Journal of Mathematical Biology

Price equation and genotype dynamics are two methods for studying the fixation of one allele by natural selection in a diploid population. There are two strict monotonicity conditions that imply the fixation of one allele. The genotype dynamics is called Haldane monotone if the relative frequency of one allele strictly increases along all solutions of the genotype dynamics, so this allele is fixed. In this paper, we show that the genotype dynamics is Haldane monotone if and only if the right-hand side of the Price equation is always strictly positive. The other strict monotonicity condition requires that the relative frequency of a homozygote strictly increase according to the genotype dynamics. For example, in a model where the genotype dynamics is governed by interactions between individuals, the cost-accepting homozygote is fixed by natural selection if the other genotypes always receive a smaller average gain from all interactions than the cost-accepting homozygote. Both monotonicity conditions require that the interaction is not well-mixed in the population. These two conditions are not equivalent. In addition, we give a non-monotonicity condition, which also implies the fixation of a homozygote. The fixation of a homozygote depends on the phenotypic payoff of the interaction, the genotype-phenotype mapping, and the interaction scheme. In a sexual population, the interaction scheme of siblings depends on the mating system, and so do the conditions of fixation of the cost-accepting homozygote. We present examples showing that if we only change the monogamous mating system, assuming panmixing or mating assortativity, then the condition for the fixation of the cooperator homozygote is b > 2c and b > c, respectively.

The Poisson-Nernst-Planck (PNP) system is an accurate model of electrodiffusion of ionic species. It is commonly used in situations where nanoscale resolution is required, for instance close to ion channels in the membranes of biological cells. The inherent stiffness of the equations has made them challenging to solve and has limited the applicability of the system. In particular, the time step required for stable solutions has typically needed to be very short (nanoseconds), which makes simulations on the time scale of an action potential (milliseconds) difficult. Recently, it has been observed that avoiding operator splitting and instead solving the concentration equations and the electrostatic equation in a coupled manner relaxes the time-step limitation considerably. However, no theoretical explanation of this observation has been provided. Here, we aim to explain why the coupled scheme allows much larger time steps. We illustrate the mechanism by considering special cases that define necessary, but not sufficient, conditions for stability. We also show that these conditions remain relevant for the fully coupled PNP model in 3D.

1Although resources are typically distributed continuously in space, species distributions often organize into discrete clusters. In his seminal paper [36], Turing demonstrated that such clusters can spontaneously arise in population densities, even when populations evolve in environments with continuously varying conditions. This phenomenon is known as Turing instability. In this work, we focus on two models grounded in population dynamics: a one-dimensional model based on the nonlocal Fisher-KPP equation, and a two-dimensional model involving an environmental gradient. We show that phenotypic clusters (sometimes referred to as "species") emerge in these models. We prove that they do not emerge because of Turing instability, but because of stochasticity, and that they disappear when stochasticity is reduced. First, for both models, we start our simulations with initial populations uniformly distributed in the state space. We show that phenotypic clusters quickly emerge and that the distances between them depend on the population size, that is, on the degree of stochasticity. Next, we start from already clearly defined phenotypic clusters. We identify three regimes in the connection between population size, the initial distances between clusters, and the distances between clusters at equilibrium. Last, on the two-dimensional model, we relax the hypothesis of complete clonality by varying the effective recombination rate, explore its effect on phenotypic clustering, and show that phenotypic clustering decays drastically with slight recombination.

AO_SCPLOWBSTRACTC_SCPLOWOncolytic viruses provide cancer therapy using replication-competent viruses that selectively infect and lyse tumour cells. Their tumour-specific replication also enables the delivery of targeted, virus-encoded gene products, such as enzymes that activate prodrugs. This dual functionality offers the potential for synergistic effects by combining direct oncolysis with localised drug activation. The interplay between infection, replication, lysis, and gene product delivery remains poorly understood. Here, we introduce a spatially structured, multi-patch model of cancer cells infected by an oncolytic virus engineered to deliver a prodrug-activating enzyme. The spatial system is first represented as a microscopic model and subsequently reduced via spectral dimension reduction techniques. This reduction yields an ordinary differential equation model for a set of coarse-grained variables, which we analyze both without the transgene (OV model) and with the transgene (TOV model). For each case, we compute the basic reproduction number, R0, which determines the conditions for viral spread. Both models exhibit three regimes via transcritical bifurcations: (i) R0 < 0, extinction of both cancer and infected cells; (ii) 0 < R0 [&le;] 1, persistence of cancer cells only; and (iii) R0 > 1, coexistence as a stable node or as a focus. The TOV model, as a difference form the OV model, can undergo periodic oscillations arising from a Hopf-Andronov bifurcation. Notably, oscillation amplitudes can be controlled such that cancer cells largely decrease when drug-induced death is stronger in non-infected cells than in infected ones, enabling effective cancer cells killing while maintaining viral replication and prodrug activation. The qualitative behaviour of the coarse-grained model is shown to be preserved in both the microscopic and spatially explicit models.

