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Transcalvarial brain herniation volume after decompressive craniectomy is the difference between two spherical caps
Medical Hypotheses 84 (2015) 183–188
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Medical Hypotheses journal homepage: www.elsevier.com/locate/mehy
Transcalvarial brain herniation volume after decompressive craniectomy is the difference between two spherical caps Chun-Chih Liao a,b, Yi-Hsin Tsai a,c, Yi-Long Chen d, Ke-Chun Huang a, I-Jen Chiang a,e, Jau-Min Wong a, Furen Xiao a,f,⇑ a
Institute of Biomedical Engineering, National Taiwan University, No.1, Sec.1, Ren’ai Rd., Taipei City 10051, Taiwan Department of Neurosurgery, Taipei Hospital,Ministry of Health and Welfare, No.127, Siyuan Rd., New Taipei City 24213, Taiwan Department of Surgery, Far Eastern Memorial Hospital, No.21, Sec.2, Nanya S. Rd., New Taipei City 22060, Taiwan d Institute of Biomedical Engineering, National Yang-Ming University, No.155, Sec.2, Linong St., Taipei City 11221, Taiwan e Graduate Institute of Medical Informatics, Taipei Medical University, No.250, Wuxing St., Taipei City 11031, Taiwan f Clinical Center for Neuroscience, National Taiwan University Hospital, No.7, Zhongshan S. Rd., Taipei City 10002, Taiwan b c
a r t i c l e
i n f o
Article history: Received 19 June 2014 Accepted 22 December 2014
a b s t r a c t Decompressive craniectomy (DC) is a surgical procedure used to relieve severely increased intracranial pressure (ICP) by removing a portion of the skull. Following DC, the brain expands through the skull defect created by DC, resulting in transcalvarial herniation (TCH). Traditionally, people measure only changes in the ICP but not in the intracranial volume (ICV), which is equivalent to the volume of TCH (VTCH), in patients undergoing DC. We constructed a simple model of the cerebral hemispheres, assuming the shape of the upper half of a sphere with a radius of 8 cm. We hypothesized that the herniated brain following DC also conforms to the shape of a spherical cap. Considering that a circular piece of the skull with a radius of a was removed, VTCH is the volume difference between 2 spherical caps at the operated side and the corresponding non-operated side, which represents the pre-DC volume underneath the removed skull due to the bilateral symmetry of the skull and the brain. Subsequently, we hypothesized that the maximal extent of TCH depends on a because of the biomechanical limitations imposed by the inelastic scalp. The maximum value of VTCH is 365.0 mL when a is 7.05 cm and the height difference between the spherical caps (Dh) at its maximum is 2.83 cm. To facil^ TCH ¼ 1 A2 Dh, where A ¼ 2a. itate rapid calculation of VTCH, we proposed a simplified estimation formula, V 2 ^ With the a value ranging between 0 and 7 cm, the ratio between V TCH and VTCH ranges between 0.77 and 1.27, with different Dh values. For elliptical skull defects with base diameters of A and C, the formula ^ TCH ¼ 1 AC Dh. changes to V 2 If our hypothesis is correct, surgeons can accurately calculate VTCH after DC. Furthermore, this can facilitate volumetric comparisons between the effects of DCs in skulls of varying sizes, allowing quantitative comparisons between ICVs in addition to ICPs. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction In 1783, Alexander Monro deduced that the cranium was a ‘‘rigid box’’ filled with a ‘‘nearly incompressible brain’’ and that its total volume, the intracranial volume (ICV), tends to remain constant [1]. The doctrine states that any increase in the volume of the cranial contents (e.g., brain, blood, cerebrospinal fluid [CSF], or pathologies such as tumor or hematoma) will increase the intracranial pressure (ICP). Furthermore, the increase in the ⇑ Corresponding author at: No.7, Chung Shan S. Rd., Zhongzheng Dist., Taipei City 10002, Taiwan. Tel.: +886 2 2312 3456; fax: +886 2 2395 7986. E-mail address: [email protected] (F. Xiao). http://dx.doi.org/10.1016/j.mehy.2014.12.018 0306-9877/Ó 2014 Elsevier Ltd. All rights reserved.
volume of any of these 4 elements must occur at the expense of the volume of the remaining 3 elements. In 1824, George Kellie confirmed many of Monro’s early observations [2]. Decompressive craniectomy (DC) is a neurosurgical procedure, wherein a portion of the skull is removed to decrease ICP [3]. The rationale for this procedure is based on the Monro–Kellie doctrine; expanding the physical space that confines edematous brain tissue after traumatic brain injury will decrease the ICP. The process of expanding the brain through a skull defect created by DC is called transcalvarial herniation (TCH). Considerable debate has been generated over the efficacy of DC despite its sound rationale and historical significance [4]. In addition, the DC procedure involves certain risks [5]. Of note, considerable technical variations exist
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in the employment of this potentially valuable technique. One way to address these concerns is to establish a consistent methodology that can be applied to evaluate and quantitatively compare the decompressive effort and effects of this procedure. The size of the removed skull plate is an objective measure of the decompressive effort. In our previous studies, we proposed methods for automatically or manually measuring the volume of the skull defect (VSD) using a simple estimation formula, ^ SD ¼ ABC [6,7]. By contrast, the decompressive effect is currently V assessed on the basis of the postoperative ICP alone and not the ICV. Because TCH is usually the only difference in the ICV after DC, measuring its volume (VTCH) would provide a volumetric measure of the decompressive effect. ICVs before and after DC can be compared on computed tomography (CT) or magnetic resonance (MR) images by image registration and computer-aided volumetry [8]. However, the ideal conditions for image processing are not always met because the qualities of these studies are often unsatisfactory due to unstable patient status and motion artifacts. In this paper, we propose a hypothesis for estimating the VTCH using our simplified geometric model of the brain. This hypothesis can be used to quantitatively evaluate the decompressive effects created by DC on the basis of postoperative CT or MR images alone. Subsequently, we provide a simple approximation formula that can be easily applied for the accurate estimation of postoperative VTCH. The hypothesis On the basis of craniometric data obtained from the normal CT images of patients with minor head injury, we constructed a simple model of the supratentorial brain or cerebral ‘‘hemispheres’’ [9]. This brain model assumes the shape of the upper half of a sphere with a radius (R) of 80 mm. The convexity of the skull is modeled as a spherical dome of a given thickness covering the brain. To simplify the condition, we assume that a round piece of bone with a base diameter of A (i.e., a radius of a) is removed, until indicated otherwise in this paper. The resulting defect can be considered as the surface of a spherical cap of the convexity skull. We hypothesized VTCH as the volume difference between the spherical caps at the operated and non-operated sides. Furthermore, VTCH can be approximated by A and the spherical cap height difference ^ TCH ¼ 1 A2 Dh. For craniectomies involv(Dh) by using the formula V 2 ing removal of an elliptical piece of bone with perpendicular axes of A and C, VTCH can be approximated using the formula ^ TCH ¼ 1 AC Dh. V 2 In geometry, a spherical cap is a portion of a sphere cut off by a plane. The removed skull can be considered as the outer surface of the spherical cap, wherein the margin of the skull defect forms a plane separating the cap from the other portion of the sphere, as illustrated in Fig. 1. The portion of the ICV between the cutting plane and the inner surface of the skull represents the volume of the spherical cap, defined as
V SC ¼
ph 6
2
ð3a2 þ h Þ;
Fig. 1. The model of the skull and its defect (dotted arc). The circular segment (shaded area) denotes the spherical cap intersected with the equatorial plane across its base diameter. Modified from [6].
