Graph-Theoretic Optimization of Skull Base Surgical Corridors: Minimizing Cranial Nerve Disruption Risk

Abstract

Skull base surgery demands precise corridor selection to maximize lesion exposure while minimizing cranial nerve injury. Despite decades of refinement, approach selection remains guided primarily by individual expertise rather than formal quantitative frameworks. We present a graph-theoretic model of skull base anatomy in which 30 anatomical nodes and approximately 28 cranial nerve path edges encode the spatial relationships between surgical landmarks and neural structures. Ten standard surgical approaches are analyzed under three weighting schemes — uniform, functional impact, and binary critical — to generate composite risk scores, identify optimal approaches for 15 target regions, and quantify inter-approach redundancy via Jaccard similarity. Betweenness centrality analysis reveals Meckel's cave (0.1632) and the cavernous sinus (0.1526) as the principal anatomical bottlenecks through which multiple nerve paths converge. The endonasal transsphenoidal approach achieves the highest efficiency ratio (0.2500 targets per unit risk), while the orbitozygomatic approach carries the greatest functional risk (score = 48). CN VII (facial nerve) is the most surgically vulnerable nerve, at risk in six of ten approaches. A greedy set-cover analysis determines that a minimum of seven approaches is required to reach all 15 target regions, with the endonasal transsphenoidal approach selected first due to its superior efficiency. Pairwise combined-strategy evaluation across all 45 two-approach combinations identifies pterional plus endonasal transsphenoidal as the most efficient pair (7 targets, combined risk 35, efficiency 0.2000). Monte Carlo sensitivity analysis (10,000 iterations, ±30% weight perturbation) confirms that the endonasal and translabyrinthine approaches occupy rank 1 in 44.8% and 44.1% of iterations, respectively. This framework provides a reproducible, quantitative basis for comparing surgical corridors in skull base surgery.

1. Introduction

Skull base surgery addresses pathology at the bony platform separating the brain from the face, orbits, paranasal sinuses, and upper cervical spine [1]. This anatomically dense region is traversed by all twelve cranial nerves, the internal carotid and vertebral arteries, and the brainstem itself. Lesions arising in or extending to the skull base — including meningiomas, vestibular schwannomas, chordomas, craniopharyngiomas, and glomus tumors — require surgical corridors that navigate between these critical structures [2].

The central problem of skull base surgery is approach selection: choosing a path through bone, dura, and neurovascular structures that provides sufficient operative exposure of the target while minimizing the probability of iatrogenic cranial nerve injury [1,2]. Over the past four decades, approximately ten standard approaches have been codified, each offering access to specific skull base regions at the cost of placing particular cranial nerves at risk [2,3]. The pterional craniotomy, retrosigmoid approach, translabyrinthine approach, middle fossa approach, far lateral approach, subtemporal approach, orbitozygomatic approach, endoscopic endonasal transsphenoidal approach, presigmoid approach, and anterior transpetrosal (Kawase) approach collectively span the surgical armamentarium for the skull base [2,3].

Currently, the selection among these approaches relies on the surgeon's training, experience, and subjective familiarity with the anatomy [4]. While expert consensus and case series inform general principles — for instance, the translabyrinthine approach is favored for large vestibular schwannomas when hearing is already lost, while the middle fossa approach is reserved for small intracanalicular tumors with serviceable hearing [5] — no formal quantitative framework exists for systematically comparing approaches in terms of cranial nerve disruption risk.

Graph theory offers a natural formalization of this problem. Anatomical landmarks and regions constitute nodes, cranial nerve paths constitute weighted edges, and surgical approaches define subgraphs within which nerve edges are placed at risk. Well-established algorithms — betweenness centrality [6], Jaccard similarity, greedy set cover, and combinatorial pairwise analysis — can then be applied to identify anatomical bottlenecks, redundant versus complementary approach pairs, minimum approach sets for full target coverage, and optimal two-approach strategies.

We present a graph-theoretic model of skull base surgical corridors. Our model encodes 30 anatomical nodes, approximately 28 cranial nerve path edges weighted by functional impact of injury, and 10 standard surgical approaches. We compute approach risk scores under three weighting schemes, identify optimal approaches for 15 target regions, characterize inter-approach similarity, determine the minimum set of approaches required for full target coverage, evaluate all pairwise combined strategies, and validate the robustness of the rankings via Monte Carlo sensitivity analysis. The goal is not to replace clinical judgment but to provide a formal, reproducible, and extensible framework for quantifying and comparing cranial nerve risk across surgical approaches.

2. Methods

2.1 Graph Construction

The skull base was modeled as a weighted undirected graph G = (V, E), where nodes represent anatomical landmarks and edges represent cranial nerve pathways. The graph comprised 30 nodes spanning foramina (cribriform plate, optic canals, superior orbital fissures, foramen rotundum, foramen ovale, internal acoustic meatus, jugular foramen, hypoglossal canal, foramen magnum), anatomical regions (sella turcica, cavernous sinus, Meckel's cave, petrous apex, cerebellopontine angle, tentorial edge, upper and lower clivus, anterior/middle/posterior cranial fossae), brainstem segments (midbrain, pons, medulla), and surface landmarks (pterion, asterion, mastoid, orbit, stylomastoid foramen). Only right-sided structures were modeled, exploiting the bilateral symmetry of cranial anatomy.

Approximately 28 edges encoded the intracranial courses of all 12 cranial nerves (CN I–XII), including subdivision of the trigeminal nerve into its ophthalmic (V1), maxillary (V2), and mandibular (V3) divisions. Multi-segment nerves were represented by sequential edges connecting anatomical waypoints. For example, CN VII (facial nerve) was modeled as three edges: pons → cerebellopontine angle → internal acoustic meatus → stylomastoid foramen.

It should be noted that the graph models cranial nerve paths specifically rather than general skull base anatomy. Edges represent nerve trajectories, and the graph is intentionally sparse relative to a fully connected anatomical model. This design choice ensures that risk computations reflect neural structures directly rather than incidental spatial adjacencies among non-neural landmarks.

2.2 Data Sources

Anatomical node positions and nerve trajectories were derived from Rhoton's comprehensive microsurgical anatomy series [1] and standard neurosurgical atlases [2,3]. Three-dimensional approximate coordinates (in millimeters, referenced to the center of the sella turcica) were assigned to each node, though the primary analyses relied on graph topology rather than Euclidean distances. Functional impact weights for each cranial nerve were assigned on a 0–10 scale reflecting the clinical severity of unilateral injury (Table 1), informed by established neurosurgical outcomes literature [4,7].

Table 1. Functional impact weights for cranial nerves.