Experimental studies show that tumor cells adopt migratory or proliferative phenotypes depending on the local extracellular matrix (ECM). In this work, we propose a minimal go-or-grow invasion model, comprising two specialist cell phenotypes: proliferating and migratory, with phenotypic switching and cell migration depending on local ECM density. Numerical simulations of this model reveal that the spatial arrangement of proliferative and migratory cells depends on the choice of phenotypic switching function. We then ask whether this specialist cell-phenotype model can be reduced to a generalist cell-phenotype model. We derive a relationship between the reduced model and go-or-grow model in the fast phenotypic switching regime. We observe that the reduced model captures the dynamics of the original model, for a range of realistic phenotypic switching functions. We analytically derive the minimum traveling wave speed of the reduced model in a homogeneous ECM bed. Moreover, using linear stability analysis on the go-or-grow model, we recover the same wave speed expression. In addition, we numerically explore how the key parameters influence the traveling wave speed profile. Our analysis indicated the counter-intuitive result that the wave speed is independent of the matrix degradation rate, and our simulations show that, at most, the speed is weakly dependent on this parameter.

Minimum viable populations (MVPs) are population levels large enough to surmount risk from demographic, environmental, and genetic stochasticity. MVPs are estimated by biologists to guide conservation practices. However, MVPs are generally estimated for a target population without regard for how they interact with intra- and inter-species population dynamics in the broader ecological community. Thus, how and why population dynamics interact with MVPs imposed by conservation biologists remain unclear. When MVPs are imposed on a continuous population model, traditional analyses fail to capture the range of possible outcomes those MVPs create. Here, we describe viability space decomposition (VSD) as a mathematical tool to systematically analyze the potential crossing of MVPs during population dynamics. We demonstrate that different extinction and survival outcomes can be recovered from a model with imposed MVPs using three VSD concepts in junction with a traditional phase portrait: mortality manifolds which separate conditions that lead to different existential outcomes, ordering manifolds which determine the order of extinction events for multiple populations, and collapse manifolds which determine the survival or extinction of one species given the loss of another. We employ these methods with a standard consumer-resource model, and the methods can be scaled to systems with more species. VSD is a useful tool for conservation biologists and community ecologists concerned with boundary crossing problems in any dynamical system.

This work focuses on the global and partial identification problem for fractional differential equations. We provide a general numerical procedure based on global and local optimization algorithms with two refinements for biological systems that ensure solution positivity and homogeneous parameter units. The method is applied to a new fractional model of Dengue outbreak called the Fractional Homogeneous Nishiura (FHN) model, calibrated using data of newly infected people in Cape Verde. We show that our identification method yields a better fit between data and model solutions than previous approaches and that our FHN model captures the dynamics of Dengue more closely than existing systems.