procedures such as CSF drainage [11]. Assuming that the brain and the skull are bilaterally symmetric, we hypothesized that V SC beneath the skull defect before performing DC is the same as V SC on the non-lesion side with a height of hN, which is not affected by DC. On postoperative images, VTCH can be calculated using the V SC on the operated (lesion) side, denoted as V SCL , and that on the non-lesion side, denoted as V SCN .
V TCH ¼ V SCL V SCN ¼
phL 6
2
ð3a2 þ hL Þ
phN 6
2
ð3a2 þ hN Þ;
ð2Þ
where hL is the height of the spherical cap on the lesion side. The measurement process is illustrated in Fig. 2. The CT or MR slice with the largest dimension of the skull defect is selected. First, the base diameter A is measured. Second, the axis of symmetry is drawn, which is denoted as the intact midline (iML) or the intact midsagittal plane (iMSP) in 3 dimensions [12]. Subsequently, 2 parallel lines are drawn perpendicular to the iML (dotted lines in Fig. 2). Connecting the intersections between these lines and the inner surface of the skull on each side of the iML provides 2 line segments with lengths of A. Finally, hL and hN can be measured on the lesion and non-lesion sides, perpendicular to their base diameters, respectively. VTCH can be obtained after measuring a, hL, and hN using (2).
ð1Þ
where h denotes the height of the spherical cap [6,10]. Given a and R, h can be calculated using the relationship 2 R2 ¼ a2 þ ðR hÞ (Fig. 1). Thus, the height of the spherical cap for the intact skull model before DC, denoted as hN, is determined by the base diameter a. Alternatively, a and hN can be expressed as functions of R and h, a ¼ R sin h and hN ¼ Rð1 cos hÞ. DC usually comprises removal of the skull bone and opening the dura, the tough membrane covering the brain [3]. In comparison with its preoperative convex shape, the underlying brain surface after DC may become more protruded, less protruded, or even depressed, depending on the disease course and the associated
Fig. 2. Measurement of A, hL and hN after identification of the intact midline. VTCH can be calculated using these parameters.
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After removing the skull and opening the dura during DC, the swollen brain can expand through the skull defect beyond the thickness of the removed bone, as observed in Fig. 2. The height of the spherical cap after DC, hL, may be greater than that of hN plus the thickness of the skull, but it has certain limits. To our knowledge, no study has described the maximal possible extent of TCH or hL. However, the extent of TCH remains limited by the inelastic scalp, with the galea aponeurotica being poorly elastic [13,14]. In extreme cases of TCH, the brain ‘‘mushrooms’’ or ‘‘booms’’ through the skull defect to create a spherical cap with a diameter greater than that of the skull defect, a. The herniated portion of the brain then assumes the shape of a greater cap of another sphere out of the brain, thereby prohibiting adequate closure of the scalp, which is associated with failure to decrease the ICP and poor chance of survival. We considered the ability to close the scalp as the upper limit of ‘‘technically successful’’ DC and thus hypothesized hL < a to reflect this constraint. Thus, VTCH is always smaller than the volume of a half sphere with a radius of a. Let Dh ¼ hL hN , wherein the maximum value of Dh with a given a, dhMax ðaÞ, is a hN . The relationship between dhMax ðaÞ and a is not linear, as depicted in Fig. 3. With a ¼ 0 or a ¼ R, a equals to hN, and maximum of Dh is pffiffi pffiffiffi Dh becomes zero. The global Rð 2 1Þ ¼ 3:314 cm, when h ¼ 45 and a ¼ 22 R ¼ 5:657 cm. On combining the 2 aforementioned constraints, we have a hN > Dh > 2hN . Fig. 4 summarizes the values of VTCH versus a across all possible positive Dh values as well as negative Dh values up to 3 cm. The global maximum of VTCH is 365.0 mL, when a is 7.05 cm and Dh is allowed to have its maximum, 2.83 cm. In addition, the isovolume curves of VTCH are plotted in Fig. 4. With the same VTCH, Dh values decreased quadratically with decreasing a values (Pearson’s correlation coefficient >0.99 for all curves). The simplified formula Although the VTCH estimation formula proposed in the previous sections is geometrically straightforward, it is still too complicated for clinicians to use routinely without certain calculations. To resolve this problem, we proposed an approximation formula for measuring the volume of TCH. We hypothesized that VTCH is approximately Dh multiplied by the area of the rectangle with an edge length of A and divided by 2.
1 V^ TCH ¼ A2 Dh ¼ 2a2 Dh 2
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Fig. 4. The values of VTCH plotted against a (horizontal axis) and Dh (vertical axis) shown in gray scale overlaid with iso-volume curves, units in mL. The values of a and Dh range from 0 to 8 cm and from 3 to 4 cm, respectively.
