Topological complexity determines the frequency of branching variations in the celiac trunk–superior mesenteric artery system

Topological complexity determines the frequency of branching variations in the celiac trunk–superior mesenteric artery system | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Systematic Review Topological complexity determines the frequency of branching variations in the celiac trunk–superior mesenteric artery system Satoru Muro, Akimoto Nimura, Keiichi Akita This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8637430/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Background Variations in the anatomical branching patterns of the celiac trunk and superior mesenteric artery (SMA) directly affect vascular identification, surgical planning, and risk of intraoperative vascular injury. However, despite the extensive classification of these variations, the principles governing the occurrence of certain branching patterns remain unclear. Methods We analyzed 28 branching subtypes of the celiac trunk–SMA system described by Adachi. Each branching pattern was quantified relative to the standard configuration by using Node-Shift (relative displacement of branching nodes), Edge-Gain (number of additional branches), and Edge-Loss (number of missing branches). The relationships between these topological parameters and branching frequency were examined using linear, log-linear, and exponential regression models. Results Branching frequency consistently decreased as topological complexity increased. In subtypes without branch loss, the branching frequency showed a strong exponential decline with increasing Node-Shift and Edge-Gain (R² = 0.979). When all subtypes were included, the exponential model maintained a similarly high explanatory power (R² = 0.979), whereas the linear and log-linear models showed a limited fit. Branch loss was exclusively associated with very-low-frequency patterns. Conclusions Arterial branching variations do not arise in a purely random manner and are consistent with the probabilistic constraints of developmental and anatomical pattern formation. Celiac Trunk Superior Mesenteric Artery Anatomical Variation Arterial Branching Topology Hepatobiliary Surgery Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 INTRODUCTION Variations in the branching patterns of the celiac trunk and superior mesenteric artery (SMA) directly influence the identification of major vessels and the safety of vascular management during hepatobiliary–pancreatic surgery. Accordingly, the morphological diversity of these arterial branches has attracted the attention of anatomists and surgeons [1, 2]. Anatomical studies have classified the branching patterns of the celiac trunk–SMA system on the basis of large case series and documented their occurrence frequency in detail [3–6]. These data form an essential foundation for understanding the diversity of human vascular anatomy. A re-examination of the course and distribution of abdominal vessels in relation to membranous structures, such as the peritoneum, mesentery, and fusion fasciae, suggests that the spatial constraints imposed by these membranes influence vascular trajectories [7–9]. However, within these constraints, the extent to which specific branching patterns are determined or whether they arise in an essentially random manner remains unclear. In particular, no quantitative or systematic theoretical framework has been established to explain the differences in the occurrence frequency of branching patterns. Arterial branching patterns are defined more by branch connectivity (i.e., which arteries connect to which points) than by geometric features, such as vessel length or branching angles. In line with this framework, classical classifications and contemporary descriptions of anatomical variations have characterized branching types primarily on the basis of topological structures [1, 2, 10]. More recently, we reported that the “topological distance” between aortic arch branching patterns shows an exponential relationship with their occurrence frequency [11]. Therefore, the variations in arterial branching patterns are not governed by complete randomness but instead follow inherent regularities embedded within the connectivity structure itself. This observation motivated the hypothesis that a similar mathematical principle might also apply to the branching variations of the celiac trunk and SMA. In this study, we aimed to elucidate the underlying regularities governing the occurrence of branching variations in the celiac trunk–SMA system. We developed novel scoring parameters to quantitatively characterize branching patterns from a topological perspective and formally defined each branching type by using these indices. Thereafter, we analyzed