Image Anal Stereol 2021;40:63-69  doi: 105566/ias.2581 
Short Research Communication 
63 
 
QUANTIFICATION OF THE INTERVENTIONAL APPROACHES INTO 
THE PTERYGOPALATINE FOSSA BY SOLID ANGLES USING VIRTUAL 
REALITY 
ANŽE JERMAN
1
, JIŘÍ JANÁČEK
2
, ŽIGA SNOJ
3,4
, NEJC UMEK
,5
 
1Faculty of Medicine, Ljubljana, University of Ljubljana, Korytkova 2, 1000 Ljubljana, Slovenia; 2Department 
of Biomathematics, Institute of Physiology, Czech Academy of Science, Vídeňská 1083, 14220 Prague 4, 
Czech Republic; 3Department of Radiology, Faculty of Medicine, Ljubljana, University of Ljubljana, Vrazov 
trg 2, 1000 Ljubljana, Slovenia; 4Clinical Institute of Radiology, University Medical Center Ljubljana, Zaloška 
cesta 7, 1000 Ljubljana, Slovenia; 5Institute of Anatomy, Faculty of Medicine, Ljubljana, University of 
Ljubljana, Korytkova 2, 1000 Ljubljana, Slovenia 
e-mail: anze.jerman@gmail.com, Jiri.Janacek@fgu.cas.cz, ziga.snoj@gmail.com, nejc.umek@mf.uni-lj.si 
(Received June 1, 2021; revised June 24, 2021; accepted June 27, 2021) 
ABSTRACT 
Virtual reality is increasingly used in medicine for diagnostics, for visualisation of complex structures and 
for preoperative planning. In interventional radiology, minimally invasive approach could be described 
with a target point representing the desired needle tip position and an array of all possible trajectories 
leading to it resembling irregular “cone” or “pyramid”. We present a pilot study of planning a minimally 
invasive posterior infrazygomatic and suprazygomatic approaches into the pterygopalatine fossa using a 
solid angle as a measure of size of the approach in five virtually reconstructed heads. The minimally 
invasive approaches were planned by manually drawing the edges of “pyramids” that described each 
approach in 3D using virtual reality program Tracer. For each head, a transverse diameter was measured 
and for each approach a solid angle size, average edge length and estimated area on the skin from where 
the target point could be reached were calculated. We found that, the solid angle of posterior infrazygo-
matic approach was significantly larger than suprazygomatic approach (p<0.001). Furthermore, the 
transverse head diameter and solid angle in posterior infrazygomatic approach were negatively correlated 
(ρ=-0.55, p=0.0002), while transverse head diameter and the estimated area on the skin from where the 
target point could be reached in the suprazygomatic approach were positively correlated (ρ=0.37, 
p=0.0206). In conclusion, our findings provide important preliminary evidence on the feasibility of 
evaluating and comparing different minimally invasive approaches using virtual reality systems, and 
affirm the validity of solid angle as a measure of the size of the approach. 
Keywords: Solid angle; virtual reality; interventional radiology; minimally invasive approach; procedure 
planning. 
INTRODUCTION  
The role of virtual reality has been increasingly 
recognised in surgical procedure planning where it can 
aid in the understanding of complex anatomical 
relationships, and its use has already been described for 
brain and abdominal tumour resections, nephrectomies, 
cerebral aneurysm clipping and others (Rizzetto et al., 
2020). Moreover, it is also increasingly used for 
diagnosis since it allows “immersion” in the image to 
better visualize target areas and therefore improves 
detection, localization and evaluation of pathologic 
formations (Elsayed et al., 2020). In interventional 
radiology the use of virtual reality has been to date 
restricted mainly to visualization of complex structures 
before the procedure to improve operator confidence 
(Devcic et al., 2018) and for practicing interventional 
procedures using phantoms (Kuhlemann et al., 2017). 
Beyond the established clinical applications, virtual 
reality could be employed to study different 
percutaneous interventions, such as biopsies, ablations, 
and various infiltrations, and for exploring new 
potential approach techniques. 3D virtual probes are 
used for interactive measurement of various 
geometrical characteristics of organs (Janáček and 
Jirák, 2019). Virtual reality can be reconstructed from 
medical image data such as computed tomography 
(CT) volumes depicting bony structures, CT 
angiography volumes depicting bony structures and 
vessels, magnetic resonance imaging (MRI) volumes 
 JERMAN A ET AL: Quantification of the interventional approaches 
64 
 