The q value varies with different a and Dh values (Fig. 5). In general, the q value increases with decreasing Dh values, being close to 1 with Dh of approximately 1 cm in addition to a medium tolarge craniectomy radius, a. Theoretically, the full interval of q is p2 ; p4 , which equals to (0.64,1.27). However, under most clinical conditions, the q values range between 0.8 and 1.2. The median value of q ranges from 1.11, with an a value close to 0, to 0.97, with an a value of 7 cm (Fig. 6). When the skull defect created by DC assumes an elliptical shape with base diameters of A and C, linear transformation can be ^ TCH ¼ 1 AC Dh. applied to transform the simplified formula into V 2 The C value can be most accurately measured on coronal images obtained on planes perpendicular to the anteroposterior anatomical axis [6]. Our simplified formula for VTCH follows the same format as the other ‘‘ABC’’ style formulas for intracranial lesion volumetry, such
ð3Þ
Let q denote the accuracy of the simplified formula (3) in com^ parison with the full estimation formula (2), q ¼ VV TCH . TCH
Fig. 3. The relationship between hN, a hN and a using our model. Units in cm.
Fig. 5. The values of q plotted against a and Dh. Iso-value curves are also shown. With a between 0 and 7 cm, q ranges between 0.77 and 1.27.
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Fig. 6. The maxima, minima and medians of q with different values a ranging from 0 to 7 cm.
as those used for subdural hematoma (SDH), intracerebral hematoma, and arteriovenous malformation [15,16]. At a glance, it is easy to determine whether this formula is derived by subtracting 2 measurement results after applying the ‘‘ABC/2’’ formula 2 times. However, this formula is in fact derived from formula (2), and exact solutions can be demonstrated. With Dh ¼ hL hN , hL ¼ hN þ Dh and some rearrangement, (2) becomes
2 2 ðhL hN Þ 3a2 þ hL þ hL hN þ hN 6 p 2 2 2 ¼ Dh 3a þ ðhN þ DhÞ þ ðhN þ DhÞhN þ hN 6 2 2 ! p 2 hN hN Dh Dh þ ¼ a Dh 3 þ 3 þ3 ; a a 6 a a
V TCH ¼
p
Using this reformatted formula, it is clear that VTCH increased with increasing a and increasing Dh. The term Dah is not always negligible, but the exact solution of qMin with a given a can be obtained after substituting Dh with a hN and (3).
maxðV TCH Þ ¼
qMin ¼
p 6
2 Dh 4a2 þ ahN þ hN ¼
2 ! hN hN ^ TCH V þ 4þ 12 a a
p
V^ TCH 12 ¼ 2 maxðV TCH Þ hN p 4 þ a þ haN
ð4Þ
The exact solution of qMax with a given a can be obtained using factorization.
2 ! 2 ! 1 hN p hN ^ TCH minðV TCH Þ ¼ a Dh 1 þ ¼ V 4þ 4 a 2 16 a
p
2
amount of outward brain displacement (Dh) on the basis of the size of the skull plate removed (a). The authors did not reveal how they arrived at this model nor did they perform any quantitative validation. Munch et al. were the first to provide the geometric description of the herniated portion of the brain using a spherical cap model [18]. However, the formula provided was incorrect [19]. Next, Cavusoglu applied the correct spherical cap formula and attempted to calculate VTCH [20]. Although the study demonstrated certain 3dimensional reconstructions of the postoperative CT images, no volumetric verifications of their spherical cap models were attempted for unknown reasons. In addition, both aforementioned studies have equaled VTCH to VSCL and did not consider VSCN or assume VSCN to be zero. However, the images provided in these studies obviously indicated that VSCN values for such large craniectomies performed for decompression cannot be zero. Although both aforementioned studies have provided the average values of VTCH and hL, no information has been provided regarding hN. Of note, both these studies have reported a strong correlation between VSCL and a. However, it remains unclear whether such a correlation would persist when VSCL is replaced by VTCH or when hL is replaced by Dh because the extent of brain swelling would be a function of time, as described in the previous sections. To our knowledge, this is the first study to assume VTCH as the volume difference between 2 spherical caps. However, whether alternative geometric models, such as the ellipsoid model, can replace the spherical cap model remains unclear. In fact, the formula of a half ellipsoid can be rewritten into a form similar to that of a spherical cap,
p 6
A 2 Dh ¼
p
p
2
2
Dhð3a2 þ a2 Þ ðhL hN Þð3a2 þ hL þ hL hN þ hN Þ 6 6 ¼ V TCH :
These similarities indicate that the present simplified formula derived from the differences in spherical cap volumes is exactly the same as that derived from the differences in ellipsoid volumes. In fact, the simplified formula for VTCH is isomorphic to the ABC/2 formula for SDH volumetry. With SDH compressing the brain, B in the ABC/2 formula can be regarded as a negative Dh causing inward brain movement equivalent to a negative VTCH [21]. The thickness of SDH varies in different patients. Analogically, we believe that Dh can vary widely from positive to negative values with the same a value, instead of correlating with the shape and size of the portion of the bone removed during DC, as reported in several previous studies. Validity of the present model
qMax
^ TCH 16 V ¼ ¼ 2 minðV TCH Þ p 4 þ haN
ð5Þ
The minimum value of VTCH is derived when hL ¼ h2N or when the brain surface is depressed below the cutting plane of the spherical cap at a distance that is half the height of the original spherical cap. The value of qMedian can be easily calculated using (4) and (5). Evaluation of the hypothesis Alternative formulations Several previous studies have attempted to quantitatively estimate VTCH. In 1997, Wirth et al. proposed a semiquantitative geometric model of VTCH [17]; they assumed that a herniated brain assumes a cylindrical shape with gradually decreasing thickness near the craniectomy edges. In addition, they assumed a fixed
Validity of the spherical cap difference formula for VTCH estimation presented in this study depends on the accuracy of the geometric model, which, in turn, is affected by the shape and symmetry of the skull and the brain in addition to the shape of the brain surface at the region of skull defect. Li et al. analyzed the critical geometric characteristics of living Chinese human skulls in a large study population using CT images [22]. They reported that the thickness and breadth of the human skull are not normally distributed. Women exhibit thicker skulls than men do, but men exhibit wider breadth and longer anterior-posterior (AP) lengths than women do. The ratio between the breadth and length of the skull varied widely, ranging between 0.684 and 0.992, with an average of 0.826. Moreover, this study provided quantitative metrics for the skull symmetry but did not provide a cutoff point indicating the ‘‘critical’’ asymmetry beyond which our formulas may fail. Using the spherical cap difference model, we attempted to calculate VTCH with different a and Dh values that can be
