research article quantification of stretching in the...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 350359 10 pageshttpdxdoiorg1011552013350359

Research ArticleQuantification of Stretching in the Ventricular Wall and CorpusCallosum and Corticospinal Tracts in Hydrocephalus before andafter Ventriculoperitoneal Shunt Operation

Hans von Holst12 and Xiaogai Li2

1 Department of Neurosurgery Karolinska University Hospital 171 76 Stockholm Sweden2Division of Neuronic Engineering School of Technology and Health Royal Institute of Technology (KTH)KTH-Flemingsberg Alfred Nobels Alle 10 Huddinge 141 52 Stockholm Sweden

Correspondence should be addressed to Hans von Holst hansvonholstkarolinskase

Received 6 March 2013 Accepted 10 April 2013

Academic Editor Hang Joon Jo

Copyright copy 2013 H von Holst and X Li This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this study we establish a quantitativemodel to define the stretching of brain tissue especially in ventricularwalls corpus callosum(CC) and corticospinal (CS) fiber tracts and to investigate the correlation between stretching and regional cerebral blood flow(rCBF) before and after ventriculoperitoneal shunt operations A nonlinear image registration method was used to calculate thedegree of displacement and stretching of axonal fiber tracts based on the medical images of six hydrocephalus patients Also therCBF data from the literature was analyzed and correlated with the strain level quantified in the present study The results showedsubstantial increased displacement and strain levels in the ventricular walls as well as in the CC and CS fiber tracts on admissionFollowing shunt operations the displacement as well as the strain levels reduced substantially A linear correlation was found toexist between strain level and the rCBF The reduction in postoperative strain levels correlated with the improvement of rCBFAll patients improved clinically except for one patient due to existing dementia These new quantitative data provide us with newinsight into the mechanical cascade of events due to tissue stretching thereby provide us with more knowledge into understandingof the role of brain tissue and axonal stretching in some of the hydrocephalus clinical symptoms

1 Introduction

Hydrocephalus is the consequence of various causes such ascongenital malformations subarachnoid hemorrhage trau-matic brain injury and benign and malignant brain tumorsRegardless of etiology the common denominator amongthe patients is obstruction of cerebrospinal fluid interferingwith the brain tissue function The symptoms and signs aresomewhat different depending on whether the patient is aninfant is child or receives the hydrocephalus in mature age

The expansion of the ventricular system influences thesurrounding areas in the brain tissue in various intensitiesThe clinical condition is usually defined as the Hakim triadof symptoms including gait apraxia cognitive disturbanceand incontinence Initially the white matter suffers morethan the grey matter with reversible changes followingshunt operation which reduces the expanding ventricles

Depending on the age at which hydrocephalus developstogether with the magnitude of ventricular expansion theneuropathological pattern is different in severityWhen long-standing the mechanical distortion results in more advancedneuropathological signs such as damage to the axons andmyelin Also when the cerebral arteries and capillaries aresubstantially distorted there is a potential of reduced regionalcerebral blood flow which may further interfere with theclinical condition [1 2]

Stretchingcompression of the ventricular wall [3] thecorpus callosum (CC) [4 5] and corticospinal (CS) [67] fiber tracts in hydrocephalus patients has been widelyrecognized in many previous research investigations Alsoan early hypothesis suggested that the compression andordeformation of the CS tracts due to enlargement of theventricles in hydrocephalus patient may contribute to the gaitdisturbance [8] However there is a lack of study providing

2 Journal of Applied Mathematics

quantitative data of the axonal stretching level and how themechanical stretching may influence the axonal functionsremains unclear until now A quantitative data should alsoprovide us more insight to understand the involvement ofaxonal fiber stretching in clinical symptoms

Thus the aim of the present study was

(i) to analyze the displacement and strain levels inpatients with hydrocephalus by using a numericalsimulation method focusing on the ventricular wallsCC and CS fiber tracts before and after shunt opera-tions and

(ii) to estimate the regional cerebral blood flow aftertreatment and the correlation to the strain level afterthe neurosurgical procedure

2 Material and Methods

21 Medical Images Medical images from computed tomog-raphy (CT) and magnetic resonance (MR) imaging of the sixpatients with hydrocephaluswere investigated retrospectivelybefore (Figure 2) and after (Figure 3) a ventriculoperitonealshunt operation In this study only the geometry of the brainis of concern to the strain level quantification Thereforeboth CT and Resonance MR Images are acceptable Thisstudy was approved by the local research ethical committeeof Karolinska University Hospital Stockholm County

22 Image Registration to Obtain Displacement Field Imageregistration aims at finding a displacement field between afixed image andmoving image to align both images as accurateas possible [9] A nonlinear registration method the Diffeo-morphic Demons (DD) algorithm [9] implemented in theopen-source software Slicer 3D was used in this study whichhas been used to quantify the displacement field occurredduring decompressive craniotomy in previous studies [1011] The core components of the DD registration algorithmincluding similarity measures and the regularization modelwhich will be presented

Given a fixed image 119865(sdot) and a moving image 119872(sdot)intensity-based image registration is posed as an optimizationproblem that aims at finding a spatial mapping that will alignthe moving image to the fixed image The transformation119904(sdot) R119863 rarr R119863 119901 997891rarr 119904(119901) models the spatial mappingof a particular point 119901 from the fixed image space to themoving image space [12]The similarity criterion 119864sim(119865119872∘119904) measures the quality or the goodness of the matching of agiven transformation In this study binary images are usedand the sum of the square difference (SSD) suffices for ourapplication as defined as

119864sim (119865119872 ∘ 119904) =1

2119865 minus119872 ∘ 119904

2

=1

2Ω119901

sum

119901isinΩ119901

1003816100381610038161003816119865 (119901) minus119872(119904 (119901))1003816100381610038161003816

