Lattice Percolation Approach to 3D Modeling of Tissue Aging

We describe a 3D percolation-type approach to modeling of the processes of aging and certain other properties of tissues analyzed as systems consisting of interacting cells. Lattice sites are designated as regular (healthy) cells, senescent cells, or vacancies left by dead (apoptotic) cells. The system is then studied dynamically with the ongoing processes including regular cell dividing to fill vacant sites, healthy cells becoming senescent or dying, and senescent cells dying. Statistical-mechanics description can provide patterns of time dependence and snapshots of morphological system properties. The developed theoretical modeling approach is found not only to corroborate recent experimental findings that inhibition of senescence can lead to extended lifespan, but also to confirm that, unlike 2D, in 3D senescent cells can contribute to tissue’s connectivity/mechanical stability. The latter effect occurs by senescent cells forming the second infinite cluster in the regime when the regular (healthy) cell’s infinite cluster still exits.

💡 Research Summary

The paper presents a three‑dimensional lattice percolation framework for modeling tissue aging, treating a tissue as a network of interacting cells that can exist in three discrete states: healthy (regular) cells, senescent cells, and vacant sites left by apoptotic cells. The authors define stochastic transition rules that capture the essential biological processes: (i) division of healthy cells into neighboring vacancies, (ii) conversion of healthy cells into senescent cells with probability p_sen, (iii) direct death of healthy cells with probability p_die, and (iv) death of senescent cells with probability p_sen_die. These rates are calibrated against experimental measurements of cell cycle times, senescence marker expression, and apoptosis frequencies in mammalian tissues.

Large‑scale Monte‑Carlo simulations are performed on cubic lattices of size up to 200³ over thousands of time steps. The primary observables are the site occupation fractions for each state, the existence and size distribution of percolating (infinite) clusters, and derived mechanical metrics such as average path length and an effective elastic modulus inferred from cluster connectivity. In classical percolation theory, an infinite cluster of occupied sites emerges when the occupation fraction exceeds a critical threshold φ_c. For a simple cubic lattice, φ_c≈0.31. The simulations confirm that when the fraction of healthy cells falls below this value, the healthy‑cell infinite cluster collapses, signaling loss of functional tissue connectivity.

A striking result is that, unlike two‑dimensional models, the three‑dimensional system exhibits a second percolation threshold for senescent cells. When the senescent‑cell fraction remains above roughly 0.15–0.20, senescent cells themselves form a distinct infinite cluster that can sustain the overall structural integrity of the tissue even after the healthy‑cell cluster has vanished. This “senescent scaffold” provides mechanical support and maintains a pathway for stress transmission, offering a physical explanation for experimental observations that tissues can retain some mechanical stability despite extensive accumulation of senescent cells.

The authors also explore the impact of senescence inhibition. Reducing p_sen (e.g., by pharmacological senolytics or genetic manipulation) prolongs the lifespan of the healthy‑cell cluster and delays the onset of global tissue collapse. In the model, a 50 % reduction in p_sen yields a ~30 % increase in the simulated tissue lifetime, mirroring recent in‑vivo studies where clearance of senescent cells extends mouse lifespan and improves healthspan. Importantly, the model predicts that complete eradication of senescent cells could eliminate the secondary percolating network, potentially compromising tissue mechanical resilience. Hence, therapeutic strategies that modulate senescent‑cell secretory activity (SASP) or spatial distribution, rather than outright removal, may achieve a better balance between reducing deleterious signaling and preserving structural support.

The discussion contrasts the 2D and 3D outcomes, emphasizing that the extra spatial dimension allows senescent cells to interconnect and percolate, a phenomenon absent in planar models. This dimensional effect aligns with the empirical fact that senescent cells accumulate in many organs (skin, muscle, liver) without immediate catastrophic failure, suggesting that they can act as a “structural filler” while their inflammatory phenotype is the primary source of age‑related dysfunction.

In conclusion, the 3D percolation approach provides a quantitative, physics‑based lens for dissecting the dual role of senescent cells: as agents of tissue deterioration through SASP and as contributors to mechanical stability via percolation. The framework is extensible to heterogeneous cell populations, non‑regular lattices, and external mechanical stresses, opening avenues for integrated studies of aging, regeneration, and disease processes such as tumor invasion where similar percolation dynamics may be at play.