Toward a Consistent Framework for Describing the Free Vibration Modes of the Brain

Modal analysis has been a hallmark of structural vibration engineering for the last century as a method of estimating resonant frequencies and the associated deformation mode shapes in human-made structures [1–4]. In recent years, vibration mechanics in the human body have also received increased focus. As humans are exposed to various sources of vibrations in the natural and built environments, different tissues and organs can be susceptible to the harmful effects of mechanical resonance. Hand–arm vibration syndrome is a well-established example of harmful vibration exposure to the body [5,6], while on the other hand whole body vibration has been shown to have various positive use cases in the rehabilitation of walking and balance skills [7,8]. Humans are exposed to vibrations constantly while riding in automobiles [9,10], operating power tools [11,12], or using heavy machinery [13,14], among other situations. However, the determination of what vibration frequencies may be harmful or helpful relies on a strong understanding of vibration mechanics in the body.

Early work in vibration analysis of the head was performed by Franke with the objective of determining the natural frequencies of the skull using living subjects and fresh cadavers [15]. Due to the poor coupling between the mechanical drivers used to generate vibrations and the heads of the living subjects, impedance measurements were not possible. Using a fresh cadaver, two natural frequencies were recorded at approximately 600 and 1000 Hz [15]. Gurdjian et al. applied vibrations to a volunteer test subject and a fresh skull filled with silicone gel [16]. They found three resonant frequencies between 300 and 920 Hz. Stalnaker et al. used a mechanical shaker table to apply vibrations of different frequencies to a fresh human cadaver and found two natural frequencies at approximately 130 and 800 Hz [17]. Hakansson et al. investigated the natural damped frequencies of the skull in vivo by applying acceleration to the skull of a patient using titanium anchors implanted for hearing aids [18]. The first 20 natural frequencies measured were between 1 kHz and 10 kHz. Such early studies focused on evaluating natural frequencies of the head by measuring the response of the skull, treating the tissues of the skin, muscle, brain, and meninges as lumped masses. As a result, such studies might have only captured the resonant behavior of the skull, not the brain.

More recently, advances in noninvasive medical imaging have enabled better approaches to measuring brain vibration mechanics in vivo. Escarcega et al. used dynamic mode decomposition (DMD) to calculate the natural frequencies and vibration modes of strain in the brain tissue during linear and angular acceleration of the head using tagged magnetic resonance imaging [19,20]. They found the lowest natural frequency of the brain to be between 6.66$±$0.86 Hz for head rotation and 11.19$±$1.18 Hz for linear acceleration, and characterized the associated mode shapes. Laksari et al. used similar tagged magnetic resonance imaging data from the same group to calibrate a reduced-order rigid body dynamics model of relative skull/brain motion [21,22]. A natural frequency of approximately 15 Hz was calculated, but no applicable mode shape could be drawn from this simplified model. Alshareef et al. measured the vibration behavior of postmortem brains using sonomicrometry and identified natural frequencies in the range of 12–20 Hz, but did not pursue a rigorous description of the associated vibration modes [23].

Additional experimental work has been performed on the skull separate from the rest of the head. Khalil et al. performed an experimental natural frequency analysis using dry 50th percentile male and 5th percentile female skulls [24]. Eleven natural frequencies between 1385 and 4636 Hz were identified for the male skull, and six frequencies between 1641 and 5000 Hz were identified for the female skull. Horacek et al. performed a similar analysis on a dry and nondry male skull using a roving hammer test [25]. Ten natural frequencies between 1388 and 2762 Hz were identified. Eslaminejad et al. performed the same roving hammer test on a section of the upper cranium separated from the rest of the skull and found three natural frequencies: 497, 561, and 1246 Hz [26]. Since natural frequencies depend on skull geometry, the variation in natural frequencies observed in different studies may be due to subject-specific differences in skull size and shape. Researchers in such studies have also not pursued descriptions of the vibration modes associated with the natural frequencies identified, which severely limits comparison of results between skulls tested.

