Bidirectional Endothelial Feedback Drives Turing-Vascular Patterning and Drug-Resistance Niches: A Hybrid PDE-Agent-Based Study

Abstract

We present a hybrid partial differential equation-agent-based model (PDE-ABM). In our framework, tumor cells secrete tumor angiogenic factor (TAF), while endothelial cells chemotactically migrate and branch in response. Reaction–diffusion PDEs for TAF, oxygen, and cytotoxic drug are coupled to discrete stochastic dynamics of tumor cells and endothelial tip cells, ensuring multiscale integration. Motivated by observed perfusion heterogeneity in tumors and its pharmacokinetic consequences, we conduct a linear stability analysis for a reduced endothelial–TAF reaction–diffusion subsystem and derive an explicit finite-domain threshold for Turing instability. We demonstrate that bidirectional coupling, where endothelial cells both chemotactically migrate along TAF gradients and secrete TAF, is necessary and sufficient to generate spatially periodic vascular clusters and inter-cluster hypoxic regions. These emergent patterns produce heterogeneous drug penetration and resistant niches. Our results identify TAF clearance, chemotactic sensitivity, and endothelial motility as effective levers to homogenize perfusion. The model is two-dimensional and employs simplified kinetics, and we outline necessary extensions to three dimensions and saturable kinetics required for quantitative calibration. The study links reaction–diffusion mechanisms with clinical principles and suggests actionable strategies to mitigate resistance by targeting endothelial–TAF feedback.

1. Introduction

Tumors actively remodel their blood supply. When endothelial cells migrate toward chemical cues secreted by tumors, they generate alternating regions of high and low perfusion. These spatial patterns create treatment-resistant niches where drugs penetrate poorly and hypoxia persists. We develop a multiscale model that couples chemical diffusion of oxygen, drug, and signaling molecules with individual tumor and endothelial cells. The model identifies when and how vascular patterns emerge and how parameter changes can restore uniform perfusion and improve therapy.

Drug resistance remains a major obstacle to successful cancer treatment [1]. It results not only from genetic and epigenetic alterations within tumor cells but also from spatial heterogeneity in the tumor microenvironment [2,3,4,5,6]. Hypoxic and irregular drug diffusion create protective refugia that allow resistant clones to survive and expand [7,8,9]. Understanding how such microenvironmental heterogeneity forms is essential for designing therapies that reach and eliminate all tumor regions.

Endothelial cell migration drives much of this heterogeneity. Endothelial cells line blood vessels and generate new vascular sprouts during angiogenesis. They are not malignant but belong to the host vasculature that tumors recruit to supply oxygen and nutrients. These cells sense and move along gradients of vascular endothelial growth factor (VEGF), a signaling molecule released by hypoxic tumor cells [10]. This process, known as angiogenesis, reorganizes the vascular network and alters the spatial distribution of oxygen and drugs. Where sprouts connect, perfusion improves; where they fail, hypoxia and poor drug delivery persist. Clinically, such disorganized and uneven vasculature correlates strongly with metastasis, therapeutic failure, and relapse. Therefore, understanding how endothelial cells migrate and self-organize holds both scientific and translational importance: it provides insight into tumor progression and guides strategies to normalize vessels and homogenize treatment delivery.

Tumor cells also change dynamically within this evolving microenvironment. They divide, migrate, die, and adapt to chemical and mechanical cues generated by the vasculature. The feedback between vascular remodeling and tumor evolution links microenvironmental architecture with resistance and metastasis [11]. Vascular geometry shapes selective pressures that promote resistant phenotypes, while tumor-derived signals redirect vessel growth. This reciprocal coupling drives both spatial heterogeneity and therapeutic failure.

Mathematical modeling provides a rigorous framework by analyzing these multiscale feedback [12,13]. In mathematical oncology, three main paradigms describe tumor–vascular dynamics: continuum partial differential equation (PDE) models, agent-based models (ABMs), and hybrid PDE-ABM frameworks. Classical continuum models represent angiogenesis at the tissue scale by coupling cell density with diffusion factors [14]. The Keller–Segel formulation captures endothelial migration up VEGF gradients and predicts the conditions under which vascular patterns form [15]. These models provide analytic insight but treat cells as continuous densities and cannot represent stochastic, discrete cell behaviors.

ABMs address this limitation by explicitly simulating individual cells that divide, migrate, and interact with local microenvironments according to probabilistic rules [16]. They describe clonal diversity and drug response with high fidelity but become computationally expensive at the tissue scale. Hybrid PDE-ABM models combine the advantages of both approaches by linking reaction–diffusion equations for chemical transport with discrete cellular agents [17,18,19]. This hybridization links microscale cell behavior to macroscale spatial organization and has become a powerful tool for investigating resistance mechanisms and therapy optimization [12,13,20,21]. Appendix A summarizes representative models and highlights those that include angiogenesis, resistance, or spatial heterogeneity.

Despite significant progress, most existing angiogenesis models assume unidirectional coupling: tumor cells secrete VEGF, endothelial cells migrate up its gradient, but endothelial cells do not regulate VEGF production. In reality, endothelial cells can also modulate VEGF production, creating a feedback loop that amplifies local signaling [22]. Such bidirectional coupling may destabilize homogeneous tissue states and generate self-organized vascular patterns akin to those predicted by Turing’s reaction–diffusion theory [23].

We construct a hybrid PDE-ABM model that integrates stochastic tumor and endothelial tip dynamics with reaction–diffusion equations for oxygen, drug, and tumor angiogenic factor (TAF). The model compares unidirectional and bidirectional coupling. Analytical and numerical analyses show that bidirectional coupling is necessary and sufficient for Turing-type instabilities that generate spatially periodic vascular clusters separated by hypoxic regions. These emergent patterns explain the coexistence of vascularized and hypoxic tumor zones and demonstrate how vascular adaptation fosters drug resistance. The analysis identifies three quantitative control points—TAF clearance, chemotactic sensitivity, and endothelial motility—that can be tuned to homogenize perfusion and improve therapeutic delivery.

