Quadrilateral Segmental Analysis (QSA): A Calibration-Free Geometric Method for Detecting Lumbar Spondylolisthesis

1. Introduction

Degenerative lumbar spondylolisthesis is common in adults over 50 years of age and is associated with low back pain and neurogenic claudication [1,2,3,4]. The choice between conservative care, decompression alone and decompression with fusion is guided in part by whether — and how confidently — the slip is detected [5,6,7,8].

In routine practice, however, the slip is still judged visually or against millimeter thresholds — typically ≥3 mm [4,9] — whose validity depends on image scaling: lumbar radiographic magnification varies widely between patients and rises with body mass index [10], measurements scaled with default Digital Imaging and Communications in Medicine (DICOM) factors deviate substantially from marker-calibrated values [11] and inconsistent handling of magnification and landmarks contributes to the wide spread of published prevalence estimates [1].

Instability judgements built on these readings remain largely subjective and depend on the imaging protocol [12,13], and in secondary analyses of the NORDSTEN-DS trial neither established instability criteria nor surgeons' individual recommendations identified patients who benefited from additional fusion [14,15]. Objective classification requires reproducible metrics anchored in normative reference data [16]. Slip percentage is independent of scale, but it reduces each segment to a single ratio of translation to endplate depth and, when graded by Meyerding, to 25% steps [3].

We therefore propose quadrilateral segmental analysis (QSA): a single straight line is drawn along the superior endplate of each vertebra, and every motion segment is represented by the quadrilateral formed by two adjacent endplate lines and the connections of their anterior and posterior endpoints (Figure 1). Anterior translation skews this quadrilateral — the upper posterior interior angle M opens, the lower posterior angle N closes, and the diagonals become unequal (ratio K) — so translation is expressed by angles and dimensionless ratios that are inherently independent of calibration.

The aim of this study was to define the QSA parameters, to estimate their diagnostic accuracy against an independent two-reader reference standard, and to derive purpose-specific decision thresholds, establishing the static component of a quantitative instability work-up.

2. Materials and Methods

2.1. Design and Participants

This was a retrospective diagnostic accuracy analysis of prospectively collected data from the PROGRES cohort (Evaluation of the natural course of lumbar degenerative spine disease and factors influencing its progression, with an analysis of the impact of surgical intervention as a modifying factor in the long-term course of the disease), an ongoing prospective observational cohort study conducted at the Neurosurgical Department of the 4th Military Clinical Hospital in Wrocław, Poland. The study was approved by the Bioethics Committee of the Lower Silesian Medical Chamber in Wrocław, Poland (approval no. 09/DOBD/2025, dated 9 April 2025) and conducted in accordance with the Declaration of Helsinki. Every human participant provided written consent prior to enrolment. The study protocol was developed in accordance with the Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) guidelines for cohort studies (Supplementary Checklist S1).

The present analysis uses preoperative assessment data collected between May 2025 and May 2026. Fifty consecutive adult patients with degenerative lumbar spondylolisthesis were included. Exclusion criteria included isthmic spondylolisthesis, deformity, previous spinal surgery, spinal neoplasms, neurological or muscular diseases causing motor impairment (e.g. amyotrophic lateral sclerosis, multiple sclerosis, myopathies). Demographic and radiographic characteristics of the cohort are summarized in Table 1.

2.2. Radiographs and Reference Standard

Each patient underwent standing lateral radiography of the lumbar spine in a standardized free-standing position. No distance calibration was required because every index parameter is an angle or a dimensionless ratio.

