## Abstract

The circumstances in which we mechanically test and critically assess human calvarium tissue would find relevance under conditions encompassing real-world head impacts. These conditions include, among other variables, impact velocities, and strain rates. Compared to quasi-static loading on calvaria, there is less reporting on the impact loading of the calvaria and consequently, there are relatively fewer mechanical properties on calvaria at relevant impact loading rates available in the literature. The purpose of this work was to report on the mechanical response of 23 human calvarium specimens subjected to dynamic four-point bending impacts. Impacts were performed using a custom-built four-point impact apparatus at impact velocities of 0.86–0.89 m/s resulting in surface strain rates of 2–3/s—representative of strain rates observed in vehicle collisions and blunt impacts. The study revealed comparable effective bending moduli (11–15 GPa) to the limited work reported on the impact mechanics of calvaria in the literature, however, fracture bending stress (10–47 MPa) was relatively less. As expected, surface strains at fracture (0.21–0.25%) were less compared to studies that performed quasi-static bending. Moreover, the study revealed no significant differences in mechanical response between male and female calvaria. The findings presented in this work are relevant to many areas including validating surrogate skull fracture models in silico or laboratory during impact and optimizing protective devices used by civilians to reduce the risk of a serious head injury.

## 1 Introduction

Establishing the mechanical response of the human calvarium is required to quantify the onset of a traumatic head injury such as a skull fracture. Knowledge of such onsets allows helmet and automobile industries to then make informed decisions on enhancing protection against skull fractures in applications such as military and sports [1,2]. Furthermore, defining the mechanical onset of a skull fracture allows one to confidently validate physical surrogate and computational models to accurately simulate a fracture. As we explore novel testing methodologies on skull and brain tissue to determine mechanical properties [3], we must ensure the manner of testing contains conditions most pertinent to real-world head impacts. These conditions can include appropriate impact speeds and strain rates. With this direction of testing, engineers and scientists can be assured that subsequent mechanical properties used to verify surrogate models are derived from impact characteristics associated with real-world head injuries. The testing on full surrogate or cadaveric head models in large-scale impact testing equipment such as a linear drop tower, pendulum, pneumatic linear impactor, and oblique testing apparatuses have been well documented in the literature to simulate real-world head impacts [3–8]. This work performed controlled and small-scale impacts on human calvarium tissue with resulting strain rates applicable to rates observed in real-world head impacts [9,10].

The quasi-static bending of calvaria has been documented in the literature to investigate skull fracture mechanics [11–14]. Work by Hubbard was one of the earliest studies to perform three and four-point bending on adult parietal specimens at quasi-static deformation rates with strain rates within 0.01/s [14]. Similarly, Auperrin et al. and Delille et al. performed three-point quasi-static bending at displacement rates of 10 mm/min for fresh-frozen frontal and parietal specimens [12,13]. More recently, in a short communication, Adanty et al. performed four-point quasi-static bending on embalmed calvaria at displacement rates of 2 mm/min resulting in strain rates of 10^{−4}/s [11]. While these findings have been appropriate contributions to understanding skull fracture mechanics, quasi-static loading conditions are not associated with the sudden head impact conditions observed in real-world traumatic head impacts. Indeed, the typical strain rates observed from these studies were less than 1/s and though this rate may be appropriate to carefully note the fracture patterns of the skull and quantify mechanical properties, these strain rates are significantly less than the rates observed in real-world head impacts [9,10]. Most real-world head impacts during ballistics, falls, sports, and vehicle accidents endure intermediate to high strain rates of 1–10^{3}/s to the skull [9,10]. It is at these high-loading rates where we require further quantification of mechanical properties of skull tissue to accurately validate surrogate or computational skull models and then optimize helmet and protective devices accordingly.

The primary focus of this paper was to quantify the mechanical response of the calvarium related to impact, however, the effect of sex on mechanical response comes into question. To our knowledge, the comparison of mechanical response between sexes for the bending of calvaria has not been reported in the literature to date. Most surrogate and computational human head models utilized in injury biomechanics are typically validated from experimental data of the 50th percentile male population. This limits the conclusions of the mechanical outputs for these models to the male population as opposed to the general population consisting of males and females. Indeed, an initial understanding of any potential differences in mechanical response between sex may inform if sex-specific skull models are necessary for future design paradigms.

The purpose of this study was to report on the mechanical response of human calvaria in dynamic four-point bending impacts. A baseline understanding of skull mechanics during dynamic bending impacts can provide a critical direction toward (1) advancing helmet protection to prevent skull injuries and (2) an initial point of reference to validate physical surrogate or computational skull models to simulate fracture. In addition, the following null hypothesis was tested in this study: *no significant differences in mechanical response between male and female calvaria*.

## 2 Methods

The methods and protocols associated with this work were approved by the University of Alberta Research Ethics Board (ID: Pro00089218). All specimens came from individuals in the University of Alberta Anatomical Gift Program. All individuals were examined for pre-existing bone pathology based on provided medical records to ensure no pathological factors had influenced the subsequent results.

