Advances in computational biology approaches for digital modeling of tumor microenvironment

Abstract

Tumor progression arises from tightly coupled cellular, vascular, metabolic, immune, and biomechanical processes within the tumor microenvironment (TME), forming a heterogeneous multiscale system that cannot be fully characterized through experimental observation alone. Computational modeling provides a quantitative framework for integrating these mechanisms, but the diversity of available methods complicates model selection, parameterization, validation, and clinical translation. This review presents a problem-driven synthesis of computational approaches for TME modeling. Continuous, discrete, and discrete–continuous hybrid paradigms are compared in terms of biological resolution, data requirements, interpretability, and computational cost, and are linked to five mechanistic domains: tumor-cell dynamics, vascular transport, extracellular matrix and mechanics, metabolism and hypoxia, and immune interactions. Model credibility is examined through parameter identifiability, uncertainty quantification, numerical validity, biological fidelity, and clinical relevance, with emphasis on interpreting performance according to intended use and validation context. Machine learning is analyzed by function within mechanistic–ML integration: surrogate models accelerate forward simulation, inverse methods estimate latent parameters and states, patient-specific workflows combine multimodal clinical data with mechanistic models, and physics-informed neural networks embed governing equations into learning. We further discuss longitudinal personalization, physics-aware multiscale integration, clinically deployable digital twins, external generalization, and synthetic-data-to-clinical transfer. A translation-readiness pathway is proposed from mechanistic hypothesis generation to adaptive digital-twin decision support. Credible TME modeling should prioritize the minimum sufficient model, explicit intended use, transparent uncertainty, and evidence matched to the biological or clinical question.

Introduction

Cancer progression is increasingly recognized as an emergent systems-level phenomenon driven by complex interactions between malignant cells and the tumor microenvironment (TME) [1]. Rather than serving as a passive structural scaffold, the TME constitutes a heterogeneous and dynamically evolving ecosystem composed of extracellular matrix (ECM) architecture, vascular transport, immune surveillance, biochemical signaling, and biomechanical regulation [2–4]. These processes operate across multiple spatial and temporal scales and are governed by nonlinear feedback, stochastic variability, and adaptive remodeling. Such complexity poses challenges for purely experimental approaches and underscores the necessity of computational modeling frameworks capable of representing the TME as an integrated dynamical system. Quantitative modeling therefore plays a critical role in elucidating tumor progression mechanisms, exploring therapeutic hypotheses, and supporting predictive oncology [5, 6].

Computational investigations of the TME translate biological, chemical, and physical processes into mathematical abstractions for reproducible in silico experimentation [7]. These models support hypothesis testing, parameter exploration, and mechanistic interpretation, but the field remains fragmented across scales and methods. Discrete, continuous, and discrete–continuous hybrid models each emphasize different balances among cellular fidelity, computational tractability, and interpretability [8, 9]. Model selection should therefore be guided by the biological question, available measurements, required resolution, and intended validation target rather than by model complexity alone.

Figure 1 summarizes the relationship among these modeling paradigms and the corresponding selection criteria. Figure 1a distinguishes lattice-based discrete approaches, including lattice gas cellular automata (LGCA), cellular automata (CA), and cellular Potts models (CPMs), from off-lattice cell-based formulations. Figure 1b illustrates discrete–continuous hybrid models, in which cell-level dynamics exchange information bidirectionally with continuous transport, biochemical, mechanical, or vascular fields. Figure 1c groups continuous formulations used to describe population dynamics, reaction–diffusion and advection–diffusion transport, fluid mechanics, and compartmental pharmacokinetics. In this review, discrete–continuous hybrid modeling refers specifically to mechanistic coupling between discrete cells and continuous fields, whereas mechanistic–machine learning integration refers to the use of learning methods for simulation, parameter inference, or physics-informed approximation. Patient-specific models and digital twins are considered intended uses or deployment states rather than separate mathematical model classes. As shown in the model-selection pathway at the bottom of Fig. 1, the biological question and scale should first be matched to available data and validation targets, after which biological fidelity, interpretability, and computational cost can be balanced to identify the minimum sufficient model. Increasing coupling complexity nevertheless creates additional challenges for parameter identifiability, numerical integration, and uncertainty propagation [10].

Figure 1.

Computational modeling paradigms and problem-driven model selection for tumor microenvironment (TME) modeling. Discrete models explicitly represent individual cells and local interactions. Representative lattice-based approaches include LGCA, CA, and CPM, whereas off-lattice formulations include center-based, vertex-based, and immersed-boundary models. These approaches are suited to representing cellular heterogeneity, stochastic behavior, and local interactions. (b) Discrete–continuous hybrid models couple discrete cellular dynamics bidirectionally with continuous transport, mechanical, and vascular fields, allowing biologically meaningful variables such as oxygen, nutrients, drugs, extracellular matrix (ECM) properties, and mechanical stress to be exchanged across scales. (c) Continuous models describe population- and tissue-scale dynamics using ordinary differential equations (ODEs), reaction–diffusion and advection–diffusion partial differential equations (PDEs), fluid mechanics, and compartmental pharmacokinetic (PK) models. The lower panel presents a problem-driven model-selection pathway in which the biological question and required scale are considered together with data support and validation targets, followed by trade-offs among biological fidelity, interpretability, and computational cost. These considerations lead to selection of the minimum sufficient model, defined here as the simplest model adequate for the intended use.

Persistent limitations in mechanistic modeling, including incomplete parameter observability, computational expense of high-fidelity multiscale simulations, and simplifying assumptions required for tractability, have motivated increasing integration of machine learning into computational oncology [11]. Learning-based surrogates accelerate simulation and enable inverse parameter inference, while physics-informed neural frameworks embed governing equations within trainable architectures to balance flexibility with mechanistic consistency [12, 13]. These mechanistic–ML approaches have opened new possibilities for patient-specific modeling and digital twin systems that link individualized clinical data with predictive simulations [14]. At the same time, the convergence of data-driven and mechanistic paradigms introduces unresolved questions regarding interpretability, reliability, and uncertainty quantification, particularly in heterogeneous and data-limited clinical environments [5].

Existing reviews often emphasize individual modeling paradigms or application domains, with less attention to the assumptions that determine model choice and credibility. This review addresses that gap by comparing mechanistic formulations, computational tools, and machine learning-enhanced approaches according to their intended use, evidentiary requirements, and limitations.

The review follows a problem-driven structure. Section 2 links biological mechanisms to continuous, discrete, and discrete–continuous hybrid formulations; Section 3 examines parameterization, validation, and credibility; Section 4 distinguishes the functional roles of machine learning within mechanistic–ML integration; and Section 5 evaluates the barriers to patient-specific simulation and digital-twin development. Throughout this review, model choice is treated as conditional on the biological question, target mechanism, available data, required spatial and temporal resolution, and intended predictive use.

Key mechanisms in TME Modeling

As illustrated in Fig. 2, tumor progression emerges from the dynamic interplay between cellular states, biochemical transport, mechanical forces, and vascular-mediated exchange processes within the TME [1, 15]. Tumor cells undergo transitions between proliferative and dormant states while interacting with surrounding stromal and immune cells, ECM components, and interstitial fluid flow [16]. Concurrently, biochemical substances such as nutrients, oxygen, and metabolic waste are transported through diffusion and vascular exchange [17], while mechanical cues, including solid stresses from tissue deformation and fluid forces from interstitial flow, modulate cell behavior and spatial organization [18]. These coupled processes collectively shape tumor growth patterns, invasion dynamics, and treatment response [1, 18]. The five mechanistic domains summarized in Fig. 2 provide an organizational framework for the following sections, where each domain is discussed according to its key biological variables, mathematical representations, and available validation strategies.

Figure 2.

Mechanism-oriented organization of computational modeling in the tumor microenvironment (TME). The central schematic illustrates the coupling among tumor-cell, vascular, extracellular-matrix, metabolic, and immune processes. For each mechanistic domain, the principal biological variables, representative computational models, and commonly used validation measurements are summarized. (1) Tumor-cell dynamics describe tumor-cell proliferation, migration, dormancy, and death. In the Variables section, c denotes tumor-cell density, r is the net tumor-cell proliferation rate, and D is the effective cell motility or diffusion coefficient. Representative formulations include ordinary and partial differential equations (ODEs/PDEs), cellular automata (CA), and agent-based models (ABMs). Typical validation targets include tumor burden, the proliferation marker Ki-67, and medical imaging measurements. (2) Vascular transport represents blood-vessel development and the transport of oxygen, nutrients, therapeutic agents, and interstitial fluid. Vessel density denotes the abundance or spatial distribution of functional vessels; drug concentration denotes the local concentration of a therapeutic agent; and interstitial flow denotes the movement of extracellular fluid through tumor tissue. Transport PDEs and hybrid vascular models are commonly evaluated using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI), perfusion measurements, and drug-uptake data. (3) ECM and mechanics describe extracellular-matrix remodeling, solid stress, tissue stiffness, and deformation. Here, σ denotes mechanical or solid stress, E denotes Young’s modulus or effective tissue stiffness, and ECM density denotes the local abundance of extracellular-matrix material. Representative approaches include finite-element methods (FEM), poroelastic models, cellular Potts models (CPMs), and hybrid mechanical models, with elastography and mass-effect measurements serving as common validation targets. (4) Metabolism and hypoxia describe the transport and consumption of oxygen and nutrients and the accumulation of metabolic products. O₂, glucose, and lactate denote their respective local concentrations, whereas pH represents tissue acidity. Representative approaches include reaction–diffusion PDEs, metabolic ODE models, and FBA, with validation based on hypoxia markers and metabolomic measurements. (5) Immune–tumor interactions represent immune-cell recruitment, migration, activation, cytokine signaling, and tumor-cell killing. Immune-cell density denotes the local abundance of immune cells; killing rate denotes the rate of immune-mediated tumor-cell elimination; and cytokine concentration denotes the local level of immune-signaling molecules. ABMs, CA, and immune ODE modules are commonly validated using spatial omics and immune-cell staining. Together, these domains constitute an interconnected, multiscale system rather than independent biological modules. The biological question and available data determine which mechanisms and computational approaches should be combined.

