estimation of the craniectomy surface area by using

Research ArticleEstimation of the Craniectomy Surface Area byUsing Postoperative Images

Meng-Yin Ho1 Wei-Lung Tseng23 and Furen Xiao 1

1Department of Neurosurgery National Taiwan University Hospital Taipei Taiwan2Graduate Institute of Biomedical Electronics and Bioinformatics National Taiwan University Taipei Taiwan3Department of Neurosurgery Fu Jen Catholic University Hospital New Taipei City Taiwan

Correspondence should be addressed to Furen Xiao xfurengmailcom

Received 29 December 2017 Accepted 24 April 2018 Published 3 June 2018

Academic Editor Guowei Wei

Copyright copy 2018 Meng-Yin Ho et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Decompressive craniectomy (DC) is a neurosurgical procedure performed to relieve the intracranial pressure engendered by brainswelling However no easy and accuratemethod exists for determining the craniectomy surface area In this study we implementedand compared three methods of estimating the craniectomy surface area for evaluating the decompressive effort We collected 118sets of preoperative and postoperative brain computed tomography images from patients who underwent craniectomy proceduresbetween April 2009 and April 2011 The surface area associated with each craniectomy was estimated using the marching cube andquasi-Monte Carlo methods The surface area was also estimated using a simple AC method in which the area is calculated bymultiplying the craniectomy length (A) by its height (C)The estimated surface area ranged from 946 to 20532 cm2 with a medianof 13480 cm2 The root-mean-square deviation (RMSD) between the marching cube and quasi-Monte Carlo methods was 753cm2 Furthermore the RMSD was 1445 cm2 between the marching cube and ACmethods and 1270 cm2 between the quasi-MonteCarlo and AC methods Paired 119905-tests indicated no statistically significant difference between these methods The marching cubeand quasi-Monte Carlo methods yield similar results The results calculated using the AC method are also clinically acceptable forestimating the DC surface area Our results can facilitate additional studies on the association of decompressive effort with theeffect of craniectomy

1 Introduction

Decompressive craniectomy (DC) is a common neurosurgi-cal procedure and it involves removing part of the craniumthus relieving an edematous brain and reducing intracranialpressure (ICP) by creating extra space It is typically per-formed on patients with head injuries or stroke Although theeffectiveness of DC in controlling ICP has been demonstrated[1 2] whether this control yields more favorable clinicaloutcomes remains controversial [2 3]

A larger craniectomy might result in a more favorableICP control [4] less delayed intracranial hematoma [56] and more desirable clinical outcomes [6ndash8] howevercertain studies have reported that the craniectomy size isnot correlated with complications [9] ICP control [9] oroutcomes [10 11]

The craniectomy size can be expressed in terms of thediameter [5 8 12ndash14] area [7 9ndash13 15 16] or volume[17] of the skull flap Although researchers in many studieshave measured the craniectomy size according to its areamost of them have used a simplified formula to estimatethe area However unlike the established ABC2 formulafor estimating the hematoma volume [18ndash20] the accuracyof such estimation of craniectomy area was never verifiedprobably because of the technical difficulties associated withcomputer-assisted area analysis

In accordance with our previous study on the estimationof the skull defect volume [17] we implemented computeralgorithms for surface area estimation and used them tovalidate a simple manual method of estimating the surfacearea of the skull flap by using postoperative images To verifythe robustness of thesemethods data on small craniectomies

HindawiInternational Journal of Biomedical ImagingVolume 2018 Article ID 5237693 8 pageshttpsdoiorg10115520185237693

2 International Journal of Biomedical Imaging

(a) (b) (c) (d)

Figure 1 Illustration of image processing to generate the skull flap removed through craniectomy From (a) to (d) preoperative imagespostoperative images registered postoperative images (bone only) and the missing skull flap by subtraction

including posterior fossa and nondecompressive craniec-tomies were also collected

2 Methods

21 Patients In this retrospective observational study wecollected 118 sets of preoperative and postoperative braincomputed tomography (CT) images from patients whounderwent craniectomies between April 2009 and April 2011at the National Taiwan University Hospital We includedimage sets of 75 male and 43 female patients The indicationsfor craniectomy were trauma in 50 patients spontaneouscerebral hemorrhage in 29 cerebral infarct in 15 infection in8 dural sinus thrombosis in 1 and brain swelling after variousprocedures in 15 Posterior fossa craniectomy was performedon 19 patients and unilateral frontotemporoparietal craniec-tomy was conducted on the remaining patients

22 Data Acquisition And Preprocessing The preoperativeand postoperative imaging procedures were performedaccording to a standard protocol of the hospital All axialimages were downloaded from the picture archiving andcommunication system (PACS) of the hospital to a personalcomputer for further processing To obtain the craniectomysize we first registered the postoperative image with the pre-operative one The missing skull flap was then generated bysubtraction between the registered image pairTheprocessingis illustrated in Figure 1 Image segmentation and registrationtasks were developed using C++ and the Insight Toolkit [21]After the image analysis the computed missing skull flap ineach case was confirmed by board-certified neurosurgeonson a computer screen slice by slice