Biological pathway analysis often requires identifying interventions that block reachability to an undesirable state, such as a disease-associated module, toxic byproduct, or adverse phenotype, while preserving reachability among essential biological functions. Motivated by this setting, we study the Reachability Preserving Minimum Edge Cut (RPMEC) problem: given protected terminals s1 and s2 and a target terminal t, the goal is to remove a minimum-cost set of edges that separates s1 and s2 from t while keeping s1 and s2 connected. This formulation naturally models pathway-level intervention design, where one seeks to disrupt harmful signaling, metabolic, or interaction routes without breaking required functional connectivity. We revisit the three-terminal undirected edge-cut case and analyze a Dijkstra-style dynamic programming algorithm that is exact on planar graphs but fails on general graphs. We characterize the structural condition required for exactness, namely frontier-realizability of optimal source-side regions, and identify biological graph representations where this condition is likely to hold after appropriate preprocessing, including curated planar pathway maps, Reactome-style hierarchy trees, SCC-contracted feedback modules, metabolic building-block DAGs with dominator structure, and functional-module quotients of protein interaction networks. We further discuss directed variants, approximation strategies, and exact alternatives based on ASP, MILP, bounded-treewidth dynamic programming, and important separators. The results provide a graph-theoretic foundation for deciding when fast greedy computation is reliable for biological pathway intervention problems and when more expressive exact optimization methods are needed. Author SummaryMany real-world networks require interventions that separate harmful or undesirable states while preserving essential connectivity. This situation appears in biological pathway analysis, where one may want to block reachability to a disease-related module, toxic byproduct, or adverse phenotype without disrupting communication among essential genes, proteins, reactions, or metabolites. We study this problem through the Reachability Preserving Minimum Edge Cut formulation. Unlike ordinary minimum cut, the solution must satisfy both a separation requirement and a preservation requirement. We show why a natural Dijkstra-style algorithm works only under specific structural conditions, such as planar, laminar, or module-like pathway graphs, and why it may fail on general graphs. The results help identify when fast graph algorithms are reliable for biological intervention problems and when exact optimization tools such as Answer Set Programming or integer programming are more appropriate.

Approximate Bayesian computation sequential Monte Carlo (ABC-SMC) propagates its particles with a perturbation kernel, and with the standard Normal kernel it degrades sharply as the parameter dimension grows, a failure usually attributed to dimension itself. We show instead that it is governed by the quality of the summary statistics, with dimension entering only through a separate and milder mechanism, and that the two must act together for the Normal kernel to break. The first ingredient is covariance overinflation: the kernel covariance, estimated from the particle cloud, overshoots the true posterior covariance by a factor set by information loss in the summary statistics. We derive this overscaling factor in closed form for a Gaussian model with sufficient statistics and show that it stays modest at any dimension, shrinking toward its baseline value as the tolerance tightens; the extreme values seen in practice (of order 103) are a signature of insufficient summaries, not of dimension. The second ingredient is perturbation overconcentration: the normalised Normal step size concentrates around one as the dimension grows, so every proposal overshoots by the same factor. Either ingredient alone is harmless; only their combination breaks the Normal kernel. A Cauchy kernel (multivariate t with one degree of freedom) removes the concentration, keeping a positive acceptance rate under arbitrary overscaling at a bounded worst-case cost of 1.87x in expected squared jump distance. In a Metropolis-Hastings framework we derive closed-form acceptance rates for both kernels that illustrate the advantage of the Cauchy kernel in this limit. A series of full ABC-SMC computational experiments on five problems at d = 12, including a hierarchical gene-expression model, show the Cauchy reducing the sliced Wasserstein distance to the reference posterior by factors of up to 50 with the same simulation budget. Since the summary statistics are commonly insufficient for the models that require ABC, overinflation is structural and the Cauchy perturbation kernel is the right default for problems in higher dimensions.

The osteocyte network in bone is believed to play an important role for how bone tissues sense and respond to mechanical stimulation. Yet, bone adaptation to mechanical loads is often conceptualised as a simple response to mechanical stimuli, such as Wolffs law, which is based on mechanical variables only and takes no account of the cellular basis of mechanosensation. Wolffs law presumes the existence of a reference mechanical stimulus, the mechanical setpoint, above which bone is consolidated, and under which bone is removed. In this paper, we develop a theory of bone tissue sensing and adaptation based on osteocytes to provide new understanding of the role played by osteocyte signals in mechanical adaptation. In this theory, the mechanical setpoint of Frosts mechanostat is explicitly embodied as osteocyte properties involved in mechanotransduction. The mechanical setpoint is allowed to adapt due to the replacement of osteocytes during remodelling, making the setpoint space and time dependent. We propose a mathematical model to implement this new theory of bone adapation and present numerical simulations of this model to explore how mechanobiological response curves (effective Wolffs laws) are modulated by setpoint adaptation during remodelling. By accounting for varying osteocyte populations within bone tissue, we explore bone adaptation under osteocyte disruptions, which is particularly relevant to age-related bone loss. Our model suggests that biological disruptions of remodelling balance cannot always be compensated by mechanical feedback, and that setpoint adaptation during remodelling may have significant observable consequences, such as hysteresis in bone response signatures that resemble lazy zones.

