## Abstract

The steady-state response of a Maxwell viscoelastic cylinder to periodic sinusoidal oscillation of its boundary was studied as a simplified model of the brain responding to low-amplitude angular vibration of an idealized skull. The objectives were to identify conditions in which peak strain occurred on the interior of the cylinder, and to identify ways to scale strains from differently sized cylinders. This latter objective is motivated by the work of Holbourn to inform scaling of intracranial strains experienced under similar acceleration of skulls of different animals. The mechanical response was dictated by two dimensionless parameters that incorporate material properties and external loading frequency. The location and magnitude of maximum strain were examined with respect to these governing parameters in steady state. A frequency-dependent mapping of brain constitutive data to idealized Maxwell models was applied to predict the location and magnitude of peak strains inside a cylinder with mechanical properties representing the adult human brain. Results suggest that peak strains occur on the interior of such a cylinder for skull oscillation within a specific frequency band.

## 1. Introduction

Rapid rotation of the skull is believed to be a primary course of mild brain injury (Holbourn 1943; Ommaya & Hirsch 1971; Bailey & Gudeman 1989). As waves course through the brain in response to skull motion, cell function may become impaired when a critical strain threshold is exceeded (Ommaya *et al*. 1967; Bain & Meaney 2000; Morrison *et al*. 2000; Geddes *et al*. 2003). In the first part of this work, a solution was presented for the motion of waves in a periodically accelerated Maxwell viscoelastic cylinder as a highly simplified model of strain waves inside the brain (Massouros 2005; Massouros & Genin submitted). Although the brain is certainly not a Maxwell viscoelastic material, insight into the mechanical response of brain tissue at specific loading frequencies can be gained using procedures described herein. In this paper, the steady-state solution is explored, and its implications for brain injury are assessed.

Cell injury frequently occurs far away from the main focus of the agent that accelerates the skull, especially in the case of a type of mild traumatic brain injury known as diffuse axonal injury (e.g. Gennarelli 1983). This injury may be the result of strains localizing within the brain. The objectives of the present work are to identify (i) loading conditions in which peak strains occur on the interior of a Maxwell viscoelastic cylinder with frequency-dependent properties approximating those of the brain, and (ii) intracranial strain-based scaling laws for relating the results of animal experimentation to head injury in humans.

This latter objective is currently achieved through Holbourn-type scaling laws (Ommaya *et al*. 1967; Thibault & Margulies 1998) based upon torsional vibration of a massless, circular linear elastic shaft of shear modulus *G*, radius *R* and aspect ratio *λ*, fixed at one end, and holding an infinitely stiff circular plate of radius *R* and density *ρ* attached at the other end. The free end is accelerated rotationally by an angular acceleration of . The physiological assumption is that animals of two different species will stand equal chance of injury due to angular acceleration if the peak intracranial strains they experience are equal. Following this logic, the dimensionless constant *C* in the following expression defines cross-species scaling; when it is conserved, animals of different species will stand equal chance of sustaining brain injury:(1.1)where *M* is the mass of the brain, concentrated at the stiff plate. This type of law fails to capture the effects of viscous damping on intracranial straining, and fails to capture the effects of wave motion owing to the assumptions of Coulomb torsion theory.

Several other researchers have modelled wave motion in idealized viscoelastic bodies in similar attempts to understand the brain's response to angular skull acceleration. Axisymmetric wave motion in spheres of Kelvin/Voigt viscoelastic material was modelled analytically by Bycroft (1973), Ljung (1975) and Firoozbakhsh & DeSilva (1975), and numerically by Liu *et al*. (1975) and Misra & Chakravarty (1984). Margulies & Thibault (1989) considered approximately the periodic angular acceleration in a Kelvin/Voigt cylinder; Ljung (1975) studied a step angular loading. However, the Kelvin viscoelastic material, chosen in these models, cannot describe motion of stress waves (e.g. Flügge 1967), so scaling laws, non-dimensional parameters and comprehensive parameter studies could not be established by these authors. Lee & Advani (1970) solved the elastic case of sinusoidal loading of a sphere. Cotter *et al*. (2002) have presented a numerical solution to this problem and review the efforts of others to do so.