depicting soft tissue structures and co-registered CT 
with MRI that provide information on bony and soft 
tissue structures (Yoshinob et al., 2019; Elsayed et al., 
2020). 
Minimally invasive procedures are becoming 
increasingly important in modern medicine providing 
opportunity for targeted therapy which is a critical 
imperative for good treatment outcome (Sconfienza et 
al., 2020a, b; Snoj et al., 2020). In preparing for a 
procedure, it is of utmost importance to plan the 
optimal trajectory considering bony structures, vessels, 
and soft tissues such as nerves and glands. Target 
access is affected by the beforementioned structures, 
and this is even more pronounced where several 
structures conjointly affect the target access. This is the 
case in pterygopalatine fossa injections where several 
bony structures need to be accounted for to gain precise 
access into this narrow anatomical region on the skull 
base (Anugerah et al., 2020). The pterygopalatine fossa 
contains sympathetic and parasympathetic nerves, 
maxillary nerve and maxillary artery that supply nasal 
cavity, oral cavity and middle third of the face (Chiono 
et al., 2014). Injection of local anaesthetics into the 
pterygopalatine fossa is clinically used for treatment of 
specific headaches such as migraines, cluster 
headaches and post-dural-puncture headaches, for 
perioperative and postoperative pain management after 
maxillofacial and rhinologic procedures, and for 
management of posterior epistaxis (Chiono et al., 2014; 
Nagib et al., 2020).   
A minimally invasive approach can usually be 
described by a target point describing the desired 
needle tip position and an array of all possible 
trajectories leading from the skin surface to the target 
point considering bony structures and soft tissue 
structures that should not be injured by the needle. 
Such array of all possible trajectories usually forms an 
irregular oblique “cone” or “pyramid” with the base at 
the surface of the skin, which can be described by 
several parameters such as area of the base, height, 
length of the axis, surface and volume. However, many 
of these parameters are grossly dependent on the size 
of the individual (e.g. if comparing two individuals 
with identical intrinsic anatomy and difference in size, 
the values of parameters describing the approach would 
be significantly different), and thus influenced by age, 
sex and race, making the comparison between 
individuals very difficult. On the other hand, the solid 
angle of the “cone’s” or “pyramid’s” apex is not 
dependent on the height of the “cone” or “pyramid” 
and could be appropriate parameter to describe and 
compare the sizes of minimally invasive approaches 
avoiding the influence of confounders such is the size 
of the individual. The aims of our pilot study were to 
examine the feasibility of virtual reality to study 
minimally invasive approaches into the pterygopalatine 
fossa and to study the appropriateness of solid angle as 
a measure of size of the approach. 
MATERIAL AND METHODS  
We studied the suprazygomatic and posterior 
infrazygomatic approaches into the pterygopalatine 
fossa bilaterally in five virtually reconstructed heads. 
The heads were reconstructed from CT volumes of five 
different patients that received CT scan at the 
University Medical Centre Ljubljana. All CT scans 
were performed on a Somatom Definition Flash 
(Siemens, Germany) CT scanner. Parameters of image 
acquisition and reconstruction were 120-kV tube 
voltage and 70/150 effective mA/s, pixel size 0.383 
mm x 0.383 mm, pitch of 0.65, slice thickness of 0.75 
mm and space between slices of 0.900 mm. The study 
protocols were reviewed and approved by Republic of 
Slovenia National Medical Ethics Committee (Permit 
No: 0120-538/2019/4). 
The interventional approaches were described by 
“pyramids” with different numbers of sides. A target 
point, representing a needle tip in a clinical situation, 
was defined as a point in the pterygomaxillary fissure 
at the level of the pterygoid nerve channel and 
represented the apex of the “pyramid” for each studied 
approach. By manually drawing the lines (lateral edges 
of “pyramids”) from the target point to the point at 
surface of the skin considering bony structures, each 
approach was described by three-, four-, five- and six-
sided “pyramid” so that the lines spawn the largest 
possible solid angle in every case (Fig. 1).  
 
