Comparison between brain tissue gray and white matters in tension including necking phenomenon

INTRODUCTION

Describing mechanical behavior of brain tissue is one of the most challenging and complicated issues in biomechanics. Mechanical modeling of brain tissue is important because it has a substantial number of applications in robotic surgery, surgeon training systems, and traumatic brain injury simulation as well as in modeling of hydrocephalus and designing of helmets (1.-3) Until 1970, only a few papers were published on mechanical properties of the brain tissue (4), but just recently several groups have focused on structural properties of the brain tissue and different biomechanical models of brain tissue have been proposed.

Determining mechanical properties of brain tissue have been investigated through conducting tension (5), compression (6-7) and shear experiments usually on animal brain tissues using linear and/or nonlinear elastic, hyperelastic poroelastic or viscoelastic models.(8-11). To mention some examples, a linear viscoelastic model was proposed for brain tissue in a research that could explain the behavior of the brain tissue in lower strain rates in compression. In that study, experiments were conducted on swine brain tissue (12). Also, in a recent study, a biphasic model based on experiments on human brain was proposed for brain tissue (13)

Brain tissue white and gray matters are complex materials. Gray matter of the cerebral hemispheres consists of a mixture of neuronal cell bodies, their unmyelinated processes and neurogilia. White matter, found in subcortical regions, consists of myelinated axonal fibers surrounded by supporting cells (oligodendrocytes, astrocytes, ependyma and microglia) and blood vessels (14). Characterizing the differences between the mechanical properties of brain tissue gray and white matters is of importance in biomechanics. In a study, shear modulus of white matter of corona radiate region and thalamus gray matter of human brain were compared (15). Another study showed that white matter of bovine brain tissue is approximately 3 times greater than gray matter in shear (16).

All materials in tension test, after a meaningful time from beginning of the experiment, undergo necking phenomenon, but in tension mode of brain tissue, owing to the flexibility of tissue, necking starts from beginning of the experiment. Since necking disrupts steady manner of uniaxial stress, standard equation of stress and strain cannot be applied. In view of this, equations for post-necking have been created (17-18). It seems that in performing tension test on brain samples, cross section changes of the samples during the test would not be negligible and will affect the results. Since accurate models are needed for various applications introduced for mechanical modeling of brain tissue, it is worthwhile to consider cross section changes of the samples during the tests.

In current study, a comparison has been made between mechanical behavior of bovine brain tissue white and gray matters. For the first time, through a linear elastic theory with Bridgman method, necking phenomenon is considered for brain tissue in tension test. We use a picture analyzing approach through a computer program to trace cross section changes during the test.

MATERIALS AND METHODS

Linear Elastic Model with Considering Cross Section Change: In tension of brain tissue, owing to flexibility of tissue, necking starts from the beginning of the experiment. Since necking disrupt steady manner of uniaxial stress, standard equation of stress and strain cannot be applied and to obtain more accurate results, it is necessary to consider changes in cross section directly in the formulation. Here, a linear elastic theory with Bridgman method in tension has been applied for brain tissue.

To consider cross section changes in theory, true strain can be defined as (17-18)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [L.sub.0] is the initial length and [L.sub.f] is the ultimate length of the sample. For true strain we have:

[epsilon]= in (1+e) (2)

which e is the engineering strain.

To obtain true stresses in tension, the Bridgman method can be used. For a cylindrical beam, Bridgman hypothesized the followings (Fig 1):

[FIGURE 1 OMITTED]

1. Strain distribution in minimum area is uniform.

2. Beam's longitudinal gridline in necking zone changes to a curve with radius of curvature of 1/[rho]

1/[rho]=r/aR (3)

3. Ratio of principal stresses remains constant.

As per the Bridgman method, the equivalent uniaxial stress (the Bridgman stress) can be defined as a nominal stress ([[sigma].sub.a]).sub.av] corrected with a coefficient k as follows:

[[sigma].sub.e] = [([[sigma].sub.a]).sub.av]/[(1+2R/a)In (1+a/2R)] = k[([[sigma].sub.a]).sub.av]

where K is:

k = [[(1+2R/a)IN(1+a/2R)].sup.-1] (5)

Finally, average stress can be defined as:

[[sigma].sub.ave] = 1/2([[sigma].sub.ini]+[sigma].e]) (6)

where [[sigma].sub.ini] is the stress at sample' initial cross section.

For analyzing the pictures to trace cross section changes during the test, we prepared a computer program (we named it the Brain Test) written in visual FoxPro. For each experiment, 10 pictures were taken to compute their geometrical information. First, a picture was imported to the program. In each process, for a picture, the scale pixels were converted to 10 millimeter length (scale's length). For tension test, both the diameter of the sample [phi] and the radius of the curvature [rho] at necking zone were calculated. For computing the radius of curvature, a circle was drawn along the curvature of the sample with applying 3 points. The program running can be seen in Fig. 2.