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Research Papers

# The Relationship of Three-Dimensional Human Skull Motion to Brain Tissue Deformation in Magnetic Resonance Elastography StudiesOPEN ACCESS

[+] Author and Article Information

Biomedical Engineering,
Washington University in St. Louis,
St. Louis, MO 63105

Ruth J. Okamoto, Brent D. Efron, Sam J. Nadell

Mechanical Engineering and Materials Science,
Washington University in St. Louis,
St. Louis, MO 63105

Ramona S. Durham

Biomedical Engineering,
Washington University in St. Louis,
St. Louis, MO 63105

Curtis L. Johnson

Biomedical Engineering,
University of Delaware,
Newark, DE 19716

Philip V. Bayly

Biomedical Engineering,
Washington University in St. Louis,
St. Louis, MO 63105;
Mechanical Engineering and Materials Science,
Washington University in St. Louis,
St. Louis, MO 63105

1Corresponding author.

Manuscript received September 1, 2016; final manuscript received February 15, 2017; published online March 21, 2017. Assoc. Editor: Barclay Morrison.

J Biomech Eng 139(5), 051002 (Mar 21, 2017) (12 pages) Paper No: BIO-16-1363; doi: 10.1115/1.4036146 History: Received September 01, 2016; Revised February 15, 2017

## Abstract

In traumatic brain injury (TBI), membranes such as the dura mater, arachnoid mater, and pia mater play a vital role in transmitting motion from the skull to brain tissue. Magnetic resonance elastography (MRE) is an imaging technique developed for noninvasive estimation of soft tissue material parameters. In MRE, dynamic deformation of brain tissue is induced by skull vibrations during magnetic resonance imaging (MRI); however, skull motion and its mode of transmission to the brain remain largely uncharacterized. In this study, displacements of points in the skull, reconstructed using data from an array of MRI-safe accelerometers, were compared to displacements of neighboring material points in brain tissue, estimated from MRE measurements. Comparison of the relative amplitudes, directions, and temporal phases of harmonic motion in the skulls and brains of six human subjects shows that the skull–brain interface significantly attenuates and delays transmission of motion from skull to brain. In contrast, in a cylindrical gelatin “phantom,” displacements of the rigid case (reconstructed from accelerometer data) were transmitted to the gelatin inside (estimated from MRE data) with little attenuation or phase lag. This quantitative characterization of the skull–brain interface will be valuable in the parameterization and validation of computer models of TBI.

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## Introduction

Traumatic brain injury (TBI) is a prevalent neurological disorder where brain tissue is damaged by an external force [1]. Diffuse axonal injury, or shearing and stretching of axonal fibers, is a common type of TBI often caused by sudden linear or angular accelerations of the head [2]. While the effects of TBI are well documented, the mechanisms linking mechanical insult and neurological injury are not well understood. Mathematical modeling and computer simulation of TBI can provide insight into these mechanisms, and guide the development of new methods for injury prediction, prevention, diagnosis, and management. However, to be useful, computer models require accurate descriptions of material behavior, boundary conditions, and loading. Recent research has largely focused on the material properties of brain tissue; however, quantitative characterization of the skull–brain interface in vivo remains an important need.

Magnetic resonance elastography (MRE) is a prominent tool for assessing the properties of brain tissue in vivo. MRE is a noninvasive MRI technique that enables assessment of the dynamic mechanical properties of living biological tissues [3,4]. In MRE of the brain, vibrations are applied to the skull in order to induce shear waves throughout the brain [57]. These shear waves are observed using MRE imaging sequences with motion-encoding gradients that oscillate at the vibration frequency, producing phase contrast images proportional to displacement [3,4]. The speeds of shear waves in soft tissue are determined by its mechanical properties [57]. Previous studies have demonstrated the ability of MRE to provide local estimates of the mechanical properties of healthy brain tissue [811], as well as changes in brain material properties due to development, aging, injury, or disease [1215]. MRE is also a valuable tool for parameterizing and validating computational models of TBI [16,17]. A number of methods exist to estimate material properties in soft tissue, such as local direct inversion (LDI) [18], nonlinear inversion (NLI) [19,20], or local frequency estimation (LFE) [21].

In contrast to the substantial progress on estimation of brain material properties, the transmission of skull motion to brain tissue motion has not been extensively characterized. Inside the skull, the brain is encased in multiple membranes including the dura mater, arachnoid mater, and pia mater. As the skull moves, these membranes, together with the cerebrospinal fluid (CSF) and blood contained between the various layers, cushion the brain and limit its displacement [22,23]. Together, these membranes and fluid layers comprise the skull–brain interface and play an important role in injury mechanics. Modeling the characteristics of the skull–brain interface is challenging, however [24]. While experimental studies of ex vivo tissue samples have provided some insight into the properties of the constituent materials [5,6,8], characterization of the intact skull–brain interface in vivo is desirable [2224].

To characterize the role of the intact skull–brain interface in transmitting forces and motion to brain tissue, we record skull motion during MRE studies using an array of three MRI-safe tri-axial accelerometers, and then compare skull kinematics to brain tissue displacements estimated from MRE measurements. By contrasting the transmission of motion from skull to brain in human subjects to the analogous motion transmission from case to gel in a homogenous gelatin phantom, we quantify how the skull–brain interface attenuates and delays the brain's response to skull motion.

## Methods

###### MRE Imaging Procedures and Instrumentation.

MRE imaging of six adult, male human subjects and a cylindrical gelatin phantom was performed on Siemens Trio® 3T MRI scanners located at the Beckman Institute at University of Illinois in Urbana-Champaign and the Center for Clinical Imaging Research at Washington University in St. Louis, MO. All studies were approved by the Institutional Review Board (IRB) of the respective institutions.

Subjects lay supine with their heads centered in a Siemens 12 channel head coil. Skull vibrations were induced at a frequency of 50 Hz through a commercially available mechanical actuator (Resoundant™, Rochester, MN) using a pillow-like actuator (Mayo Clinic, Rochester, MN) positioned on the back of the subject's head at the skull occipital protuberance [12]. The head rests on the pillow, and is stabilized by pads placed laterally. Skull vibrations were induced at 18% Resoundant™ amplitude. Three-dimensional skull kinematics were measured using three MRI safe tri-axial accelerometers (TSD109C2-MRI, BIOPAC©, Goleta, CA) embedded in an array constructed from a commercial sports mouth guard and three-dimensional (3D)-printed interface (Fig. 1). This accelerometer mouth-guard array (MGA) was designed to allow for rigid placement of the accelerometers and easy retrieval from the subject during the scanning protocol. The relative positions of the MGA and the pillow actuator were identified with vitamin E capsules that appeared with high signal in the field of view of T1-weighted magnetic resonance (MR) images. Subject brain volume (1378 $±$ 192 cm3) was calculated through segmentation of gray and white matter tracts in T1-weighted MRI data [25].

A multishot spiral MRE sequence with 2 mm isotropic voxels was used to measure brain tissue displacement, which is encoded as phase in the resulting MR images [20]. Imaging parameters included 26 mT/m motion-encoding gradient strength (MEGS), 6 k-space interleaves, eight phase offsets, and twenty axial slice acquisition in the region of the corpus callosum (Fig. 2(b)), with an image volume of 240 mm × 240 mm × 40 mm and a scan duration of approximately 11 min. The accelerometers are MRI-safe, meaning they are composed of material that poses no hazard within an MRI environment. While MRI-safe, the accelerometers induced imaging artifacts, thus a single slice scan with duration of approximately 36 s was performed in order to record skull acceleration during the MRE sequence. After this scan, The MGA was removed while the subject's head remained in the head coil and the full, 20 slice MRE scan was performed.

To provide an MR phantom with simpler properties that could be used to validate techniques and compare behavior, a cylindrical phantom (∼13 cm long and 13 cm in diameter) was fabricated from a plastic case filled with 1.65 L (1.81 kg) of a gelatin-glycerol-water solution (4% gelatin, 48% glycerol, and 48% water by weight [26]). For brevity, this mixture is referred to as gelatin. The gelatin phantom was refrigerated for a minimum of 12 h to allow the gelatin to set. The gelatin phantom was equilibrated for 3 h at room temperature before imaging. The storage modulus, or real part of the viscoelastic shear modulus, of gelatin samples was 1.8 kPa, as measured with dynamic shear testing at 50 Hz [26]. A 3D printed holder was used to rigidly attach the MGA to the outer surface of the gelatin phantom's cylindrical plastic case. The gelatin phantom was positioned in the head coil, stabilized laterally with pads, and vibrated using the pillow actuator at 50 Hz. The lower weight of the gelatin phantom, 1.8 kg compared to approximately 3.5–4.5 kg for the human skull [27], necessitated a Resoundant™ actuation amplitude of 3% to minimize phase wrapping of the MRE images.

Our experimental protocol thus involved the following steps:

1. (1)Position subject outside the scanner bore with the accelerometer MGA.
2. (2)Record skull accelerations in response to the pillow actuator without the spiral MRE sequence running.
3. (3)Move subject into the scanner bore and acquire accelerometer data from a single-slice MRE sequence.
4. (4)Remove the MGA without removing the subject's head from the coil.
5. (5)Run anatomical scan (T1-weighted: 3D magnetization prepared rapid gradient, 0.9 mm isotropic voxels) and 20-slice spiral MRE sequence.

This protocol enabled comparison of skull dynamics (reconstructed from accelerometer data) and brain tissue displacement (estimated from MRE measurements) in the same subject.

###### Estimation of 3D Skull Kinematics.

Skull motion during MRE at 50 Hz is well approximated as rigid-body motion [28]; standard 3D kinematic equations for rigid-body acceleration (Eqs. (1a)(1c)) [29] are used. Here, $a$ is linear acceleration, $α$ is angular acceleration, $r$ defines the distance between some point and the skull origin (approximated as the posterior clinoid process), and $ω$ is angular velocity. Each of these vectors consists of three components, denoted as $x$, $y$, and $z$, to represent the right–left (RL), anterior–posterior (AP), and superior–inferior (SI) directions. Thus, ($axO$, ) are, respectively, the RL, AP, and SI linear accelerations of the skull origin; ($ωx$, , $ωz$) are rates of rotation (angular velocity) about the RL, AP, and SI directions, respectively; and ($αx$, $αy$, $αz$) are the components of angular acceleration. ($ax$, ) are the components of linear acceleration of a point whose position, measured from the skull origin, is (). The right-hand side of these equations will contain nine unknowns (three components of motion for each of $a$, $α$, and $ω$ at the skull origin), which can be solved using the known values of linear acceleration at the positions of the tri-axial accelerometers Display Formula

(1a)$ax=axO+(αyrz−αzry)+(ωx(ωyry+ωzrz)−rx(ωy2+ωz2))$
Display Formula
(1b)$ay=ayO+(αzrx−αxrz)+(ωy(ωzrz+ωxrx)−ry(ωx2+ωz2))$
Display Formula
(1c)$az=azO+(αxry−αyrx)+(ωz(ωxrx+ωyry)−rz(ωx2+ωy2))$

To provide additional information, rigid-body velocity expressions (Eqs. (2a)(2c)) may also be used to estimate $ω$ [29,30]. Velocity ($v$) at each accelerometer position can be estimated through integration of the acceleration signal Display Formula

(2a)$vx=vxO+ωyrz−ωzry$
Display Formula
(2b)$vy=vyO+ωzrx−ωxrz$
Display Formula
(2c)$vz=vzO+ωxry−ωyrx$

The acceleration signals are expressed as Fourier coefficients (sum of sinusoids) j, providing the ability to integrate acceleration in the frequency domain. Equations (3a)(3c) demonstrate how velocity and displacement ($u$) are calculated. Here $Ω$ is angular frequency (rad/s), $φ$ is phase shift, $D$ is amplitude, $t$ is time, the subscript $j$ denotes the Fourier harmonic, and the subscript $q$ refers to each of the three Cartesian components of motion Display Formula

(3a)$aq=∑jDqj sin(Ωjt+φj)$
Display Formula
(3b)$⇒vq=−∑jDqjΩjcos(Ωjt+φj)$
Display Formula
(3c)$⇒uq=−∑jDqjΩj2sin(Ωjt+φj)$

Displacements at four discrete points: anterior (A) near the forehead, right (R) by the right temple, left (L) by the left temple, and posterior (P) near the skull occipital protuberance (Figs. 2(a) and 2(c)) were reconstructed from acceleration data. Reconstruction points and the skull origin were located from T1-weighted anatomical scans (Fig. 2(b)). On the gelatin phantom, anterior, right, left, and posterior reconstruction points were similarly chosen as central points along each cylinder edge.

