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Hindawi Publishing Corporation

Journal of Applied Mathematics

Volume 2012, Article ID 409127, 20 pages

doi:10.1155/2012/409127

Research Article

Modeling of Brain Shift Phenomenon for Different

Craniotomies and Solid Models

Alvaro Valencia,1Benjamin Blas,1and Jaime H. Ortega2

1Department of Mechanical Engineering, Universidad de Chile, Beauchef 850, Santiago, Chile

2Department of Mathematical Engineering and Center for Mathematical Modeling, Universidad de Chile,

Av. Blanco Encalada 2120, Santiago, Chile

Correspondence should be addressed to Alvaro Valencia, alvalenc@ing.uchile.cl

Received 15 July 2011; Revised 4 October 2011; Accepted 20 October 2011

Academic Editor: Venky Krishnan

Copyright q 2012 Alvaro Valencia et al. This is an open access article distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

This study investigates the effects of different solid models on predictions of brain shift for three

craniotomies. We created a generic 3D brain model based on healthy human brain and modeled

the brain parenchyma as single continuum and constrained by a practically rigid skull. We have

used elastic model, hyperelastic 1st, 2nd, and 3rd Ogden models, and hyperelastic Mooney-Rivlin

with 2- and 5-parameter models. A pressure on the brain surface at craniotomy region was

applied to load the model. The models were solved with the finite elements package ANSYS.

The predictions on stress and displacements were compared for three different craniotomies. The

difference between the predictions of elastic solid model and a hyperelastic Ogden solid model of

maximum brain displacement and maximum effective stress is relevant.

1. Introduction

Neurosurgery requires high levels of accuracy due to the complexity of the brain. To do

this, surgeons have preoperative images that identify the exact area of operation. However,

during craniotomy, a change on loading condition occurs, that causes brain deformation.

The deformations carry a margin of error in the surgery area. The phenomenon known

as brain shift deformations will be studied in this work. We note that the brain shift is a

negative effect that occurs in the surgical opening of the skull ?craniotomy?. Brain shift is

produced by a pressure difference on the brain induced in the region of the craniotomy. This

changes the position of the pathology and healthy tissues from the calculated with high-

quality preoperative radiographic images.

Themostsurgicalnavigationsystemsuse3Dpreoperativelyacquireddataandregister

it to the patient coordinate system. This assumes that the brain is rigid and is a source of error

in the exact determination of tumor position.

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2 Journal of Applied Mathematics

There are several factors that determinate the magnitude of brain shift produced by

a craniotomy: gravity, mechanical tissue properties, loss of brain-spinal fluid, anatomical

constrains, intracranial pressure, and patient variability.

A current challenge is the determination of stress and displacements in a solid model

of the brain subject to a craniotomy. The geometry of the brain is very complex, and the

characteristics of the tissue are not easy to measure and model. The results of the solid model

will help to correct the position of the brain for the surgical navigator system.

Most solid brain models use elastic model ?1? in order to model the deformations and

stress of the brain tissue. Using an elastic solid model, the Young modulus does not affect the

displacement field if the gravity is not considered ?2?.

The effects of considering hyperelastic model of brain have been considered in few

works in the literature. The use of nonlinear solid model made it possible to obtain very good

predictions of deformation of ventricles and tumor ?3?. The same authors have supposed

that the brain deformations depend very weakly on the constitutive model and mechanical

properties of the brain tissues, and therefore simple hyperelastic model can be used ?4?.

Several authors propose the use of the linear elasticity to model the stress and

deformations of the brain tissue ?5–7?. The linear elasticity considers the determination

of some parameters as the elasticity modulus ?E?, the shear modulus, or second Lam´ e’s

parameter ?μ,λ? among others. The models consider just one brain tissue, isotropic and

incompressible, which is a simplification. Then, they assume that the brain is immersed

into the cerebrospinal liquid which is contained by the rigid skull. It is clear that this liquid

producesapressureonthebraintissue.Theskullisconsideredasanextremelyrigidstructure

which is nondeformable. The elasticity modulus is similar to the human bones, that is the

elasticity modulus of the skull is 6.5GN/m2and the Poisson constant of the skull is 0.22.

