spring is modelled as a linear compression spring; it is initially compressed and expands during device activation. It would be a relatively minor change to the mathematical model to replace the mechanical spring with a motor energy source as used in electronic autoinjectors like the EasyPod.

The individual equations describing each component are assembled into a mathematical model of the entire autoinjector system, where all the components are interdependent. We use the Simulink commercial package to solve this system of equations. To make the model user friendly for a wide audience, we built a front-end GUI in Matlab. As shown in Figure 2, the GUI allows for key design parameters to be changed easily, has a graphical representation of the syringe plunger motion, and also plots out metrics of interest such as plunger motion, drug pressure, and percentage of drug delivered as a function of time. The model allows us to look at the physical behavior of the autoinjector at various points in time. Some of the parameters we predict are easy to calculate mathematically but difficult to monitor experimentally: for instance, shear stress on the drug. However, knowledge of the shear stress on the drug is important because high levels of shear stress can damage and degrade the molecules within the drug: particularly newer biologic drugs consisting of complex chains of proteins. An example shear stress simulation is shown in Figure 3 for a drug with viscosity 6 times greater than water that is being driven by a spring with stiffness of 600 N/m with either a 27-gauge or a 25-gauge needle. Initially, the shear stress is several orders of magnitude higher than the steady state value. This is due to the impact of the initial spring release that causes rapid compression of the bubble, and subsequent high driving pressures and velocities for the drug in the needle. It is found that that the smaller diameter 27-gauge needle has a peak shear strain rate approximately 50% higher than for the 25 gauge.

The model also highlights the sensitivities in the autoinjector design. For example, consider the effect of needle diameter ±5% tolerances (nominally 27 gauge) and needle length ±5% tolerances (nominally 0.5 inch) on the time to deliver 1 ml of drug with a viscosity equivalent to water. Injection time is an important metric because the patient will prefer a shorter duration injection rather than a long one. The contour plot in Figure 4 predicts that injection time is much more sensitive to needle diameter than to needle length. A difference of 23% in injection time results from changing the needle diameter by only – 5% to +5% of nominal. This knowledge allows the designer to focus on the needle diameter as the single most powerful design parameter that can be used to adjust injection time.

A designer may have to adjust the design to compensate for changes in the drug formulation. This is particularly true for platform devices in which a standard mechanical layout is customized to suit a range of different drugs. The mathematical model can be used to predict how the design must be altered to cope with a change in drug properties such as viscosity. For instance, suppose an initial version of the device platform is developed to work with a viscosity equivalent to water (1 centiPoise) and has an injection time of 1.6 seconds. This is achieved by using a spring of stiffness 600 N/m. If a new drug formulation has a viscosity that is five times higher (5 centiPoise), and there is a requirement that the injection time should remain constant, the model predicts that to achieve this, a spring stiffness of approximately 1250 N/m will be now be required. This analysis takes minutes to perform with the mathematical model, but would require a lengthy program involving many different mechanical prototypes to do experimentally. Having determined the required higher spring force, the rest of the design can then be re-evaluated and modified if necessary to ensure it can function with the stiffer spring.


It is important to be able to confirm that the underlying equations representing each component in the model are realistic, and that the output from the mathematical model can be trusted. A way to verify the model is to compare it with experimental data of the force required to drive the plunger on a representative commercially available syringe. The measurements were made on a Mecmesin force-displacement tester, which drives the syringe plunger at constant velocity and records the required force. This is a slightly different scenario from the original mathematical model in which a known force from the spring is applied and the velocity of the plunger is calculated. Thus, a modified version of the mathematical model was developed specifically for verification in which the velocity of the plunger is specified and the plunger force is calculated. The verification model was compared against various different experimental test cases. A sample verification of the plunger force mathematical model predictions and measurements is given in Figure 5 for a test case with a 2.5-mm long air bubble in a 1-ml syringe, a 30-gauge needle, and a test liquid six times more viscous than water. Predictions and measurements are in good agreement.