Numerically, we implement an alternating direction implicit (ADI) scheme for the PDE components [24] and integrate stochastic agent-based updates for cellular events to ensure stability and efficiency.

The biological implications extend beyond oncology. Controlled vascular patterning also governs success in regenerative medicine and tissue engineering. The same bidirectional endothelial–TAF coupling that drives Turing-type patterns in tumors also provides principles for optimizing scaffold geometry, growth factor delivery, and perfusion to achieve uniform oxygenation and drug accessibility.

For clarity, Figure 1 contrasts the classical unidirectional and our bidirectional frameworks, and Figure 2 summarizes the PDE components for the four scalar state variables ($n,c,d,o$). Section 2 presents the full model, Section 3 details parameter estimation and numerical schemes, Section 4 provides the linear stability analysis, and Section 5 reports numerical simulations, bifurcation behaviors, and Turing patterns. Section 6 concludes with biological and computational implications.

2. Methods

We organize Methods into two parts: PDE equations in Section 2.1 and agent-based rules in Section 2.2.

2.1. PDE Equations

We model tumor–vasculature interactions using four scalar state variables that depend on space and time: endothelial cell density $n(x,t)$, tumor angiogenic factor (TAF or VEGF) concentration $c(x,t)$, drug concentration $d(x,t)$, and oxygen concentration $o(x,t)$. Throughout, the spatial variable $x=(x_1,x_2)\in U\subset {\mathbb{R}}^2$ represents position in a two-dimensional tissue domain. The same formulation extends directly to three dimensions with $x=(x_1,x_2,x_3)$. Endothelial cells are normal vessel-lining cells; their density is n. Tumor cells are modeled as discrete cells (agents) and are denoted by $a\in {\mathrm{\Lambda }}_t$ at time t. The growing vasculature is represented by vessel sites (endothelial tips/segments) $v\in V_t$ at time t, and these vessel sites can deliver oxygen and drugs. For clarity, ${\mathrm{\Lambda }}_t$ and $V_t$ denote the sets of tumor cells and vessel sites at time t, respectively.

We further classify tumor cells by local oxygen into normoxic (sufficient oxygen) and hypoxic (reduced oxygen) subsets, written ${\mathrm{\Lambda }}_t^n$ and ${\mathrm{\Lambda }}_t^h$ at time t. Hypoxia is defined by thresholds on $o(x,t)$ (normoxic if $o>o_{\mathrm{hyp}}$; hypoxic if $o_{\mathrm{apop}}<o\le o_{\mathrm{hyp}}$), and it drives angiogenic signaling: hypoxic tumor cells secrete TAF, which appears as a source in the c-equation, while endothelial cells bias their motion up $\nabla c$ (chemotaxis) with sensitivity $\chi \left(c\right)$, which enters the n-equation through the flux term $\nabla ·\left(\chi \right(c)n\nabla c)$. Drugs and oxygen obey transport with vascular supply and tumor uptake. Below, we write the main PDEs for the four scalar variables $n,c,d,o$ and, after each, give a brief biological meaning and how the discrete tumor cells ${\mathrm{\Lambda }}_t$ and vessel sites $V_t$ act as sources or sinks.

(i)

Endothelial dynamics n

Endothelial cell movement has two components. The first is random motility, the undirected cell motion that smooths local density and is modeled by the diffusion term $D_n\mathrm{\Delta }n$. The second is chemotaxis, directed migration up the gradient of TAF/VEGF concentration $c(x,t)$ and is modeled by the drift term $-\nabla ·\left(\chi \right(c)n\nabla c)$. Together, these processes satisfy the standard Keller–Segel conservation law [15]

$$\begin{array}{cc}\hfill & {\partial }_{t}n=D_n\mathrm{\Delta }n-\nabla ·\left(\chi \left(c\right)n\nabla c\right),\hfill \\ \hfill & \chi \left(c\right)={\displaystyle \frac{{\chi }_0{k}_1}{k_1+c}},\hfill \\ \hfill & \chi \left(c\right)\to {\chi }_0\mathrm{as}c\to 0,\hfill \\ \hfill & \chi \left(c\right)\to 0\mathrm{as}c\to \infty .\hfill \end{array}$$

(1)

Here, $D_n$ is the endothelial cell diffusion coefficient, and $\chi \left(c\right)$ is the chemotactic sensitivity. The flux of the endothelial cells is

$$\begin{array}{c}\hfill J_n=-D_n\nabla n+\chi \left(c\right)n\nabla c.\end{array}$$

The first term is random motility, and the second is drift up the TAF gradient. The decreasing, saturating function $\chi \left(c\right)$ reflects receptor kinetics: if endothelial receptors bind ligand with occupancy $f\left(c\right)=c/(k_1+c)$, then gradient sensing depends on $\nabla f$ and diminishes as receptors saturate, producing reduced responsiveness at high c. The chosen form $\chi \left(c\right)={\chi }_0{k}_1/(k_1+c)$ is a simple monotone approximation that captures this biophysical effect [25,26]. Here, ${\chi }_0$ is the maximal chemotactic coefficient and $k_1>0$ modulates TAF sensitivity. The companion equation for c (stated below) makes its kinetics explicit by including diffusion, natural clearance, secretion by hypoxic tumor cells, and endothelial uptake; production of chemoattractant therefore appears there rather than in Equation (1).