The reference standard was constructed as follows. Two experienced radiologists, blinded to the QSA output and to each other's ratings, independently reviewed every radiograph and classified each of the five motion segments from L1–L2 to L5–S1 as showing anterolisthesis or not, in accordance with recommendations for the construction and reporting of the reference standard in diagnostic accuracy studies [18,19]. Rather than relying on global impression, both readers applied the same predefined set of eight radiographic criteria: anterior translation ≥3 mm / ≥5% [20]; anterior vertebral translation graded according to Meyerding, in which a slip of 0–25% is Grade I, 25–50% Grade II, 50–75% Grade III, 75–100% Grade IV and >100% (spondyloptosis) Grade V [21]; discontinuity of the posterior vertebral body line (George's line); discontinuity of the posterior vertebral body line (Ullmann line); anterior–posterior disc height asymmetry [22]; vacuum disc phenomenon [23,24]; disc space narrowing [25]; traction spur [26]; spinous process step-off sign [27]; and Baastrup sign [28,29]. The first two are direct signs of translation, while the remaining six are indirect markers of segmental degeneration and instability that support the diagnosis when translation is borderline [30,31]. Each segment received a binary verdict (anterolisthesis present or absent). Segments on which the two readers disagreed were referred to a third, senior radiologist, who reviewed the same radiograph blinded to the index test and to the identity of the disagreeing readers, and whose verdict became final; adjudicated panel diagnosis is an accepted strategy when a single, error-free reference test is unavailable [32,33]. The adjudicated segment-level classification constituted the reference standard for all subsequent analyses.

2.3. Index Test: Quadrilateral Segmental Analysis

A single experienced spine surgeon, blinded to the radiologists’ ratings, drew one straight line along the superior endplate of L1, L2, L3, L4, L5 and S1 — six lines per patient. Annotated images were processed by a custom automated Python pipeline that detected the lines by color segmentation and fitted them by principal component analysis.

For each motion segment the quadrilateral A_sup–P_sup–P_inf–A_inf was constructed from the anterior (A) and posterior (P) endpoints of the cranial and caudal endplate lines (Figure 2). QSA parameters were: M, the interior angle at the upper posterior vertex; N, the interior angle at the lower posterior vertex; M − N, the posterior angular asymmetry; K, the ratio of the shorter to the longer diagonal (0 < K ≤ 1, with K = 1 denoting a translationally symmetric segment); and L, the angle between the diagonals at their intersection. Conventional descriptors computed from the same two lines served as comparators: segmental rotation α (the dihedral angle between the endplate lines), total rotation, wedging, tilt, and the two translation measures shear (D) and shift (F), expressed as a percentage of endplate length with anterior translation of the cranial vertebra defined as positive. For the comparison of predictors, translation measures were used as absolute values (|D|, |F|) and the upper posterior angle as its deviation from the neutral value, |M − 95°|.

Geometric consistency of every segment was verified automatically through the closure identity M + N = 180° + α, which follows from the construction and serves as built-in quality control of the annotation and computation chain.

For cohort description only, the same pipeline additionally computed the global spinopelvic parameters from the endplate lines together with a single mark on the femoral heads: lumbar lordosis (LL, L1–S1) with its proximal (PLL, L1–L4) and distal (DLL, L4–S1) arcs, sacral slope (SS), pelvic tilt (PT), pelvic incidence (PI), and the lordosis distribution index (LDI = DLL/LL × 100) [17]. These parameters were not used in any diagnostic analysis; their internal consistency was verified through the identity PI = SS + PT.

2.4. Segment Selection

Because anterolisthesis and retrolisthesis are directionally opposite deformities, segments with retrolisthesis (negative shear) were excluded a priori (n = 90 of 250; Figure 3), leaving 160 segments contributed by all 50 patients. All analyses therefore concern anterior translation only.

2.5. Statistical Analysis

Agreement between the two independent readers was quantified before adjudication and reported in accordance with the Guidelines for Reporting Reliability and Agreement Studies (GRRAS) [34]. The unit of the agreement analysis was the motion segment, and all 250 segments were analyzed together. Because a single κ can be misleading when one category dominates, complementary statistics are reported: raw observed agreement with Wilson confidence intervals (CIs); Cohen’s κ [35] with bootstrap confidence intervals (5,000 resamples) and the Landis and Koch interpretation [36]; the prevalence-adjusted bias-adjusted κ (PABAK) together with the prevalence and bias indices [37]; Gwet’s first-order agreement coefficient AC1, which is robust to the high-prevalence paradox [34]; positive and negative specific agreement; and the exact McNemar test for directional disagreement between readers.