To test the accuracy of motion reconstruction, we performed known translational and rotational motions (Fig. 3), measured accelerations with an independent sensor, and compared these data to the accelerations reconstructed from simultaneous measurements with the accelerometer array (MGA). Oscillatory rotational motion was generated by a Torsional Plant 205 (ECP©, Bell Canyon, CA) at a frequency of 10 Hz while translational motion was generated with an Electro-seis 113 Long Stroke Shaker (APS Dynamics©, San Juan Capistrano, CA) at a frequency of 50 Hz. The MGA was mounted on the moving section of each device. An additional accelerometer was placed on each device to serve as a reference. For rotational experiments, the MGA was oriented so that the motion corresponded to rotation about the RL axis. For linear motion, the MGA was oriented so translation was in the AP direction. The normalized root-mean-square error (NRMSE) was calculated from the RMS difference between the acceleration reconstructed from the MGA and the recorded signal from the reference accelerometer, normalized by the reference acceleration magnitude. In the rotational experiment, NRMSE was 0.04 in the AP direction and 0.02 in the SI direction. In the translational experiment, NRMSE was 0.04 in the AP direction and 0.08 in the SI direction. Noise, nonrigid motion, and nonlinearity can contribute to reconstruction error; these are more apparent in low-amplitude harmonic signal, such as in the translational experiment SI motion. However, since the SI component is a small fraction of the acceleration magnitude, its contribution to the overall error is small.

###### Analysis of MRE Displacement Measurements.

In MRE, motion-encoding gradients in the MRI sequence encode displacement as phase of the MR signal [20,31]. Oscillating gradients are applied along three orthogonal directions to capture motion-induced phase at each voxel in space. The resulting motion-induced MRE phase contrast ($Φ$) is proportional to the component of the oscillatory displacement ($u$) in the direction of the gradient [3,31]. MRE phase contrast includes the effects of oscillatory rigid-body motion, longitudinal wave motion, and shear wave motion. During imaging, oscillatory, rigid-body motion can be described through both translation in the $x$,$y,$ and $z$ directions along with rotation about each of those axes. Additionally, when the amplitude of motion is large, motion-induced phase wrapping may occur. This was commonly seen in the phase-encoded anterior–posterior component of motion. Wrapped phase was spatially unwrapped using FSL PRELUDE [32] and temporally unwrapped using a custom postprocessing algorithm described in Appendix A.1. A separate experiment using variable gradient strength in MRE of a single subject provided validation of the unwrapping approach used in this study (Fig. 10).

After unwrapping the MRE phase data, the three complex displacement components at each voxel were obtained by converting phase to displacement (Appendix A.2) and extracting the first harmonic using a temporal Fourier transform. Components of rigid-body displacement ($u¯$) were estimated for comparison with skull motion reconstructed from the accelerometers. A set of three rotation components () and three translation components ($u¯xO$, $u¯yO$, ) about the brain origin were used to describe 3D, rigid-body displacement in the brain at each voxel location (denoted by position $r$ with respect to the brain origin). These complex components were solved by least squares fitting of the total displacement (estimated from MRE data) at all voxels within the brain to a model of rigid-body displacement Display Formula

(4a)$u¯x=u¯xO+θyrz−θzry$
Display Formula
(4b)$u¯y=u¯yO+θzrx−θxrz$
Display Formula
(4c)$u¯z=u¯zO+θxry−θyrx$

The contribution of shear and longitudinal waves to the displacement field ($ũ$) was estimated by removing rigid-body motion from the total displacement field estimate. In addition, in an isotropic, uniform, elastic or viscoelastic material, the curl of the displacement field contains only contributions of shear waves. The curl ($Γ)$ was estimated through local polynomial fitting and differentiation of the displacement field to provide a separate estimate of the contribution of shear waves. Wave motion for both human subjects and the gelatin phantom are reported in terms of both wave displacement and curl. Strain magnitudes resulting from wave propagation are reported as octahedral shear strain ($εs$) [33]. For comparison with estimates of strain and curl (nondimensional measures of displacement per unit length), the displacement of the case or skull was normalized by the radius of the case or the major semi-axis of the skull.

To characterize brain tissue displacements in different regions and compare them with skull motion at specific locations, MRE measurements were separated into twelve regions of interest (ROI) including three shells (outer, middle, and inner) and four quadrants (anterior, A; right, R; left, L; and posterior, P) (Fig. 2(d)). MRE measurements in the gelatin phantom were separated similarly into 12 ROIs: four corresponding A, P, R, and L quadrants, and three shells.

###### Comparison of Skull and Brain Displacements.

Estimates of skull displacement at four material points in the skull (reconstructed from accelerometer measurements) were compared to analogous estimates of displacement at neighboring material points in the brain (estimated from MRE measurements). In general harmonic motion, where the three components of motion are not exactly in phase, the 3D trajectory of each point in the brain and skull is an ellipse, represented as a complex coefficient vector, $u0$, multiplied by a complex exponential in time, (Appendix A.3). The elliptical trajectories of material points in skull and brain tissue were described by their relative amplitudes, spatial orientations, and temporal phase. The amplitude of each elliptical trajectory is reported as its mean radius: the circumference of the corresponding ellipse divided by .

Consider two 3D complex coefficient vectors $u0$ and $v0$, corresponding to elliptical trajectories of skull motion and brain tissue rigid-body motion, respectively. The vectors normal to the two elliptical trajectories are used to calculate the spatial, angle, $β$. The dot product of the two complex coefficient vectors can be used to calculate temporal phase shift, $φ$ [34]. The vector normal to each ellipse is found from the cross product of the real and imaginary components of the coefficient vectors, as shown in the below equations Display Formula

(5)
Display Formula
(6)

The spatial angle, $β,$ between the two ellipses is then found from the below equation Display Formula

(7)$cos β(u0,v0)=(Nu⋅Nv)‖Nu‖‖Nv‖$

In order to compare the phases of skull and brain motion captured in their complex coefficient vectors, a common reference for t = 0 is needed to describe the relative timing between the MRE sequence and skull acceleration profile (Appendix A.2). The center of the refocusing pulse of the MRE sequence was chosen as the common reference. The temporal phase shift, $φ$, between skull and brain tissue motion was estimated as the phase angle of the complex dot product Display Formula

(8)$φ=∠(u0^*⋅v0)$

## Results

###### Motion of the Human Skull and Gelatin Phantom Case Reconstructed From Acceleration Data.

During MRE imaging of human subjects, the Resoundant™ system causes the pillow actuator to vibrate the back of the skull. Examples of accelerations reconstructed at the anterior, right, and posterior points are shown in Fig. 4. The acceleration magnitudes typically ranged from 2 to 5 m/s2. Acceleration components reconstructed at the left (L) and right (R) skull points were similar in magnitude and phase. The SI component of acceleration was largest at the posterior. AP acceleration magnitudes were similar at all reconstruction points. Data generally suggest a combination of AP linear acceleration and rotation of the skull about the RL axis. The mean magnitudes of the components of skull acceleration for the six human subjects are compared to those of the gelatin phantom case in Table 1. The gelatin phantom case had lower acceleration amplitude than the human skull, and the largest component was consistently in the AP direction, even at the posterior reconstruction point.

To characterize the uncertainty in the accelerometer measurements obtained during MRE in the absence of a reference accelerometer, we compared the accelerations reconstructed at the positions of the embedded accelerometers to the recorded accelerometer data. NRMSE for each acceleration component was obtained by averaging the NRMSE of that component for the three accelerometers in the MGA. Normalized errors for reconstructed accelerations in human studies were: NRMSE = 0.05 (RL direction), NRMSE = 0.10 (AP direction), and NRMSE = 0.07 (SI direction). In the gelatin phantom, reconstructed data also showed reasonable agreement with recorded values: NRMSE = 0.13 (RL direction), NRMSE = 0.10 (AP direction), and NRMSE = 0.16 (SI direction). The higher NRMSE in the gel phantom is likely due to the lower amplitude of acceleration, which leads to a lower signal-to-noise ratio in the data.

###### Motion of Human Brain Tissue and Phantom Gelatin Estimated From MRE Measurements.

Rigid-body motion and wave motion in both gelatin and brain tissue were estimated from MRE measurements, as described previously. In the gelatin phantom, most of the total displacement field is due to wave displacement (Figs. 5(a) and 5(b)). In human subjects, the MRE measurements reflect a relatively uniform displacement field with high amplitude in the AP direction (Fig. 5(d)), suggesting a larger contribution from rigid-body motion than from wave displacement fields. Amplitudes of wave displacement and curl in the gelatin phantom (Figs. 5(b) and 5(c)) were higher relative to total displacement than in human brain tissue.

###### Comparison of Case and Gelatin Motion in the Gelatin Phantom.

Motion trajectories at four points on the case (estimated from accelerometer measurements) were compared to trajectories at four material points inside the gelatin (estimated from MRE measurements). These comparisons were characterized by relative amplitude, orientation, and phase (Fig. 6). The rigid-body motion of the gelatin and case were similar in amplitude, particularly the dominant components of rotation ($θRL$) and translation ($u¯APO$). The spatial angle between rigid-body motion of the gelatin and rigid-body motion of the case indicated high alignment of gel and case motion (Table 2). The phase shift between case and gelatin rigid-body motion was consistently low.

The magnitudes of wave displacement in gelatin were larger than the amplitudes of rigid-body motion of both the case and gelatin (Fig. 7). No qualitative differences in curl, displacement, or strain amplitudes were seen between shells or quadrants.

###### Comparison of Skull and Brain Motion in Human Subjects.

Brain motion (estimated from MRE measurements) and skull motion (reconstructed from accelerometer measurements) were compared for six subjects. As in the gelatin phantom, elliptical trajectories of neighboring material points were used to compare brain and skull rigid-body motion (Fig. 8). In contrast to the gelatin phantom, the amplitude of rigid-body motion in the brain differed from the amplitude of nearby skull motion. At each of the brain material points, the AP component of translation was consistently the highest amplitude, while the skull experienced high amplitudes in both AP and SI directions. Rigid-body motion of the brain and skull exhibited greater differences in spatial angle than were seen between the case and contents of the gelatin phantom (Table 3). The phase shift between rigid-body motion in brain and skull was consistently longer than between case and gelatin in the gelatin phantom.