The following sections are organized around five representative mechanistic domains in TME modeling: tumor-cell dynamics, vascular transport, ECM and mechanics, metabolism and hypoxia, and immune interactions. To orient this mechanism-focused discussion introduced in Fig. 2, Table 1 maps common biological and clinical questions to the minimum mechanisms that must be represented, suitable starting models, the data required for credible use, and the outputs that can be reused in subsequent studies. The following subsections then examine the assumptions, evidence requirements, and transfer limitations associated with each mechanistic domain.

Table 1.

Question-driven mechanism-to-model map for designing tumor microenvironment models.

Biological or clinical question	Minimum mechanisms to represent	Recommended starting model	Data needed to make it credible	Reusable output for readers
How fast is the tumor burden changing?	Net proliferation, death, carrying capacity, treatment pressure	ODE growth laws or low-dimensional continuum models	Longitudinal volume, treatment timing, response endpoint	A compact baseline for forecasting and for testing whether added complexity is justified
Where will invasion occur?	Cell motility, proliferation, anisotropic tissue structure, local constraints	Reaction–diffusion PDEs or discrete–continuous hybrid invasion models	Spatial imaging, tissue boundaries, serial segmentations	A spatial risk map that can guide sampling, margin design, or mechanistic hypotheses
How do vessels affect growth or therapy?	Oxygen delivery, perfusion, angiogenic feedback, drug transport	PDE/fluid models coupled with vascular or perfusion fields	DCE-MRI, perfusion imaging, vascular density, drug schedule	A transport-aware explanation of hypoxia, delivery failure, or treatment heterogeneity
How does ECM structure shape invasion?	Matrix density, fiber orientation, degradation, cell-matrix adhesion	ABM, CA, CPM, or discrete–continuous hybrid ECM-remodeling models	Histology, matrix markers, stiffness or fiber-orientation measurements	A rule set linking matrix architecture to invasive phenotypes
Which metabolic constraint matters?	Glucose, oxygen, lactate, pathway fluxes, nutrient competition	Stoichiometric models for pathway feasibility; dynamic models for adaptation	Metabolomics, proteomics, oxygen/nutrient measurements	A prioritized list of metabolic vulnerabilities or adaptation mechanisms
Why does immune control fail or vary?	Immune recruitment, killing, suppression, cytokines, spatial access	ODEs for population-level analysis; ABMs or discrete–continuous hybrid models for spatial immunology	Immune-cell densities, spatial pathology, cytokine or treatment data	A microenvironment-aware interpretation of response, exclusion, or escape
Do mechanical forces limit transport or growth?	Solid stress, stiffness, interstitial pressure, shear, convection	Continuum mechanics, force-based cell models, or fluid–structure coupling	Elastography, pressure/perfusion surrogates, tissue geometry	A mechanical explanation for growth inhibition, invasion route, or drug-delivery barrier
Which therapy schedule should be tested?	Tumor growth, treatment effect, toxicity proxy, adaptive feedback	Mechanistic model plus surrogate, Bayesian calibration, or RL wrapper	Longitudinal response, dosing history, uncertainty estimates	A candidate treatment policy that remains mechanistically auditable

Notes: This table is intended as a reusable design aid. It starts from the biological or clinical question and identifies the minimum mechanisms, model family, data requirements, and practical output expected from a credible modeling study.

Intrinsic dynamics of tumor cells

As summarized in Fig. 2, tumor-cell dynamics represent the cellular-level foundation of TME modeling, where proliferation, migration, dormancy, and cell death are translated into mathematical variables and validated against experimental or imaging-derived observations. Together, these processes shape tumor growth, spatial expansion, and long-term persistence. Rapid proliferation and enhanced motility intensify competition for space and resources, whereas severe spatial constraints can promote dormancy or apoptosis. Stem-like tumor cells may exert a disproportionate influence because they are commonly modeled as having sustained self-renewal and greater resistance to spontaneous cell death [19, 20]. More specifically, the tumor-cell domain in Fig. 2(1) links cell concentration, growth rate and motility to ordinary differential equation (ODE) or partial differential equation (PDE), CA and agent-based models (ABMs) formulations, with tumor burden, Ki-67 and imaging providing representative validation targets.

These intrinsic behaviors are commonly represented with continuous models, in which tumor populations are described by macroscopic variables such as density, concentration, or volume. ODEs capture temporal changes without explicitly resolving spatial heterogeneity and therefore provide a compact representation of global growth dynamics. This population-level representation corresponds to the ODE and PDE branches summarized in Fig. 1c.

Tumor growth dynamics are commonly described using phenomenological continuous models that capture the macroscopic evolution of tumor burden over time. Among the earliest and most widely adopted formulations are the logistic and Gompertz growth laws. Compared with the logistic model, the Gompertz formulation provides greater flexibility in representing asymmetric sigmoidal growth, characterized by an initial exponential expansion followed by progressively decelerating proliferation and eventual saturation, a pattern frequently observed in solid tumors.

Accordingly, tumor growth is modeled using the Gompertz equation given in Equation (1):

(1)

where V(t) denotes the tumor volume at time t, r is the intrinsic tumor growth rate reflecting cellular proliferative capacity, and K represents the carrying capacity imposed by environmental constraints such as nutrient availability, vascular supply, and mechanical confinement. This formulation captures both early-stage rapid expansion and late-stage growth inhibition arising from microenvironmental limitations, and has been extensively validated in experimental and clinical tumor growth studies [21, 22]. Large-scale clinical benchmarking supports the practical value of compact growth laws. In an analysis of 1472 patients receiving chemotherapy or immunotherapy, the Gompertz model provided a favorable balance between goodness of fit and parametric parsimony [23]. The Gompertz and general Bertalanffy models also yielded the lowest mean absolute errors for forecasting final treatment response from early measurements. These results support their use as population-level baselines, but not as complete representations of spatial tumor dynamics.

Despite their conceptual simplicity and analytical tractability, ODE-based growth laws are inherently limited in representing the spatial organization of tumor tissues. Key phenomena such as invasive fronts, heterogeneous cell distributions, and nutrient gradients cannot be explicitly captured within purely temporal formulations. To overcome these limitations, continuous models based on PDEs have been introduced, which incorporate spatial variables and thereby enable explicit representation of diffusion, invasion, and other spatially dependent processes.

Within this framework, tumor invasion and spatial expansion are commonly described using reaction–diffusion equations, among which the Fisher–Kolmogorov model serves as a canonical formulation. Tumor cell dynamics are expressed in Equation (2):

(2)

where denotes the tumor cell density, D is the effective diffusion coefficient characterizing cell motility within surrounding tissue, and ρ represents the net proliferation rate. The diffusion term accounts for spatial migration of tumor cells, while the logistic reaction term captures local growth saturation arising from microenvironmental constraints. This reaction–diffusion formulation provides a mechanistic basis for modeling invasive tumor fronts and heterogeneous spatial patterns, and has served as the foundation for numerous tumor growth models and their extensions [24–28].

Recent studies have further demonstrated the quantitative predictive capability of reaction–diffusion–based tumor growth models when calibrated with imaging data. For example, Jarrett et al. developed an imaging-driven biophysical model that integrates tumor cell proliferation, diffusion, and drug delivery dynamics, and reported that the model was able to reproduce longitudinal tumor volume changes with relative errors typically below 10% when validated against in vivo measurements. In addition, spatial tumor distributions predicted by the model showed strong agreement with experimental imaging data, indicating that PDE-based formulations can achieve both quantitative accuracy and biological consistency in clinically relevant settings [26].

Validation studies illustrate both the promise and the limits of reaction–diffusion tumor models. In a diffusion-weighted magnetic resonance imaging (MRI)-constrained glioma model, Hormuth et al. reported strong in silico performance for short-term forecasts, including tumor-volume errors below 8.8%, Dice coefficients above 0.92, and concordance correlation coefficient (CCC) values above 0.80. Performance decreased in vivo, where tumor-volume errors exceeded 11.7% and CCC values fell below 0.33, reflecting the added uncertainty of real biological settings. Such metrics should therefore be read in context: cancer type, imaging modality, cohort size, prediction horizon, and validation setting all influence apparent performance [29].

Taken together, ODE and PDE formulations illustrate a common pattern in continuous modeling: compact equations can produce interpretable and quantitatively useful predictions, but their apparent simplicity should not be mistaken for biological completeness. Their strongest use is to represent dominant population-level trends or spatial fields, whereas questions involving rare cell states, local cell–cell contact, or spatially localized escape often require discrete or discrete–continuous hybrid extensions.

In contrast to continuous formulations, discrete models describe tumor growth at the level of individual cells. In this class of models, each cell is treated as an independent entity whose state evolves according to a set of predefined rules. Cellular behaviors such as proliferation, migration, dormancy, and apoptosis are evaluated at discrete time steps, typically in a probabilistic or rule-based manner. This formulation is particularly suitable for investigating cell-level heterogeneity, local interactions, and stochastic effects that cannot be explicitly represented in population-based continuous models [8, 30, 31].

Among various discrete modeling paradigms, CA models constitute one of the most widely adopted approaches. In a CA framework, the spatial domain is discretized into a lattice, and each lattice site is associated with a cell state. The evolution of the system is governed by local transition rules that depend on the states of neighboring sites. These local rules can be mathematically represented as a state transition function, where the state of each cell is updated according to its current condition, neighboring states, and predefined biological parameters:

(3)

where represents the state of cell i at time t, denotes the neighborhood information, and P represents model parameters controlling cellular behaviors such as proliferation, migration, and apoptosis. This formulation captures the discrete nature of CA models, in which tumor evolution emerges from local cell-level interactions.

A representative example is a CA model in which fundamental cellular processes such as proliferation, migration, and apoptosis were explicitly encoded as probabilistic rules [32]. The study demonstrated that the interplay among these processes alone is sufficient to reproduce a wide range of macroscopic tumor growth behaviors. This model is widely regarded as one of the early systematic attempts to investigate tumor population dynamics from a discrete, rule-based perspective.

Subsequent work extended this discrete framework by varying non-stem cell kinetics, introducing dynamically expanding domains, moving from 2D lattices to 3D simulations, and improving parallel performance [33–35]. The common lesson is not that any single CA rule set is definitive, but that a small number of biologically interpretable local rules can generate diverse macroscopic growth patterns. The main trade-off is therefore between rule expressiveness, parameter identifiability, and computational scalability.