2

(1)

where 119872 ∘ 119904 represents the morphed moving image Ω119901is

the region of overlap between 119865 and119872∘ 119904 The problem now

becomes an optimization problem to find a transformation119904 over a given space that minimizes the similarity energyfunction 119864sim

In order to end up with a global minimization of awell-posed criterion the similarity energy function in (1)was modified by introducing an auxiliary variable (iecorrespondence c) in the registration process [13] The intro-duction of this auxiliary variable 119888 decouples the complexminimization into simple and efficient steps by alternatingoptimization over 119888 and 119904 Considering a Gaussian noise onthe displacement field the global energy then becomes [12]

119864 (119888 119904) =1

1205902

119894

119864sim (119865119872 ∘ 119904) +1

1205902119909

dist (119904 119888)2

+1

1205902

119879

119864reg (119904)

(2)

where 120590119894accounts for the noise on the image intensity 120590

119909

accounts for a spatial uncertainty of the correspondencesand 120590

119879controls the amount of regularization that is needed

dist(119904 119888) = 119888 minus 119904 and 119864reg(119904) = nabla119904Themodel assumes that so-called ldquodemonsrdquo at every voxel

from the fixed image are applying forces that push the voxelsof the moving image to match up with the fixed imageThe transformation 119904 is driven by a ldquodemons forcerdquo derivedfrom the assumption of image intensity conservation [14]The original demons algorithm has beenmodified and allowsretrieving small and large dense displacement field defined as[15]

U (119901) =119865 (119901) minus119872(119904 (119901))

1003817100381710038171003817nabla119865 (119901)1003817100381710038171003817

2+ 1205722(119865 (119901) minus119872(119904 (119901)))

2nabla119865 (119901) (3)

where U(119901) is the displacement at point 119901 defined in thefixed image 120572 is a positive homogenization factor Theupdated unconstrained dense displacement field is computedbased on an optical flow computation at each voxel at everyiteration The resulting updated field is added to the globaldeformation field and the displacement field is regularizedby applying a Gaussian smoothing filter

In order to solve the displacement field U the globalenergy defined in (2) needs to be minimized This energyfunction allows the whole optimization procedure to bedecoupled into two simple steps The first step solves for thecorrespondence 119888 by optimizing (11205902

119894)119864sim + (1120590

2

119909)119904 minus 119888

2

with respect to 119888 and with s given The second stepsolves for the regularization by optimizing (11205902

119909)119904 minus 119888

2+

(11205902

119879)119864reg(119904) with respect to s and 119888 given [9]

The transformation 119904 in the original Demonsmethod [14]is not constrained and does not provide diffeomorphic trans-formations A diffeomorphic extension to the demons frame-work was proposed by adapting the optimization procedureto a space of diffeomorphic transformations It is performedby using an intrinsic update step which computes the vectorfield exponentials of the Lie group of diffeomorphims (see[9 12] for details)

23 Strain Level Quantification From the diffeomorphicDemons registration the transformation s corresponding

Journal of Applied Mathematics 3

to a displacement field U(X) is obtained from (3) Thedisplacement field is defined on every voxel in the fixedimage and morphs the fixed image to the moving image Thestrain tensor could then be calculated based on the theoryof continuum mechanics [16] A Lagrangian reference framewas assumed at the fixed image space

119904 x =X+U (X) (4)

The displacement field U(X) = [119880119883 119880119884 119880119885] for each voxel

obtained from the image registration gives a reasonablealignment to bring the fixed image at point119901with coordinatesX(119883 119884 119885) to its corresponding point in the moving imageat coordinates x(119909 119910 119911) [9] The displacement gradient isdefined as [16]

grad (U) = 119889U119889X=((

(

119889119880119883

119889119883

119889119880119883

119889119884

119889119880119883

119889119885

119889119880119884

119889119883

119889119880119884

119889119884

119889119880119884

119889119885

119889119880119885

119889119883

119889119880119885

119889119884

119889119880119885

119889119885

))

)

(5)

The deformation gradient is defined as

F = 119889x119889X=119889 (X + U (X))119889X

= I + 119889U119889X= I + grad (U) (6)

The Lagrangian finite strain tensor E is defined as

E=12(FTFminusI) (7)

The Lagrangian finite strain tensor E is a 3 times 3 symmetrictensor representing the deformation of a point Differentscalar indices that can be derived from the tensors especiallythe principal strainswhich represent themaximumandmini-mumnormal strains experienced at a point are characterizedby the tensor eigenvalues

Strain level represents the deformation of brain tissue dueto the occurrence of a disease In order to quantify the braintissue deformation a dense displacement field representingthe brain tissue motion from healthy stage to hydrocephalusstage is needed For this the main premise is the imagesacquired at two distinct states at healthy state and at diseasedstate Since the brain images of the patient under healthy statewere not available we proposed a strategy to recover a healthybrain for a specific patient by combing the information fromthe diseased brain geometry together with the brain atlas bya nonlinear image registration method [10]

The lateral ventricles and brain tissue were firstly manu-ally segmented from the recovered healthy brain and hydro-cephalic brain images A rigid registration step was first usedto centre the images about the same point which were theinput volumes to nonlinear registration process In principlethe segmented healthy brain should be chosen as the fixedimage so as to capture the motion of brain tissue fromthe healthy to hydrocephalic state In this study since theregistration performs better from healthy brain to the hydro-cephalic brain image we chose the hydrocephalic brain image

as the fixed image and the obtained displacement field wasthen inverted by an itkIterativeInverseDisplacementFieldIm-ageFilter algorithm implemented in ITK (see the descriptionfor the approach on ITKrsquos website httpwwwitkorg) Thefinal obtained displacement for further processing is theinverted displacement field defined on the recovered healthybrain space which represents the structural brain change thatoccurred from the healthy to hydrocephalus state