the relationship between these topological scores and reported occurrence frequencies by using mathematical modeling. On the basis of this approach, we propose a new explanatory framework for understanding how branching variations in the celiac trunk and SMA arise. METHODS Classification system To define the branching patterns of the celiac trunk and the SMA, we adopted the classical classification proposed by Adachi [1] (Fig. 1 ). In this system, the overall branching configuration is first categorized into six major types on the basis of the relationships between the principal arteries, namely, the common hepatic artery, splenic artery, left gastric artery, and SMA. Each type is then further subdivided into 28 subtypes (forms) according to differences in the more distal branches, including the gastroduodenal, accessory gastric, and accessory hepatic arteries. Type I (Form 1) is regarded the standard configuration. The Adachi classification represents the most comprehensive and systematically organized description of branching patterns in the celiac trunk–SMA system. Therefore, we used this classification to provide a clear and consistent structural framework for the topological analysis of the arterial branching patterns included in this study. Frequency data Data on the occurrence frequency of branching patterns were obtained from the large-scale anatomical study by Adachi [1], which examined 252 adult cadavers. In that study, the branching configurations of the celiac trunk–SMA system were identified by gross anatomical dissection and classified according to the aforementioned scheme, with the number of cases and relative frequencies documented for all 28 subtypes. In this study, these published aggregate data were used as primary frequency data and were reanalyzed to evaluate the mathematical relationship between the occurrence frequency of each subtype (form) and the topological parameters defined below. Definition of topological parameters Arterial branching structures can be represented as networks consisting of nodes (branching points) and edges (vascular segments). Even if the spatial positions of the nodes or the lengths and angles of the edges change, the topology of the structure remains unchanged as long as the connectivity between the nodes and edges is preserved. Topological differences arise only when these relationships are altered (Fig. 2 ). Therefore, this concept is well suited for capturing the essential structural characteristics of vascular branching patterns. From a developmental perspective, vascular branching is thought to arise through the elongation and remodeling of an initial vascular plexus, which involve processes such as sprouting, regression, branching, and fusion [12, 13]. The addition or loss of branches during this process reflects changes in connectivity rather than simple positional shifts, and these events can be fundamentally understood as topological events. When the most frequently observed branching pattern is defined as the standard configuration, patterns with a greater number of branch additions or losses can be regarded as having larger topological deviations from the standard. On the basis of this concept, we defined three parameters to quantify the topological complexity of the branching patterns. The Node-Shift Score follows the same concept as that used in our previous study [11], whereas the Edge-Gain and Edge-Loss Scores were derived by simplifying and redefining the Edge-Loss-Change Score from that study to better suit the celiac trunk–SMA system, which exhibits limited segmental repetition. The three parameters used in this study were defined as follows (Fig. 3 ): Node-Shift Score: The number of steps in which the position of a branching node is shifted relative to its position in the standard configuration. This reflects the changes in the relative positional relationships among nodes. Edge-Gain Score: The number of additional branches, such as accessory arteries, that are absent in the standard configuration. Edge-Loss Score: The number of arterial branches that are normally present in the standard configuration but are absent in a given subtype. By combining these three indices, we aimed to comprehensively evaluate the extent to which each branching pattern deviated topologically from the standard configuration. Statistical analysis First, the three topological parameters (Node-Shift, Edge-Gain, and Edge-Loss) were calculated for all 28 subtypes of the celiac trunk–SMA system described by Adachi [1] by using the standard configuration as the reference. Each parameter quantitatively represents the degree of topological deviation of a given subtype from the standard pattern. Thereafter, the relationships between these parameters and reported occurrence frequencies [1] were evaluated using a two-step regression analysis. In the first step, subtypes with Edge-Loss = 0 (Forms 1–23) were analyzed by using Node-Shift and Edge-Gain as explanatory variables and occurrence frequency as the dependent variable. This subset represented relatively basic topological variations without branch loss and was analyzed to examine how the frequency decreased under conditions of branch preservation. In the second step, all 28 subtypes were analyzed by adding Edge-Loss as a third explanatory variable to assess the frequency changes across a broader range of topological variations. Linear, exponential, and logarithmic regression models were applied to account for the possibility of nonlinear relationships between the topological parameters and frequency. All analyses were performed using Python (version 3.11), and model fit was evaluated using the coefficient of determination (R²). Declarations Ethical approval Institutional Review Board approval was not required because this study used previously published, aggregated data and excluded human subjects or other identifiable personal information ACKNOWLEDGMENTS None. FUNDING None. CONFLICT OF INTEREST The authors have no relevant financial relationships to disclose. DATA AVAILABILITY Data supporting the findings of this study are available from the corresponding author upon request. References B. Adachi, “Das Arteriensystem der Japaner,” Band (1928). N. A. Michels, “Newer Anatomy of the Liver and Its Variant Blood Supply and Collateral Circulation,” American Journal of Surgery 112, no. 3 (1966): 337–347, https://doi.org/10.1016/0002-9610(66)90201-7. A. Balcerzak, R. S. Tubbs, A. Waśniewska-Włodarczyk, E. Rapacka, and Ł. Olewnik, “Classification of the superior mesenteric artery,” Clinical Anatomy 35, no. 4 (2022): 501–511, https://doi.org/10.1002/ca.23841. E. Gamo, C. Jiménez, E. 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Kvietkauskas, “Anatomical variations of hepatic arteries: implications for clinical practice,” Frontiers in Surgery 12, (2025): 1593800. https://doi.org/10.3389/fsurg.2025.1593800. Table Table 1. Topological Edit Distance and Frequency of Anatomical Variations in the Celiac Trunk–Superior Mesenteric Artery System Form Node-Shift Edge-Gain Edge-Loss Frequency (Adachi 1928) (%) 1 0 0 0 55.6 2 3 0 0 2.4 3 0 1 0 9.1 4 0 1 0 9.9 5 3 1 0 0.4 6 0 2 0 0.4 7 0 1 0 1.2 8 0 1 0 4.4 9 0 2 0 0.8 10 0 2 0 0.8 11 0 2 0 2.8 12 1 0 0 3.6 13 1 1 0 0.4 14 1 1 0 1.2 15 1 2 0 0.4 16 1 1 0 0.4 17 1 2 0 0.4 18 2 0 0 0.8 19 2 1 0 0.4 20 1 0 0 1.2 21 1 1 0 0.4 22 1 1 0 0.8 23 2 0 0 0.4 24 0 2 2 0.4 25 0 3 2 0.4 26 0 2 2 0.4 27 0 3 2 0.4 28 0 2 1 0.4 Additional Declarations The authors declare no competing interests. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8637430","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Systematic Review","associatedPublications":[],"authors":[{"id":576685182,"identity":"bc56fb69-a372-43a8-9dcd-95ef123897d3","order_by":0,"name":"Satoru Muro","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzUlEQVRIiWNgGAWjYNACGyBmb2Bg4AHzDhCjJQ2IeQ6TrEUiGaaFANCdkfvwAUOCXR7/zPdHN7xhsJNnYDyL3xqzG+nGBgwJycUSt5PZbs5hSDZsYDiXQEBLGpsE4w/mxAaglts8DMxA5WcMCGthSKhPnH/zMEhLPdFaDiduuMEM0nKYCC1nnjEbJCQcT9x4Jtns5hyD44ZtBP1yPI3xwYeE6sR5xw8+u/GmolqeX4JAiIEBwlSgk9gkzhDWgQb4e0jWMgpGwSgYBcMbAABod0WBISj6QgAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-4709-6359","institution":"Institute of Science Tokyo","correspondingAuthor":true,"prefix":"","firstName":"Satoru","middleName":"","lastName":"Muro","suffix":""},{"id":576685188,"identity":"cef63cd6-241a-4275-867e-2ef70b027ece","order_by":1,"name":"Akimoto Nimura","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Akimoto","middleName":"","lastName":"Nimura","suffix":""},{"id":576685189,"identity":"6ea81a2b-bd51-489b-9885-f88e6d415a4c","order_by":2,"name":"Keiichi Akita","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Keiichi","middleName":"","lastName":"Akita","suffix":""}],"badges":[],"createdAt":"2026-01-19 09:16:33","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8637430/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8637430/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100757360,"identity":"e1d8dfac-e5c8-4c1d-9470-680e69a0e8f2","added_by":"auto","created_at":"2026-01-21 06:48:56","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":86251,"visible":true,"origin":"","legend":"","description":"","filename":"CTSMAvariationmanuscriptv11.docx","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/a38b974715fabb30074150ab.docx"},{"id":100757452,"identity":"1831aaee-3a6c-4916-8a65-072b3639f64e","added_by":"auto","created_at":"2026-01-21 