This study developed a binary logistic regression model to predict the probability of intracranial aneurysm rupture using computed tomography angiography data from 279 aneurysms in 217 patients. The model incorporated aneurysm size, shape, location and patient age. When applied to an independent prospective cohort of 49 aneurysms, the model predicted rupture status with 83% sensitivity and 80% accuracy. This validated the model as a potential tool for predicting aneurysm rupture risk to inform treatment decisions.

1. J Neurosurg 110:1–6, 2009
Predicting aneurysm rupture probabilities through the
application of a computed tomography angiography–derived
binary logistic regression model
Clinical article
CHARLES J. PRESTIGIACOMO, M.D.,1–3 WENZHUAN HE, M.D.,1 JEFFREY CATRAMBONE, M.D.,1
STEPHANIE CHUNG, B.S.,1 LYDIA KASPER, B.A.,1 LATHA PASUPULETI, B.S.,1
AND NEELESH MITTAL, M.D.1
Departments of 1Neurological Surgery and 2Radiology, and 3Neurological Institute of New Jersey,
New Jersey Medical School, University of Medicine of Dentistry of New Jersey, Newark, New Jersey
Object. The goal of this study was to establish a biomathematical model to accurately predict the probability
of aneurysm rupture. Biomathematical models incorporate various physical and dynamic phenomena that provide
insight into why certain aneurysms grow or rupture. Prior studies have demonstrated that regression models may
logistic regression model and then validated it in a distinct cohort of patients to assess the model’s stability.
Methods. Patients were examined with CT angiography. Three-dimensional reconstructions were generated and
aneurysm height, width, and neck size were obtained in 2 orthogonal planes. Forward stepwise binary logistic re-
gression was performed and then applied to a prospective cohort of 49 aneurysms in 37 patients (not included in the
original derivation of the equation) to determine the log-odds of rupture for this aneurysm.
Results. A total of 279 aneurysms (156 ruptured and 123 unruptured) were observed in 217 patients. Four of
unruptured aneurysms. Calculated volume and aneurysm location were correlated with rupture risk. Binary logistic
regression applied to an independent prospective cohort demonstrated the model’s stability, showing 83% sensitivity
and 80% accuracy.
Conclusions.
good accuracy. The use of this technique and its validation suggests that biomorphometric data and their relationships
may be valuable in determining the status of an aneurysm. (DOI: 10.3171/2008.5.17558)
KEY WORDS
S
UBARACHNOID hemorrhage secondary to the rupture Previous studies have suggested that the shape and
of an intracranial aneurysm is a life-threatening size of the aneurysm are parameters that can be used
and debilitating event with an overall morbidity to predict the risk of rupture.2 However, these studies
and mortality rate of 50–60%.2,15,20 With modern imag- only compared differences in size between unruptured
ing techniques, unruptured intracranial aneurysms can and ruptured intracranial aneurysms. Other studies have
be detected more reliably, but the management of these compared the size of the aneurysms between unruptured
lesions remains controversial.10 In part, the controversy versus ruptured groups, attempting to quantitatively as-
revolves around the fact that microsurgical and endovas- sign a critical number to aneurysm size (that is, the size
cular treatment modalities are invasive and carry some just prior to or at the time of rupture).6,9,18 However, esti-
risk to the patient.14 Consequently, the natural history of mated values of the critical size for aneurysmal rupture
any aneurysm of a given size, shape, and location must have ranged from 4.0 mm to > 10.0 mm.6,9,18 Therefore,
be balanced against the risk of complications secondary
to the treatment of the aneurysm. Of import would be the the rupture of cerebral aneurysms remains an important
ability to accurately predict the likelihood of aneurysm component of clinical decision-making in neurosurgery.
rupture such that only those patients with aneurysms that
are likely to rupture would be appropriately exposed to morbidity and mortality of intracranial aneurysms was
the risks of treatment. introduced by Richardson et al. in 1966.16 The authors
presented a discriminative function by which mortal-
Abbreviations used in this paper: ACoA = anterior communi-
cating artery; BA = basilar artery; ICA = internal carotid artery; This article contains some figures that are displayed in color
MCA = middle cerebral artery; PCoA = posterior communicating online but in black and white in the print edition.
artery; SAH = subarachnoid hemorrhage.