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encountered in actual patients. The global maximum value of VTCH is 365.0 mL when a equals to 7.05 cm and Dh equals to 2.83 cm. However, it remains unclear whether such a large volume gain can be achieved when a circular piece of skull with a base diameter of 14.1 cm is removed because the amount of outward brain displacement is limited by the scalp tension and the width of the dural graft [17]. Nevertheless, large Dh values are often encountered beyond the thickness of the removed skull after DC, as observed in the upper left part of Fig. 7. In addition, vascular compression at the edge of DC is another technical factor that may affect brain circulation and finally patient outcome [5]. Moreover, whether vascular factors limit the maximal extent of VTCH remains unclear. By contrast, the shape of the brain surface may not be perfectly concave when it shrinks from its original status. A sigmoid appearance of the brain surface inside the bone defect is common, as observed in the upper right part of Fig. 7. The present spherical cap difference model cannot be directly applied in this case. Nevertheless, volumetry for patients with negative VTCH is a less crucial concern compared with volumetry for those with positive VTCH. With the use of an appropriate surgical technique, VTCH should be the same as the change in the ICV, particularly when the brain in swollen. However, in patients with atrophic brains or subsided brain swelling after DC, the brain volume may be lower than the augmented ICV. Therefore, the brain is not always close to the inner edge of the scalp or even to that of the skull, and the space is filled with the CSF. Under such circumstances, the changes in brain volume are no longer similar to those in the ICV, and the spherical cap difference model must be modified according to the object to be measured. When the skull defect is not circular but elliptical with orthogonal base axes of A and C, there is no simple formula for calculating the volume of an ‘‘ellipsoid cap’’. The exact formula for an ellipsoid cap is extremely complicated and requires modeling the skull as
187
part of an ellipsoid rather than that of a sphere. However, a simple approximate formula using linear transformation, V TCH elliptical ¼ C V , provides an adequate estimation, similar to transforming A TCH the volume formula for a sphere into that for an ellipsoid. Practically, A and C cannot be the same when an extremely large portion of the bone is removed during DC. As indicated in the present model, the shape of the cerebral ‘‘hemispheres’’ is actually more similar to that of the quarter spheres [9]. Bilateral bone removal across the iMSP to decompress both cerebral hemispheres is limited by the tension at the superior sagittal sinus, the most crucial and essential vein that cannot be divided or stretched during virtually all neurosurgical procedures. Therefore, surgeons cannot perform DC with a ‘‘theoretically maximum value’’ of VTCH of 365.0 mL. Testing the hypothesis Although the present hypothesis has not been systematically assessed, it can be assessed in a rather straightforward manner. One can obtain preoperative and postoperative brain images from patients undergoing DC. Then, registration between the image datasets can be performed using commercially available or custom-made software. The gray level of the brain and scalp on CT images is similar, and tracing the interface between them in areas where the intervening portion of the skull has been removed by DC can be challenging. Interactive editing or even pure manual tasks may be required; this task can be performed along with the measurement of values of A, C, hL, and hN. The remaining tasks, including skull segmentation, volume summation from different slices, and subtracting the postoperative ICV from the preoperative ICV, can be performed automatically. The VTCH values obtained from image registration followed by ICV subtraction can then be compared with those calculated using the rescaled version of the formula (2) V TCH elliptical ¼ CA V TCH and the simplified formula ^ TCH ¼ 1 AC Dh. V 2
Fig. 7. Examples of post-DC images with positive and negative Dh. Upper left: a CT slice across the largest dimension of skull defect with hL and hN equal to 3.6 and 1.6 cm, respectively. The skull thickness at the craniectomy region is 0.8 cm, resulting in an additional brain protrusion 1.2 cm beyond the limit of the skull, as modeled in lower left drawing. Upper right: a CT slice taken from another patient, showing the unevenly depressed brain surface, different from the ideal model shown in lower right drawing.
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Alternatively, formulas for VTCH estimation can also be tested without preoperative images, that is, by using postoperative image datasets and brain and skull symmetries. After determining the iMSP, the postoperative ICV can be divided into the volume of the cerebral hemisphere with the lesion, ICVL , and that of the normal hemisphere, ICVN . Subsequently, VTCH can be easily calculated using the formula V TCH ¼ ICVL ICVN . Although this method is simpler and can be performed on a larger scale, it is less accurate when the brain and skull are not symmetric at the outer contour or near the iMSP. Consequences of the hypothesis If our hypothesis is correct, surgeons can easily and rapidly estimate VTCH at bedside after DC. Moreover, we can quantitatively compare the ICVs in addition to the ICP in almost all patients undergoing DC for whom postoperative images are available. This would facilitate the conduction of large-scale studies. This would thus help to establish the relationship among lesion volume, VTCH, VSD, ICP, and clinical outcomes under various neurosurgical conditions and to define which patient subgroup would benefit from DC in addition to treatment to the primary pathology itself. Furthermore, we can determine whether there is an optimal VSD suitable for a given lesion type and size by determining VTCH at various time points after DC as well as its correlation with the complications of overcraniectomy (too large VSD) or undercraniectomy (too small VSD) in addition to the effects of controlling the ICP. Conflict of interest None. Acknowledgments The research reported in this publication was supported by Taiwan Ministry of Science Technology grants 101-2314-B-002-039 and 103-2314-B-002-165. References [1] Monro A. Observations on the structure and function of the nervous system. Edinburgh: Creech & Johnson; 1783. [2] Kelly G. Appearances observed in the dissection of two individuals; death from cold and congestion of the brain. Trans Med Chir Sci Edinb 1824;1:84–169.