### 2.1 Specimens.

Calvarium specimens of curved beam geometry were extracted from the frontal or parietal regions of 23 male (*n* = 11) and female (*n* = 12) embalmed calvaria using an autopsy saw (Fig. 1). To obtain the frontal specimens, a point was landmarked approximately 1.5 cm inferior to the coronal suture and then a rectangular beam model (55 mm in length and 8 mm in width) was used to outline and extract one frontal specimen (Fig. 1) [15]. To obtain the parietal specimens, we located the center of the squamosal suture and landmarked a point approximately 1.5 cm superior to the suture to extract one specimen [15]. The orientation in which the specimens were extracted (Fig. 1) was preferred to avoid the coronal and sagittal sutures on the skull, as well as to avoid the challenges of cutting the specimens at regions with excessive curvature. Excessive curvature can be observed at bony prominences of the skull such as the metopic ridge, parietal eminence, frontal eminence, and particularly toward the supraorbital ridge and temporal ridge. Due to sample size constraints, one specimen, as opposed to multiple specimens, was extracted per calvarium. In addition, one specimen extracted per calvarium over multiple specimens avoids the challenge of cutting at regions of excessive curvature as discussed and avoids the potential effect of intraspecimen variation on the mechanical response. The specimens were then scanned using microcomputed tomography (CT) at a resolution of 18 *μ*m (Bruker-Skyscan 1176, Kontich, Belgium). Scanning parameters included a 90 kV X-ray tube voltage, 278 *μ*A X-ray current, 1 mm Al filter, 300 ms integration time, frame average of *n* = 3, and 0.7 deg angular rotation step [15]. Using the micro-CT scanned images for each specimen, the second moment of inertia (*I*) and the half-thickness (Fig. 2(a)) at the approximate center of the specimen was computed by way of third-party software (BoneJ-an ImageJ plugin) [16] and verified using a secondary software (Computed Tomography (CT)-Analyzer-version 1.10, Bruker-Skyscan). The surface radius of curvature (ROC) was computed in Geomagic software by three-dimensional systems (Rock Hill, SC) (Fig. 2(b)). Table 1 provides a geometric description of the specimens. Specimen width, thickness, and second moment of inertia were determined to estimate bending stress. Specimen ROC was computed to ensure ROC was ten times the thickness of the specimens on average to assume a straight beam when applying the Euler–Bernoulli beam theorem to compute bending stress [17]. Before testing and for the maintenance of hydration, the specimens were stored in unbuffered aqueous formaldehyde 37% embalming fluid, which is composed of 4% phenol, 4% formalin (37% concentration), 8% glycol, 8% ethyl alcohol (95% concentration), and 76% water. Additional details on the method of extraction and micro-CT imaging and analysis of calvaria are published in a recent article by the present authors [15].

Age and geometry | Male (n = 11) | Female (n = 12) | All specimens (n = 23) |
---|---|---|---|

Age | 84.6 (78.2, 90.9) | 88.0 (82.3, 93.7) | 86.4 (82.4, 90.3) |

Length (mm) | 51.68 (50.02, 53.34) | 52.12 (50.83, 53.38) | 51.90 (50.95, 52.85) |

Width (mm) | 8.46 (8.16, 8.75) | 8.76 (7.80, 9.71) | 8.61 (8.13, 9.09) |

Thickness (mm) | 6.79 (5.62, 7.96) | 7.42 (6.81, 8.02) | 7.12 (6.52, 7.72) |

Second moment of inertia (I) (m^{4}) | 2.18 × 10^{−10} (1.51 × 10^{−10}, 2.85 × 10^{−10}) | 2.74 × 10^{−10} (2.09 × 10^{−10}, 3.38 × 10^{−10}) | 2.47 × 10^{−10} (2.02 × 10^{−10}, 2.91 × 10^{−10}) |

Outer surface ROC (mm) | 69.04 (63.25, 74.83) | 62.16 (56.77, 67.56) | 65.45 (61.53, 69.37) |

Inner surface ROC (mm) | 74.98 (26.13, 123.82)^{a} | 47.7 (37.46 to 57.97) | 60.75 (38.17, 83.33) |

Age and geometry | Male (n = 11) | Female (n = 12) | All specimens (n = 23) |
---|---|---|---|

Age | 84.6 (78.2, 90.9) | 88.0 (82.3, 93.7) | 86.4 (82.4, 90.3) |

Length (mm) | 51.68 (50.02, 53.34) | 52.12 (50.83, 53.38) | 51.90 (50.95, 52.85) |

Width (mm) | 8.46 (8.16, 8.75) | 8.76 (7.80, 9.71) | 8.61 (8.13, 9.09) |

Thickness (mm) | 6.79 (5.62, 7.96) | 7.42 (6.81, 8.02) | 7.12 (6.52, 7.72) |

Second moment of inertia (I) (m^{4}) | 2.18 × 10^{−10} (1.51 × 10^{−10}, 2.85 × 10^{−10}) | 2.74 × 10^{−10} (2.09 × 10^{−10}, 3.38 × 10^{−10}) | 2.47 × 10^{−10} (2.02 × 10^{−10}, 2.91 × 10^{−10}) |

Outer surface ROC (mm) | 69.04 (63.25, 74.83) | 62.16 (56.77, 67.56) | 65.45 (61.53, 69.37) |

Inner surface ROC (mm) | 74.98 (26.13, 123.82)^{a} | 47.7 (37.46 to 57.97) | 60.75 (38.17, 83.33) |

One male specimen had an inner ROC of 291.39 mm which is a major outlier. With this outlier removed, the average inner surface ROC for the males is 53.33 mm (95% CI 46.09, 60.57).

### 2.2 Specimen Preparation.

Preceding the commencement of mechanical testing, fiber Bragg gratings (FBGs) were adhered to the outer cortical and inner cortical surfaces of each specimen to quantify surface strains during testing (Fig. 3(a)). The FBGs were glued to the cortical surfaces using cyanoacrylate to ensure the FBGs remained intimately bonded to the surface. Scotch Tape (3 M) was applied on the glued surface followed by low heat from a heat gun (100 °F for 15 s) to facilitate the drying of the cyanoacrylate. After 10 h, the cyanoacrylate was fully dried, and the Scotch Tape (3 M) was removed to begin mechanical testing. The purpose of waiting at least 10 h was to ensure the cyanoacrylate was fully cured so that the FBGs remained securely bonded to the specimens for accurate measurements of strain during testing. Preliminary examinations on adhering FBGs to swine scapula found that the cyanoacrylate did not fully cure when waiting 5 h or less after investigating FBG bonding resilience. FBGs are 1 mm strain transducers in length (Fig. 3(a)) implemented in optical fibers and perturbations on the FBG, such as surface strain, can be quantified based on proportional changes in a Bragg wavelength (Δ*λ _{B}*) [18]. The sensitivity of the FBG is 1.2 pm/

*μ*ε. The application of cyanoacrylate on FBGs has demonstrated minimal effects on the FBGs' ability to quantify strain [19]. FBGs has been used in biomedical applications for invasive biosensing in humans and animals [20,21].