J. Neurosurg. / Vol 110 / January, 2009 1

2. C. J. Prestigiacomo et al.
ity could be predicted in a type of operative approach
to ACoA aneurysms. Subsequently, they used the same
function to evaluate the prognostic factors in a series of
PCoA aneurysms.17 Since the publication of their work,
mathematical modeling of aneurysms has been used to
understand the biophysical phenomena that contribute to
aneurysm growth and rupture. Biomathematical models
can incorporate various physical and dynamic phenom-
ena that may provide insight into the potential for rupture
and possibly help predict the probability of aneurysmal
rupture.4 Our present study describes the biomorphomet-
ric properties of aneurysms in a clinical prospective se-
ries. By performing binary logistic regression analysis, a
statistical technique similar to the previously described
discriminative analysis method,16 we have derived a rela-
tional equation that describes the rupture potential for ce-
rebral aneurysms within this cohort. To assess the stabil-
ity of this equation, we then applied it to an independent
cohort of aneurysm patients to determine the rupture sta-
tus of a patient’s aneurysm. To our knowledge, this repre-
applicable mathematical formula that describes aneurysm
rupture in an independent patient population.
This study represents a retrospective review of a
prospectively maintained database of patients presenting
to the University Hospital of the University of Medicine
and Dentistry of New Jersey with SAH due to aneurysm
rupture. Between 2002 and 2005, a total of 217 patients
with 279 aneurysms (156 ruptured and 123 unruptured)
presented to our institution. Multiple aneurysms were
ruptured and unruptured aneurysms by location as well 1A and B). (The variables Y1, X1, and N1 represent mea-
as patient age and sex is summarized in Table 1. In the
setting of aneurysmal SAH in patients with multiple an- and Y2, X2, and N2 represent measurements obtained in
eurysms, the ruptured aneurysm (the index aneurysm)
such as measured and calculated aneurysmal volume, lo-
and correlated with the hemorrhage pattern on the initial cation of aneurysm, and rupture status were included in
CT scan and repeated CT scans obtained 24 hours after this initial database. This data set was used as the ini-
the initial ictus (that is, the hemorrhage), when available. tial data to generate a stepwise binary logistic regression
Cerebral aneurysms were diagnosed and evaluated with model. After having completed the registration of patients
CT angiography using a GE Systems LightSpeed 16-slice to the current study, we prospectively collected a data set
CT scanner. A total of 150 ml of contrast medium was from an independent cohort of 49 aneurysms in 37 pa-
injected intravenously via the antecubital vein at a rate tients who presented between November 2005 and June
of 4 ml/second. Images were then obtained at 0.625-mm 2006. The model was then applied to this independent co-
slice-thickness with no overlap following an 18-second hort by one of the authors who was blinded to the rupture
acquisition delay. Source images were transferred to the status of the aneurysms. The log-odds risk of rupture for
GE Advantix 3D workstation where maximal-intensity -
projections and 3D reconstructions were generated. All ity and sensitivity to predict the aneurysm status using
angiograms were analyzed by 2 investigators who were our binary logistic regression model were calculated.
clinical information including rupture status of aneu-
Statistical Analysis
- The statistical software used in this analysis was
eral biomorphometric parameters were obtained in planes SPSS version 12.0 (SPSS, Inc.) for Windows. Indepen-
dent t-tests and chi-square tests were used to compare the
the parent vessel. mean for continuous data and categorical data, respec-
Maximum aneurysm height (Y), width (X), and neck tively. Forward binary logistic regression was then used
size (N) were obtained in these orthogonal planes (Fig. to generate the model.
2 J. Neurosurg. / Vol 110 / January, 2009

3. Significance of binary logistic regression in cerebral aneurysms
* *
*
Demographic characteristics of the patients and the
location of aneurysms in the cases used to generate the
initial binary logistic regression model are demonstrated
in Table 1. The results of comparison of the mean val-
observed for the number of ruptured versus unruptured
aneurysms between men and women (p = 0.538). How-
of ruptured versus unruptured aneurysms were found
4 of the 6 biomorphometric parameters obtained in this
*
maximum width [X1] divided by the neck size [N1]), and
the N1/N2 and X1/X2 ratios (Table 2).
A stepwise binary logistic regression model was gen-
erated that incorporated the aneurysm location in addi-
tion to volume and the biomorphometric parameters. In
this equation, aneurysm location was represented as a ue of the height of the aneurysm in the plane that is per-
-
est correlation with the initial database is expressed as of the aneurysm is represented by 1 of the 4 binary “loca-
follows: tion variables” (Table 3). A value of 1 for “Location (2)”
Logit = 1.127–0.457*volume + 0.254*Y2–1.214*Location would represent a patient with an aneurysm of the BA. Of
(4) – 2.262*Location (3) – 1.184*Location (2) – 0.334* note, patients with aneurysms of the PCoA would have all
Location (1) – 0.023*patient’s age “location variables” set at 0.
