Medical & Biological Engineering & Computing

, Volume 45, Issue 10, pp 939–945

Parametric equations to represent the profile of the human intervertebral disc in the transverse plane


    • School of Engineering Systems, Institute of Health and Biomedical InnovationQueensland University of Technology
  • M. J. Pearcy
    • School of Engineering Systems, Institute of Health and Biomedical InnovationQueensland University of Technology
  • G. J. Pettet
    • School of Mathematical Sciences, Institute of Health and Biomedical InnovationQueensland University of Technology
Original Article

DOI: 10.1007/s11517-007-0242-6

Cite this article as:
Little, J.P., Pearcy, M.J. & Pettet, G.J. Med Bio Eng Comput (2007) 45: 939. doi:10.1007/s11517-007-0242-6


Computational and finite element models of the spine are used to investigate spine and disc mechanics. Subject specific data for the transverse profile of the disc could improve the geometric accuracy of these models. The current study aimed to develop a mathematical algorithm to describe the profile of the disc components, using subject-specific data points. Using data points measured from pictures of human intervertebral discs sectioned in the transverse plane, parametric formulae were derived that mapped the outer profile of the anulus and nucleus. The computed anulus and nucleus profile were a similar shape to the discs from which they were derived. The computed total disc area was similar to the experimental data. The nucleus:disc area ratios were sensitive to the data points defined for each disc. The developed formulae can be easily implemented to provide patient specific data for the disc profile in computational models of the spine.


Intervertebral discTransverse geometryAnulus boundaryNucleus boundary

1 Introduction

Computational and finite element modelling have become popular methods of analysis to investigate the biomechanical effects of intervertebral disc pathologies, spinal surgical procedures and disc/vertebral arthroplasties [7, 9, 15, 22, 25]. For such models, it is desirable to obtain a high level of accuracy in the geometry of both the soft and hard tissues. However, it is difficult to obtain comprehensive information on the dimensions and geometry of the human disc in the transverse plane using current imaging methods.

Existing approaches, which could potentially be applied to derive geometry for the intervertebral disc include the use of imaging modalities such as computed tomography (CT) or magnetic resonance imaging (MRI). While reconstruction of three-dimensional osseous geometry can be readily achieved using subject-specific CT datasets [24], selecting a threshold value to accurately derive soft tissue geometry from such datasets is difficult. Additionally, obtaining a CT dataset capable of providing such a level of soft tissue resolution would require a high radiation exposure CT protocol, which is not appropriate for in vivo studies. Alternatively, it is possible to obtain clearly demarcated boundaries for collagenous tissues using subject-specific MRI datasets, however, reconstruction of three-dimensional geometry from these requires purpose built image processing software [4].

Previous studies have provided data and algorithms to represent average or generic geometry of the vertebral endplates. Panjabi et al. [20] used plain radiographs to report data for the overall dimensions of the vertebral endplates, in both the antero-posterior and in the lateral direction. Using digitized data from CT scans, Hall et al. [13] determined best-fit curves, derived using points on the outer profile of the endplates, to define the L4–S1 vertebral endplates. Mizrahi et al. [19] developed a polar equation to define idealized geometry for the endplate and vertebral body. The equations developed by both Hall et al. [13] and Mizrahi et al. [19] were generic equations to describe an average endplate profile and thus, did not use patient specific data for anatomical landmarks on the endplate profiles. Furthermore, these equations were developed specifically to describe the vertebral profiles in the transverse plane and did not necessarily define similar profiles for the anulus and nucleus in this plane.

The aim of the current study was to develop a mathematical algorithm that parametrically represented the anulus and nucleus boundaries in the transverse plane. These formulae were to be applied on the basis of a series of patient specific points, defining turning points in the outer anulus and nucleus profiles. These formulae were to be derived using reliable geometric data taken from photographs of transversely sectioned, human intervertebral discs. It was intended that following development of the formulae using these detailed data, they could be employed with confidence in future studies, using radiographic or computed tomography images.

2 Methods

Using images obtained by Vernon-Roberts et al. [23] for the L4/5 lumbar intervertebral discs (Fig. 1), 12 points were defined on the outer boundary of the anulus and nucleus (six on each boundary) in healthy or mildly degenerate discs. Parametric equations were derived and used to generate co-ordinate points that mapped the profiles of these boundaries in two-dimensional space. Measured morphological data [23] were used for validation of the final formulae.
Fig. 1

Photograph of a transversely sectioned human intervertebral disc [23]

2.1 Anulus fibrosus outer profile

The following section details the development of formulae to map the outer boundary of the anulus fibrosus and the criteria used to determine the accuracy of these formulae.