06:50:11","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":342,"visible":true,"origin":"","legend":"","description":"","filename":"rs8637430.json","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/d72e1ecff2fb3bc4b8c3a67e.json"},{"id":100757390,"identity":"5f028b67-13d6-4ee1-9af9-78731511bc56","added_by":"auto","created_at":"2026-01-21 06:49:42","extension":"xml","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":67319,"visible":true,"origin":"","legend":"","description":"","filename":"rs86374300enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/aed97848f70b834be7bf137f.xml"},{"id":100757365,"identity":"eb3718ee-d14e-463d-8c8f-cbb2cf416356","added_by":"auto","created_at":"2026-01-21 06:49:16","extension":"xml","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":65950,"visible":true,"origin":"","legend":"","description":"","filename":"rs86374300structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/c94a3bc5424a3800cba70dd4.xml"},{"id":100757363,"identity":"45236af5-c6b5-4bd3-aec9-594fc6f202be","added_by":"auto","created_at":"2026-01-21 06:49:05","extension":"html","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":76203,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/06b290f07a09208e4d5df8e4.html"},{"id":100796797,"identity":"d0bb61e8-274f-40ff-b12c-45f9ae2ebb93","added_by":"auto","created_at":"2026-01-21 13:45:55","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":2279423,"visible":true,"origin":"","legend":"\u003cp\u003eAdachi classification of the celiac trunk–superior mesenteric artery (SMA) arterial system\u003c/p\u003e\n\u003cp\u003eSchematic overview of the classical Adachi classification (1928) of the branching patterns of the celiac trunk and SMA. The arterial configuration was classified into six major types according to the relationships between the common hepatic artery (CHA), splenic artery (SA), left gastric artery (LGA), and SMA. Each type was further subdivided into 28 forms on the basis of variations in the distal branches, including the gastroduodenal artery (GDA), accessory gastric, and accessory hepatic arteries. Type 1 (Form 1) was regarded the standard arterial configuration and used as the topological reference in the present study.\u003c/p\u003e","description":"","filename":"figure1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/326cc29e3c3208709736afd8.jpg"},{"id":100757448,"identity":"6b579689-8af4-4cc7-abbf-c83fd72bd72f","added_by":"auto","created_at":"2026-01-21 06:49:54","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":253988,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003e\u0026nbsp;Topological equivalence in arterial branching patterns\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eConceptual illustration of the topological equivalence and differences in arterial branching patterns. Arterial trees are networks comprising nodes (branching points) and edges (arterial segments). Variations in the spatial course or geometry of the arteries do not alter the topology as long as connectivity is preserved. Topological differences arise only when the branching connectivity is altered, such as by the addition, loss, or relocation of arterial branches.\u003c/p\u003e","description":"","filename":"figure2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/3d9d0f8be3d39736a3a3b550.jpg"},{"id":100757367,"identity":"ea0fef4c-d093-45d4-b66e-38b2ed1b16d6","added_by":"auto","created_at":"2026-01-21 06:49:18","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":805672,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDefinition of topological deviation parameters\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDefinition of the three parameters used to quantify topological deviation from the standard arterial configuration (Type 1 [Form 1]). Node-Shift represents the displacement of a branching point relative to its position in the standard configuration. Edge-Gain represents the presence of additional arterial branches, such as accessory arteries. Edge-Loss represents the absence of the arterial branches that are normally present in the standard configuration. Each illustrated example demonstrates a single-step change and yields a score of one for the corresponding parameter.\u003c/p\u003e","description":"","filename":"figure3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/e287b4b5bd53d7ab4efd3b4b.jpg"},{"id":100757446,"identity":"84dab44f-897d-4209-a648-ded2c9f6529a","added_by":"auto","created_at":"2026-01-21 06:49:50","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":2512342,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTopological scores and frequencies of the Adachi subtypes\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eSchematic representation of all 28 Adachi subtypes of the celiac trunk–superior mesenteric artery (SMA) system and their corresponding topological scores and reported occurrence frequencies (Adachi, 1928). For each form, the triplet indicates the scores for (Node-Shift, Edge-Gain, Edge-Loss), followed by frequency (%). This summary illustrates that increasing the topological deviation from the standard configuration is associated with progressively lower occurrence frequencies.