A mathematical model suggests that tissue growth can end at a finite size, continue indefinitely, or keep the relative sizes of its parts steady, depending on the mechanical regime built into the equations. The authors present these outcomes as versions of one continuum model, linking determinate growth—the bounded case—to indeterminate growth, in which the modeled tissue keeps expanding, and to proportionate growth in an inhomogeneous tissue. The result is a framework, not a biological finding: the work analyzes modeled tissue domains and parameter settings rather than an empirical human, animal or cell sample.
The study asks whether determinate and indeterminate growth can arise as phases of a single mechanical model, and whether that same model can produce proportionate growth in a tissue made of dissimilar parts. It treats active stress relative to elastic modulus as the switch between bounded and unbounded behavior, and mechanical impedances as the factors that set the long-run shares of a mixed tissue. That framing is an interpretation of the model, not evidence that living tissues follow the same rules.
One framework, three growth regimes
At the center of the work is an implicit-growth, coarse-grained model of a finite-size viscoelastic tissue continuum. In ordinary language, the tissue is represented as a deformable material with both elastic resistance and viscous flow, while growth is encoded in the continuum rather than tracked cell by cell. The authors examine one case driven by an imposed boundary stress and a second, autonomous case in which activity is generated from density under traction-free boundaries.
The boundary-stress problem has closed-form solutions. For the density-driven problem, the steady state and high-turnover growing state are treated analytically, while the intermediate regime is calculated numerically. Those calculations use a finite-element discretization in FEniCS with an updated-Lagrangian scheme, a way of recalculating the equations as the tissue-shaped mesh deforms.
The threshold between settling and perpetual growth
In homogeneous tissue exposed to imposed boundary stress, the model divides behavior at a simple threshold: a non-growing steady state exists only while the applied stress is below the elastic modulus. Once the stress rises above that elastic-stress limit, the model predicts perpetual flow. This is the paper’s main switch between bounded and unbounded growth, but it is a prediction of the equations rather than a measured biological threshold.
The threshold also changes how quickly the modeled tissue settles. As it is approached from below, predicted steady-state size scales as ε^-1, while the approach rate scales as ε^3. In ordinary terms, the model predicts a finite tissue size becoming arbitrarily large at the same time that relaxation becomes arbitrarily slow. These are asymptotic theoretical results within the model.
Above the threshold, the growing branch behaves differently. At long times, tissue length grows linearly, and the strain rate—the rate of local deformation—concentrates near the tissue edges. The asymptotic growth rate is set by the excess of applied stress over elastic modulus and by the tissue’s mechanical impedance. That result describes the stated long-tissue asymptotic regime, not every transient stage.
Mixed tissues do not automatically keep their starting shape
The model then turns to a two-part tissue with different material properties. When the applied stress exceeds both parts’ elastic moduli, the composite keeps growing, but its parts do not necessarily enlarge in step. Over time, their length fractions freeze at values determined by mechanical impedances and by the mismatch between their moduli. The model therefore predicts a stable division of the growing tissue without requiring the two parts to have started in the same proportion.
That long-term division is usually not inherited from the starting geometry. Frozen fractions generally differ from the initial proportions: the composite first drifts as it grows, then settles toward values set by material parameters. The paper’s proportionate-growth result is therefore conditional. It does not say that any heterogeneous tissue will automatically preserve its initial layout.
There is, however, a parameter condition under which the model can preserve those starting proportions. If the two parts have equal elastic moduli, their impedances must be inversely proportional to their initial lengths. In practical terms within the model, the mechanical resistance assigned to each part has to be tuned to its starting share. Even then, the result is asymptotic and can be disturbed by finite-size or density-gradient effects, or if the interface consumes one part.
When the activity comes from within
The second formulation asks whether a comparable threshold can emerge without imposed boundary stress. Here, the model uses logistic birth-death density dynamics to generate isotropic active stress from cell density, while the boundaries are traction-free. In numerical results, the transition appears when the activity-to-elastic-modulus ratio ζf(ρ0)/E equals 1. Increasing the turnover parameter κτ brings density closer to ρ0 and reduces the corrections caused by density gradients.
Turnover does not play the same role in every regime. Within the autonomous model, the steady-state length is independent of κτ, whereas the growth rate in the unbounded phase varies non-monotonically with turnover. The distinction matters: changing turnover can alter how quickly the model grows without changing the length it predicts in the steady state.
At sufficiently small κτ, the calculations produce a third kind of behavior: the tissue can oscillate as a whole rather than settle into either of the two special solutions. Its length and density overshoot and undershoot their steady values. This is a simulated low-turnover regime, not an observed biological subgroup, so it broadens the model’s behavior without supplying evidence that real tissues oscillate in this way.
What the preprint does—and does not—show
Taken together, the calculations support the authors’ interpretation of determinate, indeterminate and proportionate growth as regimes governed by different combinations of activity, elasticity and impedance. In the model, the phase choice comes from active stress relative to elastic modulus, while the long-run share of a heterogeneous tissue comes from mechanical impedances and modulus mismatch. The framework gives those outcomes a common mechanical description, but it does not establish that actual organisms use the proposed mechanism.
The limits are substantial. The main elongation result comes from a one-dimensional model, so growth that changes shape in multiple directions is outside that analysis. The proportion results are asymptotic and depend on the parts remaining in the long-tissue regime without an interface overtaking an edge. Intermediate-turnover behavior is handled numerically rather than fully analytically. Because the study has no empirical sample, it does not validate the framework against measured growth curves.
The document is version 1 of an arXiv preprint dated 20 August 2026. The authors acknowledge support from the Department of Atomic Energy, Government of India, under project RTI4019, and report Claude assistance with copyediting and drafting followed by author review. Those details do not change the central point: the work offers a mechanical proposal, not direct biological evidence.
Paper data and sources
Original title: Growth phases of an active tissue: determinate, indeterminate, and proportionate
Authors: Jigyasa Watwani, K. Vijay Kumar, Vishal Vasan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text