2.1.1 Disc tracing and intersection points

Photographs of sectioned intervertebral discs (Fig. 1) were used to obtain pencil tracings of the outer anulus boundaries in the L4/5 disc of 18 specimens from various age groups. Each anulus tracing was divided into six separate sectors. These sectors were defined using four tangent lines and six intersection points on the anulus boundary. The lines and points were defined as follows (Fig. 2).
  • Points 5 and 1—the 2 intersections of a line (line A) drawn tangent to the most posterior points on the anulus boundary.

  • Points 2 and 4—defined by the intersection of 2 lines drawn perpendicular to line A (called lines B and D, respectively) and tangent to the lateral-most points.

  • Point 3—defined by a line (line C) parallel to A and intersecting the most anterior point.

  • Point 0—intersection of a line (line E) parallel to line A and intersecting the most anterior point in the posterior concavity of the disc.

The x- and y-axis were parallel to the tangent lines, D and A, respectively (Fig. 3a). The rectangular coordinate system origin was located at the geometric centre of the rectangle defined by the four lines, A through D.
Fig. 2

Tangent lines creating the rectangular boundary in the transverse sectioned view of a disc
Fig. 3

a Location of anulus boundary points and radii. b Schematic illustrating the parametric formula for an elliptical arc (Note Due to the method of construction, r1 cos (θ1) = r5 cos (θ5)

Radial lines from the geometric centre to the points 0 through 5, were denoted r0, r1,…,r5. The lengths of these radii and the angle (θ) from the x-axis to the radii were measured (Fig. 3).

2.1.2 Mathematical formulae

Each point on the anulus boundary (Fig. 3a) was defined using parametric equations (Fig. 3b). The curves joining points 1 through 5 were represented using four elliptical arcs (Eq. 1).

Equation 1 shows parametric equations to represent the arcs joining points 1 through 5.

$$ \begin{aligned}{} & x{\left( \phi \right)} = x_{j} + {\left| {{\left( {x_{i} - x_{j} } \right)}} \right|}\cos {\left( \phi \right)}\quad {\text{where}},{\text{ }}\phi {\text{ }} = {\text{ }}\frac{{\theta - \theta _{i} }} {{\theta _{j} - \theta _{i} }}{\text{ }}\frac{\pi } {2} + p \\ & y{\left( \phi \right)} = y_{i} + {\left| {{\left( {y_{j} - y_{i} } \right)}} \right|}\sin {\left( \phi \right)} \\ \end{aligned} $$
The parameterized curves are given by Eq. 1 using the following values for i, j and p:
  • Sector 1:

  • Point i = point 1

  • Point j = point 2; p = 0

  • i.e. i = 1, j = 2

  • Sector 2:

  • i = 3, j = 2; \( p = \Pi \mathord{\left/ {\vphantom {\Pi 2}} \right. \kern-\nulldelimiterspace} 2 \)

  • Sector 3:

  • i = 3, j = 4; \( p = \Pi \)

  • Sector 4:

  • i = 5, j = 4; \( p = {3\Pi } \mathord{\left/ {\vphantom {{3\Pi } 2}} \right. \kern-\nulldelimiterspace} 2 \)

The curves generated to fit sectors 0 and 5 varied from the other sectors due to the inflection point present at their boundaries. Each of these sectors was fit with a cubic equation (Eq. 2). To determine the constants for the cubic equation, points 0, 1 and 5 were used as boundary conditions to satisfy the cubic equation, x = f(y), and the first derivative of this function, x′ = f(y) (Eq. 2).

Equation 2 shows cubic equation constants used to fit curves to sectors 0 and 5

$$ \begin{aligned}{} & x = a{\text{ }}y^{3} + b{\text{ }}y^{2} + c{\text{ }}y{\text{ }} + {\text{ }}d \\ & x'{\text{ }} = 3a{\text{ }}y^{2} + 2b{\text{ }}y{\text{ }} + {\text{ }}c \\ \end{aligned} $$
Boundary conditions:
  • Points, (x0, y0) and (xk, yk)

where, y0 = 0 for both sector 0 and 5, (xk, yk) = (x1, y1) for sector 0, (xk, yk) = (x5, y5) for sector 5
  • Derivative, x′ = 0

Using the above boundary conditions, the cubic constants are,
$$ a = 2\frac{{{\left( {x_{0} - x_{k} } \right)}}} {{y^{3}_{k} }},\quad b = - 3\frac{{{\left( {x_{0} - x_{k} } \right)}}} {{y^{2}_{k} }},\quad c = 0,\quad d = x_{0} $$

