Abstract
Bicycling is the leading cause of sportsrelated traumatic brain injury. Most of the current bike helmets are made of expanded polystyrene (EPS) foam and ultimately designed to prevent blunt trauma, e.g., skull fracture. However, these helmets have limited effectiveness in preventing brain injuries. With the availability of highrate microelectricalmechanical systems sensors and high energy density batteries, a new class of helmets, i.e., expandable helmets, can sense an impending collision and expand to protect the head. By allowing softer liner medium and larger helmet sizes, this novel approach in helmet design provides the opportunity to achieve much lower acceleration levels during collision and may reduce the risk of brain injury. In this study, we first develop theoretical frameworks to investigate impact dynamics of current EPS helmets and airbag helmets—as a form of expandable helmet design. We compared our theoretical models with anthropomorphic test dummy drop test experiments. Peak accelerations obtained from these experiments with airbag helmets achieve up to an 8fold reduction in the risk of concussion compared to standard EPS helmets. Furthermore, we construct an optimization framework for airbag helmets to minimize concussion and severe head injury risks at different impact velocities, while avoiding excessive deformation and bottomingout. An optimized airbag helmet with 0.12 m thickness at 72 ± 8 kPa reduces the head injury criterion (HIC) value to 190 ± 25 at 6.2 m/s head impact velocity compared to a HIC of 1300 with a standard EPS helmet. Based on a correlation with previously reported HIC values in the literature, this airbag helmet design substantially reduces the risks of severe head injury up to 9 m/s.
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Acknowledgments
The study was supported by the National Institutes of Health (NIH) National Institute of Biomedical Imaging and Bioengineering (NIBIB) 3R21EB01761101S1, Thrasher Research Fund, David and Lucile Packard Foundation 38454, Child Health Research Institute Transdisciplinary Initiatives Program, and NIH UL1 TR000093 for biostatistics consultation. Dr. Kurt is the recipient of the Thrasher Research Fund Early Career Award. A provisional patent application has been filed for a helmet design using the optimization strategy described in this paper and will be assigned to Stanford University.26 Royalties gained from any intellectual property granted for this work will be shared among the inventors, the department, and the school, according to Stanfords technology licensing policies.
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Appendix
Appendix
Helmet Impact Dynamics Modeling
The geometry we assume for the helmets is hemispherical shells, as shown in Fig. 7. In making the analysis, the following general assumptions are made for helmet dynamics:

(1)
The head and the helmet are coupled rigidly.

(2)
The head & helmet system is only moving in the vertical direction.

(3)
The normal vector of the contact area is always in the impact direction.
Airbag Impact Dynamics Modeling
In modeling the airbag impact dynamics, the following assumptons are made:

(1)
Air behaves like an ideal gas at the given conditions.

(2)
The airbag is flexible but inelastic.

(3)
Friction and the bending resistance of the airbag fabric are neglected.

(4)
The mass of the air is negligible.
The most general form of the equations of motion for a hemispherical airbag helmet is as follows:
where x represents the instanteous distance between the center of gravity (CoG) of the head and the ground, m is the total system mass, P(x) and A(x) are instantenous pressure and contact area values of the airbag and \(P_{atm}\) is atmospheric pressure. The physical constraint to the system depicted in (1) is
where \(d_{head}\) is the vertical distance of the CoG of the head to the helmet and \(d_{helmet}\) is the thickness of the helmet.
We assume a hemispherical geometry for the airbag helmet. We assume that the helmet would completely flatten out if it reached to the deformation limit. This indeed can not happen for Fig. 7 since the deformation is limited up to the skull. By using simple geometric relations, the contact area of the airbag helmet can be formulated as follows:
Note that this contact area can be adjusted with a scaling factor when the actual contact between the surface and the helmet is limited, especially in the case of complex geometries.
The initial volume of the airbag helmet is
Using (3) and (4), we find the instantenous volume of the airbag as a function of x
By using the assumption of ideal gas, we can find the instantenous pressure of the airbag helmet during the impact by simply equating P(x) V(x) with \(P_i V_i\)(initial conditions)
EPS Helmet Impact Dynamics Modeling
To model the impact dynamics of EPS helmets, we assume the same hemispherical shell geometry. The contact area of the EPS helmet is assumed to be of the form given in (3). The equation of motion for the EPS helmet is as follows:
where x represents the instantaneous distance between the center of gravity (CoG) of the head and the ground, m is the total system mass, \({\sigma _{EPS}}(\frac{x}{{{d_{helmet}}}})\) is the loading/unloading stressstrain curve depicted in Fig. 2a and A(x) is the instantenous contact area between the EPS helmet and the ground. The effective damping coefficient \(c_{EPS}\) was calculated by using relations for a bouncing elastic body on a rigid surface:36
where m is the total system mass, \(\Delta T\) represents the contact duration and \(\varepsilon \) corresponds to coefficient of restitution. In order to find \(c_{EPS}\), we need to solve for the coefficient of restitution, which is an implicit function of \(\Delta T\) as follows
where n is the number of bounces that the elastic body experiences and \(h_{0}\) is the height from which the head is dropped. We assume n to be 2 and \(\Delta T \) to be 10 ms based on our experimental observations.
Summary of Helmet Dynamics Parameters
We give an overview of the values of mass, helmet size and other critical parameters used for modeling helmet dynamics (Table 1).
Ogden Material Modeling
We use Ogden rubber material model that is widely used for modeling foams.38 Assuming a zero lateral stress and zero Poisson’s ratio, we can formulate the material stress as follows
where \(\lambda \) is defined as the stretch ratio (i.e., the ratio between the final helmet thickness to the initial helmet thickness) and α, β and μ are material constants.
We use the first order approximation of (10) to model polymeric foams. The equation reduces to the following form
For EPS20, the Ogden material parameters are given below in the table for the firstorder approximation Table 2.
For ideal foam optimization in Fig. 5a, the limits we used for the above parameters are as follows
These limits lead to a semiconstrained optimization and give the opportunity to minimize peak accelerations at varying helmet sizes, as shown in Fig. 5a.
ATD Rfor Different Impact Orientations
Below, we represent the parietal and vertex experimental results for ATD tests, which were previously depicted in Fig. 3b and Table 3.
Hövding Geometry
Since Hövding is a relatively new product, its geometry is not wellknown. Therefore, we show top, side and front views of Hövding in an inflated state in Fig. 8.
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Kurt, M., Laksari, K., Kuo, C. et al. Modeling and Optimization of Airbag Helmets for Preventing Head Injuries in Bicycling. Ann Biomed Eng 45, 1148–1160 (2017). https://doi.org/10.1007/s1043901617321
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Keywords
 Traumatic brain injury
 Concussion
 Head trauma
 Bicycle helmets
 Airbag helmets