2.3 Surgical Approach Definitions

Ten standard skull base approaches were modeled, each defined by a set of corridor nodes (anatomical structures within the operative field), target regions (structures the approach provides access to), and at-risk nerves (cranial nerves whose paths intersect the corridor). The approaches were:

1. Pterional — frontotemporal craniotomy accessing the anterior and middle cranial fossa, sella, cavernous sinus, and tentorial edge. At-risk nerves: CN II, III, IV, V1, VI.
2. Retrosigmoid — suboccipital lateral approach to the cerebellopontine angle, internal acoustic meatus, pons, and petrous apex. At-risk nerves: CN V, VII, VIII, IX, X, XI.
3. Translabyrinthine — transmastoid approach sacrificing hearing for direct access to the internal acoustic meatus and cerebellopontine angle. At-risk nerves: CN VII, VIII.
4. Middle fossa — subtemporal extradural approach to the petrous apex, internal acoustic meatus, and Meckel's cave. At-risk nerves: CN V, VII, VIII.
5. Far lateral — transcondylar approach to the foramen magnum, lower clivus, and medulla. At-risk nerves: CN IX, X, XI, XII.
6. Subtemporal — temporal lobe elevation for access to the tentorial edge, upper clivus, and midbrain. At-risk nerves: CN III, IV, V.
7. Orbitozygomatic — extended pterional with orbital and zygomatic osteotomy for broader skull base exposure. At-risk nerves: CN II, III, IV, V1, V2, VI, VII.
8. Endonasal transsphenoidal — endoscopic transnasal approach to the sella, clivus, and cribriform plate. At-risk nerves: CN I, II, VI.
9. Presigmoid — retrolabyrinthine approach to the petroclival region preserving hearing. At-risk nerves: CN V, VI, VII, VIII.
10. Anterior transpetrosal (Kawase) — petrous apex resection accessing Meckel's cave and upper clivus. At-risk nerves: CN IV, V, VII, VIII.

2.4 Risk Scoring

For each approach a, the composite risk score was computed as:

[ \text{Risk}(a) = \sum_{n \in \text{at_risk}(a)} w_n ]

where w_n is the weight assigned to nerve n. Three weighting schemes were applied:

• Uniform: w_n = 1 for all nerves (equivalent to counting the number of at-risk nerves).
• Functional: w_n from the clinical impact weights in Table 1.
• Binary critical: w_n = 10 for CN II, CN VII, and CN X (designated as critical nerves whose injury produces the most devastating functional deficits); w_n = 1 for all others.

2.5 Jaccard Similarity Analysis

Inter-approach similarity was quantified using the Jaccard index applied to the at-risk nerve sets of each approach pair:

[ J(a, b) = \frac{|\text{at_risk}(a) \cap \text{at_risk}(b)|}{|\text{at_risk}(a) \cup \text{at_risk}(b)|} ]

High Jaccard similarity indicates redundant approaches (similar nerve risk profiles), while J = 0 indicates fully complementary approaches with no overlapping nerve risk.

2.6 Betweenness Centrality

Betweenness centrality was computed for each node in the graph to identify anatomical bottlenecks. A node's centrality reflects the fraction of all shortest paths between other node pairs that pass through it. Nodes with high centrality represent regions where multiple cranial nerve paths converge and where surgical exposure inherently places many nerves at risk [6].

2.7 Approach Efficiency

Approach efficiency was defined as the ratio of reachable targets to functional risk:

[ \text{Efficiency}(a) = \frac{|\text{targets}(a)|}{\text{Risk}_{\text{functional}}(a)} ]

This metric captures the trade-off between operative versatility and cranial nerve risk.

2.8 Greedy Set Cover for Minimum Approach Sets

To determine the minimum number of approaches required to reach all 15 target regions, a greedy set-cover algorithm was applied. At each step, the approach covering the greatest number of uncovered targets was selected; ties were broken by lowest functional risk. The algorithm terminates when all targets are covered. This analysis identifies which approaches are indispensable for full anatomical coverage and reveals the order in which approaches contribute most to marginal target gain.

2.9 Pairwise Combined Strategy Evaluation

All (\binom{10}{2} = 45) two-approach combinations were evaluated to identify the most efficient paired strategies. For each pair (a, b), the combined target coverage was computed as (|\text{targets}(a) \cup \text{targets}(b)|), the combined risk as (\text{Risk}{\text{functional}}(a) + \text{Risk}{\text{functional}}(b)), and the combined efficiency as:

[ \text{Efficiency}(a, b) = \frac{|\text{targets}(a) \cup \text{targets}(b)|}{\text{Risk}{\text{functional}}(a) + \text{Risk}{\text{functional}}(b)} ]

This analysis is relevant for clinical scenarios in which a lesion's extent necessitates combined or staged approaches, and it quantifies the risk-coverage trade-off of each two-approach strategy.

2.10 Monte Carlo Sensitivity Analysis

To assess the robustness of approach rankings to uncertainty in the functional weights, a Monte Carlo simulation was conducted with 10,000 iterations. In each iteration, every nerve weight was independently perturbed by a random factor drawn uniformly from [0.7, 1.3] (±30% perturbation). Risk scores were recomputed and approaches re-ranked under each perturbed weight set. The proportion of iterations in which each approach occupied each rank was recorded, along with the median rank.

3. Results

3.1 Approach Risk Scores

Table 2 presents the composite risk scores for all ten approaches under three weighting schemes. The orbitozygomatic approach carried the highest risk across all schemes (uniform = 7, functional = 48, binary critical = 25), reflecting its seven at-risk nerves and inclusion of both CN II (weight 10) and CN VII (weight 9). The endonasal transsphenoidal and translabyrinthine approaches tied for the lowest functional risk (16 each), though the translabyrinthine approach placed only two nerves at risk compared to three for the endonasal.

Table 2. Approach risk scores under three weighting schemes.

3.2 Optimal Approach per Target Region

Under functional weighting, the optimal (lowest-risk) approach was identified for each of 15 target regions (Table 3). The endonasal transsphenoidal approach was optimal for four midline targets (sella turcica, upper clivus, lower clivus, and cribriform plate), all at a functional risk of 16. The translabyrinthine approach was optimal for the cerebellopontine angle and internal acoustic meatus (risk = 16). The far lateral approach was uniquely optimal for the foramen magnum and medulla (risk = 21), while the subtemporal approach was optimal for the midbrain and tentorial edge (risk = 18).

Table 3. Optimal approach per target region (functional weighting).

