Computational Fractal-Based Analysis of Brain Arteriovenous Malformation Angioarchitecture

Abstract

BACKGROUND:

Neuroimaging is the gold standard for diagnosis and follow-up of brain arteriovenous malformations (bAVMs), but no objective parameter has been validated for the assessment of the nidus angioarchitecture and for prognostication following treatment. The fractal dimension (FD), which is a mathematical parameter able to quantify the space-filling properties and roughness of natural objects, may be useful in quantifying the geometrical complexity of bAVMs nidus.

OBJECTIVE:

To propose FD as a neuroimaging biomarker of the nidus angioarchitecture, which might be related to radiosurgical outcome.

METHODS:

We retrospectively analyzed 54 patients who had undergone stereotactic radiosurgery for the treatment of bAVMs. The quantification of the geometric complexity of the vessels forming the nidus, imaged in magnetic resonance imaging, was assessed by means of the box-counting method to obtain the fractal dimension.

RESULTS:

FD was found to be significantly associated with the size (P = .03) and volume (P < .001) of the nidus, in addition to several angioarchitectural parameters. A nonsignificant association between clinical outcome and FD was observed (area under the curve, 0.637 [95% confidence interval, 0.49-0.79]), indicative of a potential inverse relationship between FD and bAVM obliteration.

CONCLUSION:

In our exploratory methodological research, we showed that the FD is an objective computer-aided parameter for quantifying the geometrical complexity and roughness of the bAVM nidus. The results suggest that more complex bAVM angioarchitecture, having higher FD values, might be related to decreased response to radiosurgery and that the FD of the bAVM nidus could be used as a morphometric neuroimaging biomarker.

Angioarchitecture, Arteriovenous malformation, Fractal dimension, Gamma knife radiosurgery, Image biomarker, Neuroimaging, Outcome

Brain arteriovenous malformations (bAVMs) are vascular lesions that consist of abnormal connections between veins and arteries. The connection between the feeding arteries and the draining veins is referred to as a nidus, which is the nest of vessels with no brain parenchyma interposition.¹ From an angioarchitectural point of view, the nidus is a complex vascular network formed by a tangle of coiled and tortuously enlarged vessels.^2,3

The natural history of untreated AVMs includes combined rates of major morbidity and mortality of 2.7% per year.⁴ Clinical outcomes, in terms of mortality and morbidity, depend on site, type, and size of the malformation.^2,5 Common clinical manifestations of bAVMs include seizure, hemorrhage, headache, and progressive neurological deficits.^6–9 The main cause of mortality and persistent morbidity is hemorrhage, present in 30% to 86% of patients with bAVM.¹⁰

The primary aim of treatment for bAVMs, including stereotactic radiosurgery (ie, gamma knife, linear accelerator, proton beam), microneurosurgery, and endovascular embolization techniques, or a combination of these methods, is to prevent hemorrhage and minimize the risk of postoperative complications, which may be the result of several factors including the angioarchitecture of the nidus. Following stereotactic radiosurgery, the 3-year obliteration rates range from 60% to 86.6%.^11–15

The Spetzler-Martin classification system,¹⁵ first introduced in 1986, was implemented to grade AVMs in accordance with specific factors on an ordinal scale: functional eloquence, nidus size, and venous drainage, with the final score related to the potential clinical outcomes. The system was designed to compare the long-term risks of an untreated bAVM and the potential intervention risks.^15,16 The 5-tiered system considered 3 main variables, taking into account many other factors: (1) size of the bAVM; (2) pattern of venous drainage; and (3) neurological eloquence of the brain regions adjacent to the bAVM.¹⁵ The system was later modified to a 3-tiered system, combining grade I with II (class A) and grades IV and V (class C) as a result of similar treatment and management of the combined grades.¹⁶

To provide the basis of treatment and visualization of the angioarchitecture in determining the optimal therapeutic approach, advances have been made in imaging techniques such as magnetic resonance imaging (MRI), computed tomographic angiography (CTA), and angiography, which are the most widely used neuroimaging techniques.¹⁷