These studies primarily examined intrinsic tumor dynamics without explicitly representing the microenvironment [32]. Across model variants, greater stem-cell self-renewal generally produced larger tumor masses. Competition with non-stem cells could induce stem-cell dormancy, whereas reduced non-stem-cell proliferation or increased apoptosis could facilitate stem-cell expansion. Increased motility also enabled simulated tumors to overcome spatial constraints. These mechanisms offer a model-based explanation for accelerated regrowth after incomplete radiotherapy or chemotherapy.

Biochemical microenvironment

While intrinsic cellular dynamics determine the baseline growth tendencies of tumor populations, the biochemical microenvironment introduces additional regulatory layers involving vascular transport, oxygen availability, metabolic exchange, and immune-mediated signaling (Fig. 2). These factors collectively shape the biochemical microenvironment of tumors, which plays a decisive role in modulating cell proliferation, migration, survival, and therapeutic response [1, 15, 20].

Vascularization

Vascularization is a fundamental biochemical process that regulates tumor growth by controlling the delivery of oxygen, nutrients, and therapeutic agents. As tumor cell proliferation rapidly increases metabolic demand, diffusion alone becomes insufficient to sustain viability beyond a limited spatial scale. This imbalance gives rise to hypoxic regions, which not only constrain tumor expansion but also activate angiogenic signaling pathways that promote the formation of new blood vessels. Consequently, vascular dynamics constitute a critical coupling mechanism between tumor growth and the surrounding biochemical microenvironment [17, 36]. Accordingly, the vascular-transport domain in Fig. 2(2) links vessel density, drug concentration and tissue transport properties to transport PDEs or hybrid vascular models, with DCE-MRI perfusion and drug uptake serving as validation targets.

Early computational models incorporated vascular effects primarily through simplified representations of oxygen or nutrient fields. In these formulations, blood vessels were treated as static sources, and oxygen transport was described using diffusion consumption equations. Such models successfully demonstrated how hypoxia emerges as a natural consequence of rapid tumor growth and how oxygen limitation influences proliferation, necrosis, and invasion patterns [37]. Although these approaches neglected vascular remodeling, they established a foundational link between biochemical transport processes and macroscopic tumor morphology.

Subsequent modeling efforts introduced explicit representations of angiogenesis to capture the dynamic feedback between tumor cells and the vascular network. In these models, angiogenic factors secreted by hypoxic tumor regions stimulate vessel sprouting, migration, and branching, thereby allowing the vascular structure to evolve in response to tumor progression. This class of models revealed that angiogenesis not only enhances nutrient supply but also alters spatial growth patterns and invasion dynamics by reshaping oxygen gradients within the tissue [38, 39]. Such formulations marked an important transition from static environmental assumptions to co-evolving tumor vasculature systems.

Recent vascular models increasingly focus on the abnormal structure and function of tumor-induced vessels [36]. Irregular, tortuous, and poorly organized networks generate heterogeneous perfusion, persistent hypoxia, elevated interstitial pressure, and uneven drug delivery [40, 41]. Multiscale simulations further show that these vascular features can change therapeutic outcome. For example, Nikmaneshi et al. coupled tumor growth, angiogenesis, metabolism, and drug transport to evaluate vascular normalization, finding that combined anti-angiogenic and metronomic therapy reduced viable cancer-cell burden relative to chemotherapy-only strategies while improving drug penetration and reducing normal-tissue accumulation. Across these studies, a consistent finding is that vascular normalization, rather than complete vessel destruction, may offer the best therapeutic balance, as it improves perfusion while reducing hypoxia-driven resistance [42].

Vascularization models are also increasingly constrained by clinical data. Imaging-derived vascular parameters, perfusion maps, and data-driven calibration can personalize vascular models and support treatment-response prediction. For example, a diffusion-weighted MRI (DW-MRI) and dynamic contrast-enhanced MRI (DCE-MRI)-based model jointly describing tumor cellularity and vascular dynamics achieved tumor-volume prediction errors below 12%, Dice coefficients above 0.74, and voxel-level CCC values above 0.65 for tumor and blood-volume fractions [43, 44]. These results suggest that vascularization-aware models can move from qualitative angiogenesis descriptions toward spatially resolved prediction, but their credibility depends on the imaging context and validation cohort.

Extracellular matrix

ECM serves as both a structural scaffold and a regulatory component of the TME, exerting a significant influence on tumor cell migration, invasion, and spatial organization [45]. Through its heterogeneous composition and architecture, the ECM constrains cellular movement while simultaneously providing preferential pathways that guide invasive growth. As a result, ECM-related mechanisms have been incorporated into tumor models to capture the interaction between cellular dynamics and spatial structure.

Early modeling studies represented the ECM in a simplified manner, typically as a static background or as spatial constraints that limit cell movement. In these approaches, ECM effects were introduced through predefined rules or local resistance terms that regulated cell motility and proliferation. Despite their simplicity, such models demonstrated that ECM density and spatial heterogeneity alone are sufficient to generate non-uniform invasion patterns and anisotropic tumor expansion [46, 47].

Subsequent models introduced dynamic ECM remodeling processes, allowing the matrix to evolve in response to tumor cell activity [48]. In these formulations, tumor cells actively degrade, reorganize, or align ECM components, and cell migration becomes dependent on local matrix properties [49]. This class of models revealed that feedback between ECM degradation and cell motility can substantially enhance invasive potential and promote collective migration modes, thereby providing mechanistic explanations for experimentally observed invasion phenotypes [50].

Recent ECM models improve biological fidelity by representing matrix density, fiber alignment, directional guidance, degradation, and feedback between matrix remodeling and cellular behavior [4, 51]. These processes are commonly implemented in agent-based, cellular automaton, cellular Potts, or discrete–continuous hybrid frameworks that resolve local cell-matrix interactions [48, 50].

The central modeling decision is whether the ECM acts mainly as a static structural constraint or as a dynamically remodeled component. Static representations may be sufficient for short-term invasion through a measured architecture, whereas dynamic models are required when degradation, deposition, or force-dependent remodeling is part of the biological question. Added ECM complexity should therefore be justified by corresponding structural or mechanical data [52].

Metabolism

Metabolic reprogramming is a hallmark of tumor progression and plays a central role in supporting rapid proliferation, survival under stress, and adaptation to hostile microenvironmental conditions [53, 54]. Altered glucose uptake, oxygen consumption, and byproduct accumulation not only sustain tumor growth but also reshape the biochemical landscape of the surrounding tissue [55]. Consequently, metabolic processes have been extensively incorporated into tumor models to investigate the interplay between intracellular regulation and microenvironmental constraints. This variable-to-evidence chain is summarized in Fig. 2(4), which connects oxygen, glucose, lactate and pH to reaction–diffusion, metabolic ODE and flux balance analysis (FBA) models and to validation using hypoxia markers or metabolomics.

Metabolic models are commonly divided into stoichiometric and dynamic approaches. Stoichiometric methods, including FBA, evaluate feasible metabolic routes under mass-balance and steady-state constraints [56]. They are useful for identifying dominant pathways and metabolic vulnerabilities from omics-informed reaction networks, but they do not resolve transient dynamics.

Dynamic metabolic models instead describe the temporal evolution of metabolite concentrations and reaction rates, often using ODEs for enzyme kinetics, substrate availability, and feedback regulation [57, 58]. They are better suited for hypoxia, nutrient deprivation, and treatment-induced adaptation, but require richer data and face greater parameter uncertainty.

These approaches are complementary rather than interchangeable. Stoichiometric models are most informative for pathway feasibility and vulnerability screening, whereas dynamic formulations are better suited to time-dependent metabolic adaptation and feedback with oxygen or nutrient availability. Their integration with tumor-growth and microenvironmental models has enabled investigation of how metabolic heterogeneity shapes invasion, treatment sensitivity, and tumor evolution [58–60]. However, additional metabolic detail is warranted only when the relevant fluxes, concentrations, or regulatory assumptions can be constrained by corresponding measurements.

Immune cells

Immune–tumor models differ primarily in whether immune activity is represented as a population-level pressure or as spatially resolved cellular behavior [61–63]. ODE models are useful for analyzing competition, equilibrium, oscillation, and immune escape, whereas discrete and discrete–continuous hybrid models can represent immune-cell recruitment, migration, exhaustion, and spatial exclusion. Model complexity should therefore be determined by the immune mechanism of interest and the available spatial evidence, rather than by the number of immune-cell subtypes that could be included. As shown in Fig. 2(5), immune density, killing rates and cytokine signals can be represented with ABMs, CAs or immune ODE modules and evaluated using spatial omics and immune-cell staining.

Subsequent modeling efforts introduced greater biological detail by explicitly representing different immune cell types and their functional heterogeneity. In these models, effector cells, regulatory populations, and cytokine-mediated signaling pathways were incorporated to describe complex feedback mechanisms within the immune microenvironment. Discrete–continuous hybrid frameworks enabled immune cell recruitment, migration, activation, and exhaustion to be modeled in a spatially resolved manner. These approaches revealed that spatial organization and local immune cell density critically influence tumor control efficacy and the emergence of immunosuppressive niches [64–67].

Advanced immune models increasingly couple immune dynamics with vascularization, metabolism, and ECM structure [68]. By linking immune responses to biochemical gradients, resource competition, and spatial accessibility, these models explain immune exclusion, hypoxia-induced immunosuppression, and heterogeneous immunotherapy response [69, 70]. These findings indicate that immune outcomes are governed not only by cytotoxic activity but also by the accessibility and microenvironmental state of immune cells. The appropriate formulation should therefore be selected according to whether the intended question concerns population-level regulation or spatial access and exclusion, and whether the available data can constrain the corresponding interactions.

Mechanical forces and processes

As represented in Fig. 2, ECM organization and mechanical regulation constitute a distinct mechanistic domain in TME modeling, where matrix properties, mechanical forces, computational formulations, and imaging-based validation approaches are integrated to explain tumor invasion and tissue remodeling. Mechanical processes complement biochemical descriptions by representing stress, strain, pressure, force transmission, and flow within tumors [71, 72]. Solid-mechanical models describe growth-induced compression, matrix resistance, and tissue deformation, whereas fluid-mechanical models describe interstitial pressure, convection, and shear. These domains are coupled because deformation alters transport pathways and fluid pressure contributes to tissue loading. The ECM-and-mechanics domain in Fig. 2(3) makes this mapping explicit by connecting stress, stiffness and ECM density to FEM, poroelastic, CPM and hybrid formulations, with elastography and mass effect as representative validation targets.