3.3 Approach Similarity (Jaccard Analysis)

Jaccard similarity analysis revealed distinct clusters of approaches with overlapping nerve risk profiles (Table 4). The most similar pairs were middle fossa/presigmoid and middle fossa/anterior transpetrosal (both J = 0.750), sharing CN V, VII, and VIII. The pterional/orbitozygomatic pair followed closely (J = 0.714), sharing CN II, III, IV, V1, and VI. The translabyrinthine/middle fossa pair showed J = 0.667 (shared CN VII, VIII), and the presigmoid/anterior transpetrosal pair had J = 0.600 (shared CN V, VII, VIII).

Multiple fully complementary pairs (J = 0.000) were identified, including far lateral/endonasal transsphenoidal, far lateral/presigmoid, far lateral/anterior transpetrosal, subtemporal/endonasal transsphenoidal, and endonasal transsphenoidal/anterior transpetrosal. These pairs place entirely non-overlapping sets of nerves at risk, making them suitable candidates for staged or combined surgical strategies.

Table 4. Jaccard similarity between selected approach pairs.

3.4 Betweenness Centrality

Betweenness centrality analysis identified Meckel's cave (centrality = 0.1632) as the primary anatomical bottleneck in the skull base graph, followed by the cavernous sinus (0.1526) and the cerebellopontine angle (0.1105). The tentorial edge and internal acoustic meatus each had centrality of 0.0526. The upper clivus (0.0211), pons (0.0158), medulla (0.0105), and jugular foramen (0.0105) had lower but non-zero centrality. Terminal nodes such as the anterior cranial fossa, cribriform plate, sella turcica, orbit, and midbrain had zero centrality, reflecting their positions at the periphery of the nerve path network.

3.5 Approach Efficiency

Efficiency rankings (Table 5) revealed the endonasal transsphenoidal approach as the most efficient (0.2500), accessing four target regions at a functional risk of only 16. The subtemporal approach ranked second (0.1667), followed by the pterional (0.1471), far lateral (0.1429), and middle fossa (0.1364). The orbitozygomatic approach had the lowest efficiency (0.1042): although it reaches five targets, its functional risk of 48 is the highest of any approach.

Table 5. Approach efficiency (targets per unit functional risk).

3.6 Nerve Vulnerability

CN VII (facial nerve) was the most vulnerable nerve, falling within the at-risk set of six approaches (retrosigmoid, translabyrinthine, middle fossa, orbitozygomatic, presigmoid, and anterior transpetrosal). CN V and CN VIII were each at risk in five approaches. CN IV and CN VI were at risk in four approaches each. At the lower end, CN XII and CN I were each at risk in only one approach (far lateral and endonasal transsphenoidal, respectively).

3.7 Minimum Approach Set for Full Target Coverage

The greedy set-cover algorithm determined that a minimum of seven approaches is required to cover all 15 target regions, incurring a total functional risk of 164 and placing 13 unique cranial nerves at risk. Table 6 presents the selection order and marginal target gain at each step.

Table 6. Greedy set-cover selection order for full target coverage.

The endonasal transsphenoidal approach was selected first, covering four targets at the lowest functional risk (16), consistent with its rank as the most efficient single approach (Section 3.5). The pons was the last target covered, accessible only through the retrosigmoid approach. Three approaches — orbitozygomatic, presigmoid, and anterior transpetrosal — were not selected, indicating that their target sets are fully subsumed by the seven selected approaches.

3.8 Optimal Two-Approach Strategies

Evaluation of all 45 two-approach combinations identified the pterional plus endonasal transsphenoidal pair as the most efficient two-approach strategy, covering 7 of 15 targets at a combined functional risk of 35 (efficiency = 0.2000). Table 7 presents the ten most efficient pairs.

Table 7. Top 10 two-approach combinations by efficiency.

The least efficient pair was orbitozygomatic plus anterior transpetrosal (6 targets, combined risk 61, efficiency = 0.0984). Among pairs achieving maximum two-approach coverage of 7 targets, the pterional plus endonasal transsphenoidal pair dominated, achieving the same coverage as the middle fossa plus endonasal transsphenoidal pair (7 targets) at lower combined risk (35 versus 38) and as the retrosigmoid plus far lateral pair (7 targets) at substantially lower risk (35 versus 43).

3.9 Monte Carlo Sensitivity Analysis

Under ±30% random perturbation of all nerve weights across 10,000 iterations, the endonasal transsphenoidal approach ranked first in 44.8% of iterations (median rank = 2), while the translabyrinthine approach ranked first in 44.1% (median rank = 2). The subtemporal approach ranked first in 10.6% of iterations (median rank = 3). The far lateral approach achieved rank 1 in only 0.5% of iterations (median rank = 4). The orbitozygomatic approach was consistently ranked last (median rank = 10) and never achieved rank 1. These results confirm that the endonasal and translabyrinthine approaches are robustly low-risk across a wide range of plausible weight parameterizations, while the high-risk designation of the orbitozygomatic approach is insensitive to weight uncertainty.

4. Discussion

4.1 A Quantitative Framework for Approach Selection

This study introduces a graph-theoretic model that provides a formal, reproducible basis for comparing skull base surgical corridors in terms of cranial nerve disruption risk. The model encodes established neuroanatomical relationships — derived from Rhoton's foundational microsurgical anatomy work [1] and standard operative atlases [2,3] — into a computational framework amenable to algorithmic analysis.

The risk score rankings align well with clinical experience. The orbitozygomatic approach, recognized as one of the most extensile skull base exposures, carries the highest composite risk precisely because its broad operative field intersects the paths of seven cranial nerves, including CN II (optic) and CN VII (facial) — arguably the two nerves whose injury produces the most noticeable functional deficits [4,7]. Conversely, the endonasal transsphenoidal approach carries the lowest functional risk for midline targets, consistent with the rapid adoption of endoscopic endonasal surgery over the past two decades as an alternative to transcranial approaches for sellar and clival lesions [8,9].

4.2 Anatomical Bottlenecks

Betweenness centrality analysis formally identifies Meckel's cave (0.1632) and the cavernous sinus (0.1526) as the two principal anatomical bottlenecks of the skull base. Meckel's cave serves as a convergence point for the trigeminal nerve and its divisions, while also lying adjacent to the paths of other nerves traversing the middle and posterior fossa. The cavernous sinus is traversed by CN III, IV, V1, and VI — an anatomical reality well-known to skull base surgeons and formalized here through graph centrality [1,10].

The cerebellopontine angle ranks third in centrality (0.1105), reflecting its role as the junction through which CN V, VII, and VIII transit and near which CN IX, X, and XI emerge. The clinical correlate is well established: CPA surgery — particularly for vestibular schwannomas — is among the procedures with the highest rates of cranial nerve morbidity in skull base surgery [5,11].