Fractal analysis is a novel computer-aided mathematical model that offers the fractal dimension (FD) as a measure of the geometrical complexity of natural objects, providing useful information with many applications in biology, medicine, and neuroimaging.^18–21 The quantification of the geometrical complexity of the bAVM may be assessed by means of the box-counting method, which is one of the most widely used methods of fractal analysis in the biomedical sciences,^22,23 including the neurosciences.²⁴ In his article published in Science in 1967,²² Mandelbrot introduced the main concepts of fractal geometry: (a) the scale invariance of objects, ie, the so-called self-similarity, meaning that the whole has the same shape of its parts and (b) the fractional (or fractal) dimension (FD), which is a noninteger number between the Euclidean values 0 and 3, calculated by means of the “box-counting” method, able to quantify how an object fills the space in which it is embedded (ie, its geometrical complexity). Higher FD values approaching the value 3 represent the higher capacity to fill the 3-dimensional (3-D) space. The geometrical complexity of the shape of the bAVMs may be analyzed by means of fractal analysis, and FD could be added into the existing classification systems as a computer-aided morphometric neuroimaging parameter.

Angioarchitecture of bAVMs has been assessed in relation to clinical presentation and/or outcome following radiosurgery.^12–15,25 However, the applicability of FD as a potential parameter of bAVM angioarchitecture has not been assessed. FD estimation could be added to the quantification of size and volume (Euclidean parameters) of the nidus, as well as to the other angioarchitectural parameters actually in use. FD is a non-Euclidean parameter, which quantifies the 3-D geometrical complexity of the vessels within the nidus, and this angioarchitectural parameter might be related to the clinical outcome. This study aimed to validate a computational method to use the FD of the bAVM nidus as a morphometric neuroimaging parameter, testing it as a potential clinical surrogate biomarker for prognosis.

METHODS

Patients

This study constituted a retrospective review and analysis of neuroradiological images including MRI, CT, digital subtraction angiography, as well regions of interest (ROIs) of 120 consecutive bAVM patients who had undergone stereotactic radiosurgery by 4C Gamma Knife at the University of Toronto Gamma Knife Centre between 2005 and 2009.²⁶ Patients who had previous treatment, embolization, or surgery (n = 50) or inadequate visualization of a single arteriovenous malformation (AVM) and/or quantification of imaging by FD software due to unreliable pixel normalization (n = 12) were excluded. In these 12 patients, the intensity of the pixels of the images did not allow bAVM visualization by the use of a single threshold, which was experimentally selected and standardized for the whole analysis. Four patients did not have sufficient follow-up and were excluded. The remaining 54 patients (mean age = 36.5; 24 males, 30 females) that met the inclusion criteria were reviewed. Presenting symptoms of all patients at diagnosis are summarized in Table 1. Upon review, it was found that 29 of 54 (53.7%) patients presented with a hemorrhage. Other common symptoms included headache (13.0%) and seizures (9.3%). The local ethics committee approved the study.

TABLE 1.

Presenting Symptoms at Diagnosis of All Patients Included in Study (n = 54)

MRI

For the standardization of the image analysis, our study included only patients who underwent imaging by means of the same MR machine parameters over the entire series. The 3T GE Medical Systems Sigma HDxt with an 8-channel Head Coil was used for all patients, with the following MRI parameters: axial 2-D fast-recovery fast spin-echo (FRFSE) sequence with a rectangular matrix of 320, echo train length of 15, repetition time of 5500 ms and echo time of 91 ms, thickness of 2 mm without interslice gap and with a rectangular field of view of 20 cm resulting in plane resolution of 0.625 mm square.

Images were transferred to the radiosurgery software, which is a modification of CMI software (Montreal Stereotactic Planning System; CMI Services, Montreal, QC, Canada).²⁷

Radiosurgical Procedure and Outcome

Stereotactic radiosurgery was performed by using a 4C Gamma Knife radiosurgery system.²⁷ All patients wore the Leksell frame before undergoing contrast-enhanced cross-sectional imaging of the brain (both MRI and CT), as well as stereotactic angiography. A standard dose of 25 Gy for volumes <4 cm³ and 20 Gy for bAVM volumes >4 cm³ was administered. Near eloquent areas, the dose was limited to 15 Gy.^27,28 The outcome of radiosurgery was dichotomized as complete obliteration, defined as total disappearance of the nidus on the magnetic resonance angiogram, no abnormal flow voids on MRI (T2 turbo spin-echo sequences in coronal and axial planes), and no early draining vein on the digital subtraction angiogram,²⁷ and incomplete obliteration 3 years after gamma knife radiosurgery.