 
Image Anal Stereol 2020;40:63-69 
65 
 
 
Fig. 1 Lateral edges of five-sided “pyramids” representing the suprazygomatic (blue lines) and posterior 
infrazygomatic (red lines) approaches from the anterolateral (A) and posterolateral (B) perspectives. The tip of 
“pyramids” is located in pterygomaxillary fissure at the level of pterygoid nerve channel. 
 
We tested various sided “pyramids” to find the 
minimal number of sides that describes suprazygomatic 
and posterior infrazygomatic approach sufficiently. The 
lines were manually drawn in 3D using virtual reality 
program Tracer by Janáček et al. (2011). The program 
uses stereoscopic visualization with NVIDIA 3D 
Vision glasses (NVIDIA, California, USA), haptic 
feedback using Phantom Omni haptic device (3D 
Systems, South Carolina, USA) and space mouse 
(SpaceMouse® Pro, 3Dconnexion, Poland) (Fig. 2). 
The Tracer software returns coordinates of the points in 
space representing the apex and vertices of the” 
pyramids”. The x and y coordinates of the apex and 
vertices were calibrated by pixel size, and z coordinates 
by the slice thickness.  
 
 
Fig. 2. The virtual reality setting with 3D Vision glasses, haptic feedback using haptic device and a space mouse. 
 JERMAN A ET AL: Quantification of the interventional approaches 
66 
 
 
The solid angle at the apex is equal to the area Ω of 
spherical polygon with internal angles 𝐴𝑖 that are equal 
to dihedral angles between neighboring lateral facets of 
the “pyramid“. Area of the spherical polygon can be 
calculated using the Gauss-Bonnet formula: 
 
𝛺 = ∑ 𝐴𝑖
𝑁
𝑖=1 + (2 − 𝑁)𝜋 (1) 
Let 𝑉0 be the apex and 𝑉𝑖 the vertices on the head 
surface recorded in clockwise order, then the oriented 
lateral edges are 
 
𝑃𝑖 = 𝑉𝑖 − 𝑉0 (2) 
and the normals to lateral facets are 
 
𝑄𝑖 =
𝑃𝑖−1×𝑃𝑖
|𝑃𝑖−1×𝑃𝑖|
. (3) 
Then the external angles 𝐵𝑖are 
 
𝐵𝑖 = 𝑠𝑖𝑔𝑛 𝑄𝑖 × 𝑄𝑖+1 𝑐𝑜𝑠
−1 𝑄𝑖 ∙ 𝑄𝑖+1 (4) 
and the solid angle is 
 
𝛺 = ∑ 𝐵𝑖
𝑁
𝑖=1 + 2𝜋 (5) 
For each “pyramid” solid angle Ω  and average 
length of the lateral edges were calculated: 
 
 𝑃 = 1 𝑁⁄ ∑ |𝑃𝑖|
𝑁
𝑖=1  (6) 
The area (S) on the skin from where the target 
point could be reached was estimated using: 
 