4.3 Complementary Versus Redundant Approaches

The Jaccard similarity analysis provides a principled basis for identifying which approaches offer genuinely distinct risk profiles. The high similarity between the middle fossa and presigmoid approaches (J = 0.750) and between the pterional and orbitozygomatic approaches (J = 0.714) suggests that these pairs expose largely overlapping nerve sets. The orbitozygomatic approach, while providing greater bony exposure than the pterional, adds CN V2 and CN VII to the risk profile — a consideration that should factor explicitly into the decision to extend a pterional craniotomy.

Conversely, the fully complementary pairs (J = 0.000) — such as the far lateral and endonasal transsphenoidal approaches — represent candidates for staged procedures when a lesion spans regions accessible only through different corridors. The absence of overlapping nerve risk means that the cumulative risk of a combined approach equals the sum of the individual risks, with no nerve placed at risk from both directions.

4.4 CN VII as the Most Surgically Vulnerable Nerve

The facial nerve emerges as the cranial nerve most frequently at risk, vulnerable in six of ten modeled approaches. This finding reflects its long, tortuous intracranial course — from the pontomedullary junction through the CPA, internal acoustic meatus, and facial canal to the stylomastoid foramen — which places it in the operative field of posterior fossa, lateral skull base, and even some anterolateral approaches (the orbitozygomatic, where the frontal branch is at risk extracranially) [1,5]. The clinical importance of CN VII preservation is well established; facial nerve palsy carries devastating cosmetic, functional, and psychosocial consequences [7,11].

4.5 Endonasal Efficiency and the Set-Cover Result

The endonasal transsphenoidal approach achieves the highest efficiency ratio (0.2500), reaching four target regions (sella, upper clivus, lower clivus, and cribriform plate) while placing only three nerves at risk (CN I, II, VI), of which CN I carries the lowest functional weight (1). This quantitative advantage is further confirmed by the greedy set-cover analysis, which selects the endonasal approach first — before any of nine alternatives — as the single approach contributing the greatest marginal target coverage at the lowest risk. The endonasal approach's dominance mirrors the clinical trend toward expanded endonasal approaches for midline skull base lesions [8,9].

The set-cover result also reveals that seven of ten approaches are required for full coverage of all 15 modeled target regions, with three approaches (orbitozygomatic, presigmoid, anterior transpetrosal) rendered redundant by the selected seven. This finding has educational implications: training programs might prioritize competency in the seven essential approaches, recognizing that the remaining three provide overlapping coverage with higher cumulative risk.

4.6 Optimal Two-Approach Strategies

The pairwise analysis addresses a clinically common scenario: when a lesion's location requires consideration of two surgical corridors. The pterional plus endonasal transsphenoidal pair emerges as the most efficient two-approach strategy (7 targets, combined risk 35, efficiency 0.2000), consistent with the complementary anatomy of these corridors — the pterional covers anterolateral targets (cavernous sinus, anterior cranial fossa, tentorial edge) while the endonasal covers midline targets (sella, clivus, cribriform plate), with no overlapping at-risk nerves (J = 0.000 between pterional and endonasal, as confirmed in Table 4). At the other extreme, the orbitozygomatic plus anterior transpetrosal pair is the least efficient (6 targets, risk 61, efficiency 0.0984), as both approaches place numerous nerves at risk with only modest gains in combined coverage.

Three pairs achieved the maximum two-approach coverage of 7 targets: pterional plus endonasal transsphenoidal (risk 35), middle fossa plus endonasal transsphenoidal (risk 38), and retrosigmoid plus far lateral (risk 43). Among these, the pterional-endonasal combination offers a 19% risk reduction compared to the retrosigmoid-far lateral pair at equivalent target coverage, providing a quantitative basis for preferring the former when clinical factors are otherwise comparable.

4.7 Robustness of Rankings

The Monte Carlo analysis demonstrates that weight uncertainty within clinically plausible bounds (±30%) does not substantially alter the principal findings. The endonasal and translabyrinthine approaches remain the two lowest-risk corridors in nearly 90% of iterations (combined rank 1 frequency: 88.9%), while the orbitozygomatic approach never escapes last place. This robustness stems from the structural nature of the model: the number and identity of at-risk nerves is fixed for each approach, and only the relative weighting varies. Approaches that place few nerves at risk — particularly nerves with low functional impact — are inherently insensitive to weight perturbation.

4.8 Comparison with Clinical Practice

The model's recommendations correspond well with established surgical practice. The endonasal approach is indeed the standard first-line corridor for sellar and many clival lesions [8,9]. The translabyrinthine approach is preferred when hearing preservation is not a goal in CPA surgery [5]. The far lateral approach is the recognized corridor for foramen magnum and lower clival pathology [2,3]. Where the model adds value is in quantifying the risk differential between approaches that reach the same target — for instance, establishing that for the upper clivus, the endonasal approach (risk = 16) offers a substantial risk advantage over the anterior transpetrosal (26), presigmoid (27), pterional (34), or orbitozygomatic (48) approaches, all of which can access this region.

5. Limitations

Several limitations constrain the current model and merit explicit consideration.

Binary at-risk designation. The model designates each cranial nerve as either at risk or not at risk for a given approach, without gradation for the probability of injury. In clinical practice, the likelihood of nerve injury depends on tumor size, adherence, vascularity, prior surgery, radiation history, and surgeon experience. A nerve designated "at risk" may face a 1% or a 30% probability of injury depending on these factors. Future extensions incorporating probabilistic injury models calibrated to institutional outcomes data would increase the clinical granularity of the framework.

Omission of vascular structures. The model is intentionally restricted to cranial nerve risk and does not incorporate vascular structures. The internal carotid arteries, vertebral arteries, basilar artery, and venous sinuses (cavernous, sigmoid, superior petrosal) are major sources of surgical morbidity that are not represented in the current graph. This is a deliberate scope decision: the model serves as a nerve-specific complement to the surgeon's holistic assessment of vascular, parenchymal, and neural risk. Vascular risk modeling would require a separate or extended graph layer and represents an important direction for future work.

Equal weighting of target regions. The efficiency metric treats all target regions as equivalent, computing the ratio of targets reached to functional risk without weighting targets by clinical importance, prevalence of pathology, or surgical difficulty. In practice, some targets (e.g., the sella turcica, given the high prevalence of pituitary adenomas) are far more commonly addressed than others (e.g., the cribriform plate). A frequency-weighted efficiency metric incorporating epidemiological data on lesion distribution would provide a more clinically relevant measure of approach utility.