Note: Location (1) = ACoA; Location (2) = BA; Location
(3) = ICA; Location (4) = MCA. Y2 = the height of the aneu- tested by chi-square analysis, which generated a probabil-
rysm. Volume represents the measured volume as calculated by -
the system’s software package. cation were independently correlated to rupture risk (each
The volume in this model represents the measured
volume as calculated by the system’s proprietary software in this model. Using our model, we were able to predict
package. The variable “Y2” represents the measured val- the rupture status of the 279 aneurysms with a sensitivity
J. Neurosurg. / Vol 110 / January, 2009 3

4. C. J. Prestigiacomo et al.
*
*
*
-
age accuracy of the model for correctly classifying the
aneurysm status was found to be 70%.
Most importantly, this model was then used prospec-
tively to predict aneurysm rupture in a new, independent
cohort of 49 patients. Image analysis and interrogation of
the mathematical model were performed independently *
by 2 of the investigators, each blinded to the patient’s
clinical status. The model was able to correctly pre-
dict rupture status in 39 of 49 aneurysms. The sensitivity
78%, and an overall accuracy of 80%. The results of this
to be 83 and 78%, respectively, with an accuracy of 80% mathematical analysis are in accordance with those of a
(Table 4). previous study by Hademos et al.,9 in which the correla-
tion of anatomical and morphological factors with rup-
Illustrative Example ture of intracranial aneurysms was studied in 74 patients
In the cohort of 49 patients, a 65-year-old man pre-
sented with an aneurysm located at the ACoA. After 3D and overall accuracy were 76.3% (as compared with 81%
reconstructions, we calculated the aneurysm volume at in our initial data), 61.8% (55% in our initial data), and
0.124 cm3 and measured its height in the plane perpen- 69.4% (70% in our initial data), respectively. The pub-
lished data from the International Study of Unruptured
into the equation: Intracranial Aneurysms10 has suggested that the cumula-
Log (odds of rupture of the aneurysm) = 1.127–0.457* -
Volume + 0.254*Y2–0.334*Location (1) – 0.023*Age
no history of SAH (Group 1). However, the cumulative
Log (odds of rupture of the aneurysm) = 1.127–0.457*
0.124 + 0.254*6.9–0.334*1–0.023*65 rupture rate of aneurysms of the same size was ~ 11 times
higher per year in patients who present with a history of
Log (odds of rupture of the aneurysm) = 0.9939 SAH (Group 2). The rupture rate per year in aneurysms
The probability of rupture of the aneurysm = (Odds of
rupture) / (1 + Odds of rupture) = Exp (0.9939) / (1 + Exp regardless of the SAH history. To date, our model has not
[0.9939]) = 2.702 / (1 + 2.702) = 0.7299 been used as a means of longitudinally predicting future
Thus, the probability of rupture of the aneurysm in rupture of an unruptured aneurysm. Further analysis of
this example is 0.7299. Establishing the likelihood of an- additional, more complex parameters will be forthcom-
eurysm rupture to be > 0.5, in this example, the predic- ing.
