Simulation of Mechanical Properties of the Brain Matter under Impact and Creep-Recovery Tests
A. Rezaei*, G. Karami and M. Ziejewski
Mechanical Engineering Department, North Dakota State University, Fargo, ND 58108-6050
* Corresponding Author: Mechanical Engineering Department, NDSU, email: email@example.com
Human and animal body tissues exhibit characteristics of viscoelastic materials. To simulate the behavior of viscoelastic materials, mechanical models, based on spring stiffness and dashpot damping, have been introduced. The four-element model is often used to show the behavior under creep-recovery and stress relaxation tests. The four-element model, a combination of Maxwell and Voigt models in series is composed of two springs and two dashpots. This paper presents an examination of the behavior of the porcine brainstem under a creep-recovery experiment in an effort to characterize the mechanical behavior of the brainstem. Under loading, and undergoing a large deformation, the brainstem simulated four-element viscoelastic mechanical modeling parameters were examined and extracted. Such parameters provide the data to predict the behavior of the brainstem under many other circumstances.
Keywords: brainstem, creep-recovery, four-element model, viscoelasticity
Traumatic brain injury (TBI) is an important public health problem in the United States (U.S.) and can change the ability to think, sense, speak etc. In the U.S., about 1.7 million people sustain a TBI annually and 52,000 of those injuries result in death. TBI is also a major factor of 30.5 percent of all injuries in this country . The brainstem is the lower extension of the brain, consisting of medulla oblongata, midbrain and pons, and it connects the brain to the spinal cord. Brainstem injuries are a part of diffuse axonal injury (DAI). Traumatic damage to the brainstem happens when the head experiences an impact load, causing acceleration, or force, and contributing to large deformations. Such damage can cause death because the brainstem is the pathway for most of the cranial nerves and it also plays a vital role in basic attention, consciousness, and arousal. A few studies have been conducted in this area to develop the constitutive equations and responses of the brainstem. To explain their results, Arbogast et al.  conducted experiments on porcine brain tissue, from the pons, as a part of the brainstem based on the quasi-linear viscoelasticity theory. They used this theory for relatively small strains (less than 7.5%) in order to determine the relaxation functions of the brainstem. In another work , they performed oscillatory shear tests on adult swine brainstem in three perpendicular directions. The samples were taken from the middle to the upper brainstem. The study demonstrated that the brainstem represents transversely isotropic behavior. Xinguo et al.  defined the constitutive response of the brainstem under large shear deformations. They considered the brainstem as a transversely isotropic viscoelastic material and characterized it in three different directions: 1) parallel, 2) perpendicular, and 3) cross sectional to axonal fiber orientation. They used a combined method of a genetic algorithm optimization, as well as finite element methods (FEMs), in order to find the mechanical properties of the brainstem. A four-element viscoelastic model is an appropriate model to use in finding the constitutive responses of some of the viscoelastic materials under a creep-recovery test. The overall objective of research presented here was to characterize the mechanical properties of the brainstem under the creep-recovery loading as a four-element viscoelastic model. This research provided information on how to find the springs and dashpot constants so they could predict the brainstem behavior.
In order to understand the behavior of the viscoelastic materials under different loading conditions, several types of mechanical models were introduced including a spring, an elastic substance, a dashpot, and a viscose substance, or a linear combination of them. (It is important to note that the arrangement of the elements differed and could be modeled as electrical circuits. Stress, in this electrical equivalent system, was simply represented by voltage and the rate of strain by current.) While there are a number of models, this paper reviews only the four most commonly known types.
Maxwell model: The Maxwell model is represented by an elastic spring and a viscous dashpot connected in a series. This model predicts that stress decreases exponentially with time and is acceptable for polymers. It cannot predict, however, the response of polymers under the creep.
Voigt model: The model consists of a dashpot and a spring in parallel. It is used to show the behavior of polymers under the creep and it is accurate. Its limitation, however, is that it cannot explain the material response while it is under constant strain (stress relaxation).
Standard linear solid model: The linear model is the combination of the Maxwell model and an elastic spring in parallel. It is more acceptable than the Maxwell and Voigt model for predicting the response of the material under stress and strain.
Four-element or Burger model: This model is a combination of the Maxwell and Voigt models in series. This model of various elements is used with success in order to predict linear behavior of the substance . The four-element viscoelastic model is a good predictor for creep and recovery tests. Figure 1 is a schematical representation of the four different types of mechanical models.
When a sudden constant stress is applied to the Burger model, the model represents an immediate response as a vertical jump in strain axis. This jump occurs because of the response of the spring . For some materials, this area is very small and shows the elastic deformation of the sample. After fully extending the spring, the rest of the elements can react. The curve, beginning at the end of elastic response, represents the behavior of the Voigt element of the four-element viscoelastic model. It is difficult to separate it into individual components of the Voigt model. The dashpot also contributes to this curve. The dashpots make the response of the material slowly, until the spring is fully extended . Figure 2 depicts, in order, the performance of the elements in the four-element viscoelastic model under creep-recovery.
