Breastfeeding efficiency relies on coordinated tongue motion, sustained tissue contact and maintenance of an effective intraoral seal. Current assessments of seal formation mainly use local kinematic descriptors or pressure recordings, which do not capture the global structural continuity of the sealing region. We introduce a systolic geometry based approach in which each sagittal ultrasound frame is modeled as a two dimensional deformable domain bounded by tongue, palate and nipple contours. Global seal continuity is formalized through the shortest closed curve that cannot be contracted to a point because of the overall geometry of the domain. The nipple defines a central region that must be circumferentially enclosed by a contact band to maintain suction. Within this band, closed curves encircling the nipple exactly once can be identified; the shortest of these curves defines a normalized systolic index representing the tightest admissible sealing loop. Simulations of symmetric thinning, localized discontinuities and cyclic perturbations reveal feasibility boundaries separating seal preserving from seal breaking configurations. Notably, admissible encircling curves may transiently disappear even when overall geometric motion remains smooth. By capturing global circumferential continuity that cannot be inferred from local metrics alone, our approach generates testable hypotheses linking the existence and temporal stability of admissible encircling curves to milk transfer efficiency and vacuum stability. Applied to segmented ultrasound data and integrated with pressure measurements, our systolic approach could provide a quantitative framework for objective assessment of seal integrity and longitudinal monitoring of latch stability.
Breastfeeding efficiency depends on coordinated interactions among tongue deformation, nipple positioning, intraoral pressure modulation and maintenance of an effective oral seal (Wen et al. 2022;Hill, Richard, and Pados 2023). Quantitative investigations rely primarily on ultrasound-based analysis, pressure measurements and anatomical descriptors like tongue excursion amplitude, nipple compression depth and palate-tongue distance (Elad et Adjerid et al. 2023). These approaches have improved the understanding of suck-swallow-breathe coordination and latch mechanics, clarifying differences associated with prematurity or ankyloglossia (Ito 2014;Costa-Romero et al. 2021;Cordray et al. 2023;Rossato 2025). However, most current metrics are local and scalar, focusing on displacement magnitudes or gap distances at selected points. They do not explicitly account for the contact region's global geometric continuity that sustains negative intraoral pressure. Seal integrity is often inferred indirectly from pressure traces or visual inspection rather than assessed through structural invariants (Martin-Harris et al. 2020; Aoyagi et al. 2021;Jiang et al. 2021; Kabakoff et al. 2021; Kabakoff et al. 2023). Indeed, locally similar configurations may differ in their global capacity to maintain a continuous encircling contact band. A formal description of oral seal formation integrating geometry and topology remains underdeveloped, particularly in two-dimensional ultrasound representations where the seal is usually inferred from time-varying contours.
Our theoretical study aims to simulate oral seal formation during breastfeeding using systolic geometry principles. Systolic geometry is a branch of mathematics assessing the shortest closed loop that cannot be “shrunk away” because of the shape of the surrounding space (Katz 2007;Columbus et al. 2022;Bähni 2026). In our context, this means identifying the smallest closed path that fully surrounds the nipple while remaining inside the contact area between tongue and palate. If such a loop exists, the seal is structurally intact; if no such loop can be drawn, the seal is broken. For each ultrasound frame, we would represent the oral cavity as a two-dimensional shape defined by the contours of the tongue, palate and nipple. Within this shape, we would identify the region where the tongue and surrounding tissues are sufficiently close to sustain suction. We would then search for closed curves inside this region wrapping once around the nipple. Among all admissible curves, we would select the shortest one. This “minimal encircling curve” would define a measurable length scale that reflects how tight and continuous the seal is. Using simulated deformable shapes, we would examine how small geometric changes, such as thinning of the contact band or localized gaps, could affect the presence and size of this encircling curve. The aim is to describe seal integrity as a structural, measurable property rather than relying only on local movement or pressure values and to provide quantitative descriptors that can later be tested in clinical studies.
We will proceed as follows. First, we formalize the geometric representation of the oral domain and define admissible encircling curves. Second, we introduce the systolic invariant and its normalization. Third, we present simulation results illustrating seal-preserving and seal-breaking perturbations. Finally, we compare our invariant with conventional kinematic descriptors and describe the practical outcomes of our theoretical approach.
We describe the mathematical construction, simulation procedures and computational techniques to formalize oral seal formation as a topology-constrained geometric problem in two dimensions. All results derive from analytically defined deformable domains and numerical simulations; no empirical ultrasound data were processed.
Geometric representation of the oral domain. Each instantaneous sagittal configuration of the infant oral cavity was modeled as a bounded planar domain Ω 𝑡 ⊂ ℝ 2 . The nipple cross-section was represented as a compact, simply connected subset 𝑁 𝑡 ⊂ Ω 𝑡 , approximated by a disk of radius 𝑟 𝑡 centered at 𝑐 𝑡 ∈ ℝ 2 such that 𝑁 𝑡 = {𝑥 ∈ ℝ 2 : ∥ 𝑥 -𝑐 𝑡 ∥≤ 𝑟 𝑡 }. The effective contact band 𝐴 𝑡 ⊂ Ω 𝑡 ∖ int(𝑁 𝑡 ) was defined as a closed annular-type region surrounding 𝑁 𝑡 , representing the seal-supporting domain. Topologically, 𝐴 𝑡 was constructed to admit either homotopy type 𝑆 1 × [0,1] (seal-preserving configuration) or a disconnected or simply connected variant (seal-breaking configuration), consistent with the schematic shown in Figure 1. The metric structure on Ω 𝑡 was taken to be the Euclidean metric inherited from ℝ 2 , with arc length defined as
for any rectifiable curve 𝛾: [0,1] → Ω 𝑡 . Discretization was performed on a uniform grid of resolution 300 × 300points over a square domain [-𝐿, 𝐿] 2 , with 𝐿 = 3𝑟 𝑡 . Binary masks represented membership in 𝐴 𝑡 . Connectivity was defined usin
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