Hope folks enjoy the read. Such situations represent a serious health issue. With the link between dental disease and cardiovascular disease being present but poorly understood, such models have importance. I suspect that it is these types of teeth in particular that represent a disproportionate amount of issues. While the growth of such lesions around teeth are slow, the development of mobility and transition to softer, fast growing biofilms is likely a critical phase transition. Not interested in publishing, but perhaps this could make an interesting youtube video if anyone wants to collab? Thanks.

Please see addendum:

Addendum: Kinematic model (revised §4) — definitions, prior art, and derivation

Why this addendum exists. The original §4 jumped straight to a lever-arm equation. It never defined the quantities it used (center of rotation, contact centroid, apical sweep) and never connected the single-hinge idealization to the existing literature on how teeth actually move under load. That makes it unfollowable and makes the assumptions look arbitrary when they aren't. This addendum (1) defines every term, (2) grounds each modeling choice in published biomechanics, and (3) re-derives the apical sweep so each number is traceable.

4.1 Definitions

• Periodontal ligament (PDL) — thin viscoelastic tissue (~0.15–0.38 mm wide [K1]) suspending the root in its socket; permits small physiologic tooth movement.
• Long-axis length (L) — length of the tooth along its principal axis (18.54 mm here, from the STL fit, Fig 11).
• Center of resistance (CRes) — the restrained-body analogue of center of mass: the point through which a single force produces pure translation, no rotation [K5, K6].
• Center of rotation (CRot) — the point that stays still while the tooth rotates; the fulcrum of tipping. Its location is set by the loading [K5].
• Force (F) — the applied occlusal load (here the 54 N lateral component, §5).
• Moment (M) — rotational tendency of an off-axis force, M = F·d, where d is the perpendicular distance from CRes (N·mm).
• Moment-to-force ratio (M/F) — selects the type of movement: low M/F → tipping; high M/F → bodily translation or root movement (units mm) [K5, K6].
• Tipping — rotation about a fulcrum apical to the crown; produced by an off-axis force with low M/F.
• Contact centroid — the load-weighted center of the occlusal contact patch (STL height-map, §3).
• Apex–contact distance (r_c) — distance from apex to contact centroid along the long axis (17.22 mm).
• Apex–fulcrum distance (r_a) — distance from apex to CRot (the apex's lever arm).
• Static interference (Δz) — vertical overlap of the two unloaded STL meshes (1.11–1.27 mm, §3).
• Contact excursion (x_c) — how far the contact point actually travels per cycle under load.
• Apical sweep (A_a) — how far the apex travels per cycle (the quantity that feeds the §6.1 pump).
• Envelope of motion — the bounded set of all extreme ("border") mandibular positions; classically Posselt's sagittal teardrop [K2].
• Premature / deflective contact — a contact reached before maximum intercuspation on closure, deflecting the mandible and concentrating load on that tooth.
• Fremitus — palpable/visible movement of a tooth when the patient taps into occlusion; a clinical sign of functional (often traumatic) mobility [K9].
• Initial vs secondary mobility — Mühlemann's distinction [K3]: light loads move the tooth within the PDL space (initial, nonlinear); heavier loads engage alveolar-bone deformation (secondary, ~linear).

4.2 Prior art: the three frameworks this model borrows from

The single-hinge reduction isn't novel physics — it's a coarse-grained version of three well-developed fields. Naming them makes the assumptions auditable.

(a) Tooth-movement biomechanics (orthodontics). A tooth in bone is a rigid body restrained by the PDL, characterized by its center of resistance and, under a given load, a center of rotation. Christiansen & Burstone showed CRot shifts predictably with the force system [K5]; Smith & Burstone formalized how M/F selects tipping vs translation vs root movement [K6]; standard treatments are in Nanda [K7]. For a single-rooted tooth with intact support, CRes lies ~one-third of the root length apical to the crest. Crucially here: as bone support is lost, CRes migrates toward the remaining bone [K7].

(b) Periodontal mobility research. Coolidge established PDL width [K1]; Mühlemann's periodontometry [K3] and Parfitt's axial measurements [K4] quantified physiologic mobility — tenths of a mm horizontally in health, rising sharply with attachment loss. This is the literature saying a tooth doesn't rigidly intrude; it rotates within a compliant ligament, and a tooth with heavy bone loss is markedly more mobile.

(c) Gnathologic occlusion theory. Mandibular motion is governed by two TMJs, idealized in articulators as condylar spheres guided down inclined fossae [K8, K10]. Posselt's envelope [K2] bounds all extreme positions; function happens inside it, combining vertical seating with horizontal, condylar-guided sliding. Occlusal trauma and the first-point-of-contact mechanism are described in the periodontal/occlusion literature [K9, K11]. Modern measurement of the loaded, moving contact path uses jaw tracking, axiography, and T-Scan, not static scans [K10].

What the model keeps and drops. From (a) and (b) it keeps "rigid tooth tipping about a CRot in a compliant ligament." From (c) it keeps only a 1 Hz sinusoidal drive — it collapses the two-joint, condylar-guided, sliding path into a single vertical hinge. That's the model's main kinematic limitation and the reason its output is a floor (§4.5).