23 Surface Area Estimation Three methods were used toestimate the surface area of the skull flap marching cubequasi-Monte Carlo and our simple AC method (Figure 2)

The marching cube method is a computer graphicsalgorithm that creates polygonal models of constant-densitysurfaces from three-dimensional (3D) data [22] We used theVisualization Toolkit [23] to create a triangulated representa-tion of the missing skull flap and then summed the areas ofsmall triangles A skull flap mainly has two sides (inner andouter) therefore we considered only triangles demonstratinga surface normal pointing outside the image center

The quasi-Monte Carlo method is based on the Cauchy-Crofton formula from integral geometry Liu et al publishedthe detailed methodology for applying this method for thesurface area estimation of digitized 3D objects [24] Themethod essentially entails estimating the surface area of avolumetric object by counting the number of intersectionpoints between the objectrsquos boundary surface and a set ofuniformly distributed lines generated using low-discrepancysequences Liu et al further found that a desirable surfacearea estimate can be obtained using few thousand lines ifa favorable low-discrepancy sequence is used We followedthis method to estimate the surface area of the missing skullflap After generating a ball with a known surface area (119878

119887)enclosing the skull flap we used a four-dimensional Sobolsequence [25] implemented in the GNU Scientific Library[26] to generate 20000 lines in the ball Furthermore wecounted the total intersection points between these linesand the skull flap (119899) The skull flap surface could then beestimated by 119878 = (119899119899119887)119878119887 where 119899119887 is the number ofintersection points between these lines and the enclosing balland its value was 20000 times 2 = 40000 in our implementationThe total surface area was then halved as the estimate of theouter surface area

The proposed manual method can be termed the ACmethod according to the ABC convention used to estimateintracerebral hemorrhage volumes [20] The greatest dimen-sion of the skull defect was identified after reviewing the axialbrain CT images and the linear distance between corners

International Journal of Biomedical Imaging 3

Table 1 Surface area results obtained using different methods

Surface area(cm)2 Marching cube(Sm) Quasi-Monte Carlo (Sq) AC method(Sac) Mean of Sm Sq and SacRange 939 sim 20546 1014 sim 20398 885 sim 21386 946 sim 20532Median 13452 13173 13328 13480Mean 12058 plusmn 5423 11957 plusmn 5328 11977 plusmn 5732 11997 plusmn 5454

(a) (b)

Figure 2 Illustration of the surface area estimation methods (a) The marching cube method entails estimating the surface area by splittingthe surface into small polygons (triangles in our implementation) The quasi-Monte Carlo method involves estimating the surface area bycounting the number of intersections between the skull flap and lines generated using a low-discrepancy sequence (b) AC method 119860 isthe longest length of craniectomy on the axial slices and C is the product of the number of axial cuts induced by craniectomy and slicethicknessspacing

of the outer table of the skull defect was used to determinethe craniectomy length (A) (Figure 2(b)) Furthermore thecraniectomy height (C) was determined by adding the inter-slice distance at which the full-thickness skull defect wasvisible on the postoperative CT images The mathematicalanalysis involved in this method is highly similar to thatreported in our previous study [17] and is included in theappendix because the mathematical part is subordinate to thetheme of this study In our study one neurosurgeon estimatedthe surface areas of all postoperative images obtained usingthis method

24 Data Analysis After surface areas estimated throughthree methods were recorded data were analyzed using R[27] The Pearson correlation coefficient 119903 was calculatedusing linear regression We used the root-mean-square devi-ation (RMSD) to measure the pairwise differences in thesurface areas obtained using two methods Bland-Altmananalysis [28] was also performed to assess the agreementbetween these methods To conduct comparisons with pre-vious estimation studies calculating the hematoma volume[18 19] we also calculated the mean deviation (in percentage)between the various methods

3 Results

Table 1 presents a summary of the surface areas estimatedusing the marching cube (Sm) quasi-Monte Carlo (Sq) and

AC (Sac) methods The estimated median surface areas were13452 cm2 for Sm 13173 cm2 for Sq and 13328 cm2 for SacConsidering their mean as metaestimates we determinedthat the craniectomy size ranged from 946 to 20532 cm2

The data plots obtained using the line of equality andBland-Altman analyses were our main results and are shownin Figure 3 Sm and Sq were similar and their correlationcoefficient was 0990 and RMSD was 753 cm2 The meandeviation was 365 Using Bland-Altman analysis we deter-mined that the 95 confidence interval (CI) for the bias wasminus1366 to 1569 cm2

