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. Author manuscript; available in PMC: 2015 Oct 14.
Published in final edited form as: Biomech Model Mechanobiol. 2008 Jul 24;8(4):253–262. doi: 10.1007/s10237-008-0131-4

Residual stress in the adult mouse brain

Gang Xu 1, Philip V Bayly 2, Larry A Taber 3,✉
PMCID: PMC4605564  NIHMSID: NIHMS729137  PMID: 18651186

Abstract

This work provides direct evidence that sustained tensile stress exists in white matter of the mature mouse brain. This finding has important implications for the mechanisms of brain development, as tension in neural axons has been hypothesized to drive cortical folding in the human brain. In addition, knowledge of residual stress is required to fully understand the mechanisms behind traumatic brain injury and changes in mechanical properties due to aging and disease. To estimate residual stress in the brain, we performed serial dissection experiments on 500-μm thick coronal slices from fresh adult mouse brains and developed finite element models for these experiments. Radial cuts were made either into cortical gray matter, or through the cortex and the underlying white matter tract composed of parallel neural axons. Cuts into cortical gray matter did not open, but cuts through both layers consistently opened at the point where the cut crossed the white matter. We infer that the cerebral white matter is under considerable tension in the circumferential direction in the coronal cerebral plane, parallel to most of the neural fibers, while the cerebral cortical gray matter is in compression. The models show that the observed deformation after cutting can be caused by more growth in the gray matter than in the white matter, with the estimated tensile stress in the white matter being on the order of 100–1,000 Pa.

Keywords: Biomechanics, Morphogenesis, Cortical folding, Axon, Stiffness

1 Introduction

During human brain development, the cerebral cortex undergoes substantial folding, which leads to a highly convoluted brain surface. Proper cortical folding is critical to normal brain function, as anomalies in this process are linked to a number of disorders, including schizophrenia, autism, epilepsy, and mental retardation (Nordahl et al. 2007; Porter et al. 2002; Sallet et al. 2003; Welker 1990). Although intriguing neuroscientists for more than a century, the mechanisms of cortical folding remain incompletely understood.

Clearly, cortical folding involves biomechanical forces, and some researchers have speculated that folding results from stresses induced by differential or constrained growth (Richman et al. 1975). More recently, investigators have postulated that tension in neural axons of cerebral white matter causes specific folding patterns (Hilgetag and Barbas 2005, 2006; Van Essen 1997). However, while experiments have shown that axons originating from chick sensory neurons can sustain significant tension (Dennerll et al. 1989), axons from embryonic chick brain neurons sustain considerably less tension (Chada et al. 1997). Hence, it is uncertain if significant tension exists in axons in both the developing and mature brain.

The primary objective of this work is to determine whether residual stress exists in the brain and to explore the distributions and source of this stress, if it exists. In living tissues, residual stresses may arise from growth, remodeling, or active force generation by cytoskeletal components (Fung 1993, 1998). Here, we use a variation of the method commonly used to characterize residual stress in soft tissue, whereby the tissue is cut and the resulting deformation measured as the stress is released (Fung 1993).

In this initial study, adult mouse brains were used. Although the brains of mice, unlike those of many large mammals, do not undergo cortical folding, they are widely used and readily available for brain tissue studies due to their well-documented structural and functional information. Indeed, mouse brains can develop aberrant folding patterns ex vivo if cortical growth is enhanced by lysophosphatidic acid (Kingsbury et al. 2003), making them a potential candidate for studying abnormal folding mechanisms. Because axon tension has been postulated to play a role in brain folding, we herein focus on the regions of the mouse brain that contain most of the cortical white matter. A typical coronal section of the mouse brain containing cerebral white matter (corpus callosum) has three major components: cortical gray matter, white matter tract, and inner gray matter of thalamus (Fig. 1c). Gray matter is composed of unmyelinated neurons, including nerve cell bodies, glial cells, and short nerve cell extensions. White matter is composed largely of myelinated nerve cell processes, or parallel axons. The white matter tract contains three well-recognized regions: corpus callosum, cingulum, and external capsule (Fig. 1c). White matter conducts signals between different regions of gray matter, and, in particular, the corpus callosum connects structurally the two cerebral hemispheres of the brain.