Darwinian fitness is equated here with invasion fitness and defined as the quantity determining the fate--certain extinction or possible spread--of a single mutant type. We derive it, together with its phenotypic derivative, for evolution in group-structured populations under limited genetic mixing, where the demography of the focal species and its environment is modeled as a discrete-time stochastic process. Reproduction, physiological development, dispersal, and survival are influenced by interactions within and between groups and by environmental fluctuations within and across generations. Using multitype branching processes in random environments, we show that invasion fitness is predicted by a stochastic growth rate that can be represented biologically in two meaningful genealogical ways. First, as the long-term geometric mean of the expected per-capita number of mutant copies produced per time step by a representative member of the mutant lineage. Second, as the the long-term geometric mean of the expected reproductive-value-weighted per-capita number of mutant copies produced by such an individual. This latter representation is useful for computing the phenotypic directional derivative of invasion fitness. Moreover, this derivative can be written as an actor-centered inclusive-fitness effect derived from properties of the resident population process. This effect depends on class-specific fitness differentials, relatedness, reproductive values, and class frequencies. However, unless generation- and class-specific fitness defines a stochastic matrix, the derivative does not separate stochastic reproductive values from relatedness and class frequencies, and must be evaluated by simulations. In summary, we formalize invasion fitness biologically quite generally and show how Hamiltons marginal rule is deduced from it.

AnnotationMitochondrial heteroplasmy --the coexistence of both wild-type and mutant copies of mitochondrial DNA (mtDNA) within a cell--is a key factor in the pathogenesis of mitochondrial diseases. Classical approaches, which rely solely on the scalar fraction of mutant DNA, fail to fully account for threshold effects, the stochastic nature of heteroplasmy dynamics, and tissue specificity. The aim of the work is to construct a complex stochastic model of heteroplasmy dynamics, which for the first time combines the effects of selection, genetic drift, migration of mitochondrial genomes between tissues and threshold mechanisms of pathology development, for a quantitative assessment of the risk of mitochondrial diseases. In this paper, we propose a complex-phase formalism in which the state of a cells mitochondrial genome is described by a complex number Z = a + ib, where a and b are the absolute numbers of normal and mutant mtDNA copies, respectively. This approach naturally combines information on copy number and heteroplasmy level, and the argument{phi} = arctan (b / a) is interpreted as a phase characterizing the mutant load. Based on this formalism, we developed a stochastic model of tissue dynamics that includes the processes of selection, genetic drift, and intertissue migration of mitochondrial genomes. Using Monte Carlo methods (1000 simulations), we demonstrated that neuronal tissues are characterized by high heteroplasmy variability and a significant probability of reaching a pathological threshold even with a relatively low systemic mutant load. Kaplan-Meier survival analysis demonstrates that the development of pathology is probabilistic and can be described as a time -to-event process . The proposed approach enables quantitative assessment of the individual risk of developing mitochondrial diseases and opens the door to personalized prognosis.

We describe a statistical agent-based model (SABM) for binary spatio-temporal data in which the occupancy of each cell evolves as a Bernoulli mixture of three mechanistically distinct processes: local persistence, anisotropic neighborhood dispersal, and long-distance dispersal. The model is embedded in a hierarchical Bayesian framework with conjugate Beta full-conditionals for the persistence and long-distance parameters and a Dirichlet prior on the directional dispersal kernel. A nonstationary extension links the dispersal kernel to a latent habitat-suitability surface through directional gradients of a Gaussian process. We show that, in the small-step regime, the Lagrangian recurrence for the dispersal kernel scales to a classical two-dimensional advection-diffusion partial differential equation whose drift and dispersion coefficients are the first and second moments of the dispersal probabilities. We provide an MCMC algorithm exploiting the exact full-conditionals and demonstrate parameter recovery and PDE-scaling agreement in a simulated example.

Ordinary differential equation models of biochemical reactions are often formulated as stoichiometric systems in which the dynamics arise from a collection of interacting processes. A central challenge is that the functional form of each process is rarely known a priori and may be difficult to infer from data. We propose biochemically informed neural ordinary differential equations (BINODEs), a neural-ODE framework that retains the stoichiometric structure of mechanistic models while representing individual processes by neural networks. In BINODEs, the outputs of neural network processes (NNPs) are mapped to state derivatives through a linear layer analogous to a stoichiometric matrix. This architecture allows biological side information, such as process-specific inputs, sign constraints, and monotonicity assumptions, to be built directly into the model. We characterize the approximation properties of NNPs for several standard biochemical rate laws and show that the proposed framework recovers both trajectories and process-level structure in Monod, Lotka-Volterra, pharmacokinetic, and ultradian endocrine models. These results suggest that BINODEs offer a useful compromise between mechanistic interpretability and data-driven flexibility for modeling partially known biochemical or biological dynamical systems.