(ii)

TAF dynamics c

The TAF concentration $c(x,t)$ evolves through diffusion, clearance, secretion by hypoxic tumor cells, and uptake by vessels:

$$\begin{array}{c}\hfill {\partial }_{t}c=D_c\mathrm{\Delta }c-{\xi }_{c}c+\eta \sum _{a\in {\mathrm{\Lambda }}_t^h}{\varphi }_a-\lambda c\sum _{v\in V_t}{\varphi }_v.\end{array}$$

(2)

Here, $D_c$ is the diffusion coefficient, ${\xi }_c$ the clearance rate, $\eta$ the secretion rate per hypoxic cell, and $\lambda$ the uptake rate near vessels. Tumor cells $a\in {\mathrm{\Lambda }}_t$ and vessel sites $v\in V_t$ occupy positions $a^{\mathrm{x}}\left(t\right),v^{\mathrm{x}}\left(t\right)\in U$. We emphasize that the positions $a^{\mathrm{x}}\left(t\right),v^{\mathrm{x}}\left(t\right)$ are not input parameters but dynamical states that enter Equation (2) as moving sources and sinks. Each influences nearby tissue through a normalized spatial kernel:

$$\begin{array}{c}\hfill {\varphi }_a(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\pi R_c^2}}\hfill & if∥x-a^{\mathbf{x}}\left(t\right)∥\le R_c,\hfill \\ 0\hfill & \mathrm{otherwise},\hfill \end{array}\right.\\ \hfill {\varphi }_v(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\pi R_c^2}}\hfill & if∥x-v^{\mathbf{x}}\left(t\right)∥\le R_c,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

The normalization ensures ${\int }_U{\varphi }_a(x,t)dx={\int }_U{\varphi }_v(x,t)dx=1$. The parameter $R_c>0$ defines the cell radius.

Biological interpretation. The diffusive term $D_c\mathrm{\Delta }c$ captures the passive spread of angiogenic factors in tissue. The term $-{\xi }_{c}c$ models natural clearance or degradation. The localized source $\eta {\sum }_{a\in {\mathrm{\Lambda }}_t^h}{\varphi }_a$ adds TAF in regions surrounding hypoxic tumor cells, while the sink term $-\lambda c{\sum }_{v\in V_t}{\varphi }_v$ removes TAF near endothelial vessels that absorb it. Together, these processes establish angiogenic gradients that drive the directed migration of endothelial cells in Equation (1), coupling tumor hypoxia to vascular response.

(iii)

Drug dynamics (d)

Drug concentration $d(x,t)$ changes by diffusion, clearance, uptake by tumor cells, and vascular delivery:

$$\begin{array}{c}\hfill {\partial }_{t}d=D_d\mathrm{\Delta }d-{\xi }_{d}d-{\rho }_{d}d\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a+S_d\left(t\right)\sum _{v\in V_t}{\varphi }_v.\end{array}$$

(3)

Here, $D_d$ is the drug diffusion coefficient (spatial dispersal), ${\xi }_d$ is systematic/interstitial clearance, ${\rho }_d$ is tumor uptake per cell (a sink proportional to local d and the nearby cell density), and $S_d\left(t\right)$ is the vascular delivery at vessel sites. We encode dosing regimens via

$$\begin{array}{c}\hfill S_d\left(t\right)=\left\{\begin{array}{cc}{S}_d,\hfill & \mathrm{treatment}\mathrm{on},\hfill \\ 0,\hfill & \mathrm{treatment}\mathrm{off},\hfill \end{array}\right.\end{array}$$

for pulsed therapy, and $S_d\left(t\right)\equiv S_d$ for continuous low dose administration.

Biological interpretation. The term $D_d\mathrm{\Delta }d$ governs spatial dispersal of drug. The term $-{\xi }_{d}d$ captures clearance. The sink $-{\rho }_{d}d{\sum }_{a\in {\mathrm{\Lambda }}_t}{\varphi }_a$ models tumor uptake. The source $S_d\left(t\right){\sum }_{v\in V_t}{\varphi }_v$ represents time dependent vascular delivery. Collectively, Equation (3) captures the spatiotemporal distribution of therapeutic agents under distinct dosing regimens.

(iv)

Oxygen dynamics (o)

We represent oxygen with diffusion, decay, tumor consumption, and feedback-limited vascular supply:

$$\begin{array}{c}\hfill {\partial }_{t}o=D_o\mathrm{\Delta }o-{\xi }_{o}o-{\rho }_o\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a+S_o(o_{max}-o)\sum _{v\in V_t}{\varphi }_v.\end{array}$$

(4)

Here, $D_o$ denotes oxygen diffusion coefficient, ${\xi }_o$ denotes natural degradation, ${\rho }_o$ denotes tumor uptake, $S_o$ denotes vessel supply rate, and $o_{max}$ denotes vessel saturation concentration. Therefore, the term $S_o(o_{max}-o)$ implies that if local oxygen o is near $o_{max}$, the source from vessels is small.

Biological interpretation. The term $D_o\mathrm{\Delta }o$ models passive oxygen spread. The term $-{\xi }_{o}o$ captures oxygen decay. The term $-{\rho }_o{\sum }_{a\in {\mathrm{\Lambda }}_t}{\varphi }_a$ captures cellular consumption. The feedback-controlled source term $S_o(o_{max}-o){\sum }_{v\in V_t}{\varphi }_v$ ensures oxygen saturates near physiological levels. This formulation captures hypoxia in poorly vascularized regions and homeostatic oxygen regulation in well-perfused zones.