The skull is a rigid structure, which contains three elements, the brain tissue ?86%?,

blood ?4%?, and cerebrospinal liquid ?CRL, 10%?. The interaction among these elements

produces a pressure called intracranial pressure. Normally, this pressure in a health adult is

around 10mmHg ?1332.8Pa? and must not be higher than 15mmHg. The density of the CRL

is 1007kg/m3. Furthermore, the brain tissue corresponds only to the 2% of the total weight

but is the element with highest intracranial volume. The weight of the brain is between 1300

and 1600gr, and its volume is around 1000 to 1500cc. Its density is closer to the water density,

that is 1040kg/m3.

We can see that, about the properties of the brain tissue, in particular, the values of the

elasticity constants, there are several differences among the authors ?8, 9?.

It is important to remark that the linear elasticity has a suitable behavior for small

deformations, and it is clear from several authors that the relationship between stress and

deformation for soft tissue is not linear ?10, 11?.

In the present investigation, we report the effects of hyperelastic solid models on

maximal displacement and effective stress of the brain. We have calculated the brain shift

for three craniotomies.

2. Mathematical Models

The linear elasticity theory is the study of linear elastic solids undergoing small deformations.

The linearity means that the components of the stress tensor are a linear combination of the

deformations.

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Journal of Applied Mathematics3

The relationship that defines each element of the strain tensor is shown in ?2.1?. This

tensor is known as infinitesimal tensor of Green-Cauchy:

εij?1

2

?

∂ui

∂xj

?∂uj

∂xi

?

,

?2.1?

with i,j ? 1,2,3.

The constitutive equations of linear elasticity for an elastic solid are represented by

generalized Hooke’s law:

σij? Cijkmεkm,

?2.2?

where i,j ? 1,2,3.

However, if the material is assumed homogeneous and isotropic, we obtain the consti-

tutive equation of Lam´ e-Hooke.

2.1. Constitutive Equation of Lam´ e-Hooke

Itiswellknownthat,consideringahomogeneous andisotropicmaterial,weobtaintheLam´ e-

Hooke constitutive equations. That means that the components of the elastic tensor depend

on two particular constants of each material, these constants are the so-called Lam´ e modulus

?λ,μ?. The relation between the elastic coefficients and the Lam´ e modulus is the following:

Cijkm? λδijδkm? μ?δikδjm? δim? δjk

?.

?2.3?

Finally, after some simplifications, we have

σij? λδijεkk? 2μεij,

?2.4?

where εkkis the trace of the deformation tensor.

2.2. Nonlinear Elasticity

The nonlinear elasticity is an observed phenomenon in elastomeric material ?rubber? and

porous media. The origin of both materials is different, for instance, the elastomeric materials,

which are polymers, can be synthetic or natural rubber, and, on the other hand, porous

media exist in the nature in form of organic materials, vegetal and animal tissue. The main

characteristic of this material is their deformation capacity, which can arrive from 200% to

300%. Nevertheless, these large deformations can be recovered, and the material comes back

to its natural state. It is important to note that the human tissues behave as a nonlinear elastic

material.

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4Journal of Applied Mathematics

In what follows, we will present some ideas on the nonlinear elastic models. Firstly, we

will define the relationship between the strains and the displacement vector, which is defined

in ?2.5?, which is a nonlinear relationship:

?

∂xj

∂xi

εij?1

2

∂ui

?∂uj

?∂uk

∂xi

∂uk

∂xj

?

,

?2.5?

with i,j ? 1,2,3.

From the above equation, we can obtain the strain tensor or Green-Lagrange strain

tensor. This tensor helps to quantify the length changes of the material and the variation of

the angle between the material fibers.

The deformation energy is a useful function in order to define a hyperelastic material.

This function gives a relation between the stored energy with the strain and deformations

generated in the solid. Moreover, its derivatives with respect to stretch give us the stress

producedfortheappliedforce?Cauchystress?.Inordertocomputethedeformationenergies,

it is necessary to introduce the deformation gradient

?F? ?

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

∂x

∂X

∂y

∂X

∂z

∂X

∂x

∂Y

∂y

∂Y

∂z

∂Y

∂x

∂Z

∂y

∂Z

∂z

∂Z

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.

?2.6?

This tensor represents the variation of a deformed material point with respect to its

initial state. To simplify the computations, it is interesting to obtain the Green-Cauchy

left deformation tensor ??B?? and the Green-Cauchy right deformation tensor ??C??, both

can be recovered from the deformation gradient tensor, and the Green-Cauchy invariant

deformation tensor can be easily obtained:

?B? ? ?F??F?T,

?C? ? ?F?T?F?.