The motion segment was the unit of analysis; within-patient clustering (up to five segments per patient) was addressed by patient-level cluster bootstrap (2,000 resamples, percentile 95% CIs [38]) for all area under the curve (AUC) estimates and AUC differences. Continuous parameters are summarized as median (interquartile range, IQR) and compared with the Mann–Whitney U test, interpreted descriptively. For the classical single-threshold analysis of K, the full receiver operating characteristic (ROC) curve was computed and operating characteristics were evaluated for 61 candidate thresholds spanning [0.70, 1.00] in steps of 0.005; sensitivity, specificity, positive and negative predictive values (PPV, NPV), likelihood ratios (LR+, LR−), accuracy and the Youden index (J) were tabulated for every candidate, and the optimum was reported both on the full grid and restricted to one-decimal thresholds. Predictive performance of K was compared with |D|, |F| and |M − 95°| by paired ΔAUC with cluster-bootstrap CIs. Multivariable logistic models were fitted on standardized predictors and validated by leave-one-out cross-validation (LOO-CV), with the AUC computed on pooled out-of-fold predictions; leave-one-patient-out cross-validation is reported alongside as the more conservative estimate under clustering. Proportions carry Wilson 95% CIs; balanced thresholds were selected by the Youden index [39]. Reporting follows the Standards for Reporting of Diagnostic Accuracy Studies (STARD) 2015 [18]. No a priori sample-size calculation was performed. Analyses used Python 3.12.3 (NumPy 2.4.4, pandas 3.0.2, SciPy 1.17.1, scikit-learn 1.8.0, statsmodels 0.14.6 [40]); α = 0.05, two-sided.

3. Results

3.1. Segments and Prevalence

Fifty patients contributed 250 segments; 90 retrolisthetic segments were excluded, leaving 160 for analysis (Figure 3, Table 2). Anterolisthesis was present in 54 segments (33.8%), located at L4–L5 in 45 cases (83.3%), L3–L4 in 5 (9.3%) and L5–S1 in 4 (7.4%). Every patient contributed at least one positive and at least one negative segment, so each patient served as their own control. The closure identity held within 1.0° in all segments (mean absolute deviation 0.3°, r = 0.996).

Global sagittal alignment in the cohort was that of a degenerative, largely compensated lumbar spine. Lumbar lordosis averaged 52.5 ± 9.7° and pelvic incidence 50.6 ± 9.9°, so that the PI–LL mismatch was minimal (median −2.5°, IQR −10.8 to 3.8) and fell within ±10° in 30 patients (60%), while pelvic tilt averaged 18.0 ± 7.0° and exceeded 20° in 22 patients (44%). Lordosis was nevertheless apportioned unevenly between its two arcs — proximal 22.0 ± 7.5° and distal 30.6 ± 7.4° — giving a lordosis distribution index (LDI, lower-arc L4–S1 lordosis as a percentage of total L1–S1 lordosis [17]) of 58.5 ± 11.6% (median 57.7%, range 40.5–84.8). By the established categories, 37 patients (74%) were within the aligned range of 50–80%, 10 (20%) showed moderate hypolordotic maldistribution (40–49%) and 3 (6%) hyperlordotic maldistribution (> 80%), with none below 40%. The cohort thus combined preserved global balance with a loss of lordosis from the lower arc in one fifth of patients, the alignment pattern expected in degenerative rather than deformity-driven disease.

3.2. Reference Standard: Inter-Reader Agreement

Before adjudication, the two readers agreed on 243 of the 250 segments (97.2%, 95% CI 94.3–98.6), with 49 concordant positive and 194 concordant negative verdicts (Table 3). Cohen’s κ was 0.916 (95% CI 0.848–0.974), corresponding to almost perfect agreement [36]. Because negative verdicts predominated (prevalence index 0.580), the prevalence-robust coefficients are reported alongside κ: PABAK 0.944 and Gwet’s AC1 0.958 (0.923–0.988). Specific agreement was 93.3% (86.9–96.7) for positive and 98.2% (96.4–99.1) for negative verdicts.