Fig. 1.

Fig. 1

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Schematic of experimental procedure for mouse brain dissection (not drawn to scale). a The whole brain was removed from 12–15-week-old adult mice. b The brain was then mounted on the platform of a sectioning vibratome and 500-μm thick coronal slices were cut. c Only the slices containing corpus callosum were collected. A typical brain slice contains three major regions: cortical gray matter (cGM), thalamus gray matter (tGM), and white matter tract consisting of corpus callosum (cc), cingulum (cg), and external capsule (ec). In the experiments, a razor blade was used to make the first radial cut only through the cortical gray matter, and the second radial cut was made deeper through either the white matter tract or the inner gray matter of thalamus. The cartoons shown in a and b were adapted from http://pages.slc.edu/~krader/animals/index.htm, and http://www.hms.harvard.edu/ research/brain/atlas.html, respectively

The results of the dissection experiments described below suggest that the cerebral white matter tract is under considerable tension in the circumferential direction in the coronal plane, parallel to most of the neural fibers, while the cerebral cortical gray matter is in compression. Using computational modeling, we investigated whether the tension in the white matter can be caused by gray matter growing at a faster rate than, and hence stretching, the white matter. A finite element model for a brain slice indicated that such differential growth can produce deformations that agree well with our cutting experiments. These results are consistent with the idea that cortical growth constrained by axonal tension causes brain folding.

2 Materials and methods

2.1 Brain slice preparation

All animal procedures were approved by the Washington University Animal Studies Committee. We used 12–15-week-old adult female C57BL/6J mice (Jackson Laboratory, Bar Harbor, ME, USA). Methods were modified from a previous report for acute mouse brain slice preparation (Tekkok and Goldberg 2001). Briefly, each mouse (n = 5) was anesthetized with 5% isoflurane in air. The whole brain was immediately removed from the skull (Fig. 1a) and placed into ice-cold artificial cerebrospinal fluid (aCSF; 25 mM NaHCO3, 122 mM NaCl, 1.3 mM CaCl2, 1.2 mM MgSO4, 3 mM KCl, and 0.4 mM KH2PO4; pH 7.35) (Alexander et al. 2000). The brain was then mounted on the platform of a sectioning vibratome (Vibratome 1000, Technical Products, St Louis, MO, USA) and 500-μm thick coronal slices were cut (Fig. 1b). In the sectioning process, the brain was immersed in ice-cold aCSF. Only the slices (about 8/brain) containing corpus callosum were collected and included in the following serial dissection experiments.

2.2 Tissue dissection

Brain slices were allowed to recover for about 15 min at room temperature (~22°C) in aCSF. To probe the circumferential (in the coronal plane) residual stresses in different cerebral layers, radial cuts were made with a razor blade completely through the slice thickness at three levels of depth into the brain (Fig. 1c): (1) through the cerebral cortical gray matter, (2) deeper into the white matter tract, and (3) further into inner gray matter of thalamus. The degree of opening at the cutting site served as an indicator of the magnitude of the residual stress in the circumferential direction. Tensile stresses would pull the cut open while compressive stress would keep the cut closed. To minimize tissue damage by the blade and ensure a straight cut, only one cut at a specified depth was made at each location. To ensure clean cuts, a new blade was used for each brain. In general, radial cuts of different depths were made on each of two symmetric sides (hemispheres) of the same brain slice for comparison (Fig. 1c).

To probe the deformation caused by surrounding gray matter, some white matter tracts were isolated (with a small amount of residual gray matter) from intact brain slices. Inner gray matter was removed first by a series of small cuts with a pair of fine spring scissors (Fine Science Tools, Foster City, CA, USA). Most of the cortical gray matter was then removed by several straight cuts with a razor blade. The overall morphology of the cortical white matter tract was characterized by the angle (φ) between two radii joining the midpoint of the inner wall to the tips of the white matter tract (see Fig. 4a1, a2). To compare slices between different regions in different subjects, the measured angles were normalized relative to their original values from the intact brain slices (φ* = φcut/φintact).