We replace linear uptake by Michaelis–Menten saturation kinetics when data justify nonlinear consumption [27]. This formulation reduces to linear uptake at low concentrations while preventing unbounded consumption at high concentrations. For drug and oxygen consumption, we write

$$\begin{array}{c}\hfill {\displaystyle \frac{{\rho }_{d}d}{K_d+d}}\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a,{\displaystyle \frac{{\rho }_{o}o}{K_o+o}}\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a.\end{array}$$

Here, ${\rho }_d,{\rho }_o$ denote maximum consumption rates and $K_d,K_o$ denote half-saturation concentrations. For $d\ll K_d,o\ll K_o$, the kinetics reduce to the linear form ${\displaystyle \frac{{\rho }_{d}d}{K_d+d}}\approx {\displaystyle \frac{\rho }{K_d}}d,{\displaystyle \frac{{\rho }_{o}o}{K_o+o}}\approx {\displaystyle \frac{{\rho }_o}{K_o}}o$. In contrast, consumption saturates at ${\rho }_d,{\rho }_o$ when concentrations are large. See Ojwang et al. [28] and Ginneken et al. [29] for Michaelis–Menten kinetics of tumor oxygenation and pharmacokinetics, respectively.

Our drug equation includes diffusion, uptake, and decay, but not efflux, an active resistance mechanism in which tumor cells pump drug back into the extracellular space. This omission is deliberate for tractability, though Appendix B illustrates how efflux can be incorporated via a source term proportional to the tumor cell number.

Given the limited availability of reliable $K_m$ values for our system, we adopt the linear form as a parsimonious approximation within the biologically relevant range. To further confirm that this simplification does not alter our conclusions, we performed a parallel linear stability analysis analogous to the comprehensive analysis in Section 4. We replace the oxygen supply term $S_o(o_{max}-o){\sum }_v{\varphi }_v$ by Michaelis–Menten kinetics. Appendix G provides a detailed derivation and demonstrates that the qualitative stability results are robust: all perturbation modes remain stable without any spurious pattern-forming instabilities.

Homogeneous Neumann boundary conditions approximate conservation absent sources in the isolated tissue domain:

$$\begin{array}{c}\hfill \overrightarrow{n}·\nabla \varphi {|}_{\partial U}=0,\varphi \in \{n,c,d,o\}.\end{array}$$

Here, $\overrightarrow{n}$ is the unit outward normal on the boundary $\partial U$. Real tumor microenvironment and tissue may receive external signals, violating the Neumann boundary condition. Dirichlet or Robin boundary conditions could be choices, and we leave such an extension for future work. We discretize the chemotaxis Equation (1) with an explicit time-stepping scheme and solve the remaining reaction–diffusion equations with an ADI scheme for numerical efficiency and stability.

Table 1. Mechanisms encoded in each partial differential equation (PDE) for endothelial cells n, tumor angiogenic factor (TAF) c, drug d, and oxygen o. Diffusion coefficients are denoted by $D_{\varphi }$, decay rates by ${\xi }_{\varphi }$, uptake rates by $\lambda$ or ${\rho }_{\varphi }$, and source terms by $\eta$ or $S_{\varphi }$. Uptake terms are sinks (negative signs in PDEs). Decay terms also come with negative signs in PDEs.

Field	Diffusion	Decay	Uptake	Supply
n	$D_n$	None	None	None
c	$D_c$	${\xi }_c$	$\lambda c$	$\eta$ from hypoxic cells
d	$D_d$	${\xi }_d$	${\rho }_{d}d$	$S_d\left(t\right)$ at vessels
o	$D_o$	${\xi }_o$	${\rho }_o$	$S_o(o_{max}-o)$

summarizes the mechanisms encoded in each PDE. We model the system on a two-dimensional tissue slice of effective thickness $h=1\times {10}^{-4}$ m, representative of thin in vitro assays [30]. We convert dimensional parameters reported in standard volumetric (3D) units as in the literature to the 2D formulation using this thickness. Appendix E gives the conversion details.

We nondimensionalize using characteristic length scale L, diffusion time $\tau =L^2/D$, and concentration scales $n_0,c_0,d_0,o_{max}$.

Table 2. Characteristic quantities used for nondimensionalization.

Symbol	Quantity	Rationale
L	Length	Spatial extent of parent vessel to tumor distance
$\tau =L^2/D$	Time	Typical diffusion time scale or cell cycle duration
$n_0$, $c_0$, $d_0$, $o_{max}$	Field concentrations	Normalization of PDE variables: $n_0,c_0,d_0$ are endothelial, TAF, drug scales, respectively, and the maximum concentration $o_{max}$ is the oxygen scale

lists these characteristic scales. Here, D denotes a representative diffusion coefficient, chosen such that $\tau =L^2/D=5.76\times {10}^4$ s for a spatial scale $L=5\times {10}^{-3}$ m. All parameter values are taken from the authoritative set in

Table 3. The table lists all model parameters, providing their dimensional values (D-values, in SI units), corresponding nondimensional values (ND-values), and sources or justifications; see also the machine-readable files params.json and params.csv in the GitHub Repository in the Data Availability Statement Section. Nondimensionalization is performed with respect to the characteristic length (L), time scale ($\tau =L^2/D_o$), and concentration ($c_0$). Parameters labeled as “calibrated” were chosen to ensure consistency with established biological behavior, whereas those marked “(n/a)” correspond to quantities without currently available empirical measurements. Table 4 provides a complete description of all parameters and their modeling roles.