\u003c/p\u003e","description":"","filename":"figure4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/136b2e7f17e62b8ad80fd05e.jpg"},{"id":100757435,"identity":"0bf4e1d0-cae1-4dac-9653-5d5b3a9c2a4b","added_by":"auto","created_at":"2026-01-21 06:49:44","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":535683,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eDistribution of occurrence frequency across topological parameter space\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eHeat maps illustrating the relationship between topological parameters and occurrence frequency. The Node-Shift and Edge-Gain are plotted on the horizontal and vertical axes, respectively, and the frequency (%) is indicated by color intensity. Separate panels show Edge-Loss Scores of = 0, 1, and 2. High-frequency patterns are concentrated near the standard configuration, whereas patterns with greater topological deviation are rare.\u003c/p\u003e","description":"","filename":"figure5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/a71b9cb5f889fd8e44b2bce4.jpg"},{"id":100757387,"identity":"0882ca69-cf5c-4fae-aa12-791fd3acc28c","added_by":"auto","created_at":"2026-01-21 06:49:35","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":944655,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eExponential relationship between topological deviation and occurrence frequency\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e(A) Three-dimensional plot showing the observed occurrence frequencies of branching patterns without branch loss (Edge-Loss = 0; Forms 1–23) as a function of Node-Shift and Edge-Gain, together with the fitted exponential regression surface.\u003c/p\u003e\n\u003cp\u003e(B) Comparison between observed and predicted frequencies derived from the exponential regression model with a constant term. The dashed line represents the line of identity. The close correspondence indicates that even modest increases in topological deviation from the standard arterial configuration are associated with a rapid decline in occurrence frequency.\u003c/p\u003e","description":"","filename":"figure6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/7b68276a56358e96afb71181.jpg"},{"id":100798030,"identity":"5bec3435-4925-4071-8bf2-aad329c3b4e3","added_by":"auto","created_at":"2026-01-21 13:52:17","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":7932985,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8637430/v1/4b235518-1cc1-4752-915d-d8a8c3d9e2c1.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eTopological complexity determines the frequency of branching variations in the celiac trunk–superior mesenteric artery system\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"INTRODUCTION","content":"\u003cp\u003eVariations in the branching patterns of the celiac trunk and superior mesenteric artery (SMA) directly influence the identification of major vessels and the safety of vascular management during hepatobiliary\u0026ndash;pancreatic surgery. Accordingly, the morphological diversity of these arterial branches has attracted the attention of anatomists and surgeons [1, 2]. Anatomical studies have classified the branching patterns of the celiac trunk\u0026ndash;SMA system on the basis of large case series and documented their occurrence frequency in detail [3\u0026ndash;6]. These data form an essential foundation for understanding the diversity of human vascular anatomy.\u003c/p\u003e \u003cp\u003eA re-examination of the course and distribution of abdominal vessels in relation to membranous structures, such as the peritoneum, mesentery, and fusion fasciae, suggests that the spatial constraints imposed by these membranes influence vascular trajectories [7\u0026ndash;9]. However, within these constraints, the extent to which specific branching patterns are determined or whether they arise in an essentially random manner remains unclear. In particular, no quantitative or systematic theoretical framework has been established to explain the differences in the occurrence frequency of branching patterns.