A common problem in injection device development is to understand the effects of manufacturing tolerances on the design. The designer must ensure the device performs within specification over the range of component tolerances, whilst considering that relying on excessively tight manufacturing tolerances can make a design uneconomic. We can apply the mathematical model to predict the sensitivities of an autoinjector to realistic manufacturing process tolerances to see whether a design will always perform to specification.

For example, the mathematical model can be used to aid the designer in understanding how differences in spring stiffness or shape arising from production variations might affect drug delivery performance. For example, for a 5-centiPoise formulation, a 10% reduction in spring force from 1250 N/m increases injection time by a similar percentage, thus the dependence is not so critical.

As another example, consider a device with a 1-ml BD HyPak syringe with a nominal 27-gauge needle of 0.5-inch length, and investigate the effect of tolerances on injection time. We assume that each needle diameter, needle length, and syringe diameter vary with a process variation of Cp = 1.33. Using a Monte Carlo approach, we simulate a sample of 1000 syringes that embody this distribution of tolerances, and the results are shown in Figure 6. The mathematical model is run once for each of these syringes, and the injection times are predicted. This takes a matter of hours. Doing this experimentally would take weeks, and would require a lot of test specimens to be made at representative tolerances that would be expensive.

This Monte Carlo approach is more realistic and offers a narrower range of injection times than a simpler worst case tolerance analysis in which all tolerances are assumed to be at the maximum permissible extremes. Our combination of the mathematical model with a Monte Carlo approach means the designer has a more realistic knowledge of the effects of tolerances and can make statements, such as “90% of all devices will have an injection time between 1.5 and 1.9 seconds.” That the real distribution of tolerances are narrower than the worst case values means the designer can achieve a given outcome and still specify less-restrictive tolerances in the manufacturing processes. This will result in considerable cost savings, particularly with an autoinjector that is likely to be made in quantities of millions per year.


This article has introduced a mathematical model of an autoinjector. The fundamental approach considers the generic physical features of a device, and can be tailored to model specific commercial products as required. The model has been written in the commercial package Simulink to make it quick to adapt and to run. Simulink solves the detailed mathematics describing the physics of the device. To make the model intuitive and easy to use, we created a graphical user interface, which means it is accessible to designers who do not necessarily need to know the specifics of the physics behind the device.

We have shown how the model can be used at the early design stage to define key components, such as the drive spring or needle in order to meet the product requirements specification. The model also facilitates platform solutions by allowing the designer to predict rapidly the changes needed in an existing device to meet a change in drug specification. Much of the behavior the model can predict is difficult to measure experimentally, such as shear stress in the drug or behavior over short timescales, but gives valuable insight into how the device functions. The model is also valuable during the manufacturing scale-up process, as it uses a Monte Carlo approach to simulate the effect of tolerance interactions on device performance. This is particularly expensive to do experimentally, as it would require a large range of samples to be manufactured and tested.

We believe more extensive use of mathematical modelling in combination with experimental testing throughout the development process can lead to more robust platform injection devices hence reducing the risk of product recalls and allowing more reliable and cost-effective adaption of device designs for the delivery of new therapies.

Dr. Jonathan Wilkins led a range of drug delivery device developments as Senior Consultant at Cambridge Consultants: from inhalers through to novel injection systems. He has an interest in applying mathematical modelling techniques to speed up the development and optimization times of new products. He has an engineering degree and PhD from Imperial College, University of London, specializing in fluid mechanics. He is currently Qualification Manager at Magma Global, developing novel materials and processes for the oil and gas industry.

Dr. Iain Simpson is Associate Director of Drug Delivery in the Global Medtech Practice at Cambridge Consultants. He has a 20- year track record of multidisciplinary technology and product development, the last 12 of which have been spent mainly in drug delivery covering parenteral, pulmonary, nasal drug delivery as well as more invasive device technology of which he has mainly worked in program management and review roles. He has written and presented papers on a range of drug delivery topics, including developing inhalers for children, technology licensing, improving device compliance, and usability. He also lectures on drug delivery to the Cambridge University Masters in Bioscience Programme and is a past Chairman of the R&D Society. Dr. Simpson can be reached at iain.simpson@cambridgeconsultants.com and to whom correspondence should be addressed.