From the obtained displacement field a quantitativedescription of the nerve tissue deformation that occurredduring hydrocephalus development can be derived in theform of the finite Lagrange strain tensor [16 17] Differentscalar indices can be derived from the strain tensor especiallythe 1st principal strain defined as the maximum eigenvalueof Lagrange strain tensor E representing the maximumstretching was used to describe the stretching of brain tissue

24 Axonal Fiber Tract Extraction and Healthy Brain ShapefromAtlas Froma series of diffusionweighted (DW) imagesthe effective diffusion tensor D can be estimated using therelationship between the measured signal intensity at eachvoxel and the applied magnetic field gradient sequence [1819] The diffusion tensor is symmetric (ie 119863

119894119895= 119863119895119894) and

thus it contains only six unique values Given this at leastsix noncollinear diffusion gradient directions are required todetermine the diffusion tensor [20 21]

Once the diffusion tensor D is determined differentindices can be derived to provide information for each voxelin the imageThediffusion tensor is a real symmetric second-order tensor Mathematically this entails a linear rotationof the diffusion tensor to diagonalize it thereby setting off-diagonal elements to zero [22]

D = [[

119863119909119909119863119909119910119863119909119911

119863119910119909119863119910119910119863119910119911

119863119911119909119863119911119910119863119911119911

]

]

= [e1e1e3] [

[

12058210 0

0 12058220

0 0 1205823

]

]

[e1e1e3]119879

(8)

where 1205821 1205822 and 120582

3are eigenvalues (120582

1gt 1205822gt 1205823) and

the corresponding eigenvectors are e1 e2 e3The eigenvalues

of the diffusion tensor provide diffusion coefficients alongthe orientations defined by its respective eigenvectors [22]1205821represents the greatest diffusion value along a fiber axis

denoted by the direction vector e1 1205822 1205823represent the

diffusion value along two axes perpendicular to e1 Because

D is symmetric and positive definite its three eigenvectors(principal coordinate directions) e

1 e2 and e

3are orthogonal

[23]The Fractional Anisotropy (FA) is the most commonly

used anisotropy measure and is a normalized expression ofthe tensor eigenvalues [24] according to

FA = radic3 ((1205821minus 120582)2

+ (1205822minus 120582)2

+ (1205823minus 120582)2

)

2 (1205822

1+ 1205822

2+ 1205822

3)

(9)

where 120582 is the mean of the eigenvalues of the diffusion tensor


Table 1 Summary of six hydrocephalic patients including age clinical symptoms before treatment with a ventriculoperitoneal shuntpostoperative image performance and outcome evaluated in days after the operation

Patientage Clinical symptomsGait disturbance Cognitive deficit Incontinence Improved outcomedays after treatment

KP45 Yes Yes No Yes63KA47 Yes Yes No Yes96KB72 Yes Yes Yes Yes30KF70 No Yes Yes Yes27PN57 No Yes No Yes19SB71 Yes Yes No No21

Once the fractional anisotropy (FA) and the preferreddiffusion direction have been created it is possible to performfiber tracking or diffusion tensor tractography Streamlinetractography is one of the methods for fiber tract estimation[25 26] Streamline tractography uses the maximum orien-tation described by the eigenvector e

1associated with the

largest eigenvalue 1205821 as an estimate of local tract orientation

It assumes that 1205821associated with the maximum eigenvector

is the main fiber directionIn this study whitematter fiber tracts were extracted from

ICBMDTI-81 atlas provided by the international consortiumfor brainmapping (ICBM) [27]TheT2-weighted image fromthe same dataset which represents an average adult brainwas used for recovery of a healthy brain as described in aprevious section The atlas is based on probabilistic tensormaps obtained from 81 normal adults ranging from 18 to59 years of age From the diffusion tensor imaging theFractional Anisotropy (FA) could be calculated which is ascalar value representing the anisotropic propertyThe color-coded FA value calculated from the DTI tensor is displayedin (Figure 1 left) where the color reflects the white matterorientation (red (right-left) green (anterior-posterior) andblue (superior-inferior)) The fiber tracts which pass throughthe predefined region of interests (ROI) are CC and CS fibertracts The corresponding sagittal and coronal sections ofthe T2-weighted images from the same dataset are shown inFigure 1 upper row on the right

Streamline method [26 28] was used to extract whitematter fiber tracts The final extracted white matter tracts inthe whole brain contain the polylines with a correspondingdiffusion tensor at each fiber point (Figure 1 lower row leftand middle) from which the CC and CS fiber tracts wereisolated (Figure 1 lower row right) The obtained fiber tractsfrom the atlas were then adapted to the patient brain byapplying the obtained displacement field which represent thecranial structural shape difference between the patient andatlasThis gave the fiber tracts for the recovered healthy brainfor a specific patient

3 Results

31 Patient Information The average age of the six patientswith hydrocephalus was calculated to 60 years Clinicalsymptoms among the six patients showed gait disturbances infour cases and cognitive deficits in all patients while only two

had urinary incontinence (Table 1) Five patients improvedin their clinical symptoms individually evaluated between 19and 96 days after the shunt operation while the sixth patientdid not probably due to the presence of dementia

32 Strain Level at Ventricular Wall CC and CS Fiber TractsThe increased strain levels in the six patients are presented forpreoperative (Figure 2) and postoperative stages (Figure 3)Before and after the operation all patients had a substantialdilatation of the lateral ventricles in various degrees althoughin less degree after the treatment The ventricular dilatationcaused increased strain levels not only in the ventricularwalls ranging from 99 to 333 but also in the braintissue surrounding the ventricles in different degrees As aconsequence the increased strain levels found in the CC fibertracts ranged from58 to 139and in theCSfiber tracts from29 to 74 among the patients (Figure 7)

