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The moment of inertia I , however, is always specified with respect to that axis and is defined as the sum of the products obtained by multiplying the mass of each particle of matter in a given body by the square of its distance from the axis.
In calculating angular momentum for a rigid body, the moment of inertia is analogous to mass in linear momentum. The unit of moment of inertia is a composite unit of measure. In the International System SI , m is expressed in kilograms and r in metres, with I moment of inertia having the dimension kilogram-metre square. In the U. Black Ice Impact. Similar artists. Listen Create Services Support. All rights reserved. Damasio AR.
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This indicated that the brain's motion can be approximated by a single degree of freedom along the first eigendirection figure 2 a. The exponential decay in the magnitude of the eigenvalues of the 3DOF motion further justified this reduction figure 2 b. The kinematic relations between the three DOF the first eigendirection are given in table 1. The resulting 1DOF rotation showed that the brain lags behind in the initial motion of the skull but continues to oscillate even after the skull has settled figure 2 c.
We use the 1DOF kinematic approximation for all further dynamic analyses of the brain—skull system. Results for PCA fits to the brain's relative rigid-body motion.
Principal component analysis of rigid-body motion based on tagged MRI data [ 24 ]: a 3DOF motion of brain in the sagittal plane is shown for the three subjects. There is a strong correlation between the 3DOF such that it can be approximated by a single DOF, b first few principal components for the rigid-body motion are shown as the ratio of total variance. Online version in colour.
Given the clear periodic oscillatory behaviour of relative brain motion figure 2 c , we speculated that a linear second-order discrete system could explain the relative brain—skull motion. In the small angle regime, this can be physically interpreted as the brain rolling along the inner surface of the skull.
We chose to use the brain's relative angle as the independent degree of freedom because of the importance of brain rotation in diffuse axonal injury [ 15 , 16 ]. We subsequently used the constrained kinematics of relative brain motion derived separately for each subject to develop a coupled linear dynamic model with the skull's linear and rotational accelerations as the inputs. The model consists of a parallel torsional spring-damper element connecting the brain's CoR to the skull's CoM figure 3 a.
Equations of motion for the 1DOF dynamic model are given in the electronic supplementary material, Dynamic modelling section. Dynamic model fit of experimental brain motion data. The representative system could be described by rolling kinematics, where the translational motion is coupled to rotation through a kinematic constraint CoR path in figure 1 c.
The model parameters include the stiffness k and damping c of the torsional spring-damper element, and the brain's mass m and moment of inertia I , which were fitted separately for each subject. The fitted parameters for each subject and their corresponding R 2 values are reported in table 2.
The fitted mass and moment of inertia values are close to the values reported in the literature for the brain, indicating the physical relevance of the model [ 19 , 23 ]. It should be noted that while this simple model is not anatomically detailed, it closely represents the coupled motion of the brain and skull observed in vivo , whereas physical parameters such as mass and inertia are preserved and the compliance of the brain—skull interface subarachnoid space, bridging veins and trabeculae is represented by the torsional spring-damper.
An advantage of the model developed above is that while it captures the dynamic properties of the brain—skull system, it is based on the rolling kinematics of the brain, i.
To estimate the accuracy of the model's predictions, we calculated the error introduced at each stage of added modelling constraint: unconstrained kinematics 3DOF , constrained kinematics 1DOF and constrained dynamics 1DOF.
We calculated the distance errors between every experimental nodal value in MRI and the corresponding model prediction through all time steps for all three subjects. In order to study the frequency response of the brain—skull system, we reduced the 1DOF dynamic model to a linear single-input—single-output SISO system, by using PCA on the skull motion and assuming an equivalent stiffness, damping and torque in the constrained dynamic equations of motion electronic supplementary material, Frequency modelling section.
This step is feasible given the coupling between the DOF for both the brain and skull. Resonance frequency: non-dimensionalized frequency domain fit of electronic supplementary material, equation A 11 based on the empirical transmissibility amplitudes of MRI data for the three subjects.
As a reference, the temporal fit to the dynamic model is also shown. Although the dynamic parameters determined in the time domain above are expected to represent the system in the frequency domain as well, in order to study the system's frequency behaviour, it is customary to fit the parameters to the frequency response of the system. The frequency fit marked as MRI frequency fit closely replicates the experimental transmissibility function figure 4.
The transmissibility function calculated using the parameters from the time-domain fit marked as MRI temporal fit also shows a similar trend with a small difference in resonance frequency prediction, which might be due to the linearization of the transmissibility function as detailed in the electronic supplementary material, Frequency modelling section figure 4.
In this section, we verify the 1DOF dynamic model both in frequency and time domains with brain displacement data derived from X-ray measurements during PMHS head impacts [ 25 ]. This step is crucial to examine the applicability of our model to loadings that are more representative of typical on-field head impacts. Given the slightly higher density of the brain compared with the cerebrospinal fluid CSF and possible sinking in the inverted PMHS, we re-fitted the brain's kinematic path parameters for PMHS figure 5 a.
Assuming symmetry of forces, we used the same dynamic parameters as determined from the in vivo data. Several factors might cause the discrepancy in the maximum amplification frequency in the brain response between the MRI and PMHS measurements, including variations in experimental protocols or changes in the material properties of the ex vivo tissue.
Finally, as an additional step to verify the predictions of the constrained dynamic model, we compared its predictions with simulated injury monitor SIMon software, which is an FE model of the human head developed and used by the US National Highway Traffic and Safety Administration [ 29 ].
Despite the much fewer degrees of the freedom, the 1DOF model developed in this study fared well. The details of this verification are reported in electronic supplementary material, Model verification section electronic supplementary material, figure S9. Image in a is adopted from [ 25 ]. We have developed the first reduced-order dynamical brain model based on in vivo human data.
Owing to research ethics considerations, it is not yet possible to acquire in vivo brain data from concussive-level impacts. Thus, we developed our model based on mild impacts in an MRI study and subsequently investigated the applicability of the model in moderate impacts using PMHS brain displacement data. Our analysis of these datasets revealed that relative brain motion owing to head impacts can be described by an under-damped second-order system with linear dynamics that is driven to resonance at around 15 Hz.
While modal behaviour of the brain was not the primary focus of most of previous lumped-parameter models, the reported or calculated values for natural frequency were either much smaller less than 5 Hz [ 20 ] or much greater than more than 50 Hz the values reported above [ 22 , 23 ]. In another study, we have collected head kinematic measurements from a number of contact sports using an instrumented mouthguard capable of recording 6DOF motion [ 30 , 31 ].
The MRI 1. Of field football head impacts, we selected impacts where sagittal rotation was dominant the sagittal component in rotational acceleration had the largest contribution to the peak magnitude of acceleration. With these impacts, we performed a fast Fourier transform FFT on the sagittal rotational acceleration time trace.
The ms time trace, sampled at Hz, was padded with zeros to increase the resolution for the FFT. The frequency with the largest FFT amplitude was identified as the primary frequency of the impact. The striking uniformity of impacts oscillating around 20 Hz indicates that a substantial portion of sports head impacts are exciting a mechanical resonance of the skull—brain system and causing amplified skull—brain relative motions.
In previous studies using our instrumented mouthguard, we showed that the natural frequency of the mouthguard is around Hz and that head impacts had much higher energy levels at lower frequencies, suggesting that the measured frequency content has not been biased by the instrumentation [ 30 , 32 ]. The above information again prompts the long-ignored question: can mechanical resonance lead to mTBI? Field measurements of head impact kinematics: a comparison of translational top and rotational bottom input levels for MRI and PMHS studies against field measurements, b histogram of primary frequencies in rotational acceleration from field measurements.
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