?2.7?

ThedeformationenergyofthematerialisafunctionoftheinvariantsoftheleftGreen-Cauchy

deformation tensor ??B??. If we assume isotropy of the material, the energy depends on the

first three invariants of the tensor

W ? W?I1,I2,I3?.

?2.8?

The invariants for an isotropic material are as follows.

First Invariant:

I1? tr?B?.

?2.9?

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Journal of Applied Mathematics5

Second Invariant:

I2?1

2

?

tr?B?2− tr

?

B2??

.

?2.10?

Third Invariant:

I3? det?B?.

?2.11?

If the normal forces are parallel to the principal direction of the material, we have that

the invariants only depend on the principal elongations of the solid. It is important to remark

that this condition can occur in every isotropic material, and this is due to that in all directions

the measurements are equal.

The principal stretches are defined as the quotient between the final length and the

initial length in the direction of the deformation. The invariants are functions of the principal

stretches:

I1? λ2

I2? λ2

1? λ2

2? λ2

I3? λ2

2? λ2

2λ2

1λ2

3,

1λ2

3? λ2

2λ2

1λ2

3

3,

,

?2.12?

where λ1, λ2, and λ3are the stretch in the principal directions.

If we suppose that the solid is almost incompressible or with a high compression

modulus, the deformation energy depends only on the first and second invariant, since the

third invariant verifies I3 ? 1. The Cauchy strains are calculated from the derivative with

respect to the deformations of the deformation energy, that is,

σij? −pδij?∂W

∂I1Bij?∂W

∂I2B−1

ij,

?2.13?

where p represents the pressure produced in the principal directions.

In what follows, we will present several solid models, such as, their deformation

energies and principal stress obtained under different assumptions as hyperelasticity,

isotropy, incompressibility and, under uniaxial tension. For the uniaxial tension, we have

that

λ1? λ,

λ2? λ3?

1

√λ.

?2.14?

Thus, we obtain that

I1? λ2? 2λ−1,

I2? 2λ ? λ−2.

?2.15?

In what follows, we will describe some different hyperelastic models used for brain tissue

modeling.

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6 Journal of Applied Mathematics

2.2.1. Neo-Hooke Material Model

In this case, the deformation energy model is given by

W ? C1?I3− 3?,

?2.16?

where C1is a given constant. The stress in the direction of the principal stretch is

σ ? C1

?

2λ −2

λ2

?

.

?2.17?

2.2.2. Mooney-Rivlin Material Model

In this case, the deformation energy model, with 5 parameters, is the following:

w ? C1?I1− 3? ? C2?I2− 3? ? C3?I1− 3?2? C4?I1− 3??I2− 3? ? C5?I2− 3?2,

?2.18?

where C1to C5are material constants. In this case, the stress direction for the principal stretch

corresponds to

?

? C4

λ2

?

σ ? C1

2λ −2

??

λ2

2λ −2

2λ ?1

?

? C2

?

2 −2

2λ ?1

??

λ3

?

? 2C3

?

?

?

?

λ2?2

λ− 3

??

??

2λ −2

2 −2

λ2

?

?? ??

λ2− 3

2 −2

λ3

?

λ2?2

λ− 3

λ3

? 2C5

λ2− 3

.

?2.19?

2.2.3. Odgen Material Model

In this case, the deformation energy is given by

W ?

n ?

k?1

μk

αk

?

λαk?

?1

√λ

?αk

− 2

?

,

?2.20?

where αkand μkare constants of the material. The stress direction of the principal stretch is

σ ?

n ?

k?1

μk

?

λαk−1? ?λ?−0.5αk−1?

.

?2.21?

From the Cauchy tensor, it is possible to compute the equivalent stresses. The

equivalent stress. is computed using the Von Misses formula:

?

σvm?

?σ1− σ2?2? ?σ2− σ3?2? ?σ1− σ3?2

2

,

?2.22?

where σ1, σ2, and σ3are the principal stresses.

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Journal of Applied Mathematics7

The equivalent strain is defined as

εeq?

1

√2?1 ? υeq

?

??εx− εy

?2??εy− εz

?2? ?εz− εx?2?3

2

?

γ2

xy? γ2

yz? γ2

xz

??1/2

,

?2.23?

where ε and γ are the components of the deformation tensor of Hencky.