Model selection should begin with the quantity of interest, such as solid stress, matrix deformation, interstitial pressure, shear stress, or drug convection, because each requires different constitutive assumptions, boundary conditions, measurements, and validation endpoints. This distinction organizes the following discussion into solid- and fluid-stress formulations.

Solid stress

Solid stress refers to the mechanical forces transmitted through the solid and elastic components of tumor tissues, including cells and the surrounding ECM. These forces arise from cell proliferation, matrix remodeling, and geometric confinement, and they collectively determine how mechanical loads are distributed within the tissue. Solid stress is closely associated with mechanical properties such as stiffness, elasticity, and viscoelasticity, all of which have been shown to differ markedly between malignant and healthy tissues [73, 74]. Such biomechanical alterations not only reflect pathological remodeling of the microenvironment but also actively regulate tumor cell migration, invasion, and growth.

From a modeling perspective, solid stress is commonly described using two conceptually distinct approaches [75]. The first class of models explicitly formulates mechanical forces through governing equations, whereas the second class represents mechanical effects implicitly through energy-based formulations. These two strategies differ in their mathematical structure, biological interpretability, and computational complexity, and they are therefore suited to different research objectives.

The force-based approach explicitly defines the interaction forces between cells and between cells and the matrix, often within vertex-based or center-based frameworks. In these models, mechanical interactions such as adhesion, repulsion, and elastic deformation are directly incorporated into the equations of motion, and the collective behavior of the tissue emerges from local force balance. Cellular behaviors such as migration and proliferation are subsequently derived from the resulting interaction network [76, 77]. In this formulation, the displacement of a cell is determined by the net mechanical force acting on it over a given time interval. A representative formulation is given in Equation (4), in which the total interaction force acting on cell i is first computed, and the effective displacement of this cell over a time interval \varDelta t is then obtained through the corresponding force–motion relationship.

(4)

In this expression, the force term typically accounts for contributions from cell–cell adhesion, elastic repulsion, and matrix resistance, while the displacement term describes the mechanically induced motion of the cell. Such force-resolved models are particularly suitable for investigating how local mechanical interactions give rise to collective invasion patterns, tissue compaction, and growth-induced compression [76, 77].

In contrast, energy-based models represent tissue mechanics through an effective energy function rather than explicitly resolving each interaction force. In cellular Potts and related formulations, candidate changes in cell position or configuration are accepted or rejected according to the associated change in system energy. This formulation allows cell–cell and cell–matrix adhesion, together with geometric constraints on cell size and shape, to regulate collective tissue organization and cell movement [30, 78]. A general decomposition of the effective energy is given by Equation (5):

(5)

Here, H denotes the total effective energy of the system, represents cell–cell and cell–matrix adhesion, penalizes deviations from the prescribed cell volume, and represents constraints associated with cell area or surface area. During model evolution, local state changes that decrease the total energy are preferentially accepted, whereas energetically unfavorable changes are accepted with a lower probability. Tissue-scale organization therefore emerges from repeated local updates. Compared with force-based models, energy-based formulations are particularly suitable for studying cell sorting, shape regulation, tissue morphology, and mechanically constrained invasion, although their energy terms usually represent effective model penalties rather than directly measured physical energies.

Fluid stress

Fluid stress refers to the mechanical forces exerted by fluid flow within the TME, including shear stress acting on cell surfaces and the ECM, as well as pressure-related effects associated with interstitial fluid pressure. These forces play a central role in regulating nutrient transport, metabolic exchange, and cellular mechanotransduction. Unlike solid stress, which primarily arises from tissue deformation and elastic resistance, fluid stress emerges from hydrodynamic transport processes and is inherently coupled to vascular architecture, permeability, and interstitial structure. Consequently, fluid mechanical effects are essential for understanding how biochemical substances, therapeutic agents, and even tumor cells themselves are redistributed within tissues [79–81].

From a modeling perspective, fluid stress is commonly described using continuum-based formulations that treat the interstitial space and vascular compartments as fluid domains governed by conservation laws. In these models, fluid velocity, pressure, and shear stress are determined by PDEs derived from fluid mechanics. This class of approaches enables the explicit characterization of transport processes, including convection, diffusion, and pressure-driven flow, thereby providing a quantitative description of how hydrodynamics shapes the spatial distribution of oxygen, nutrients, and drugs within tumors [82–84].

For explicitly resolved vascular lumina, microchannels, or other fluid domains, incompressible Newtonian flow can be described using the Navier–Stokes equations:

(6)

where u represents fluid velocity, p denotes pressure, ρ and μ are fluid density and dynamic viscosity, respectively, and f represents external forces. This formulation provides spatially resolved predictions of velocity, pressure, and shear stress, but requires detailed geometric reconstruction and appropriate boundary conditions.

However, tumor interstitium is often treated as a porous medium rather than a free-flowing fluid domain. In this context, Darcy-based formulations are more commonly used to describe pressure-driven interstitial fluid transport:

(7)

where u denotes interstitial fluid velocity and κ represents effective hydraulic conductivity of the porous tissue. Compared with Navier–Stokes formulations, Darcy models provide a simplified description of interstitial transport and are therefore widely used for modeling tumor interstitial fluid pressure and drug convection.

A second class of models focuses on the interaction between fluid flow and cellular or tissue-scale mechanics. In these formulations, fluid stress is not treated as an isolated physical quantity but is instead coupled to vessel deformation, matrix remodeling, and cellular responses. By incorporating feedback mechanisms between fluid transport and tissue structure, these models have demonstrated that abnormal vascular geometry and heterogeneous permeability can induce localized shear stress and pressure gradients, which in turn promote tumor cell extravasation, invasion, and metastatic dissemination [85].

Beyond fluid velocity and pressure, transported substances such as oxygen, nutrients, and therapeutic agents are commonly modeled using convection–diffusion–reaction equations:

(8)

where c represents the concentration of transported substances, D denotes the diffusion coefficient, and represents source or sink terms associated with production, consumption, or degradation processes. Coupling this equation with fluid velocity fields enables simulation of how hydrodynamic transport regulates oxygen availability, drug penetration, and biochemical gradients within tumors.

Fluid-stress models therefore emphasize convection, pressure gradients, and shear-induced responses, whereas solid-stress models emphasize elastic resistance and compressive loading. Coupled formulations are warranted when deformation and transport are expected to influence each other and when data can constrain both components [71, 73, 79, 86].

Mechanistic discrete–continuous hybrid Modeling

While individual modeling approaches often focus on specific mechanisms, realistic TME simulations commonly require several interacting processes to be represented simultaneously. Table 2 compares representative mathematical and computational models across the major mechanisms discussed above, placing their physiological scope, methodological formulation, advantages, and limitations on a common basis. This comparison shows why no single paradigm is uniformly optimal and motivates discrete–continuous hybrid architectures that couple complementary descriptions across spatial and temporal scales [77, 87, 88].

Table 2.

Representative mathematical and computational models of tumor microenvironment mechanisms.

Reference	Mechanic	Method	Physiological process	Pros	Cons
Rockne [120]	Intrinsic dynamics	PDE	Population-level tumor growth and invasion dynamics	Continuous formulation; scalable; analytically tractable	Limited cellular heterogeneity
Enderling, H [32]		CA	Cell-level rule-based tumor dynamics	Discrete; intuitive; captures heterogeneity	Difficult parameter calibration
Valentim, Carlos A [35]		CA	Discrete cellular proliferation and migration dynamics	High single-cell resolution; flexible rules	Grid dependence; computational cost
Poleszczuk J [34]		CA + CPM	Cancer stem cell hierarchy and clonal evolution	Multiscale representation of stem cell dynamics	High computational burden
Rahmati [121]	Biochemistry	LBM + PDE	Hemodynamic shear stress–mediated tumor–vessel interactions	Accurate fluid–cell coupling; shear stress resolution	Complex numerical implementation
Swanson, Kristin R [122]		PDE	Angiogenesis-driven tumor progression	Efficient large-scale vascular modeling	Limited single-cell resolution
Juan Uriel Legaria-Peña [93]		CA + ODE	Therapy-induced tumor response dynamics	Dynamic coupling of treatment and tumor response	Many uncertain parameters
Colson, Chloé [123]		PDE	ECM-mediated tumor invasion	Quantitative invasion front modeling	Simplified ECM microstructure
Xie, Hang [124]		CA + PDE	Chemotherapy–modulated invasive tumor growth	Integrates therapy and invasion dynamics	Computationally intensive
M. A. Henson [125]		PDE	Energy-metabolism–regulated tumor growth	Quantitative metabolic modeling	Limited intracellular resolution
Tripathi, Shubham [59]		PDE	Metabolic adaptation and pathway reprogramming	Captures metabolic pathway switching	Parameter sensitivity
Ruscone, Marco [126]		CBM	Immune cell migration and molecular regulation	Force-based immune cell interaction modeling	Simplified immune signaling
Raeisi, Elham [127]		PDE	Immune–tumor interaction dynamics	Continuous immune–tumor coupling	Limited immune heterogeneity
Van Genderen [128]		ABM	Hormone-modulated tumor microenvironment regulation	Captures heterogeneous therapy responses	High computational demand
Leschiera E [129]		IBM	Psychosocial stress–immune–tumor coupling	Flexible representation of systemic regulation	Difficult biological validation
Tsingos E [130]	Mechanical Forces	CPM + ODE	Cell–ECM mechanical interaction and solid stress regulation	Morphological–mechanical coupling	Approximate mechanical fidelity
Phillips, Caleb M [39]		ABM + PDE	VEGF-driven angiogenic sprouting and vascular pattern formation	High-resolution vascular growth modeling	High computational cost
Chen, H [87]		CBM	Solid stress–mediated tumor–stroma mechanics	Force-resolved stress modeling	High computational complexity
Carvalho, Joao [131]		CPM	Mechanically driven epithelial invasion	Efficient tissue-scale simulation	Reduced physical realism
Simon [132]		VBM	Cell deformation and morphology under mechanical constraints	High-fidelity shape representation	Computationally expensive
Bhandari, Ajay [133]		PDE	Interstitial fluid pressure and flow–mediated transport	Quantitative transport modeling	Coarse-grained cellular detail
Lavi, Ido [134]		PDE	Drug transport in fluid-mediated tumor environments	Scalable drug transport simulation	Simplified tissue heterogeneity

Notes: Representative studies are selected to illustrate commonly used mathematical and computational approaches for modeling key processes in the tumor microenvironment (TME), including cellular dynamics, vascularization, metabolism, immune regulation, and mechanical interactions. Models are categorized according to their dominant methodological framework (e.g. PDE-based, agent-based, cellular Potts, or discrete–continuous hybrid approaches), while the reported advantages and limitations reflect general methodological characteristics rather than exhaustive benchmarking across studies.