### 2.3 Mechanical Testing: Four-Point Dynamic Bending Impacts.

The specimens were placed in a guided custom-built four-point testing apparatus to perform impacts (Figs. 3(b) and 3(c)). The four-point bending configuration was chosen because this configuration produces as close as possible a state of pure bending at the midregion of the specimens. Therefore, failure was mainly attributed to bending stress and less so with a complex stress-state encompassing bending and shear that can arise in three-point bending. The velocities of the top impact fixture ranged between 0.86 and 0.89 m/s just prior to impact. These velocities were captured using a high-speed camera at 5000 frames per second (Phantom v61-1280 × 800 CMOS sensor) and verified using phantommulticam software. Given the mass of the top impact fixture attached to the guide rail was 2.62 kg (Fig. 3(b)), the impact kinetic energy was 0.97–1.04 J. The instrumentation for the guided top impact fixture included two inertially compensated piezo-electric force transducers (PCB model 208C05) for each impact fixture (Fig. 3(c)). Each specimen was set on the bottom fixtures and was free to move horizontally to avoid horizontal shear stresses within the specimen as much as possible during bending deformation. The free horizontal movement of the specimens holds no clinical relevance regarding injury, rather the intention was to carry out testing conditions related to pure bending and ensure fracture was associated with bending stress.

The mechanical response variables determined for each bend test are outlined in Table 2. All fracture properties of the specimens were documented at the observation of fracture initiation, where initiation of fracture was determined visually from instant playback video recorded from the high-speed camera. The authors applied the Euler–Bernoulli beam theory to make a simplified gross estimation of the stress-state of the calvaria [17]. One major assumption when applying the theorem is that the specimens are straight and if curved the specimen must have a thickness of ten times the ROC—the average specimens in our work satisfied this assumption for both inner and outer ROC thus the specimens were assumed to be straight (Table 1) [17]. A second major assumption is that plane sections of the specimens remain perpendicular to the neutral axis before deformation and remain perpendicular to the neutral axis during deformation [17,22]. A third major assumption is that the strains the specimens experience are small [17,22]. Both these assumptions were met since the specimens fractured at small strains of less than 0.5% [23] and we did not observe any noticeable changes in geometry during deformation just as observed with other brittle materials. Indeed, the calvarium is a heterogeneous structure, of nonuniform cross section, and anisotropic, which provide some limitations to the use of the beam theory [17]. Though, previous studies have applied the beam theorem to make parallel assumptions to estimate stress and thus draw comparisons between studies [12,13,24–27]. The FBGs detected a linear relationship between surface strain and time following the application of the force from the impact prongs on the outer cortical surface of the specimens (Fig. 4). At the initial stages of the impact where the impact prongs first came into sudden contact with the specimen, surface strain remained in an unstrained state and several milliseconds later strain increased linearly with time. Therefore, strain rates and effective bending moduli-slope of the stress–strain plot (Fig. 4) were reported for all the specimens at corresponding linear strain-time variations.

Mechanical response variables | Description |
---|---|

Force at fracture (N) | The force measured at time of fracture |

Bending moment at fracture (N·m) | The calculated bending moment endured by the specimen at time of fracture |

Tensile surface strain at fracture (%) Compressive surface strain at fracture (%) | The surface strain of the specimen at the time of fracture, measured using FBGs. Tensile strain was quantified on the inner cortical surface and compressive strain was quantified on the outer cortical surface (Fig. 3(a)) |

Tensile bending stress at fracture (MPa) Compressive bending stress at fracture (MPa) | The calculated bending stress endured by the specimen at the time of fracture, estimated using the Euler–Bernoulli beam theorem by assuming specimens were homogenous and of simple beam geometry [17] |

$\sigma tensile=\u2009M(c2)I$ | |

$\sigma compression=\u2009M(c1)I$ | |

where σ = stress at fracture (MPa)M = bending moment at fracture (N·m)I = second moment of inertia (m^{4})c = half-thickness-the distance from the centroid to the surface of the specimen (see Fig. 2(a)) | |

Tensile effective bending modulus (GPa) Compressive effective bending modulus (GPa) | The slope of the stress–strain plot (Fig. 4) associated with initial linear region of the strain-time plot. The term effective signifies that the calvaria are indeed three-layer composite structures and that the modulus is a bulk estimate assuming the specimen is homogenous [17] |

Tensile strain rate (1/s) Compressive strain rate (1/s) | The slope at the initial linear region for each strain-time plot (see Fig. 4) |

Mechanical response variables | Description |
---|---|

Force at fracture (N) | The force measured at time of fracture |

Bending moment at fracture (N·m) | The calculated bending moment endured by the specimen at time of fracture |

Tensile surface strain at fracture (%) Compressive surface strain at fracture (%) | The surface strain of the specimen at the time of fracture, measured using FBGs. Tensile strain was quantified on the inner cortical surface and compressive strain was quantified on the outer cortical surface (Fig. 3(a)) |

Tensile bending stress at fracture (MPa) Compressive bending stress at fracture (MPa) | The calculated bending stress endured by the specimen at the time of fracture, estimated using the Euler–Bernoulli beam theorem by assuming specimens were homogenous and of simple beam geometry [17] |

$\sigma tensile=\u2009M(c2)I$ | |

$\sigma compression=\u2009M(c1)I$ | |

where σ = stress at fracture (MPa)M = bending moment at fracture (N·m)I = second moment of inertia (m^{4})c = half-thickness-the distance from the centroid to the surface of the specimen (see Fig. 2(a)) | |

Tensile effective bending modulus (GPa) Compressive effective bending modulus (GPa) | The slope of the stress–strain plot (Fig. 4) associated with initial linear region of the strain-time plot. The term effective signifies that the calvaria are indeed three-layer composite structures and that the modulus is a bulk estimate assuming the specimen is homogenous [17] |

Tensile strain rate (1/s) Compressive strain rate (1/s) | The slope at the initial linear region for each strain-time plot (see Fig. 4) |

### 2.4 Statistical Analysis.

Descriptive statistics on the mechanical response variables were reported. To test the following null hypothesis: *no significant differences in mechanical response between male and female calvaria*, an independent-samples t-test was performed using IBM SPSS version 25 (IBM Inc., Armonk, NY). The independent variables were male and female and the dependent variables were the mechanical response variables (Table 2). When a dependent variable was not normally distributed (verified using a Shapiro–Wilk test) for an independent variable a Mann–Whitney U test was performed. Therefore, the null hypothesis tested for this nonparametric test was *no significant differences in distributions or medians in mechanical response between male and female calvaria*. Using preliminary data from a pilot project, a power analysis determined a total minimum sample size of *N* = 12 specimens to ensure the probability of committing a type 2 error was 20% [28]. The alpha level was set at 0.05 where a *p*-value less than 0.05 was statistically significant.