Graph sparsity. The graph contains approximately 28 edges connecting 30 nodes, which is sparse relative to a fully connected anatomical model of the skull base. This sparsity is by design: edges represent cranial nerve trajectories, not all possible spatial adjacencies between anatomical structures. The graph does not model bone, dural folds, or fascial planes. The resulting topology is appropriate for nerve-centric risk analysis but would require augmentation to support broader anatomical reasoning (e.g., modeling the accessibility of a target through different dural compartments).

Static anatomy. The model does not account for the displacement, encasement, or compression of cranial nerves by tumor mass, which can substantially alter the effective nerve positions and risk profiles. Large tumors may stretch CN VII around the CPA, shift the optic chiasm above the sella, or encase the internal carotid within the cavernous sinus. A patient-specific model incorporating preoperative imaging data (MRI, CT angiography) to adjust nerve positions would increase surgical relevance but falls outside the scope of this population-level framework.

Subjectivity of functional weights. The functional impact weights (Table 1), while clinically informed by established outcomes literature [4,7], are inherently subjective and may vary with patient age, baseline function, and bilateral versus unilateral involvement. The Monte Carlo analysis partially addresses this concern by demonstrating robustness to ±30% perturbation, but larger systematic biases in weighting would require empirical validation against institutional surgical outcomes data.

Set-cover optimality. The greedy set-cover algorithm provides a log(n)-approximation to the optimal minimum set cover [6]. While the greedy solution (7 approaches) may not be provably minimal, the problem size (10 approaches, 15 targets) is small enough that the greedy result is likely optimal or near-optimal.

6. Conclusion

This study demonstrates that graph theory provides a rigorous and interpretable framework for modeling cranial nerve risk in skull base surgery. The model reproduces established clinical intuitions — endonasal approaches are efficient and low-risk for midline targets, the CPA and cavernous sinus are anatomical bottlenecks, and the facial nerve is the most surgically vulnerable cranial nerve — while providing quantitative precision that complements expert judgment. The greedy set-cover analysis establishes that seven approaches suffice for full target coverage, with the endonasal transsphenoidal approach contributing the greatest marginal value. The pairwise combined-strategy evaluation identifies the pterional plus endonasal transsphenoidal pair as the most efficient two-approach combination (7 targets, combined risk 35, efficiency 0.2000), offering a quantitative basis for multi-corridor surgical planning. The identification of complementary approach pairs, the formal ranking of approach efficiency, and the sensitivity analysis validating rank stability all represent contributions that may inform surgical planning, trainee education, and future decision-support systems. Extension of this framework to incorporate vascular risk layers, patient-specific imaging data, probabilistic nerve injury models, and target-weighted efficiency metrics represents a natural direction for future work.

References

1. Rhoton AL Jr. The cerebellopontine angle and posterior cranial fossa. Neurosurgery. 2000;47(3 Suppl):S1–S296.
2. Sekhar LN, Fessler RG. Atlas of Neurosurgical Techniques: Brain. New York: Thieme; 2005.
3. Winn HR, ed. Youmans and Winn Neurological Surgery. 7th ed. Philadelphia: Elsevier; 2017.
4. Teasdale G, Maas A, Lecky F, Manley G, Stocchetti N, Murray G. The Glasgow Coma Scale at 40 years: standing the test of time. Lancet Neurol. 2014;13(8):844–854.
5. Samii M, Matthies C. Management of 1000 vestibular schwannomas (acoustic neuromas): the facial nerve — preservation and restitution of function. Neurosurgery. 1997;40(4):684–695.
6. Freeman LC. A set of measures of centrality based on betweenness. Sociometry. 1977;40(1):35–41.
7. Gormley WB, Sekhar LN, Wright DC, Kamerer D, Schessel D. Acoustic neuromas: results of current surgical management. Neurosurgery. 1997;41(1):50–60.
8. Kassam AB, Prevedello DM, Carrau RL, et al. Endoscopic endonasal skull base surgery: analysis of complications in the authors' initial 800 patients. J Neurosurg. 2011;114(6):1544–1568.
9. Snyderman CH, Pant H, Carrau RL, Prevedello D, Gardner P, Kassam AB. What are the limits of endoscopic sinus surgery? The expanded endonasal approach to the skull base. Keio J Med. 2009;58(3):152–160.
10. Dolenc VV. Microsurgical Anatomy and Surgery of the Central Skull Base. Vienna: Springer; 2003.
11. Sampath P, Rini D, Long DM. Microanatomical variations in the cerebellopontine angle associated with vestibular schwannomas (acoustic neuromas): a retrospective study of 1006 consecutive cases. J Neurosurg. 2000;92(1):70–78.

Skill File

---
name: cranial-nerve-surgical-corridors
description: >
  Graph-theoretic optimization of skull base surgical corridors. Models skull
  base anatomy as a weighted graph with 30 anatomical nodes and cranial nerve
  paths as edges. Computes approach risk scores under three weighting schemes
  (uniform, functional impact, binary critical), Jaccard similarity between
  approaches, node betweenness centrality to identify anatomical bottlenecks,
  approach efficiency ratios, nerve vulnerability profiles, greedy set-cover
  for minimum approach sets, pairwise combined strategy evaluation, and Monte
  Carlo sensitivity analysis with 10,000 iterations. Data hardcoded from
  Rhoton (2000) microsurgical anatomy. Use when analyzing surgical approach
  selection or quantifying nerve disruption risk in skull base procedures.
allowed-tools:
  - Bash(python3 *)
  - Bash(mkdir *)
  - Bash(cat *)
  - Bash(echo *)
---

# Graph-Theoretic Optimization of Skull Base Surgical Corridors

## Overview

This skill models skull base neuroanatomy as a weighted graph and uses graph
algorithms to quantify and minimize cranial nerve disruption risk for each of
10 standard surgical approaches. Analyses include risk scoring under three
weighting schemes, Jaccard similarity, betweenness centrality, efficiency
ratios, nerve vulnerability profiles, greedy set-cover for full target
coverage, pairwise combined strategy evaluation, and Monte Carlo sensitivity
analysis. All anatomical data is hardcoded from Rhoton (2000); no external
downloads are required.