Discrete–continuous mechanistic models can couple three levels of description: intracellular regulation, cell-level behavior, and tissue-scale transport. This organization allows molecular states to influence proliferation, adhesion, or migration, while extracellular fields represent nutrients, drugs, and mechanical cues. PhysiBoSS2.0 illustrates this architecture by linking Boolean regulatory networks to ABMs and continuous transport [88]. Its prostate-cancer application simulated ~1000 LNCaP cells over seven days under multiple drug concentrations. More broadly, discrete–continuous coupling is most informative when each exchanged variable has a defined biological meaning and can be calibrated or experimentally evaluated. This architecture corresponds to the bidirectional coupling in Fig. 1b, where discrete cells both respond to and modify continuous transport, ECM mechanics and vascular fields.

Implementation platforms differ mainly in which components they resolve explicitly. Continuum solvers efficiently represent transport and tissue-scale fields, whereas cell-based solvers preserve individual-cell heterogeneity and local interactions [30, 78]. Discrete–continuous hybrid platforms combine these components when information must pass between scales, although the resulting coupling increases calibration and computational demands.

Beyond biochemical coupling, mechanics-oriented discrete–continuous hybrid frameworks further incorporate cell deformation, tissue elasticity, and fluid–structure interactions, providing dedicated tools for studying solid stress, mechanically regulated invasion, and growth-induced compression [87]. These approaches extend conventional discrete–continuous hybrid models by explicitly resolving force transmission and structural remodeling within the TME.

Complementing the study-level comparison in Table 2 and Table 3 summarizes reusable open-source platforms for multiscale TME simulation. The table distinguishes agent-based, cellular, continuum, and mechanics-oriented environments according to their principal capabilities, advantages, and typical applications. This distinction is important because a mathematical formulation and a software platform are not interchangeable: the former defines the biological abstraction, whereas the latter determines how that abstraction can be implemented, coupled, and scaled.

Table 3.

Open-source modeling frameworks for multiscale tumor microenvironment simulation.

Representative open-source frameworks for TME modeling paradigms	Representative tools	Targeted processes	Key advantages	Typical applications
Agent-based / discrete	NetLogo [135]; Repast [136]; PhysiCell [77]	Cell proliferation; migration; immune interaction	Flexible cell-level rules; captures heterogeneity	Tumor growth; immune–tumor dynamics
Cellular Potts / discrete–continuous hybrid	CompuCell3D [78]; Morpheus [137]; Microvessel Chaste [138]	Tissue morphogenesis; angiogenesis; ECM interaction	Intuitive model construction; multiscale coupling	Tumor invasion; vascular remodeling
PDE-based / continuum	Chaste [139]; BioFVM [140]	Diffusion; proliferation; nutrient transport	High numerical accuracy; efficient field simulation	Macroscopic tumor growth; transport processes
Mechanics-focused	Tyssue [141]; LBIBCell [132]	Cell deformation; tissue mechanics	Explicit biomechanical representation	Solid stress; epithelial mechanics
Fluid dynamics	SunlightLB [142]; VirtualFluids [143]	Blood flow; drug delivery	High-performance hydrodynamics; GPU support	Hemodynamics; intravascular transport

Notes: This table summarizes representative open-source computational frameworks commonly used for multiscale tumor microenvironment simulation. Frameworks are grouped according to their dominant modeling paradigm, including agent-based, discrete–continuous hybrid cellular, continuum PDE-based, and mechanics-oriented platforms. Listed tools represent widely adopted examples in computational oncology, while targeted processes and applications reflect typical modeling scenarios reported in the literature.

Discrete–continuous hybrid modeling should therefore be justified by a clear coupling hypothesis rather than by the desire to include every available process. Each exchanged variable, such as oxygen, growth factors, drug concentrations, mechanical stress, or immune-cell accessibility, should have a defined biological meaning and be amenable to calibration or falsification. Table 4 concludes Section 2 by translating this principle into a scale- and data-dependent decision guide that identifies suitable computational strategies and the conditions under which particular methods should be avoided.

Table 4.

Decision-oriented guideline for selecting computational modeling strategies in tumor microenvironment research under varying biological scales and data constraints.

Modeling objective	Biological scale	Data availability	Recommended modeling strategy	Rationale	Key limitations and when to avoid
Mechanistic understanding of cell behavior	Cellular / microscale	Low–moderate	Agent-based models (ABM), Cellular automata (CA), Cellular Potts model (CPM)	Explicit representation of individual cells and local interactions enables detailed study of heterogeneity, clonal evolution, and cell–cell communication	High computational cost; difficult calibration; limited scalability. Avoid for very large cell populations or population-level endpoints only.
Spatial tumor growth and invasion dynamics	Tissue scale	Moderate	PDE-based continuous models (reaction–diffusion, continuum mechanics)	Efficient simulation of macroscopic processes such as tumor expansion, nutrient transport, and diffusion-driven pattern formation	Limited single-cell resolution; simplified assumptions. Avoid when discrete cell events or heterogeneity are central.
Multiscale coupling of cellular and microenvironmental processes	Cellular → tissue	Moderate–high	Discrete–continuous hybrid models (ABM + PDE, CPM + transport)	Integrates cell-level decision-making with tissue-scale transport, enabling realistic multiscale interactions	Complex implementation; identifiability and computational burden. Avoid when data or computational resources are limited.
Vascular dynamics and hemodynamics	Tissue / vascular scale	Moderate–high	PDE + fluid dynamics (e.g. Navier–Stokes, LBM)	Captures blood flow, shear stress, and vascular transport critical for angiogenesis and drug delivery	Numerically demanding; requires detailed vascular data. Avoid without reliable perfusion or vascular imaging.
Biomechanical regulation (solid/fluid stress)	Cellular → tissue	Moderate	Force-based models (CBM, vertex-based, continuum mechanics)	Enables modeling of mechanical interactions, deformation, and stress-induced tumor progression	Difficult parameterization; simplified physical assumptions. Avoid without relevant mechanical measurements.
Imaging-informed mechanistic models with patient-specific calibration	Multiscale	High (multimodal data)	Imaging-informed mechanistic models with patient-specific calibration	Incorporates patient geometry and biological heterogeneity for personalized simulation	High data-integration and calibration burden. Avoid without individual-level multimodal data.
Fast simulation and large-scale parameter exploration	Any	Moderate–high	Machine learning surrogates (CNN, regression, emulators)	Provides orders-of-magnitude acceleration for repeated simulations and sensitivity analysis	Approximation and extrapolation errors. Avoid outside the training domain or without mechanistic cross-checks.
Inverse modeling and parameter inference	Multiscale	Moderate	Bayesian inference, Approximate Bayesian Computation (ABC)	Enables estimation of unobservable parameters and uncertainty quantification	High computational cost; sensitivity to summary statistics. Avoid when suitable observations or informative statistics are unavailable.
Real-time or near real-time prediction	Multiscale	High	End-to-end ML models or mechanistic–ML integration frameworks	Enables rapid prediction from imaging or clinical data	Reduced interpretability; training-data dependence. Avoid when regulatory traceability is essential without mechanistic support.
Physics-consistent learning under limited data	Multiscale	Low–moderate	Physics-informed neural networks (PINNs)	Embeds governing equations into learning, improving data efficiency and physical consistency	Training instability in stiff or high-dimensional systems. Avoid without appropriate stabilization strategies.
Adaptive therapy optimization and decision-making	Multiscale	Moderate–high	Reinforcement learning + mechanistic modeling	Supports dynamic optimization of treatment strategies based on simulated feedback	Sensitive to reward design; limited clinical validation. Avoid when actions or rewards are poorly defined.
Digital twin development and longitudinal prediction	Multiscale (full system)	High (longitudinal data)	Mechanistic–ML integration frameworks with longitudinal updating	Enables patient-specific, continuously updated predictive models for clinical decision support	High integration and validation burden. Avoid without longitudinal data, uncertainty quantification, and prospective validation.

Notes: This table provides a decision-oriented guideline for selecting computational modeling strategies in tumor microenvironment studies. Modeling objectives define the primary research questions, while biological scale and data availability represent key practical constraints that influence method selection. Recommended strategies are summarized based on commonly adopted approaches in the literature, with corresponding rationale and limitations reflecting their typical strengths and trade-offs. The categories are not mutually exclusive. Discrete–continuous coupling and mechanistic–ML integration may be combined when a research question requires both multiscale simulation and learning-based acceleration or calibration. Data availability is classified qualitatively (low, moderate, high) to reflect the general level of accessible experimental or clinical information rather than precise quantitative thresholds. This guideline is intended as a conceptual reference to support method selection rather than a prescriptive rule set.

The practical value of mechanistic TME models depends on how their parameters are estimated, how predictions are validated, and how uncertainty is communicated. Section 3 therefore examines model credibility, while Section 4 considers whether surrogate, inverse, and physics-informed learning can reduce computational and observational bottlenecks without erasing mechanistic interpretability.