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Transcalvarial brain herniation volume after decompressive craniectomy is the difference between two spherical caps Chun-Chih Liao a,b, Yi-Hsin Tsai a,c, Yi-Long Chen d, Ke-Chun Huang a, I-Jen Chiang a,e, Jau-Min Wong a, Furen Xiao a,f,⇑ a
Institute of Biomedical Engineering, National Taiwan University, No.1, Sec.1, Ren’ai Rd., Taipei City 10051, Taiwan Department of Neurosurgery, Taipei Hospital,Ministry of Health and Welfare, No.127, Siyuan Rd., New Taipei City 24213, Taiwan Department of Surgery, Far Eastern Memorial Hospital, No.21, Sec.2, Nanya S. Rd., New Taipei City 22060, Taiwan d Institute of Biomedical Engineering, National Yang-Ming University, No.155, Sec.2, Linong St., Taipei City 11221, Taiwan e Graduate Institute of Medical Informatics, Taipei Medical University, No.250, Wuxing St., Taipei City 11031, Taiwan f Clinical Center for Neuroscience, National Taiwan University Hospital, No.7, Zhongshan S. Rd., Taipei City 10002, Taiwan b c
a r t i c l e
i n f o
Article history: Received 19 June 2014 Accepted 22 December 2014
a b s t r a c t Decompressive craniectomy (DC) is a surgical procedure used to relieve severely increased intracranial pressure (ICP) by removing a portion of the skull. Following DC, the brain expands through the skull defect created by DC, resulting in transcalvarial herniation (TCH). Traditionally, people measure only changes in the ICP but not in the intracranial volume (ICV), which is equivalent to the volume of TCH (VTCH), in patients undergoing DC. We constructed a simple model of the cerebral hemispheres, assuming the shape of the upper half of a sphere with a radius of 8 cm. We hypothesized that the herniated brain following DC also conforms to the shape of a spherical cap. Considering that a circular piece of the skull with a radius of a was removed, VTCH is the volume difference between 2 spherical caps at the operated side and the corresponding non-operated side, which represents the pre-DC volume underneath the removed skull due to the bilateral symmetry of the skull and the brain. Subsequently, we hypothesized that the maximal extent of TCH depends on a because of the biomechanical limitations imposed by the inelastic scalp. The maximum value of VTCH is 365.0 mL when a is 7.05 cm and the height difference between the spherical caps (Dh) at its maximum is 2.83 cm. To facil^ TCH ¼ 1 A2 Dh, where A ¼ 2a. itate rapid calculation of VTCH, we proposed a simplified estimation formula, V 2 ^ With the a value ranging between 0 and 7 cm, the ratio between V TCH and VTCH ranges between 0.77 and 1.27, with different Dh values. For elliptical skull defects with base diameters of A and C, the formula ^ TCH ¼ 1 AC Dh. changes to V 2 If our hypothesis is correct, surgeons can accurately calculate VTCH after DC. Furthermore, this can facilitate volumetric comparisons between the effects of DCs in skulls of varying sizes, allowing quantitative comparisons between ICVs in addition to ICPs. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction In 1783, Alexander Monro deduced that the cranium was a ‘‘rigid box’’ filled with a ‘‘nearly incompressible brain’’ and that its total volume, the intracranial volume (ICV), tends to remain constant [1]. The doctrine states that any increase in the volume of the cranial contents (e.g., brain, blood, cerebrospinal fluid [CSF], or pathologies such as tumor or hematoma) will increase the intracranial pressure (ICP). Furthermore, the increase in the ⇑ Corresponding author at: No.7, Chung Shan S. Rd., Zhongzheng Dist., Taipei City 10002, Taiwan. Tel.: +886 2 2312 3456; fax: +886 2 2395 7986. E-mail address: [email protected] (F. Xiao). http://dx.doi.org/10.1016/j.mehy.2014.12.018 0306-9877/Ó 2014 Elsevier Ltd. All rights reserved.
volume of any of these 4 elements must occur at the expense of the volume of the remaining 3 elements. In 1824, George Kellie confirmed many of Monro’s early observations [2]. Decompressive craniectomy (DC) is a neurosurgical procedure, wherein a portion of the skull is removed to decrease ICP [3]. The rationale for this procedure is based on the Monro–Kellie doctrine; expanding the physical space that confines edematous brain tissue after traumatic brain injury will decrease the ICP. The process of expanding the brain through a skull defect created by DC is called transcalvarial herniation (TCH). Considerable debate has been generated over the efficacy of DC despite its sound rationale and historical significance [4]. In addition, the DC procedure involves certain risks [5]. Of note, considerable technical variations exist
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in the employment of this potentially valuable technique. One way to address these concerns is to establish a consistent methodology that can be applied to evaluate and quantitatively compare the decompressive effort and effects of this procedure. The size of the removed skull plate is an objective measure of the decompressive effort. In our previous studies, we proposed methods for automatically or manually measuring the volume of the skull defect (VSD) using a simple estimation formula, ^ SD ¼ ABC [6,7]. By contrast, the decompressive effect is currently V assessed on the basis of the postoperative ICP alone and not the ICV. Because TCH is usually the only difference in the ICV after DC, measuring its volume (VTCH) would provide a volumetric measure of the decompressive effect. ICVs before and after DC can be compared on computed tomography (CT) or magnetic resonance (MR) images by image registration and computer-aided volumetry [8]. However, the ideal conditions for image processing are not always met because the qualities of these studies are often unsatisfactory due to unstable patient status and motion artifacts. In this paper, we propose a hypothesis for estimating the VTCH using our simplified geometric model of the brain. This hypothesis can be used to quantitatively evaluate the decompressive effects created by DC on the basis of postoperative CT or MR images alone. Subsequently, we provide a simple approximation formula that can be easily applied for the accurate estimation of postoperative VTCH. The hypothesis On the basis of craniometric data obtained from the normal CT images of patients with minor head injury, we constructed a simple model of the supratentorial brain or cerebral ‘‘hemispheres’’ [9]. This brain model assumes the shape of the upper half of a sphere with a radius (R) of 80 mm. The convexity of the skull is modeled as a spherical dome of a given thickness covering the brain. To simplify the condition, we assume that a round piece of bone with a base diameter of A (i.e., a radius of a) is removed, until indicated otherwise in this paper. The resulting defect can be considered as the surface of a spherical cap of the convexity skull. We hypothesized VTCH as the volume difference between the spherical caps at the operated and non-operated sides. Furthermore, VTCH can be approximated by A and the spherical cap height difference ^ TCH ¼ 1 A2 Dh. For craniectomies involv(Dh) by using the formula V 2 ing removal of an elliptical piece of bone with perpendicular axes of A and C, VTCH can be approximated using the formula ^ TCH ¼ 1 AC Dh. V 2 In geometry, a spherical cap is a portion of a sphere cut off by a plane. The removed skull can be considered as the outer surface of the spherical cap, wherein the margin of the skull defect forms a plane separating the cap from the other portion of the sphere, as illustrated in Fig. 1. The portion of the ICV between the cutting plane and the inner surface of the skull represents the volume of the spherical cap, defined as
V SC ¼
ph 6
2
ð3a2 þ h Þ;
Fig. 1. The model of the skull and its defect (dotted arc). The circular segment (shaded area) denotes the spherical cap intersected with the equatorial plane across its base diameter. Modified from [6].