Parameter	Description	D-Value (SI Units)	ND-Value	Provenance
$\mathrm{\Delta }x$	Spatial discretization	$2.5\times {10}^{-5}$ m	0.005	Calculated
$\mathrm{\Delta }t$	Temporal discretization	CFL condition (Equation (13))	CFL condition (Equation (13))	Stability constraint
h	Slab thickness for dimension conversion	$1\times {10}^{-4}$ m	0.02	[30]
$R_c$	Tumor and vessel cell radius	$1.25\times {10}^{-5}\mathrm{m}$	0.005	[31]
$D_c$	TAF diffusion coefficient	$5.21\times {10}^{-11}{\mathrm{m}}^2/\mathrm{s}$	0.12	[32,33]
${\xi }_c$	TAF decay rate	$3.47\times {10}^{-8}{\mathrm{s}}^{-1}$	0.002	[34]
$\eta$	TAF production rate	$1.7\times {10}^{-22}\mathrm{mol}/(\mathrm{cell}·\mathrm{s})$	$6.27\times {10}^3$	[35]
$\lambda$	TAF uptake rate	$2.71\times {10}^{-20}{\mathrm{m}}^3/(\mathrm{cell}·\mathrm{s})$	0.1	[25]
$D_d$	Drug diffusion coefficient	$2.17\times {10}^{-10}{\mathrm{m}}^2$/s	0.5	[36]
${\xi }_d$	Drug decay rate	$1.74\times {10}^{-7}{\mathrm{s}}^{-1}$	0.01	[18]
${\rho }_d$	Drug uptake rate	$1.36\times {10}^{-19}{\mathrm{m}}^3/(\mathrm{cell}·\mathrm{s})$	0.5	[18]
$S_d$	Drug supply rate	$3.94\times {10}^{-20}\mathrm{mol}/(\mathrm{cell}·\mathrm{s})$	2	[18]
$p_r$	Damage clearance rate	(n/a, nondimensionalized)	0.2	[18]
$D_o$	Oxygen diffusion coefficient	$2.78\times {10}^{-10}{\mathrm{m}}^2/\mathrm{s}$	0.64	[37]
${\xi }_o$	Oxygen decay rate	$4.34\times {10}^{-7}{\mathrm{s}}^{-1}$	0.025	[38]
${\rho }_o$	Oxygen uptake rate	$6.25\times {10}^{-17}\mathrm{mol}/(\mathrm{cell}·\mathrm{s})$	34.39	[18]
$S_o$	Oxygen supply rate	$9.33\times {10}^{-19}{\mathrm{m}}^3/(\mathrm{cell}·\mathrm{s})$	3.44	[39]
$\epsilon$	Tumor motility intensity	(n/a)	0.0215	[40]
$o_{max}$	Maximum oxygen concentration	$6.7\mathrm{mol}/{\mathrm{m}}^3$	1	[35]
$o_{\mathrm{hyp}}$	Hypoxia threshold	$1.675\mathrm{mol}/{\mathrm{m}}^3$	0.25	[35]
$o_{\mathrm{apop}}$	Apoptosis threshold	$0.335\mathrm{mol}/{\mathrm{m}}^3$	0.05	[35]
$D_n$	Endothelial diffusion coefficient	$2.00\times {10}^{-13}{\mathrm{m}}^2/\mathrm{s}$	$4.61\times {10}^{-4}$	[25]
${\chi }_0$	Chemotaxis coefficient	$2.60\times {10}^{-4}{\mathrm{m}}^5/(\mathrm{s}·\mathrm{mol})$	0.0599	[25]
$\alpha$ or $k_1$ ($\alpha =c_0/k_1$)	Chemotaxis saturation parameter	$1.6667\times {10}^{-7}\mathrm{mol}/{\mathrm{m}}^3$	0.6	[25]
$\psi$	Minimum branching age	$6.48\times {10}^4$ s	1.125	[25]
$c_{br}$	Baseline branching rate	(n/a)	1	[19]
$a_S^{death}$	Death threshold (sensitive cells)	(n/a)	0.5	[18]
$T{h}_{\mathrm{multi}}$	Death threshold ratio (resistant cells)	5	5	[18]
${\wp }_{age}$	Cell cycle duration	$\mathrm{Uniform}[3.24\times {10}^4,3.96\times {10}^4]\mathrm{s}$	0.56–0.69	[41,42]
${\alpha }_n$	Proliferation rate	Derived from $log\left(2\right)/{\wp }_{age}$	1.0082–1.2323	Derived
$F_{max}$	Maximum neighbor cell count	10	10	[19]

; see also the machine-readable files params.json and params.csv in the GitHub Repository in the Data Availability Statement Section.

The full nondimensionalization details are (Note that the nondimensional form reparameterizes the saturation as $\alpha =c_0/k_1$)

$$\begin{array}{cc}\hfill & {\tilde{D}}_n={\displaystyle \frac{D_n}{D}},{\tilde{\chi }}_0={\displaystyle \frac{{\chi }_0{c}_0}{D}},\alpha ={\displaystyle \frac{c_0}{k_1}},\hfill \\ \hfill & {\tilde{D}}_c={\displaystyle \frac{D_c}{D}},{\tilde{\xi }}_c=\tau {\xi }_c,\tilde{\eta }={\displaystyle \frac{\eta \tau n_0}{c_0}},\tilde{\lambda }=\lambda \tau n_0,\hfill \\ \hfill & {\tilde{D}}_d={\displaystyle \frac{D_d}{D}},{\tilde{\xi }}_d=\tau {\xi }_d,{\tilde{\rho }}_d={\rho }_d\tau n_0,{\tilde{S}}_d={\displaystyle \frac{S_d\tau n_0}{d_0}},\hfill \\ \hfill & {\tilde{D}}_o={\displaystyle \frac{D_o}{D}},{\tilde{\xi }}_o=\tau {\xi }_o,{\tilde{\rho }}_o={\displaystyle \frac{{\rho }_o\tau n_0}{o_{max}}},{\tilde{S}}_o=S_o\tau n_0.\hfill \end{array}$$

(5)