Infectious disease time series often show signs of epidemic transitions, such as the peaks and troughs of the time series. In these time series, key system parameters can lead to catastrophic changes in the dynamical system behaviour (often called critical transitions). Modellers have increasingly shown that early warning signals can anticipate these transitions, both critical and non-critical, in infectious disease time series. Existing methods, however, generally focus on univariate time series data, or ignore spatiotemporal patterns that may be present as a disease spreads through a population. Recent ecological literature developments expand existing temporal and spatial methods to consider the covariance matrix of multiple, related time series. However, many of these proposed signals still make an assumption of stationary time series/system equilibrium. Whilst often true in ecological modelling, disease systems are seldom at equilibrium. In this paper, we propose the usage of the eigendecomposition of the non-stationary covariance matrix as a more suitable early warning signal for epidemiological data. We first analyse the expected trends in the eigenvalues and eigenbasis of the covariance matrix on approach to a transition. Next we apply these methods to a spatially-structured susceptible-infectious-recovered model to explore how the eigenbasis may provide extra information to modellers. Finally, we test these methods on SARS-CoV-2 case data during the 2020-2021 pandemic period in England.

Diffuse gliomas are deadly because the individual tumor cells invade - they travel far from the imageable mass, so it is impossible to remove the tumor completely. On the cellular level, glioma cells seem to be in either a "go" state (in which they do not divide) or a "grow" state (in which they do not migrate). We investigate what this tiny choice has to say about the large-scale speed of the invasion front and whether the implication is sufficiently strong to rule out the classical description of the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) type, in which a single phenotype migrates and proliferates. We derive a two-phenotype reaction-diffusion model with density-dependent switching, and we prove the cooperative (quasi-monotone) structure and the associated comparison principle and study travelling-wave solutions of the model. A leading-edge linearization gives minimal front speed as minimizer of an explicit dispersion relation, and direct simulation verifies the predicted speed. In the experimentally relevant fast switching limit, we find a closed-form expression for the speed, that is, we obtain an effective Fisher-KPP equation with rescaled diffusivity and growth rate, with the fractions of the phenotypes. The "go-or-grow" (GoG) front can move at a maximum speed of half the Fisher speed for the same single-cell motility $D$ and proliferation rate $r$, which occurs only when the cells divide their time equally between the two phenotypes. This bound is directly testable: measurement of the front speed, plus independent determination of $D$ and $r$, discriminates the two hypotheses, and in the GoG case, yields recovery of the phenotype balance. We then extend the result to anisotropic (DTI-informed) invasion along white-matter tracts and discuss implications for understanding clinical measurements of growth rate.

Environmental variability can strongly alter coexistence among competing species and their extinction risk, particularly when population dynamics are shaped by behavioral interactions, such as fear. In this work, we develop a novel stochastic differential equation competition model that incorporates both non-consumptive fear effects and environmental variability to investigate how behavioral interactions influence species coexistence under random fluctuations. Our result reveals that environmental stochasticity can drive species to extinction even when the corresponding deterministic system admits coexistence. In particular, under an explicit stability condition on the fear and competition parameters and sufficiently strong averaged noise intensities, we prove that both competing species become extinct exponentially almost surely. Conversely, we derive a stochastic persistence criterion in terms of fear, competition, and noise-induced suppression parameters for the fearful species. We further demonstrate that environmental noise may reverse classical competition-exclusion outcomes, leading to qualitatively different long-term dynamics from those predicted deterministically. These results provide rigorous thresholds separating stochastic extinction from persistence and highlight the critical role of environmental variability in fear-mediated competitive ecosystems. From an applied perspective, these results provide insight into how behavioral interactions and environmental variability influence species survival, with potential applications in ecological management and conservation.

