Severe maxillary atrophy removes the alveolar bone that conventional implant dentistry depends on, and contemporary full-arch reconstruction answers with remote cortical anchorage: pterygoid, nasal-crest, transnasal, trans-sinus, transpalatal, and zygomatic trajectories that borrow bone the resorptive process spares [2][3]. The PATZi protocol organizes these options into posterior (P), anterior (A), tilted (T), and zygomatic (Z) families and has been described clinically for the moderately and severely atrophic maxilla [2][3]. What the clinical literature has not yet supplied is a quantitative planning layer: a reproducible way to convert a chosen trajectory into a predicted implant length, angulation, and focused inventory before freehand surgery begins.

The conceptual rationale for such a geometric layer was introduced by our group in a brief perspective article [4]. The format of that report could accommodate only the premise, a classification table, and a single geometric figure. The present article is its methodological companion and contains the material a clinician or software developer actually needs to apply or test the method: the complete anatomic definition of all eleven trajectory subtypes with original multiplanar illustrations, the full set of geometric variables and calculation rules, a step-by-step clinical protocol, worked numerical examples, a subtype-specific safety matrix, a reference software specification with regulatory context, and a staged validation framework with defined outcomes. None of this content appears in the perspective, and the two reports are intended to be read as concept and method, respectively.

Two commitments run through the method. First, determinism where determinism is possible: the entry-to-target distance is an exact computation once E and X are selected, and that computation must remain auditable even when artificial intelligence (AI) assists the surrounding workflow. The clinically selected implant length, by contrast, remains conditional on landmark selection, device geometry, the surgeon-defined safety adjustment, and intraoperative changes; the method does not remove that judgment but makes each conditioning step explicit and recordable. Second, honesty about evidence: no data yet demonstrate that this planning method predicts the implant ultimately placed or reduces inventory, and every anticipated benefit in this article is framed as a hypothesis attached to a defined validation outcome.

The classification comprises eleven trajectories in four families, spanning the degrees of alveolar resorption described by Cawood and Howell [5]. Each subtype is defined by a planned entry region, a cortical target, and the multiplanar views required to evaluate its vector; the summary appears in Table 1.

Table 1. Geometric PATZi classification with required multiplanar evaluation.

E, planned post-reduction entry/platform coordinate; X, planned cortical target coordinate; 3D, three-dimensional. Tooth positions follow the Universal Numbering System.

The surgeon first defines the restorative envelope and the final post-alveoloplasty platform; this coordinate is designated E. The desired cortical engagement point is designated X. On a correctly oriented oblique two-dimensional (2D) reformat containing both points, the Pythagorean theorem gives the straight-line distance S2D = sqrt(a² + b²). In the native CBCT coordinate system the preferred calculation is the generalized three-dimensional form, S3D = sqrt((delta x)² + (delta y)² + (delta z)²), which is mathematically exact for the straight-line distance between the two selected voxel coordinates. CBCT is the appropriate modality when three-dimensional anatomy affects management [36].

The method is applied as a structured preoperative worksheet in eight steps (Figure 5): 1) plan the prosthesis and restorative envelope first, establishing the required restorative space and occlusal scheme; 2) plan the alveoloplasty and mark the final platform level, fixing E for each implant site; 3) assign a PATZi subtype to each site from Table 1, based on residual anatomy and the intended arch configuration; 4) place X within the subtype-specific cortical target using the required multiplanar views; 5) calculate S3D and record delta x, delta y, delta z, the unit direction, and the associated angulations; 6) apply the safety review in Table 3 and complete the endpoint-selection record of Section 3.2 for each remote cortical target; 7) map the adjusted length to the implant catalog, selecting a primary implant plus a deliberately narrow backup set, typically one adjacent shorter and one adjacent longer length; and 8) record the plan, including the pre-reduction crest, planned reduction, all coordinates, and the selected inventory, so the prediction can later be audited against the implant actually placed.