The differences between Sm and Sac were significantlyhigher than those between Sm and Sq The correlationcoefficient and RMSDwere 0968 and 1445 cm2 respectivelyIn addition the mean deviation was 1212 and the 95 CIfor the bias was minus2557 to 2920 cm2

The differences between Sq and Sac were slightly smallerthan those between Sm and Sac The correlation coefficientand RMSD were 0976 and 1270 cm2 respectively Moreoverthe mean deviation was 1141 and the 95 CI for the biaswas minus2518 to 2479 cm2

The paired 119905-test revealed that Sm and Sq demonstratedno statistically significant difference (119901 = 0144) No statisti-cally significant difference was observed between Sm and Sac(119901 = 0543) or Sq and Sac (p = 0865) In brief the surface areameasured using these three methods showed no statisticallysignificant differences


50 100 150 200

5010

015

020

0 r = 0990RMSD = 753

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0968RMSD = 1445

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0976RMSD = 1270

50 100 150 200

minus20

020

40

minus20

020

40

minus20

020

40

60

Surfa

ce ar

ea b

y qu

asindash

Mon

te C

arlo

(Sq

＝Ｇ2)

Surface area by quasindashMonte Carlo (Sq ＝Ｇ2)

Surface area by marching cube (Sm ＝Ｇ2)


Diff

eren

ce S

mndashS

q (＝Ｇ

2)

Average (＝Ｇ2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Diff

eren

ce S

mndashS

ac (＝

Ｇ2)

Average (＝Ｇ2)

Average (＝Ｇ2)

Diff

eren

ce S

qndashSa

c (＝Ｇ

2)

Figure 3 Data plots with the line of equality and Bland-Altman analyses between the craniectomy area estimated using the marching cube(Sm) quasi-Monte Carlo (Sq) and AC (Sac) methods RMSD root-mean-square deviation


Figure 4 Previous attempts to measure the craniectomy area The surface area can be estimated using the spherical cap formula 119860119904=120587[(1198892)2 + ℎ2] [12 13] The base area can be estimated using 119860

119887= (1205874)119889 times 119863 [7]

4 Discussion

DC is a neurosurgical procedure to relieve brain swellingThe craniectomy size might be correlated with ICP control[4] complications [5 6] and clinical outcomes [6ndash8 11]therefore the measurement or estimation of the craniectomysize is clinically relevant and should be routinely performedfor every patient undergoing DC

The simplest measurement of the craniectomy size is theskull flap diameter estimated either during the operation oron postoperative images Because of bone loss engenderedby craniotomy and possible additional bone removal inducedby the use of a rongeur size estimation on postoperativeimages might be a more accurate approach considering theadvancement of PACSs However the simple anteroposteriordiameter or largest diameter represents only one dimensionof the craniectomy size Therefore measuring both theanteroposterior (AP) and superoinferior (SI) diameters is alsocommon in clinical practice although notations such as 15 times12 cm complicate comparisons and statistics

The most common measurement of the craniectomy sizeis its area either the base or surface area If the craniectomybase is modeled as an ellipse (Figure 4) then its base area canbe estimated by (1205874)119889 times119863 where 119889 and119863 represent the APand SI diameters respectively [7] Although this estimationmethod is straightforward validating it is difficult becausethe craniectomy base cannot be precisely defined to facilitatecomputer-assisted planimetry

Studies have proposed estimating the craniectomy sur-face area by considering it as a spherical cap [12 13] (Figure 4)The surface area of such a cap is 120587[(1198892)2 +ℎ2] where 119889 is theAP diameter and h is the longest distance from 119889 to the duralflap The formula ignores the SI dimension and estimatesthe protruding brain surface area instead of the craniectomysurface area therefore it is prone to change over the entireclinical course

As per our review of the relevant literature the presentstudy is the first to validate the estimation formula throughcomputerized methods Our results revealed that the surface

areas estimated using the marching cube and quasi-MonteCarlo methods were highly close because these two com-puterized methods are adequately established in the surfacearea estimation of digitized 3D objects However outliersexist of which the difference may increase to 40 cm2 Theseimages were typically degraded by metallic artifacts whichproduced greater errors during the surface area estimationWe did not discard these data because we aimed to examinethe robustness of thesemethods Our results confirm the highlevel of agreement between these digital methods

Even when the computerized methods were used theestimated surface area of the digitized object is slightlydifferent from that of the original object [24] Our implemen-tation of these two digital methods was also not perfect Inthe marching cube method we summed the area of smalltriangles exhibiting outward-pointing surface normals thiswould inevitably include triangles on the edge in additionto the outer surface of craniectomy Similarly in the quasi-Monte Carlo method dividing the total surface area bytwo may overestimate the surface area This should not bea problem in large craniectomies but would contribute toinaccuracies in small craniectomies

