The motion segment is the smallest functional unit of the spine[1]. The spinal motion segment (SMS) is deformed under the various loads of daily activities, typically described as flexion-extension, torsion, lateral bending, shear, and compression. The mechanical properties of spinal motion segments can be derived from load-deflection curves (Figure 1). One example of such a property is the range of motion (ROM), which is defined as the deflection difference between the maximum applied loads in each direction (Figure 1). Another quantification used in the load-deflection curve is the neutral zone (NZ), described as the range over which a motion segment moves with minimal resistance [1, 2]. This range increases with (mild) degeneration of the intervertebral disc[3]. Excessive intervertebral motion may result in pain due to overstretching of soft tissues and/or entrapment of nerve roots. Clinically, excessive motion is referred to as spinal instability, which may result from, among others, degenerative disc disease and spondylolisthesis. Degeneration of the intervertebral disc is routinely assessed radiologically, but it appears to be poorly related to pain and instability[4]. The NZ offers a more direct measure of spinal instability and more recently techniques have been developed to estimate this parameter in vivo[5]. The NZ was found to be more sensitive than the ROM in quantifying spinal destabilization (e.g. caused by spondylolisthesis[5, 6]).

Panjabi [2] operationally defined the NZ based on the finding that a motion segment does not return to its initial position after loading in a particular direction. This residual displacement, which was measured 30 seconds after removal of the load, was equated with the magnitude of the NZ. The NZ is thus seen as play in the SMS when moving from one direction to the other, which is essentially different from the original definition as the zone of minimal stiffness[2]. Residual displacement results from the visco- and poro-elastic (i.e. time-dependent) properties of the intervertebral disc, which cause the SMS to creep under load. This implies that the magnitude of the NZ, given this operational definition, depends on the measurement time after removal of the load as well as on the loading history before relaxation. Both the maximum load applied, and thus the loading history, and the point in time at which the residual displacement is measured, are arbitrarily chosen.

Where Panjabi used static residual displacements for determining the magnitude of the NZ, Wilke and colleagues determined the NZ in dynamic loading experiments[7]. The NZ was defined as the angulation difference at zero load between the two directions of motion (Figure 1). The continuous motion also allows defining the neutral zone compliance, which was defined as the tangent to the curve at zero load in both directions. The advantage of this method is that the NZ is derived from the load-deflection curves without involving any arbitrary choices, but the deformation at zero load does not per se mark the borders of minimal segment stiffness. For example, the spinal motion segment may not have been embedded accurately in its neutral position, thereby shifting the zero-load condition towards one end of the ROM, resulting in a different-presumably smaller- NZ region. In the case of sagittal bending where asymmetry exists in flexion and extension, one may in fact question where the neutral position of a SMS actually is. More importantly, it is not clear what the angulation difference at zero load physically means: it is tempting to call it the backlash of the SMS, but that would imply that the SMS can move freely between the given boundaries of the NZ. This is not the case, because the NZ stiffness is not zero. Instead, like Panjabi's operational definition for the NZ, the angulation difference is a consequence of the hysteresis of the SMS, which in turn depends on the loading history of the poro-visco-elastic structure.

Spenciner and colleages [8] defined the borders of the neutral zone as the intersection of the y-axis (zero load) and the tangent line to the load-deflection curve drawn at the point of movement reversal (maximum load). With this method, the NZ obviously depends on the maximum load applied and thus the ROM. When the applied maximum load is smaller, the slope of the curve will be larger, causing a steeper slope of the tangent line, which results in a smaller NZ. Thus, the stiffness at the reversal points is taken into account but not the stiffness of the NZ.

Sarver and Elliott[9] defined the neutral zone for tension-compression loading using a tri-linear fit to the load-deflection curve. The middle region of the best fit, with the steepest slope (high compliance) was defined as the neutral zoned. However, the kind of data obtained under bending and torsion loading cannot often be fitted well with a tri-linear function, as clear transition zones are present in load-deflection curves under bending. (Figure 1).

Thompson and colleagues[10] explicitly defined the NZ in terms of minimal stiffness. They fitted a fourth degree polynomial over the load-deflection data. The stiffness was then simply obtained as the first derivative of this polynomial and the borders of the NZ were defined at 0.05 Nm/° and -0.05 Nm/°. There are two points of concern with this approach. First, arbitrary boundaries for the minimal stiffness region are used. Secondly, it appears that the borders are defined in such a way that a NZ cannot always be defined because the first derivative does not always exceed the thresholds of 0.05 Nm/° and -0.05 Nm/° (data not shown).