In the authors’ six-implant full-arch workflow, the composite (cumulative) insertion torque is defined as the arithmetic sum, in Ncm, of the final seating insertion-torque values of all implants intended to support the immediate provisional prosthesis; in the standard configuration this is all six implants, and any implant excluded from the provisional is excluded from the sum. A composite value greater than 120 Ncm supports consideration of immediate loading, always taken together with implant distribution, cross-arch splinting, per-site stability, bone quality, and prosthetic judgment; the benchmark is part of the authors’ clinical routine and is not proposed as a validated universal criterion.

Figure 5. Geometric PATZi planning workflow linking restorative planning, subtype selection, coordinate definition, vector calculation, safety review, focused inventory, and intraoperative stability assessment.

Table 3. Subtype-specific planning checks.

CBCT, cone-beam computed tomography.

Table 4 applies the calculation to four illustrative sites using schematic coordinates on a model maxilla; the values demonstrate the arithmetic and the catalog-mapping step and are not patient data. Consider the P3 example in detail. The post-reduction platform is set at E = (58.0, 34.0, 18.0) mm and the target within the medial pterygoid plate at X = (66.0, 46.0, 28.0) mm, giving delta x = 8.0, delta y = 12.0, delta z = 10.0 mm. Then S3D = sqrt(64 + 144 + 100) = sqrt(308) = 17.5 mm. If the surgeon’s system offers 15-, 18-, and 20-mm lengths and the plan calls for engagement to, but not beyond, the cortical stop, the 18-mm implant is selected as primary with 15- and 20-mm backups, and the 0.5-mm difference is absorbed within the planned depth of cortical engagement and safety adjustment m. The same arithmetic yields a short intra-alveolar calculation for P2 and a long vector for Z1, which is precisely the point: one data structure and one equation serve every subtype, and only the coordinates change.

The proposed architecture separates deterministic geometry from AI inference (Figure 6). A machine-learning layer may segment the maxilla, sinus, nasal cavity, orbit, pterygoid plates, zygomatic body, palatal cortex, and other exclusion zones;

Table 4. Worked examples on schematic coordinates (illustrative; not patient data).

E, planned entry coordinate; X, cortical target coordinate; S3D = sqrt((delta x)² + (delta y)² + (delta z)²). Catalog lengths are examples; each system’s actual offerings, drill-tip geometry, and the surgeon-defined safety adjustment determine the final selection.

detect candidate E and X landmarks; classify the PATZi subtype; and rank feasible corridors, consistent with published evidence on AI implant planning, navigation, and robotics [37]-[40]. The normative calculation, however, remains auditable: delta = X − E; S3D = sqrt((delta x)² + (delta y)² + (delta z)²), with the unit direction vector and axial, coronal, and sagittal angulations reported alongside length. Minimum inputs are calibrated DICOM (Digital Imaging and Communications in Medicine) CBCT data with voxel spacing and orientation, the post-reduction coordinate E, the target X, catalog dimensions, planned diameter, and surgeon-defined safety constraints. Minimum outputs are subtype, coordinates, S3D, vector angles, nearest-anatomy clearances, primary catalog candidate, backup lengths, uncertainty, and an auditable overlay (Figure 7). Software claiming to implement the method should identify the algorithm version, preserve the defined variables and equations, document deviations, and cite this specification; the specification does not authorize autonomous placement or replace surgeon review.

A clinical application that analyzes patient-specific CBCT data and recommends surgical trajectories may fall within clinical decision-support or Software as a Medical Device frameworks depending on intended use and claims [41]-[44]. Development should therefore include traceable requirements, cybersecurity, dataset governance, external validation, uncertainty reporting, change control, and human-in-the-loop review. This article is a scientific specification, not a cleared medical device.

Three benefits motivate the method, and each is a hypothesis. Dimensional discipline: virtual pterygoid planning already demonstrates that CBCT can define prosthetically driven entry, angulation, and the length actually engaging the target, with substantial individual variation [9][13]—exactly why a patient-specific coordinate method should outperform selecting a long implant from experience alone. Inventory rationalization: a calculated plan identifies a primary implant and a deliberately narrow backup set instead of stocking a broad catalog for each freehand case; the economic benefit remains to be measured prospectively. Communication: the labels E, X, S3D, subtype, and target cortex give the surgeon, restorative dentist, laboratory, implant representative, and software developer one unambiguous vocabulary, and the same annotation scheme structures future datasets for AI development [39][40].