As expected the agreement between the results of theAC method and those of the other two methods was notas high Between Sm and Sac the Pearson 119903 value was0968 and mean deviation was 1212 and between Sq andSac 119903 was 0976 and the mean deviation was 1141 Forcomparison a study reported that the correlation coefficient(119903) between the ABC2 method and volumetric analysis was0842 for subdural hematoma and 0929 for intraparenchymalhematoma [18] The mean deviation between the ABC2method and volumetric analysis was reported to be 1454[19] Therefore the effectiveness of the AC method in esti-mating the craniectomy surface area is similar to that of theABC2 method in estimating the hematoma volume if notmore favorable

Certain factors render the AC method more useful thanthe digital methods First the AC method requires only thePACS for estimating the craniectomy surface area Even if


the CT image is placed in the traditional view box we canstill obtain an estimate from the film by using only axialslices More accurate digital methods require extra processesincluding segmentation and registration which are typicallynot available on the commercial PACS

The AC method requires only postoperative images toestimate the craniectomy surface area By contrast if preoper-ative images are unavailable for digital methods we can onlyprovide the mirror image of postoperative CT to estimate thesurface area However the estimation may be error prone ifthe patient has an asymmetric skull or if pathology or artifactexists on the other side of the skull

The AC method is the product of two length measure-ments therefore it should not be affected by most motion-and metal-related artifacts in images Although digital meth-ods are quite accurate they might fail if severe image artifactsexist

Our previous simpleABCmethod for estimating the skullflap volume is the product of the estimated surface area andbone thickness However the bone flap thickness might beerroneously measured because the thickness is usually thesmallest dimension and is not even around the skull flap Forinstance we can expect a significant difference in the bonethickness measured between the bone and brain windowsWe consider the AC method as a more robust methodfor estimating the craniectomy surface area however wealso encourage investigators to record the skull flap volumeestimated using the ABC method When available digitalmethods can be used to achieve a more accurate estimationof the craniectomy volume and surface area If they are notreadily available we consider that the AC and ABC methodsshould be used to report the craniectomy size for associatedstudies If only one measurement of the craniectomy size canbe obtained we recommend using the surface area estimatedthrough theACmethod which is a simple robust andwidelyapplicable method

The association between the decompressive effort andcraniectomy effect has been described in numerous previousstudies Studies have considered that DC may reduce themedical refractory ICP [29ndash32] However other studiesconclude that the adverse effect of DC results in poor patientoutcomes [13 31] Little evidence describing the necessityof craniectomy appropriate timing and clinical outcomesof different craniectomy sizes is available Future studiesregarding the immediate postcraniectomy effect such as thechange in ICP reduction of the midline shift or neurologicalimprovement can determine the correlation between thecraniectomy size and treatment outcomes according to ourapproach Studies may also evaluate the complication ratethrough various craniectomy sizesThey can also evaluate theideal craniectomy size after further clinical investigation andindividualized preoperative plans can be devised in diversecircumstances

The adequacy of craniectomy does not solely depend onits size Other factors including the extent of brain injuryspace-occupying lesions and systemic disorders may alsocontribute to the effects of this procedure However thesefactors do not mean that a quantification tool for estimatingthe craniectomy size is not necessary or is not valuable The

methods we presented for estimating the craniectomy surfacearea provide tools for further investigation on the effects ofsize

Our study had limitations First we examined the pos-sible inaccuracy in small craniectomy procedures by usingdigital methods Only one neurosurgeon evaluated all thecraniectomy surface area by using the ACmethod thereforeinterrater agreement could not be established Finally thisstudy focused on the craniectomy size therefore the effect ofcraniectomy and its association with size were not evaluated

5 Conclusion

In this study we compared three methods for estimating thecraniectomy surface area We confirmed that the marchingcube and quasi-Monte Carlo methods were consistent Theaccuracy of the simple AC method was also evaluated Thesemethods provide a quantitative evaluation for postoperativeassessment in patients who have undergone craniectomy

Appendix

Mathematical Analysis of the AC Method

The mathematical analysis of this method is highly similarto that reported in our previous study [17] To simplify thederivation we first model the convexity of the skull as aspherical dome with radius 119877 and diameter 119863 We alsoassume that the craniectomy procedure is performed on around piece of skull bone with diameter 119860 Therefore theskull defect can be considered a spherical cap of the skull(Figure 5(a)) The apex angle denoted as 2120579 is defined asfollows

sin 120579 = 1198602119877 =119860119863 (A1)

An estimate of the surface area of the skull defect (S) isconsidered as that of the spherical cap with a base diameterof A [33]

119878 = 2120587119877ℎ = 21205871198772 (1 minus cos 120579) (A2)

where ℎ is the cap height When (A1) and (A2) are used theestimated surface area of the skull defect can be expressed asfollows

119878 = 2120587( 1198602 sin 120579)2 (1 minus cos 120579) = 120587 (1 minus cos 120579)2 sin2 120579 1198602