3. Numerical Methods

3.1. Modeling of the Brain Shift

In order to make the numerical simulations of the brain shift, we will consider the experi-

mental data of Mehdizadeh et al. ?8?. In the experiments, the gray matter is obtained from

the parietal lobe and the white matter is obtained from the corpus callosum from a one-year-

old bovine. The tissue obtained corresponds to discs of 15mm diameter and 5mm of height.

The tests were realized with a uniform rate of deformation of 1mm/min in order to avoid

inertial forces. The used machine was a dynamic testing machine, Hct/25–400 with servo

hydraulic valve PID controller. The elastic modulus obtained was E ? 24.6kPa and ν ? 0.49.

For the Neo-Hooke model, the constant is C1? 7903Pa.

To study the hyperelastic solids, we use the data obtained for the gray matter. The

curve for the uniaxial traction for the gray matter is showed in Figure 1.

The cerebral cortex consists of gray matter, and this region is the most affected by brain

shift. Also, the ratio of the volume of grey matter to white matter in the cerebral hemisphere

for a 20-year-old man is 1.3 ?12?. The mechanical properties of gray and white matters

measured by Mehdizadeh et al. ?8? show differences for gray matter, true Young modulus

of 24.6kPa and, for the white matter, true Young modulus of 19kPa have been derived. On

the other hand, it is practically impossible to build a simple solid model considering the real

white and gray matter distribution in a human brain. Considering these reasons, we have

used the mechanical properties of gray matter for the complete brain model. Due to the facts

that the larger brain displacements are near brain surface and the brain cortex is composed

only principally of gray matter, the model predicts brain shift with acceptable precision.

The Odgen material model was studied considering the first, second, and third order.

The Mooney-Rivlin model was studied considering two of its forms, with two and five

parameters ?see Tables 1 and 2?.

3.2. CAD Geometry

To quantify accurately the deformations and stresses produced in the phenomenon of brain

shift during a brain craniotomy, the CAD model of brain is relevant. The CAD geometry

used in the present work is an approximation with characteristics similar to a real brain. We

modeled the characteristics of a healthy male brain of 35 years. The brain is approximately

a ball whose surface geometry is characterized by irregular folds, see Figure 2. In this area

circulate most of the blood vessels, veins, and arteries. The width of the brain is variable;

however, the average value is 140mm. The average length is 170mm. The height of the

brain varies with respect to the observed cross-section up to 120mm. Considering the

above measures as a reference and using MRI images of the brain, a CAD 3D is generated.

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8 Journal of Applied Mathematics

Unia

Unia

Biax

Shea

Uniaxial test data

Stress (×103) (Pa)

2

1.5

1

0.5

0

0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Strain (mm−1)

Figure 1: Stress-strain curves for gray matter.

Table 1: Odgen constants.

First order

8743

1.8127

—

—

—

—

Second order

629,5

10.499

657.22

10.188

—

—

Third order

423.62

9.8787

445.45

10.213

459.1

10.156

μ1?Pa?

α1

μ2?Pa?

α2

μ3?Pa?

α3

Table 2: Mooney-Rivlin constants.

MR 2 parameters ?Pa?

3922.2

30.838

—

—

—

MR 5 parameters ?Pa?

−620.52

4466.1

3.85E ? 06

−8.71E ? 06

4.93E ? 06

C1

C2

C3

C4

C5

The CAD software used is solid edge, as it provides the necessary tools to model complex

nonparameterized curves. The methodology is to build a hemisphere from the outer contours

of the brain. To obtain these, contours are sectioned in coronal three-dimensional models,

then the contours are drawn to generate the solid model. Once the model is built in, with the

option Mirror, the second hemisphere is created ensuring the model symmetry. The last step

is to use the option Swept Protrusion to create the final CAD model of the brain. The model

obtained is showed in Figure 3.

The cerebral cortex is characterized as a cortical layer with a convoluted topology,

Figure 2. This complex geometry is modeled as simple hemisphere as in all previously

reported investigations about brain shift, see Wittek et al. ?3, 4?. The model must be relatively

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Journal of Applied Mathematics9

c

Figure 2: Coronal view of the brain.

?a?

?b?