The amplitude of wave motion in brain tissue was lower than the amplitude of reconstructed skull motion (Fig. 9) in all quadrants and shells. The displacement amplitudes in the outer and middle shells were typically higher than in the inner shell. The amplitudes of curl and strain also decreased progressively in regions closer to the center of the brain.

## Discussion and Conclusion

In this study, we directly characterized skull kinematics using accelerometers in order to relate skull motion to brain tissue motion. To describe brain tissue motion, MRE measurements of displacement were decomposed into wave motion and rigid-body motion. Comparing skull kinematics with brain tissue motion in vivo enables assessment of relative motion and deformation. Furthermore, comparison of data from human subjects to data from a gelatin phantom (in which the gelatin adheres to the cylindrical case) provides insight into how the skull–brain interface (a more complex material interface) affects motion transmission.

The effects of the skull–brain interface are illuminated by comparison to the coupling of case to gelatin in the gelatin phantom. In the phantom, harmonic, rigid-body displacements of both gelatin and case were similar. Material points in the case and material points in gelatin exhibited elliptical trajectories due to rigid-body motion. The elliptical trajectories of neighboring points on opposite sides of the adherent interface exhibited similar amplitude, direction (spatial angle), and temporal phase. While we expect particles in the case and in the gelatin at the interface to have almost identical trajectories, discrepancies in the rigid-body motions are apparent (for example, Fig. 6). There are several likely reasons: (i) gelatin motion includes both rigid-body and wave motion; (ii) the case is not perfectly rigid (which introduces some error into the estimate of case motion); (iii) motion estimates from MRE are average values within a finite region of interest. Displacements in the gelatin due to waves were consistently higher than the displacements due to rigid-body motion of the gelatin and case. It is likely that harmonic excitation at 50 Hz induces constructive wave interactions in the weakly dissipative gelatin, causing higher displacement fields in the interior.

In contrast, displacement fields in the human brain exhibited higher amplitudes of rigid-body motion than wave motion. Furthermore, rigid-body motion of the brain differed strongly from rigid-body motion of the skull. Considering first the rigid-body components of motion, elliptical trajectories of material points in the brain exhibited significant differences in amplitude, orientation (spatial angle), and temporal phase compared to elliptical trajectories of neighboring points on the skull. Wave components of displacement in the brain were even smaller relative to skull motion; displacements due to waves at material points in the brain were an order of magnitude smaller than displacements of neighboring material points in the skull. In addition, waves were attenuated significantly as they propagated inward from the outer cortical surface toward the midbrain.

These results indicate significant mechanical compliance and dissipation at the skull–brain interface. These properties are attributable to the dural membranes, blood vessels, and fluids that comprise the interface. In contrast, the gelatin phantom has none of these structures, and the adherent interface between gelatin and case exhibits dramatically simpler behavior.

Some experimental limitations are acknowledged. All analyses performed in this study assumed perfect coupling between the accelerometers and the skull. In practice, this is unlikely; slip or compliance will affect the transmission of motion from the jaw, via the commercial mouth guard, to the accelerometer array. However, by using the same array in the gel phantom, we demonstrate that these effects are not dominant. Accelerations were assumed to be harmonic; Fourier components at a single frequency were converted to displacement in the frequency domain for comparison with MRE measurements. As noted in the description of experimental methods, and in the Appendix, the methods for measurement and reconstruction of skull kinematics from accelerometer data were validated in separate experiments.

Substantial variation in acceleration (Table 1) and spatial angle (Table 3) between skull and brain motion is seen over the cohort of six subjects. These variations likely reflect intersubject differences in head size, shape, and anatomy, as well as individual responses to actuation and variations in subject and actuator positioning. Despite intersubject variability, subjects consistently showed higher amplitudes of skull translation in the AP and SI directions, relative to RL translation, and higher amplitudes of rotation about the RL axis, relative to other rotation components. Additionally, recorded human MRE data are consistent with results reported by other groups using a similar actuation system [20].

Future work should explore the possibility of frequency-dependent, anisotropic, and nonlinear mechanical behavior of the skull–brain interface. Different actuation locations and frequencies could be used to investigate such behavior. Additionally, MRE data from the entire brain could be compared to a more extensive set of reconstruction points on the skull.

This study confirms that the skull–brain interface is a critical determinant of brain tissue deformation due to skull loading, and thus is a major factor in TBI. The properties of this interface are measurable; the quantitative data from this study should be of immediate use in the parameterization and validation of computer models and simulations of TBI. The properties of the skull–brain interface are also critical to the development of strategies for MRE of the brain, in which enhanced transmission of waves into the brain is desirable. The compliant and dissipative nature of the skull–brain interface clearly protects the brain during mild skull accelerations. Further studies are needed to determine the conditions under which this protective system may fail.

## Acknowledgements

Financial support for this study was provided by National Institutes of Health (NIH) Grant No. R01 NS055951.

## Nomenclature

• $a$ =

linear acceleration (m/s2)

• $D$ =

wave amplitude (m/s2, m/s, m)

• $N$ =

vector normal to an ellipse (m)

• $r$ =

position (m)

• $u$ =

displacement (m)

• $u¯$ =

rigid-body displacement (m)

• $ũ$ =

wave displacement (m)

• $x,y,z$ =

vector components: right-left, anterior–posterior, and superior–inferior

• $α$ =

• $β$ =

• $θ$ =

• $v$ =

velocity (m/s)

• $ϕ$ =

• $Φ$ =

• $ω$ =

• $Ω$ =

## Appendices

###### Unwrapping MRE Phase Contrast

MRE phase contrast ($Φ$) is originally constrained to the principal range $(−π,π]$, which can cause wrapping and complicate recovery of the original amplitude. To validate our unwrapping approach, an echo planar imaging (EPI) sequence was used to acquire MRE data from a single human subject at motion-encoding gradient strengths (MEGS) between 4 and 25.9 mT/m (Fig. 10). Imaging parameters included 3 mm isotropic voxels, 240 mm field of view, and 24 axial slices. As in the two-dimensional spiral sequence, MRE data that encoded displacements in the AP direction were often wrapped when acquired at high motion-encoding gradient strengths. In addition to phase wrapping in space (which is already a challenging problem [35,36]), the harmonic data may also be wrapped in time. Unwrapping in time is further challenging due to the very coarse sampling: eight points per period. Therefore, to accurately recover the harmonic time series, we used a custom temporal postprocessing algorithm following initial spatial unwrapping of 3D volumes with a widely used, third-party algorithm (FSL PRELUDE [32]). The temporal postprocessing algorithm consisted of tabulating all possible permutations of the motion-encoded phase at a selected voxel, with a value of added at each time point. This resulted in 38, or 6561 permutations. The eight-point FFT of each permutation was calculated, and the permutation that contained the highest fraction of power at the fundamental frequency FFT, was then selected as the unwrapping solution. As Fig. 10 shows, the rigid-body motion is homogeneous across the field of view, meaning that the same unwrapping scheme was used for each voxel. We then acquired MRE data at lower MEGS to view an unwrapped AP phase contrast waveform in the same subject. Data from this protocol confirmed that MRE phase contrast was proportional to the strength of the motion encoding gradients. The MRE phase waveform acquired at lower MEGS was used as a template to check the unwrapped solution for the waveform at higher MEGS.

###### Relating MR Sequence Timing With Accelerometer Recordings

Artifacts such as aliasing and sudden changes in amplitude in the accelerometer trace corresponded with landmarks in the MRE sequence such as the spiral gradients and refocusing pulse (Fig. 11). To relate the phases of skull motion and brain motion, the center of the 180 deg refocusing pulse was chosen as a common temporal landmark for synchronizing the accelerometer and MRE signals. The relative phase of the accelerometer recordings was determined from the FFT of the accelerometer time series, setting the initial time point at the center of the refocusing pulse. The relative phase shift of the MRE signal was found directly from the FFT of the time series of MRE phase contrast, which is already referenced to the center of the refocusing pulse.

The MRE phase contrast, $Φ,$ for time-varying (square wave) motion-encoding gradients, $G(t),$ as shown in Fig. 11, and harmonic displacement field $u(r,t)$ is Display Formula

(A1)$Φ=Nγ∫0TG(t)⋅u(r,t)dt$

where $γ$ is the gyromagnetic ratio of 1H, T is the period, $N$ is the number of motion-encoding gradients, and $u(r,t)=uoei(Ωt−ψ)$ is the vector of displacement at position $r$ and time $t$. Here, $Ω$ is angular frequency (rad/s) and $ψ=ψ(r)$ is the temporal phase shift, $2πΔt(r)/T$, of the displacement at that voxel. Evaluation of this integral yields an expression for the MRE phase contrast $Φ$, including the temporal phase shift relative to the refocusing pulse Display Formula

(A2)

Here, $uG$ is the component of $uo$ in the direction of the gradient: $uG=G⋅uo/G$ and $G$ is the motion encoding gradient strength. From this equation, we find that the temporal phase of the MRE measurements must be shifted $π/2$ rad to reference them to the center of the refocusing pulse. After computing this shift, the relative phase of the accelerometer data and MRE measurements is then calculated using Eq. (8).

The sensitivity of the sequence was doubled by computing the phase difference between acquisitions using positively ($P$) and negatively ($N$) polarized motion-encoding gradients. Display Formula

(A3)

The magnitudes of the total (reported) phase $ΦT$ and of the displacement $uG$ are linearly related to each other by the ratio $uG$/$ΦT=Ω/(8NγG).$ For the spiral sequence used in this study, , $N=2$, and $γ$ of 1H is $2.68(105)rad/(s mT)$, so that $uG$ / $ΦT$ per radian of MRE phase contrast.

###### Elliptical Trajectories Described by Complex Coefficient Vectors

Consider a complex vector $u0=(u0,v0,w0)$ describing 3D harmonic motion at a frequency, Ω: $u(t)=Re[u0eiΩt]$. The components of this vector can be described by Display Formula

(A4)$u0=r1eiψ1$
Display Formula
(A5)$v0=r2eiψ2$
Display Formula
(A6)$w0=r3eiψ3$
where $rn$ is the amplitude and $ψn$ is the phase of each component. In harmonic motion, the trajectory shape is determined by the relative amplitudes and phases of these components as shown in Fig. 12. When all three components are exactly in phase, that is $ψ1=ψ2=ψ3$, the trajectory collapses to a straight line (Fig. 12(a)). If the three components have different temporal phase, however, then the resulting trajectory is an ellipse (Fig. 12(b)). For our analysis, we determined the relative spatial orientation of two elliptical trajectories by finding the spatial angle between the vectors normal to the plane of each ellipse (Fig. 12(c)). The normal vectors are calculated from the cross product of the real and imaginary parts of the coefficient vectors.