Given this study was limited to 23 specimens, the frontal (*n* = 18) and parietal (*n* = 5) regions were grouped together as one region. Therefore, to justify omitting region as a second independent variable, an additional statistical test was performed to test the following null hypothesis: *no significant differences in mechanical response between frontal and parietal regions.* Due to an uneven sample size between the two regions, parametric and nonparametric tests were performed to test the null hypothesis.

## 3 Results

The descriptive statistics of the mechanical response with respect to sex are presented in Table 3. In addition, descriptive statistics of the mechanical response averaged across both sexes are presented in Table 4. No significant differences were reported in mechanical response between sex. All tests performed using a t-test are displayed as bar charts (Fig. 5). All tests performed using a Mann–Whitney U Test are displayed as histograms to present the distribution of the mechanical variables concerning sex (Fig. 6). By visual inspection, the distributional shapes between male and female compressive effective bending modulus were not comparable (Fig. 6). Thus, mean ranks were assessed between males (mean rank: 10.91) and females (mean rank: 13.00) for compressive effective bending modulus (*p* = 0.49). Distributional shapes between male and female tensile effective bending modulus were comparable, therefore, medians were assessed between males (8.58 GPa) and females (7.52 GPa). No significant differences between sex were reported for the geometric properties (Table 5). Table 6 provides a comprehensive comparison between this study's findings to the quasi-static and dynamic bending results of adult calvaria reported in the literature. From independent-sample t-tests and Mann–Whitney U tests, the differences in mechanical response between regions (frontal and parietal) were not significant (*p* > 0.05), thus, including region as a second independent variable was not necessary for this study.

Mechanical response variables | Male (M) and female (F) mean | 95% CI | Male (M) and female (F) median |
---|---|---|---|

Force at fracture (N) | M:224.75 ± 83.82 | M:168.43, 281.06 | M:200.76 |

F:222.93 ± 79.71 | F:172.28, 273.58 | F:217.62 | |

Bending moment at fracture (N·m) | M:1.70 ± 0.56 | M:1.32, 2.08 | M:1.51 |

F:1.70 ± 0.64 | F:1.29, 2.11 | F:1.65 | |

Tensile surface strain at fracture (%) | M:0.24 ± 0.08 | M:0.19, 0.30 | M:0.26 |

F:0.25 ± 0.09 | F:0.19, 0.31 | F:0.25 | |

Compressive surface strain at fracture (%) | M:0.22 ± 0.10 | M:0.15, 0.28 | M:0.25 |

F:0.20 ± 0.11 | F:0.13, 0.27 | F:0.20 | |

Tensile bending stress at fracture (MPa) | M:29.89 ± 9.10 | M:23.77, 36.00 | M:29.60 |

F:24.29 ± 7.01 | F:19.84, 28.74 | F:24.21 | |

Compressive bending stress at fracture (MPa) | M:29.73 ± 9.78 | M:23.16, 36.31 | M:28.47 |

F:23.54 ± 7.72 | F:18.63, 28.45 | F:23.45 | |

Tensile effective bending modulus (GPa) | M:11.74 ± 11.07 | M:4.30, 19.17 | M:8.58 |

F:11.03 ± 8.39 | F:5.71, 16.37 | F:7.52 | |

Compressive effective bending modulus (GPa) | M:10.83 ± 6.09 | M:6.74, 14.92 | M:10.15 |

F:18.27 ± 18.09 | F:6.78, 29.76 | F:11.14 | |

Tensile strain rate (1/s) | M:3.27 ± 2.56 | M:1.55, 4.99 | M:3.50 |

F:2.93 ± 1.40 | F:2.04, 3.82 | F:3.10 | |

Compressive strain rate (1/s) | M:2.89 ± 2.14 | M:1.46, 4.33 | M:2.75 |

F:1.87 ± 1.39 | F:0.98, 2.75 | F:1.43 |

Mechanical response variables | Male (M) and female (F) mean | 95% CI | Male (M) and female (F) median |
---|---|---|---|

Force at fracture (N) | M:224.75 ± 83.82 | M:168.43, 281.06 | M:200.76 |

F:222.93 ± 79.71 | F:172.28, 273.58 | F:217.62 | |

Bending moment at fracture (N·m) | M:1.70 ± 0.56 | M:1.32, 2.08 | M:1.51 |

F:1.70 ± 0.64 | F:1.29, 2.11 | F:1.65 | |

Tensile surface strain at fracture (%) | M:0.24 ± 0.08 | M:0.19, 0.30 | M:0.26 |

F:0.25 ± 0.09 | F:0.19, 0.31 | F:0.25 | |

Compressive surface strain at fracture (%) | M:0.22 ± 0.10 | M:0.15, 0.28 | M:0.25 |

F:0.20 ± 0.11 | F:0.13, 0.27 | F:0.20 | |

Tensile bending stress at fracture (MPa) | M:29.89 ± 9.10 | M:23.77, 36.00 | M:29.60 |

F:24.29 ± 7.01 | F:19.84, 28.74 | F:24.21 | |

Compressive bending stress at fracture (MPa) | M:29.73 ± 9.78 | M:23.16, 36.31 | M:28.47 |

F:23.54 ± 7.72 | F:18.63, 28.45 | F:23.45 | |

Tensile effective bending modulus (GPa) | M:11.74 ± 11.07 | M:4.30, 19.17 | M:8.58 |

F:11.03 ± 8.39 | F:5.71, 16.37 | F:7.52 | |

Compressive effective bending modulus (GPa) | M:10.83 ± 6.09 | M:6.74, 14.92 | M:10.15 |