Reader 1 classified 55 segments as positive and reader 2 classified 50, with no evidence of systematic directional disagreement (bias index 0.020; exact McNemar p = 0.125). The seven discordant segments (2.8%) were adjudicated by the third radiologist, who sided with reader 1 in six and with reader 2 in one; the final reference identified 54 positive segments.

3.3. Parameter Distributions

All translation-encoding parameters separated the two groups strongly (Table 4, Figure 4). K was lower in anterolisthesis (median 0.861 vs 0.948), N smaller (87° vs 98°), M larger (103.5° vs 94°) and M − N reversed in sign (+16° vs −5°); |D|, |F| and |M − 95°| were correspondingly increased (all p < 0.001). Neither L (p = 0.085) nor the rotational and shape descriptors (rotation p = 0.091, total rotation p = 0.151, wedging p = 0.223, tilt p = 0.179) differed between groups.

3.4. Single-Threshold Discrimination of K

K discriminated anterolisthesis with an AUC of 0.892 (95% CI 0.841–0.937). Across the 61 candidate thresholds in [0.70, 1.00], sensitivity rose from 1.9% at K ≤ 0.70 to 100% at K ≤ 0.95, while specificity fell from 100% to 47.2%; the Youden index peaked at K ≤ 0.905 (J = 0.590, sensitivity 74.1%, specificity 84.9%) and displayed a broad plateau between 0.88 and 0.91, so nearby thresholds perform almost identically (Figure 5). When thresholds are restricted to one decimal — the resolution most likely to be used in routine reporting — the Youden optimum is K ≤ 0.9 (J = 0.516, sensitivity 64.8%, specificity 86.8%, accuracy 79.4%).

Because a single operating point cannot serve every purpose, seven representative thresholds are given in Table 5. The trade-off is monotone and steep at both ends: K ≤ 0.85 confirms a slip (specificity 99.1%, PPV 96.0%, LR+ 47.1) at the cost of missing more than half of them, whereas K ≤ 0.95 excludes a slip entirely (sensitivity 100%, NPV 100%, LR− 0.00) but calls positive more than half of the normal segments.

3.5. K versus Other Geometric Predictors

K was a good but not the strongest single predictor (Table 6, Figure 6). The direct translation measures |D| (AUC 0.943) and |F| (0.941) and the posterior angles N (0.948) and M − N (0.966) all exceeded it, with cluster-bootstrap ΔAUC excluding zero; the deviation of the upper posterior angle from neutral, |M − 95°| (0.851), was statistically indistinguishable from K (ΔAUC −0.041, 95% CI −0.106 to 0.021). L and the rotational descriptors carried essentially no information (AUC 0.56–0.58), confirming that the detectable deformity is translational rather than angular. K correlated only moderately with the translation measures (Spearman ρ = −0.51 with |F|, −0.53 with |D|, −0.56 with |M − 95°|) and weakly with N (−0.35), indicating that it encodes partly independent information — the direction-blind magnitude of translational asymmetry.

3.6. Multivariable Models with Leave-One-Out Validation

Combining K with a posterior angle produced a marked, cross-validated gain (Table 7). K alone reached a LOO-CV AUC of 0.886 (0.833–0.934); adding N raised it to 0.972 (0.951–0.990) and adding shift as well to 0.977 (0.960–0.992), with leave-one-patient-out estimates virtually identical (0.972 and 0.978), indicating that the improvement is not an artefact of within-patient correlation. At a probability cut-off of 0.5, the K + N model classified segments with 85.2% sensitivity, 95.3% specificity and 91.9% accuracy. Since the increment from the two- to the three-parameter model was small (ΔAUC 0.005), K + N is the parsimonious choice for routine use.