\u003c/p\u003e \u003cp\u003eArterial branching patterns are defined more by branch connectivity (i.e., which arteries connect to which points) than by geometric features, such as vessel length or branching angles. In line with this framework, classical classifications and contemporary descriptions of anatomical variations have characterized branching types primarily on the basis of topological structures [1, 2, 10]. More recently, we reported that the \u0026ldquo;topological distance\u0026rdquo; between aortic arch branching patterns shows an exponential relationship with their occurrence frequency [11]. Therefore, the variations in arterial branching patterns are not governed by complete randomness but instead follow inherent regularities embedded within the connectivity structure itself. This observation motivated the hypothesis that a similar mathematical principle might also apply to the branching variations of the celiac trunk and SMA.\u003c/p\u003e \u003cp\u003eIn this study, we aimed to elucidate the underlying regularities governing the occurrence of branching variations in the celiac trunk\u0026ndash;SMA system. We developed novel scoring parameters to quantitatively characterize branching patterns from a topological perspective and formally defined each branching type by using these indices. Thereafter, we analyzed the relationship between these topological scores and reported occurrence frequencies by using mathematical modeling. On the basis of this approach, we propose a new explanatory framework for understanding how branching variations in the celiac trunk and SMA arise.\u003c/p\u003e"},{"header":"METHODS","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eClassification system\u003c/h2\u003e \u003cp\u003eTo define the branching patterns of the celiac trunk and the SMA, we adopted the classical classification proposed by Adachi [1] (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). In this system, the overall branching configuration is first categorized into six major types on the basis of the relationships between the principal arteries, namely, the common hepatic artery, splenic artery, left gastric artery, and SMA. Each type is then further subdivided into 28 subtypes (forms) according to differences in the more distal branches, including the gastroduodenal, accessory gastric, and accessory hepatic arteries. Type I (Form 1) is regarded the standard configuration.\u003c/p\u003e \u003cp\u003eThe Adachi classification represents the most comprehensive and systematically organized description of branching patterns in the celiac trunk\u0026ndash;SMA system. Therefore, we used this classification to provide a clear and consistent structural framework for the topological analysis of the arterial branching patterns included in this study.\u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003eFrequency data\u003c/h3\u003e\n\u003cp\u003eData on the occurrence frequency of branching patterns were obtained from the large-scale anatomical study by Adachi [1], which examined 252 adult cadavers. In that study, the branching configurations of the celiac trunk\u0026ndash;SMA system were identified by gross anatomical dissection and classified according to the aforementioned scheme, with the number of cases and relative frequencies documented for all 28 subtypes.\u003c/p\u003e \u003cp\u003eIn this study, these published aggregate data were used as primary frequency data and were reanalyzed to evaluate the mathematical relationship between the occurrence frequency of each subtype (form) and the topological parameters defined below.\u003c/p\u003e\n\u003ch3\u003eDefinition of topological parameters\u003c/h3\u003e\n\u003cp\u003eArterial branching structures can be represented as networks consisting of nodes (branching points) and edges (vascular segments). Even if the spatial positions of the nodes or the lengths and angles of the edges change, the topology of the structure remains unchanged as long as the connectivity between the nodes and edges is preserved. Topological differences arise only when these relationships are altered (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). Therefore, this concept is well suited for capturing the essential structural characteristics of vascular branching patterns.\u003c/p\u003e \u003cp\u003eFrom a developmental perspective, vascular branching is thought to arise through the elongation and remodeling of an initial vascular plexus, which involve processes such as sprouting, regression, branching, and fusion [12, 13]. The addition or loss of branches during this process reflects changes in connectivity rather than simple positional shifts, and these events can be fundamentally understood as topological events. When the most frequently observed branching pattern is defined as the standard configuration, patterns with a greater number of branch additions or losses can be regarded as having larger topological deviations from the standard.\u003c/p\u003e \u003cp\u003eOn the basis of this concept, we defined three parameters to quantify the topological complexity of the branching patterns. The Node-Shift Score follows the same concept as that used in our previous study [11], whereas the Edge-Gain and Edge-Loss Scores were derived by simplifying and redefining the Edge-Loss-Change Score from that study to better suit the celiac trunk\u0026ndash;SMA system, which exhibits limited segmental repetition.