F:18.27 ± 18.09 | F:6.78, 29.76 | F:11.14 | |

Tensile strain rate (1/s) | M:3.27 ± 2.56 | M:1.55, 4.99 | M:3.50 |

F:2.93 ± 1.40 | F:2.04, 3.82 | F:3.10 | |

Compressive strain rate (1/s) | M:2.89 ± 2.14 | M:1.46, 4.33 | M:2.75 |

F:1.87 ± 1.39 | F:0.98, 2.75 | F:1.43 |

Mechanical response variables | Mean | 95% CI | Median |
---|---|---|---|

Force at fracture (N) | 223.80 ± 79.82 | 189.28, 258.32 | 208.41 |

Bending moment at fracture (N·m) | 1.70 ± 0.59 | 1.45, 1.96 | 1.60 |

Tensile surface strain at fracture (%) | 0.25 ± 0.08 | 0.21 to 0.28 | 0.25 |

Compressive surface strain at fracture (%) | 0.21 ± 0.10 | 0.16 to 0.25 | 0.21 |

Tensile bending stress at fracture (MPa) | 26.97 ± 8.39 | 23.34, 30.60 | 24.84 |

Compressive bending stress at fracture (MPa) | 26.50 ± 9.13 | 22.55, 30.45 | 25.73 |

Tensile effective bending modulus (GPa) | 11.37 ± 9.54 | 7.25, 15.50 | 7.95 |

Compressive effective bending modulus (GPa) | 14.71 ± 13.96 | 8.68, 20.75 | 10.21 |

Tensile strain rate (1/s) | 3.09 ± 2.00 | 2.23, 3.96 | 3.40 |

Compressive strain rate (1/s) | 2.35 ± 1.82 | 1.57, 3.15 | 2.09 |

Mechanical response variables | Mean | 95% CI | Median |
---|---|---|---|

Force at fracture (N) | 223.80 ± 79.82 | 189.28, 258.32 | 208.41 |

Bending moment at fracture (N·m) | 1.70 ± 0.59 | 1.45, 1.96 | 1.60 |

Tensile surface strain at fracture (%) | 0.25 ± 0.08 | 0.21 to 0.28 | 0.25 |

Compressive surface strain at fracture (%) | 0.21 ± 0.10 | 0.16 to 0.25 | 0.21 |

Tensile bending stress at fracture (MPa) | 26.97 ± 8.39 | 23.34, 30.60 | 24.84 |

Compressive bending stress at fracture (MPa) | 26.50 ± 9.13 | 22.55, 30.45 | 25.73 |

Tensile effective bending modulus (GPa) | 11.37 ± 9.54 | 7.25, 15.50 | 7.95 |

Compressive effective bending modulus (GPa) | 14.71 ± 13.96 | 8.68, 20.75 | 10.21 |

Tensile strain rate (1/s) | 3.09 ± 2.00 | 2.23, 3.96 | 3.40 |

Compressive strain rate (1/s) | 2.35 ± 1.82 | 1.57, 3.15 | 2.09 |

Geometry | p-value | F-statistic |
---|---|---|

Length (mm) | 0.65 | 0.32 |

Width (mm) | 0.83^{a} | — |

Thickness (mm) | 0.31^{b} | — |

Second moment of inertia (I) (m^{4}) | 0.20 | <0.01 |

Outer surface ROC (mm) | 0.07 | 0.12 |

Inner surface ROC (mm) | 0.13^{a} | — |

Geometry | p-value | F-statistic |
---|---|---|

Length (mm) | 0.65 | 0.32 |

Width (mm) | 0.83^{a} | — |

Thickness (mm) | 0.31^{b} | — |

Second moment of inertia (I) (m^{4}) | 0.20 | <0.01 |

Outer surface ROC (mm) | 0.07 | 0.12 |

Inner surface ROC (mm) | 0.13^{a} | — |

Mann–Whitney U test exact *p*-value.

Welch t-test for violating homogeneity of variance.

Authors | Q = quasi-static or D = dynamic impacts | E = embalmed or UE = unembalmed | M = male or F = female | Age | Loading rate (LR) and strain rate (SR) | Regions sampled | Force (N) | Bending moment (N·m) | Stress (MPa) | Strain (%) | Modulus (GPa) |
---|---|---|---|---|---|---|---|---|---|---|---|

Hubbard [14] | Q, three-point Q, four-point | E | — | N/A | SR: 0.01 s^{−1} | P | — | Four-point: 3.08 | — | Four-point: 0.51 | Three-point: 9.69 |

Delille et al. [13] | Q, three-point | UE | M, F | 70.95 | LR: 10 mm/min | F, P, T | — | — | — | — | • RP: 5.00 |

• LP: 4.90 | |||||||||||

• F: 3.80 | |||||||||||

• T:11.30 | |||||||||||

Auperrin et al. [12] | Q, three-point | UE | M | 74.8 | LR: 10 mm/min | F, P, T | — | — | — | — | • P: 5.00 |

• F: 3.81 | |||||||||||

• T: 9.70 | |||||||||||

Rahmoun et al. [29] | Q, three-point | — | M | 88 | LR: 10 mm/min | F, P, T coronal suture | — | — | — | — | • RP: 3.74 |

• LP: 4.53 | |||||||||||

• F: 3.28 | |||||||||||

• RT: 5.22 | |||||||||||

• LT: 6.00 | |||||||||||

• Coronal suture: 2.04 | |||||||||||

Lee et al. [26] | Q, three-point | E | M | M = 61 F = 86 | LR: 10 mm/min | F, P, T, O | — | — | • Bare bone skull 1 and 2: 42 and 53 | — | • Bare bone skull 1 and 2: 1.70 and 2.74 |

• Bone with periosteum attached skull 1 and 2: 68 and 99 | • Bone with periosteum attached skull 1 and 2: 2.28 and 3.95 | ||||||||||

Motherway et al. [30] | D, three-point | UE | M, F | 81 | • LR: 0.5, 1, 2.5 m/s | F, P | • 0.5 m/s at RP, LP and F: 734.6, 721.7, 1062.3 | — | • 0.5 m/s at RP, LP and F: 84.50, 82.13, 90.80 | — | • 0.5 m/s at RP, LP and F: 10.33, 5.70, 4.35 |

• SR at 0.5 m/s: 19–22 s^{−1} | • 1 m/s at RP, LP and F: 793.6, 584.3, 1035.9 | • 1 m/s at RP, LP and F: 82.98, 78.15, 102.60 | • 1 m/s at RP, LP and F: 9.44, 17.69, 4.87 | ||||||||

• SR at 1 m/s: 25–31 s^{−1} | • 0.5 m/s at RP, LP and F: 1161.9, 1228.6, 1315.9 | • 0.5 m/s at RP, LP and F: 123.12, 133.61, 126.91 | • 0.5 m/s at RP, LP and F: 12.80, 18.12, 16.34 | ||||||||