3.7. Three-Tier Reading Rule

Rounding K to one decimal, as it would be reported in practice, yields three tiers: both extreme tiers were error-free in this cohort (Table 8). K < 0.9 contained 24 segments, all with anterolisthesis (PPV 100%, 95% CI 86.2–100), and K = 1.0 contained 51 segments, none with anterolisthesis (NPV 100%, 93.0–100), together classifying 46.9% of segments without error. The intermediate tier (K = 0.9) held 85 segments (53.1%) with a 35.3% slip rate, within which N retained full discriminative power (AUC 0.931). Applying N ≤ 93° in that tier gave a two-step rule with 94.4% sensitivity, 95.3% specificity and 95.0% accuracy (leave-one-patient-out 95.6%), improving on N alone by 5.0 percentage points of accuracy (cluster bootstrap 1.9–8.6; McNemar p = 0.008, 8 errors corrected, none introduced). All eight residual errors fell inside the intermediate tier.

Reporting K to one decimal is what makes this tiered reading possible, and it is the reason the rule is built on K rather than on the stronger single predictors N or M − N. A ratio bounded by 1 collapses onto three values a reader can assign at a glance — 0.9 or below, 0.9, and 1.0 — whereas the posterior angles are read on a continuous degree scale on which no comparable rounding exists: 93° and 94° are one degree apart but sit on opposite sides of the decision boundary. The three tiers of Table 8 therefore carry an explicit action, and the error-free extremes are reached without any measurement finer than the first decimal place.

4. Discussion

4.1. Principal Findings

QSA turns a translation problem into an angular one. Anterior displacement skews the segmental quadrilateral, so the posterior angles and the diagonal ratio encode the slip without distance measurement. These calibration-free descriptors discriminated as well as percentage translation computed from the same lines. L and all rotational and wedging descriptors were uninformative. This contrast is partly built into the design, because the reference standard defines anterolisthesis by translation. It is consistent, however, with flexion–extension data in which translation, rather than angulation, was related to symptoms [41]. The standing film records the position of the segment under axial load, and QSA quantifies it. Unlike flexion–extension imaging, it requires no effort from the patient; in one series, detected instability rose from 6% to 38% after analgesia [42].

4.2. Decision Thresholds

The threshold sweep argues against a single cut-off for K. The Youden optimum (0.905) is a compromise; the clinically useful thresholds lie at the extremes, 0.85 for confirmation and 0.95 for exclusion. Rounded to one decimal, the same parameter yields a three-tier rule that resolved 75 of 160 segments (47%) without error (Table 8). All 24 segments in the positive tier were reference-positive (PPV 100%; 95% CI 86.2–100), and all 51 in the negative tier were reference-negative (NPV 100%; 95% CI 93.0–100). Zero observed errors remain compatible with a true error rate of up to 4.8% [43]. The 85 intermediate segments were passed to a second step (N ≤ 93°). Reporting them in a 3 × 2 classification, rather than discarding them, preserves this information [44]. The cross-validated K + N model (AUC 0.972) shows that the two descriptors are complementary: K measures translational asymmetry regardless of direction, whereas N is signed. That N and M − N discriminate better as continuous variables (Table 6) does not make them better reading rules: they are read in degrees and carry no natural rounding, whereas one decimal of K defines the three tiers of Table 8 directly.

4.3. Comparison with Existing Methods

QSA avoids the scaling dependence of millimeter thresholds. It shares scale independence with slip percentage and with endplate-normalized motion metrics [11]. It differs from both by reading the slip from angles alone and by returning a tiered decision with an internal geometric check. Deep-learning detectors reach a pooled sensitivity of 94.7% and specificity of 97.1%, with substantial heterogeneity [45]. A segmentation pipeline published in this journal reported accuracies of 96.1% and 94.4% on internal and external data [46]. QSA reaches comparable discrimination without a trained network, although cross-study comparisons are indirect. Because it uses the endplates and corners that such networks already extract, QSA can serve as an interpretable decision layer on top of automated landmarking.