Other researchers have used mathematical models and numerical simulations to calculate the vibration mechanics of the brain. Tseng et al. performed modal analysis on a brain tissue model without the constraint of the skull and found six natural frequencies between 1.87 and 14.57 Hz [27]. Li and Luo used a two-dimensional (2D) model of the skull, cerebrospinal fluid (CSF), and brain tissue in the transverse plane and found that the fundamental natural frequency was 137 Hz [28]. However, such studies have reported only the natural frequencies observed, and have not comprehensively described the associated mode shapes. Tse et al. performed a modal analysis on a subject-specific model of the head–neck system, which includes the skull, spine, cartilage, brain tissue, and CSF [29]. Twenty-five natural frequencies between 36 and 357 Hz were identified; however, the mode shapes are only described in terms of which tissues in the model were excited. El Baroudi et al. developed a simplified analytical model of the brain, CSF, and skull, treating the system as a set of concentric annuli [30]. Eight frequencies between 26 and 73 Hz were identified and the associated mode shapes were described, but the simplified geometry limited applicability to the brain.

Researchers have also extensively used DMD to describe the vibration behavior of numerical head models. McLean performed a DMD of the natural frequencies of a finite element (FE) head model at different angles along the midsagittal plane [31]. Three natural frequencies between 7 and 425 Hz were identified but a description of the mode shapes was not provided. Laksari et al. used DMD on a FE model of sports impacts and found two natural frequencies at approximately 28 and 42 Hz, with a description of the time evolution of the associated modes [32]. Rezayaraghi et al. used DMD on simulations of impacts on various types of sporting helmets and observed a fundamental natural frequency in the range of 7–15 Hz, but did not report the associated vibration mode [33]. It is unclear how much of the variation in natural frequencies reported in the literature is due to different choices in material model selection or experimental procedure versus inherent biological traits such as tissue variation and subject-specific geometry. Additionally, many of the studies discussed provide insufficient or no description of the vibration modes observed in the brain which limits the ability to compare modes and their associated frequencies across different studies.

Most of the studies discussed above have focused on identifying the natural vibration frequencies of the brain with limited attention paid to the associated vibration modes. Such an approach has limitations. The deformation associated with each mode directly influences the tissue deformations at specific locations in the brain at that frequency, providing insight into whether the tissue is under shear, dilatational, or combined modes of deformation. This helps with the identification of the thresholds of acceptable deformation levels and helps providing likely maps of tissue damage in the brain matter. Additionally, without a description of the vibration mode associated with an observed or calculated natural frequency, it is difficult to establish whether the vibration behavior reported in a given natural frequency in one study is the same in another. For these reasons, a thorough understanding of the mode shapes should be part of the dynamic analysis of brain tissue.

In this technical brief, modal analysis of a FE model of the brain was used to identify the vibration modes of the brain. A framework for describing the vibration modes decomposing the brain into three basic planes and identifying classical two-dimensional vibration modes was then established. We propose the descriptions used as an approach for more unified communication of brain vibration dynamics which relies on established, simple notations to describe the complex vibration behavior of the brain.

For a system with N degrees-of-freedom, the characteristic equation will yield N eigenvalues, or natural frequencies, and N corresponding eigenvectors, or mode shapes. For the modal analyses discussed in the present study, only the lowest 20 vibration modes and their corresponding natural frequencies were evaluated.

Modal analysis was performed using the commercial FE code ls-dyna (ANSYS, Inc., Canonsburg, PA) and a FE model of the head. The Total Human Model for Safety (thumsv7) was selected for the analysis presented herein (Toyota Central R&D Labs, Inc., Nagakute, Japan). Various finite element head models have been developed to analyze the kinematics of the head and assess the risk of traumatic brain injury, a comprehensive review of which can be found in Ref. [35]. The thums model is more anatomically faithful than the head models used in previous numerical studies of head vibration mechanics, incorporating the white and gray matter of the brain, meningeal layers, CSF, skull and skull sutures, muscle, and skin with a higher mesh resolution than many previous models used. thumsv7 includes three models: 5th percentile female, 50th percentile male, and 95th percentile male. Analysis was performed on all three models to attain an estimate of the highest, lowest, and average natural frequencies expected of the general population.

To evaluate the natural frequencies of the head model alone, the full-body model was segmented at the base of the skull. A separate analysis was performed segmenting the model at the base of the neck. Inclusion of the neck did not have an effect on the lowest 20 vibration modes of the brain, so the head-only model was used for all subsequent analyses to reduce computational cost. The head model consisted of 256,823 elements, with the brain model consisting of 118,840 of those elements.