• SR at 2.5 m/s: 102–110 s^{−1} | |||||||||||

Zwirner et al. [31] | D, three-point | E | M,F | 48 | LR:2.5, 3.0, 3.5 m/s | F, P, T, O | • 2.5 m/s: 716 | — | • 2.5 m/s: 98 | — | — |

3.5 m/s: 1264 | • 3.0 m/s: 119 | ||||||||||

• 2.5 m/s at T: 638 | • 3.5 m/s: 130 | ||||||||||

• 3.5 m/s at T: 1136 | |||||||||||

This study | D, four-point | E | M, F | 86.4, (82.4, 90.3) | • LR: 0.86–0.89 m/s | F, P | 223.80 (189.28, 258.32) | 1.70 (1.45, 1.96) | • Tensile: 26.97 (23.34, 30.60) | • Tensile: 0.25 (0.21, 0.28) | • Tensile: 11.37 (7.25, 15.50) |

• SR: tensile = 3.09 s^{−1} (2.23, 3.96) | • Compression: 26.50 (22.55, 30.45) | • Compression: 0.21 (0.16, 0.25) | • Compression: 14.71 (8.68, 20.75) | ||||||||

• SR: compressive = 2.35 s^{−1} (1.57, 3.15) |

Authors | Q = quasi-static or D = dynamic impacts | E = embalmed or UE = unembalmed | M = male or F = female | Age | Loading rate (LR) and strain rate (SR) | Regions sampled | Force (N) | Bending moment (N·m) | Stress (MPa) | Strain (%) | Modulus (GPa) |
---|---|---|---|---|---|---|---|---|---|---|---|

Hubbard [14] | Q, three-point Q, four-point | E | — | N/A | SR: 0.01 s^{−1} | P | — | Four-point: 3.08 | — | Four-point: 0.51 | Three-point: 9.69 |

Delille et al. [13] | Q, three-point | UE | M, F | 70.95 | LR: 10 mm/min | F, P, T | — | — | — | — | • RP: 5.00 |

• LP: 4.90 | |||||||||||

• F: 3.80 | |||||||||||

• T:11.30 | |||||||||||

Auperrin et al. [12] | Q, three-point | UE | M | 74.8 | LR: 10 mm/min | F, P, T | — | — | — | — | • P: 5.00 |

• F: 3.81 | |||||||||||

• T: 9.70 | |||||||||||

Rahmoun et al. [29] | Q, three-point | — | M | 88 | LR: 10 mm/min | F, P, T coronal suture | — | — | — | — | • RP: 3.74 |

• LP: 4.53 | |||||||||||

• F: 3.28 | |||||||||||

• RT: 5.22 | |||||||||||

• LT: 6.00 | |||||||||||

• Coronal suture: 2.04 | |||||||||||

Lee et al. [26] | Q, three-point | E | M | M = 61 F = 86 | LR: 10 mm/min | F, P, T, O | — | — | • Bare bone skull 1 and 2: 42 and 53 | — | • Bare bone skull 1 and 2: 1.70 and 2.74 |

• Bone with periosteum attached skull 1 and 2: 68 and 99 | • Bone with periosteum attached skull 1 and 2: 2.28 and 3.95 | ||||||||||

Motherway et al. [30] | D, three-point | UE | M, F | 81 | • LR: 0.5, 1, 2.5 m/s | F, P | • 0.5 m/s at RP, LP and F: 734.6, 721.7, 1062.3 | — | • 0.5 m/s at RP, LP and F: 84.50, 82.13, 90.80 | — | • 0.5 m/s at RP, LP and F: 10.33, 5.70, 4.35 |

• SR at 0.5 m/s: 19–22 s^{−1} | • 1 m/s at RP, LP and F: 793.6, 584.3, 1035.9 | • 1 m/s at RP, LP and F: 82.98, 78.15, 102.60 | • 1 m/s at RP, LP and F: 9.44, 17.69, 4.87 | ||||||||

• SR at 1 m/s: 25–31 s^{−1} | • 0.5 m/s at RP, LP and F: 1161.9, 1228.6, 1315.9 | • 0.5 m/s at RP, LP and F: 123.12, 133.61, 126.91 | • 0.5 m/s at RP, LP and F: 12.80, 18.12, 16.34 | ||||||||

• SR at 2.5 m/s: 102–110 s^{−1} | |||||||||||

Zwirner et al. [31] | D, three-point | E | M,F | 48 | LR:2.5, 3.0, 3.5 m/s | F, P, T, O | • 2.5 m/s: 716 | — | • 2.5 m/s: 98 | — | — |

3.5 m/s: 1264 | • 3.0 m/s: 119 | ||||||||||

• 2.5 m/s at T: 638 | • 3.5 m/s: 130 | ||||||||||

• 3.5 m/s at T: 1136 | |||||||||||

This study | D, four-point | E | M, F | 86.4, (82.4, 90.3) | • LR: 0.86–0.89 m/s | F, P | 223.80 (189.28, 258.32) | 1.70 (1.45, 1.96) | • Tensile: 26.97 (23.34, 30.60) | • Tensile: 0.25 (0.21, 0.28) | • Tensile: 11.37 (7.25, 15.50) |

• SR: tensile = 3.09 s^{−1} (2.23, 3.96) | • Compression: 26.50 (22.55, 30.45) | • Compression: 0.21 (0.16, 0.25) | • Compression: 14.71 (8.68, 20.75) | ||||||||

• SR: compressive = 2.35 s^{−1} (1.57, 3.15) |

Verified through playback of high-speed video, fracture initiation originated at the tensile (inner cortical) surface and then propagated through the diploë and finally through the compressive (outer cortical) surface of the specimens as expected. Fracture did not always initiate at the location of the FBGs (center of the specimen) but rather anywhere between the two points of loading which was expected in four-point bending since bending moment is constant and maximum between the two points of loading. As a result of this, the strain response imposed on the FBGs when a fracture occurred elsewhere was measured.

## 4 Discussion

To our knowledge, this is the first set of works to study the mechanics of calvaria in a four-point bending impact modality. The mechanical response of 23 calvaria at strain rates (2 to 3/s on average) associated with real-world head impacts was reported [9,10]. No significant differences in mechanical response between sex were revealed.