ROI Segmentation

The ROI, or “target” for the radiosurgical procedure, was developed by consensus of 6 experts (2 neurosurgeons, 1 neuroradiologist, 1 radiation oncologist, and 2 physicists, with an average experience of 15 years). To develop the ROI, we used the MRI and magnetic resonance angiogram, CTA, and cerebral angiogram. We outlined the ROI on each image modality separately and then we combined the images into 1 stereotactic space. The ROIs from angiogram and CTA were ultimately superimposed onto the space from the axial FRFSE T2 post-gadolinium MR images to signify the final target ROI that was used for treatment planning. Therefore, the fractal-based analysis was performed exclusively on the axial FRFSE T2 post-gadolinium images in order to standardize the procedure. The normalization of the intensity of the pixels within the ROI was performed to standardize the image analysis, by means of the Brightness Progressive Normalization algorithm, described by Di Ieva et al²⁹ and published shareware in http://www.fractal-lab.org/Downloads/bpn_algorithm.html. This algorithm normalized the intensity of the set of images and allowed the use of a single gray scale threshold for the computer-aided automatic extraction of the vessels within the nidus. Because this was a retrospective analysis, the experts choosing the ROIs for radiosurgery planning were blinded to fractal analysis results and subsequent outcome.

Computation of the FD of bAVM Angioarchitecture

To investigate whether the FD could measure the complexity of AVM nidus angioarchitecture, the quantification of the geometric complexity of the vessels forming the nidus was assessed by means of the box-counting dimension, and FD has been compared with other angioarchitectural parameters currently used in diagnosis and treatment planning. The nidus angioarchitecture was analyzed as a whole, without taking into account individual components forming it. The box-counting method applies the equation:

 

where dim_box(S) is the computed box-counting FD of the bAVM surface area on each slice, ε refers to the side length of the box, and N(ε) is the smallest number of nonoverlapping boxes of side length ε required to cover bAVM surface completely.^30–32 However, because the limit or ε cannot equal zero in biological structures, the FD was estimated as dim_box(S) = d, where d is the slope of the graph of log [N(ε)] against log (1/ε).³³ Because natural objects are “scale invariant,” the FD of the biological component (the angioarchitecture of the nidus, in our study) is self-similar within a fixed range referred to as the fractal window (ε_min-ε_max).^32,34,35 Natural fractal objects show a self-similar pattern in a range of at least 2 orders of magnitude^18,20 and in our analysis we chose a 2-order scale starting at the highest resolution of the MR images (ε_min = 0.86 mm-ε_max = 86 mm) (Figure 1).

FIGURE 1.

Method for the estimation of the fractal dimension (FD) of the nidus. A, bAVM region of interest (ROI) selection and segmentation of the nidus. (Exemplary slice selected from the whole stack of axial sections of the AVM. However, FD was measured in the 3-D reconstruction of the entire AVM) (see methods in di Ieva et al⁴⁹). B, FD plot [logN(ε)/log(1/ε)] in the box-counting method. AVM, arteriovenous malformation; bAVM, brain arteriovenous malformation.

Statistical Analysis

The association between the AVM angioarchitectural parameters, FD, and outcome (in terms of radiosurgical obliteration at 3 years) was examined by using Fisher exact, χ², and Mann-Whitney tests, where appropriate. AVM and angioarchitecture associations were examined using Mann-Whitney and Kruskal-Wallis tests or Spearman rank correlation. For FD and variables that were found to be associated with outcome, logistic regression analysis was used to examine the strength and nature of their relationship with outcome, whereas receiver operating characteristic curve analysis was used to examine their predictive ability. All statistical comparisons were 2-sided, and a P value of less than .05 was considered an indicator of a statistically significant association. Analyses were performed using software (IBM SPSS Statistics for Windows, Version 20.0. IBM Corp, Armonk, New York).