Fig. 4.

Fig. 4

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White matter tract of mouse brain with and without the constraints of gray matter. a1 Brain slice obtained by vibratome sectioning. The overall morphology is typical for a slice just posterior to the one shown in Fig. 3a1. The angle (φintact = 150°) characterizing the white matter tract is shown. a2 Same slice with inner thalamus gray matter dissected away by scissors and cortical gray matter cut away with several straight cuts by a blade. In this specimen, the angle (φcut = 99°) of the white matter tract decreased by 33% (φ* = 0.66) after these cuts. b1, b2 Normalized circumferential stress distributions in finite element models before and after dissections. The model for the intact slice (b1) is the same as that in Fig. 3c1. Similar to the experimental result, after removal of the majority of the gray matter, the white matter angle decreased dramatically (φ* = 0.70), and straight cuts on the cortical gray matter became curved (b2)

Each slice was imaged immediately and 15 min after each dissection with a video camera (COHU, Model 4915, San Diego, CA, USA) mounted on a dissecting microscope (Leica Microsystems, Model MZ8, Bannockburn, IL, USA). Images were acquired by a frame grabber (FlashBus MV, Integral Technologies, Indianapolis, IN, USA) and imaging software (SigmaScan Pro V.5.0, Systat Software, San Jose, CA, USA).

2.3 Finite element model

To explore the possible cause of tension in the white matter and to estimate the magnitude of the residual stress, a nonlinear finite element model for the mouse brain slice was created using the commercial finite element software COM-SOL Multiphysics (V.3.3, COMSOL AB). Resembling the overall morphology of real mouse brain slices, the plane-stress model was composed of three layers representing cortical gray matter, white matter tract, and inner gray matter (Fig. 2a). Due to symmetry, only half of the slice needed to be analyzed. Symmetry conditions were always specified along the straight vertical boundary. To simulate a cut of specific depth, a thin (10-μm wide) rectangular region along the radial direction was removed (Fig. 2b). The model geometry was partitioned into triangular mesh elements in COMSOL (Fig. 2b). The mesh was further refined near the boundaries of the cut (see close-up of the regional mesh in Fig. 2c). The sufficiency of the mesh density was confirmed by solving the problem for increasing mesh densities.

Fig. 2.

Fig. 2

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Finite element model for the mouse brain in the coronal plane. a Schematic of model for intact brain slice. Due to symmetry, only half of the slice is analyzed. Symmetry conditions (represented by rollers) are specified along the straight vertical boundary (one point was fixed). Other boundaries are free. b Model geometry partitioned into triangular mesh elements. A narrow cut is simulated by removing a thin (10-μm wide) rectangular region. Note denser mesh for the white matter track and the cut region. c Close-up of mesh near the cut (dashed rectangle in b)

We speculate that residual stress in the brain is generated by differential growth during development. Volumetric growth was modeled using the theory of Rodriguez et al. (1994). Briefly, starting with the zero-stress configuration, each material element grows according to a specified growth deformation gradient tensor Fg. Maintaining geometric compatibility between elements generally requires elastic deformation Fe that causes stress. Thus, the total deformation is described by F = Fe · Fg, where Fg = I for no net growth with I being the identity tensor. In our computations, the main deformation variable is F, Fg is specified, and stress depends on Fe=F·Fg-1. In the current models, we considered only isotropic growth with Fg = λgI, where λg is the growth ‘stretch ratio’ (Taber and Perucchio 2000; Taber 2001).

To a first approximation, both gray matter and white matter are assumed to be isotropic, nearly incompressible, pseudoelastic materials characterized by the modified neo-Hookean strain-energy density function W=12μ(I1Je-23-3)+p(1-Je-p2κ), where μ and κ are the shear and bulk modulus, respectively, I1 is the first invariant of the right Cauchy–Green deformation tensor (given by FeT·Fe), Je = detFe is the elastic volume ratio, and p is a penalty variable introduced for nearly incompressible materials. The Cauchy stress tensor is given by the constitutive relation σ=Je-1Fe·∂W/∂FeT. More details on the implementation of growth in COMSOL are given in Taber (2007).