The CSF, superior sagittal sinus, and the white and gray matter of the brain in the thumsv7 head were modeled using a viscoelastic model, implemented as MAT_006 and MAT_061 in ls-dyna [36]. The plates and sutures of the skull were modeled using an isotropic elastic–plastic model, implemented as MAT_012. Where more comprehensive yield data were available, sections of the skull were modeled using piecewise variations of the isotropic plasticity formulation: MAT_024 and MAT_105. The meninges, scalp, and dental alveoli are modeled as linear elastic using ls-dyna MAT_001. Originally, the skin in the thumsv7 model used a fabric model (MAT_034), which is incompatible with implicit modal analysis. As such, the material model for the skin was changed to the linear elastic model as well. The validation testing originally performed on the thumsv7 model was repeated following the material model change, and negligible differences in brain motion were observed. All material properties of the thums model are listed in Table 1.

In the original thumsv7 model, the brain is modeled using a one-point numerical integration element formulation (ls-dyna ELFORM 1). The hourglass controls available for implicit analysis [37] were evaluated in the present study, but could not reduce the hourglass deformation to a satisfactory threshold during the modal analysis. As a result, the element formulation for all parts in the model was changed to an eight-point numerical integration (ELFORM 2), thus eliminating the source of nonphysical hourglass vibration modes. Such a formulation is less stable for large deformation problems, but produced results for the small-vibration modal analysis problem with less numerical error.

Modal analysis was performed on the thumsv7 model to compare the calculated frequencies and mode shapes with prior results from the numerical and experimental literature. The first ten natural frequencies were calculated for four different models: the brain alone; brain with falx, tentorium, meningeal layers, and CSF; the skull alone; and the full head. The subsection analysis was performed on a model with 50th percentile male head dimensions. Modal analysis was also performed on full head models with 5th percentile female and 95th percentile male dimensions to determine if head size changed the characteristic vibration modes.

To define a consistent descriptive framework for the three-dimensional deformation modes associated with each natural frequency, we created transverse, coronal, and parasagittal section views of each mode through the thickest part of the cerebrum as shown in Fig. 1. The vibration modes in each section were characterized as a superposition of rotation, translation, shear, and out-of-plane deformation (OPD) modes. The OPD modes were classified using the notation for circular and rectangular plate bending vibration modes depicted in Fig. 2 [38]. The model was found to exhibit bilateral symmetry, so rotation, translation, and shearing modes are described in terms of their location in one half of the brain.

The range of the first ten natural frequencies for different sections of the head is summarized in Table 2. The brain alone has the lowest natural frequencies due to its low stiffness relative to the other tissues of the head. The addition of the surrounding support structures of the meningeal layers, falx, tentorium, and CSF results in a higher range of natural frequencies. The natural frequencies of the skull are 2 orders of magnitude higher, consistent with the skull being a much more rigid structure. The addition of skull, muscles, eyes, and skin acts similarly to the addition of meninges, slightly increasing the frequency response by further constraining the brain matter.

The first 20 natural frequencies of the full head for the 5th percentile female and 50th and 95th percentile male heads are shown in Fig. 3, and the corresponding vibration modes are provided in the Supplemental Materials on the ASME Digital Collection. For each mode number, the natural frequency decreases as the head size increases due to the larger mass of the brain. There is a range of 6–7 Hz between the first and twentieth natural frequency for all three head models. For each mode number, the natural frequency of the 5th percentile female is 18–19% larger than that of the 95th percentile male.

Mode shapes for the 5th percentile female, 50th percentile male, and 95th percentile male models were the same for each mode. The vibration modes for the first, tenth, and twentieth modes can be seen in Fig. 4. Each mode shape for the full head model is described in Table 3, and animated images of the section views of each mode can be found in the Supplemental Materials. While the lower mode numbers exhibit simple behavior generally consisting of a single shearing, OPD, or rotational behavior—most commonly around the center of mass (COM) of the brain or the COM of each hemisphere—the higher mode numbers demonstrate a complex superposition of multiple behaviors. In the cases where multiple behaviors were present in the same plane and mode number, the dominant behavior was selected as a descriptor.