RESULTS

The FD was computed for all patients (n = 58, mean FD = 1.68). However, 4 patients did not have clinical outcome data and were excluded from further analyses (n = 54). Fisher exact test, unless otherwise indicated as χ² test, was used to correlate the angioarchitectural parameters to FD and clinical outcome, dichotomized as complete obliteration, indicating response to radiosurgery (mean FD = 1.61 ± 0.296) or incomplete obliteration (mean FD = 1.75 ± 0.263) (Tables 2 and 3, respectively). Complete obliteration was achieved in 23 patients (43%), and incomplete obliteration was found for 31 patients (57%). AVMs of 42 patients (72%) were in eloquent areas and received a lower dose (15 Gy), which may have contributed to the relatively lower rate of obliteration in our cohort. The log-log plots obtained from the box-counting dimension gave a curve with a monofractal behavior (with a very high “goodness of fit”: R² > 0.92) for all bAVMs analyzed. The slope of the log-log curve represents the FD of the nidus (Figure 1B).

TABLE 2.

bAVM Angioarchitectural Parameters Associated With Fractal Dimension^a

TABLE 3.

bAVM Angioarchitecture Parameters Associated With complete Obliteration Outcome^a

The association between FD and angioarchitectural parameters²⁸ is summarized in Table 2. Venous drainage (P < .001) and size were found to be positively associated with FD (P = .03) (Figure 2A). Arterial factors such as arterial enlargement (P = .002), presence of both nonsprouting (P = .001) and sprouting angiogenesis (P = .01), presence of pseudo phlebitic pattern (P = .01), and flow pattern (P = .03) (Figure 2B) were also significantly associated with higher FD values. Other angioarchitectural variables, including venous ectasia (P = .02), the number of draining veins (P < .001), and presence of venous rerouting (P = .009), were significantly correlated to FD. With the use of Spearman rank correlation, volume was found to be strongly associated with FD (ρ = 0.977, P < .001).

FIGURE 2.

A, box plot of FD change with respect to size, showing a strong association between increased size and higher FD (P = .03). B, box plot of FD change with respect to flow pattern, showing a strong association between moderate- to high-flow and higher FD (P = .03). FD, fractal dimension.

Angioarchitectural parameters of the bAVMs and their relation to obliteration outcome are summarized in Table 3. In contrast to associations with the FD, the outcome was related to 5 of the 10 parameters that were associated. Eloquence was not associated; however, both size and venous drainage (P = .001) were strongly associated. The size of the bAVM was associated with the outcome (P = .03), suggesting the bAVM size between 3 and 6 cm is related to incomplete obliteration. Other flow determinant parameters associated with outcome included the presence of sprouting angiogenesis (P = .04), moderate- to high-flow pattern (P = .001), and the presence of a venous pouch (P = .02).

In addition, we further assessed the significant relationships between FD, as a noncategorical factor, moderate- to high-flow pattern, venous drainage, sprouting angiogenesis, and the presence of a venous pouch with the outcome by using a binary logistic regression, summarized in Table 4. Flow pattern was the found to be the only angioarchitectural parameter that would be useful for prediction of outcome (flow pattern: AUC for obliteration, 0.752 [95% confidence interval, 0.616-0.889]). Deep or mixed venous drainage was also found to be strongly associated with an incomplete obliteration (0.734 [95% confidence interval, 0.601-0.868]). The AVMs with higher FD, having a nidus geometrically more complex (with higher grade of roughness), were associated to the lower response to radiosurgery. For the association between FD and outcome, the odds of having an incomplete obliteration outcome increased by a factor of 1.21 for every increase in FD by 0.1. However, the result may be interesting and clinically important, but was not statistically significant (0.637 [95% confidence interval, 0.49-0.79]). Figure 3 shows some exemplary cases.

TABLE 4.

Logistic Regression and ROC Analysis of Significant Angioarchitectural Parameters Associated With Obliteration Outcome^a

FIGURE 3.

Exemplary cases of AVMs with different FD values. The outcome of case 1 (A) was a complete obliteration of the nidus 3 years after stereotactic gamma knife radiosurgery, while an incomplete obliteration occurred in cases 2 (B) and 3 (C) (SM, Spetzler-martin score). AVM, arteriovenous malformation; FD, fractal dimension.