Figure 6. Reference software architecture. AI modules perform segmentation, landmark proposal, and corridor ranking; the deterministic geometry engine calculates the entry-to-target vector, length, angulation, clearances, and catalog mapping, with every recommendation subject to surgeon review and an auditable output report.

Figure 7. Proposed AI-assisted planning and inventory workflow. The inventory-reduction benefit is a hypothesis requiring prospective validation.

The limitations are equally explicit. No data yet demonstrate predictive accuracy, time savings, or inventory reduction. Error can enter through voxel size, patient positioning, metal artifact, segmentation, landmark selection, planned reduction depth, and honest disagreement about where a target should end; regional landmarks such as the Z-point and golden triangle must be defined operationally on CBCT, and interobserver agreement in assigning P1 versus P2 versus P3 must be measured. The subtype labels are planning categories, not permission, and immediate-loading decisions rest on distribution, splinting, bone quality, and prosthetic judgment [45][46], not on any single composite torque number. Clinical validation involving patient imaging or outcomes will require institutional review board approval or exemption and appropriate consent.

A staged pathway is specified (Figure 8). Phase 1 is retrospective: independent surgeons select E and X on de-identified CBCTs of completed full-arch cases, and two co-primary comparisons are prespecified and kept distinct. The first is catalog agreement: whether the catalog length predicted from S3D (the primary candidate and, separately, the primary-plus-backup set) matches the catalog length ultimately placed. The second is coordinate-plan accuracy: the continuous difference between predicted S3D and the achieved entry-to-apex distance measured on postoperative imaging where available. Interobserver agreement on subtype, E, X, and S3D is the co-primary reliability outcome. Every case record documents intraoperative deviations—any change of trajectory subtype, endpoint, or implant system—on a structured form; the deviation rate is reported as an outcome, and the prespecified analysis presents both an intention-to-plan comparison, in which deviated cases count against the prediction, and a per-protocol comparison restricted to cases executed as planned. Angular and apical deviation, insertion torque, immediate-loading decision, inventory pulled versus used, procedural

Figure 8. Proposed four-phase validation pathway: retrospective model development, prospective clinical validation, inventory/cost evaluation, and AI-assisted implementation.

Table 5. Proposed validation outcomes.

RMSE, root-mean-square error; ICC, intraclass correlation coefficient; SKU, stock-keeping unit; 3D, three-dimensional; CBCT, cone-beam computed tomography.

time, and complications are secondary outcomes (Table 5). Phase 2 compares standard versus geometric inventory preparation prospectively. Phase 3 evaluates inventory and cost at scale, and Phase 4 introduces AI-assisted planning in silent mode before any decision support is surfaced, following published AI-validation practice [39][40] and regulatory guidance [41]-[44]. The validation study will be submitted as a full research article; the present specification defines the reference against which it will be judged.

The geometric PATZi method converts a clinically useful cortical-anchorage taxonomy into measurable CBCT coordinates: eleven subtypes, two coordinates per implant, one exact equation, and a defined path from continuous measurement to catalog selection. This article supplies the complete operational protocol, worked calculations, safety matrix, and software reference specification that the introductory perspective [4] could not accommodate. Its anticipated benefits—clearer freehand planning, rational inventory, a shared vocabulary, and a foundation for AI-assisted decision support—remain hypotheses attached to the defined validation outcomes, and the authors invite colleagues who treat the atrophic maxilla to test the framework against their own cases.

During preparation of this work, the authors used AI-assisted writing tools for literature organization, language editing, document formatting, and figure ideation. The authors reviewed and edited all content and take full responsibility for the content of this article.

Conceptualization, A.R.E. and J.G.; methodology, A.R.E., C.H. and J.G.; visualization, C.H.; writing—original draft preparation, A.R.E.; writing—review and editing, C.H. and J.G.; supervision, J.G. All authors have read and agreed to the published version of the manuscript.