\u003c/p\u003e \u003cp\u003eThe three parameters used in this study were defined as follows (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e):\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eNode-Shift Score: The number of steps in which the position of a branching node is shifted relative to its position in the standard configuration. This reflects the changes in the relative positional relationships among nodes.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eEdge-Gain Score: The number of additional branches, such as accessory arteries, that are absent in the standard configuration.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eEdge-Loss Score: The number of arterial branches that are normally present in the standard configuration but are absent in a given subtype.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eBy combining these three indices, we aimed to comprehensively evaluate the extent to which each branching pattern deviated topologically from the standard configuration.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003eStatistical analysis\u003c/h2\u003e \u003cp\u003eFirst, the three topological parameters (Node-Shift, Edge-Gain, and Edge-Loss) were calculated for all 28 subtypes of the celiac trunk\u0026ndash;SMA system described by Adachi [1] by using the standard configuration as the reference. Each parameter quantitatively represents the degree of topological deviation of a given subtype from the standard pattern.\u003c/p\u003e \u003cp\u003eThereafter, the relationships between these parameters and reported occurrence frequencies [1] were evaluated using a two-step regression analysis. In the first step, subtypes with Edge-Loss\u0026thinsp;=\u0026thinsp;0 (Forms 1\u0026ndash;23) were analyzed by using Node-Shift and Edge-Gain as explanatory variables and occurrence frequency as the dependent variable. This subset represented relatively basic topological variations without branch loss and was analyzed to examine how the frequency decreased under conditions of branch preservation. In the second step, all 28 subtypes were analyzed by adding Edge-Loss as a third explanatory variable to assess the frequency changes across a broader range of topological variations.\u003c/p\u003e \u003cp\u003eLinear, exponential, and logarithmic regression models were applied to account for the possibility of nonlinear relationships between the topological parameters and frequency. All analyses were performed using Python (version 3.11), and model fit was evaluated using the coefficient of determination (R\u0026sup2;).\u003c/p\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthical approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eInstitutional Review Board approval was not required because this study used previously published, aggregated data and excluded human subjects or other identifiable personal information\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eACKNOWLEDGMENTS\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNone.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFUNDING\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNone.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCONFLICT OF INTEREST\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no relevant financial relationships to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDATA AVAILABILITY\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData supporting the findings of this study are available from the corresponding author upon request.\u003cstrong\u003e\u003cbr\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eB. 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Ikenaga, et al., \u0026ldquo;Precision anatomy for safe approach to pancreatoduodenectomy for both open and minimally invasive procedure: A systematic review,\u0026rdquo; \u003cem\u003eJournal of Hepato-Biliary-Pancreatic Sciences\u003c/em\u003e 29, no. 1 (2022): 99\u0026ndash;113, https://doi.org/10.1002/jhbp.901.\u003c/li\u003e\n \u003cli\u003eD. Ban, H. Nishino, and T. Ohtsuka, \u0026ldquo;International Expert Consensus on Precision Anatomy for minimally invasive distal pancreatectomy: PAM-HBP Surgery Project,\u0026rdquo; \u003cem\u003eJournal of Hepato-Biliary-Pancreatic Sciences\u003c/em\u003e 29, no. 1 (2022): 161\u0026ndash;173, https://doi.org/10.1002/jhbp.1071.\u003c/li\u003e\n \u003cli\u003eD. Coco and S. Leanza, \u0026ldquo;Celiac Trunk and Hepatic Artery Variants in Pancreatic and Liver Resection Anatomy and Implications in Surgical Practice,\u0026rdquo; \u003cem\u003eOpen Access Macedonian Journal of Medical Sciences\u003c/em\u003e 7, no. 15 (2019): 2563\u0026ndash;2568, https://doi.org/10.3889/oamjms.2019.328.