Because regional material properties during growth have not yet been measured in the mouse brain, we assume that the constitutive relations and growth are uniform throughout the white and gray matter, although the modulus and growth magnitudes can differ. In fact, for nearly incompressible materials, the behavior of the model depends only on the ratios μ* = μG/μW and λg∗=λgG/λgW, where the subscripts W and G denote white and gray matter, respectively. Hence, the specified differential growth ( λg∗≠1) is the only driving force in the model. (For λg∗=1, no residual stresses develop even when μ* ≠ 1.) The values of these parameters and how they affect the results are discussed below.

Residual stresses also are normalized relative to the white matter shear modulus ( σθ∗=σθ/μW). Moreover, simulations revealed that when κ is large relative to μ, i.e., the material is nearly incompressible, the results are relatively insensitive to the value of the bulk modulus. For numerical stability, we took κ to be about six orders of magnitude larger than μ for both white and gray matter.

3 Results

3.1 Gray matter is in compression and white matter tract is in tension

Following a radial cut in the cortical gray matter, the cut did not open (Fig. 3a2, b2). This behavior was consistent for all the coronal brain slices (containing corpus callosum) from the adult mouse (n = 5). This result indicates the presence of little residual stress or compressive stress in the circumferential direction of the coronal plane in the cortical gray matter of the intact slice.

Fig. 3.

Fig. 3

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Serial cuts made on a coronal slice of an adult mouse brain. a1, b1 Brain slice obtained by vibratome sectioning. a2, b2 The first radial cut (indicated by a pair of solid arrowheads) was made only through the cortical gray matter. The cut did not open. a3, b3 The second radial cut (indicated by a pair of open arrowheads) was made through the underlying white matter tract (a3) or deeper into inner thalamus gray matter (b3). The cut opened at the site of the white matter tract or inner gray matter. a4, b4 After about 15 min, the opening in the white matter tract became wider, but cuts through cortical gray matter stayed closed. The overall morphology of the slices in a1 and b1 is typical for the first and third corpus callosum-containing slices, respectively, obtained during coronal sectioning (from anterior to posterior part of the brain). c1–c4 Normalized circumferential stress ( σθ∗) distribution in finite element models for the dissection experiments (μ* = 0.5, λg∗=1.3). Due to relatively more growth in gray matter, gray matter is in compression and white matter is in tension. Radial cuts were made into cortical gray matter (c2), white matter (c3), or inner gray matter (c4), respectively. The smaller outline shown in each plot is the same slice before growth

Since little deformation occurred after the first cut, this cut likely had little effect on the stress distribution in the white matter tract and the rest of the brain tissue. Thus, another deeper cut was next made on the other side (hemisphere) of the brain slice (Fig. 3a3, b3). Cutting radially through both the cortical gray matter and the white matter tract resulted in considerable opening of the white matter tract in the circumferential direction (Fig. 3a3). The opening is a clear signature of tensile residual stresses in the circumferential direction in the coronal cerebral plane, parallel to the most of the neural fibers along the white matter tract (Larvaron et al. 2007; Olivares et al. 2001; Valverde 2004). The edges in the white matter tract from a straight cut always assumed a concave shape with a wider opening or separation in the middle of the tract and much smaller separation in the adjacent gray matter (Fig. 3a3). The largest cut separation distance was approximately 100 μm right after cutting (Fig. 3a3), and it became nearly twice as large after about 15 min (Fig. 3a4), illustrating the viscoelastic nature of living brain tissue. However, the cut in the cortical gray matter remained mostly closed (Fig. 3a3, a4), indicating the presence of compressive residual stresses in this region.

When deeper cuts were made into the inner gray matter of the thalamus, the inner gray matter opened considerably and resulted in wider opening of the white matter tract (Fig. 3b3). Similarly, these openings enlarged after about 15 min (Fig. 3b4), presumably due to viscoelastic deformation.