## References

Coronado, V. G. , Haileyesus, T. , Cheng, T. A. , Bell, J. M. , Haarbauer-Krupa, J. , Lionbarger, M. R. , Flores-Herrera, J. , McGuire, L. C. , and Gilchrist, J. , 2015, “ Trends in Sports-and Recreation-Related Traumatic Brain Injuries Treated in U.S. Emergency Departments: the National Electronic Injury Surveillance System-All Injury Program (NEISS-AIP) 2001-2012,” J. Head Trauma Rehabil., 30(3), pp. 185–197. [PubMed]
Smith, D. H. , and Meaney, D. F. , 2000, “ Axonal Damage in Traumatic Brain Injury,” Neuroscientist, 6(6), pp. 483–495.
Muthupillai, R. , Lomas, D. J. , Rossman, P. J. , and Greenleaf, J. F. , 1995, “ Magnetic Resonance Elastography by Direct Visualization of Propagating Acoustic Strain Waves,” Science, 269(5232), pp. 1854–1857. [PubMed]
Kruse, S. A. , Smith, J. A. , Lawrence, A. J. , Dresner, M. A. , Manduca, A. J. F. G. , Greenleaf, J. F. , and Ehman, R. L. , 2000, “ Tissue Characterization Using Magnetic Resonance Elastography: Preliminary Results,” Phys. Med. Biol., 45(6), pp. 1579–15790. [PubMed]
Kruse, S. A. , Rose, G. H. , Glaser, K. J. , Manduca, A. , Felmlee, J. P. , Jack, C. R. , and Ehman, R. L. , 2008, “ Magnetic Resonance Elastography of the Brain,” Neuroimage, 39(1), pp. 231–237. [PubMed]
Sack, I. , Beierbach, B. , Hamhaber, U. , Klatt, D. , and Braun, J. , 2008, “ Non-Invasive Measurement of Brain Viscoelasticity Using Magnetic Resonance Elastography,” NMR Biomed., 21(3), pp. 265–271. [PubMed]
Streitberger, K. J. , Wiener, E. , Hoffmann, J. , Freimann, F. B. , Klatt, D. , Braun, J. , Lin, K. , McLaughlin, J. , Sprung, C. , Klingebiel, R. , and Sack, I. , 2011, “ In Vivo Viscoelastic Properties of the Brain in Normal Pressure Hydrocephalus,” NMR Biomed., 24(4), pp. 385–392. [PubMed]
Clayton, E. H. , Genin, G. M. , and Bayly, P. V. , 2012, “ Transmission, Attenuation and Reflection of Shear Waves in the Human Brain,” J. R. Soc. Interface, 9(76), pp. 2899–2910. [PubMed]
Murphy, M. C. , Huston, III, J. , Jack, Jr., C. R. , Glaser, K. J. , Senjem, M. L. , Chen, J. , Manduca, A. , Felmlee, J. P. , and Ehman, R. L. , 2013, “ Measuring the Characteristic Topography of Brain Stiffness With Magnetic Resonance Elastography,” PLoS One, 8(12), p. e81668. [PubMed]
Guo, J. , Hirsch, S. , Fehlner, A. , Papazoglou, S. , Scheel, M. , Braun, J. , and Sack, I. , 2013, “ Towards an Elastographic Atlas of Brain Anatomy,” PloS One, 8(8), p. e71807. [PubMed]
Johnson, C. L. , Schwarb, H. , McGarry, M. D. J. , Anderson, A. T. , Huesmann, G. R. , Sutton, B. P. , and Cohen, N. J. , 2016, “ Viscoelasticity of Subcortical Gray Matter Structures,” Human Brain Mapping, 37(12), pp. 4221–4233.
Murphy, M. C. , Huston, J. , Jack, C. R. , Glaser, K. J. , Manduca, A. , Felmlee, J. P. , and Ehman, R. L. , 2011, “ Decreased Brain Stiffness in Alzheimer's Disease Determined by Magnetic Resonance Elastography,” J. Magn. Reson. Imaging, 34(3), pp. 494–498. [PubMed]
Arani, A. , Murphy, M. C. , Glaser, K. J. , Manduca, A. , Lake, D. S. , Kruse, S. A. , Jack, C. R. , Ehman, R. L. , and Huston, J. , 2015, “ Measuring the Effects of Aging and Sex on Regional Brain Stiffness With MR Elastography in Healthy Older Adults,” Neuroimage, 111, pp. 59–64. [PubMed]
Streitberger, K. J. , Sack, I. , Krefting, D. , Pfüller, C. , Braun, J. , Paul, F. , and Wuerfel, J. , 2012, “ Brain Viscoelasticity Alteration in Chronic-Progressive Multiple Sclerosis,” PloS One, 7(1), p. e29888. [PubMed]
Romano, A. , Guo, J. , Prokscha, T. , Meyer, T. , Hirsch, S. , Braun, J. , Sack, I. , and Scheel, M. , 2014, “ In Vivo Waveguide Elastography: Effects of Neurodegeneration in Patients With Amyotrophic Lateral Sclerosis,” Magn. Reson. Med., 72(6), pp. 1755–1761. [PubMed]
Cloots, R. J. H. , Van Dommelen, J. A. W. , Nyberg, T. , Kleiven, S. , and Geers, M. G. D. , 2011, “ Micromechanics of Diffuse Axonal Injury: Influence of Axonal Orientation and Anisotropy,” Biomech. Model Mechanobiol., 10(3), pp. 413–422. [PubMed]
Giordano, C. , Cloots, R. J. H. , Van Dommelen, J. A. W. , and Kleiven, S. , 2014, “ The Influence of Anisotropy on Brain Injury Prediction,” J. Biomech., 47(5), pp. 1052–1059. [PubMed]
Oliphant, T. E. , Manduca, A. , Ehman, R. L. , and Greenleaf, J. F. , 2001, “ Complex‐Valued Stiffness Reconstruction for Magnetic Resonance Elastography by Algebraic Inversion of the Differential Equation,” Magn. Reson. Med., 45(2), pp. 299–310. [PubMed]
McGarry, M. D. J. , Van Houten, E. E. W. , Johnson, C. L. , Georgiadis, J. G. , Sutton, B. P. , Weaver, J. B. , and Paulsen, K. D. , 2012, “ Multiresolution MR Elastography Using Nonlinear Inversion,” Med. Phys., 39(10), pp. 6388–6396. [PubMed]
Johnson, C. L. , McGarry, M. D. J. , Van Houten, E. E. W. , Weaver, J. B. , Paulsen, K. D. , Sutton, B. P. , and Georgiadis, J. G. , 2013, “ Magnetic Resonance Elastography of the Brain Using Multishot Spiral Readouts With Self‐Navigated Motion Correction,” Magn. Reson. Med., 70(2), pp. 404–412. [PubMed]
Manduca, A. , Oliphant, T. E. , Dresner, M. A. , Mahowald, J. L. , Kruse, S. A. , Amromin, E. , Felmlee, J. P. , Greenleaf, J. F. , and Ehman, R. L. , 2001, “ Magnetic Resonance Elastography: Non-Invasive Mapping of Tissue Elasticity,” Med. Image Anal., 5(4), pp. 237–254. [PubMed]
Hutson, M. , and Speed, C. , 2011, Sports Injuries, Oxford University Press, Oxford, UK.
Woolsey, T. A. , Hanaway, J. , and Gado, M. H. , 2013, The Brain Atlas: A Visual Guide to the Human Central Nervous System, Wiley, Hoboken, NJ.
Mazumder, M. M. G. , Bunt, S. , Mostayed, M. , Joldes, G. , Day, R. , Hart, R. , and Wittek, A. , 2013, “ Mechanical Properties of the Brain–Skull Interface,” Acta Bioeng. Biomech., 15(2), pp. 3–11. [PubMed]
Steen, R. G. , Hamer, R. M. , and Lieberman, J. A. , 2007, “ Measuring Brain Volume by MR Imaging: Impact of Measurement Precision and Natural Variation on Sample Size Requirements,” Am. J. Neuroradiol., 28(6), pp. 1119–1125.
Okamoto, R. J. , Clayton, E. H. , and Bayly, P. V. , 2011, “ Viscoelastic Properties of Soft Gels: Comparison of Magnetic Resonance Elastography and Dynamic Shear Testing in the Shear Wave Regime,” Phys. Med. Biol., 56(19), pp. 6379–6400. [PubMed]
Kleiven, S. , and von Holst, H. , 2002, “ Consequences of Head Size Following Trauma to the Human Head,” J. Biomech., 35(2), pp. 153–160. [PubMed]
Gurdjian, E. S. , Hodgson, V. R. , and Thomas, L. M. , 1970, “ Studies on Mechanical Impedance of the Human Skull: Preliminary Report,” J. Biomech., 3(3), pp. 239–247. [PubMed]
Ginsberg, G. , and Genin, J. , 1995, Dynamics, Wiley, New York.
Naunheim, R. S. , Bayly, P. V. , Standeven, J. , Neubauer, J. S. , Lewis, L. M. , and Genin, G. M. , 2003, “ Linear and Angular Head Accelerations During Heading of a Soccer Ball,” Med. Sci. Sports Exercise, 35(8), pp. 1406–1412.
Atay, S. M. , Kroenke, C. D. , Sabet, A. , and Bayly, P. V. , 2008, “ Measurement of the Dynamic Shear Modulus of Mouse Brain Tissue In Vivo by Magnetic Resonance Elastography,” ASME J. Biomech. Eng., 130(2), p. 021013.
Smith, S. M. , Jenkinson, M. , Woolrich, M. W. , Beckmann, C. F. , Behrens, T. E. , Johansen-Berg, H. , Bannister, P. R. , De Luca, M. , Drobnjak, I. , Flitney, D. E. , and Niazy, R. K. , 2004, “ Advances in Functional and Structural MR Image Analysis and Implementation as FSL,” Neuroimage, 23, pp. S208–S219. [PubMed]
McGarry, M. D. J. , Van Houten, E. E. W. , Perrinez, P. R. , Pattison, A. J. , Weaver, J. B. , and Paulsen, K. D. , 2011, “ An Octahedral Shear Strain-Based Measure of SNR for 3D MR Elastography,” Phys. Med. Biol., 56(13), pp. N153–N164. [PubMed]
Scharnhorst, K. , 2001, “ Angles in Complex Vector Spaces,” Acta Appl. Math., 69(1), pp. 95–103.
Wang, H. , Weaver, J. B. , Perreard, I. I. , Doyley, M. M. , and Paulsen, K. D. , 2011, “ A Three-Dimensional Quality-Guided Phase Unwrapping Method for MR Elastography,” Phys. Med. Biol., 56(13), pp. 3935–3952. [PubMed]
Barnhill, E. , Kennedy, P. , Johnson, C. L. , Mada, M. , and Roberts, N. , 2015, “ Real‐Time 4D Phase Unwrapping Applied to Magnetic Resonance Elastography,” Magn. Reson. Med., 73(6), pp. 2321–2331. [PubMed]
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## References