For notational simplicity, unless otherwise stated, we drop the tildes and write all variables in their dimensionless form from here on. After applying the nondimensionalization defined above, the governed PDEs take the form

$$\begin{array}{cc}\hfill {\partial }_{t}n& =D_n\mathrm{\Delta }n-\nabla ·\left({\displaystyle \frac{{\chi }_0}{1+\alpha c}}n\nabla c\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\partial }_{t}c& =D_c\mathrm{\Delta }c-{\xi }_{c}c+\eta \sum _{a\in {\mathrm{\Lambda }}_t^h}{\varphi }_a-\lambda c\sum _{v\in V_t}{\varphi }_v,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\partial }_{t}d& =D_d\mathrm{\Delta }d-{\xi }_{d}d-{\rho }_{d}d\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a+S_d\sum _{v\in V_t}{\varphi }_v,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\partial }_{t}o& =D_o\mathrm{\Delta }o-{\xi }_{o}o-{\rho }_o\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a+S_o(1-o)\sum _{v\in V_t}{\varphi }_v,\hfill \end{array}$$

(9)

All fields $\varphi \in \{n,c,d,o\}$ satisfy homogeneous Neumann (no-flux) boundary conditions in dimensionless form:

$$\begin{array}{c}\hfill \overrightarrow{n}·\nabla \varphi {|}_{\partial U}=0.\end{array}$$

The dimensionless coefficients $D_n,D_c,\dots$ represent ratios relative to the chosen characteristic scales D and $\tau$. For example, $D_c$ is the diffusion coefficient of TAF normalized by D, and $D_n$ is the normalized endothelial cell motility coefficient. The nondimensionalization $\alpha =c_0/k_1$ provides one such example of this scaling. Since all variables are in their dimensionless form, unless otherwise stated, $t,x,y,n,c,\dots$ are understood to be dimensionless quantities. Accordingly, the numerical time steps $\mathrm{\Delta }t$ for PDE and $\mathrm{\Delta }{t}^{\prime }$ for ABM in Section 3.2 are also nondimensional.

Most of our work is performed under the nondimensionalized system (Equations (6)–(9)). All PDEs displayed in Section 2 and Section 4 are in dimensionless form unless explicitly labeled as dimensional; Table 3 lists dimensional parameter values and units.

Table 4. Summary of all parameters used in the model, grouped by type: PDE system and agent-based model (ABM). This table provides an overview of the variables and their modeling roles. For specific numerical values (dimensional and nondimensional) and their units, refer to Table 3.

Parameter	Meaning
PDE-related parameters
$D_n,D_c,D_d,D_o$	Diffusion coefficients of endothelial cells (n), TAF (c), drug (d), and oxygen (o)
${\chi }_0$	Chemotactic sensitivity coefficient
$\alpha$	Saturation parameter for chemotaxis
${\xi }_c,{\xi }_d,{\xi }_o$	Natural decay rates of TAF, drug, and oxygen, respectively
${\rho }_d,{\rho }_o$	Cellular uptake rates of drug and oxygen
$S_d,S_o$	Vessel supply rates of drug and oxygen
$\eta ,\lambda$	TAF production rate by hypoxic cells and uptake rate by endothelial cells
${\varphi }_a,{\varphi }_v$	Normalized indicator functions for tumor agents and vessel locations
$R_c$	Tumor and vessel cell radius
ABM-related parameters
${\mathrm{\Lambda }}_t$, ${\mathrm{\Lambda }}_t^n$, ${\mathrm{\Lambda }}_t^h$, $V_t$, $T_t$	Sets of all tumor cells, normoxic tumor cells, hypoxic tumor cells, vessel cells, and endothelial tip cells at time t
$A_t$	Angiogenic network at time t
$i{d}_a$, $i{d}_b$	Lineage identifiers for tumor and endothelial tip cells
$a^{\mathbf{x}}\left(t\right)$, $b^{\mathbf{x}}\left(t\right)$, $v^{\mathbf{x}}\left(t\right)$	Spatial coordinates of agents $a\in {\mathrm{\Lambda }}_t$, $b\in T_t$, $v\in V_t$ at time t
$a^o\left(t\right)$, $a^d\left(t\right)$, $a^{dam}\left(t\right)$, $a^{death}\left(t\right)$, $a^{age}\left(t\right)$, $a^{mat}$	Local oxygen, drug level, accumulated DNA damage, death threshold, age, and maturation time for tumor cell $a\in V_t$
$b^{age}\left(t\right)$	Age of endothelial tip cell $b\in T_t$
$\mu$	Mutation intensity for the Poisson process
$p_r$	DNA damage repair or clearance rate
$\epsilon$	Tumor cell motility coefficient
$o_{max}$	Maximum oxygen concentration
$o_{\mathrm{hyp}},o_{\mathrm{apop}}$	Hypoxia threshold and apoptosis threshold for oxygen concentration
$P_0,P_1,P_2,P_3,P_4$	Probabilities of endothelial cell remaining stationary or moving left, right, down, or up
$\psi$	Minimum age required for tip branching
$c_{br}$	Branching intensity coefficient
$a_S^{death},a_R^{death}$	Death thresholds for sensitive and resistant tumor cells
$T{h}_{\mathrm{multi}}$	Multiplicative factor defining resistance death threshold ($a_R^{death}=T{h}_{\mathrm{multi}}·a_S^{death}$)
${\wp }_{age}$	Tumor cell cycle duration
${\alpha }_n$	Proliferation rate of normoxic tumor cells
$F_{max}$	Crowding threshold above which proliferation is suppressed

gives an overview of all model parameters used in the study, both for the PDE system, the ABM, and nondimensionalization. The table lists each parameter, along with a brief description of how our modeling framework incorporates it.