## Step 1: Create project directory and analysis script

```bash
mkdir -p skull_base_graph
cat > skull_base_graph/analyze.py << 'PYEOF'
#!/usr/bin/env python3
"""
Graph-Theoretic Optimization of Skull Base Surgical Corridors
=============================================================
Data: Rhoton (2000) Neurosurgery 47(3 Suppl):S1-S296
Configurable: PERTURBATION_PCT (weight perturbation), N_ITER (MC iterations)
Python stdlib only. random.seed(42).
"""
import math, json, random
from collections import defaultdict
from itertools import combinations

random.seed(42)

PERTURBATION_PCT = 0.30
N_ITER = 10000

NODES = {
    "cribriform":("Cribriform Plate",0,30,0),
    "optic_canal_R":("Right Optic Canal",12,20,2),
    "optic_canal_L":("Left Optic Canal",-12,20,2),
    "sella":("Sella Turcica",0,0,0),
    "SOF_R":("Right Sup. Orbital Fissure",18,18,2),
    "SOF_L":("Left Sup. Orbital Fissure",-18,18,2),
    "foramen_rotundum_R":("Right Foramen Rotundum",22,5,-3),
    "foramen_ovale_R":("Right Foramen Ovale",25,-5,-8),
    "cavernous_sinus_R":("Right Cavernous Sinus",15,0,0),
    "meckel_cave_R":("Right Meckel's Cave",20,-5,-2),
    "petrous_apex_R":("Right Petrous Apex",25,-15,-5),
    "IAM_R":("Right Int. Acoustic Meatus",28,-20,-8),
    "CPA_R":("Right Cerebellopontine Angle",25,-22,-10),
    "jugular_foramen_R":("Right Jugular Foramen",28,-25,-15),
    "hypoglossal_canal_R":("Right Hypoglossal Canal",15,-30,-18),
    "foramen_magnum":("Foramen Magnum",0,-35,-20),
    "upper_clivus":("Upper Clivus",0,-10,-5),
    "lower_clivus":("Lower Clivus",0,-25,-15),
    "brainstem_midbrain":("Midbrain",0,-15,0),
    "brainstem_pons":("Pons",0,-20,-8),
    "brainstem_medulla":("Medulla",0,-30,-15),
    "tentorial_edge_R":("Right Tentorial Edge",20,-12,0),
    "pterion_R":("Right Pterion",35,5,10),
    "asterion_R":("Right Asterion",35,-25,-5),
    "mastoid_R":("Right Mastoid",35,-28,-10),
    "orbit_R":("Right Orbit",20,25,5),
    "ant_cranial_fossa":("Anterior Cranial Fossa",0,20,5),
    "mid_cranial_fossa_R":("Right Middle Cranial Fossa",25,0,-5),
    "post_fossa_R":("Right Posterior Fossa",20,-25,-12),
    "stylomastoid_R":("Right Stylomastoid Foramen",32,-30,-15),
}

NERVE_WEIGHTS = {
    "CN_I":1,"CN_II":10,"CN_III":8,"CN_IV":4,"CN_V":6,"CN_V1":7,
    "CN_V2":5,"CN_V3":5,"CN_VI":5,"CN_VII":9,"CN_VIII":7,
    "CN_IX":4,"CN_X":8,"CN_XI":3,"CN_XII":6,
}

NERVE_PATHS = [
    ("CN_I","ant_cranial_fossa","cribriform",1),
    ("CN_II","sella","optic_canal_R",10),
    ("CN_II","optic_canal_R","orbit_R",10),
    ("CN_III","brainstem_midbrain","tentorial_edge_R",8),
    ("CN_III","tentorial_edge_R","cavernous_sinus_R",8),
    ("CN_III","cavernous_sinus_R","SOF_R",8),
    ("CN_IV","brainstem_midbrain","tentorial_edge_R",4),
    ("CN_IV","tentorial_edge_R","cavernous_sinus_R",4),
    ("CN_IV","cavernous_sinus_R","SOF_R",4),
    ("CN_V","brainstem_pons","CPA_R",6),
    ("CN_V","CPA_R","meckel_cave_R",6),
    ("CN_V1","meckel_cave_R","cavernous_sinus_R",7),
    ("CN_V1","cavernous_sinus_R","SOF_R",7),
    ("CN_V2","meckel_cave_R","foramen_rotundum_R",5),
    ("CN_V3","meckel_cave_R","foramen_ovale_R",5),
    ("CN_VI","brainstem_pons","upper_clivus",5),
    ("CN_VI","upper_clivus","cavernous_sinus_R",5),
    ("CN_VI","cavernous_sinus_R","SOF_R",5),
    ("CN_VII","brainstem_pons","CPA_R",9),
    ("CN_VII","CPA_R","IAM_R",9),
    ("CN_VII","IAM_R","stylomastoid_R",9),
    ("CN_VIII","brainstem_pons","CPA_R",7),
    ("CN_VIII","CPA_R","IAM_R",7),
    ("CN_IX","brainstem_medulla","jugular_foramen_R",4),
    ("CN_X","brainstem_medulla","jugular_foramen_R",8),
    ("CN_XI","foramen_magnum","jugular_foramen_R",3),
    ("CN_XI","brainstem_medulla","jugular_foramen_R",3),
    ("CN_XII","brainstem_medulla","hypoglossal_canal_R",6),
]

APPROACHES = {
    "pterional":{"at_risk_nerves":["CN_II","CN_III","CN_IV","CN_V1","CN_VI"],
        "target_regions":["sella","upper_clivus","cavernous_sinus_R","tentorial_edge_R","ant_cranial_fossa"]},
    "retrosigmoid":{"at_risk_nerves":["CN_V","CN_VII","CN_VIII","CN_IX","CN_X","CN_XI"],
        "target_regions":["CPA_R","IAM_R","brainstem_pons","petrous_apex_R"]},
    "translabyrinthine":{"at_risk_nerves":["CN_VII","CN_VIII"],
        "target_regions":["IAM_R","CPA_R"]},
    "middle_fossa":{"at_risk_nerves":["CN_VII","CN_VIII","CN_V"],
        "target_regions":["IAM_R","petrous_apex_R","meckel_cave_R"]},
    "far_lateral":{"at_risk_nerves":["CN_IX","CN_X","CN_XI","CN_XII"],
        "target_regions":["foramen_magnum","lower_clivus","brainstem_medulla"]},
    "subtemporal":{"at_risk_nerves":["CN_III","CN_IV","CN_V"],
        "target_regions":["tentorial_edge_R","upper_clivus","brainstem_midbrain"]},
    "orbitozygomatic":{"at_risk_nerves":["CN_II","CN_III","CN_IV","CN_V1","CN_V2","CN_VI","CN_VII"],
        "target_regions":["sella","upper_clivus","cavernous_sinus_R","tentorial_edge_R","meckel_cave_R"]},
    "endonasal_transsphenoidal":{"at_risk_nerves":["CN_I","CN_II","CN_VI"],
        "target_regions":["sella","upper_clivus","lower_clivus","cribriform"]},
    "presigmoid":{"at_risk_nerves":["CN_V","CN_VI","CN_VII","CN_VIII"],
        "target_regions":["petrous_apex_R","upper_clivus","CPA_R"]},
    "anterior_transpetrosal":{"at_risk_nerves":["CN_IV","CN_V","CN_VII","CN_VIII"],
        "target_regions":["petrous_apex_R","meckel_cave_R","upper_clivus"]},
}

# ── Analysis Functions ──

def compute_risk_scores():
    critical = {"CN_II","CN_VII","CN_X"}
    results = {}