= 12058711986022 (1 + cos 120579) =

12058711986024 cos2 (1205792)

(A3)

119878 = 120587( 1198602 cos (1205792))

2

(A4)

The formula is similar to that of the area of a disk (119860119903119890119886 =1205871199032) where 119903 = 1198602 cos(1205792) in our caseThe skull is not a true sphere If wemodel it as an ellipsoid

we can also obtain a similar estimate of 119878 However theexact formula for the surface area of the assumed ellipsoid or


S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)

Figure 5 Schematic of the skull defect (a)The skull is modeled as a spherical dome with the diameterD = 2RThe base diameter of the skulldefect (S dashed) is 119860 The apex angle of the spherical cap is 2120579 (b) Coronal view of the skull dome The defect starts from the skull baseand the apex angle is 2120593 Although the height is C the actual base diameter is 1198621015840

spheroid cap is extremely complicated [34 35] and beyondour need for a simple approximate formula If we can simplyobtain two orthogonal base axes of 119860 and 1198601015840 of the skullflap as an extremely intuitive generalization of (A4) wecan estimate the surface area of such a defect by using thefollowing formula

119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)

where sin 1205791015840 = 1198601015840119863Most craniotomy procedures are performed close to the

skull base therefore we assume that the skull defect reachesthe base of an imaginary dome and derive another axis fromthe coronal section (Figure 5(b)) The height (C) that weobtained from the axial slices of CT is shorter than the secondaxis (1198621015840) of the ellipsoid cap that we aimed to investigate(Figure 5(b))

sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)

From (A5) we know that

119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)

where 120573 = 1205874 cos (1205792) cos (1205932) cos120593The value of 120573 is dependent on 120579 and 120593 It can also be

expressed in terms of AD and CR If both ratios are closeto 08 as in most large craniotomy procedures the factor 120573

is close to 1 [17] Even if the skull defect is extremely smallor large 120573 rarely falls out of the range 08ndash12 Hence we canassume that 120573 asymp 1 and estimate the surface area by using S =120573119860119862 asymp 119860119862 equiv S119886119888Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by Taiwan Ministry of Science andTechnology (Grant nos 101-2314-B-002-039 and 106-2314-B-002-082)

References

[1] E Bor-Seng-Shu E G Figueiredo R L O Amorim et alldquoDecompressive craniectomy A meta-analysis of influences onintracranial pressure and cerebral perfusion pressure in thetreatment of traumatic brain injuryrdquo Journal of Neurosurgeryvol 117 no 3 pp 589ndash596 2012

[2] D James Cooper J V Rosenfeld L Murray et al ldquoDecom-pressive craniectomy in diffuse traumatic brain injuryrdquoTheNewEngland Journal ofMedicine vol 364 no 16 pp 1493ndash1502 2011

[3] J Sahuquillo F Martınez-Ricarte and M-A Poca ldquoDecom-pressive craniectomy in traumatic brain injury after theDECRAtrialWhere dowe standrdquoCurrent Opinion in Critical Care vol19 no 2 pp 101ndash106 2013

[4] C P Gao and B T Ang ldquoBiomechanical modeling of decom-pressive craniectomy in traumatic brain injuryrdquo Acta Neu-rochirurgica Supplementum vol 102 pp 279ndash282 2008

[5] S Wagner H Schnippering A Aschoff J A Koziol S Schwaband T Steiner ldquoSuboptimum hemicraniectomy as a cause ofadditional cerebral lesions in patientswithmalignant infarctionof the middle cerebral arteryrdquo Journal of Neurosurgery vol 94no 5 pp 693ndash696 2001

[6] J-Y Jiang W Xu W-P Li et al ldquoEfficacy of standardtrauma craniectomy for refractory intracranial hypertension


with severe traumatic brain injury A multicenter prospectiverandomized controlled studyrdquo Journal of Neurotrauma vol 22no 6 pp 623ndash628 2005

[7] J ChungO Y Bang Y C Lim S K Park andY S Shin ldquoNewlysuggested surgical method of decompressive craniectomy forpatientswithmiddle cerebral artery infarctionrdquoTheNeurologistvol 17 no 1 pp 11ndash15 2011

[8] C L Sedney T Julien J Manon and A Wilson ldquoThe effectof craniectomy size on mortality outcome and complicationsafter decompressive craniectomy at a rural trauma centerrdquoJournal of Neurosciences in Rural Practice vol 5 no 3 pp 212ndash217 2014

[9] L Tanrikulu A Oez-Tanrikulu C Weiss et al ldquoThe bigger thebetter About the size of decompressive hemicraniectomiesrdquoClinical Neurology and Neurosurgery vol 135 article no 4052pp 15ndash21 2015

[10] W T Curry Jr M K Sethi C S Ogilvy and B S CarterldquoFactors associated with outcome after hemicraniectomy forlarge middle cerebral artery territory infarctionrdquoNeurosurgeryvol 56 no 4 pp 681ndash692 2005