Figure 3: Principal and Isometric views of the brain model.

simple to be used as predictive tool for the clinicians with a minimum error. The comparison

of model predictions with clinical results of brain shift ensures that the approximation of the

complex brain structure is correct for the goal of the present model. We will try to use this

model in brain surgery to predict brain shift after clinical validation in the future; for this

reason, the model must produce computational results in short time. Models that consider

the topology of cerebral cortex as highly convoluted sheet for investigation of the gray matter

deformation have been reported by Chung et al. ?13?. However, the model is too complex to

be applied during a surgery.

The skull is made similarly to the brain. To do this, we use the option Offset tool in the

CAD software. The goal is to keep the skull around the brain model but with a separation

between these elements. According to what was observed in the MRI images of the middle-

aged male patient, the gap between the elements was determined. Figure 4 shows the process

of generating the skull geometry.

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Figure 4: Design for the skull model.

Figure 5: Views of the brain and skull model.

Figure 5 shows a view of the brain and the skull used in the present investigation.

Although the models are a simplification of the real, it is important to note that they retain

morphological similarity.

In the present investigation, we do not consider the cerebrospinal fluid CSF and the

brain can be deformed in this space. in a brain craniotomy, CSF is extracted during surgery,

and, therefore, this model restriction has low effect on brain displacement. The subarachnoid

space between brain and skull is small compared with nominal brain diameter, also variation

of model distance between brain and skull was considered as second-order effect. The present

brain model intends to describe a methodology to predict brain shift.

3.3. Boundary Conditions

For the simulation of the brain shift effects, we consider two boundary conditions.

Fix of spinal cord: in order to limit movement of the brain and allow greater defor-

mations only in the area affected by the change in pressure.

Pressure variation in area of operation: intracranial pressure caused by brain, blood,

and CSF is approximately 770mmHg. The atmospheric pressure is 760mmHg. Upon opening

the skull, there is trickle of CSF and blood, mainly affecting the operation area and leaving it

exposed to atmospheric pressure. This condition results in a negative pressure in the opening

area equivalent to 10mmHg or 1333Pa.

Figure 6 shows the application of the pressure boundary conditions for the different

craniotomies. Figure 6 shows the skull with the brain inside, and indicated with red color the

region where the pressure boundary condition is applied in each case.

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Journal of Applied Mathematics 11

?a?

J: craneotomia lateral

Static structural

Time: 6 s

09-11-2010 13 : 07

A

B

Fixed support

0 0.050.1

0.025

0.075

Pressure: −1333 Pa

(m)

Image to file...

XY

Z

?b?

Static structural

Time: 6 s

09-11-2010 12 : 56

Pressure: −1333 Pa

Fixed support

A

B

0 0.05 0.1

0.025

0.075

(m)

I: craneotomia delantera

XY

Z

?c?

Static structural

Time: 6 s

09-11-2010 13 : 21

Pressure: −1333 Pa

Fixed support

A

B

00.05 0.1

0.025

0.075

(m)

K: craneotomia trasera

Z

Y

X

?d?

Figure 6: Boundary condition for parietal, frontal, and posterior craniotomy.

The region size to apply the pressure boundary condition in the three brain crani-

otomies was very difficult to choose. The first idea was to apply the pressure on the same

area for the three cases; however, personal communications with neuroradiologists from the

Instituto de Neurocirugia Asenjo, that help us in this project, indicated that the affected area

is different for the three investigated craniotomies. The red areas showed in Figure 6 show

the chosen areas to apply the pressure boundary condition. The areas were not the same for

the tree cases, because the goal was to try to reproduce clinical results.

For the brain shift, the small distance between skull and the brain is the most relevant

parameter that induces pressure differences on brain surface. During a craniotomy, CSF

moves outside the skull and CSF flow does not produces pressure difference on brain surface.

3.4. Computational Method

The model was solved by a commercial finite element package ANSYS v12.1. The finite

element method ?FEM? is used to solve the governing equations. The FEM discretizes the

computational domain into finite elements that are interconnected by element nodal points.

We have used the static structural formulation with a maximum time of 6s. Incompressible

material behavior may lead to some difficulties in numerical simulation, such as volumetric

locking, inaccuracy of solution, checkerboard pattern of stress distributions, or divergence.

We used the mixed u-P elements available in ANSYS to overcome these problems.

The unstructured grids were composed of tetrahedral SOLID187 with 10 nodes

available in ANSYS. Figure 7 shows the details of the three different grids used for the

parietal, posterior, and frontal craniotomies. For the parietal craniotomy, the grid was more

refined in the middle brain region. For the frontal craniotomy, the grid was refined nearer

than the frontal region of the brain.