 𝑆 =  𝛺 × 𝑃
2
 (7) 
The size of the head was estimated by the transver-
sal head diameter (D) measured as the distance 
between most lateral points of the mandibular 
condyles.  
STATISTICAL ANALYSIS 
Statistical analysis was performed using the 
GraphPad Prism 8 (GraphPad Software, San Diego, 
CA, USA). Shapiro–Wilk test was used to test the 
numerical data for normal distribution. The numeric 
parameters between two approaches, left and right side 
and different sided “pyramids” were compared with 
repeated measures ANOVA with Tukey’s post hoc 
tests. The correlation between solid angle, length of 
lateral edges and area on the skin that represent the 
base of the “pyramid” was tested using Pearson’s 
correlation coefficient. Data are represented as means 
and standard deviations. The differences were deemed 
statistically significant at p<0.05.  
RESULTS 
In the present pilot study, we studied suprazygo-
matic and posterior infrazygomatic pterygopalatine 
ganglion block approach in reconstructed heads of 
three male and two female adult patients bilaterally 
(n=10) with transversal head diameter (D) of 15.2 ± 1.1 
cm, ranging from 14.1 to 16.8 cm.  
The solid angle of the posterior infrazygomatic 
approach was significantly larger than suprazygomatic 
approach when measured using four- (p=0.0002), five- 
(p<0.0001), and six-sided (p<0.0001) “pyramids” 
(Table 1). In the suprazygomatic approach, there was a 
significant difference in solid angle size between three- 
and six-sided “pyramids” (p=0.0267). In the posterior 
infrazygomatic approach, the three-sided “pyramid” 
had significantly smaller solid angle than four- 
(p=0.0045), five- (p=0.0018) and six-sided (p=0.0024) 
“pyramids”. There were no significant differences 
between any parameters between left and right side. 
There was a significant positive correlation 
between transversal head diameter and average lateral 
edge length in the suprazygomatic approach (ρ=0.58, 
p<0.0001) and posterior infrazygomatic approach 
(ρ=0.53, p=0.0004). We also noted a significant 
negative correlation (ρ=-0.55, p=0.0002) between 
transverse head diameter and solid angle in the 
posterior infrazygomatic approach, while there was no 
significant correlation between transverse head 
diameter and solid angle in the suprazygomatic 
approach. Furthermore, there was a significant positive 
correlation between transverse head diameter and the 
estimated area on the skin from where the target point 
could be reached in the suprazygomatic approach 
(ρ=0.37, p=0.0206), while there was no significant 
correlation in the posterior infrazygomatic approach. 
 
 
Image Anal Stereol 2020;40:63-69 
67 
 
Table 1. Solid angle (Ω), average lateral edge length (𝑃) and estimated area on the skin from where the target area 
could be reached (S) for three-, four-, five- and six-sided “pyramids” for suprazygomatic and posterior 
infrazygomatic approaches. 
 Suprazygomatic approach  Posterior infrazygomatic approach 
 Ω 𝑃 (cm) S (cm
2)  Ω 𝑃 (cm) S (cm
2) 
3-sided 0.098±0.039⸸ 5.3±0.6 2.8±1.3  0.089±0.035*,#,⸸ 7.0±1.5 4.2±1.8*,#,⸸ 
4-sided 0.106±0.040Ψ 5.1±0.6 2.9±1.2⸸  0.135±0.028 6.6±1.0 5.9±1.7 
5-sided 0.108±0.043 Ψ 5.3±0.5 3.0±1.3  0.140±0.033 6.6±1.0 6.1±2.0 
6-sided 0.114±0.041 Ψ 5.3±0.6 3.2±1.3  0.146±0.036 6.5±1.0 6.2±1.9 
*p<0.05 vs. 4-sided, 
#
p<0.05 vs. 5-sided, 
⸸
p<0.05 vs. 6-sided “pyramid”, 
Ψ
p<0.05 vs. solid angle of posterior 
infrazygomatic approach (repeated-measures ANOVA). Data are means ± standard deviations. 
 