This paper furthers the study of intracranial wave motion and cross-species scaling by analysing the problem of a sinusoidally loaded Maxwell viscoelastic cylinder in closed form. Scaling laws are established for the model and appropriate non-dimensional parameters are identified. A class of problems addressing a similar physical model encompasses efforts to derive viscoelastic storage and loss moduli from the measurements on Goldberg & Sandvik (1947)-type coaxial oscillatory rheometers. Solutions to this problem (Oldroyd 1951; Markovitz 1952; Oka 1960; Bird *et al*. 1987) involve analytical steady-state expressions for the oscillation of an annulus of viscoelastic material between two coaxially vibrating cylinders. These solutions, however, cannot accommodate a zero radius of the internal cylinder, so a useful comparison with the present results cannot be made.

An additional use of the results in this paper lies in the interpretation of computational and experimental estimates of brain motion during repeated skull acceleration. Computer models of brain mechanics exist for the human for rapid (e.g. Ruan *et al*. 1994; Zhang *et al*. 2001) and slow (Miga *et al*. 1999, 2000; Ho & Kleiven 2007; Wittek *et al*. 2007) deformation. Computer models of the infant pig brain (Thibault & Margulies 1998) and movies of accelerated, gel-filled skulls (Margulies *et al*. 1990; Meaney *et al*. 1995) are also highly informative in interpreting the response of the human brain. Magnetic resonance imaging has been used to image human brain dynamics (Bayly *et al*. 2005, 2006; Sabet *et al*. in press) and the response of physical models (Bayly *et al*. 2004), and bi-planar X-ray imaging has been applied to track the displacement of markers in accelerated cadaver brains (Hardy *et al*. 2001). An objective of the current work is to provide qualitative results to aid in the interpretation of computational and experimental investigations of repeated skull acceleration.

## 2. Steady-state expression and scaling laws

The analytical expression for strain derived in the first part of this paper describes the behaviour of strain waves in a Maxwell viscoelastic cylinder under a sinusoidal perturbation of its outer boundary (Massouros 2005; Massouros & Genin submitted). The model considered was a rigid, infinitely long cylindrical shell of radius *a* filled with an incompressible, homogeneous Maxwell material of density *ρ*, shear modulus *μ* and viscosity *η*. The applied displacement on the outer boundary of the cylinder was . The tensorial linear shear strain *ϵ*_{rθ} was given by the expression(2.1)where *λ*_{k} are the positive roots of Bessel's function of order 1 (*J*_{1}(*z*)); *J*_{0}(*z*) and *J*_{2}(*z*) are Bessel's function of order 0 and 2, respectively; and . The following parameters are used in expressions (2.1) and throughout this work: the dimensionless space variable *x*=*r*/*a*, where *r* is the radial distance from the centre of the cylinder; the dimensionless time variable , where *τ*=*η*/*μ*; the dimensionless external frequency of oscillation *Ω*=*ωτ* (a Deborah number of the problem); the dimensionless wave speed ; and the ratio of the loading time scale to the time required for a reflected wave to return to the outer boundary, (a second Deborah number).

The nature and severity of the strain experienced by an accelerated Maxwell-type viscoelastic cylinder is characterized completely by the two dimensionless parameters *v*_{M} and *Ω* in equation (2.1). These parameters describe a family of such cylinders for which the balance between damping (*η*), stiffness (*μ*) and inertia (*ρa*^{2}) will result in identical dynamic behaviour for loadings with the same frequency *ω*, or, as can be shown, for the same step loading. For a particular cylinder, in the limiting case when *v*_{M} is very small, the ratio *μ*/*η*^{2} will be very large and the behaviour of the material approaches that of a viscous fluid. Similarly, for very large *v*_{M}, the behaviour of the material approaches that of an elastic solid. Large values of inertia also enhance the viscous aspect of the model. Thus, a material with lower *v*_{M} will behave more viscously than one of larger *v*_{M} and vice versa.

As described in the first part of this paper, the terms in equation (2.1) represent exponential decay and approach zero as steady state is reached. The term that describes steady state is the following:(2.2)

Over a broad range of values *v*_{M} and *Ω*, the location of the peak strain is the outer boundary of the cylinder, *x*=1. In such cases, asymptotic limits for equation (2.2) can be found for cases in which the quantity is either very small or very large. The derivations of these limits are presented in appendix A. For the case of small , the peak strain scales as(2.3)For the case of large , the asymptotic form is listed in appendix A. For *Ω*≫1, the peak strain scales as(2.4)

These expressions dictate how strains, and possibly injury, scale from animals to humans for the cases in which the peak strain occurs at the outer boundary. The range of values of *v*_{M} and *Ω* for which the peak strains do indeed occur at the outer boundary in the steady state is established in §3, along with a graphical representation of the scaling of shear strains over regimes for which a meaningful asymptotic form cannot be found.