It is generally accepted that the NZ of a spinal segment is defined as the region of minimal stiffness. However, only the methods by Sarver and Elliot and Thompson et al. correctly consider the SMS stiffness (or its inverse: the compliance) to determine the boundaries of the NZ region. We built on these methods by developing an operational definition of the NZ based on a mathematical model of the load-deflection data, that is: free from the limitations mentioned above. Thus, the general aim of the paper is to present and evaluate a strict mathematical definition of the Neutral Zone of a spinal motion segment. In order to evaluate the new method for determining NZ magnitude and stiffness, we performed experimental biomechanical testing on porcine lumbar motion segments. The specific aims of these experiments were [1] to determine the degree of fit of a newly defined function to the load-deflection data of SMS and the degree of linear fit over the load-deflection data within the NZ to estimate NZ stiffness; [2] to establish the robustness of the method for the direction of loading (flexion-extension vs. extension-flexion); [3] to compare calculated NZ and NZ stiffness to the angulation difference at zero load and the related stiffness as defined by Wilke; [4] to explore whether NZ magnitude as defined by the angulation difference at zero load indeed reflects hysteresis; and [5] to determine the effect of compressive loading on NZ, NZ stiffness, and hysteresis. The general hypothesis is that the two methods will provide different results, since they represent different underlying concepts. In addition, we hypothesized that NZ magnitude as defined by the angulation difference at zero load is correlated to hysteresis, because this actually measures the distance between the flexion-extension and extension-flexion curves at zero load. With respect to the effect of compressive loading, we hypothesized in line with literature[11] that a period of axial compression decreases the NZ magnitude of the SMS as estimated with our method, while it increases the hysteresis. Given the assumed relationship of the angulation difference at zero load with hysteresis, we also hypothesized the NZ as determined with Wilke's method to be increased after axial loading.

In this study, we propose a new operational definition of the neutral zone (NZ), the region of minimal stiffness or maximal compliance of a spinal motion segment, with an objective, mathematical method. Building on existing methods, this definition is based on the mechanical properties of the spinal motion segment and it does not depend on arbitrary choices in the analysis or on orientation of the segment after embedding. A summed sigmoid is fitted over the data and the maximum and minimum of the second derivative are used as the unique and objective borders of the NZ. The summed sigmoid function provided an excellent fit to experimental load-deflection data of porcine SMS. The neutral zone defined according to the new method was characterized by a nearly linear load-deflection relationship. When applied to freshly thawed lumbar spines, the magnitude of the neutral zone according to the new mathematical definition was significantly larger than and not correlated to the angulation difference at zero load, the most commonly used operational definition of the NZ proposed by Wilke[14–18]. Our measurements showed that the NZ magnitude is strongly influenced by loading history: a 7-hour continued axial compression decreased the NZ magnitude as determined with the new method, but increased the angulation difference at zero load as determined by Wilke's method, resulting in a remarkably similar NZ magnitude for both methods. The NZ stiffness was not dependent on method, nor on loading history.

It should be emphasised that no theory exists which prescribes the type of curve to be used for fitting the load-deflection data. However, it also should be noted that the NZ exists by virtue of the S-shaped load-deflection curve typically observed in bending- and torsion experiments. Several other mathematical functions (e.g.: polynomials, exponential functions) could (and in fact have been[10]) tested as well, but the summation of two sigmoids captures both the S-shape and the asymmetry of the typical SMS load-deflection curve and provides coefficients that are interpretable in mechanical terms. Despite the excellent fit of the summed sigmoid function (r² > 0.976), it should be emphasized that it is still no more than an approximation of the real load-deflection curve. Its advantages are that it disregards the noise of the experimental data and it allows for an analytical calculation of the zone with the least stiffness. Other curve types may do that as well or even better, crucial is that by curve fitting, the NZ and its stiffness can objectively be derived from the experimental data in a way that is consistent with the definition of the neutral zone and with the typical asymmetrical S-form of the load-deflection curve.