DISCUSSION

In our series, we assessed clinical outcomes as complete obliteration or incomplete obliteration following stereotactic gamma knife radiosurgery. We also assessed the angioarchitecture of bAVMs according to the parameters described in the literature, which correlate to clinical outcomes including obliteration rates. To date, there have not been any proposed parameters that can objectively predict the likelihood of obliteration of bAVMs following radiosurgery. In addition, here, we address the limitations of previous work that emphasized that the quantification of bAVMs using neuroimaging remains difficult owing to some degree of interobserver variability.

The high variability in radiosurgical response and prognosis of bAVMs originates from the differences in angioarchitecture, pattern of venous drainage, number of feeding arteries, and the size of the nidus.³⁶ Schuster et al³⁷ analyzed the hemodynamics of bAVMs and proposed that bAVM volume and excessive shunt volumes strongly correlate with the blood flow in the feeding extracranial artery on the side of the lesion and the cerebral blood flow, respectively.³⁷ Owing to high flow³⁸ and the lack of interposing capillaries,³⁹ bAVMs are characterized by low resistance in intranidal vascular structures, as well as increased artery diameter,⁴⁰ impaired venous drainage,^41–43 which increased the risk of vessel rupture, and cerebral hemorrhage.²⁷ In accordance with the literature, our results indicate a positive correlation with flow pattern, suggesting that high flow is associated with decreased response to radiosurgery and a lower rate of obliteration.

In addition to flow parameters, the number of draining veins has also been assessed as a factor by several studies. Niu et al⁴⁴ reported that, in the nonruptured AVM (nrAVM) group in their study, 1 patient (6.3%) had a single draining vein compared with 6 (37.5%) in the ruptured AVM group, concluding that the number of draining veins was higher in the nrAVM group than in the ruptured AVM group. The number of draining veins was the factor most associated with rupture in this study,⁴⁴ showing that AVMs with a single draining vein tended to rupture. In the study conducted by Kubalek et al,⁴⁰ 55 of 77 bAVMs (71.4%) with up to 2 draining veins had bled (P < .001), whereas 5 or more draining veins showed a lower bleeding risk. However, in the current series, mixed venous drainage was strongly associated with incomplete obliteration, whereas the number of initially draining veins was not. As noted in previous literature,³⁷ a limited draining system suggests that the pressure within the nidus and its feeding and draining vessels rises, resulting in bleeding. It is assumed that the intravascular pressure within the nidus and its related veins is a decisive factor concerning the bleeding risk.⁴²

The significance of lesion size has been variable in the studies that assess angioarchitecture in AVMs. All studies, including the current series, assessed bAVM size by using the Spetzler-Martin classification system. However, 2 studies that assessed hemorrhage presentation following radiosurgery measured bAVM size by volume.^26,46 Niu et al⁴⁴ found that in the group of AVMs that did not present with hemorrhage, 13 cases had medium lesions (81.3%) and 1 had a large lesion (6.3%). In the nrAVM group, 3 cases (20%) had small lesions; however, in accordance with other studies reviewed,^12,25,47 lesion size was not significantly different between the 2 groups. Two other studies also found small nidal size to be an independent predictor of bleeding.^42,48 In contrast to other reviewed studies, Hirai et al⁴⁵ found that large AVMs presented with hemorrhage. However, the potential to estimate the risks associated with treatment of bAVMs is limited owing to unaccounted features of the variables, including the vascular complexity within the nidus when considering its size. The current study assessed the relationship between bAVM size and obliteration and found that increased size was positively associated with incomplete obliteration (P = .04).

The present study assessed the parameters analyzed by Taeshineetanakul et al,²⁶ who concluded that a low-flow pattern, no arterial enlargement, and no perinidal angiogenesis were strongly related to complete obliteration. In accordance with previous literature, bAVM volume and the number of initially draining veins were also identified as significant parameters affecting the outcome of radiosurgery. However, the limitations contributing to the heterogeneity of the patient population (in regard to type of radiosurgical treatment, treatment before radiosurgery, follow-up imaging modality) may have resulted in conflicting reports.²⁸