Coronado, V. G. , Haileyesus, T. , Cheng, T. A. , Bell, J. M. , Haarbauer-Krupa, J. , Lionbarger, M. R. , Flores-Herrera, J. , McGuire, L. C. , and Gilchrist, J. , 2015, “ Trends in Sports-and Recreation-Related Traumatic Brain Injuries Treated in U.S. Emergency Departments: the National Electronic Injury Surveillance System-All Injury Program (NEISS-AIP) 2001-2012,” J. Head Trauma Rehabil., 30(3), pp. 185–197. [PubMed]
Smith, D. H. , and Meaney, D. F. , 2000, “ Axonal Damage in Traumatic Brain Injury,” Neuroscientist, 6(6), pp. 483–495.
Muthupillai, R. , Lomas, D. J. , Rossman, P. J. , and Greenleaf, J. F. , 1995, “ Magnetic Resonance Elastography by Direct Visualization of Propagating Acoustic Strain Waves,” Science, 269(5232), pp. 1854–1857. [PubMed]
Kruse, S. A. , Smith, J. A. , Lawrence, A. J. , Dresner, M. A. , Manduca, A. J. F. G. , Greenleaf, J. F. , and Ehman, R. L. , 2000, “ Tissue Characterization Using Magnetic Resonance Elastography: Preliminary Results,” Phys. Med. Biol., 45(6), pp. 1579–15790. [PubMed]
Kruse, S. A. , Rose, G. H. , Glaser, K. J. , Manduca, A. , Felmlee, J. P. , Jack, C. R. , and Ehman, R. L. , 2008, “ Magnetic Resonance Elastography of the Brain,” Neuroimage, 39(1), pp. 231–237. [PubMed]
Sack, I. , Beierbach, B. , Hamhaber, U. , Klatt, D. , and Braun, J. , 2008, “ Non-Invasive Measurement of Brain Viscoelasticity Using Magnetic Resonance Elastography,” NMR Biomed., 21(3), pp. 265–271. [PubMed]
Streitberger, K. J. , Wiener, E. , Hoffmann, J. , Freimann, F. B. , Klatt, D. , Braun, J. , Lin, K. , McLaughlin, J. , Sprung, C. , Klingebiel, R. , and Sack, I. , 2011, “ In Vivo Viscoelastic Properties of the Brain in Normal Pressure Hydrocephalus,” NMR Biomed., 24(4), pp. 385–392. [PubMed]
Clayton, E. H. , Genin, G. M. , and Bayly, P. V. , 2012, “ Transmission, Attenuation and Reflection of Shear Waves in the Human Brain,” J. R. Soc. Interface, 9(76), pp. 2899–2910. [PubMed]
Murphy, M. C. , Huston, III, J. , Jack, Jr., C. R. , Glaser, K. J. , Senjem, M. L. , Chen, J. , Manduca, A. , Felmlee, J. P. , and Ehman, R. L. , 2013, “ Measuring the Characteristic Topography of Brain Stiffness With Magnetic Resonance Elastography,” PLoS One, 8(12), p. e81668. [PubMed]
Guo, J. , Hirsch, S. , Fehlner, A. , Papazoglou, S. , Scheel, M. , Braun, J. , and Sack, I. , 2013, “ Towards an Elastographic Atlas of Brain Anatomy,” PloS One, 8(8), p. e71807. [PubMed]
Johnson, C. L. , Schwarb, H. , McGarry, M. D. J. , Anderson, A. T. , Huesmann, G. R. , Sutton, B. P. , and Cohen, N. J. , 2016, “ Viscoelasticity of Subcortical Gray Matter Structures,” Human Brain Mapping, 37(12), pp. 4221–4233.
Murphy, M. C. , Huston, J. , Jack, C. R. , Glaser, K. J. , Manduca, A. , Felmlee, J. P. , and Ehman, R. L. , 2011, “ Decreased Brain Stiffness in Alzheimer's Disease Determined by Magnetic Resonance Elastography,” J. Magn. Reson. Imaging, 34(3), pp. 494–498. [PubMed]
Arani, A. , Murphy, M. C. , Glaser, K. J. , Manduca, A. , Lake, D. S. , Kruse, S. A. , Jack, C. R. , Ehman, R. L. , and Huston, J. , 2015, “ Measuring the Effects of Aging and Sex on Regional Brain Stiffness With MR Elastography in Healthy Older Adults,” Neuroimage, 111, pp. 59–64. [PubMed]
Streitberger, K. J. , Sack, I. , Krefting, D. , Pfüller, C. , Braun, J. , Paul, F. , and Wuerfel, J. , 2012, “ Brain Viscoelasticity Alteration in Chronic-Progressive Multiple Sclerosis,” PloS One, 7(1), p. e29888. [PubMed]
Romano, A. , Guo, J. , Prokscha, T. , Meyer, T. , Hirsch, S. , Braun, J. , Sack, I. , and Scheel, M. , 2014, “ In Vivo Waveguide Elastography: Effects of Neurodegeneration in Patients With Amyotrophic Lateral Sclerosis,” Magn. Reson. Med., 72(6), pp. 1755–1761. [PubMed]
Cloots, R. J. H. , Van Dommelen, J. A. W. , Nyberg, T. , Kleiven, S. , and Geers, M. G. D. , 2011, “ Micromechanics of Diffuse Axonal Injury: Influence of Axonal Orientation and Anisotropy,” Biomech. Model Mechanobiol., 10(3), pp. 413–422. [PubMed]
Giordano, C. , Cloots, R. J. H. , Van Dommelen, J. A. W. , and Kleiven, S. , 2014, “ The Influence of Anisotropy on Brain Injury Prediction,” J. Biomech., 47(5), pp. 1052–1059. [PubMed]
Oliphant, T. E. , Manduca, A. , Ehman, R. L. , and Greenleaf, J. F. , 2001, “ Complex‐Valued Stiffness Reconstruction for Magnetic Resonance Elastography by Algebraic Inversion of the Differential Equation,” Magn. Reson. Med., 45(2), pp. 299–310. [PubMed]
McGarry, M. D. J. , Van Houten, E. E. W. , Johnson, C. L. , Georgiadis, J. G. , Sutton, B. P. , Weaver, J. B. , and Paulsen, K. D. , 2012, “ Multiresolution MR Elastography Using Nonlinear Inversion,” Med. Phys., 39(10), pp. 6388–6396. [PubMed]
Johnson, C. L. , McGarry, M. D. J. , Van Houten, E. E. W. , Weaver, J. B. , Paulsen, K. D. , Sutton, B. P. , and Georgiadis, J. G. , 2013, “ Magnetic Resonance Elastography of the Brain Using Multishot Spiral Readouts With Self‐Navigated Motion Correction,” Magn. Reson. Med., 70(2), pp. 404–412. [PubMed]
Manduca, A. , Oliphant, T. E. , Dresner, M. A. , Mahowald, J. L. , Kruse, S. A. , Amromin, E. , Felmlee, J. P. , Greenleaf, J. F. , and Ehman, R. L. , 2001, “ Magnetic Resonance Elastography: Non-Invasive Mapping of Tissue Elasticity,” Med. Image Anal., 5(4), pp. 237–254. [PubMed]
Hutson, M. , and Speed, C. , 2011, Sports Injuries, Oxford University Press, Oxford, UK.
Woolsey, T. A. , Hanaway, J. , and Gado, M. H. , 2013, The Brain Atlas: A Visual Guide to the Human Central Nervous System, Wiley, Hoboken, NJ.
Mazumder, M. M. G. , Bunt, S. , Mostayed, M. , Joldes, G. , Day, R. , Hart, R. , and Wittek, A. , 2013, “ Mechanical Properties of the Brain–Skull Interface,” Acta Bioeng. Biomech., 15(2), pp. 3–11. [PubMed]
Steen, R. G. , Hamer, R. M. , and Lieberman, J. A. , 2007, “ Measuring Brain Volume by MR Imaging: Impact of Measurement Precision and Natural Variation on Sample Size Requirements,” Am. J. Neuroradiol., 28(6), pp. 1119–1125.
Okamoto, R. J. , Clayton, E. H. , and Bayly, P. V. , 2011, “ Viscoelastic Properties of Soft Gels: Comparison of Magnetic Resonance Elastography and Dynamic Shear Testing in the Shear Wave Regime,” Phys. Med. Biol., 56(19), pp. 6379–6400. [PubMed]
Kleiven, S. , and von Holst, H. , 2002, “ Consequences of Head Size Following Trauma to the Human Head,” J. Biomech., 35(2), pp. 153–160. [PubMed]
Gurdjian, E. S. , Hodgson, V. R. , and Thomas, L. M. , 1970, “ Studies on Mechanical Impedance of the Human Skull: Preliminary Report,” J. Biomech., 3(3), pp. 239–247. [PubMed]
Ginsberg, G. , and Genin, J. , 1995, Dynamics, Wiley, New York.
Naunheim, R. S. , Bayly, P. V. , Standeven, J. , Neubauer, J. S. , Lewis, L. M. , and Genin, G. M. , 2003, “ Linear and Angular Head Accelerations During Heading of a Soccer Ball,” Med. Sci. Sports Exercise, 35(8), pp. 1406–1412.
Atay, S. M. , Kroenke, C. D. , Sabet, A. , and Bayly, P. V. , 2008, “ Measurement of the Dynamic Shear Modulus of Mouse Brain Tissue In Vivo by Magnetic Resonance Elastography,” ASME J. Biomech. Eng., 130(2), p. 021013.
Smith, S. M. , Jenkinson, M. , Woolrich, M. W. , Beckmann, C. F. , Behrens, T. E. , Johansen-Berg, H. , Bannister, P. R. , De Luca, M. , Drobnjak, I. , Flitney, D. E. , and Niazy, R. K. , 2004, “ Advances in Functional and Structural MR Image Analysis and Implementation as FSL,” Neuroimage, 23, pp. S208–S219. [PubMed]
McGarry, M. D. J. , Van Houten, E. E. W. , Perrinez, P. R. , Pattison, A. J. , Weaver, J. B. , and Paulsen, K. D. , 2011, “ An Octahedral Shear Strain-Based Measure of SNR for 3D MR Elastography,” Phys. Med. Biol., 56(13), pp. N153–N164. [PubMed]
Scharnhorst, K. , 2001, “ Angles in Complex Vector Spaces,” Acta Appl. Math., 69(1), pp. 95–103.
Wang, H. , Weaver, J. B. , Perreard, I. I. , Doyley, M. M. , and Paulsen, K. D. , 2011, “ A Three-Dimensional Quality-Guided Phase Unwrapping Method for MR Elastography,” Phys. Med. Biol., 56(13), pp. 3935–3952. [PubMed]
Barnhill, E. , Kennedy, P. , Johnson, C. L. , Mada, M. , and Roberts, N. , 2015, “ Real‐Time 4D Phase Unwrapping Applied to Magnetic Resonance Elastography,” Magn. Reson. Med., 73(6), pp. 2321–2331. [PubMed]

## Figures

Fig. 1

Schematic illustration of the accelerometer mouth guard array (MGA) and its positioning for a human subject. (a) Spacing of the three accelerometers within the MGA. (b) The subject bites down firmly on the mouth guard while lying supine on the pillow actuator. (c) Three-dimensional rendering of the relative placement of the subject, pillow actuator, and MGA inside the head coil.

Fig. 3

Validation of accelerometer motion reconstruction using constrained angular and linear motion. (a-i) The accelerometer MGA was placed on the bottom disk of a torsional vibration demonstration system (ECP 205 Torsional Plant, ECP©, Bell Canyon, CA) and used to reconstruct in-plane motion at the location of a fourth reference accelerometer placed at the top of the platform. Oscillation frequency: 10 Hz. (a-ii) Normalized reconstruction RMS error in the AP direction, NRMSE = 0.04; (a-iii) normalized RMS error in the SI direction, NRMSE = 0.02. (b-i) A horizontal shaker (APS Electro-seis 113 Long Stroke Shaker, APS Dynamics ©, San Juan Capistrano, CA) was used to validate translational motion reconstruction. The MGA was placed on the posterior of the platform while the reference accelerometer was placed on the side. Oscillation frequency: 50 Hz. (b-ii) AP direction NRMSE = 0.04, (b-iii) SI direction NRMSE = 0.08.

Fig. 2

Three-dimensional kinematics are estimated at four points around the skull. (a) Schematic diagram of the accelerometer positions relative to the origin. (b) Sagittal MR image showing the relative distance, rA1/O, between the skull origin, approximated as the posterior clinoid process (circle), and accelerometer 1 (A1) within the MGA. The relative location of the pillow actuator is outlined as the rectangle. The MRE acquisition planes, encompassing twenty slices at 2 mm voxel resolution, are shown. (c) Anatomical (T1-weighted) images are used to determine the position of the four reconstruction points anterior (A), right (R), left (L), and posterior (P) relative to the origin. (d) Regional analysis of MRE measurements is performed by segmenting the MRE images into three rings and four quadrants. A mask based on image amplitude was applied to isolate brain tissue from the skull (dotted oval).