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12Journal of Applied Mathematics

00.05 0.1

0.0250.075

(m)

X

Y

Z

?a?

00.05 0.1

0.025 0.075

(m)

X

Y

Z

?b?

0

0.050.1

0.025 0.075

(m)

X

Y

Z

?c?

Figure 7: Isometric and superior views of the different computational grids of the brain model used for ?a?

parietal, ?b? posterior, and ?c? frontal craniotomy.

Table 3: Comparison of maximum displacement, strain, and stress for three different grid sizes.

Grid Elements

Maximum displacement

?mm?

3.845

3.83

3.806

Maximum equivalent

strain %

59.14

59.31

59.85

Maximum equivalent

stress ?Pa?

14527

14591

14724

1

2

3

49453

34356

11730

The three grids used are similar, and the variations of element size in the brain depend

also on model construction, see Figure 5.

Grid independence study was performed for three grid sizes; maximum displacement,

equivalent strain, and equivalent stress were compared in Table 3. For the comparison, we

have used the frontal craniotomy with the elastic brain. The differences between the results

are very small. Therefore, the middle grid size was used to perform all the computational

simulations. This test ensures that the grid density does not affect the expected results about

brain shift.

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Journal of Applied Mathematics13

Table 4: Comparison of maximum displacement and effective stress using seven brain models for a frontal

pressure boundary condition without skull.

ElasticNeo-Hookean

8.631

16920

Ogden 1

8.625

16401

Ogden 2

9.0894

47729

Ogden 3

9.0667

46639

M-R 2

8.630

16880

M-R 5

7.935

56375

Maximum displacement ?mm? 8.923

Maximum effective stress ?Pa? 37045

4. Results and Discussion

The predictions of maximum displacement and effective stress of brain under a pressure

boundary condition similar to frontal craniotomy but without skull were compared for

elastic, neo-Hookean, Ogden first-order, Ogden second-order, Ogden third-order, Mooney-

Rivlin with two-parameter, and Mooney Rivlin with five-parameter models. The results are

showed in Table 4. The low-order neo-Hookean model, the first-order Ogden model, and the

Mooney-Rivlin model with two parameters predict similar displacement and effective stress.

Theelasticmodelpredictssimilardisplacementasthehigh-orderhyperelasticOgdenmodels;

however, the prediction of maximum effective stress is 66% lower than the prediction of the

second-order Ogden model.

The predictions of hyperelastic second-order Ogden and third-order Ogden models

are similar. The Mooney-Rivlin model with five parameters predicts lower brain displace-

ment compared with all models included in the elastic model and therefore is discarded. With

these considerations, the hyperelastic second-order Ogden model is selected as adequate

solid model for comparison with an elastic model for prediction of brain shift phenomena.

In this work, we have considered three brain craniotomies, a parietal, a posterior, and

a frontal case as showed in Figure 6, and the maximum displacement and effective stress

were investigated. The solid models of brain are the elastic and the hyperelastic second-order

Ogden models.

Figure 8 shows in logarithmic scale the distribution of the effective stress for the

parietal craniotomy, and an isometric and an inferior views of the brain are showed. The

effective stress on brain surface shows large areas with values around 1000Pa. The maximum

effective stress is 53286Pa. The maximum is on the brain base near the spinal cord, where the

model is fixed.

Figure 9 shows in logarithmic scale the distribution of the effective stress for the

posterior craniotomy, and an isometric and an inferior views of the brain are showed. The

effective stress on brain surface shows large areas with values around 500Pa. The maximum

effectivestress isnow38073Pa.Themaximum isonthebrainbase nearthespinal cord,where

the model is fixed. Areas on the brain base are under relatively high stress compared with the

rest of brain.

Figure 10 shows in logarithmic scale the distribution of the effective stress for the

frontal craniotomy, and an isometric and a inferior views of the brain are showed. The

effective stress on brain surface shows large areas with values around 200Pa. The maximum

effective stress is now only 10049Pa. The maximum is on the brain base near the spinal cord,

where the model is fixed. Areas on the brain base are under relatively high stress compared

with the rest of brain.

A comparison between the three craniotomies shows that the parietal produces higher

effective stress on brain than the posterior and frontal interventions. High stress values are

distributed principally on brain base.