Parameterization, validation, and model credibility in TME modeling

Mechanism-based TME models depend critically on appropriate parameterization and rigorous evaluation to ensure biological relevance and predictive reliability. Model parameters are typically derived from a combination of in vitro measurements, clinical observations, and prior literature, reflecting both intrinsic cellular properties and microenvironmental conditions. However, owing to experimental limitations and strong inter-patient heterogeneity, many parameters cannot be directly measured. This limited observability introduces fundamental challenges in parameter identifiability, where multiple parameter configurations may yield indistinguishable model behaviors, thereby complicating model calibration and interpretation. Consequently, parameterization and evaluation should be considered jointly within a unified framework of model credibility rather than as independent technical steps [10, 11, 43, 89]. Parameter calibration can therefore be formulated as an inverse problem, where model parameters are inferred by minimizing the discrepancy between model predictions and experimental or clinical observations:

(9)

where denotes the optimal parameter estimate, represents the model predictions generated using parameters θ, and Y denotes the corresponding experimental or clinical observations. This formulation highlights that parameter estimation is not merely a numerical optimization problem but a process constrained by biological observability.

In practice, parameter calibration commonly begins with values inferred from controlled in vitro experiments. Physical and biological quantities, such as proliferation rates, diffusion coefficients, or vascular-to-matrix ratios, are frequently adopted from previously reported measurements and incorporated into computational frameworks. Representative examples include cellular automaton models calibrated using experimentally derived kinetic parameters and image-informed vascular models grounded in physical measurements of tissue composition [44]. While such approaches provide biologically plausible initial conditions, their generalizability to in vivo settings remains inherently limited. Sensitivity analysis is therefore often employed to identify dominant parameters and reduce effective dimensionality, partially mitigating identifiability issues. Nevertheless, parameter uncertainty remains unavoidable in multiscale models, particularly when nonlinear coupling across components amplifies uncertainty propagation.

Model evaluation strategies can be broadly categorized into three complementary classes: experimental validation, clinical outcome validation, and qualitative or indirect validation, each serving distinct purposes and operating under different practical constraints [10]. These validation strategies can be further interpreted as forming a hierarchical structure of evidence, ranging from mechanistic consistency to clinical applicability.

When quantitative metrics are reported, they should therefore be read as context-specific evidence rather than as universal performance guarantees. A Dice coefficient from simulated glioma infiltration, an area under the receiver operating characteristic curve (AUC) from a single-center breast cancer cohort, and a relative volume error from an animal imaging study reflect different endpoints, data quality, and validation standards, even when the numerical values appear similar.

Experimental validation relies on direct comparison between model predictions and controlled laboratory observations, typically involving tumor spheroids, cell cultures, or animal models. Common quantitative endpoints include tumor volume, cell population size, spatial density distributions, and growth trajectories. For instance, long-term spheroid simulations have been assessed by matching predicted cell counts against experimental measurements, while spatial invasion patterns and growth curves have been compared with previously reported in vitro data. This form of validation offers the highest degree of mechanistic interpretability but is often restricted by experimental cost, limited throughput, and simplified biological conditions [10, 39, 44, 57].

Clinical outcome validation asks whether model predictions correspond to patient-derived endpoints such as imaging biomarkers, treatment response, or prognosis. Recent biophysical breast cancer simulations illustrate the potential of this approach: one platform achieved 88.0% sensitivity and 89.4% specificity for pathological complete response (pCR) prediction in 144 tumors, and an independent 80-patient validation reported 91.2% overall pCR prediction accuracy [90, 91]. Mechanistic models calibrated to serial MRI have also identified alternative dosing regimens predicted to outperform standard-of-care protocols in 12 of 13 patients [92]. These results are promising, but they remain context-dependent because clinical validation is shaped by cohort composition, imaging protocol, endpoint definition, and confounding in real-world data.

When direct quantitative validation is not feasible, qualitative or indirect validation is commonly employed. In this setting, models are assessed based on their ability to reproduce observed phenomena or explain previously unexplained clinical behaviors. Examples include mechanistic explanations of accelerated tumor recurrence following treatment, simulation-based analysis of resistance dynamics under radiotherapy, chemotherapy, and immunotherapy [23], and inference based on internal model indicators such as virtual tumor size evolution or entropy changes during simulation [93]. Related approaches have also utilized surrogate indices, such as radiation severity metrics derived from model-predicted lung volume and functional subunit survival, to characterize treatment-induced fibrosis [94]. While informative for hypothesis generation, such indirect validation lacks quantitative rigor and is primarily exploratory.

Table 5 compares experimental, clinical-outcome, and qualitative validation strategies in terms of their data sources, typical endpoints, strengths, and limitations. The comparison makes clear that these strategies are complementary rather than interchangeable: mechanistic plausibility, quantitative prediction, and clinical relevance provide different forms of evidence and should not be treated as equivalent. Robust model assessment typically requires integrating multiple levels of evidence, where consistency across experimental observations, simulation outputs, and clinical data serves as a key indicator of model reliability.

Table 5.

Evaluation strategies for tumor microenvironment models.

Evaluation type	Data source	Typical endpoints	Strengths	Limitations
Experimental validation	In vitro / animal models	Cell count, volume, spatial distribution	High mechanistic interpretability	Simplified biology, low throughput
Clinical outcome validation	Imaging, prognosis, therapy response	Biomarkers, pCR, survival	Clinically relevant	Strong heterogeneity, confounders
Qualitative / indirect validation	Simulation indicators	Growth patterns, entropy, virtual size	Explains complex phenomena	Limited quantitative rigor

Notes: Evaluation strategies are categorized into experimental validation, clinical outcome validation, and qualitative or indirect validation according to the type of evidence used to assess model performance. Data sources, endpoints, strengths, and limitations summarize commonly reported evaluation practices in tumor microenvironment modeling studies and are intended to highlight methodological trade-offs rather than provide an exhaustive taxonomy.

Beyond validation, uncertainty quantification has emerged as a critical component of TME modeling, particularly in the context of patient-specific prediction [43]. Sources of uncertainty include parameter estimation errors, model structural assumptions, and stochastic biological variability. Bayesian inference frameworks, ensemble simulations, and probabilistic modeling approaches provide principled tools for quantifying uncertainty and propagating it to model outputs. However, these methods are often computationally expensive in large-scale multiscale models. Recent advances in machine learning, including surrogate modeling and physics-informed neural networks, offer promising solutions by enabling efficient approximation of model behavior while preserving key physical constraints, thereby facilitating scalable uncertainty-aware simulation.

Successful translational medicine applications require model evaluation to be aligned with the intended use of the model. We organize model credibility into three linked dimensions. Numerical validity concerns implementation correctness and computational stability. Biological fidelity concerns agreement with observed mechanisms and dynamics. Clinical relevance concerns predictive performance, robustness, and generalization across patients. Together, these dimensions expose the trade-offs among model complexity, interpretability, and computational cost.

Standardized benchmarks, multimodal datasets, and unified evaluation protocols will be essential for improving reproducibility and translational credibility. Model credibility is not an intrinsic property of a formulation but an outcome of concordant evidence across parameterization, numerical behavior, biological plausibility, validation, uncertainty quantification, and reproducibility. Building on the validation strategies compared in Table 5 and Table 6 translates these broader requirements into a practical checklist for designing, reporting, and reviewing TME modeling studies [10, 43, 95].

Table 6.

Credibility checklist for interpreting and reporting tumor microenvironment modeling studies.

Credibility checkpoint	What should be reported?	Why does it matter?	Common red flag
Question and intended use	Whether the model is for explanation, prediction, optimization, or clinical decision support	The acceptable level of complexity and validation depends on intended use	A model claims clinical relevance but is only validated as a qualitative simulation
Model boundary	State variables, spatial scale, time scale, excluded mechanisms, and coupling assumptions	Readers need to know what the model can and cannot represent	Important biological processes are omitted without explanation
Parameter provenance	Which parameters are measured, fitted, borrowed, or assumed	Parameter source determines transferability across tumors and cohorts	Large parameter sets are borrowed from unrelated systems
Identifiability and sensitivity	Sensitive parameters, non-identifiable parameters, and robustness to parameter variation	Good fit can arise from many parameter combinations	Only one best-fit parameter set is reported
Validation context	Tumor type, cohort size, imaging modality, endpoint, prediction horizon, and validation setting	Metrics cannot be compared without context	Dice, CCC, AUC, or error values are presented as universal evidence
Uncertainty quantification	Confidence intervals, posterior distributions, ensembles, or prediction bands	Clinical and experimental users need to know when not to trust a prediction	Point predictions are reported without uncertainty
External generalization	Independent cohort, cross-site testing, prospective setting, or distribution-shift analysis	A model useful in one dataset may fail elsewhere	Only retrospective internal validation is available
Reproducibility	Code, solver settings, training data, preprocessing, and validation split	Reusable models require transparent implementation details	Methods cannot be reproduced from the manuscript

Notes: The checklist emphasizes that model credibility depends on intended use, model boundaries, parameter provenance, validation context, uncertainty quantification, generalization, and reproducibility.

Mechanistic–machine learning integration

Machine learning is increasingly used to address three limitations of mechanistic TME models: expensive simulation, incomplete parameter observability, and heterogeneous patient data. Rather than treating these methods as a single category, this section distinguishes them by function [11, 12, 89, 96].

Surrogates accelerate forward simulation, inverse methods estimate hidden parameters or states, physics-informed networks constrain learning with governing equations, and patient-specific workflows combine these tools with individual clinical data. Table 7 compares these forms of mechanistic–ML integration according to modeling task, contribution, advantages, limitations, and representative evidence, whereas Fig. 3 summarizes how these complementary components can be integrated into a patient-specific modeling workflow that ultimately supports clinically deployable digital twins.

Table 7.

Functional categories of mechanistic–machine learning integration in tumor modeling.