Fig. 4

Reconstructed acceleration profile (RL, AP, and SI components of motion) at skull anterior (A), right (R), and posterior (P) points during MRE of a typical human subject (actuator frequency: 50 Hz, actuator amplitude = 18%)

Fig. 8

Rigid-body motion of human brain tissue estimated from MRE measurements, compared with skull motion reconstructed from accelerometer data. (a) Trajectories at individual reconstruction points on the skull anterior (A), right (R), left (L), and posterior (P) are compared with analogous trajectories of neighboring brain tissue material points selected from the outer shell ROI. Filled circles on the elliptical trajectories indicate the reference time, t = 0. (b) The amplitude of the rigid-body motion coefficients for translation and rotation about the skull origin are compared between the skull and brain (n = 6). Error bars indicate standard deviation between subjects. (c) The component of displacement in the AP direction is plotted for the skull and brain posterior reconstruction points.

Fig. 5

(a) and (d) Representative total MRE displacement fields in (a) the gelatin phantom and (d) a human subject at the first of eight time points; (b) and (e) corresponding wave displacement fields; (c) and (f) corresponding curl fields. Wave displacement is obtained by subtracting rigid-body motion from the total displacement field. Note the different displacement amplitudes in panels (d) and (e). The bar at the bottom of each image indicates the approximate location of the pillow actuator.

Fig. 6

Rigid-body motion of the gelatin phantom (estimated from MRE measurements) compared with motion of the case (reconstructed from accelerometer data). (a) Motion reconstructed at individual points on the case anterior (A), right (R), left (L), and posterior (P) are compared with corresponding material points within the outermost shell of corresponding gelatin regions. Filled circles on the elliptical trajectory indicate the reference time, t = 0. b) Rigid-body motion coefficients for translation and rotation about the gelatin phantom origin are compared between the phantom case and gelatin. (c) The component of displacement in the AP direction for the case and gelatin posterior reconstruction points is shown as a function of time.

Fig. 7

Magnitudes (RMS) of (b) wave displacement; (c) curl; and (d) octahedral shear strain in the gelatin phantom. Values are shown for each quadrant-shell ROI (a). For comparison to curl and strain (dimensionless measures of displacement/length), the case displacement magnitude is normalized by the radius of the case (striped). Error bars in all three plots represent the standard deviation of each quantity within the shell ROI.

Fig. 9

Magnitudes (RMS) of (a) wave displacement; (b) curl; and (c) octahedral shear strain, in the human brain. Values are shown for each quadrant-shell ROI. For comparison with curl and strain (dimensionless measures of displacement/length), skull displacement magnitude is normalized by the major semi-axis of the skull (striped). Error bars in all three plots show the standard deviation among subjects (n = 6).

Fig. 10

Comparison of MRE phase contrast (Φ)  captured by different motion-encoding gradient strengths (MEGS). (a) Visualization of the wrapped and unwrapped phase contrast encoded by the maximum MEGS of 25.9 mT/m at time point 2. (b) Comparison of wrapped and unwrapped MRE phase contrast at a single voxel (circle) between EPI-MRE trials at 4, 7, 10, and 25.9 mT/m. (c) Line plot of MRE phase contrast, normalized to the unwrapped phase contrast at 25.9 mT/m. Unwrapped data points were fitted to a linear model (R2 = 0.99).

Fig. 12

During 3D harmonic motion, the trajectory of each point depends on the amplitude and phase of each component. (a) The trajectory is a straight line when all three components have the same temporal phase, ψ, even if the component magnitudes, rn, are different. (b) The trajectory becomes an ellipse when the three components have different temporal phase. (c) The spatial angle, β, between the vectors normal to the plane of each ellipse (arrows) was used to describe the difference in spatial orientation of motion (β  = 0.9 rad in this example).

Fig. 11

Artifact in the accelerometer trace can be used to locate landmarks in the MR pulse sequence, such as the RF pulse, refocusing gradient, and spiral gradients. A two-dimensional spiral MRE pulse sequence diagram is shown in Ref. [20]. Bipolar motion-encoding gradients (dashed boxes) are applied sequentially.

## Tables

Table 1 Reconstructed acceleration components for human skull and gelatin phantom case at the four reconstruction points. NRMSE characterizes the accuracy of reconstruction using the difference between the measured accelerations and the accelerations reconstructed at the accelerometer locations.
Table 2 Spatial angle ($β$) and temporal phase delay ($ϕ$) between harmonic motion of the gelatin phantom case and harmonic, rigid-body motion of gelatin
Table 3 Spatial angle ($β$) and temporal phase delay ($ϕ$) between harmonic motion of the human skull and harmonic, rigid-body motion of brain tissue

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text:
# The average translational kinetic energy of O2 (molar mass 32) molecules at a particular temperature is…

Q: The average translational kinetic energy of O2 (molar mass 32) molecules at a particular temperature is 0.048 eV. The translational kinetic energy of N2 (molar mass 28) molecules in eV at the same temperature is

(a) 0.0015

(b) 0.003

(c) 0.048

(d) 0.768

Ans: (c)

Sol: The average Translational $\large K.E = \frac{3}{2}kT$ depends on temperature only .  Here temperature is same , hence  translational kinetic energy  will be same .

format: plain


text:
# SoCraTes UK 2016: the passion awakens

## They may say I’m a dreamer, but I’m not the only one.

It often starts like a subtle sickness, one day you wake up less motivated than usual, you know your code could be better but there’s a lot to be done, and you start with a

// TODO: refactor this later!

Time passes by, and you get used to a daily routine made of commit, push, release, fix, clean, build, install. A developer, this is what they call you, and you start believing that this is who you are… or not?

Few days ago I watched a documentary titled “The Watchmaker’s Apprentice”, and I found the story of George Daniels very fascinating. The attention, passion and dedication that he put into his watches, the energy despite his age, the resolution to go against a world that runs fast (often headed nowhere), just walking towards perfection and enjoying the results of his work, are all invaluable examples for any person who aims to become the best she can, no matter what others think or say about her.

I’ve been introduced to the concept of Software Craftsmanship by a colleague of mine, and of course by the universally recognised must read book “Clean Code” (R.C. Martin). London completed the induction giving me the chance to participate to few meet ups organised by the London Software Craftsmanship Community.

The natural evolution of this - for now - short journey was my participation to SoCraTes UK, the International Software Craftsmanship Gathering that this year was held in Dorking, Surrey, in Wotton House:

Amazing picture from Twitter

I didn’t really know what to expect, I joined this group of around 80 people on Wednesday the 2nd, we gathered in the main conference room in a circle, and a positive energy immediately filled the space: roughly 30% of participants was new to this event, and Rachel, facilitator for the event, introduced everyone to the concept of open space, marketplace, lightning talk, and so on and so forth. This is what it looked like:

From Twitter

## Open Space Principles

• Whoever comes is the right people

If people decide to participate, you can assume they are interested in what you are going to say. If they are not, they will probably leave at some point.

• Whenever it starts is the right time

There’s no pressure on when to start, and the presentation is not governed by a clock

• Wherever it happens is the right place

Every space is valid… conference rooms, bar, tea room, garden, hallway, corridors

• Whatever happens is the only thing that could have

There’s no predefined outcome: if the discussion ends up on a completely different topic, it’s fine, as long as participants are happy and engaged.

• When it’s over, it’s over

Ideally we organised in 1h (or 2h) slots, but the session could end whenever participants felt that it was appropriate.

## Marketplace

Friday and Saturday morning, participants had the chance of briefly introducing a topic to everyone, and block a time slot on the board, the marketplace.

From Twitter

The result for the two days was this:

Details here

As a young padawan on the path that leads to the Force, I attended as many session as I could, and I gradually felt more and more part of this sort of Jedi council where there’s no right or wrong, and passion and will to learn new things are both fuel of this incredible engine.

I attended sessions on

One of the best things of SoCraTes is the incredible diversity of skills and nationalities, but mostly the passion that everyone demonstrates towards software development, which for a fortunate coincidence happens to be their job.

## Conclusions

Do you want to become just a developer, or the finest watchmaker? Are you ready to turn down the dark side, and embrace Software Craftsmanship? If the answer is yes, then you’re halfway there. Meanwhile, follow @SoCraTes_UK and get ready for 2017 edition! I’ll see you there ;-)

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text:
## Elementary Geometry for College Students (6th Edition)

There is an angle in-between these two angles. $\angle$CAD is between these two angles.


text:
## Toward better GHC syntax errors

As a neophyte to functional programming and Haskell, with full appreciation over the succinctly outputted type errors, for a long time I felt something was remiss with the output of syntax errors. In this post I present a preliminary fix to GHC that improves in the aformentioned arena.

Like many other compilers who have not yet forsaken the annotated BNF approach of describing a syntax and parser, Haskell’s prominent compiler GHC uses Happy in order to generate efficient parsing tables for the syntax.

# The problem

While there are some hand-coded hints about errors being detected while compiling a module with GHC, other errors sometimes don’t provide any extra information.

## Example

 1 2  test i = case i of 2 main = return ()

Output:

 example1.hs:2:1: parse error (possibly incorrect indentation or mismatched brackets)

## Example

 1 2 3 4  test i = i where e main = return ()

Output:

 example2.hs:4:1: parse error (possibly incorrect indentation or mismatched brackets)

As you can see, we have the same error on both occasions. Is it possible to generate an error message that differentiates between the two, automatically? Apparently yes.

# Making it happier

Thanks to the parsing tables, the parser does know which tokens can follow at the place of the error. So, I took the time to modify Happy and modify GHC to produce the following results. The gist of these changes is to allow passing the list of next possible tokens to the user-provided error function. The results are not perfect, but they are interesting nonetheless.

Let’s repeat the previous examples and add some more.

## Example from section 1.1, revisited

 1 2  test i = case i of 2 main = return ()

Output:

 example1.hs:2:1: parse error (possibly incorrect indentation or mismatched brackets), possible tokens: '|' '->'

## Example from section 1.2, revisited

 1 2 3 4  test i = i where e main = return ()

Output:

 example2.hs:4:1: parse error (possibly incorrect indentation or mismatched brackets), possible tokens: '=' '|'

## Example

 1 2 3 4  data X = X { bla :: Int test :: Int }

Output:

 example3.hs:3:10: parse error on input, possible tokens: '}' ‘::’

The avid reader would wonder at this point, why ‘::’ is mentioned and ‘,’ is not mentioned in the possible tokens list. We observe that the error is after ‘Int test’, probably much deeper and in a different syntax production than the one that defines each part of a record. With more time I would examine the Action and Goto tables generated by Happy from the extensive definition of the syntax in GHC.

## Example

 1 2 3 4 5 6  data X = X { bla :: Int , test :: Int } deriving main = return ()

Output:

 example4.hs:6:1: parse error (possibly incorrect indentation or mismatched brackets), possible tokens: '(' CONID QCONID PREFIXQCONSYM

Here it gets more interesting, because the parser now outputs token identifiers that are not just punctuation or operators, but user-defined identifier strings. These are the same names used when defining the syntax.