An examination of the literature (see Table 6) disclosed two studies that conducted bending impacts on cranial specimens in a three-point configuration [30,31]. Motherway et al. did not clarify their approach in computing strain rates but reported a range between 20 and 100/s for impact velocities between 0.5 and 2.5 m/s [30]. Zwirner et al. conducted impacts at 2.5, 3.0, and 3.5 m/s but failed to disclose strain rates [31]. Nonetheless, at 2.5 m/s, Motherway et al. recorded greater fracture forces of 1161 N to 1315 N compared to the findings of Zwirner et al. which averaged 716 N [30,31]. In addition, the fracture forces of Motherway et al. at 1.0 m/s (584 N to 1035 N) [30] were greater compared to this study which documented an average of 223 N at 0.86–0.89 m/s. There is no single factor to explain the variation of fracture forces between this work and the literature, however, there are a few characteristics from each study that may contribute to differences in mechanical properties [30,31]. The first characteristic is the difference in bending configuration between this and previous works [30,31]. In the three-point configuration, the peak stress is located directly below the single point of impact on the specimen, whereas the peak stress is distributed over the distance between two points of impact in the four-point configuration. Additionally, in the four-point configuration, the greater number of pores located in the specimens' diploë distributed between the two points of impact may contribute to reducing overall bending strength compared to three-point loading [32]. This notion may be a contributing factor for observing lower stresses between this work's findings at 27 MPa for four-point loading and previous findings at 78–103 MPa in three-point loading [30]. Two studies that examined polymer and wood-based materials found that three-point bending yielded greater bending strength compared to four-point bending [33,34]. An additional consideration worth noting is that the kinetic energy (0.5·m·v^{2}) and geometry of the impacting mass may be distinct across studies despite similar impact velocities. This may also influence mechanical outcomes such as force and stress. A second characteristic between studies is morphometry, donor age, and treatment of specimens [15]. In this study, specimens were extracted from the frontal and parietal regions in which density, porosity, and diploë or cortical morphometry may be distinctive to temporal and occipital specimens employed by Zwirner et al. [15,31]. Zwirner et al. also sampled specimens from cadavers with an average age of 48 years old over a range of 3 weeks to 94 years old, however, this work sampled from donors at an average age of 86 years old and Motherway et al. [30] from 81 years old. Age may influence the biomechanical response of the calvarium and warrants further investigation [30,31]. Concerning tissue treatment, this study extracted embalmed tissue, Zwirner et al. [31] extracted samples at a median of 70-h postmortem with a freeze-thaw procedure prior to testing, and Motherway et al. [30] obtained samples from fresh-frozen cadavers. According to previous studies, different tissue treatments or preservation methods may or may not significantly affect the mechanical response of bone [35–45], nevertheless, future work should investigate if fresh, fresh-frozen, and embalmed tissue influence the calvarium's mechanical response.

Despite differing characteristics between this work and previous findings, one optimistic observation is that the bending moduli from the work of Motherway et al. (5–19 GPa) are in the range of effective bending moduli reported in this work (7.25–20.75 GPa), see Table 6. Conversely, and as shown in Table 6, the average bending moduli derived from three-point quasi-static bending (1.70–11.30 GPa) [12,14,26,29] falls in the lower end of the spectrum for the range of bending moduli presented in this study (7.25–20.75 GPa) and the findings of Motherway et al. (5–19 GPa) [30]. This can simply be attributed to contrary loading and strain rates between the dynamic and quasi-static testing which also verifies the viscoelastic nature of biological specimens—strain rate dependent. In any case, one must cautiously compare information gathered from quasi-static and impact testing. In quasi-static loading, the force applicator is preloaded on the specimen, the load is applied in a way that inertial effects of the specimen can be ignored, and strain rates remain relatively constant until fracture. Conversely, as observed in this study, the force applicator is not preloaded on the specimen resulting in a sudden impact. As shown in Fig. 4, the strain-time curve for one sample exhibits a nonlinear behavior until about 0.05%, this could be the region where the impact prongs are first adapting to full contact with the specimen. After about 0.05% the strain then begins a gradual linear response as well as its corresponding stress–strain curve. It was in this linear region modulus was determined such that the strain-rate remained constant (Fig. 4). Since such a sudden impact is a defining characteristic of real-world head impacts, future work should consider the effect of inertial response such as the specimen acceleration during impact on mechanical properties. This study is the first set of works to document strains on both the inner cortical and outer cortical surfaces of calvaria using FBGs. The strains at fracture in this study are marginally less (0.21–0.25%) than the tensile surface strains reported in quasi-static bending by Hubbard [14] (0.33–0.76%) and Adanty et al. [11] (mean: 0.31%). This may suggest that greater strain rates observed during impact bending can initiate a fracture at lower strains compared to specimens subjected to quasi-static bending at lower strain rates [46]. To support this suggestion, a study on tensile loading of human cranial bone found a negative linear regression coefficient deemed significant between breaking strain and strain rate [47]. Therefore, from a logical standpoint, the strain at fracture attributed to a strain rate of 2–3 s^{−1} in this study should be less than the strain reported by Hubbard that yielded a strain rate of 0.01 s^{−1}.

There was no evidence to reject the null hypothesis: *no significant differences in mechanical response between male and female calvaria*. Likewise, recent work by Zwirner et al. produced sex-independent force and stress measurements [31]. Compared to work on load-bearing bones in humans, there is scarce literature differentiating the mechanical properties of male and female crania [48]. One probable factor for observing no differences was the comparable geometry in specimens between sex since specimen thickness and second moment of inertia are associated with the computation of bending stress and modulus. Similarly, surface strain is a measure proportional to the specimen geometry [14]. Since the calvarium is a three-layered composite structure, Hubbard demonstrated that surface strain on the cortical layer is a function of calvarium thickness and bending stiffness, where stiffness is related to second moment of inertia and elastic modulus [14]. For surrogate and computational modelers, this work's findings on sex-independent mechanical response may infer that future calvarium models irrespective of sex are appropriate to model injury for the general population. However, a greater sample size across different age groups and greater impact speeds are required to support this work's findings.