2.1.3 Validation of the anulus formulae

Two criteria were used to validate the final formulae. Firstly, a visual validation was used. If the general shape of the disc matched that of the original specimen, in terms of curvature, gradients and turning points, then the formulae were considered to be suitable. Secondly, the area bounded by the outer anulus profile was determined by numerical integration using the sum of areas method. This area was compared with the disc area data from Vernon-Roberts et al. [23]. Statistical analyses of the results were carried out using Matlab 7.0.1 (Mathworks Inc., Massachusetts, USA). The continuous outcome variable, anulus area, was tested to determine if it was normally distributed using a normal probability plot. If the outcome variable was normally distributed, a Pearson’s correlation co-efficient, ‘r’, was used to determine associations between this variable and the continuous independent variable, experimentally determined area [23]. In the event that the outcome variable was not normally distributed, associations were determined using the non-parametric statistic, the Spearman correlation co-efficient, ‘rho’.

2.2 Nucleus pulposus outer profile

This section details the development of formulae to map the outer profile of the nucleus pulposus and the criteria used to validate this technique.

2.2.1 Disc tracing and intersection points

Measurements taken for the nucleus boundary were similar to those obtained for the anulus boundary. A local x–y co-ordinate system was constructed using the same approach as was outlined for the anulus. Three additional dimensions were obtained, which allowed the nucleus location and orientation relative to the anulus to be defined. These additional dimensions included:
  • the radial distance from the geometric centre of the anulus to the geometric centre of the nucleus, R;

  • the angle from the x-axis of the anulus to a line between the anulus geometric centre and the nucleus geometric centre, θR; and

  • the angle between the x-axis of the anulus and the x-axis of the nucleus, θaxes.

Fewer specimens were used to determine the formulae for the nucleus profile because only specimens with a distinct boundary between the anulus and nucleus were chosen.

2.2.2 Mathematical formulae

The equations used to define the nucleus boundary were the same as for the anulus boundary. However, the co-ordinate points calculated were in the local, two dimensional co-ordinate system defined by the six points on the nucleus boundary. These data points were transformed into the anulus co-ordinate system using a rigid body transformation matrix derived from the parameters R, θR and θaxes.

2.2.3 Validation of the nucleus equations

Three criteria were used for the validation of the nucleus boundary equations. Both the visual and the area validations outlined for the anulus were applied to the nucleus. The nucleus areas determined by Vernon-Roberts et al. [23] were expressed in terms of the ratio of the nucleus area to the total disc area. In presenting this ratio for the computed areas, the computed nucleus areas were compared to the total disc areas from the results of Vernon-Roberts et al. [23]. This avoided inclusion of the error present in the computed total disc area. The computed and experimentally determined nucleus ratios were compared statistically using a similar approach to that employed for the anulus profile. Additionally, the offset between the centre of the nucleus and anulus was compared to experimentally measured values [23].

3 Results

3.1 Validation of the anulus formulae

The formulae produced plots with outer boundaries very similar to the specimens from which they were taken (Fig. 4). However, each sector was slightly more curved outward than the original tracing of the disc. This was due to the initial assumption that the sector formulae were of an elliptical form.
Fig. 4

Photograph of a transversely sectioned human disc [23] with calculated anulus profile overlaid in light grey and nucleus profile overlaid in dark grey. Black squares show the points used to calculate the profiles

The discrepancy between the areas calculated mathematically and experimentally was low (Fig. 5). Also, this error was generally negative, indicating that the areas calculated from the formulae were conservative and the discrepancy between the two profiles due to the assumption of an overall elliptical shape, did not have a considerable impact on the final disc shape.
Fig. 5

a Comparison of total disc area with experimental results [23]; b percentage variation in disc area compared to the area values from Vernon-Roberts et al. [23] (The horizontal axes in both figures shows specimen number)

The anulus areas, both computed and experimentally determined, were not normally distributed. The Spearman correlation co-efficient, rho, was 0.967 for a comparison of the computed and experimental anulus areas, which was highly significant at the 0.01 level.