procedures such as CSF drainage [11]. Assuming that the brain and the skull are bilaterally symmetric, we hypothesized that V SC beneath the skull defect before performing DC is the same as V SC on the non-lesion side with a height of hN, which is not affected by DC. On postoperative images, VTCH can be calculated using the V SC on the operated (lesion) side, denoted as V SCL , and that on the non-lesion side, denoted as V SCN .
V TCH ¼ V SCL V SCN ¼
phL 6
2
ð3a2 þ hL Þ
phN 6
2
ð3a2 þ hN Þ;
ð2Þ
where hL is the height of the spherical cap on the lesion side. The measurement process is illustrated in Fig. 2. The CT or MR slice with the largest dimension of the skull defect is selected. First, the base diameter A is measured. Second, the axis of symmetry is drawn, which is denoted as the intact midline (iML) or the intact midsagittal plane (iMSP) in 3 dimensions [12]. Subsequently, 2 parallel lines are drawn perpendicular to the iML (dotted lines in Fig. 2). Connecting the intersections between these lines and the inner surface of the skull on each side of the iML provides 2 line segments with lengths of A. Finally, hL and hN can be measured on the lesion and non-lesion sides, perpendicular to their base diameters, respectively. VTCH can be obtained after measuring a, hL, and hN using (2).
ð1Þ
where h denotes the height of the spherical cap [6,10]. Given a and R, h can be calculated using the relationship 2 R2 ¼ a2 þ ðR hÞ (Fig. 1). Thus, the height of the spherical cap for the intact skull model before DC, denoted as hN, is determined by the base diameter a. Alternatively, a and hN can be expressed as functions of R and h, a ¼ R sin h and hN ¼ Rð1 cos hÞ. DC usually comprises removal of the skull bone and opening the dura, the tough membrane covering the brain [3]. In comparison with its preoperative convex shape, the underlying brain surface after DC may become more protruded, less protruded, or even depressed, depending on the disease course and the associated
Fig. 2. Measurement of A, hL and hN after identification of the intact midline. VTCH can be calculated using these parameters.
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After removing the skull and opening the dura during DC, the swollen brain can expand through the skull defect beyond the thickness of the removed bone, as observed in Fig. 2. The height of the spherical cap after DC, hL, may be greater than that of hN plus the thickness of the skull, but it has certain limits. To our knowledge, no study has described the maximal possible extent of TCH or hL. However, the extent of TCH remains limited by the inelastic scalp, with the galea aponeurotica being poorly elastic [13,14]. In extreme cases of TCH, the brain ‘‘mushrooms’’ or ‘‘booms’’ through the skull defect to create a spherical cap with a diameter greater than that of the skull defect, a. The herniated portion of the brain then assumes the shape of a greater cap of another sphere out of the brain, thereby prohibiting adequate closure of the scalp, which is associated with failure to decrease the ICP and poor chance of survival. We considered the ability to close the scalp as the upper limit of ‘‘technically successful’’ DC and thus hypothesized hL < a to reflect this constraint. Thus, VTCH is always smaller than the volume of a half sphere with a radius of a. Let Dh ¼ hL hN , wherein the maximum value of Dh with a given a, dhMax ðaÞ, is a hN . The relationship between dhMax ðaÞ and a is not linear, as depicted in Fig. 3. With a ¼ 0 or a ¼ R, a equals to hN, and maximum of Dh is pffiffi pffiffiffi Dh becomes zero. The global Rð 2 1Þ ¼ 3:314 cm, when h ¼ 45 and a ¼ 22 R ¼ 5:657 cm. On combining the 2 aforementioned constraints, we have a hN > Dh > 2hN . Fig. 4 summarizes the values of VTCH versus a across all possible positive Dh values as well as negative Dh values up to 3 cm. The global maximum of VTCH is 365.0 mL, when a is 7.05 cm and Dh is allowed to have its maximum, 2.83 cm. In addition, the isovolume curves of VTCH are plotted in Fig. 4. With the same VTCH, Dh values decreased quadratically with decreasing a values (Pearson’s correlation coefficient >0.99 for all curves). The simplified formula Although the VTCH estimation formula proposed in the previous sections is geometrically straightforward, it is still too complicated for clinicians to use routinely without certain calculations. To resolve this problem, we proposed an approximation formula for measuring the volume of TCH. We hypothesized that VTCH is approximately Dh multiplied by the area of the rectangle with an edge length of A and divided by 2.
1 V^ TCH ¼ A2 Dh ¼ 2a2 Dh 2
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Fig. 4. The values of VTCH plotted against a (horizontal axis) and Dh (vertical axis) shown in gray scale overlaid with iso-volume curves, units in mL. The values of a and Dh range from 0 to 8 cm and from 3 to 4 cm, respectively.