Our calculated range of 1.1–2.3 kHz for the natural frequencies of the skull is consistent with the results of previous experimental modal analysis studies using dry skulls and in vivo testing [18,24,25]. Natural frequencies are geometry dependent, so subject specificity may account for part of the difference observed between the experimental results and the present study. The lowest natural frequencies for the full head, calculated in the range of 14–17 Hz, are similar to those of previous finite element studies performed by Tseng et al. [27] and Ruan and Prasad [39]. The addition of the falx, tentorium, meningeal layers, and cerebrospinal fluid stiffened the response of brain tissue, increasing the natural frequencies, also consistent with the results observed by Ruan and Prasad [39]. Additionally, both the first natural frequency we calculated and the first vibration mode—tissue rotation in the sagittal plane—are similar to the reduced-order model results reported by Laksari et al. [21]. The lowest natural frequencies are within the range of the frequency response found by Alshareef et al. [23] using postmortem tissue.

While the natural frequencies calculated from the thums model are similar to those calculated from previous finite element and analytical models, the range is higher than the in vivo results produced by Escarcega et al. [19,20]. Such a difference suggests a discrepancy between the models and the actual vibration mechanics of the brain. Researchers have previously shown the sensitivity of finite element head model results to the selection of brain matter and CSF material models and parameters, as well as the selection of brain/CSF and CSF/skull-interface interaction [40–42]. Such studies have focused on time-domain results and on shorter timescales applicable only to high frequencies. Similar analysis in the frequency domain may yield valuable insight into the appropriate conditions for modeling vibration mechanics in the brain. However, such analysis cannot be conducted without a frame of reference for comparing the behavior associated with given vibration modes, which the work herein aims to provide.

The thumsv7 head used a simplified viscoelastic material model for both white and gray matter in the brain. Such a model may not fully capture the mechanical response of the brain tissue, which exhibits hyperelastic, anisotropic, and hysteretic tendencies. The eigenvalue analysis reported herein is conducted by using the tangent modulus of the tissue, assuming small deformations. The full range of the hyperelastic material behavior encountered in large deformations are not captured in this first order approach; the effects of nonlinear material behavior can be captured by using asymptotic expansions or transient solutions. On the other hand, material anisotropy would likely affect the resonant frequencies and mode shapes as the tangent modulus would be location and orientation dependent; this can be captured, to the first order, by using an eigenvalue analysis. Accurate characterization of the material properties and the mechanical modeling of brain tissue is still a developing field [43]. Material anisotropy and higher order effects will be investigated as part of future work as described above.

As seen in Fig. 4 and the Supplemental Materials on the ASME Digital Collection provided, the vibration modes associated with the lower mode numbers consist of a single rotation or translation in one of the basic planes, with corresponding motion in the other planes. At higher mode numbers, the calculated vibration modes show complex, three-dimensional deformation in the brain tissue. The vibration modes of the full head model in the coronal and transverse planes exhibited bilateral symmetry in out-of-plane deformation. In-plane rotation was observed in multiple vibration modes with one or more centers of rotation distributed with bilateral symmetry. The symmetry observed in our results is likely a product of the simplified, symmetric brain model analyzed. It is possible that incorporating geometric asymmetry present in many individual heads may affect such results. Although the vibration modes were characterized in two-dimensional planes, the observed deformations in each plane were interrelated. For example, the rotation in the sagittal plane observed in mode 1, seen in Fig. 4, was directly related to the OPD mode observed in the transverse and coronal planes. The higher modes were characterized by the superposition of multiple different behaviors in the same plane.

The characteristic modes we have calculated are similar to the modes reported in the literature. For example, the third vibration mode we have calculated—at a frequency of 15.25–18.17 Hz—exhibited similar in-plane shearing behavior reported by Escarcega et al. with a frequency of 6.66$±$0.86 Hz for the case of rotational head motion [19]. Qualitative similarities can also be drawn between vibration modes 2, 8, and 20 from our results and modes 5, 9, and 19 from Tse et al. [29], albeit with natural frequencies different by an order of magnitude. The behavior of mode 5 (16.31–19.42 Hz) in our model was similar to the dominant mode at 28 Hz described in Ref. [32]. Such results underscore the importance of a consistent descriptive framework. As discussed in the Introduction, there is a wide range in natural frequencies of the brain, and the head more broadly, reported in both the experimental and numerical literature. It is unclear how much of that spread is due to selection of material models, experimental conditions, or analysis techniques. Without a framework for relating results between studies, the uncertainty due to such conditions cannot be effectively quantified across different bodies of work. One of the aims of this work is to draw attention to this gap in interpreting the vibration simulations and provide a way to interpret the future work.