[11] C RWirtz T Steiner A Aschoff et al ldquoHemicraniectomywithdural augmentation in medically uncontrollable hemisphericinfarctionrdquo Neurosurgical Focus vol 2 no 5 p E7 1997

[12] PDe Bonis A Pompucci AMangiola L Rigante andCAnileldquoPost-traumatic hydrocephalus after decompressive craniec-tomy An underestimated risk factorrdquo Journal of Neurotraumavol 27 no 11 pp 1965ndash1970 2010

[13] E Munch P Horn L Schurer A Piepgras T Paul and PSchmiedek ldquoManagement of severe traumatic brain injury bydecompressive craniectomyrdquo Neurosurgery vol 47 no 2 pp315ndash323 2000

[14] T J Kenning R H Gandhi and J W German ldquoA comparisonof hinge craniotomy and decompressive craniectomy for thetreatment of malignant intracranial hypertension early clinicaland radiographic analysisrdquo Neurosurgical Focus vol 26 no 6p E6 2009

[15] M Olivecrona M Rodling-Wahlstrom S Naredi and L-O D Koskinen ldquoEffective ICP reduction by decompressivecraniectomy in patients with severe traumatic brain injurytreated by an ICP-targeted therapyrdquo Journal of Neurotraumavol 24 no 6 pp 927ndash935 2007

[16] C M Schirmer D A Hoit and A M Malek ldquoDecompressivehemicraniectomy for the treatment of intractable intracra-nial hypertension after aneurysmal subarachnoid hemorrhagerdquoStroke vol 38 no 3 pp 987ndash992 2007

[17] F Xiao I-J Chiang T M-H Hsieh et al ldquoEstimating post-operative skull defect volume from CT images using the ABCmethodrdquo Clinical Neurology and Neurosurgery vol 114 no 3pp 205ndash210 2012

[18] J M Gebel C A Sila M A Sloan et al ldquoComparison of theABC2 estimation technique to computer-assisted volumetricanalysis of intraparenchymal and subdural hematomas compli-cating the GUSTO-1 trialrdquo Stroke vol 29 no 9 pp 1799ndash18011998

[19] H B Huttner T Steiner M Hartmann et al ldquoComparison ofABC2 estimation technique to computer-assisted planimetricanalysis in warfarin-related intracerebral parenchymal hemor-rhagerdquo Stroke vol 37 no 2 pp 404ndash408 2006

[20] R U Kothari T Brott and J P Broderick ldquoThe ABCs ofmeasuring intracerebral hemorrhage volumesrdquo Stroke vol 27no 8 pp 1304-1305 1996

[21] T S Yoo M J Ackerman W E Lorensen et al ldquoEngineeringand algorithm design for an image processing API a technicalreport on ITK-the insight toolkitrdquo Studies in health technologyand informatics pp 586ndash592 2002

[22] W E Lorensen and H E Cline ldquoMarching cubes a high res-olution 3D surface construction algorithmrdquo ACM SIGGRAPHComputer Graphics vol 21 no 4 pp 163ndash169 1987

[23] W Schroeder K Martin and W Lorensen The VisualizationToolkit Kitware New York NY USA 2004

[24] Y-S Liu J Yi H Zhang G-Q Zheng and J-C Paul ldquoSurfacearea estimation of digitized 3D objects using quasi-MonteCarlomethodsrdquo Pattern Recognition vol 43 no 11 pp 3900ndash39092010

[25] I A Antonov and V M Saleev ldquoAn economic method ofcomputing LP120591-sequencesrdquo USSR Computational Mathematicsand Mathematical Physics vol 19 no 1 pp 252ndash256 1979

[26] B Gough GNU scientific library reference manual NetworkTheory Ltd GNU scientific library reference manual NetworkTheory Ltd 2009

[27] Team RC R A Language And Environment for StatisticalComputing R Foundation for Statistical Computing ViennaAustria 2012

[28] J Martin Bland and D Altman ldquoStatistical methods for assess-ing agreement between two methods of clinical measurementrdquoThe Lancet vol 327 no 8476 pp 307ndash310 1986

[29] T S Skoglund C Eriksson-Ritzen C Jensen and B RydenhagldquoAspects on decompressive craniectomy in patients with trau-matic head injuriesrdquo Journal of Neurotrauma vol 23 no 10 pp1502ndash1509 2006

[30] J S Huh H S Shin J J Shin T H Kim Y S Hwang and SK Park ldquoSurgical Management ofMassive Cerebral InfarctionrdquoJournal of Korean Neurosurgical Society vol 42 no 4 p 3312007

[31] S-H Chen Y Chen W-K Fang D-W Huang K-C Huangand S-H Tseng ldquoComparison of craniotomy and decompres-sive craniectomy in severely head-injured patients with acutesubdural hematomardquo Journal of Trauma - Injury Infection andCritical Care vol 71 no 6 pp 1632ndash1636 2011