The q value varies with different a and Dh values (Fig. 5). In general, the q value increases with decreasing Dh values, being close to 1 with Dh of approximately 1 cm in addition to a medium tolarge craniectomy radius, a. Theoretically, the full interval of q is p2 ; p4 , which equals to (0.64,1.27). However, under most clinical conditions, the q values range between 0.8 and 1.2. The median value of q ranges from 1.11, with an a value close to 0, to 0.97, with an a value of 7 cm (Fig. 6). When the skull defect created by DC assumes an elliptical shape with base diameters of A and C, linear transformation can be ^ TCH ¼ 1 AC Dh. applied to transform the simplified formula into V 2 The C value can be most accurately measured on coronal images obtained on planes perpendicular to the anteroposterior anatomical axis [6]. Our simplified formula for VTCH follows the same format as the other ‘‘ABC’’ style formulas for intracranial lesion volumetry, such
ð3Þ
Let q denote the accuracy of the simplified formula (3) in com^ parison with the full estimation formula (2), q ¼ VV TCH . TCH
Fig. 3. The relationship between hN, a hN and a using our model. Units in cm.
Fig. 5. The values of q plotted against a and Dh. Iso-value curves are also shown. With a between 0 and 7 cm, q ranges between 0.77 and 1.27.
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Fig. 6. The maxima, minima and medians of q with different values a ranging from 0 to 7 cm.
as those used for subdural hematoma (SDH), intracerebral hematoma, and arteriovenous malformation [15,16]. At a glance, it is easy to determine whether this formula is derived by subtracting 2 measurement results after applying the ‘‘ABC/2’’ formula 2 times. However, this formula is in fact derived from formula (2), and exact solutions can be demonstrated. With Dh ¼ hL hN , hL ¼ hN þ Dh and some rearrangement, (2) becomes
2 2 ðhL hN Þ 3a2 þ hL þ hL hN þ hN 6 p 2 2 2 ¼ Dh 3a þ ðhN þ DhÞ þ ðhN þ DhÞhN þ hN 6 2 2 ! p 2 hN hN Dh Dh þ ¼ a Dh 3 þ 3 þ3 ; a a 6 a a
V TCH ¼
p
Using this reformatted formula, it is clear that VTCH increased with increasing a and increasing Dh. The term Dah is not always negligible, but the exact solution of qMin with a given a can be obtained after substituting Dh with a hN and (3).
maxðV TCH Þ ¼
qMin ¼
p 6
2 Dh 4a2 þ ahN þ hN ¼
2 ! hN hN ^ TCH V þ 4þ 12 a a
p
V^ TCH 12 ¼ 2 maxðV TCH Þ hN p 4 þ a þ haN
ð4Þ
The exact solution of qMax with a given a can be obtained using factorization.
2 ! 2 ! 1 hN p hN ^ TCH minðV TCH Þ ¼ a Dh 1 þ ¼ V 4þ 4 a 2 16 a
p
2
amount of outward brain displacement (Dh) on the basis of the size of the skull plate removed (a). The authors did not reveal how they arrived at this model nor did they perform any quantitative validation. Munch et al. were the first to provide the geometric description of the herniated portion of the brain using a spherical cap model [18]. However, the formula provided was incorrect [19]. Next, Cavusoglu applied the correct spherical cap formula and attempted to calculate VTCH [20]. Although the study demonstrated certain 3dimensional reconstructions of the postoperative CT images, no volumetric verifications of their spherical cap models were attempted for unknown reasons. In addition, both aforementioned studies have equaled VTCH to VSCL and did not consider VSCN or assume VSCN to be zero. However, the images provided in these studies obviously indicated that VSCN values for such large craniectomies performed for decompression cannot be zero. Although both aforementioned studies have provided the average values of VTCH and hL, no information has been provided regarding hN. Of note, both these studies have reported a strong correlation between VSCL and a. However, it remains unclear whether such a correlation would persist when VSCL is replaced by VTCH or when hL is replaced by Dh because the extent of brain swelling would be a function of time, as described in the previous sections. To our knowledge, this is the first study to assume VTCH as the volume difference between 2 spherical caps. However, whether alternative geometric models, such as the ellipsoid model, can replace the spherical cap model remains unclear. In fact, the formula of a half ellipsoid can be rewritten into a form similar to that of a spherical cap,
p 6
A 2 Dh ¼
p
p
2
2
Dhð3a2 þ a2 Þ ðhL hN Þð3a2 þ hL þ hL hN þ hN Þ 6 6 ¼ V TCH :
These similarities indicate that the present simplified formula derived from the differences in spherical cap volumes is exactly the same as that derived from the differences in ellipsoid volumes. In fact, the simplified formula for VTCH is isomorphic to the ABC/2 formula for SDH volumetry. With SDH compressing the brain, B in the ABC/2 formula can be regarded as a negative Dh causing inward brain movement equivalent to a negative VTCH [21]. The thickness of SDH varies in different patients. Analogically, we believe that Dh can vary widely from positive to negative values with the same a value, instead of correlating with the shape and size of the portion of the bone removed during DC, as reported in several previous studies. Validity of the present model
qMax
^ TCH 16 V ¼ ¼ 2 minðV TCH Þ p 4 þ haN
ð5Þ
The minimum value of VTCH is derived when hL ¼ h2N or when the brain surface is depressed below the cutting plane of the spherical cap at a distance that is half the height of the original spherical cap. The value of qMedian can be easily calculated using (4) and (5). Evaluation of the hypothesis Alternative formulations Several previous studies have attempted to quantitatively estimate VTCH. In 1997, Wirth et al. proposed a semiquantitative geometric model of VTCH [17]; they assumed that a herniated brain assumes a cylindrical shape with gradually decreasing thickness near the craniectomy edges. In addition, they assumed a fixed
Validity of the spherical cap difference formula for VTCH estimation presented in this study depends on the accuracy of the geometric model, which, in turn, is affected by the shape and symmetry of the skull and the brain in addition to the shape of the brain surface at the region of skull defect. Li et al. analyzed the critical geometric characteristics of living Chinese human skulls in a large study population using CT images [22]. They reported that the thickness and breadth of the human skull are not normally distributed. Women exhibit thicker skulls than men do, but men exhibit wider breadth and longer anterior-posterior (AP) lengths than women do. The ratio between the breadth and length of the skull varied widely, ranging between 0.684 and 0.992, with an average of 0.826. Moreover, this study provided quantitative metrics for the skull symmetry but did not provide a cutoff point indicating the ‘‘critical’’ asymmetry beyond which our formulas may fail. Using the spherical cap difference model, we attempted to calculate VTCH with different a and Dh values that can be