The summed sigmoid function approached the experimental data quite well, but not always for the full range of motion (Figure 7a). This means that Eq.1 does not fully cover the SMS load-deflection behaviour, especially at higher bending loads. It is unclear, however, what this practically means for the calculation of the NZ magnitude and stiffness: even for the worst fit in our series, NZ magnitude and stiffness seem reasonably well estimated (Figure 7a). Apparently, the double sigmoid function correctly describes the inflection points of compliance, which define both magnitude and stiffness of the NZ. Higher order functions like a triple sigmoid would probably improve the goodness of fit, but also increases the risk of fitting artefacts.

The application of our model to data drawn from experiments with young pigs may mask a limitation of the method: experienced experimenters know that older and more pathological (human) specimens may exhibit a much more discontinuous response (Figure 7b), as would specimens that have been experimentally destabilized, with clear and abrupt inflection points in the loading and unloading curves. Such data, however, may be difficult to interpret. Discontinuities may arise from measurement artefacts like suboptimal embedding or slip-stick within the testing device. While such errors should be avoided, they may be difficult to recognize even for an experienced experimenter. Furthermore, discontinuities may occur as a result of impingement of the vertebrae, or due to tissue damage in the specimen. It appears that curves still may fit very well, but the interpretation becomes more questionable: for example, what does a local maximum within the neutral zone really mean? The newly proposed method nor any other method can answer such questions. The neutral zone as also defined by others assumes a more or less smooth, asymmetrical S-shaped load-deflection curve. As such, the new method presents an excellent way of quantifying magnitude and stiffness of the neutral zone. Lack of fit can easily be flagged and if data cannot be fitted properly, results should be interpreted with care.

The load-deflection curves before and after a seven-hour period of axial compression showed an interesting difference between the NZ as defined by the double sigmoid function and the angular deflection at zero load as defined by Wilke et al.[7] NZ magnitude decreased when based on stiffness considerations, but increased when based on angular difference at zero loads. Both parameters thus represent another aspect of SMS behavior. As can be appreciated from Figure 4, the distance between the flexion-extension curve and the extension-flexion curve has increased. Accordingly, we found a consistent increase in hysteresis, despite the strong decrease in the range of motion. The angular deflection at zero load thus appears as a measure of hysteresis of the load-deflection curve and in fact a moderate positive correlation was found. Why the hysteresis increases while the NZ magnitude decreases, is not quite understood, but it is in line with earlier reports[11]. Presumably, axial compression released interstitial fluid from the specimens, thereby leaving the intervertebral disc more viscous than before axial compression was applied.

Interestingly, the NZ magnitude after the seven hour compression period was remarkably similar for both methods: the magnitude was the same (1.96° p = 0.98) and they were highly correlated (r = 0.893; p = 0.003; Figure 5). This would indicate that the methods are interchangeable, but the other findings in the current study show that this is not the case: the angulation difference is both conceptually and numerically different from the stiffness derived NZ magnitude. In contrast to other reports, we only found an insignificant increase in NZ stiffness after axial compression when determined by Wilke's method, and no increase when determined by the double sigmoid function. We suggest that this occurred because we released compression just before testing; which resulted in a situation in which the disc contained less fluid, but the fibers of the disc were released as in the original situation before axial compression had been applied.

All methods reviewed in the literature define only one NZ, whereas we defined a NZ for both directions of the load-deflection curve. This is inherent to the fact that poro-visco-elastic behaviour of the spinal segment is reflected in a hysteresis loop. However, NZ magnitudes in flexion and extension appeared to be similar and quite well correlated (see Figure 5a for an illustration of this and Figure 5b for a counterexample). Practically, one thus might define the NZ magnitude as the NZ magnitude in either direction, or alternatively as their average. The same applies to the NZ stiffness. We urge future users to report both NZ magnitude and NZ stiffness, as these are more or less independent values (r² = 0.24) and provide valuable information on the mechanical behaviour of the spinal motion segment.

A strict mathematical definition was proposed to objectively quantify the size of the neutral zone as well as its stiffness. In contrast to existing methods, this definition is based solely on the mechanical properties of the spinal motion segment and it does not depend on arbitrary choices in thresholds or orientation of the segment after embedding. Furthermore, we showed that this method is sensitive to changes in the loading history of the segment. We hope this method will prove useful to more correctly quantify spinal segment behaviour.