3.2 Stresses in white and gray matter are coupled

On a coronal slice of an adult mouse brain, the cortical white matter tract is relatively thick in the center region (corpus callosum) and gradually attenuates (cingulum and external capsule) in moving along the circumferential direction (Figs. 3a1, b1, 4a1). To examine the mechanical coupling between the white and gray matter, we removed most of the inner and cortical gray matter and examined how the overall shape (characterized by the angle φ defined in Fig. 4a1, a2) of the white matter tract changed. On average, the angle decreased by about 30% after removal of the gray matter (φ* = φcut/φintact = 0.67 ± 0.04; p < 0.0001; four slices from four mice) (see Fig. 4a1, a2). These results indicate that residual tension in the white matter is closely coupled with compression in the gray matter. Other noteworthy features include the change in curvature of the corpus callosum and the curved edges of the cortical gray matter following the straight cuts (Fig. 4a2).

3.3 Quantitative estimate of residual stress in white matter

The cutting experiments have demonstrated that residual stresses exist in both gray and white matter in the adult mouse brain. The white matter tract contains considerable tensile stress in the axonal fiber directions along the white matter tract in the coronal plane, while the gray matter is in a state of circumferential compression. To explore the possible source of these stresses, as well as to estimate their magnitude, we created finite element models for the above experiments (Figs. 3, 4). The overall morphology of our models (Figs. 3c1, 4b1) is similar to that of actual mouse brain slices (Figs. 3a1, b1, 4a1). The results in these figures are based on the parameter values μ* = 0.5 and λg∗=1.3. (The effects of varying these parameters are discussed below.)

The results from the model indicate that differential growth can generate considerable residual stress with tension in white matter and compression in gray matter (Fig. 3c1). For the selected model parameters, simulating cuts of certain depths into the brain slice yields results in good agreement with those given by our cutting experiments (compare experimental and computational results for each row in Fig. 3). In addition, for nearly isolated white matter (with some remaining cortical gray matter), our model produces similar shapes (Fig. 4b2) to those in our experiments (see Fig. 4a2). The model further predicts the curved surfaces following straight cutting of the cortical gray matter. Taken together, these results indicate that differential growth between gray and white matter can generate the observed residual tension in the white matter tract.

Because of the wide range of material properties reported for brain tissue (see Table 1 of Thibault and Margulies 1998), and the lack of prior information on mechanical properties and growth in the mouse brain, we conducted a sensitivity analysis to examine the effects of the relative (gray/white) modulus (μ*) and growth ( λg∗) on the normalized angle (φ*) and residual circumferential stress in the white matter tract. (The circumferential stress σθ∗ represents the average value across the symmetry plane of the corpus callosum.) In general, φ* decreases with increasing relative gray matter growth ( λg∗) or decreasing relative modulus (μ*), while σθ∗ increases with increasing λg∗ or increasing μ* (Fig. 5). In other words, the larger the relative gray matter growth, the larger the change in the white matter angle after isolation and the larger the residual stress in the white matter in the intact slice. Also, the larger the relative gray matter shear modulus, the smaller the change in the white matter angle after isolation (gray matter is harder to deform) and the larger the residual stress in the white matter in the intact slice (gray matter stretches white matter more easily).

Fig. 5.

Fig. 5

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Dependence of the normalized white matter tract angle (φ*) and residual circumferential stress ( σθ∗) in the white matter on relative growth ( λg∗) and stiffness (μ*) from the finite element model. a Dependence of φ* on λg∗ and μ*. The shadowed area is the relative growth range for gray matter when the normalized white matter angle matches the experimental data (φ* = 0.67 ± 0.04). b Dependence of σθ∗ on λg∗ and μ*. The white matter stress σθ∗ represents the average value across the symmetry plane of the corpus callosum. The shadowed area corresponds to the same growth range for gray matter from a and covers the scope of expected residual stress level in the white matter