## 3. Behaviour of an ideal Maxwell-type viscoelastic cylinder in steady state

The application of the Maxwell viscoelastic model to this problem enables the exploration of wave propagation behaviour. Of key interest are the peak strain magnitude and the location at which this peak strain occurs. A discussion of the family of steady-state wave patterns that can exist is presented, along with a mapping over parameter space of the location and severity of peak strains resulting from the periodic loading.

The behaviour of the Maxwell-type viscoelastic cylinder is best viewed through three-axis plots (figure 1). In these, shear strain magnitude is plotted with respect to both position, *x*=*r*/*a*, and dimensionless time, =*t*/*τ*. The strain is renormalized by *U*/*a*. The sinusoidal loading is evident at the boundary *x*=1; the strain remains at 0 for all values of at the centre of the cylinder, *x*=0. A first strain wave, of discontinuous strain rate, is initiated at the boundary at =0 and propagates through the material. A series of subsequent continuous waves are thereafter initiated periodically at the boundary. After reflecting from the centre, each wave interferes with the subsequent waves, while losing energy exponentially due to the viscous aspect of the material. Since each wave starts with the same energy and speed, at some time after enough waves interfere, the strain field will reach a steady oscillatory state. Note that these conditions are applicable to vibration of the cylinder, and not to impact of the cylinder.

Three aspects of the steady-state solution are of interest. First is the variation of the magnitude of the maximum strain with respect to the governing parameters of the model *v*_{M} and *Ω*. Second, to understand situations in which the most severe injury to the brain from angular acceleration occurs on the brain's interior, the position of the maximum strain in relation to *v*_{M} and *Ω* needs to be established. Finally, to identify conditions in which less severe injury might be expected on the brain's interior, the number of standing waves that appear in the steady-state solution must be understood.

From the form of equation (2.2), it is apparent that the Bessel function of order 2 defines the steady-state spatial pattern of the waves along the interior of the material. This pattern is modulated sinusoidally over time with a frequency *Ω*.

Sample cases of steady-state wave patterns that this combination yields are displayed in figure 1, in which steady-state shear strain fields are plotted with respect to position *x* and dimensionless time for three periods. Three cases of *v*_{M} and for each one three cases of *Ω* are illustrated, showing the variety of material response. The variations in magnitude and location of peak strains throughout the *v*_{M}–*Ω* space are explored in figures 2 and 3.

Figure 2*a* is a plot of the position of the maximum strain in the steady state as a function of the two governing parameters of this model (*v*_{M} and *Ω*). The maximum strain is shown in figure 2*b* as a function of these parameters. Note that the axes are reversed in the maximum strain plot. Figure 3 shows these as a function of *v*_{M} and *ζ*.

From figures 2*a* and 3*a*, one can observe that for sufficiently small values of *v*_{M}, the maximum shear strain in steady state will always occur at the outermost boundary of the cylinder (the plateau at *x*=1). In the cases of large *ζ* (small *Ω*), the waves travelling inside the material lose energy too quickly for the location of the peak strain to shift towards the centre. In these cases, the viscous aspect of the Maxwell material dominates. Figure 2*b* shows that the magnitude of the peak strain for values of *v*_{M} beneath this threshold scales nearly linearly with log(*v*_{M}) and *Ω*.

As values of *ζ* decrease, a threshold in *v*_{M} exists which varies with *ζ*; beyond this threshold, the position of the maximum strain progresses from the outermost boundary to a location that approaches the centre of the cylinder. For the case of very small *Ω* (large *ζ*), the maximum strain occurs at the outer boundary in steady state. At higher frequencies, more energy is input to the system and the position of the maximum strain ‘slides’ towards the centre forming standing half-waves. As more waves appear at even higher frequencies, the wave closest to the centre remains the dominant one. This qualitative behaviour occurs for all higher values of *v*_{M} as well, and for the elastic case, in which the maximum strain always occurs at the innermost wave. For high values of *v*_{M} and low values of *Ω*, the peak strain shown in figure 2*b* is given by the asymptotic form in equation (2.3). The position of the peak strain is directly proportional to *ζ* when the peak strain occurs on the interior of the cylinder.

For values of *v*_{M} slightly lower than 0.5, there exists an intermediate region in which the location of the maximum strain suddenly ‘shifts’ away from outer boundary with increasing *Ω*. At a critical frequency, constructive interference causes the peak strain of the inner wave to exceed the peak strain on the outer boundary, resulting in the sudden ‘shift’ mentioned above, in contrast to the more gentle ‘slide’ observed for larger values of *v*_{M}. As *Ω* increases further and more standing waves are added, the maximum strain continues to occur at the peak of the innermost wave. This transition corresponds to the crevice in figures 2*b* and 3*b*.