After the ventriculoperitoneal shunt had been installedthe lateral ventricles decreased although not to an extentas could be expected after such an operation Thus thepostoperative strain levels of the ventricular walls decreasedin all six patients ranging from 9 to 60 based on thepreoperative values (Figure 7)

In a similar way although decreased after the shuntoperation the strain levels in the CC and CS fiber tracts werestill on a surprisingly high level ranging from 9 to 56 forCC fiber tracts and from 5 to 20 in the CS fiber tracts(Figure 7) The strain level data showed normal distributionand therefore a paired sample t-test was used to comparethe CC and CS fiber strain levels The results were evaluatedby a 1-tailed t-test at 95 confidence level It was found thatthe strain levels in CC fiber tracts are significantly higherthan those of CS fiber tracts for both pre- (119875 lt 005) andpostcraniotomy period (119875 lt 005) This indicates a morestretched scenario for the CC fiber tracts

The trend was that those patients with the highest ven-tricular dilatation and increased strain levels also showed thehighest strain levels in the CC and CS fiber tracts both beforeand after treatment A similar trend was found between CCand CS fiber tracts Also since the CC fiber tracts are closerto the expanded ventricles it seems logical that the increasedstrain levels are higher in these fibers compared to thosefound in the CS fiber tracts which are anatomically locatedfurther away from the ventricles


2

1

Figure 1 Upper row color-coded FAmap calculated from the ICBMDTI-81 atlas (left) Sagittal and coronal sections of T2-weighted imagesof the atlas (right) Lower row extracted fiber tracts from the ICBM DTI-81 atlas in the whole brain Rear view (left) and top view of theextracted fiber tracts in the whole brain (middle) Note the isolated CC and CS tracts (right)

PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)

Figure 2 Overview of the 6 patients at their preoperative stages Each row from top to bottom represents the medical images used fornonlinear image registration from which the strain level was quantified in the entire brain and the calculated strain levels are illustrated atcoronal sections and the ventricular walls At bottom the strain level at CS and CC fiber tracts is presented

The individual follow-up time ranged from 19 to 96 daysThe increased strain levels should be expected to decreasemore in those patients with longer follow-up times comparedto those with a short follow-up time However there was nosuch pattern found among the patients in this study

The result is exemplified with one of the six patients(patient PK) which is presented in more detail (Figure 4)As defined previously the dilatation of the lateral ventricles

causes distortion of most of the surrounding axonal fibertracts and where the CC and CS fiber tracts are addressedhere Thus during the hydrocephalus development the CSfiber tracts were displaced outwards by the arrows represent-ing the displacement vector obtained from image registration(Figure 4 upper row)Themaximumdisplacement evaluatedwas 107mm (Figure 4 upper row left) The closer to thelateral ventricles the larger the displacement found in the CS


PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)

Figure 3 Overview of the 6 patients at their postoperative stages Each row from top to bottom represents the medical images used fornonlinear image registration from which the strain level was quantified in the entire brain and the calculated strain levels are illustrated atcoronal sections and the ventricular walls At bottom the strain level at CS and CC fiber tracts is presented

Figure 4 Upperrow displacement of the CC and CS fibertracts during hydrocephalus development overplayed with the pre-operative brain image at coronal cross-section (left) and sagittalcross-section (right) The axonal tracts with white color representthe tracts at the healthy stage the purple arrow shows the displace-ment occurred during hydrocephalus development with the arrowlength in proportion to the displacementmagnitude and the coloredfiber tracts represent the distorted tracts at postoperative stage withcolor scale legend indicating magnitude of displacement Lowerrow the 1st principal strain levels at the CS and the CC fiber tractsNote that the color scale for the CS and CC fiber tracts is differentin order to have a better illustration

fiber tracts This resulted into compression of the CS fibertracts close to the lateral ventricles For axonal CCfiber tractsthe displacement field showed a more dispersed pattern witha maximum value up to 154mm located at the anterior hornof lateral ventricle (Figure 4 upper row right)

The strain levels for both CS andCCfiber tracts (Figure 4lower row) showed that the CS fiber tracts presented a lowervalue compared to that found in the CC fiber tractsThus themaximum strain level increase in the CS fiber tracts was 05or 50 while that in the CC fiber tracts was as high as 12 or120

When comparing the average strain levels found in theventricles the CC and CS fibers the trend seemed clear thatthe longer distance from the ventricles the less the strain levelsboth before and after the neurosurgical procedure (Figure 7)Hence the difference was the least in fiber tracts in theperiphery of the brain tissue for both pre-and postoperativestages

To evaluate the strain level at brain tissue with furtherdistance from the lateral ventricles the original ventricularwalls were scaled outwards until it reached the cranialboundary and which was taken as the maximum distancefor evaluation Between the original normal ventricles andthe maximum dilated ventricles ten different-sized lateralventricles were created by equally interpolating the twoextreme cases The obtained different-sized ventricles are


Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB

Distance from ventricle Distance from ventricle Distance from ventricle


Figure 5 Relationship between the distance from the lateral ventricles and the mean strain level of brain tissue The picture on the leftillustrates the interpolated different-sized ventricles which were used for average strain level calculation at different distances A decreasedstrain level was found after the neurosurgical procedure in all of the 6 patients in various degrees

illustrated in Figure 5 left with only three ventricles shownaiming at a better illustration The obtained interpolatedventricles were then used to resample the strain levels in theentire brain and based on which the average strain levels atdifferent distances to the ventricles were calculated As thebrain sizes are different across the patients in the presentstudy the distance from the ventricles was normalized foreach patient by dividing the brain size to the largest distancein that particular patientThis yielded distances range from 0that is brain tissue adjacent to the ventricles to 1 that is theoutermost brain tissue which were binned in an increment of01 A profile of the strain levels in brain tissue as a functiondistance from the ventricles could then be plotted Thusthe relationship between the distance from the ventriclesand the average strain levels showed a logarithmic profilein all patients both at the pre- and postoperative periods(Figure 5) The evaluation showed decreasing strain levelswith increasing distances from the ventricles for each patientalthough in different degrees