The influence of arbitrary randomness in cell division times on the variability of protein copy numbers within a lineage ensemble has been recently studied, going beyond the contributions of noisy gene expression and partitioning error. However, variability of protein concentrations need separate study, since cell size growth between cell divisions dilute protein concentrations at the same rate as size growth, which also determines mean division times. Here for a model of bursty protein production, we present exact moments (of all orders) of protein concentrations in the cyclo-stationary state, comparing: (i) population and lineage cell ensembles, and (ii) statistics at different cell ages. Two interesting results emerge. While the variance of protein concentration changes with the degree of division time heterogeneity at any cell age, the age-averaged variance is independent of it within lineage ensemble but stays dependent within population ensemble. The skewness within population ensemble is higher in younger cells than within lineage ensemble, and this behavior reverses at older ages. Such a feature vanishes for the age-averaged distribution, with population based skewness always dominating over that of lineage. We also show that mother-daughter correlations in generation times, do not add any significant difference to the results.

A phylogenetic network represents evolutionary relationships involving hybridization, gene flow, or admixture. While the full network may not be identifiable from genomic data under common coalescent models, its tree of blobs, depicting only the tree-like portions of the network structure, is. We introduce ECToBlob (Edge Contraction for Tree of Blobs), a new statistically-consistent algorithm to estimate the tree of blobs from quartet concordance factors. Starting from a resolved tree, ECToBlob successively contracts edges which statistical tests indicate do not belong in the tree of blobs, due to reticulate or polytomous signal. We show that ASTRAL provides a valid starting tree under common assumptions, in that, asymptotically in the number of loci, trees optimizing ASTRALs criterion refine the tree of blobs. We describe several algorithm variants, differing in how evidence from multiple tests are combined to determine if the edge should be contracted, and provide software implementations. Relevance to Life SciencesHybridization, gene flow, or admixture are now recognized as important aspects of evolutionary history, but their genomic signal is confounded with that from a coalescent process, creating substantial challenges for inferring phylogenetic networks. The networks tree of blobs identifies areas where reticulation occurred, separated by tree-like branching. ECToBlob quickly estimates the tree of blobs using quartet concordance factors from gene trees, and provides a measure of statistical support for its result. Performance is illustrated through simulation and on empirical data, using an implementation in the R package MSCquartets. While the presence of a blob may be all that can be inferred in some cases, in others ECToBlob offers a robust and principled way to focus further analyses on more local reticulate structure. Mathematical ContentThis work makes contributions to mathematical phylogenetics in optimization, combinatorics, and statistics. We show that any tree maximizing quartet support (the criterion underlying ASTRAL) is a refinement of the networks tree of blobs under the coalescent model. Second, we give a concise proof that whether a network has a cut-edge corresponding to a given split is determined by information in certain subcollections of its 4-taxon subnetworks (quarnets). Finally, we propose valid statistical approaches for combining p-values across multiple quarnet hypothesis tests, proving that their use with specific decreasing test levels leads to statistically consistent inference as the number of loci grows. MSC codes05C90, 60J95, 62-04, 62F07, 92D15

Curvature evolution on a deforming surface is governed by the full change in the surface metric, but on biological surfaces captured by serial three-dimensional imaging, only the local area change is observable. The loss of the shear component leaves prediction of curvature evolution underdetermined from imaging alone. On the thoracic aorta, where curvature change marks disease progression, we derive a closed-form equation that predicts the change in integrated Gaussian curvature from the area dilation and initial geometry. The equation combines a conformal term in the area dilation with a leading anisotropy correction from the initial geometry. These two analytic levels, augmented by multi-scale spatial features at neighboring regions and a graph neural network trained on residuals, form a four-level nested predictor. On a synthetic aortic geometry under prescribed isotropic expansion, the equation recovers the analytic coefficient exactly. Across a continuum from pure expansion to pure shear, it holds R2 [&ge;] 0.71. On 236 paired thoracic aortic surfaces spanning dissection, aneurysm, traumatic injury, and non-pathologic controls, the equation recovers within-surface curvature change patterns with per-patient median Pearson [Formula] and pooled R2 = +0.238 [+0.225, +0.250], matching the graph neural network on the same inputs. The residual is a direct measurement of how far the observed growth field departs from conformality. HighlightsO_LIClosed-form equation predicts aortic curvature change from paired computed tomography scans. C_LIO_LIRecovers analytic predictions exactly on synthetic aortic geometries. C_LIO_LIAnisotropy proxy holds R2 [&ge;] 0.71 from pure expansion to pure shear. C_LIO_LICoefficients tie to geometric mechanisms ensuring interpretability. C_LIO_LIAnisotropy term, computable from one CT, is twice as large on diseased aortas. C_LI