ML approach	Modeling task	Main contribution	Advantages / limitations	Representative refs
High-performance sensitivity analysis	Forward simulation	Accelerates large-scale parameter exploration in discrete models	+ Simple integration; effective for low -dimensional spaces - Poor scalability in high-dimensional parameter spaces	[98]
Surrogate modeling (CNN, regression, ML emulators)	Forward approximation	Replaces repeated mechanistic simulations with fast ML predictors	+ Orders-of-magnitude speedup; preserves dominant dynamics - Approximation error; limited extrapolation	[12, 144, 145]
Approximate Bayesian computation	Parameter inference	Enables likelihood-free calibration of tumor growth models	+ Handles complex models without explicit likelihood - Computationally intensive; sensitive to summary statistics	[99]
Neural PDE solvers / supervised surrogates	PDE acceleration	Learns neural representations of PDE solutions	+ Continuous differentiable fields; flexible geometries - Requires large training data; weak physical guarantees	[146–148]
Bayesian learning and uncertainty quantification	Inverse modeling	Improves robustness and parameter identifiability	+ Explicit uncertainty estimation; principled inference - High computational cost; slow convergence	[89, 149–151]
End-to-end inverse learning	Imaging-driven inversion	Directly maps medical images to biophysical parameters	+ Near real-time inference; avoids iterative optimization - Reduced interpretability; depends on synthetic training data	[97]
Multimodal ML integration	Patient-specific modeling	Integrates imaging and clinical data for personalized models	+ Enables individualized prediction - Data heterogeneity and alignment challenges	[11, 38, 91, 152, 153]
Reinforcement learning	Phenotype optimization	Learns adaptive cellular and vascular evolution strategies	+ Dynamic decision-making - Reward design sensitive; training instability	[154, 155]
Digital twin frameworks	Personalized simulation	Couples ML and mechanistic models for patient-specific replicas	+ End-to-end personalization; supports therapy planning - High implementation complexity; limited clinical validation	[11]
Physics-informed neural networks (PINNs)	Forward + inverse PDE modeling	Embeds governing equations into neural learning	+ Data-efficient; physically consistent; interpretable - Training instability; sensitive to PDE stiffness	[13, 146, 156–158]
KAN-enhanced PINNs	Multiscale physics learning	Improves representation of sharp gradients and multiscale dynamics	+ Higher expressivity; improved interpretability - Early-stage; limited biomedical validation	[159]

Notes: ML, machine learning; CNN, convolutional neural network; PINN, physics-informed neural network. This table summarizes representative forms of mechanistic–ML integration in tumor microenvironment modeling, including surrogate-based simulation, inverse parameter inference, patient-specific calibration, and physics-informed learning. The listed methods are selected to illustrate major methodological categories rather than to provide an exhaustive survey. Reported advantages and limitations reflect commonly observed characteristics in the literature and may vary depending on specific implementations and application contexts.

Figure 3.

Workflow of patient-specific computational modeling toward clinically deployable digital twins. Multimodal clinical data, including magnetic resonance imaging (MRI), computed tomography (CT), histopathology, molecular profiles, and clinical records, undergo preprocessing, feature extraction, segmentation, multimodal alignment, region-of-interest (ROI) definition, and data harmonization. These data are integrated to construct a patient-specific representation of anatomy, tumor geometry, tissue compartments, and the surrounding tumor microenvironment (TME). Mechanistic and hybrid models combine governing equations, agent-based cellular dynamics, transport fields, and immune and stromal components to simulate tumor progression within this individualized context. Calibration and inference use patient-specific observations for parameter estimation, inverse modeling, posterior characterization, and uncertainty-aware calibration. The calibrated model supports personalized prediction of tumor evolution, treatment response, and disease risk, while longitudinal data integration enables adaptive updating toward a predictive and updatable patient-specific digital twin. The lower panel illustrates the conceptual translational progression from basic mechanistic modeling through treatment-response prediction, therapy planning, longitudinal monitoring, and adaptive model updating to a clinically deployable digital twin.

Machine learning for forward simulation and inverse inference

Forward and inverse problems assign fundamentally different roles to learning. In a forward problem, model parameters and initial conditions are specified, and the objective is to predict tumor evolution or associated microenvironmental fields. Machine learning is used mainly to emulate an expensive numerical solver, thereby accelerating repeated simulation, sensitivity analysis, or uncertainty propagation. The surrogate should therefore be evaluated against the mechanistic solver across the parameter domain in which it will be used [12, 89, 96].

An inverse problem starts from observations, such as MRI, histopathology, or longitudinal tumor measurements, and seeks latent model quantities that could have generated those observations. These quantities may include diffusion coefficients, proliferation rates, initial conditions, or spatial cell-density fields. The central challenge is not only computational cost but also identifiability, because different parameter combinations may yield similar observable behavior [89, 97].

Two broad strategies address this inverse task. Optimization- and Bayesian-based methods repeatedly evaluate the forward model to minimize a data-model discrepancy or characterize a posterior distribution. Learning-based methods instead train an amortized inverse map from observations to parameters or latent states, often using simulated examples. The latter can greatly reduce inference time, but its validity depends on whether the synthetic training distribution adequately represents the clinical cases encountered at deployment.

Forward problems: Surrogate modeling and simulation acceleration

In discrete mechanistic models, estimated parameter values are typically used for forward simulation, and sensitivity analysis is often required to explore the impact of different parameter combinations on model behavior. To address the computational burden associated with large-scale parameter exploration, Niida et al. developed MASSIVE, which uses high-performance parallel computing to accelerate the exploration of multiple free parameters [98]. While effective for models with limited dimensionality, contemporary tumor models often involve substantially larger parameter spaces, rendering exhaustive exploration increasingly impractical and motivating the development of more efficient computational strategies.

Surrogate modeling reduces the cost of repeated simulations by learning fast approximations of discrete or differential equation-based models. Across CA models, regression approaches, approximate Bayesian computation, and Bayesian tracking studies, the common goal is not to replace the mechanistic model but to emulate the response surface needed for sensitivity analysis, calibration, or prediction [96, 99]. These approaches can substantially reduce runtime while preserving dominant dynamics, but their reliability depends on whether the training data cover the parameter regimes and biological scenarios in which the surrogate will be used.

Parallel efforts have focused on accelerating the numerical solution of PDEs by representing PDE solutions with neural networks trained through physics-informed or supervised learning paradigms. Kisbenedek constructed neural surrogates to approximate in vivo parameter-dependent PDE solutions and iteratively refined the networks using prediction errors [100], significantly reducing computation time compared with conventional solvers. Related approaches similarly exploit neural representations to replace repeated numerical integration, enabling rapid approximation of complex spatiotemporal fields [101].

Inverse problems: Parameter inference and model calibration

Optimization- and Bayesian-based inverse methods

Optimization-based inverse methods estimate parameters by minimizing an objective function that measures disagreement between simulated and observed tumor characteristics. Gradient-based, adjoint, or Newton-type methods can be efficient when the forward model is differentiable, whereas Bayesian approaches additionally represent parameter uncertainty through posterior distributions [89, 102]. Their main advantage is explicit coupling to the governing model and likelihood assumptions. Their main limitation is computational cost, because repeated forward simulations or posterior sampling may be required, particularly for high-dimensional and weakly identifiable models.

Learning-based amortized inverse methods

Learning-based inverse methods replace case-by-case iterative calibration with an amortized mapping learned across many simulated observation-parameter pairs. Learn-Morph-Infer, e.g. trains a neural network on synthetic tumors and then infers patient-specific biophysical parameters from image-derived tumor segmentations [97]. This strategy can reduce inference from repeated optimization to a single network evaluation and is therefore attractive for rapid personalization. However, the predicted parameters remain conditional on the simulator, parameter priors, observation model, and anatomical variability represented during training. External validation should therefore test both parameter recovery and downstream predictive performance under clinical distribution shift.

Patient-specific modeling

Patient-specific modeling combines mechanistic formulations with individual-level data to estimate model states, parameters, or geometry. As summarized in Table 7, these approaches use machine learning at different stages of the personalization pipeline, including image processing, parameter inference, surrogate evaluation, and uncertainty-aware prediction. Imaging, histopathology, molecular profiles, treatment history, and longitudinal measurements are informative only when each data stream constrains a defined model component. The value of personalization should therefore be judged by the intended prediction and its validation, rather than by the number of modalities included [103].

An imaging-informed biomechanistic model of prostate cancer during active surveillance illustrates this distinction. Calibrated using longitudinal multiparametric MRI, the model represented and forecast global tumor burden in seven patients, with patient-level CCCs of 0.93–0.99. The mean proliferation activity derived from the model was associated with higher-risk disease (P = .041), and a logistic-regression classifier based on this biomarker achieved an AUC of 0.83 [11, 104]. These findings demonstrate feasibility in a small pilot cohort and should not be interpreted as population-level clinical validation.

As shown in Fig. 3, a credible personalization workflow preprocesses patient data, reconstructs individualized anatomy, calibrates mechanistic parameters, propagates uncertainty, and evaluates predictions against longitudinal observations. Machine learning can accelerate individual steps in this pipeline, but sparse sampling, intratumoral heterogeneity, protocol variation, and parameter non-identifiability remain important limitations. Each derived feature should therefore be linked to a measurable model quantity and evaluated against an endpoint that matches the intended use [11, 95, 103, 105, 106].

Patient-specific models may form components of a digital twin, but the terms are not interchangeable [95]. In this review, a digital twin denotes a patient-specific model that can be updated with longitudinal data and used for counterfactual simulation. A static image-derived reconstruction or a one-time calibrated growth model remains a patient-specific model rather than a deployable clinical twin. This distinction provides a natural bridge to physics-informed learning, which can couple sparse observations to governing equations during patient-specific calibration.

Physics-informed learning for biomedical modeling

Physics-informed learning is treated separately in Table 7 because governing equations, boundary conditions, or constitutive relations directly constrain the learning objective. Physics-informed neural networks (PINNs) represent solution fields with neural networks and use automatic differentiation to penalize violations of the governing equations. This framework can support forward simulation, inverse parameter inference, and data assimilation when observations are sparse [13, 107].

In TME modeling, the encoded constraints may describe tumor growth, molecular transport, vascular flow, or tissue mechanics. As summarized in Fig. 4, the loss function combines data mismatch with residuals of the governing equations and the relevant initial or boundary conditions. This objective can be expressed as:

Figure 4.

Physics-informed mechanistic–machine learning framework for tumor modeling, parameter inference, and longitudinal updating toward patient-specific digital twins. Patient-specific observations, including serial magnetic resonance imaging (MRI), histopathology, longitudinal biomarker measurements, and tumor microenvironment (TME) context, provide data for constraining a reaction–diffusion tumor model. A neural network approximates the spatiotemporal tumor field, while automatic differentiation provides the derivatives required to evaluate the governing partial differential equation (PDE). Training jointly minimizes residuals associated with observed data, the governing PDE, the initial condition (IC), and the boundary condition (BC), thereby integrating observational evidence with mechanistic constraints. The trained mechanistic–machine learning framework supports two parallel tasks: (A) forward prediction of spatiotemporal tumor evolution and (B) inverse inference of spatially varying model parameters. (C) New longitudinal observations are subsequently assimilated to update the model state and posterior estimates. The feedback pathway from longitudinal data assimilation to the optimizer represents iterative updating of model states, parameters, and network weights as new patient-specific information becomes available, providing a basis for continuously updated patient-specific modeling and digital-twin development.