An earlier study has shown a logarithmic correlationbetween the regional cerebral blood flow (rCBF) and thedistance from the lateral ventricles in hydrocephalic patients[1] Thus the displacement and increased strain levels pre-sented in this study may have a substantial impact on therCBF among these patients By combining the rCBF valuesfrom that earlier study at a distance from the lateral ventricles

together with the increased strain levels found in this study atthe same distance the relationship between CBF and strainlevels was plotted (Figure 6) showing a linear correlationbetween the increased strain level in this study and the rCBF(119903 = 09905 119875 lt 00001) from the earlier study [1] Thus asillustrated at one point in the figure by reducing the increasedstrain level from preoperative 176 to postoperative 130shown on the horizontal axis the rCBF increased from about65 to 114 mL100mgmin (ΔrCBF in Figure 6)

4 Discussions

All six patients had increased displacement as well asprofound increased strain levels in the surrounding braintissue both before and although less pronounced after theneurosurgical procedure Also all patients were defined withcognitive disturbances Incontinence was defined in four ofthe patients and without any correlation to displacement orstrain level However the neurosurgical procedure resulted ina definite reduction of both displacement and strain level Inthose two cases with the lowest increase of displacement andstrain level there was no gait ataxia This study to the bestof our knowledge is the first to provide a quantitative viewregarding the stretching of ventricular wall CC and CS tractsin hydrocephalic patients Further a close correlation wasfound to exist between the increased strain level and the rCBF


0

5

10

15

20

25

05 1 15 2Mean strain level

119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost

Figure 6 A linear relationship (119903 = 09905 119875 lt 00001) wasfound to exist between the preoperative strain levels (x-axis where1 = 100 strain level increase) and the regional cerebral blood flow(rCBF) (black dots data from a previous publication [1] One pointwas illustrated to show improved rCBF due to shunt operation (lightdots ΔrCBF)

The presented results provide direct evidence that Hakimtriad could to some extent be explained by the increasedstrain levels

Those patients with the largest expansion of the ventriclesshowed the most profound strain level increase in the fibertracts of CC and CS while those with smaller ventricularincrease presented a more modest strain level increase Thismay indicate that patients with larger ventricles should beexpected to present a worse outcome However this was notthe case in this study Instead it is suggested that largerdisplacement of the ventricles will influence the surroundingbrain tissuemore distant compared to those patients with lessexpanded ventricles Also the larger the ventricle expansionthe more severe the strain level in CC fibers tracts andthis could tentatively interfere negatively with connectionsbetween the two hemispheres thereby prolonging the clinicalimprovement for such patients Since the CS fiber tracts areanatomically located further away from the ventricles theyhad a less pronounced strain level increase which was to beexpected

Experimental studies of axonal stretching have showna clear correlation between the strain level and axonalfunction An increased strain level of 5 was found toalter neuronal function while 10 may cause cell deathin rapid axonal stretching models [29] Bain and Meaney(2000) demonstrated in an animal model that a strain levelof approximately 21 will elicit electrophysiological changeswhile a strain of approximately 34will causemorphologicalsigns of damage to the white matter [30] Under slow loadingrates however axons could tolerate much higher strain levelsand with stretch of up to twice of their original length

(a strain of 100) with no evidence of damage [31] Usinga model of sciatic nerve stretch Fowler et al reported thateven minimal tension if maintained for a significant amountof time may result in loss of neuronal function [32] Basedon these it should be expected that an increased strain levelin both CC and CS fiber tracts in hydrocephalus patientsfound in the present study very well interferes with thenormal conductivity in the axons hence also interferes withthe patientsrsquo rehabilitation When the strain level increasereaches a critical point or threshold it may very well initiatea biochemical response which may further alter the CC andCS fiber tracts From a clinical aspect the sustained increaseddisplacement and increased strain levels of axons may alsoresult in gait disturbances due to the potential interferencewith the long corticospinal fiber tracts connecting motorand sensory areas with the spinal cord as well as thosein the corpus callosum connecting the electrophysiologicalfunctions of both hemispheres with each other [33]

The displacement and strain level in hydrocephaluspatients belong to a static or semistatic increase therebyallowing the fiber tracts to accept the changes during a longerperiodThismay favor the rehabilitation even if the increasedstrain level is substantial due to the very low deformationrate [31] However of special interest are elderly peoplewho already have an atrophied brain tissue and which maysuffer more than those in employment age Many of thehydrocephalus patients improve after ventriculoperitonealshunt treatment However there is no treatment for thosewho do not improve due to a lack of explanation of the causesIt was suggested that the increased strain levels may causea disturbance in the electrophysiological function in thesefiber tracts and consequently a tailoredmatrix of conductiveorganic bioelectrodes could potentially be implanted in thearea of corpus callosum fiber tracts thereby counterbalancingthe electrophysiological dysfunction as proposed in a previ-ous study [33]

In the present study all patients had indeed a reduction inthe strain level postoperatively However the reduced strainlevel was still on a surprisingly high level and could thereforehave a negative influence on the anatomy and histology ofthe nervous tissue together with a sustained alteration in theconductivity of the axons in thewhitematter It seems that thereduction of rCBF clearly correlated with that of increasedstrain levels Even a small reduction in rCBF may influencethe rehabilitation especially among the vulnerable elderlypatients and which should be taken into consideration

5 Conclusions

A numerical method based on nonlinear image registrationwas used to quantify the stretching of ventricular wall corpuscallosum and corticospinal axonal fiber tracts in six patientswith hydrocephalus both before and after shunt operationThese data provide new insight into the mechanical cascadeof events due to tissue stretching thereby to provide us withmore knowledge into understanding of the role of braintissue and axonal stretching in some of the hydrocephalusclinical symptoms Moreover a linear correlation was found