## Example

 1  main = = return ()

Output:

 example5.hs:1:8: parse error on input, possible tokens: '_' 'as' 'case' 'do' 'hiding' 'if' 'let' 'qualified' 'forall' 'export' 'label' 'dynamic' 'safe' 'interruptible' 'unsafe' 'mdo' 'family' 'role' 'stdcall' 'ccall' 'capi' 'prim' 'javascript' 'proc' 'group' 'static' '{-# CORE' '{-# SCC' '{-# GENERATED' '\\' '~' '-' '[' '[:' '(' '(#' '(|' SIMPLEQUOTE VARID CONID QVARID QCONID PREFIXQVARSYM PREFIXQCONSYM IPDUPVARID CHAR STRING INTEGER RATIONAL PRIMCHAR PRIMSTRING PRIMINTEGER PRIMWORD PRIMFLOAT PRIMDOUBLE '[|' '[p|' '[t|' '[d|' '[||' TH_ID_SPLICE '$(' TH_ID_TY_SPLICE '$\$(' TH_TY_QUOTE TH_QUASIQUOTE TH_QQUASIQUOTE ‘=’

Obviously, there are many things that could come instead of the second =. Perhaps it would require some community debate regarding how to present this behemoth list of possible tokens to the user in this case.

April 6, 2015


text:
• 453

Goal

International Standard Book Number (ISBN) is a unique numeric commercial book identifier.

Before year 2007 ISBNs were 10-digit long. After that year ISBNs extended to 13 digits. In both ISBN-10 and ISBN-13 standards, the last digit is the check digit, for error detection.

ISBN-10 check digit is calculated by Modulus 11 with decreasing weights on the first 9 digits.
Example: 030640615?
0×10 + 3×9 + 0×8 + 6×7 + 4×6 + 0×5 + 6×4 + 1×3 + 5×2 = 130.
130 / 11 = 11 remainder 9.
Check digit is the value needed to add to the sum to make it dividable by 11. In this case it is 2.
So the valid ISBN is 0306406152.
In case 10 being the value needed to add to the sum, we use X (upper case X) as the check digit instead of 10.

ISBN-13 check digit is calculated by Modulus 10 with alternate weights of 1 and 3 on the first 12 digits.
Example: 978030640615?
9×1 + 7×3 + 8×1 + 0×3 + 3×1 + 0×3 + 6×1 + 4×3 + 0×1 + 6×3 + 1×1 + 5×3 = 93.
93 / 10 = 9 remainder 3.
Check digit is the value needed to add to the sum to make it dividable by 10. So the check digit is 7. The valid ISBN is 9780306406157.

A valid ISBN should contain the digits 0 to 9 only, except for the check digit X which is determined as explained above. X or other improper characters appearing in the wrong place will render an ISBN invalid.
Input
Line 1: An integer N for the number of ISBNs to verify.
Next N lines: One line for each ISBN to verify.

In this puzzle we assume all ISBNs have been normalized - have spaces and dash characters removed.
Output
Line 1: A string Y invalid: where Y is the number of invalid ISBNs.
Next Y lines: One line for each invalid ISBN, in the same order as given in the inputs.
Constraints
1 ≤ N ≤ 500
1 ≤ YN (i.e. there will always be at least one invalid ISBN in each case)
Example
Input
6
0306406152
013603599X
1145185215X
9780306406157
9780306406154
978043551907X

Output
3 invalid:
1145185215X
9780306406154
978043551907X

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text:
# Question #a9231

Oct 13, 2017

Here's why that is the case.

#### Explanation:

The idea here is that in order for an atom to be an atom of boron, it must have $5$ protons inside its nucleus. That is a must.

In other words, if an atom has an atomic number equal to $5$, you can say for a fact that you're dealing with an atom of boron.

However, the number of neutrons that an atom can have and still be an atom of boron can vary. The problem tells you that boron has two isotopes, boron-10 and boron-11.

These two atoms are said to be isotopes of boron because

• they have the same number of protons inside the nucleus
• they have different numbers of neutrons inside the nucleus

The number added to the name of the isotope tells you the mass number of the isotope, i.e. the number of protons and neutrons located inside the nucleus.

So you can say that

• $\text{boron-10: " "protons + neutrons} = 10$
• $\text{boron-11: " "protons + neutrons} = 11$

Since both of them must have $5$ protons inside the nucleus, you can say that

• $\text{boron-10: " "5 protons + 5 neutrons}$
• $\text{boron-11: " "5 protons + 6 neutrons}$

So, to sum this up, boron-10 and boron-11 are isotopes because they have the same number of protons in the nucleus, i.e. $5$, and different numbers of neutrons in the nucleus, i.e. $5$ for boron-10 and $6$ for boron-11.


text:
# Tag Info

Accepted

### asymptotic of a determinant

The matrix $M_n$ is the Hilbert matrix $H_n = \left(\dfrac{1}{i+j+1}\right)_{0 \le i,j \le n-1}$ but flipped horizontally. Hence, $|\det(M_n)| = |\det(H_n)|$ for all $n$. From the Wikipedia article on ...
• 51.9k
Accepted

### Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?

We have \begin{align*} \prod\limits_{2<p \le n} {\left( {1 - \frac{2}{p}} \right)^{ - 1} } & = \exp \left( { - \sum\limits_{2<p \le n} {\log \left( {1 - \frac{2}{p}} \right)} } \right) \\ &...
• 20.4k
Accepted

### Asymptotic behavior of integral with Laplace's method

Elaborating on Maxim's comment. If $v=xT$, then \begin{align*} I(n)&=\int_0^1\!\! {\int_0^1 {e^{ - nxT} \sqrt {1 - (1 - \sqrt x + \sqrt {xT} )^2 } dT} dx} \\& = \int_0^1\!\! {\int_0^x {e^{ - ...
• 20.4k
Accepted

### sequence such that $x_{n+1}=n(x_n-n)$

By some experimentation from the starting point $x_n=an+b+y_n$ and reporting in the equation I quickly found that $x_n=n+1+2y_n$ lead to the following simplification $$y_{n+1}=ny_n-1$$ So we have ...
• 24.6k
Accepted

### Asymptotics of a two dimensional integral

It turns out that @Gary 's suggestion to reverse the order of integration is a fruitful first step because the integral over $\epsilon$ is analytically tractable. Exchanging the order results in the ...
• 7,978
Accepted

### Bounding a convergent sum

For each $n$, let $f_n(x) = \frac{\exp(-n/2^x)}{2^x x}$. Then we can show that(1),(2) $$\left| \sum_{i=1}^{n} f_n(i) - \int_{1}^{n} f_n(x) \, \mathrm{d}x \right| \leq \max_{1\leq x \leq n} f_n(x).$$ ...
• 143k

### Is this a sound line of reasoning to conclude that $\sqrt[n]{n!} \sim \frac{n}{e}$?

You may just use the following Lemma if $\{a_n\}_{n\geq 1}$ is a sequence of positive numbers such that $\lim_{n\to +\infty}\frac{a_{n+1}}{a_n}=L$, then $\lim_{n\to +\infty}\sqrt[n]{a_n}=L$. Apply ...
• 340k

### Slight variation of Mertens' third theorem. Do we have an estimate of $\sum_{d \mid n\#}(-1)^{\omega(d)}\dfrac{\sigma_0(d)}{d}$?

Asymptotically, $$g(n) = \prod_{2 < p < n}^{p \text{ prime}} 1-\frac2p \sim \frac{4C_2e^{-2\gamma}}{(\log n)^2} \approx \frac{0.83244}{(\log n)^2}$$ And presumably your expression is simply the ...
• 1,385
Accepted

• 217k
Accepted

• 33.5k

• 6,340

• 20.4k
1 vote

• 1,110
1 vote

### Expansion of logarithm with little o

When you have on $o(z)$ in a development it means that anything smaller than order $z$ is simply ignored. When $z\to 0$ we have to ignore any terms in $z^a$ where $a>1$, in particular $z^2$ and of ...
• 24.6k
1 vote

### Expansion of logarithm with little o

You were too aggressive with your $o$s, and confused an $O(|z+o(z)|^2)$ term with $o(|z+o(z)|^2)$. You could have instead expanded as: $$z+o(z)-\frac{1}{2}(z+o(z))^2+o(|z+o(z)|^2)$$ We can clean this ...
• 10.6k

Only top scored, non community-wiki answers of a minimum length are eligible


text:
# Is there anyway to define an Euler Characteristic for infinite CW complexes, like $\mathbb{R}P^{\infty}$?

I ask this because of the following thing I have been thinking of:

We know $\mathbb{R}P^{\infty}$ can be thought of as an infinite CW complex with one cell in each dimension.

If you try to use the same formula as the normal Euler Characteristic, you see you have $1-1+1-1+1-....$

This really doesn't converge to a number, however there is that way in which you can view the sum of this series as being $1/2$.

What I am wondering is if there is any way to interpret the answer of $1/2$ in terms of the topological properties of the space $\mathbb{R}P^{\infty}$.

Thanks!

-
There's also "that way" where you can view the sum of this series as being $0$, I'm not sure what an undefined number could tell you about the topology. The Poincaré series is interesting though. –  Najib Idrissi Feb 20 '14 at 20:40

I'm not sure if this directly answer your question, but:

I don't believe you can really define an Euler characteristic. But one thing is actually interesting: the Poincaré series of the space. In general, for a graded vector space $V$, you define it as $L(t) = \sum \dim(L_n) t^n$. This is a formal power series. For a space $X$, you can use its cohomology with coefficients in $A$: $$L_X(t) = \sum \dim(H^n(X; A)) t^n$$

If $X$ is a finite CW-complex, then this is related to the Euler characteristic by $\chi(X) = L_X(-1)$.

$\mathbb{RP}^\infty$ is an Eilenberg-MacLane of type $K(\mathbb{Z/2Z}, 1)$. In his article Cohomologie modulo $2$ des complexes d'Eilenberg-MacLane (French), Serre described how the behavior of the Poincaré series (mod 2) of an Eilenberg-MacLane space can give you information about the space in question.

You should read it (I'm not about to sum up the whole article in this comment box), but there are very interesting results. In particular this theorem (Théorème 10) is proved through the use of the Poincaré series of Eilenberg-MacLane spaces, and more specifically their behavior at $1$:

Theorem: Let $X$ be a simply connected space, such that:

1. $H_i(X; \mathbb{Z})$ is a finitely generated abelian group for all $i > 0$;
2. $H_i(X; \mathbb{Z/2Z}$ is zero for big enough $i$;
3. $H_i(X; \mathbb{Z/2Z}$ is nonzero for at least one $i$.

Then there is an infinite number of integers $i$ such that $\pi_i(X)$ contains a subgroup isomorphic to either $\mathbb{Z}$ or $\mathbb{Z/Z}$.

-


text:
# suppress page numbers in document's front matter

I have a problem and I want that someone tell me how I can resolve that. I want that the number of page in my report start in the page of introduction but in reality it starts in the didecas and liste of contents. I want that these pages have not number. This is my code

\documentclass[12pt,a4paper]{report}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[french]{babel}
\usepackage{color}
\usepackage{xcolor} %les couleurs
\definecolor{myblue}{rgb}{0.2,0.4,1}
\usepackage{layout}
\usepackage[top=2 cm, bottom=2 cm, left=2 cm, right=2 cm]{geometry}
\usepackage{setspace}
\usepackage[pdftex]{graphicx}
\usepackage{array}
%\documentclass[landscape]{scrartcl}
\usepackage{geometry}
\usepackage{multirow}

\usepackage[justification=centering]{caption}

\usepackage{pgfplots}

\usepackage{fancyhdr}
\fancyhf{} % clear all header and footers

\begin{document}
\pagestyle{empty}
\include{title-content}
\color{black}
\doublespacing
\include{dedicaces-content}
%\include{remerciement}

\tableofcontents

\listoffigures
\listoftables

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagestyle{plain}
\pagenumbering{arabic}
\chapter{Introduction générale}

-
Please provide a bit more information regarding the LaTeX document class you use (report? book? something else?) and whether you use any LaTeX packages that may affect the page style and page numbering style that's in effect. You should certainly try (a) issuing the command \pagestyle{empty} immediately after \begin{document} and (b) issuing the commands \pagestyle{plain} and \pagenumbering{arabic} immediately before \chapter{Introduction}. That said, I can't help but remark that it would be a bit unusual not to have any page numbers in the document's front matter area. –  Mico Jun 25 '12 at 20:58
@Mico: have tried what you suggest but the same problem last. –  boutheina Jun 25 '12 at 21:19
Thanks for providing the additional pieces if information. You may have to include the instruction \thispagesttle{empty} in each of your included .tex files and also have to insert that instruction after the \tableofcontents command (and issue a \clearpage after the front matter). –  Mico Jun 25 '12 at 21:36
@Mico: thank you for your answer can you put this in my code please? –  boutheina Jun 25 '12 at 21:40

(Note: I've rewritten this answer after the OP explained more details of the document's setup.)