tion would be that the aneurysm had ruptured. Clinical Our study revealed that aneurysm location is one of
-
aneurysm. eurysm, which is consistent with previous studies.1,2,7,9,23
Although previous studies support that location of the
aneurysm is a valid predictor of rupture, a correlation
between location and rupture of the aneurysm has not
The results of the present study indicate that our been established to date. By using this logistic regression
binary logistic regression model generated from an in- model, we were able to correlate the likelihood of rup-
dependent cohort of patients accurately determined the ture of an aneurysm with different locations as well as
rupture status of aneurysms within a second prospective with other parameters. For instance, careful analysis of
the equation demonstrates that, when keeping all other
4 J. Neurosurg. / Vol 110 / January, 2009

5. Significance of binary logistic regression in cerebral aneurysms
parameters unchanged, the odds ratio of an aneurysm
rupture at the ACoA to that of an aneurysm rupture at the
BA would be 2.34. In other words, if all parameters were
equal except the location, an ACoA aneurysm has a prob-
ability of rupture 2.34 times greater than an aneurysm of
the BA of equal size. Similarly, the model suggests that
an aneurysm has the least probability of rupture when it
is located at the ICA, while the same aneurysm located
at the PCoA has the greatest probability of rupture. This
-
miological study of aneurysm size and location.3 Other
studies have demonstrated similar results indicating that
PCoA and ACoA aneurysms are more prone to rupture
than aneurysms in other sites.1,8,9
was found to be the measured height of the aneurysm in
-
fully analyzing the algorithm above, one can note that,
for every unit of increasing height of the aneurysm, the
odds of rupture increase by a factor of 1.29, suggesting increase in volume. Thus, the function of the odds risk to
that a positive correlation exists between aneurysm size the volume may be written as a segment function. In our
and the risk of rupture. Many studies have advocated the -
importance of the size of aneurysms in association with
rupture and have suggested a linear relationship between groups based on aneurysm rupture status. We next intro-
aneurysm size and rupture.9,11,23 Several studies have at- duced additional categorical data to stratify volume in
tempted to determine the threshold or critical size at 3 levels. During logistic regression analysis, the volume
which an aneurysm becomes likely to rupture. 5,10,15,21,24
Nevertheless, results to date have been extremely variable some volume ranges, the odds of rupture increases with
with a wide range of critical sizes from 4 mm to > 10 the volume increase, while in other ranges, it decreases
mm.9 Beck et al.2 studied the size and location of ruptured as the volume increases (Fig. 2 plots the rupture prob-
and unruptured aneurysms and concluded that a critical ability of aneurysms versus volumes of aneurysms). This
-
and Heros22 suggested in a review that rupture can and esis, although we were unable to demonstrate enhanced
does occur at any size. Taken together, these studies indi- -
-
An additional interesting observation in our study
was the revelation of patient age as a factor in predict-
were observed between ruptured and unruptured aneu- ing the risk of aneurysm rupture. When we compared
rysm groups in several measurements (Table 2), including
measured heights and widths (Y1, Y2, X1, and X2) in both between-group differences were observed, in agreement
planes that are parallel and planes that are perpendicular with previous reports.23 Our logistic regression model,
however, suggests that patient age at diagnosis is inverse-
by the authors of previous studies.9,19 Based on our model, ly correlated with the risk of rupture (p = 0.031), which is
the odds of rupture of an aneurysm are positively corre- also consistent with previously published data.12 For ev-
lated with the height of the aneurysm measured in a plane ery 1 year of additional age, the statistical odds of rupture
decrease by a factor of 1.023. These results suggest that
Prior studies have also suggested that there is a sig- age should not be treated as an isolated predictive fac-
- tor for the risk of rupture, but rather should be combined
rysms and those of unruptured aneurysms. Although we with factors like aneurysm location and size.
hypothesized that larger volumes correlated with higher
risk of rupture, our binary logistic regression model dem-
onstrated a negative relationship between aneurysm vol-
ume and the odds of rupture. One explanation for this may Using a new binary logistic regression model of
be that the likelihood of aneurysm rupture is not linearly aneurysm rupture and basic biomorphometric data and
related to the volume of the aneurysm; there may be a relationships obtained from CT angiography in orthogo-
critical volume for which rupture risk begins to decrease. nal dimensions, we were able to accurately identify the
Some early observational data and recent biomathemati- status of an aneurysm with a sensitivity of 83% and an
cal modeling lend support to this hypothesis.10,13 Interest- overall accuracy of 80% in a prospectively obtained in-
ingly, however, within the largest range of aneurysm vol- dependently derived cohort of 37 patients with 49 aneu-
umes, the odds of rupture once again increases with an rysms. This cohort was distinct from the original cohort
J. Neurosurg. / Vol 110 / January, 2009 5

6. C. J. Prestigiacomo et al.
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tomical and morphological factors correlating with rupture of Please include this information when citing this paper: published
intracranial aneurysms in patients referred for endovascular online October 17, 2008; DOI: 10.3171/2008.5.17558.
treatment. 40:755–760, 1998 Address correspondence to: Charles J. Prestigiacomo, M.D.,
10. International Study of Unruptured Intracranial Aneurysms Departments of Neurological Surgery and Radiology, New Jersey
Investigators: Unruptured intracranial aneurysms—risk of Medical School, University of Medicine and Dentistry of New
rupture and risks of surgical intervention. 339: Jersey, 90 Bergen Street, Suite 8100, Newark, New Jersey, 07101.
1725–1733, 1998 email: c.prestigiacomo@umdnj.edu.
6 J. Neurosurg. / Vol 110 / January, 2009