Also of interest is the small region near *Ω*=1 and *v*_{M}=0.4. The ‘teeth’ in figure 3*a* indicate a series of slides and shifts over a very small range of the parameters. This is the only region of parameter space in which shifts occur both away from and back to the outer boundary with increasing *Ω*.

The transition line in parameter space where the maximum strain moves from the outermost boundary is apparent in figures 2*b* and 3*b* as well. In the graph of the peak strain magnitude, the competition between the strain at the boundary and the peak of the innermost wave is apparent. For low frequencies, in which only a single standing wave exists, the variation of peak strain with the parameters is smooth. For larger values of *Ω* (smaller values of *ζ*), the shift is visible where the maximum strain of the innermost wave peak exceeds the strain at the boundary.

The ‘ripples’ in the maximum strain (figure 3*b*) appear as new standing waves are introduced. The number of standing waves inside the cylinder in the radial direction increases with *Ω* and decreases with *v*_{M} (figure 4). Figure 4 shows the number of steady-state standing waves plotted as a function of these two governing parameters, as well as the boundary where the maximum strain shifts from the outer boundary of the cylinder. The contour lines that describe cases with equal number of waves follow the ridges on the maximum strain figure: the peak strain is slightly higher when an integral number of half-waves exist in the steady state. In terms of *ζ*, the number of standing waves at every time is simply 2/*ζ*. For *ζ*=1, there is a half-wave travelling in steady state at every time, for *ζ*=0.5 a full wave, for *ζ*=0.25 two full waves and so on. This follows from the definition of *ζ* and underlines its importance as a more intuitive parameter in understanding the problem.

## 4. Application to human brain data

### (a) The brain as a Maxwell viscoelastic material

The analytical expression for the tensorial shear strain *ϵ*_{rθ} is now fit to data for the human brain in an effort to understand the brain's response to applied harmonic oscillation. Data from harmonic shear tests are available (e.g. Margulies & Meaney 1998), which give values for the complex shear modulus *G*^{*} for different applied frequencies *ω*. In such experiments, a harmonic shear stress *σ*=*σ*_{0} e^{iωt} is applied, resulting in harmonic shear strain *ϵ*=*ϵ*_{0} e^{iωt}. From the stress–strain relation of the Maxwell model, it readily follows that:(4.1)where *G*_{1}(*ω*) and *G*_{2}(*ω*) are called storage and loss moduli, respectively. Equating the real and imaginary parts in equation (4.1), we obtain the following expressions that relate the Maxwell model characteristic parameters *τ* and *μ* with the storage and loss moduli:(4.2)

The density of the average adult human brain is found to be 1030–1041 kg m^{−3} and for further calculation it will be taken as *ρ*=1035 kg m^{−3}; the average radius of an adult human brain is *a*=14 cm in the horizontal plane (Margulies & Meaney 1998).

Data presented by Margulies & Meaney (1998) are presented as values of *v*_{M} and *Ω* in figure 5. Note that for the Maxwell model, *Ω*=*G*_{1}/*G*_{2}. If the brain material behaved exactly like a Maxwell-type material, the dimensionless parameter, *v*_{M}, which depends only on material properties, would be expected to be a constant with respect to the dimensionless frequency *Ω*, which, in turn, would be expected to be proportional to the frequency of loading *ω*. However, this is not the case. Figure 5 defines a mapping of *v*_{M} versus *Ω*, which is necessary if the brain is to be approximated as a Maxwell material.

The brain is described by a different Maxwell material at each loading frequency. As shown in figure 5*b*, the dimensionless wave speed in the brain approaches 0 at very high values of applied frequency *ω*, meaning that viscous behaviour dominates at high frequencies. Figure 5*a* shows that the appropriate value of *Ω* scales as log(*v*_{M}).

### (b) Behaviour of a cylinder of white matter

To identify the peak strains that would occur inside a sinusoidally loaded cylinder of white matter, slices through parameter space that corresponds to the mapping in figure 5*a* must be examined in figure 2*a*,*b*. These slices are shown in figure 6; the horizontal axis of figure 6*a*,*b* is mapped to the applied frequency using figure 5*b*. In the steady-state case, all standing waves in the entire cylinder oscillate at the applied frequency *ω*; therefore, the entire cylinder is described by a single Maxwell model; that is, by a single point on the line in figure 5*a* at each loading frequency.

