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Cognitive Neurodynamics

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01.09.2009 | Research Article

New mechanics of traumatic brain injury

verfasst von: Vladimir G. Ivancevic

Erschienen in: Cognitive Neurodynamics | Ausgabe 3/2009

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Abstract

The prediction and prevention of traumatic brain injury is a very important aspect of preventive medical science. This paper proposes a new coupled loading-rate hypothesis for the traumatic brain injury (TBI), which states that the main cause of the TBI is an external Euclidean jolt, or SE(3)-jolt, an impulsive loading that strikes the head in several coupled degrees-of-freedom simultaneously. To show this, based on the previously defined covariant force law, we formulate the coupled Newton–Euler dynamics of brain’s micro-motions within the cerebrospinal fluid and derive from it the coupled SE(3)-jolt dynamics. The SE(3)-jolt is a cause of the TBI in two forms of brain’s rapid discontinuous deformations: translational dislocations and rotational disclinations. Brain’s dislocations and disclinations, caused by the SE(3)-jolt, are described using the Cosserat multipolar viscoelastic continuum brain model.

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A closed injury occurs when the head suddenly and violently hits an object but the object does not break through the skull.

A penetrating injury occurs when an object pierces the skull and enters brain tissue.

SI is a vector quantity indicating the direction and magnitude of power flow inside a dynamically loaded structure.

The mechanical SE(3)-jolt concept is based on the mathematical concept of higher-order tangency (rigorously defined in terms of jet bundles of the head’s configuration manifold) (Ivancevic and Ivancevic 2006c, e), as follows: When something hits the human head, or the head hits some external body, we have a collision. This is naturally described by the SE(3)-momentum, which is a nonlinear coupling of three linear Newtonian momenta with three angular Eulerian momenta. The tangent to the SE(3)-momentum, defined by the (absolute) time derivative, is the SE(3)-force. The second-order tangency is given by the SE(3)-jolt, which is the tangent to the SE(3)-force, also defined by the time derivative.

Recall that the cross product ${\mathbf{u}\times \mathbf{v}}$ of two vectors ${\mathbf{u}} $ and ${\mathbf{v}}$ equals ${\mathbf{u}\times \mathbf{v}}=uv\sin\theta {\mathbf{n}},$ where θ is the angle between ${\mathbf{u}}$ and ${\mathbf{v}},$ while $ {\mathbf{n}}$ is a unit vector perpendicular to the plane of ${\mathbf{u}}$ and $ {\mathbf{v}}$ such that ${\mathbf{u}}$ and ${\mathbf{v}}$ form a right-handed system.

In reality, mass and inertia matrices (M, I) are not diagonal but rather full 3 × 3 positive-definite symmetric matrices with coupled mass- and inertia-products. Even more realistic, fully-coupled mass–inertial properties of a brain immersed in (incompressible, irrotational and inviscid) cerebrospinal fluid are defined by the single non-diagonal 6 × 6 positive-definite symmetric mass–inertia matrix $ {\mathcal{M}}_{SE(3)},$ the so-called material metric tensor of the SE(3)-group, which has all nonzero mass–inertia coupling products. In other words, the 6 × 6 matrix ${\mathcal{M}}_{SE(3)}$ contains: (i) brain’s own mass plus the added mass matrix associated with the fluid, (ii) brain’s own inertia plus the added inertia matrix associated with the potential flow of the fluid, and (iii) all the coupling terms between linear and angular momenta. However, for simplicity, in this paper we shall consider only the simple case of two separate diagonal 3 × 3 matrices (${\mathbf{M,I}}).$

In reality, $\varvec{\omega }$ is a 3 × 3 attitude matrix (see Appendix). However, for simplicity, we will stick to the (mostly) symmetrical translation-rotation vector form.

In a fully-coupled Newton–Euler brain dynamics, instead of Eq. 4 we would have brain’s kinetic energy defined by the inner product:

$$ E_{k}=\frac{1}{2}\left[\frac{{\mathbf{p}}}{\varvec{\pi}} {\mathcal{M}}_{SE(3)}\frac{{\mathbf{p}}}{\varvec{\pi}}\right]. $$

Note that the derivative of the cross-product of two vectors follows the standard calculus product-rule: $\frac{\text{d}}{{\text{d}}t}({\mathbf{u\times v}})={\dot{\mathbf{u}}\times {\mathbf{v}}+{\mathbf{u}}\times {\dot{\mathbf{v}}.}}$

In this paragraph the overdots actually denote the absolute Bianchi (covariant) time-derivative (1), so that the jolts retain the proper covector character, which would be lost if ordinary time derivatives are used. However, for the sake of simplicity and wider readability, we stick to the same overdot notation.

Differential p-forms are totally skew-symmetric covariant tensors, defined using the exterior wedge-product and exterior derivative. The proper definition of exterior derivative d for a p-form β on a smooth manifold M, includes the Poincaré lemma (Ivancevic and Ivancevic 2006c, 2007e): d(dβ) = 0, and validates the general Stokes formula

$$ \int_{\partial M}\beta =\int_{M}d\beta , $$

where M is a p-dimensional manifold with a boundary and ∂M is its (p − 1)-dimensional boundary, while the integrals have appropriate dimensions.

A p-form β is called closed if its exterior derivative is equal to zero,

$$ \text{d}\beta =0. $$

From this condition one can see that the closed form (the kernel of the exterior derivative operator d) is conserved quantity. Therefore, closed p-forms possess certain invariant properties, physically corresponding to the conservation laws.

A p-form β that is an exterior derivative of some (p − 1)-form α,

$$ \beta =\text{d}\alpha , $$

is called exact (the image of the exterior derivative operator d). By Poincaré lemma, exact forms prove to be closed automatically,

$$ \text{d}\beta =\text{d}(\text{d}\alpha )=0. $$

This lemma is the foundation of the de Rham cohomology theory (Ivancevic and Ivancevic 2006c, 2007e).

One practical application of the proposed model is in design of helmets. Briefly, a ‘hard’ helmet saves the skull but not the brain; alternatively, a ‘soft’ helmet protects the brain from the collision jolt but does not protect the skull. A good helmet is both ‘hard’ and ‘soft’. In other words, if a human head covered with a solid helmet collides with a massive external body, the skull will be protected by the helmet—but the brain will still be shocked by the SE(3)-jolt, and a TBI will be caused. With or without the ‘hard’ helmet, brain’s inertia tensor will be moved and rotated by the external SE(3)-jolt, and this will cause a brain injury, proportional to the jolt-collision with the head. Therefore, while protecting the skull is a necessary condition for protecting the brain, it is not enough. Brain’s inertia tensor needs another kind of protection from the external collision-jolt. Contrastingly, if a human head covered with a ‘soft’ helmet collides with a massive external body, the helmet will dissipate the energy from the collision jolt, but will not necessarily protect the skull. As a result, a proper helmet would have to have both a hard external shell (to protect the skull) and a soft internal part (that will dissipate the energy from the collision jolt by its own destruction, in the same way as a car saves its passengers from the collision jolt by its own destruction). Note that, hypothetically speaking, an ideal shock-absorber is not a classical spring-damper system (with the distance-dependent spring and velocity-dependent damper), but rather a constant-resistance damper.

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Titel: New mechanics of traumatic brain injury
verfasst von: Vladimir G. Ivancevic
Publikationsdatum: 01.09.2009
Verlag: Springer Netherlands
Erschienen in: Cognitive Neurodynamics / Ausgabe 3/2009
Print ISSN: 1871-4080
Elektronische ISSN: 1871-4099
DOI: https://doi.org/10.1007/s11571-008-9070-0

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