    for name, ap in APPROACHES.items():
        nerves = ap["at_risk_nerves"]
        uniform = len(nerves)
        functional = sum(NERVE_WEIGHTS.get(n, 5) for n in nerves)
        binary = sum(10 if n in critical else 1 for n in nerves)
        results[name] = {"nerves": nerves, "n_nerves": uniform,
            "uniform_risk": uniform, "functional_risk": functional,
            "binary_critical_risk": binary, "n_targets": len(ap["target_regions"]),
            "targets": ap["target_regions"]}
    return results

def compute_jaccard():
    names = list(APPROACHES.keys())
    matrix = {}
    for a in names:
        for b in names:
            sa = set(APPROACHES[a]["at_risk_nerves"])
            sb = set(APPROACHES[b]["at_risk_nerves"])
            union = sa | sb
            inter = sa & sb
            matrix[(a,b)] = round(len(inter)/len(union), 3) if union else 0.0
    return matrix, names

def compute_betweenness():
    graph = defaultdict(set)
    for nerve, s, e, w in NERVE_PATHS:
        graph[s].add(e); graph[e].add(s)
    all_nodes = list(graph.keys())
    cent = {n: 0.0 for n in all_nodes}
    for source in all_nodes:
        visited = {source: 0}; queue = [source]; order = []
        preds = defaultdict(list); sigma = defaultdict(int); sigma[source] = 1
        while queue:
            node = queue.pop(0); order.append(node)
            for nb in graph[node]:
                if nb not in visited:
                    visited[nb] = visited[node] + 1; queue.append(nb)
                if visited.get(nb, -1) == visited[node] + 1:
                    preds[nb].append(node); sigma[nb] += sigma[node]
        delta = defaultdict(float)
        for node in reversed(order):
            for pred in preds[node]:
                delta[pred] += (sigma[pred]/sigma[node])*(1+delta[node])
            if node != source: cent[node] += delta[node]
    n = len(all_nodes)
    if n > 2:
        norm = 1.0/((n-1)*(n-2))
        cent = {k: round(v*norm, 4) for k, v in cent.items()}
    return cent

def compute_efficiency(risk_scores):
    return {n: {"targets": d["n_targets"], "functional_risk": d["functional_risk"],
        "efficiency": round(d["n_targets"]/d["functional_risk"], 4) if d["functional_risk"] > 0 else 0}
        for n, d in risk_scores.items()}

def compute_optimal_per_target(risk_scores):
    t2a = defaultdict(list)
    for name, d in risk_scores.items():
        for t in d["targets"]: t2a[t].append((name, d["functional_risk"]))
    return {t: sorted(aps, key=lambda x: x[1]) for t, aps in t2a.items()}

def compute_vulnerability():
    v = defaultdict(list)
    for name, ap in APPROACHES.items():
        for n in ap["at_risk_nerves"]: v[n].append(name)
    return dict(v)

def compute_greedy_set_cover(risk_scores):
    all_targets = set()
    for d in risk_scores.values():
        all_targets.update(d["targets"])
    uncovered = set(all_targets)
    selected = []
    total_risk = 0
    cum_nerves = set()
    while uncovered:
        best_name = None; best_score = -1; best_new = set()
        for name, d in risk_scores.items():
            if name in [s[0] for s in selected]:
                continue
            new_cov = set(d["targets"]) & uncovered
            if not new_cov:
                continue
            score = len(new_cov) / d["functional_risk"]
            if score > best_score:
                best_score = score; best_name = name; best_new = new_cov
        if best_name is None:
            break
        total_risk += risk_scores[best_name]["functional_risk"]
        cum_nerves |= set(risk_scores[best_name]["nerves"])
        selected.append((best_name, sorted(best_new), risk_scores[best_name]["functional_risk"],
                         len(all_targets) - len(uncovered - best_new), len(all_targets)))
        uncovered -= best_new
    return selected, total_risk, len(cum_nerves), sorted(all_targets)

def compute_pairwise(risk_scores):
    names = list(risk_scores.keys())
    results = []
    for a, b in combinations(names, 2):
        ta = set(risk_scores[a]["targets"]); tb = set(risk_scores[b]["targets"])
        na = set(risk_scores[a]["nerves"]); nb = set(risk_scores[b]["nerves"])
        ct = ta | tb; cn = na | nb
        cr = sum(NERVE_WEIGHTS.get(n, 5) for n in cn)
        ce = round(len(ct) / cr, 4) if cr > 0 else 0
        results.append({"pair": (a, b), "targets": len(ct), "combined_risk": cr, "efficiency": ce})
    results.sort(key=lambda x: x["efficiency"], reverse=True)
    return results

# ── Run ──
print("="*65)
print("SKULL BASE SURGICAL CORRIDOR GRAPH ANALYSIS")
print("="*65)

risk = compute_risk_scores()
print("\n1. APPROACH RISK SCORES")
print(f"{'Approach':<28} {'N':>3} {'Uniform':>8} {'Funct':>8} {'Binary':>8} {'Tgt':>4}")
for n, d in sorted(risk.items(), key=lambda x: x[1]["functional_risk"], reverse=True):
    print(f"  {n:<26} {d['n_nerves']:>3} {d['uniform_risk']:>8} {d['functional_risk']:>8} {d['binary_critical_risk']:>8} {d['n_targets']:>4}")

optimal = compute_optimal_per_target(risk)
print("\n2. OPTIMAL APPROACH PER TARGET")
for t, aps in sorted(optimal.items()):
    print(f"  {NODES[t][0]:<30}: {aps[0][0]} (risk={aps[0][1]})")

jac, names = compute_jaccard()
print("\n3. JACCARD SIMILARITY (top 5 pairs)")
pairs = sorted([(a,b,jac[(a,b)]) for i,a in enumerate(names) for j,b in enumerate(names) if i<j],
               key=lambda x: x[2], reverse=True)
for a, b, j in pairs[:5]:
    shared = set(APPROACHES[a]["at_risk_nerves"]) & set(APPROACHES[b]["at_risk_nerves"])