PK KA KB KF PN SBStrain_Pre 103 333 161 99 140 222

Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)

Figure 7 Diagram of average strain levels at the lateral ventricle (a) CC fiber (b) and CS fiber (c) for all the 6 patients The horizontal axis isthe patientrsquos identification (id) and the vertical axis shows the strain levels for the corresponding patient at precraniotomy stage (Strain Prewith darker gray color) and postcraniotomy stage (Strain Post with lighter gray color)The exact strain level values are presented in the lowertable

to exist between strain level and the rCBF The combinationof increased displacement of the ventricular walls sustainedincreased strain levels and derangement in electrophysiologyactivity in the white matter together with a reduced rCBFmay under certain circumstances has a profound impact onthe outcome following hydrocephalus

Conflict of Interests

The authors declare that they have no conflict of interests

Acknowledgment

The present study was supported by research funds from theRoyal Institute of Technology Stockholm Sweden

References

[1] S Momjian B K Owler Z Czosnyka M Czosnyka APena and J D Pickard ldquoPattern of white matter regionalcerebral blood flow and autoregulation in normal pressurehydrocephalusrdquo Brain vol 127 no 5 pp 965ndash972 2004

[2] B K Owler and J D Pickard ldquoNormal pressure hydrocephalusand cerebral blood flow a reviewrdquo Acta Neurologica Scandinav-ica vol 104 no 6 pp 325ndash342 2001

[3] J M Miller and J P McAllister ldquoReduction of astrogliosis andmicrogliosis by cerebrospinal fluid shunting in experimentalhydrocephalusrdquo Cerebrospinal Fluid Research vol 4 article 52007

[4] M Mataro M Matarın M A Poca et al ldquoFunctional andmagnetic resonance imaging correlates of corpus callosum innormal pressure hydrocephalus before and after shuntingrdquoJournal of Neurology Neurosurgery and Psychiatry vol 78 no4 pp 395ndash398 2007

[5] S Roricht B U Meyer C Woiciechowsky and R LehmannldquoCallosal and corticospinal tract function in patients withhydrocephalus a morphometric and transcranial magneticstimulation studyrdquo Journal of Neurology vol 245 no 5 pp 280ndash288 1998

[6] E Hattingen A Jurcoane J Melber et al ldquoDiffusion ten-sor imaging in patients with adult chronic idiopathic hydro-cephalusrdquo Neurosurgery vol 66 no 5 pp 917ndash924 2010

[7] T Hattori T Yuasa S Aoki et al ldquoAltered microstructurein corticospinal tract in idiopathic normal pressure hydro-cephalus comparison with Alzheimer disease and Parkinsondisease with dementiardquo The American Journal of Neuroradiol-ogy vol 32 no 9 pp 1681ndash1687 2011

[8] A Marmarou M Bergsneider P Klinge N Relkin and P ML Black ldquoINPH guidelines part III the value of supplementalprognostic tests for the preoperative assessment of idiopathicnormal-pressure hydrocephalusrdquoNeurosurgery vol 57 no 3 pS2 2005

[9] T Vercauteren X Pennec A Perchant and N Ayache ldquoDiffeo-morphic demons efficient non-parametric image registrationrdquoNeuroImage vol 45 no 1 pp S61ndashS72 2009

[10] H von Holst X Li and S Kleiven ldquoIncreased strain levels andwater content in brain tissue after decompressive craniotomyrdquoActa Neurochirurgica vol 154 no 9 pp 1583ndash1593 2012

[11] X Li Finite Element and Neuroimaging Techniques to ImproveDecision-Making in Clinical Neuroscience Royal Institute ofTechnology (KTH) 2012

[12] T Vercauteren X Pennec E Malis A Perchant and NAyache ldquoInsight into efficient image registration techniques andthe demons algorithmrdquo in Information Processing in MedicalImaging pp 495ndash506 Springer New York NY USA 2007

[13] P Cachier E Bardinet D Dormont X Pennec and NAyache ldquoIconic feature based nonrigid registration the PASHAalgorithmrdquo Computer Vision and Image Understanding vol 89no 2-3 pp 272ndash298 2003

[14] J PThirion ldquoImagematching as a diffusion process an analogywith Maxwellrsquos demonsrdquo Medical Image Analysis vol 2 no 3pp 243ndash260 1998

[15] P Cachier X Pennec and N Ayache Fast Non Rigid Match-ing by Gradient Descent Study and Improvements of theldquoDemonsrdquo Algorithm Rapport de Recherche-Institut Nationalde Recherche en Informatique et en Automatique 1999

[16] G A Holzapfel Nonlinear Solid Mechanics A ContinuumApproach for Engineering JohnWiley amp SonsWest Sussex UK2000


[17] J Gee T Sundaram I Hasegawa H Uematsu and H HatabuldquoCharacterization of regional pulmonarymechanics from serialmagnetic resonance imaging datardquoAcademic Radiology vol 10no 10 pp 1147ndash1152 2003

[18] P J Basser and D K Jones ldquoDiffusion-tensor MRI theoryexperimental design and data analysismdasha technical reviewrdquoNMR in Biomedicine vol 15 no 7-8 pp 456ndash467 2002

[19] P J Basser J Mattiello and D Lebihan ldquoEstimation of theeffective self-diffusion tensor from the NMR spin echordquo Journalof Magnetic Resonance B vol 103 no 3 pp 247ndash254 1994

[20] T L Chenevert ldquoPrinciples of diffusion-weighted imaging(DW-MRI) as applied to body imagingrdquo in Diffusion-WeightedMR Imaging pp 3ndash17 2010