## 5. Discussion

Excessive shear strain inside the brain, induced by the angular acceleration of the skull, is believed to lead to brain injury. To shed light on the scaling of strains in brain tissue, a cylindrical viscoelastic medium attached to the interior of a rigid shell was studied, and generalized scaling laws for shear strain inside such a cylinder could be established. Brain material was approximated as Maxwell viscoelastic, which enabled the study of wave motion.

The steady-state behaviour for the tensorial shear strain was explored in relation with the governing parameters: the dimensionless wave speed, *v*_{M}; the dimensionless frequency, *Ω* (which can be viewed as a Deborah number for the problem); and a second time-scale ratio *ζ*. These describe families of Maxwell models which behave identically in cylinders of different sizes and densities. The model itself describes a range of viscoelastic behaviour, the limits of which are the behaviour of a viscous fluid and that of an elastic solid. Since the magnitude of strain is the basis of the criterion used to predict brain injury for closed head injury, maximum shear strain and the position at which it occurs were emphasized. The asymptotic expressions extracted from the steady-state expression describe scaling laws for the maximum shear strain for some limiting cases of the dimensionless parameters *v*_{M} and *Ω* or *ζ*. From the steady-state expression for the strain, wave motion was studied, and in many cases, peak strains were found very close to the centre. Maximum strain and its position inside the cylinder were plotted over the entire parameter space (figures 2 and 3), leading to a better understanding of scaling of shear strain inside a viscoelastic cylinder.

The application of the results to the human brain must be interpreted in the context of an idealized model problem. The brain is certainly not a Maxwell linear viscoelastic solid (e.g. Miller 2002) and can be approximated as such only for steady-state, small amplitude, oscillatory loading at one specific frequency; nonlinearity can be expected to lead to a spreading out of the wavefront with time (Nekouzadeh *et al*. 2005). Additionally, recent imaging work indicates that the boundary between the brain and skull in mild skull motion is characterized by a combination of smooth sliding with restraint from anatomical features such as vessels, nerves, membranes and vasculature (Ji *et al*. 2004; Bayly *et al*. 2005; Sabet *et al*. in press). In the impact testing of cadaver heads, sliding appears to give way to a fixed boundary condition only at higher levels of acceleration (Zou *et al*. 2007).

When applied to white matter in the human brain, whose material properties defined a cutting plane through parameter space, a prediction was obtained for the severity and location of the maximum strain that occurs for oscillation at a particular frequency (figure 6*a*,*b*). The peak strain occurred on the interior of the cylinder over only a small band of applied frequencies; one could surmise that, if oscillatory sinusoidal boundary conditions were applied to a volume of brain tissue, a similar band of critical frequencies might exist that protective gear might be designed to block preferentially. The specific band identified here is dependent upon the choice of material properties for the brain tissue; using other available data (e.g. Miller 1999; Miller *et al*. 2000; Miller & Chinzei 2002; Prange & Margulies 2002; Lippert *et al*. 2004; Nicolle *et al*. 2004) would shift this band in frequency space. However, as can be seen in figure 2, values of high strain can exist close to the centre even in cases where the maximum strain occurs at the boundary.

## 6. Conclusions

The steady-state response of a Maxwell viscoelastic cylinder to periodic sinusoidal oscillation is governed by two dimensionless parameters: a dimensionless wave speed, *v*_{M}, and one of two time-scale ratios, *Ω* or *ζ*. The parameter *Ω* can be viewed as a Deborah number and *ζ* represents the number of standing waves that persist in the steady-state solution. These dimensionless quantities describe families of Maxwell models that behave identically in cylinders of different sizes and densities. Scalings and asymptotic forms for peak strains were presented for specific ranges of these parameters.

The solution was studied to identify situations in which the peak strain would be expected to occur on the interior of an oscillating Maxwell viscoelastic cylinder. Studies of parameter space identified specific values of *v*_{M} for which this occurs; this threshold value decreases non-monotonically as *Ω* increases. When applied to cylinders with mechanical response representative of white matter, the result was that peak strains occurred on the interior of such a cylinder only for a specific band of loading frequencies.

## Acknowledgments

The authors gratefully acknowledge Philip V. Bayly for his helpful discussions. This work was sponsored in part by the Johanna D. Bemis Trust and the National Institutes of Health through grant no. NS55951.

## Footnotes

- Received June 15, 2007.
- Accepted September 27, 2007.

- © 2007 The Royal Society