[32] B M Eberle B Schnuriger K Inaba J Peter Gruen DDemetriades and H Belzberg ldquoDecompressive craniectomySurgical control of traumatic intracranial hypertension mayimprove outcomerdquo Injury vol 41 no 9 pp 894ndash898 2010

[33] J W Harris and H Stocker Handbook of Mathematics andComputational Science Springer New York NY USA 1998

[34] H Y Erbil and R A Meric ldquoEvaporation of sessile dropson polymer surfaces Ellipsoidal cap geometryrdquo The Journal ofPhysical Chemistry B vol 101 no 35 pp 6867ndash6873 1997

[35] S T Wong A K Ball and J G Sivak ldquoModel of retinalsurface area and neuron distribution in the avian eyerdquo Journalof Neuroscience Methods vol 123 no 1 pp 1ndash9 2003

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(a) (b) (c) (d)

Figure 1 Illustration of image processing to generate the skull flap removed through craniectomy From (a) to (d) preoperative imagespostoperative images registered postoperative images (bone only) and the missing skull flap by subtraction

including posterior fossa and nondecompressive craniec-tomies were also collected

2 Methods

21 Patients In this retrospective observational study wecollected 118 sets of preoperative and postoperative braincomputed tomography (CT) images from patients whounderwent craniectomies between April 2009 and April 2011at the National Taiwan University Hospital We includedimage sets of 75 male and 43 female patients The indicationsfor craniectomy were trauma in 50 patients spontaneouscerebral hemorrhage in 29 cerebral infarct in 15 infection in8 dural sinus thrombosis in 1 and brain swelling after variousprocedures in 15 Posterior fossa craniectomy was performedon 19 patients and unilateral frontotemporoparietal craniec-tomy was conducted on the remaining patients

22 Data Acquisition And Preprocessing The preoperativeand postoperative imaging procedures were performedaccording to a standard protocol of the hospital All axialimages were downloaded from the picture archiving andcommunication system (PACS) of the hospital to a personalcomputer for further processing To obtain the craniectomysize we first registered the postoperative image with the pre-operative one The missing skull flap was then generated bysubtraction between the registered image pairTheprocessingis illustrated in Figure 1 Image segmentation and registrationtasks were developed using C++ and the Insight Toolkit [21]After the image analysis the computed missing skull flap ineach case was confirmed by board-certified neurosurgeonson a computer screen slice by slice

23 Surface Area Estimation Three methods were used toestimate the surface area of the skull flap marching cubequasi-Monte Carlo and our simple AC method (Figure 2)

The marching cube method is a computer graphicsalgorithm that creates polygonal models of constant-densitysurfaces from three-dimensional (3D) data [22] We used theVisualization Toolkit [23] to create a triangulated representa-tion of the missing skull flap and then summed the areas ofsmall triangles A skull flap mainly has two sides (inner andouter) therefore we considered only triangles demonstratinga surface normal pointing outside the image center

The quasi-Monte Carlo method is based on the Cauchy-Crofton formula from integral geometry Liu et al publishedthe detailed methodology for applying this method for thesurface area estimation of digitized 3D objects [24] Themethod essentially entails estimating the surface area of avolumetric object by counting the number of intersectionpoints between the objectrsquos boundary surface and a set ofuniformly distributed lines generated using low-discrepancysequences Liu et al further found that a desirable surfacearea estimate can be obtained using few thousand lines ifa favorable low-discrepancy sequence is used We followedthis method to estimate the surface area of the missing skullflap After generating a ball with a known surface area (119878

119887)enclosing the skull flap we used a four-dimensional Sobolsequence [25] implemented in the GNU Scientific Library[26] to generate 20000 lines in the ball Furthermore wecounted the total intersection points between these linesand the skull flap (119899) The skull flap surface could then beestimated by 119878 = (119899119899119887)119878119887 where 119899119887 is the number ofintersection points between these lines and the enclosing balland its value was 20000 times 2 = 40000 in our implementationThe total surface area was then halved as the estimate of theouter surface area

The proposed manual method can be termed the ACmethod according to the ABC convention used to estimateintracerebral hemorrhage volumes [20] The greatest dimen-sion of the skull defect was identified after reviewing the axialbrain CT images and the linear distance between corners




(a) (b)




3 Results








50 100 150 200

5010

015

020

0 r = 0990RMSD = 753

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0968RMSD = 1445

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0976RMSD = 1270

50 100 150 200

minus20

020

40

minus20

020

40

minus20

020

40

60

Surfa

ce ar

ea b

y qu

asindash

Mon

te C

arlo

(Sq

＝Ｇ2)




Diff

eren

ce S

mndashS

q (＝Ｇ

2)