Figure 1: Four different types of viscoelastic models
Figure 2: Schematic creep and recovery behavior test and the effect of each element in the Burger model
Through only a few calculations, the total strain can be obtained as follows:
Where is the retardation time, which is the required amount of time for the Voigt model to deform to 63.21% of its total deformation .
Figure 3: Mechanical behavior of the four-element model under creep and recovery 
The brainstem cross section is populated by axons which can be relatively assumed as unidirectional. Materially, the axons can be assumed as transversely isotropic materials inside the extracellular matrix (ECM). Due to the impact on the head, or neck, some of the injuries may contribute to elongate the brainstem in axons direction. It was logical, therefore, to prepare the sample in an appropriate direction parallel to the axons.
In this experiment, the fresh porcine brain, and brainstem, was prepared in the slaughter house of the Animal Science Department at North Dakota State University (NDSU). The pig was healthy and the brain, as well as the brainstem, was carefully removed from the head. Both parts were placed immediately in a physiological saline solution to prevent dehydration. They were then transported to the lab and kept frozen. The test was conducted on the first day after preparation. The part of the brainstem that was harvested for this experiment was the medulla oblongata, or the lower part of the brainstem. (The medulla oblongata is responsible for involuntary functions, such as heart rate and breathing. It also deals with the respiratory system, vomiting etc.) Figure 4 shows the brainstem and its parts. One sample was cut carefully in the area in order to have a constant cross section.
Figure 4: Brainstem and some of its important parts .
The test was performed using a Bose 3200 Electroforce machine. (The machine is powerful and accurate for microscale mechanical analysis and is able to perform creep-recovery, stress relaxation and dynamic mechanical analysis (DMA) tests.) The capacity of the load cell used for this experiment was 22 N. The device could easily be switched to the force, or displacement, control, necessary for different loading conditions. As depicted in Figure 5, in order to conduct the creep recovery test in tension, two glass plates were attached to the grippers and the sample was fixed between them using a type of cyanoacrylate glue.
Figure 5: Electroforce machine and its grippers for tensile test.
The behavior of the brainstem under creep-recovery was to be determined and to be examined in order to find the parameters of the four-element viscoelastic model. The constant stress applied to the tissue and the response of the material is shown in Figure 6.
Figure 6: Brainstem under creep-recovery test
It needs be noted that, in practice, it is very difficult to apply a small sudden force, with some of the testing devices, in order for the stress to reach the predetermined value in a very short period of time. As can be seen, in Figure 6, the brainstem behaved like a viscoelastic substance during creep loading and the recovery region is clearly shown. This is the characteristic of a viscoelastic material. It is important to emphasize that mechanical models such as the Maxwell, Voigt and four-element viscoelastic model, can only predict the behavior of viscoelastic materials in linear regions. In the nonlinear regions, the molecular structure of the sample will be changed and the performance of the substance depends on the strain rate as well as the strain . In such circumstances, the change in stress causes the creep-recovery curve to vary and, thus, the parameters of the springs and dashpots will be changedaccordingly. As can be observed in Figure 6, the tissue, in this present study, underwent a large deformation (on average 18%). The tissue was exhibited as a nonlinear viscoelastic substance and its behavior was examined in this region. Based on the presented formulations, the constants for the Burgers elements were calculated (Table 1). The slope of the curve before recovery response (Figure 6) is the strain rate of the substance under the loading.
Table 1: Creep-recovery test results of the brainstem as a Burgers model
A simple MATLAB code was written in order to draw the curve of the four-element viscoelastic model and its behavior, as depicted in Figure 7.
Figure 7: Creep-recovery test of the brainstem and four-element model (green color) obtained by experimental data of the brainstem (blue line).
In Figure 7, there is a discrepancy between the four-element viscoelastic model and the brainstem response because it took a short period of time in order for the stress to reach a constant value. The structure of the brainstem under large deformations can be damaged, resulting, therefore, in some differences between the two curves, especially in the recovery response.
The focus of this work is the mechanical properties of the porcine brainstem. A constant impact tensile stress was applied on the medulla oblongata of the brainstem in axonal direction to predict the creep behavior. The stress was then removed and the brainstem experienced recovery behavior. The creep-recovery curve was simulated by the four-element viscoelastic mechanical model, the parameters related to spring stiffness, and the dashpot coefficients of damping were extracted. The results showed that the brainstem has a close four-element viscoelastic model resemblance behavior similar to many viscoelastic substances. The recovery curve of the brainstem, however, is slightly different from the curve obtained by a four-element viscoelastic model. This explains why the mechanical models are able to predict the behavior of the viscoelastic materials in linear regions. The brainstem samples, used for the experiments, tolerated much greater strain than the linear behavior regions. The inaccuracy of the testing, and the sample damage that occurred, may have contributed to this discrepancy. While it seems that the behavior of the brain tissue can be much more closely simulated by such a model in small strain regions, the conclusion of this research is that the four-element model can be an appropriate model for tissue behavior in a large deformation. The researchers plan to conduct further studies, with a number of samples, to verify how a four-element model can resemble the small and large deformation behavior in order to obtain the constitutive response of the brainstem and other parts of the brain. They will also examine to what extent, the tissue can tolerate deformation under impact loading.
The authors would like to acknowledge the Army Research Office (ARO) for the financial support of this research.