[21] P J Basser and C Pierpaoli ldquoA simplified method to measurethe diffusion tensor from seven MR imagesrdquo Magnetic Reso-nance in Medicine vol 39 no 6 pp 928ndash934 1998

[22] D Le Bihan J F Mangin C Poupon et al ldquoDiffusion tensorimaging concepts and applicationsrdquo Journal of Magnetic Reso-nance Imaging vol 13 no 4 pp 534ndash546 2001

[23] P J Basser J Mattiello and D LeBihan ldquoMR diffusion tensorspectroscopy and imagingrdquo Biophysical Journal vol 66 no 1pp 259ndash267 1994

[24] P J Basser ldquoInferring microstructural features and the physi-ological state of tissues from diffusion-weighted imagesrdquo NMRin biomedicine vol 8 no 7-8 pp 333ndash344 1995

[25] S Mori and J Zhang ldquoPrinciples of diffusion tensor imagingand its applications to basic neuroscience researchrdquoNeuron vol51 no 5 pp 527ndash539 2006

[26] S Mori B J Crain V Chacko and P van Zijl ldquoThree-dimensional tracking of axonal projections in the brain bymagnetic resonance imagingrdquo Annals of Neurology vol 45 no2 pp 265ndash269 1999

[27] S Mori K Oishi H Jiang et al ldquoStereotaxic white matteratlas based on diffusion tensor imaging in an ICBM templaterdquoNeuroImage vol 40 no 2 pp 570ndash582 2008

[28] P J Basser S Pajevic C Pierpaoli J Duda and A AldroubildquoIn vivo fiber tractography using DT-MRI datardquo MagneticResonance in Medicine vol 44 no 4 pp 625ndash632 2000

[29] Z Yu and B Morrison ldquoExperimental mild traumatic braininjury induces functional alteration of the developing hip-pocampusrdquo Journal of Neurophysiology vol 103 no 1 pp 499ndash510 2010

[30] A C Bain andD FMeaney ldquoTissue-level thresholds for axonaldamage in an experimental model of central nervous systemwhite matter injuryrdquo Journal of Biomechanical Engineering vol122 no 6 pp 615ndash622 2000

[31] M D Tang-Schomer A R Patel P W Baas and D HSmith ldquoMechanical breaking of microtubules in axons duringdynamic stretch injury underlies delayed elasticity microtubuledisassembly and axon degenerationrdquo FASEB Journal vol 24no 5 pp 1401ndash1410 2010

[32] S S Fowler J P Leonetti J C Banich J M Lee RWurster andM R I Young ldquoDuration of neuronal stretch correlates withfunctional lossrdquo OtolaryngologymdashHead and Neck Surgery vol124 no 6 pp 641ndash644 2001

[33] H von Holst ldquoOrganic bioelectrodes in clinical neurosurgeryrdquoBiochimica et Biophysica Acta 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


quantitative data of the axonal stretching level and how themechanical stretching may influence the axonal functionsremains unclear until now A quantitative data should alsoprovide us more insight to understand the involvement ofaxonal fiber stretching in clinical symptoms

Thus the aim of the present study was

(i) to analyze the displacement and strain levels inpatients with hydrocephalus by using a numericalsimulation method focusing on the ventricular wallsCC and CS fiber tracts before and after shunt opera-tions and

(ii) to estimate the regional cerebral blood flow aftertreatment and the correlation to the strain level afterthe neurosurgical procedure

2 Material and Methods

21 Medical Images Medical images from computed tomog-raphy (CT) and magnetic resonance (MR) imaging of the sixpatients with hydrocephaluswere investigated retrospectivelybefore (Figure 2) and after (Figure 3) a ventriculoperitonealshunt operation In this study only the geometry of the brainis of concern to the strain level quantification Thereforeboth CT and Resonance MR Images are acceptable Thisstudy was approved by the local research ethical committeeof Karolinska University Hospital Stockholm County

22 Image Registration to Obtain Displacement Field Imageregistration aims at finding a displacement field between afixed image andmoving image to align both images as accurateas possible [9] A nonlinear registration method the Diffeo-morphic Demons (DD) algorithm [9] implemented in theopen-source software Slicer 3D was used in this study whichhas been used to quantify the displacement field occurredduring decompressive craniotomy in previous studies [1011] The core components of the DD registration algorithmincluding similarity measures and the regularization modelwhich will be presented

Given a fixed image 119865(sdot) and a moving image 119872(sdot)intensity-based image registration is posed as an optimizationproblem that aims at finding a spatial mapping that will alignthe moving image to the fixed image The transformation119904(sdot) R119863 rarr R119863 119901 997891rarr 119904(119901) models the spatial mappingof a particular point 119901 from the fixed image space to themoving image space [12]The similarity criterion 119864sim(119865119872∘119904) measures the quality or the goodness of the matching of agiven transformation In this study binary images are usedand the sum of the square difference (SSD) suffices for ourapplication as defined as

119864sim (119865119872 ∘ 119904) =1

2119865 minus119872 ∘ 119904

2

=1

2Ω119901

sum

119901isinΩ119901

1003816100381610038161003816119865 (119901) minus119872(119904 (119901))1003816100381610038161003816

2

(1)

where 119872 ∘ 119904 represents the morphed moving image Ω119901is

the region of overlap between 119865 and119872∘ 119904 The problem now

becomes an optimization problem to find a transformation119904 over a given space that minimizes the similarity energyfunction 119864sim

In order to end up with a global minimization of awell-posed criterion the similarity energy function in (1)was modified by introducing an auxiliary variable (iecorrespondence c) in the registration process [13] The intro-duction of this auxiliary variable 119888 decouples the complexminimization into simple and efficient steps by alternatingoptimization over 119888 and 119904 Considering a Gaussian noise onthe displacement field the global energy then becomes [12]

119864 (119888 119904) =1

1205902

119894

119864sim (119865119872 ∘ 119904) +1

1205902119909

dist (119904 119888)2

+1

1205902

119879

119864reg (119904)