The current study provided the first assessment of FD as a quantifier of vascular complexity of the bAVM nidus. According to our findings, it can be concluded that bAVM angioarchitecture is quantifiable by means of FD, and the FD correlated to several angioarchitectural parameters assessed in the literature. All parameters that were found to be associated with outcome were also correlated to FD. However, FD maintained a strong correlation with other parameters including arterial enlargement, presence of both sprouting and non-sprouting angiogenesis, as well as both venous ectasia and venous rerouting. The vascular complexity of bAVMs was analyzed by Reishofer et al,² who conducted the only study using fractal analysis and a combination of imaging techniques to correlate the FD with the complexity of bAVMs.² The results showed a higher FD of the hemisphere containing an AVM compared with the contralateral unaffected hemisphere, suggesting also a relation between vascular complexity and nidus size.² The FD has been also found to strongly correlate with the maximum slope of contrast media transit, because it indicates a higher number of feeding vessels to the nidus²; however, in their study, the vasculature of the nidus itself was not considered. The correlations made by Reishofer and colleagues² are limited to physiological and anatomic properties of an AVM, lacking considerations on the angioarchitecture of the nidus and clinical outcomes following treatment. Additional studies with a larger sample size are required to assess the validity of FD correlation to clinical outcome, as well as angioarchitectural parameters. Moreover, FD as an angioarchitectural imaging parameter could be tested in AVMs undergoing different treatments, such as embolization and microneurosurgical techniques, as well as in longitudinal studies assessing the morphological changes of the nidus over time after radiosurgery.

Limitations

The major limitations of this study are the selection of the imaging series, the choice of the ROI (ie, the edges of the nidus), considering that it is still operator dependent, as well as the retrospective study design. At our institution, the ROI selection and segmentation is usually done by a multidisciplinary team.²⁷ In the future, computational techniques of automatic extraction of the ROI may be implemented. Within the image analysis, the choice of the threshold for the automatic extraction of the vessels can be operator dependent, then introducing a further variable in the analysis. As in our previous neuroimaging computer-aided analysis,^29,49 we introduced a new algorithm (the Brightness Progressive Normalization) to normalize the intensity of the pixels to provide homogeneous images on which to easily apply a single threshold for the extraction of the nidus. Moreover, we tested the technique only on a specific MRI sequence to standardize the methodology, but it is well known that fractal analysis can be performed on every kind of imaging.^21,24 Further studies are in progress for the multiparametric fractal analysis and comparison of several imaging techniques and sequences.

CONCLUSION

In the present series, we present a novel approach for the quantitative measure of vascular complexity of bAVMs by means of the FD. Our results suggest that FD assessed by using the box-counting method is related to structural vascular complexity due to the increased number and tortuosity of vessels forming the nidus in patients with bAVMs. Our computational fractal-based analysis found that FD is positively associated with several angioarchitectural parameters. A nonstatistical relation was found with the clinical outcome, in terms of postradiosurgical obliteration rate, indicating that more complex bAVM angioarchitecture might be related to decreased response to radiosurgery. These findings should be investigated in further studies, before rejecting them in light of the nonstatistical significance in a small retrospective analysis. The results propose the FD of the AVM nidus as an objective computational morphometric neuroimaging biomarker, as well as a potential surrogate biomarker for treatment response and prognostication. However, further studies with additional data monitoring the change in FD over the 3-year follow-up and a larger sample size would strengthen the conclusions.

ABBREVIATIONS:

• AVM

  arteriovenous malformation

• bAVM

  brain arteriovenous malformation

• CTA

  computed tomographic angiography

• FD

  fractal dimension

• nrAVM

  nonruptured arteriovenous malformation

• ROI

  region of interest

COMMENT

The architecture of arteriovenous malformations (AVMs) can be visualized by modern imaging technologies with more detail and reliability. Fractal analysis is a method to quantify the geometrical complexity. The authors applied the parameter of fractal dimension (FD) to assess the angioarchitecture of AVM. This is an innovative approach for image analysis and might show a possibility to get information out of the images beyond of just describing them. It will be interesting to see whether the FD will become a reliable parameter that will be used during decision making for AVM therapy or whether it has a potential to be a parameter to evaluate clinical course investigations after therapy. It will be interesting to see whether FD provides additional information that can not be obtained by standard classification systems or descriptive evaluation strategies of the angioarchitecture. The authors are to be encouraged to continue their investigations, as well as their method to determine if the FD parameter should be made freely available so that it can be applied by other groups in a comparable way.

Christopher Nimsky

Marburg, Germany