To match the experimental value of φ* (0.67) within one standard deviation after isolation (see Fig. 4a1, a2), the value of λg∗ must fall in a relatively narrow range of about 1.25–1.47, which also depends on the value of μ* (Fig. 5a). This is consistent with data showing that the increase in overall size of the mouse brain during postnatal maturation is roughly 20–30% (Larvaron et al. 2007). For this amount of growth, the value of σθ∗ lies in the range of about 0.5–1.7, depending on the value of μ* (Fig. 5b). Unfortunately, the mechanical properties of the mouse brain have not yet been measured. However, if we assume that the overall stiffness of white matter is similar to that of gray matter, as has been shown for the porcine brain (μ* =~1.3) (Prange and Margulies 2002), then Fig. 5b indicates that the value of σθ∗ likely is close to 1. Hence, if the mouse brain has properties similar to those of the rat brain, which has a bulk shear modulus of 100–1,000 Pa (Elkin et al. 2007; Gefen et al. 2003; Georges et al. 2006), then the residual stress ( σθ=μWσθ∗) in the mouse brain would be on the order of 100–1,000 Pa.

4 Discussion

In this work, we provide evidence that axons in the mature brain sustain considerable tension. Some investigators have speculated that axon tension plays a crucial role in cortical folding during brain development (Hilgetag and Barbas 2005, 2006; Van Essen 1997). One possibility is that axons constrain the expansion of local regions of the growing brain cortex, causing it to form specific folding patterns. Our findings support the feasibility of this idea. An important caveat, however, is that developing brains may have different mechanical properties than mature brains (Gefen et al. 2003).

Previous studies with single neurons have shown that developing axons can bear considerable axial tension (Lamoureux et al. 1989). In general, a chick sensory axon elongates and sustains tensile forces on the order of 100 μdyn (or 1 nN) when pulled by a needle (Dennerll et al. 1989; Lamoureux et al. 1989), while neurites from embryonic chick forebrain neurons apparently sustain much smaller tensile forces that are on the order of 10 μdyn (or 0.1 nN) (Chada et al. 1997). In addition, axons from hippocampal neurons of embryonic rats can also grow in response to tensions on the order of 10–100 μdyn (Lamoureux et al. 2002), although the magnitude of tensile force that these axons can maintain is unknown.

If we assume that axons in the mature mouse brain can sustain tensions similar to those for developing chick forebrain axons (0.1 nN), the stress can be estimated by noting that the diameter of axon fibers in the mammalian corpus callosum is conserved across species and is generally in the range of 0.1–1 μm (Faisal et al. 2005; Olivares et al. 2001). Hence, the magnitude of the tensile stress (force/area) in an axon of the adult mouse brain is in the range of 102–104 Pa. The tensile stress in the white matter tract from our finite element model (on the order of 100–1,000 Pa) agrees well with this estimate, suggesting that a similar level of stress may be maintained in axons in both the developing and mature central nervous system. The magnitude of the residual stress given by the models depends on the assigned material properties, as well as the specified growth in different regions. The parameter values used are based on available data in the literature, but these data are incomplete. The models, however, predict quite well the results from the cutting experiments (see Figs. 3, 4), suggesting that the chosen values are reasonable.

As a first approximation, our model includes isotropic growth of gray matter relative to white matter without any change of material properties. However, the material properties of the brain might change significantly during postnatal development or maturation due to alterations in water content and myelination of fiber tracts (Larvaron et al. 2007; Neil et al. 2002). Therefore, to refine our finite element models much work is needed to determine the regional material properties (and residual stresses) of the mouse brain at different developmental stages.

Our experiments indicate that, while the white matter tract is in tension, the cortical plate and inner region of gray matter are in a state of compression in the circumferential direction in the coronal cerebral plane. The results from our finite element models agree with this conclusion (Fig. 3). In addition, stresses in different regions are closely coupled as manifested by the significant change of the white matter angle after isolation from the brain slice (Fig. 4). Equilibrium of an unloaded object requires that tension in some regions must be balanced by compression in other regions. Thus, there may be no physiological reason for the compression in the gray matter. However, from a mechanics perspective, the coupling between tension in white matter and compression in gray matter may provide superior structural stability for normal brain functioning, as the white matter tract connects structurally the two cerebral hemispheres of the brain. This finding is also relevant to studies on traumatic brain injury that use mouse brain models.