(10)

where represents the discrepancy between neural network predictions and available observations, measures the residuals of governing equations, and and enforce initial and boundary conditions, respectively. The weighting coefficients determine the relative contribution of observational data and mechanistic constraints during optimization.

These constraints can reduce the space of admissible solutions, but they do not ensure accurate predictions. Model misspecification, loss weighting, optimization stiffness, and limited observations can still constrain performance [108].

Evidence from cardiovascular modeling illustrates the type of inverse problem that is relevant to vascular components of the TME. Kissas et al. used a physics-informed network to reconstruct pressure, velocity, and wall-displacement waveforms from noisy and spatially sparse 4D flow MRI data in an arterial network [109]. This study supports the feasibility of inferring unmeasured spatial fields from limited imaging observations. It should be regarded as a methodological analogue, rather than evidence of tumor-specific accuracy.

Cancer-specific studies are beginning to provide more directly relevant quantitative evidence. Camacho-Gomez et al. combined an initial magnetic resonance image with longitudinal prostate-specific antigen measurements to reconstruct prostate tumor growth over 2.5 years. Relative errors in tumor volume ranged from 0.8% to 12.28% across the evaluated patients [106]. The framework is an early patient-specific validation study and should not yet be interpreted as a clinically deployable digital twin.

Related physics- and data-informed approaches can also support treatment planning. In a retrospective cohort of 152 patients with glioblastoma, the GliODIL framework integrated multimodal imaging with a tumor-growth constraint. For equal-volume treatment plans, mean recurrence coverage increased from 63.59% using the standard uniform-margin plan to 67.80% using the individualized plan [105]. This result concerns recurrence coverage rather than generic predictive accuracy, and external prospective validation remains necessary.

Architectural alternatives to conventional multilayer perceptrons may further broaden physics-informed learning. Kolmogorov-Arnold-informed neural networks have shown higher accuracy and faster convergence than multilayer-perceptron-based PINNs in several computational solid-mechanics PDE benchmarks, although they performed less well for complex geometries [110]. No direct TME validation was identified. KAN-based methods should therefore be presented as a methodological opportunity, not as evidence that they currently improve tumor modeling.

The available quantitative studies measure different endpoints, including field reconstruction, tumor-volume error, and radiotherapy recurrence coverage. They cannot be placed on a single performance scale because their diseases, observations, governing models, cohorts, and intended uses differ. For clinical translation, the most informative studies will be those that align the validation metric with the intended decision and assess performance in independent patient cohorts [5].

Challenges and future directions

The preceding sections identify four linked barriers to clinically credible TME models: longitudinal personalization, multiscale integration, deployable digital-twin infrastructure, and external validation. Progress therefore depends not only on more complex models, but also on explicit intended-use definitions, calibrated uncertainty, reproducible data-assimilation procedures, and task-specific validation.

Longitudinal patient personalization

A major challenge lies in model parameterization. Many existing frameworks rely on population-averaged parameters or values inferred from simplified experimental settings, which may not accurately reflect individual tumor biology. Bayesian sequential learning has been introduced to progressively calibrate continuum tumor models using patient data [111], while recent approaches further incorporate learnable priors to improve parameter identifiability in inverse tumor growth modeling [112]. Despite these advances, uncertainties arising from sparse observations and evolving tumor states remain significant obstacles. Future research should move beyond static calibration toward longitudinal personalization strategies that assimilate sequential imaging and clinical measurements, enabling continuous updating of model states and parameters over the disease course.

Physics-aware multiscale integration

Another key challenge concerns the representation of tumor heterogeneity across spatial and temporal scales. Mechanical regulation of immune cell activity and macrophage activation highlights the importance of biomechanical interactions within the TME [113, 114], yet such multiscale couplings are often simplified in current models. Recent physics-aware inverse modeling strategies, including discrete-loss optimization [115], self-supervised learning for physiologically based pharmacokinetic modeling [116], and learning-based inverse solvers such as Learn-Morph-Infer [97], demonstrate the potential of mechanistic–ML integration to combine mechanistic consistency with data-driven flexibility. However, developing unified frameworks that systematically couple biochemical, mechanical, and immune processes across scales remains an open research direction.

Clinically deployable digital twins

Digital-twin frameworks combine patient data with mechanistic simulations for individualized prediction, but most current oncology applications remain research prototypes limited by batch calibration, poorly characterized latency, inconsistent uncertainty reporting, and infrequent data assimilation. A clinically deployable workflow must specify when new observations update the model, how uncertainty affects the output, and when a recommendation should be withheld [14, 43, 95, 117].

A useful maturity distinction is between a patient-specific model, a predictive twin, and an actionable twin. The first individualizes parameters or geometry; the second assimilates longitudinal observations and forecasts future states with calibrated uncertainty; and the third compares counterfactual interventions under human oversight and explicit safety constraints. This is a proposed analytical framework for this review, not a universally adopted clinical standard.

Table 8 formalizes this progression by separating mechanistic hypothesis generation, retrospective calibration, patient-specific prediction, prospective decision support, and adaptive digital-twin use according to their required evidence and permissible claims. The three categories proposed above should therefore be interpreted as a concise conceptual grouping of these readiness levels rather than as a competing classification.

Table 8.

Translation-readiness matrix for patient-specific simulation and digital twin development.

Readiness level	Main purpose	Required evidence	What users can safely do?	What should still be avoided?
Level 1: Mechanistic hypothesis	Explain a biological mechanism or generate hypotheses	Qualitative agreement, sensitivity analysis, plausible parameters	Use the model to organize mechanisms and design experiments	Do not claim patient-level prediction
Level 2: Retrospective calibration	Fit existing experimental or clinical data	Transparent parameter fitting and internal validation	Compare mechanisms and identify dominant drivers	Do not assume generalization beyond the fitted dataset
Level 3: Spatial or longitudinal prediction	Predict future tumor state or spatial distribution	Independent validation, prediction horizon, uncertainty bands	Support risk mapping, trial design, or model-informed hypotheses	Do not use without reporting uncertainty and failure modes
Level 4: Patient-specific simulation	Personalize parameters or geometry for an individual patient	Multimodal data integration, calibrated uncertainty, clinical endpoint testing	Explore counterfactual treatment scenarios under supervision	Do not replace clinical judgment or validated protocols
Level 5: Adaptive digital twin	Update the model as new patient data arrive	Latency budget, online data assimilation, prospective evaluation, audit trail	Support adaptive planning as a decision-support tool	Do not deploy without governance, interpretability, and safety thresholds

Notes: The matrix separates useful intermediate modeling outputs from clinically deployable digital twins. Higher readiness requires stronger evidence, explicit uncertainty, prospective validation, data assimilation, and governance.

Translational validation and generalization

Rigorous validation remains the main barrier to translation. Many studies still emphasize retrospective fitting or qualitative agreement, whereas prospective evaluation across diverse cohorts is limited. Three gaps are especially important: prospective, pre-registered testing on independent cohorts is rare; cross-site generalization is seldom assessed, leaving models vulnerable to distribution shift in imaging protocols and patient populations; and shared benchmarks with common endpoints and validation splits remain insufficient [43]. These gaps echo the metric-comparability concerns raised in Section 3.

Such benchmarks should specify not only input data and outcome labels but also prediction horizons, permissible training information, calibration procedures, and uncertainty reporting formats. Without these details, reported improvements may reflect differences in validation design rather than genuine gains in biological or clinical prediction [43].

Synthetic data and Sim-to-real generalization

Mechanism-generated synthetic data may alleviate clinical data scarcity by producing virtual cohorts across plausible tumor geometries, growth rates, transport conditions, treatment schedules, and measurement noise [118]. Such data can support surrogate training or amortized inverse inference before fine-tuning on smaller real-world datasets, as illustrated by learning-based calibration methods trained on simulated tumors [119].

Synthetic abundance does not, however, guarantee clinical generalization. Misspecified mechanisms, restricted parameter ranges, simplified observation models, and unrealistic noise can all be inherited by the trained network. Performance should therefore be evaluated separately on held-out simulations and external clinical cohorts, with sensitivity analysis, real-data fine-tuning, and prospective cross-site testing used to reduce the simulation-to-reality gap [117, 119].

Taken together, the field is moving from isolated proof-of-concept simulations toward credibility-aware and data-assimilative modeling systems. Progress will depend on explicit intended-use definitions, transparent model boundaries, calibrated uncertainty, external validation, and longitudinal updating. Together, these requirements define a path from mechanistic hypothesis generation to adaptive, clinically governed digital-twin decision support.

Key Points

• The tumor microenvironment is a complex, multiscale system whose progression can be formalized through continuous, discrete, and discrete–continuous hybrid computational paradigms.

• This review systematically synthesizes mathematical formulations for TME modeling, covering cellular dynamics, angiogenesis, metabolism, immune regulation, and biomechanical processes.

• Machine learning integration, including surrogate-based simulation, inverse parameter inference, and physics-informed neural networks, substantially enhances computational efficiency and model consistency.

• Benchmark evaluation across six classical ODE-based growth models fitted to 1472 solid tumor patients demonstrates that the Gompertz formulation achieves the best balance between goodness of fit and parametric parsimony.

• Practical decision tools link biological questions and data availability to model selection and credibility assessment, and identify patient-specific modeling, digital twins, and adaptive therapy optimization as priorities for clinical translation.

Author contributions

W. M. provided direction and guidance throughout the preparation of this manuscript. W. M., H. L. and T. Z. collected and interpreted the studies, and wrote the manuscript with the assistance of J. X. and H. L. Both J. X. and H. L. reviewed and made revisions to the manuscript. All authors read and approved the submission and publication.

Conflicts of interest

None declared.

Funding

None declared.

Data availability

This article is a methodological and literature-based review and does not generate new datasets. All data used for analysis consist of publicly available manuscripts, software repositories, and benchmark resources, which are cited throughout the text.