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14 Journal of Applied Mathematics

Equivalent stress

Type: equivalent (von-Mises) stress

Unit: Pa

Time: 6

Custom

11-11-2010 14 : 34

53286

5328.6

532.86

53.286

5.3286

0.53286

0.053286

0.0053286

0.00053286

5.3286e−5

5.3286e−6

5.3286e−7

5.3286e−8

5.3286e−9

Min

0

J: craneotomia lateral

0

0.050.1

0.0250.075

(m)

Max

X

Y

Z

?a?

53286

5328.6

532.86

53.286

5.3286

0.53286

0.053286

0.0053286

0.00053286

5.3286e−5

5.3286e−6

5.3286e−7

5.3286e−8

5.3286e−9

0

Min

0

0.050.1

0.0250.075

(m)

Equivalent stress

Type: equivalent (von-Mises) stress

Unit: Pa

Time: 6

Custom

11-11-2010 14 : 27

J: craneotomia lateral

Max

X

Y

?b?

Figure 8: Effective stress for parietal craniotomy, hyperelastic second-order Ogden model.

Comparing the maximum effective stress for the frontal craniotomy with the data of

Table 4 can be concluded that the effect of the skull is very important, and the stress for

the craniotomy is considerably lower as the value obtained only with a pressure boundary

condition. This indicates that modeling of brain shift must consider the skull to obtain more

realistic values. Ji et al. ?5? reported the relevance of brain-skull contact in the determination

of brain shift compensation.

The maximum effective stress is high compared with the values of the stress strain

curve showed in Figure 1; therefore, the use of hyperelastic model for the brain is relevant for

better prediction of brain shift.

The distribution of brain displacement for parietal craniotomy calculated with the

hyperelastic second-order Ogden model is showed in Figure 11. Figure 11 shows the

displacement of brain surface and the displacement in a plane through the craniotomy. The

upper surface shows displacements around 7mm. The brain area with large displacement is

important. Also, in the brain center, the displacements are around 3mm. Figure 11 shows the

relevance of brain shift for parietal craniotomy.

Figure 11 shows that a part of the brain is displaced out of the skull in the craniotomy

area. The zones with low displacement are near the spinal cord, due to the fix condition in

this area. From a neurological point of view, this result is realistic. For the brain, the zone

with maximum stress does not coincide with the location of maximum brain shift or brain

displacement.

Page 15

Journal of Applied Mathematics 15

Equivalent stress

Type: equivalent (von-Mises) stress

Unit: Pa

Time: 6

Custom

11-11-2010 15 : 01

38073

3807.3

380.73

38.073

3.8073

0.38073

0.038073

0.0038073

0.00038073

3.8073e−5

3.8073e−6

3.8073e−7

3.8073e−8

3.8073e−9

Min

0

K: craneotomia trasera

0 0.05 0.1

0.0250.075

(m)

Max

X

Z

?a?

Equivalent stress

Type: equivalent (von-Mises) stress

Unit: Pa

Time: 6

Custom

11-11-2010 15 : 01

38073

3807.3

380.73

38.073

3.8073

0.38073

0.038073

0.0038073

0.00038073

3.8073e−5

3.8073e−6

3.8073e−7

3.8073e−8

3.8073e−9

Min

0

K: craneotomia trasera

0 0.050.1

0.025 0.075

(m)

Max

X

Y

?b?

Figure 9: Effective stress for posterior craniotomy, hyperelastic second-order Ogden model.

The distribution of brain displacement for the posterior craniotomy calculated with

the hyperelastic second-order Ogden model is showed in Figure 12. The figure shows the

displacement of brain surface and the displacement in a middle brain plane. The frontal

region shows displacements around 12mm. The brain area with large displacement is

important. Also, in the brain center, the displacements are around 5mm. Figure 12 shows

the relevance of brain shift for posterior craniotomy. The zones with low displacement are

near the spinal cord, due to the fix condition in this area.

Finally, the distribution of brain displacement for the frontal craniotomy calculated

with the hyperelastic second-order Ogden model is showed in Figure 13. The figure shows

the displacement of brain surface and the displacement in a middle brain plane. The

superior region shows displacements around 4mm. The brain area with large displacement

is important. Also, in the brain center, the displacements are around 2mm. Figure 13 shows

the relevance of brain shift for frontal craniotomy. The zones with low displacement are near

the spinal cord, due the fix condition in this area. For the brain, the zone with maximum

stress does not coincide with the location of maximum brain shift or brain displacement and

a craniotomy.