In addition to differential growth, tension in axons could be generated by active contraction. However, myosin-based cytoskeletal contraction is likely minimal in most white matter of the adult mouse brain due to relatively low levels of myosin II in this region (Kioussi 2007; Miller et al. 1992). Hence, differential growth is the more likely mechanism. This conclusion agrees with previous speculation by Richman et al. (1975), as well as recent experiments showing that accelerated growth can lead to aberrant folding in the mouse brain (Kingsbury et al. 2003).

Quantitative biomechanical models for the brain, like those used in this work, are useful for testing theories for cortical folding (Richman et al. 1975; Todd 1982; Toro and Burnod 2005; Van Essen 1997). In addition, there is an increasing demand for computational models in the areas of neurosurgery (Kyriacou et al. 2002; Miller 1999) and traumatic brain injury (Zhang et al. 2001), where material properties of brain tissue change due to injury, aging, and diseases (Drapaca et al. 2006; Gefen et al. 2003; Prange and Margulies 2002; Taylor and Miller 2004; Thibault and Margulies 1998). Mechanical properties of brain tissues are needed for these models. Accurate determination of these properties requires knowledge of the zero-stress state of the tissue (Fung 1993; Zamir and Taber 2004). Residual stress also can greatly affect the stress distributions in loaded tissue (Fung 1998), e.g., during impact.

To date, however, residual stresses have generally been ignored in studies of brain biomechanics. Material properties for various animal brains have been reported using conventional testing methods such as shear (Coats and Margulies 2006; Georges et al. 2006; Hrapko et al. 2006; Prange and Margulies 2002; Thibault and Margulies 1998), compression (Cheng and Bilston 2007; Elkin et al. 2007; Miller 1999; Miller and Chinzei 1997; Prange and Margulies 2002), indentation (Elkin et al. 2007; Gefen et al. 2003; Gefen and Margulies 2004; Miller et al. 2000), and extension (Miller and Chinzei 2002). These measurements have been performed on either the whole brain or certain regions of the brain. However, residual stress has been ignored in all of these reports. The present study provides information that can be used to improve these measurements of material properties.

Limitations in our models mainly reside in the assumptions regarding material properties. First, white matter probably is better modeled as a transversely isotropic material due to the orientation of neural fibers. Second, although better than linear elasticity on which many reported measurements are based, a neo-Hookean material may not be an accurate representation for mouse brain tissue. Indeed, other forms of strain-energy density function for brain tissue have been proposed in the literature, e.g., the first order Ogden hyperelastic material for the porcine brain (Coats and Margulies 2006; Prange and Margulies 2002). Third, material properties may vary regionally within gray or white matter. Finally, brain tissue is clearly viscoelastic, as demonstrated by the time-dependent changes after cutting (Fig. 3a4, b4). To address these issues, as well as to improve our model, further mechanical testing on regional material properties of the mouse brain is needed.

In conclusion, we have found that the adult mouse brain contains significant residual stress. The white matter tract and gray matter are in states of tension and compression, respectively, in the circumferential direction in the coronal cerebral plane. These results could have important implications for studies of brain folding, mechanical properties, and injury due to trauma. We hope this study motivates further work, especially on the mechanical properties of the developing brain of mammals with gyrencephalic (folding) brains. Such data are needed to help evaluate the tension-based theory of cortical folding.

Acknowledgments

We gratefully acknowledge technical assistance from Edward Kim, William McCarran, and Christine McDonald in harvesting the mouse brains used in this work. We also thank Paul Bridgman, Krikor Dikranian, Jeffrey Neil, and David Van Essen for their helpful discussions and suggestions. This work was supported by the National Science Foundation (grant #DMS-0540701).

Contributor Information

Gang Xu, Department of Biomedical Engineering, Washington University, One Brookings Drive, Campus Box 1097, St Louis, MO 63130-4899, USA.

Philip V. Bayly, Department of Biomedical Engineering, Washington University, One Brookings Drive, Campus Box 1097, St Louis, MO 63130-4899, USA. Department of Mechanical, Aerospace, and Structural Engineering, Washington University, St Louis, MO 63130, USA

Larry A. Taber, Email: lat@wustl.edu, Department of Biomedical Engineering, Washington University, One Brookings Drive, Campus Box 1097, St Louis, MO 63130-4899, USA

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