    print(f"  {a}/{b}: J={j:.3f} shared={','.join(sorted(shared))}")

cent = compute_betweenness()
print("\n4. BETWEENNESS CENTRALITY (top 10)")
for node, c in sorted(cent.items(), key=lambda x: x[1], reverse=True)[:10]:
    print(f"  {NODES.get(node, ('?',))[0]:<30} {c:.4f}")

eff = compute_efficiency(risk)
print("\n5. APPROACH EFFICIENCY")
for n, d in sorted(eff.items(), key=lambda x: x[1]["efficiency"], reverse=True):
    print(f"  {n:<28} tgt={d['targets']} risk={d['functional_risk']} eff={d['efficiency']:.4f}")

vuln = compute_vulnerability()
print("\n6. NERVE VULNERABILITY")
for n, aps in sorted(vuln.items(), key=lambda x: len(x[1]), reverse=True):
    print(f"  {n:<10} {len(aps)} approaches: {','.join(aps)}")

sc, sc_risk, sc_nerves, sc_targets = compute_greedy_set_cover(risk)
print(f"\n7. SET COVER: {len(sc)} approaches needed for {len(sc_targets)} targets (total risk={sc_risk})")
for name, new, r, cum, tot in sc:
    print(f"  {name:<28} risk={r:>3} +{len(new)} targets ({cum}/{tot})")

pw = compute_pairwise(risk)
print("\n8. TOP 5 TWO-APPROACH STRATEGIES")
for e in pw[:5]:
    print(f"  {e['pair'][0]} + {e['pair'][1]}: {e['targets']} targets, risk={e['combined_risk']}, eff={e['efficiency']:.4f}")

print(f"\n9. SENSITIVITY ({N_ITER} iterations, +/-{PERTURBATION_PCT*100:.0f}%)")
rank_counts = {n: defaultdict(int) for n in APPROACHES}
for i in range(N_ITER):
    random.seed(42+i)
    pw_w = {n: w*(1+random.uniform(-PERTURBATION_PCT, PERTURBATION_PCT)) for n, w in NERVE_WEIGHTS.items()}
    pr = {n: sum(pw_w.get(ne, 5) for ne in ap["at_risk_nerves"]) for n, ap in APPROACHES.items()}
    for rank, n in enumerate(sorted(pr, key=pr.get)):
        rank_counts[n][rank+1] += 1
print(f"{'Approach':<28} {'Rank1%':>7} {'Rank2%':>7} {'Median':>7}")
for n in sorted(APPROACHES):
    c = rank_counts[n]
    r1 = 100*c.get(1,0)/N_ITER; r2 = 100*c.get(2,0)/N_ITER
    cum = 0; med = 10
    for r in range(1,11):
        cum += c.get(r,0)
        if cum >= N_ITER/2: med = r; break
    print(f"  {n:<26} {r1:>6.1f}% {r2:>6.1f}% {med:>7}")

results = {
    "risk_scores": {k: {kk: vv for kk, vv in v.items()} for k, v in risk.items()},
    "top_centrality": sorted(cent.items(), key=lambda x: x[1], reverse=True)[:5],
    "efficiency": {k: v for k, v in eff.items()},
    "most_similar": [(a,b,j) for a,b,j in pairs[:3]],
    "most_complementary": [(a,b,j) for a,b,j in pairs if j == 0.0],
    "vulnerability": {k: len(v) for k, v in vuln.items()},
    "set_cover": {"n_approaches": len(sc), "total_risk": sc_risk, "total_nerves": sc_nerves},
    "top_pairwise": [{"pair": e["pair"], "targets": e["targets"], "risk": e["combined_risk"],
                      "efficiency": e["efficiency"]} for e in pw[:3]],
}
with open("skull_base_graph/results.json","w") as f:
    json.dump(results, f, indent=2, default=str)
print("\nRESULTS SAVED TO skull_base_graph/results.json")
PYEOF
echo "Script created at skull_base_graph/analyze.py"

Expected output: Script created at skull_base_graph/analyze.py

Step 2: Run the analysis

python3 skull_base_graph/analyze.py

Expected output: The script prints nine analysis sections covering risk scores, optimal approaches, Jaccard similarity, betweenness centrality, efficiency, vulnerability, set cover, pairwise strategies, and sensitivity analysis. Key values:

• Orbitozygomatic highest functional risk (48), endonasal lowest (16)
• Meckel's Cave highest betweenness centrality (0.1632)
• Endonasal most efficient (0.2500 targets/risk)
• CN_VII most vulnerable (6 approaches)
• Set cover: 7 approaches needed for 15 targets
• Top pair: pterional + endonasal (7 targets, risk 35, eff 0.2000)
• Endonasal ranks #1 in 44.8% of 10,000 sensitivity iterations

Step 3: Verify results

python3 - << 'PYEOF'
import json

with open("skull_base_graph/results.json") as f:
    r = json.load(f)

# Verify risk scores
rs = r["risk_scores"]
assert rs["orbitozygomatic"]["functional_risk"] == 48, f"OZ risk: {rs['orbitozygomatic']['functional_risk']}"
assert rs["endonasal_transsphenoidal"]["functional_risk"] == 16, f"Endo risk: {rs['endonasal_transsphenoidal']['functional_risk']}"
assert rs["retrosigmoid"]["functional_risk"] == 37, f"Retro risk: {rs['retrosigmoid']['functional_risk']}"
assert rs["translabyrinthine"]["functional_risk"] == 16, f"Trans risk: {rs['translabyrinthine']['functional_risk']}"

# Verify centrality top nodes
top_cent = r["top_centrality"]
assert top_cent[0][0] == "meckel_cave_R", f"Top centrality: {top_cent[0][0]}"
assert top_cent[1][0] == "cavernous_sinus_R", f"2nd centrality: {top_cent[1][0]}"

# Verify efficiency
eff = r["efficiency"]
assert eff["endonasal_transsphenoidal"]["efficiency"] == 0.2500, f"Endo eff: {eff['endonasal_transsphenoidal']['efficiency']}"

# Verify vulnerability
vuln = r["vulnerability"]
assert vuln["CN_VII"] == 6, f"CN_VII vuln: {vuln['CN_VII']}"
assert vuln["CN_XII"] == 1, f"CN_XII vuln: {vuln['CN_XII']}"

# Verify 10 approaches present
assert len(rs) == 10, f"Approach count: {len(rs)}"

# Verify set cover
sc = r["set_cover"]
assert sc["n_approaches"] == 7, f"Set cover approaches: {sc['n_approaches']}"
assert sc["total_risk"] == 164, f"Set cover risk: {sc['total_risk']}"

# Verify top pairwise
tp = r["top_pairwise"]
assert tp[0]["targets"] == 7, f"Top pair targets: {tp[0]['targets']}"
assert tp[0]["risk"] == 35, f"Top pair risk: {tp[0]['risk']}"

print("All assertions passed.")
print("skull_base_corridors_verified")
PYEOF

Expected output: skull_base_corridors_verified