4.4. Clinical Implications

QSA is intended as an add-on to the routine standing radiograph. The positive tier supports anterolisthesis, the negative tier argues against it, and intermediate segments warrant a second parameter or dynamic imaging. Slip thresholds drive several decisions. Appropriate use criteria separate static from dynamic spondylolisthesis when choosing between decompression and fusion [6]. Trials of basivertebral nerve ablation excluded slips greater than 2 mm [47]. Given lumbar magnification of 1.09–1.63 [10], an uncorrected true 2-mm slip may measure 2.2–3.3 mm. QSA could also provide an objective static input to instability classifications [12,48]. It requires six lines per patient, no calibration and no landmark other than the endplate. The closure identity M + N = 180° + α provides internal quality control.

On flexion–extension or standing–supine pairs, ΔN and ΔK would quantify the dynamic component of instability. This extension requires standardized acquisition, because instability classification varies with imaging protocol and pain control [13,42]. It also requires validation against outcomes. In NORDSTEN-DS, neither established instability criteria nor surgeons' recommendations identified patients who benefited from fusion [14,15].

4.5. Limitations

The study has several strengths: a prospectively enrolled cohort, a predefined criterion-based reference standard with blinded double reading and adjudication, and an index test that is calibration-free by construction and transparent at every step.

This is a single-center analysis with a case-mix typical of degenerative spondylolisthesis, in which 90% of L4–L5 segments were reference-positive; level-specific estimates at that level are therefore imprecise, overall accuracy partly reflects between-level contrasts, and predictive values will be lower where prevalence is lower. Segments are nested within patients, and intervals that ignore this clustering overstate precision [49]; likewise, leave-one-out cross-validation at the segment level allows segments from the same patient to inform both training and testing, and a patient-level scheme would better approximate prospective use [50].

The reference standard was expert visual judgement: criterion-based, double-read (κ 0.916) and adjudicated, it was not verified against dynamic imaging or computed tomography, so part of the residual disagreement — four of the five false positives showed geometric translation above 15% — may reflect the limits of the reference rather than of the index test. Because the reference and the index test read the same radiograph, and the reference includes direct translation criteria, some shared information is unavoidable and may favour translational descriptors.

Above all, a single film captures only the static component of instability: QSA quantifies the loaded position, not motion, and its dynamic extension remains untested. Scale independence also does not imply projection independence: axial rotation, beam obliquity and double endplate contours, as well as irregular endplates, transitional anatomy and pelvic overlap at L5–S1, can affect line placement.

The endplate lines were drawn by a single observer, and a reliability study following GRRAS recommendations [34] with intraclass correlation coefficient (ICC) analysis [51] is required before clinical use; that study must also show that the measurement error of K is small relative to the 0.1-wide intermediate band on which tier assignment depends. Thresholds derived here, including the three-tier boundaries, are data-driven, an approach known to inflate accuracy estimates, particularly in small samples [52], and need external validation. Finally, the analysis was restricted by design to anterior translation (retrolisthesis excluded). The restriction removed no symptomatic segment, as all 54 symptomatic segments lay in the M > 90° domain, but performance in retrolisthesis remains to be established.

5. Conclusions

Quadrilateral segmental analysis detects degenerative lumbar spondylolisthesis — the static hallmark of segmental instability — on a single standing lateral radiograph without any distance calibration.

The diagonal ratio K translates the radiologist's assessment of anterolisthesis into a single number: it discriminates well (AUC 0.892) with purpose-dependent thresholds — 0.85 to confirm, 0.9 for balanced reading, 0.95 to exclude — and, rounded to one decimal, classifies nearly half of all segments without error. The intermediate tier is resolved by the lower posterior angle N, which combined with K reaches a cross-validated AUC of 0.972. QSA is simple enough for routine reporting and fully automatable. Reliability testing, external validation and extension to dynamic image pairs for complete instability assessment are the next steps.