We state model assumptions compactly. We assume fixed per cell oxygen and drug consumption rates, linear isotropic diffusion and decay for drug, chemotactic guidance of angiogenic tip cells by TAF gradients, neutral stochastic mutation processes, and a simplified DNA damage repair model without detailed biochemical kinetics. These assumptions restrict scope but keep the model analytically and computationally tractable.

This PDE model gives a rigorous mathematical framework for tumor growth and resistance to therapy, grounded in biological mechanisms. Reaction–diffusion equations capture microenvironmental heterogeneity through diffusion, decay, and local production or consumption. They account for effects such as hypoxic cores, where blood vessels poorly supply certain regions, limited drug penetration, and TAF-induced chemotactic angiogenesis. This continuum PDE formulation sets the stage for the agent-based rules described next. The bidirectional PDE–ABM coupling enables multiscale modeling of cellular heterogeneity. Cells respond to local signaling fields while dynamically altering them through production and consumption. This framework is computationally efficient and readily extends to incorporate advanced cellular behaviors and microenvironmental interactions.

2.2. Agent-Based Rules

2.2.1. ABM Tumor Cell Rules

We implement tumor cells as discrete agents embedded in a two-dimensional lattice framework. These agents interact with continuum microenvironmental fields governed by PDEs in Section 2.1. Each cell follows biologically motivated and mathematically explicit rules for proliferation, apoptosis, mutation, and motility. The choice of the lattice neighborhood structure constrains spatial interactions.

Figure 3 illustrates two common configurations. The Von Neumann neighborhood comprises the four adjacent lattice sites (left, right, down, up), while the Moore neighborhood additionally incorporates the four diagonal sites. Tumor cells use the Von Neumann neighborhood when computing movement probabilities, whereas branching checks Moore neighbors for available space.

Tumor cells are not confined to lattice positions, as continuous Brownian dynamics describe their motion (Equation (10)). In contrast, lattice positions constraint tip and vessel agents. Therefore, only the four cardinal directions in the Von Neumann neighborhood are allowed for tip motion.

Table 5. Neighborhood structures and spatial rules used for different cellular processes.

Cell Type	Process	Spatial Rule/Neighborhood
Tumor cell	Migration	Continuous Brownian motion (not lattice-confined)
Tumor cell	Branching (daughter placement)	Continuous, off-lattice positioning
Tumor cell	Crowding effect	$F_{max}$ check within $R_c$ neighborhood
Tip cell	Migration	Von Neumann neighborhood (4 sites)
Tip cell	Branching (new tip placement)	Moore neighborhood (8 sites)
Tip cell	Occupancy/anastomosis check	Von Neumann neighborhood (4 sites)

summarizes the neighborhood structures used for different cellular processes in the ABM component.

(i)

Cell trait

We represent each tumor cell by a state vector tracking lineage identifier ${\mathrm{id}}_a$ (dimensionless), its position $a^{\mathbf{x}}\left(t\right)$ (unit: m), local oxygen concentration $a^o\left(t\right)$ (unit: mol/$m^3$) accumulated drug exposure $a^d\left(t\right)$ (unit: $(\mathrm{mol}·\mathrm{s})/{\mathrm{m}}^3$) and DNA damage $a^{dam}\left(t\right)$ (unit: $(\mathrm{mol}·\mathrm{s})/{\mathrm{m}}^3$), death threshold $a^{death}\left(t\right)$ (unit: $(\mathrm{mol}·\mathrm{s})/{\mathrm{m}}^3$), the time elapsed since the last division $a^{age}\left(t\right)$ (unit: s), and the cell cycle duration $a^{mat}\left(t\right)$ (unit: s):

$$\begin{array}{c}\hfill a=\left\{{\mathrm{id}}_a,a^{\mathbf{x}}\left(t\right),a^o\left(t\right),a^d\left(t\right),a^{dam}\left(t\right),a^{death}\left(t\right),a^{age}\left(t\right),a^{mat}\left(t\right)\right\}.\end{array}$$

For clarity, we provide the complete definition of the state vector in Appendix C.

(ii)

Cell motility

At the individual-cell level, tumor motion follows independent and identically distributed (i.i.d.) Brownian dynamics:

$$\begin{array}{c}\hfill d{a}^{\mathbf{x}}=\epsilon d{W}_t^a,{\left\{W_t^a\right\}}_{a\in {\mathrm{\Lambda }}_t}\mathrm{i}.\mathrm{i}.\mathrm{d}.\mathrm{Brownian}\mathrm{motions}.\end{array}$$

(10)

Here, the parameter $\epsilon =0.0215$ (dimensionless) directly controls the strength of random fluctuations in cell trajectories. Positions update via Euler–Maruyama:

$$\begin{array}{c}\hfill a^{\mathbf{x}}(t+\mathrm{\Delta }t)=a^{\mathbf{x}}\left(t\right)+\epsilon \sqrt{\mathrm{\Delta }t}{Z}_t^a,Z_t^a\sim \mathcal{N}(0,1)\mathrm{i}.\mathrm{i}.\mathrm{d}.\end{array}$$

Random movement, combined with displacement from daughter cell division, constitutes the only migration mechanisms for tumor cells. Cells move in continuous space and freely occupy positions off the lattice. We enforce reflective boundaries to prevent tumor agents from leaving the domain.

(iii)

Cell sensing

At each ABM time step, tumor cells sense local chemical fields:

$$\begin{array}{c}\hfill a^o(t+\mathrm{\Delta }t)=o(a^{\mathbf{x}}\left(t\right),t),a^d(t+\mathrm{\Delta }t)=a^d\left(t\right)+d(a^{\mathbf{x}}\left(t\right),t)\mathrm{\Delta }t.\end{array}$$

The drug level $a^d$ represents cumulative exposure rather than instantaneous intracellular drug concentration. It also implies that the drug enters and remains in the cell linearly. We omit explicit intracellular efflux and degradation, which contribute to drug resistance in many cancers [43]. This death threshold abstraction implicitly encompasses these resistance-related processes, and we left the incorporation of these mechanisms for future work.