To suppress page numbering in a part of the document, one has to both set \pagenumbering{empty} and take care of commands -- such as \tableofcontents, \listoffigures, and \listoftables -- that internally invoke a command (specifically, the \chapter command) which (re)sets the page style of the current page to plain. In the example below, I use the \patchcmd macro of the etoolbox package to "patch" the chapter command. To keep the scope of the modification local to the front matter of the document, I've put a \begingroup ... \endgroup pair around the relevant part of the document. Finally, after the end of the frontmatter, it's necessary to restore plain pagenumbering and to restart the page numbering by issuing the command \pagenumbering{arabic}. Happy TeXing!

\documentclass[12pt,a4paper]{report}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[french]{babel}
\usepackage{xcolor}
\definecolor{myblue}{rgb}{0.2,0.4,1}
\usepackage{layout}
\usepackage[margin=2cm]{geometry}
\usepackage{setspace}
\usepackage[pdftex]{graphicx}
\usepackage{array,multirow,pgfplots,filecontents}
\usepackage[justification=centering]{caption}

% Load the 'etoolbox' package to provide to
% patch the '\chapter command (locally, in this case)
\usepackage{etoolbox}

\begin{document}
\pagestyle{empty}    % switch to "empty" page style for front matter of document
\begingroup
% In this TeX group, the first page of new "chapters" should obey
% '\thispagestyle{empty}' rather than the default '\thispagestyle{plain}'.
\patchcmd{\chapter}{plain}{empty}{}{}

\include{title-content}
\color{black}
\doublespacing
\include{dedicaces-content}
\include{remerciement}

\tableofcontents
\listoffigures
\listoftables
\endgroup % end of TeX group

\clearpage
\pagestyle{plain}      % revert to "plain" page style
\pagenumbering{arabic} % restart page numbering
\chapter{Introduction générale}
% 19 more "dummy" chapters, just to make the ToC span 2 pages
\chapter{Bla 2}
\chapter{Bla 3}
\chapter{Bla 4}
\chapter{Bla 5}
\chapter{Bla 6}
\chapter{Bla 7}
\chapter{Bla 8}
\chapter{Bla 9}
\chapter{Bla 10}
\chapter{Bla 11}
\chapter{Bla 12}
\chapter{Bla 13}
\chapter{Bla 14}
\chapter{Bla 15}
\chapter{Bla 16}
\chapter{Bla 17}
\chapter{Bla 18}
\chapter{Bla 19}
\chapter{Bla 20}
\end{document}
`
-


text:
# Dakota's travel toothbrush case, shaped like an equilateral triangular prism, has perpendicular height of 2.3 in. and depth of 8.4 in. How much plastic is the case made out of, in sq. in.?

Oct 5, 2017

About $28.42$ square inches.

#### Explanation:

Assuming this includes plastic to cover the three long sides as well as the two triangular ends, Let's look at the ends first:
The perpendicular height of the triangle is an altitude drawn from the top vertex, which gives you two 30-60-90 triangles on either end.
On questions like this, always draw a picture first!!

You can use the law of sines to set it up like this: $\frac{\sin {60}^{o}}{2.3} = \frac{\sin {90}^{o}}{x}$, with $x$ representing the length of a side of the triangle.

Or you can use your 30-60-90 rules, which say that the long leg (the perpendicular height) is $\frac{\sqrt{3}}{2} \cdot$the hypotenuse and solve for $x$ that way. $2.3 \sqrt{3} = 2 x \implies \frac{2.3 \sqrt{3}}{2} = x \approx 2.66$.
Either method gives you a value where $x = \frac{2.3}{\sqrt{3}} \approx 2.66$.
To find the area of the triangles, use the formula Area=1/2 base*height. The area of each triangle ends up being $2.3 \frac{x}{2} \approx 3.054$.

For the long sides, you just have three congruent rectangles. Taking that value of $x$ from before, the rectangles have short sides$= x$ in. and long sides$= 8.4$ in. To find the area, length$\cdot$width, you have $8.4 x \approx 22.31$ in.

Three of these plus the two triangles on the end, the surface area is about $28.42$ square inches.


text:
Home > Research > Publications & Outputs > Bayesian estimation of agent-based models

### Electronic data

• 1-s2.0-S0165188917300222-main

Rights statement: This is the author’s version of a work that was accepted for publication in Journal of Economic Dynamics and Control. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Economic Dynamics and Control, 77, 2017 DOI: 10.1016/j.jedc.2017.01.014

Accepted author manuscript, 1.72 MB, PDF document

## Bayesian estimation of agent-based models

Research output: Contribution to journalJournal article

Published

### Standard

Bayesian estimation of agent-based models. / Grazzini, Jakob; Richiardi, Matteo; Tsionas, Efthymios.

In: Journal of Economic Dynamics and Control, Vol. 77, 04.2017, p. 26-47.

Research output: Contribution to journalJournal article

### Harvard

Grazzini, J, Richiardi, M & Tsionas, E 2017, 'Bayesian estimation of agent-based models', Journal of Economic Dynamics and Control, vol. 77, pp. 26-47. https://doi.org/10.1016/j.jedc.2017.01.014

### APA

Grazzini, J., Richiardi, M., & Tsionas, E. (2017). Bayesian estimation of agent-based models. Journal of Economic Dynamics and Control, 77, 26-47. https://doi.org/10.1016/j.jedc.2017.01.014

### Vancouver

Grazzini J, Richiardi M, Tsionas E. Bayesian estimation of agent-based models. Journal of Economic Dynamics and Control. 2017 Apr;77:26-47. https://doi.org/10.1016/j.jedc.2017.01.014

### Author

Grazzini, Jakob ; Richiardi, Matteo ; Tsionas, Efthymios. / Bayesian estimation of agent-based models. In: Journal of Economic Dynamics and Control. 2017 ; Vol. 77. pp. 26-47.

### Bibtex

@article{4f4bea9e612144d6bcf155f536b8ed80,
title = "Bayesian estimation of agent-based models",
abstract = "We consider Bayesian inference techniques for Agent-Based (AB) models, as an alternative to simulated minimum distance (SMD). Three computationally heavy steps are involved: (i) simulating the model, (ii) estimating the likelihood and (iii) sampling from the posterior distribution of the parameters. Computational complexity of AB models implies that efficient techniques have to be used with respect to points (ii) and (iii), possibly involving approximations. We first discuss non-parametric (kernel density) estimation of the likelihood, coupled with Markov chain Monte Carlo sampling schemes. We then turn to parametric approximations of the likelihood, which can be derived by observing the distribution of the simulation outcomes around the statistical equilibria, or by assuming a specific form for the distribution of external deviations in the data. Finally, we introduce Approximate Bayesian Computation techniques for likelihood-free estimation. These allow embedding SMD methods in a Bayesian framework, and are particularly suited when robust estimation is needed. These techniques are first tested in a simple price discovery model with one parameter, and then employed to estimate the behavioural macroeconomic model of De Grauwe (2012), with nine unknown parameters.",
keywords = "Agent-based, Estimation, Bayes, Approximate bayesian computation, Likelihood",
author = "Jakob Grazzini and Matteo Richiardi and Efthymios Tsionas",
note = "This is the author{\textquoteright}s version of a work that was accepted for publication in Journal of Economic Dynamics and Control. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Economic Dynamics and Control, 77, 2017 DOI: 10.1016/j.jedc.2017.01.014",
year = "2017",
month = apr
doi = "10.1016/j.jedc.2017.01.014",
language = "English",
volume = "77",
pages = "26--47",
journal = "Journal of Economic Dynamics and Control",
issn = "0165-1889",
publisher = "Elsevier",

}

### RIS

TY - JOUR

T1 - Bayesian estimation of agent-based models

AU - Grazzini, Jakob

AU - Richiardi, Matteo

AU - Tsionas, Efthymios

N1 - This is the author’s version of a work that was accepted for publication in Journal of Economic Dynamics and Control. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Economic Dynamics and Control, 77, 2017 DOI: 10.1016/j.jedc.2017.01.014

PY - 2017/4

Y1 - 2017/4

N2 - We consider Bayesian inference techniques for Agent-Based (AB) models, as an alternative to simulated minimum distance (SMD). Three computationally heavy steps are involved: (i) simulating the model, (ii) estimating the likelihood and (iii) sampling from the posterior distribution of the parameters. Computational complexity of AB models implies that efficient techniques have to be used with respect to points (ii) and (iii), possibly involving approximations. We first discuss non-parametric (kernel density) estimation of the likelihood, coupled with Markov chain Monte Carlo sampling schemes. We then turn to parametric approximations of the likelihood, which can be derived by observing the distribution of the simulation outcomes around the statistical equilibria, or by assuming a specific form for the distribution of external deviations in the data. Finally, we introduce Approximate Bayesian Computation techniques for likelihood-free estimation. These allow embedding SMD methods in a Bayesian framework, and are particularly suited when robust estimation is needed. These techniques are first tested in a simple price discovery model with one parameter, and then employed to estimate the behavioural macroeconomic model of De Grauwe (2012), with nine unknown parameters.

AB - We consider Bayesian inference techniques for Agent-Based (AB) models, as an alternative to simulated minimum distance (SMD). Three computationally heavy steps are involved: (i) simulating the model, (ii) estimating the likelihood and (iii) sampling from the posterior distribution of the parameters. Computational complexity of AB models implies that efficient techniques have to be used with respect to points (ii) and (iii), possibly involving approximations. We first discuss non-parametric (kernel density) estimation of the likelihood, coupled with Markov chain Monte Carlo sampling schemes. We then turn to parametric approximations of the likelihood, which can be derived by observing the distribution of the simulation outcomes around the statistical equilibria, or by assuming a specific form for the distribution of external deviations in the data. Finally, we introduce Approximate Bayesian Computation techniques for likelihood-free estimation. These allow embedding SMD methods in a Bayesian framework, and are particularly suited when robust estimation is needed. These techniques are first tested in a simple price discovery model with one parameter, and then employed to estimate the behavioural macroeconomic model of De Grauwe (2012), with nine unknown parameters.

KW - Agent-based

KW - Estimation

KW - Bayes

KW - Approximate bayesian computation

KW - Likelihood

U2 - 10.1016/j.jedc.2017.01.014

DO - 10.1016/j.jedc.2017.01.014

M3 - Journal article

VL - 77

SP - 26

EP - 47

JO - Journal of Economic Dynamics and Control

JF - Journal of Economic Dynamics and Control

SN - 0165-1889

ER -