DISCUSSION 
We demonstrated that virtual reality can be used to 
study minimally invasive approaches and that solid 
angle with the distance from the skin to the target area 
can be used as a measure to quantify the size of a 
minimally invasive approach, which allows the 
comparison between different approaches. Moreover, 
we showed that posterior infrazygomatic approach is 
larger than suprazygomatic approach into the 
pterygopalatine fossa. 
The virtual reality has already been used for 
planning several surgical and interventional approaches 
in individual patients (Ong et al., 2018; Cao and 
Cerfolio, 2019; Ayoub and Pulijala, 2019; Ghaednia et 
al., 2021), however, its importance in studying 
different minimally invasive approaches systematically 
has not yet been commonly exploited (Deora et al., 
2020). In the present pilot study, we showed the 
feasibility of virtual reality to systematically study two 
minimally invasive approaches into the pterygopalatine 
fossa. There are several advantages of virtual reality to 
study minimally invasive approaches. It can be easily 
reconstructed from CT or MRI volumes. Individual 
approach can be studied in many patients, with 
different morphometric characteristics. It allows the 
correlation of different morphometric characteristics 
with the size of the minimally invasive approach; and 
enables more accurate manual measurements of 
distances and angles of 3D structures. By reconstruct-
ing virtual reality from co-registered CT angiographic 
and MRI volumes, it is easier to take into consideration 
several important anatomic structures such as bones, 
vessels, nerves and other important structures, and to 
study how different structures affect the size and path 
of minimally invasive approach (Knez and Vrtovec, 
2020; Deora et al., 2020; Elsayed et al., 2020). 
However, virtual reality also has some limitations, such 
as ergonomic limitations from prolonged use and 
relatively high costs for hardware and software 
adoption and use (Porcino et al., 2017; Sutherland et 
al., 2019).  
We demonstrated that the minimally invasive 
approach can be represented by different-sided 
“pyramid” like objects with the apex at the target point 
and the base on the skin. Such representation of 
approach is very convenient, since it can be described 
by only a small number of points in 3D space, allowing 
calculation of different parameters describing these 
objects from the coordinates of these points (e.g. solid 
angle, lateral edge lengths, surface area of the base). 
The number of points and thus sidedness of “pyramids” 
that sufficiently describe a concrete minimally invasive 
approach should be preliminarily tested for every 
studied approach. We noted that minimal sided 
“pyramid” that sufficiently describes the suprazygo-
matic approach is three-sided, while the posterior 
infrazygomatic approach is sufficiently described by 
minimally four-sided “pyramid”. Note that increasing 
the number of sides increases the solid angle as the 
“pyramid” better fits the shape of complex anatomical 
structures. However, the observed increase from the 
minimal sufficient number of sides is probably 
clinically insignificant.  
There was a positive correlation between the 
average lateral edge length and transverse diameter of 
the head, reaffirming that body size and thus the 
distance from the skin to the target point could 
influence the parameters describing minimally invasive 
approach that are in relationship with the distance from 
the skin to the target point (e.g. area on the skin from 
where the target point could be reached and volume of 
 JERMAN A ET AL: Quantification of the interventional approaches 
68 
 
the tissue through which the target point could be 
reached). This makes the latter parameters less useful 
for quantifying minimally invasive approach size, since 
the differences in body size which can be due to 
different age, sex, race and body weight could 
significantly affect their value. Solid angle on the other 
hand is not dependent on the distance from the skin to 
the target point but is dependent only on the intrinsic 
anatomic relationship of structures that limit the 
interventional approach. Note that the average distance 
from the skin to the target point is also important and 
should be reported in the studies because it determines 
the needle length that should be used.  
The posterior infrazygomatic approach was 
significantly larger than suprazygomatic approach, 
however, this does not necessarily mean that posterior 
infrazygomatic approach is superior to suprazygomatic 
approach because in the present pilot study we did not 
take the course of maxillary artery into account and the 
number of included subjects was only five. In clinical 
setting it is very important to avoid puncture of the 
maxillary artery to prevent unnecessary complications 
(Anugerah et al., 2020), therefore, further studies using 
CT angiographic volumes to reconstruct virtual reality 
and determine the size of solid angles for each 
approach considering the course of maxillary artery are 
warranted.  
 
CONCLUSION 
We employed virtual reality methods to study the 
spatial geometrical characteristics of two clinically 
important minimally invasive approaches to the 
pterygopalatine fossa, using solid angle with the 
distance from the skin to the target area as a 
quantification index to compare the size of the different 
minimally invasive approaches. Our findings provide 
important preliminary evidence on the feasibility of 
evaluating and comparing different minimally invasive 
approaches using virtual reality systems, and affirm the 
validity of solid angle as a measure of the size of the 
approach.  
ACKNOWLEDGMENTS 
This study was supported by Slovenian research 
agency (Grant No. P3-0043) and the Czech-
BioImaging large RI project (LM2015062 funded by 
MEYS CR) and European Regional Development Fund 
Project - "National infrastructure for biological and 
medical imaging" (No. CZ.02.1.01/0.0/0.0/16_013/ 
0001775). We are thankful to Thomas Puhov for 
technical support, and to Chiedozie K. Ugwoke for 
manuscript proofreading. 
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