3.2 Validation of the nucleus pulposus formulae

Based on a visual comparison of the original specimens, it was apparent that the nucleus was subject to a much greater degree of variation in its profile than the outer anulus boundary. A special case for the nucleus boundary was encountered for a specimen with a near circular nucleus boundary, resulting in only four sectors. Because the equations for the nucleus generated points for 6 sectors, it was necessary to define two arbitrary points on the nucleus boundary which were very close to the posterior-most point ‘0’. The final calculated area of the nucleus for this specimen was similar to that determined by Vernon-Roberts et al. [23] (Fig. 6, Specimen 7).
Fig. 6

Comparison of the computed and experimental nucleus areas with the total experimental disc area, expressed as an area ratio. (The horizontal axis shows specimen number)

Visually, the shape, location and orientation of the nuclei were similar to that of the original specimens. The only discrepancy was the increased curvature on the computed nucleus, which was attributed to the elliptical formulae used. However, this increased curvature was not as pronounced over the small sector lengths of the nucleus.

Given how sensitive the nucleus:disc area ratios were to variation, there was reasonable agreement between the experimental results and the computed ratios (Fig. 6). The nucleus area ratios were not normally distributed. The Spearman correlation co-efficient, rho, was 0.943 for a comparison of the calculated and experimentally determined nucleus ratios, which was significant at the 0.02 level.

The error in the offset between the centroids of the nucleus and anulus ranged from 0.63 to 1.9 mm with a mean of 1.3 mm and standard deviation 0.52 mm (Table 1).
Table 1

Comparison between the nucleus offset determined in the current study and the nucleus offset determined experimentally [15]


Computed offset (mm)

Experimental offset (mm)

Absolute error in the offset (mm)










0. 844










0. 638


0. 392


0. 628



Standard deviation

0. 521

4 Discussion

In the current study, a mathematical algorithm was developed to map the outer profiles of the anulus and nucleus using images of the transverse plane through the intervertebral disc. Computational and finite element models have been developed in previous studies, to simulate varying levels of complexity in the spinal geometry—from individual intervertebral discs [8] and spinal motion segments [9, 10, 14, 21] to whole spine osseous/osseoligamentous models [2, 6, 11, 15]. Increasingly, it is becoming desirable to base these models on patient specific rather than average or idealized data, in order to simulate the specific response of a patient’s spinal anatomy to surgical procedures [3, 5, 6, 11, 12, 15]. Thus, in this study of cadaveric specimens, subject specific geometry for the intervertebral disc was obtained.

This algorithm may be easily implemented in models of the spine (although it has only been validated for the lumbar spine) and involves minimal computational time and resources. Using this algorithm, the series of calculated coordinate points mapping the disc profile may be used to define nodal points in a finite element mesh [16, 17]. This process has been automated using Matlab 7.0.1, the methods and results for which are detailed elsewhere [1618]. Thus, by incorporating the parametric algorithm derived in the current study into purpose developed pre-processing software, a subject-specific finite element mesh can be derived [1, 18]. Using the developed parametric equations, modelling of individual patients would be possible by obtaining patient specific data points for the intervertebral disc from MRI images taken transversely through the disc or using points obtained from CT slices through the vertebral endplates.

A limitation for the use of these formulae in defining the profile of the nucleus pulposus was the necessity to obtain transverse images of intervertebral discs with a well-defined outer boundary for the nucleus pulposus. This would likely only occur in relatively healthy intervertebral discs or discs with minimal degeneration of the nucleus.

Modelling the anulus boundary using the six formulae showed good agreement between the experimental [23] and computed anulus areas. The computed anulus profiles were visually similar to the specimens on which they were based. Comparing the computed anulus profile with the idealized vertebral endplate profile calculated using the algorithm from Mizrahi et al. [19] showed similar contours laterally. However, the posterior concavity observed in the transverse disc profile was not reproduced by the equation derived by Mizrahi et al. [19].

While visual comparison of the nucleus profiles with photographs showed a similar boundary and statistical comparison of computed and experimentally determined nucleus area ratios showed significant correlation, there was notable variation in the computed nucleus offset when compared to the experimental data. This was due to the inherent inter-observer error in locating a precise boundary for the nucleus in a healthy and hydrated intervertebral disc. The errors in the computed nucleus offsets were comparatively small given the region over which the nucleus boundary could have reasonably been interpreted was approximately 2.5–3.5 mm on some tracings. Given the difficulty in correctly locating the nucleus in the transverse plane and the large difference in the nucleus centroid compared to experimental values, it may be possible to provide just as accurate geometry for the nucleus by offsetting the profile for the anulus boundary radially inward.

5 Conclusions

A mathematical algorithm was developed to map the outer, transverse profile of the anulus fibrosus and nucleus pulposus, using data points derived from patient-specific imaging. Based on the low variation between the experimental and the computed anulus area, it was considered that the formulae could adequately model the true form of the disc in the transverse plane.

Copyright information

© International Federation for Medical and Biological Engineering 2007