A Dynamic Method to Develop Nonlinear Viscoelastic Model of Brain Tissue
Authors: Darvish Kurosh , Takhounts Erik G. , Crandall Jeff R. 
Published: January 2022

Abstract
The objective of this paper is to present an instrumentation and data analysis method developed to determine a nonlinear viscoelastic model of brain tissue using a forced vibration technique. The application of this model is mainly for studying the injury mechanisms of brain tissue resulting from impacts to the head.
Since the early 1950s several attempts have been recorded to model the mechanical behavior of brain tissue. Investigators have used a variety of quasi-static and dynamic experimental techniques in their studies and there is a general agreement that brain exhibits viscoelastic characteristics (Galford and McElhaney, 1969). However, there are three major limitations associated with most of the previous studies. First is the limitation of boundary and environmental conditions. By applying small strains (less than 10%), most researchers have explained their experimental results with linear models (Shuck and Advani, 1972). However, biomechanical models show that shear strains up to 100% occur in brain in impact situations (Ueno et al., 1996). To include finite deformation, a nonlinear model is needed to characterize the biomechanics of impact and injury of the brain tissue. Although viscoelastic material properties are generally very sensitive to temperature (Haddad, 1995), the effect of temperature on brain material properties has not been investigated in the previous studies. Most in vitro tests have been performed in the room temperature and only a few in vivo studies have been reported (Fallenstein et al., 1969; Wang and Wineman, 1972). The effect of gravity has not been addressed in the previous studies. The brain sample is so soft that it creeps under its own weight, which causes pre-stress and pre-strain in the sample (Takhounts, 1998).
The second limitation is with regard to the constitutive models. A few researchers, who have investigated the nonlinear behavior of the brain tissue, have presented the results of their studies in some special forms of stress-strain relationships (Donnely, 1993). These relationships are generally dependent on the type of experiment and can not be used for other types of loading and deformation. In order to develop an analytical or numerical model of brain, a three-dimensional constitutive relation is required that is independent of the type of experiment.
The third limitation is the experimental methods. In the stress relaxation test method, due to hardware and inertial limitations, the jump of ideal step input is usually simulated with high-speed ramp input (Takhounts, 1998). Therefore, time constants that are shorter than the ramp time interval (about 0.04 s) can not be identified in the relaxation response. The small time constants have a significant effect in the short time response of the material, which is of primary interest in the biomechanics of impact and injury. On the other hand, due to inertial effects, the transient vibration of the system perturbs the first portion (about 400 ms) of the relaxation curve (Takhounts, 1998). In the ramp test method also, due to the initial acceleration period, the short time constants can not be measured correctly (Donnely, 1993). Previous studies based on the forced vibration technique, due to hardware limitations, have been performed in the frequency range of 5–350 Hz with strain levels of up to 35% (Shuck and Advani, 1972; Arbogast et al., 1997). These results have been used to develop linear viscoelastic models for shear with time constants in the range of 0.4 to 32 milliseconds.
In the method presented in this paper, the goal was to develop a viscoelastic model of brain tissue that is free from the constraints discussed above. The samples are taken from fresh human and bovine brain tissues (maximum 24 hours after death or slaughter). Samples are cut with cylindrical metal cores (5–20 mm diameter) from different parts of the brain tissue and in transverse and vertical anatomic directions. Using the same experimental apparatus, cylindrical samples with the length of 5–30 mm are studied in both simple extension and in simple shear modes. In order to study the effect of temperature, canceling the effect of gravity and minimizing material deterioration, each sample is placed in a temperature controlled slow flow of saline solution throughout the experiment. An electromechanical vibrator with frequency response of dc-6500 Hz and maximum force of 65lb is used to apply the input displacement to one end of the specimen. The characteristics of the vibrator allow the identification of a wide range of time constants of the brain tissue (from 80 μs to 1.6 s) in a wide range of strain inputs (infinitesimal to 100%). The reaction force at the other end of the specimen is recorded via a miniature high precision load cell. As shown in figure 1, the analog signals of the load cell, an accelerometer that measures the motion of the vibrator, and a thermocouple that measures the temperature of the sample are collected via an isolated analog to digital converter in a personal computer. Via a digital to analog converter, the computer also controls the motion of the vibrator. The whole system works as a closed loop control system. The resultant forces of a simple harmonic displacement input and also the superposition of a series of simple harmonic inputs are analyzed in the frequency domain to generate linear, quasilinear and nonlinear third order Green-Rivlin viscoelastic models of the brain tissue (Fung, 1993 and Lockett, 1972). In addition, square wave and triangular wave inputs are applied to study the relaxation and hysteresis phenomena. The lateral movement of the samples is recorded with a high-speed camera and digital image analysis. The results obtained from the samples in transverse and vertical directions are used to develop three-dimensional transversely isotropic models. Preliminary experiments, as shown in figure 2, show that for low strain levels below 10%, linear viscoelastic model describes the short time behavior of brain tissue to a high degree of accuracy. For strain levels between 10% to 40% and short relaxation times below 100 ms, a quasilinear model can be used that only considers the strain nonlinearity of the material. Assuming that the effect of a single relaxation exponential function, after passing four time constants, is negligible, 100 ms relaxation time corresponds to the frequency of 6.4 Hz. For higher strain levels (up to 100%) and longer relaxation times (up to 5 s) or lower frequencies (below 6.4 Hz), a third order Green-Rivlin model, which includes both strain and time nonlinearity, should be used. The discrete spectrum approximation is used to represent the relaxation functions. It is shown that by using this form, the nonlinear models can be easily implemented in numerical algorithms that can be used in finite element programs (Puso and Weiss, 1998). A complete set of tests on a single specimen takes between 15–30 minutes. Therefore, multiple sections from a whole brain can be analyzed in a few hours, which minimizes the effect of material deterioration.