\u003c/li\u003e\n \u003cli\u003eA. Samuolyte, R. Luksaite-Lukste, and M. Kvietkauskas, \u0026ldquo;Anatomical variations of hepatic arteries: implications for clinical practice,\u0026rdquo; \u003cem\u003eFrontiers in Surgery\u003c/em\u003e 12, (2025): 1593800. https://doi.org/10.3389/fsurg.2025.1593800.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Table","content":"\u003cp\u003e\u003cstrong\u003eTable 1.\u0026nbsp;\u003c/strong\u003eTopological Edit Distance and Frequency of Anatomical Variations in the Celiac Trunk\u0026ndash;Superior Mesenteric Artery System\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003eForm\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003eNode-Shift\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003eEdge-Gain\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003eEdge-Loss\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003eFrequency\u003c/p\u003e\n \u003cp\u003e(Adachi 1928) (%)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e55.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e2.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e9.1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e9.9\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e1.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e4.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e2.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e3.6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e1.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e1.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 63px;\"\u003e\n \u003cp\u003e28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 121px;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 197px;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Institute of Science Tokyo","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Celiac Trunk, Superior Mesenteric Artery, Anatomical Variation, Arterial Branching, Topology, Hepatobiliary Surgery","lastPublishedDoi":"10.21203/rs.3.rs-8637430/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8637430/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cb\u003eBackground\u003c/b\u003e\u003c/p\u003e \u003cp\u003eVariations in the anatomical branching patterns of the celiac trunk and superior mesenteric artery (SMA) directly affect vascular identification, surgical planning, and risk of intraoperative vascular injury. However, despite the extensive classification of these variations, the principles governing the occurrence of certain branching patterns remain unclear.\u003c/p\u003e\u003cp\u003e\u003cb\u003eMethods\u003c/b\u003e\u003c/p\u003e \u003cp\u003eWe analyzed 28 branching subtypes of the celiac trunk\u0026ndash;SMA system described by Adachi. Each branching pattern was quantified relative to the standard configuration by using Node-Shift (relative displacement of branching nodes), Edge-Gain (number of additional branches), and Edge-Loss (number of missing branches). The relationships between these topological parameters and branching frequency were examined using linear, log-linear, and exponential regression models.\u003c/p\u003e\u003cp\u003e\u003cb\u003eResults\u003c/b\u003e\u003c/p\u003e \u003cp\u003eBranching frequency consistently decreased as topological complexity increased. In subtypes without branch loss, the branching frequency showed a strong exponential decline with increasing Node-Shift and Edge-Gain (R\u0026sup2; = 0.979). When all subtypes were included, the exponential model maintained a similarly high explanatory power (R\u0026sup2; = 0.979), whereas the linear and log-linear models showed a limited fit. Branch loss was exclusively associated with very-low-frequency patterns.\u003c/p\u003e\u003cp\u003e\u003cb\u003eConclusions\u003c/b\u003e\u003c/p\u003e \u003cp\u003eArterial branching variations do not arise in a purely random manner and are consistent with the probabilistic constraints of developmental and anatomical pattern formation.\u003c/p\u003e","manuscriptTitle":"Topological complexity determines the frequency of branching variations in the celiac trunk–superior mesenteric artery system","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-21 06:39:14","doi":"10.21203/rs.3.rs-8637430/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a6b2f1ad-5d20-417a-8e8a-cf1a47eeb553","owner":[],"postedDate":"January 21st, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-21T06:39:14+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-21 06:39:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8637430","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8637430","identity":"rs-8637430","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}