Average (＝Ｇ2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Diff

eren

ce S

mndashS

ac (＝

Ｇ2)

Average (＝Ｇ2)

Average (＝Ｇ2)

Diff

eren

ce S

qndashSa

c (＝Ｇ

2)




119887= (1205874)119889 times 119863 [7]

4 Discussion



















5 Conclusion


Appendix



sin 120579 = 1198602119877 =119860119863 (A1)


119878 = 2120587119877ℎ = 21205871198772 (1 minus cos 120579) (A2)



= 12058711986022 (1 + cos 120579) =

12058711986024 cos2 (1205792)

(A3)

119878 = 120587( 1198602 cos (1205792))

2

(A4)




S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)



119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)



sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)


119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)





Acknowledgments


References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





(a) (b)




3 Results








50 100 150 200

5010

015

020

0 r = 0990RMSD = 753

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0968RMSD = 1445

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0976RMSD = 1270

50 100 150 200

minus20

020

40

minus20

020

40

minus20

020

40

60

Surfa

ce ar

ea b

y qu

asindash

Mon

te C

arlo

(Sq

＝Ｇ2)




Diff

eren

ce S

mndashS

q (＝Ｇ

2)

Average (＝Ｇ2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Diff

eren

ce S

mndashS

ac (＝

Ｇ2)

Average (＝Ｇ2)

Average (＝Ｇ2)

Diff

eren

ce S

qndashSa

c (＝Ｇ

2)




119887= (1205874)119889 times 119863 [7]

4 Discussion



















5 Conclusion


Appendix



sin 120579 = 1198602119877 =119860119863 (A1)


119878 = 2120587119877ℎ = 21205871198772 (1 minus cos 120579) (A2)



= 12058711986022 (1 + cos 120579) =

12058711986024 cos2 (1205792)

(A3)

119878 = 120587( 1198602 cos (1205792))

2

(A4)




S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)



119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)



sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)


119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)





Acknowledgments


References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



50 100 150 200

5010

015

020

0 r = 0990RMSD = 753

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0968RMSD = 1445

50 100 150 200

50 100 150 200

5010

015

020

0 r = 0976RMSD = 1270

50 100 150 200

minus20

020

40

minus20

020

40

minus20

020

40

60

Surfa

ce ar

ea b

y qu

asindash

Mon

te C

arlo

(Sq

＝Ｇ2)




Diff

eren

ce S

mndashS

q (＝Ｇ

2)

Average (＝Ｇ2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Surfa

ce ar

ea b

y AC

(Sac

＝Ｇ

2)

Diff

eren

ce S

mndashS

ac (＝

Ｇ2)

Average (＝Ｇ2)

Average (＝Ｇ2)

Diff

eren

ce S

qndashSa

c (＝Ｇ

2)




119887= (1205874)119889 times 119863 [7]

4 Discussion



















5 Conclusion


Appendix



sin 120579 = 1198602119877 =119860119863 (A1)


119878 = 2120587119877ℎ = 21205871198772 (1 minus cos 120579) (A2)



= 12058711986022 (1 + cos 120579) =

12058711986024 cos2 (1205792)

(A3)

119878 = 120587( 1198602 cos (1205792))

2

(A4)




S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)



119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)



sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)


119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)





Acknowledgments


References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119887= (1205874)119889 times 119863 [7]

4 Discussion



















5 Conclusion


Appendix



sin 120579 = 1198602119877 =119860119863 (A1)


119878 = 2120587119877ℎ = 21205871198772 (1 minus cos 120579) (A2)



= 12058711986022 (1 + cos 120579) =

12058711986024 cos2 (1205792)

(A3)

119878 = 120587( 1198602 cos (1205792))

2

(A4)




S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)



119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)



sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)


119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)





Acknowledgments


References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











5 Conclusion


Appendix



sin 120579 = 1198602119877 =119860119863 (A1)


119878 = 2120587119877ℎ = 21205871198772 (1 minus cos 120579) (A2)



= 12058711986022 (1 + cos 120579) =

12058711986024 cos2 (1205792)

(A3)

119878 = 120587( 1198602 cos (1205792))

2

(A4)




S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)



119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)



sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)


119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)





Acknowledgments


References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



S

Rr

Ah

2

2 D = 2R

(a)

S

CC

2

R = D2

(b)



119878 = 120587( 1198602 cos (1205792))(

11986010158402 cos (12057910158402)) (A5)



sin 2120593 = 119862R

1198621015840 = 119862cos120593

(A6)


119878 = 120587( 1198602 cos (1205792))(

11986210158402 cos (1205932))

= 120587( 1198602 cos (1205792))(

1198622 cos (1205932) cos120593)

= 1205874 cos (1205792) cos (1205932) cos120593119860119862 = 120573119860119862

(A7)





Acknowledgments


References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


estimation of the craniectomy surface area by using

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