The development of FE head models involves balancing biofidelity with computational efficiency. The thumsv7 model used in this simulation relied on the simplification of the geometry and materials of the head to make simulation more computationally efficient. Geometrically, the details of the sulci and gyri on the surface of the brain were smoothed, and the boundaries between white and gray matter were discretized into coarse elements. While broadly the behavior described by our results matches many of the previous studies performed, it is likely that there are additional local details of the brain vibration modes that are not captured by the geometrically simplified model we have used.

In this technical brief, we have proposed a method of describing the three dimensional vibration mode shapes of the brain using conventions adapted from existing 2D vibration mechanics literature. The natural frequencies calculated using the thumsv7 model are similar to those reported in the experimental and numerical literature. We found that the mode shapes do not change significantly with head size. We have comprehensively described the observed mode shapes and outlined a method for future researchers to describe any additional mode shapes not observed in the present study. Accurate description of the vibration modes associated with frequency-dependent behavior is critical for comparing results between different studies. As research interest increases in brain vibration mechanics and the volume of manuscripts describing natural frequencies and their associated mode shapes increases, such a uniform method of communicating results will help to reduce uncertainty from choices in experiment or numerical model setup. We believe that the use of a consistent framework for describing the vibration behavior of the brain will promote faster convergence of results across different methodologies toward a holistic description of brain vibration mechanics.

The authors thank Northeastern University Research Computing for their computational resources and expertise. We would also like to acknowledge the Alsaif Doctoral Fellowship in the Department of Mechanical and Industrial Engineering at Northeastern University in partial support of this work. The views presented in this paper are those of the authors and do not necessarily represent the views of the sponsors.

• National Science Foundation CAREER Award (No. 2049088; Funder ID: 10.13039/100000001).

The data and information that support the findings of this article are freely available at this link.²

$[C]$ =

finite element damping matrix

{D} =

finite element nodal displacement vector

${D˙}$ =

finite element nodal velocity vector

${D¨}$ =

finite element nodal acceleration vector

$[K]$ =

finite element stiffness matrix

$[M]$ =

finite element consistent mass matrix

${R}$ =

finite element nodal force vector

t =

time

Greek Symbols

$ωn$ =

natural frequency

Abbreviations

COM =

center of mass

CSF =

cerebrospinal fluid

DMD =

dynamic mode decomposition

FE =

finite element

OPD =

out-of-plane deformation

thums =

Total Human Model for Safety

We have developed an educational component related to the methods presented in this research paper similar to our previous efforts [44–49], as part of our continuous commitment to broadening the impact of our scientific investigation. This “homework assignment” is appropriate for an undergraduate-level introductory course in vibrations.

Problem:

The first vibration mode of the human brain is characterized by rotation of the brain matter in the sagittal plane (see Fig. 5(a)). The behavior of the brain around this natural frequency can be modeled as a one-degree-of-freedom lumped system (see Fig. 5(b)), where the brain is treated as a body with lumped mass M and radius of gyration R. Resistance to rotation applied by the cerebrospinal fluid and skull is modeled as a torsional spring-and-dashpot system with a stiffness $Kt$ and damping constant $Ct$⁠.

1. Write the equation of motion for the system shown in Fig. 5(b).

2. Assuming undamped, free vibration, calculate the natural frequency of the first vibration mode of the brain.

3. A person is seated in a moving vehicle. The vibration of the vehicle applies a periodic force to the brain via the spinal column with a line of action through the radius of gyration (see Fig. 5(c)). Derive an expression for the amplitude of forced vibration as a function of the applied frequency $ω$⁠.

For all questions, calculate the result analytically, then substitute in values found in Table 4.