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encountered in actual patients. The global maximum value of VTCH is 365.0 mL when a equals to 7.05 cm and Dh equals to 2.83 cm. However, it remains unclear whether such a large volume gain can be achieved when a circular piece of skull with a base diameter of 14.1 cm is removed because the amount of outward brain displacement is limited by the scalp tension and the width of the dural graft [17]. Nevertheless, large Dh values are often encountered beyond the thickness of the removed skull after DC, as observed in the upper left part of Fig. 7. In addition, vascular compression at the edge of DC is another technical factor that may affect brain circulation and finally patient outcome [5]. Moreover, whether vascular factors limit the maximal extent of VTCH remains unclear. By contrast, the shape of the brain surface may not be perfectly concave when it shrinks from its original status. A sigmoid appearance of the brain surface inside the bone defect is common, as observed in the upper right part of Fig. 7. The present spherical cap difference model cannot be directly applied in this case. Nevertheless, volumetry for patients with negative VTCH is a less crucial concern compared with volumetry for those with positive VTCH. With the use of an appropriate surgical technique, VTCH should be the same as the change in the ICV, particularly when the brain in swollen. However, in patients with atrophic brains or subsided brain swelling after DC, the brain volume may be lower than the augmented ICV. Therefore, the brain is not always close to the inner edge of the scalp or even to that of the skull, and the space is filled with the CSF. Under such circumstances, the changes in brain volume are no longer similar to those in the ICV, and the spherical cap difference model must be modified according to the object to be measured. When the skull defect is not circular but elliptical with orthogonal base axes of A and C, there is no simple formula for calculating the volume of an ‘‘ellipsoid cap’’. The exact formula for an ellipsoid cap is extremely complicated and requires modeling the skull as
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part of an ellipsoid rather than that of a sphere. However, a simple approximate formula using linear transformation, V TCH elliptical ¼ C V , provides an adequate estimation, similar to transforming A TCH the volume formula for a sphere into that for an ellipsoid. Practically, A and C cannot be the same when an extremely large portion of the bone is removed during DC. As indicated in the present model, the shape of the cerebral ‘‘hemispheres’’ is actually more similar to that of the quarter spheres [9]. Bilateral bone removal across the iMSP to decompress both cerebral hemispheres is limited by the tension at the superior sagittal sinus, the most crucial and essential vein that cannot be divided or stretched during virtually all neurosurgical procedures. Therefore, surgeons cannot perform DC with a ‘‘theoretically maximum value’’ of VTCH of 365.0 mL. Testing the hypothesis Although the present hypothesis has not been systematically assessed, it can be assessed in a rather straightforward manner. One can obtain preoperative and postoperative brain images from patients undergoing DC. Then, registration between the image datasets can be performed using commercially available or custom-made software. The gray level of the brain and scalp on CT images is similar, and tracing the interface between them in areas where the intervening portion of the skull has been removed by DC can be challenging. Interactive editing or even pure manual tasks may be required; this task can be performed along with the measurement of values of A, C, hL, and hN. The remaining tasks, including skull segmentation, volume summation from different slices, and subtracting the postoperative ICV from the preoperative ICV, can be performed automatically. The VTCH values obtained from image registration followed by ICV subtraction can then be compared with those calculated using the rescaled version of the formula (2) V TCH elliptical ¼ CA V TCH and the simplified formula ^ TCH ¼ 1 AC Dh. V 2
Fig. 7. Examples of post-DC images with positive and negative Dh. Upper left: a CT slice across the largest dimension of skull defect with hL and hN equal to 3.6 and 1.6 cm, respectively. The skull thickness at the craniectomy region is 0.8 cm, resulting in an additional brain protrusion 1.2 cm beyond the limit of the skull, as modeled in lower left drawing. Upper right: a CT slice taken from another patient, showing the unevenly depressed brain surface, different from the ideal model shown in lower right drawing.
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Alternatively, formulas for VTCH estimation can also be tested without preoperative images, that is, by using postoperative image datasets and brain and skull symmetries. After determining the iMSP, the postoperative ICV can be divided into the volume of the cerebral hemisphere with the lesion, ICVL , and that of the normal hemisphere, ICVN . Subsequently, VTCH can be easily calculated using the formula V TCH ¼ ICVL ICVN . Although this method is simpler and can be performed on a larger scale, it is less accurate when the brain and skull are not symmetric at the outer contour or near the iMSP. Consequences of the hypothesis If our hypothesis is correct, surgeons can easily and rapidly estimate VTCH at bedside after DC. Moreover, we can quantitatively compare the ICVs in addition to the ICP in almost all patients undergoing DC for whom postoperative images are available. This would facilitate the conduction of large-scale studies. This would thus help to establish the relationship among lesion volume, VTCH, VSD, ICP, and clinical outcomes under various neurosurgical conditions and to define which patient subgroup would benefit from DC in addition to treatment to the primary pathology itself. Furthermore, we can determine whether there is an optimal VSD suitable for a given lesion type and size by determining VTCH at various time points after DC as well as its correlation with the complications of overcraniectomy (too large VSD) or undercraniectomy (too small VSD) in addition to the effects of controlling the ICP. Conflict of interest None. Acknowledgments The research reported in this publication was supported by Taiwan Ministry of Science Technology grants 101-2314-B-002-039 and 103-2314-B-002-165. References [1] Monro A. Observations on the structure and function of the nervous system. Edinburgh: Creech & Johnson; 1783. [2] Kelly G. Appearances observed in the dissection of two individuals; death from cold and congestion of the brain. Trans Med Chir Sci Edinb 1824;1:84–169.
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