## 5 Limitations

### 5.1 Specimens.

The findings from this study are biased toward an older age population in the province of Alberta, Canada, therefore, future work is encouraged to perform testing on a younger age cohort or a specific population more vulnerable to head injury. Despite performing a power analysis, the authors of this work acknowledge limitations on the sample size. The specimens in this work were sampled by convenience, meaning it was the maximum number of specimens accessible to the authors at the time of the study. The authors limited the extraction of the specimens on each calvarium in a horizontal orientation (parallel to sutures) at the frontal (medial to lateral) and parietal (anterior to posterior) regions as displayed in Fig. 1. This is one orientation out of many in which the specimens could have been extracted, however, bony prominence regions can vary in size and geometry between individual calvaria. For example, extracting specimens in the vertical orientation at the frontal region would require carefulness when cutting through the metopic ridge and the frontal eminence as the curvature considerably changes when cutting superior to inferior or vice versa. The result of this would then be calvarium beam specimens with a complex curvature that would then be challenging to configure for impact in four-point bending. Nonetheless, the authors suggest future studies consider harvesting specimens at multiple orientations to account for the anisotropic nature of bone during mechanical testing.

This study employed embalmed calvarium specimens. In biomechanical research, tissue is preserved with embalming fluid for the following reasons: (1) if fresh tissue is inaccessible and sufficient sample size is required, (2) if additional time and care are necessary to apply instrumentation and prepare for mechanical testing, and (3) to prevent the transmission of infectious diseases such as AIDS (HIV), Hepatitis, and more recently SARS-CoV-2 virus [37]. To determine the mechanical effects embalming may have on fresh tissue, we investigated a dissertation by Crandall that mechanically tested 150 bovine ribs [37]. Similar to the geometry of calvaria employed in this study, the ribs were curved and comprised cortical and trabecular tissue. The ribs were also tested under bending conditions like the calvaria in this study, except under three-point quasi-static loading. The calvaria used in this study were subject to formaldehyde embalming fluid, similarly, two groups of ribs from Crandall's work were subject to formaldehyde solution by Michigan Anatomical Fluid and formaldehyde by University of Virginia Fluid [37]. Both groups of ribs treated with formaldehyde were found to have a less than 12% difference in elastic modulus, yield stress, ultimate stress, and ultimate strain compared to the fresh ribs which were not statistically significant [37]. Rather, the frozen group of ribs had up to a 25% and 28% difference compared to fresh and embalmed ribs, respectively, which were both statistically significant [37]. Meanwhile, previous studies cited by Crandall showed that the freezing process was an appropriate preservation technique that did not affect material properties [38,39]. In addition to Crandall's findings, Nazarian et al. found that formalin altered the viscoelasticity of bone significantly, but bending stiffness, modulus of elasticity, yield displacement, yield load, yield strain, and yield strength was not significantly different between frozen, formalin-fixed, and fresh murine bone [40]. Mick et al. [41] and Topp et al. [35] demonstrated that there were no significant differences between embalmed and fresh-frozen tissue of the human femur. Wilke et al. showed the range of motion for L1-L2 spinal segments from 16-week-old calves was significantly reduced in the embalmed group compared to the fresh group [42]. Earlier studies by Evans and Carothers et al. showed that formalin can significantly increase the strength of human long bones under tensile and bending loading [43,44]. Burkhart et al. showed that the axial stiffness of human femora significantly increased by 14% in the embalmed group compared to the fresh group [45]. In summary, embalming of tissue is indeed a limitation but its mechanical effect on fresh tissue is inconsistent across the literature and we cannot be certain on how much it influences calvarium exclusively. Therefore, when it comes to the human calvarium it is unknown how much of a biomechanical effect embalming has on fresh calvaria. We encourage future studies to explore different forms of tissue preservations and their mechanical effect on skull or calvarium tissue to formulate if the use of embalmed skull tissue is truly a viable source or not for biomechanical testing.

### 5.2 Mechanical Testing.

As discussed earlier, most specimens experienced fracture initiation between the two points of loading, however, few specimens experienced fracture initiation under one of the two points of loading. One possible reason for this occurrence is because, at that point of loading, the specimen contained a considerable number of pores in the diploë that weakened the specimen at that area and thus initiated fracture at that point. This may not necessarily be a limitation with respect to the specimen's natural morphology, however, it may be a limitation to consider when using four-point bending for impact on a nonhomogeneous structure. The pattern or shape of the fracture was not reported as this study was concerned with quantifying mechanical response at fracture. However, future work may consider documenting qualitative observations on fracture shapes and patterns—butterfly fracture, linear fracture, or oblique fracture as these are important characteristics to consider when developing models to mimic skull fracture. The authors acknowledge the limitations of modeling the specimens as an Euler–Bernoulli beam to estimate bending stress. The authors attempted to model the bending stress using the curved beam theorem based on equations proposed by Roark et al. [17]. The stress at fracture derived from the curved beam theorem had an average percent difference of 1.66% in tensile stress at fracture and a difference of 2.13% in compressive stress at fracture with respect to fracture stress derived from the Euler–Bernoulli beam theorem. This percent difference is less than the 4–5% error advised by Roark et al. when applying the Euler–Bernoulli beam theorem on a curved beam [17]. One possible explanation for this small difference between the two theorems is that the calvarium specimens fractured at small strains of less than 0.5% which satisfies the assumption of small deformation when applying the Euler–Bernoulli beam theorem.

## 6 Conclusion

To express the importance of testing human calvaria under conditions most applicable to real-world head impacts (strain rates > 1 s^{−1}), the mechanical response of calvaria subjected to four-point bending impacts was determined. This study documented effective bending moduli that were in line with previous studies that performed three-point impact bending on calvaria, however, fracture bending stress was less. Surface strains were relatively less compared to previous findings that quantified surface strains during quasi-static bending. No significant differences in mechanical response between male and female calvaria were established.

## Acknowledgment

The authors would like to thank Hugh Barrett, Jason Papirny, and Dr. Daniel Livy of the University of Alberta's Division of Anatomy for providing access to human cadavers and the necessary aid during dissection. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. The authors declare that the roles of this study's sponsor included the financial resources to complete the study, the analysis and interpretation of data, the writing of the paper, and the decision to submit the paper for publication.

## Funding Data

U.S. Army Combat Capabilities Development Command—Army Research Laboratory and the International Technical Center Americas (Cooperative Agreement No. W911NF-19-2-0336; Funder ID: 10.13039/100006754).