(2)

where 120590119894accounts for the noise on the image intensity 120590

119909

accounts for a spatial uncertainty of the correspondencesand 120590

119879controls the amount of regularization that is needed

dist(119904 119888) = 119888 minus 119904 and 119864reg(119904) = nabla119904Themodel assumes that so-called ldquodemonsrdquo at every voxel

from the fixed image are applying forces that push the voxelsof the moving image to match up with the fixed imageThe transformation 119904 is driven by a ldquodemons forcerdquo derivedfrom the assumption of image intensity conservation [14]The original demons algorithm has beenmodified and allowsretrieving small and large dense displacement field defined as[15]

U (119901) =119865 (119901) minus119872(119904 (119901))

1003817100381710038171003817nabla119865 (119901)1003817100381710038171003817

2+ 1205722(119865 (119901) minus119872(119904 (119901)))

2nabla119865 (119901) (3)

where U(119901) is the displacement at point 119901 defined in thefixed image 120572 is a positive homogenization factor Theupdated unconstrained dense displacement field is computedbased on an optical flow computation at each voxel at everyiteration The resulting updated field is added to the globaldeformation field and the displacement field is regularizedby applying a Gaussian smoothing filter

In order to solve the displacement field U the globalenergy defined in (2) needs to be minimized This energyfunction allows the whole optimization procedure to bedecoupled into two simple steps The first step solves for thecorrespondence 119888 by optimizing (11205902

119894)119864sim + (1120590

2

119909)119904 minus 119888

2

with respect to 119888 and with s given The second stepsolves for the regularization by optimizing (11205902

119909)119904 minus 119888

2+

(11205902

119879)119864reg(119904) with respect to s and 119888 given [9]

The transformation 119904 in the original Demonsmethod [14]is not constrained and does not provide diffeomorphic trans-formations A diffeomorphic extension to the demons frame-work was proposed by adapting the optimization procedureto a space of diffeomorphic transformations It is performedby using an intrinsic update step which computes the vectorfield exponentials of the Lie group of diffeomorphims (see[9 12] for details)

23 Strain Level Quantification From the diffeomorphicDemons registration the transformation s corresponding



119904 x =X+U (X) (4)



grad (U) = 119889U119889X=((

(

119889119880119883

119889119883

119889119880119883

119889119884

119889119880119883

119889119885

119889119880119884

119889119883

119889119880119884

119889119884

119889119880119884

119889119885

119889119880119885

119889119883

119889119880119885

119889119884

119889119880119885

119889119885

))

)

(5)


F = 119889x119889X=119889 (X + U (X))119889X

= I + 119889U119889X= I + grad (U) (6)


E=12(FTFminusI) (7)







119894119895= 119863119895119894) and



D = [[

119863119909119909119863119909119910119863119909119911

119863119910119909119863119910119910119863119910119911

119863119911119909119863119911119910119863119911119911

]

]

= [e1e1e3] [

[

12058210 0

0 12058220

0 0 1205823

]

]

[e1e1e3]119879

(8)

where 1205821 1205822 and 120582


1gt 1205822gt 1205823) and






1 e2 and e

3are orthogonal



FA = radic3 ((1205821minus 120582)2

+ (1205822minus 120582)2

+ (1205823minus 120582)2

)

2 (1205822

1+ 1205822

2+ 1205822

3)

(9)













3 Results








2

1


PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)






PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)








Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB







4 Discussions



0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 x =X+U (X) (4)



grad (U) = 119889U119889X=((

(

119889119880119883

119889119883

119889119880119883

119889119884

119889119880119883

119889119885

119889119880119884

119889119883

119889119880119884

119889119884

119889119880119884

119889119885

119889119880119885

119889119883

119889119880119885

119889119884

119889119880119885

119889119885

))

)

(5)


F = 119889x119889X=119889 (X + U (X))119889X

= I + 119889U119889X= I + grad (U) (6)


E=12(FTFminusI) (7)







119894119895= 119863119895119894) and



D = [[

119863119909119909119863119909119910119863119909119911

119863119910119909119863119910119910119863119910119911

119863119911119909119863119911119910119863119911119911

]

]

= [e1e1e3] [

[

12058210 0

0 12058220

0 0 1205823

]

]

[e1e1e3]119879

(8)

where 1205821 1205822 and 120582


1gt 1205822gt 1205823) and






1 e2 and e

3are orthogonal



FA = radic3 ((1205821minus 120582)2

+ (1205822minus 120582)2

+ (1205823minus 120582)2

)

2 (1205822

1+ 1205822

2+ 1205822

3)

(9)













3 Results








2

1


PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)






PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)








Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB







4 Discussions



0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















3 Results








2

1


PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)






PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)








Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB







4 Discussions



0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

1


PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)






PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)








Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB







4 Discussions



0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










PK KA

Image

KB KF PN SB

(a)

Strain12

08

04

0

15

(b)

0

1

2

Strainat ventricle

(c)

Strainat CS fibers

(d)

Strainat CC fibers

(e)








Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB







4 Discussions



0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

Stra

in

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0

1

2

3

4

0 05 1 0 05 1 0 05 1

0 05 1 0 05 1 0 05 1

PrePost

PrePost

PrePost

KP KA KB

KF PN SB







4 Discussions



0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

20

25


119910 = minus10511119909 + 25069

Mea

n rC

BF (m

L10

0 mg

min

)

ΔrC

BF

Δ strain

PreFitted prePost








5 Conclusions




Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Strain_Post 60 273 122 90 53 180

0

50

100

150

200

250

300

350

Stra

in (

)

(a)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 27 115 55 49 28 77

(b)

0

50

100

150

200

250

300

350

Stra

in (

)


Strain_Post 14 61 30 24 12 36

(c)





Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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