4.3 The fulcrum for this tooth

This tooth has heavy coronal bone loss: bony support is confined to the apical ~8 mm. By framework (a), the center of resistance of a reduced periodontium sits within the remaining bone [K7]. Taking the clinical proxy that the tipping fulcrum lies at the midpoint of the residual bony support:

r_a ≈ (8 mm residual apical bone) / 2 ≈ 4 mm from the apex

So the apex sits ~4 mm below the fulcrum, at the apical edge of the remaining bone, and swings through the periapical compartment (the granuloma/biofilm region) as the tooth tips. The 21° inclined contact (§5) delivers the 54 N lateral component off-axis, producing a moment (M = F·d) with a low M/F ratio — i.e. uncontrolled tipping, not intrusion. (A strict pure-tipping CRot sits slightly apical to CRes [K5], which would shorten the apical lever arm; functional load concentration (§4.5) pushes the other way. The 4 mm midpoint is a reasonable center value between these.)

4.4 Lever-arm derivation

Model the loaded tooth as a rigid body in small rotation about CRot. Let x_a = apex displacement, x_c = contact-centroid displacement. From the STL:

r_c ≈ 17.22 mm                          (2)

Displacement scales linearly with distance from the fulcrum, so the apex (lever arm r_a) and contact (lever arm r_c − r_a) move in the ratio:

x_a / x_c = r_a / (r_c − r_a)            (3)

ratio = 4 / (17.22 − 4) = 0.30

The apex moves ~30% of the contact excursion. Using the static interference as a first estimate (x_c ≈ Δz = 1.11–1.27 mm) gives A_a,simulated ≈ 0.34–0.38 mm. The working value carried into §6.1 is set slightly above this to allow for load concentration (§4.5):

A_a ≈ 0.5 mm   (peak-to-peak)            (4)

Plain language: the tooth see-saws about a point 4 mm above its apex. The contact, far out on the long arm, travels ~1 mm; the apex, on the short arm, travels ~0.3–0.5 mm the opposite way, sweeping through the soft lesion.

4.5 Static interference vs functional excursion — the first-point-of-contact correction

Δz is the interference of two unloaded meshes — a maximum-intercuspation snapshot. Two facts from §4.2(c) make the real per-cycle x_c larger:

1. Function adds horizontal slide. Within Posselt's envelope the antagonist cusp travels a condylar-guided path, adding lateral excursion on top of the vertical overlap, so x_c,functional ≳ Δz.
2. A deflective contact concentrates load. A supererupted/malpositioned tooth acting as a first point of contact is struck before the arch shares the load — the mechanism behind clinical fremitus [K9]. Under that concentrated load, x_c exceeds the static interference.

Because A_a = 0.30·x_c, the apical fulcrum caps the gain — even heavy deflective contact yields ~1 mm at the apex, not several:

x_c (excursion)   Clinical reading                      A_a = 0.30·x_c   v_a = π·f·A_a (1 Hz)
---------------   -----------------------------------   --------------   --------------------
1.1–1.3 mm        static STL interference               0.34–0.38 mm     1.1–1.2 mm/s
2 mm              mild fremitus / early premature        0.60 mm         1.9 mm/s
3 mm              moderate first-point-of-contact load   0.91 mm         2.9 mm/s
4 mm              heavy deflective contact               1.21 mm         3.8 mm/s

The working A_a ≈ 0.5 mm corresponds to x_c ≈ 1.6 mm and v_a ≈ 1.6 mm/s (§6.1). The dominant kinematic uncertainty is thus x_c under first-contact loading, not the fulcrum, which is fixed by the residual bone.

4.6 Conjecture: eccentric loading and the spiral biofilm pattern (speculative — needs more modeling)

The biofilm forms a helical band around the root, apparently occupying surface regions not routinely compressed in open function. A purely planar tip would produce horizontal bands, not a helix — so the spiral hints at an axial-rotation (torsion) component in the functional load, layered on the tipping. Proposed mechanism: an inclined, off-center contact applies both a tipping moment and a torque about the long axis; as the load vector rotates around the chewing loop (framework (c)), the instantaneous compression/shear locus migrates around and along the root, leaving a helical minimum in time-integrated mechanical load — and biofilm persists in those low-stress refugia. A minimal test: take the real 3D contact-force trajectory (jaw tracking / articulator settings), compute the rigid-body displacement field about CRot, integrate a per-point PDL compression/shear "dose" over the surface, and check whether the predicted low-dose region is helical and co-registers with the fluorescence map. Flagged speculative: fluorescence isn't diagnostic, and the pattern interpretation is unconfirmed.

References (kinematics)

[K1]  Coolidge ED. Thickness of the human periodontal membrane. J Am Dent Assoc. 1937;24:1260–1270.
[K2]  Posselt U. Studies in the mobility of the human mandible. Acta Odontol Scand. 1952;10(Suppl 10).
[K3]  Mühlemann HR. Tooth mobility: clinical aspects and research findings. J Periodontol. 1967;38:686–713.
[K4]  Parfitt GJ. Measurement of physical mobility of single teeth in an axial direction. J Dent Res. 1960;39:608–618.
[K5]  Christiansen RL, Burstone CJ. Centers of rotation within the periodontal space. Am J Orthod. 1969;55(4):353–369.
[K6]  Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod. 1984;85(4):294–307.
[K7]  Nanda R (ed). Biomechanics in Clinical Orthodontics. WB Saunders; 1997.
[K8]  Gysi A. Articulation and the principles of articulator design. (verify exact citation)
[K9]  American Academy of Periodontology. Glossary of Periodontal Terms.
[K10] Okeson JP. Management of Temporomandibular Disorders and Occlusion. Elsevier/Mosby.
[K11] Lang NP, Lindhe J (eds). Clinical Periodontology and Implant Dentistry.
[K12] The Glossary of Prosthodontic Terms, 9th ed (GPT-9). J Prosthet Dent. 2017;117(5S):e1–e105.