(iv)

Damage accumulation and death criteria

DNA damage evolves through the following:

$$\begin{array}{c}\hfill a^{dam}(t+\mathrm{\Delta }t)=a^{dam}\left(t\right)+\left[d\left(a^{\mathbf{x}}\left(t\right),t\right)-p_r{a}^{dam}\left(t\right)\right]\mathrm{\Delta }t.\end{array}$$

Here, $p_r$ denotes the DNA repair rate. Cell death occurs when $a^{dam}\left(t\right)>a^{death}\left(t\right)$, and death occurs immediately at the current time step. The death threshold follows:

$$\begin{array}{c}\hfill a_R^{death}=T{h}_{\mathrm{multi}}·a_S^{death}.\end{array}$$

Here, $T{h}_{\mathrm{multi}}$ is the multiplicative resistance factor that scales the death threshold in resistant clones. We set baseline death threshold value $a_S^{death}=0.5$ in nondimensional units. In the preexisting resistance scenario, we fix $T{h}_{\mathrm{multi}}=5$. In the mutation-acquired resistance scenario, $T{h}_{\mathrm{multi}}$ evolves according to the neutral mutation rule and is updated by a random multiplier $r\sim \mathrm{Uniform}[0.7,1.7]$. Since we impose phenotypic constraint in Equation (12), $T{h}_{\mathrm{multi}}\in [0.5,4]$.

(v)

Division rules

Cells are classified by the oxygen level $a^o\left(t\right)$. Cells are normoxic ($a\in {\mathrm{\Lambda }}_t^n$) if the local nondimensional oxygen concentration satisfies $a^o\left(t\right)>o_{\mathrm{hyp}}$. Cells are hypoxic ($a\in {\mathrm{\Lambda }}_t^h$) if $o_{\mathrm{apop}}<a^o\left(t\right)\le o_{\mathrm{hyp}}$, and apoptotic (removed immediately from ${\mathrm{\Lambda }}_t$) if $a^o\left(t\right)\le o_{\mathrm{apop}}$ during the current time step. Here, $o_{\mathrm{apop}}<o_{\mathrm{hyp}}$ are the critical thresholds. In our nondimensionalization, we set $o_{max}=1$ as the reference oxygen concentration. The hypoxia and apoptosis thresholds are set to $o_{\mathrm{hyp}}=0.25$ and $o_{\mathrm{apop}}=0.05$, respectively (see Table 3).

Cells age only when normoxic:

$$\begin{array}{c}\hfill a^{age}(t+\mathrm{\Delta }t)=\left\{\begin{array}{cc}{a}^{age}\left(t\right)+\mathrm{\Delta }t\hfill & \mathrm{if}{a}^o\left(t\right)>o_{\mathrm{hyp}},\hfill \\ a^{age}\left(t\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Normoxic cells divide when $a^{age}\left(t\right)\ge a^{mat}\left(t\right)$, where maturation time depends on division rate ${\alpha }_n$:

$$\begin{array}{c}\hfill a^{mat}\left(t\right)={\displaystyle \frac{log\left(2\right)}{{\alpha }_n}}.\end{array}$$

Local density is

$$\begin{array}{c}\hfill F(x,t)=\sum _{\tilde{a}\in {\mathrm{\Lambda }}_t}{\chi }_{B_{R_c}}\left(x-{\tilde{a}}^{\mathbf{x}}\left(t\right)\right),\end{array}$$

which counts the number of cells within radius $R_c$ of a given cell position $a^{\mathbf{x}}$, and it regulates crowding during proliferation. If $F(x,t)$ at a cell’s location exceeds the threshold $F_{max}$, the cell is considered overcrowded and cannot divide at that time step. Instead, it retains its current age and retries in the next time step.

In two dimensions, when we model cells as disks that interact via repulsive forces such as the Lennard–Jones potential, they spontaneously form close-packed hexagonal arrangements [19]. In this ideal packing, each cell has roughly six immediate neighbors, suggesting a natural cutoff of $F_{max}=6$. Tumor tissue, however, is typically more densely packed than the hexagonal arrangement. To reflect this biological observation while still preventing unrealistic overlap, we define

$$\begin{array}{c}\hfill F_{max}=10.\end{array}$$

This value sets the maximum allowable local occupancy. In other words, $F_{max}$ defines the crowding threshold, specifying the largest number of neighboring cells within radius $R_c$ that still permits cell division.

When $F(a^{\mathbf{x}}\left(t\right),t)\le F_{max}$, the mother divides into two daughters, $a_1$ and $a_2$. Their positions are

$$\begin{array}{cc}\hfill a_1^{\mathbf{x}}\left(t\right)& =a^{\mathbf{x}}\left(t\right),\hfill \\ \hfill a_2^{\mathbf{x}}\left(t\right)& =a^{\mathbf{x}}\left(t\right)+0.1(cos\left(2\pi \theta \right)\mathrm{\Delta }x,sin\left(2\pi \theta \right)\mathrm{\Delta }y),\hfill \end{array}$$

(11)

where $\theta \sim \mathrm{Uniform}[0,1]$ and $\mathrm{\Delta }x,\mathrm{\Delta }y$ are spatial discretizations. The small displacement $0.1\mathrm{\Delta }x$ prevents perfect overlap while remaining negligible relative to the cell diameter, effectively positioning the two daughter cells nearly on top of each other. A slightly larger displacement or random perturbation could better mimic mechanical pushing, but overlap is negligible because the $F_{max}$ check is in place. If $a_2^{\mathbf{x}}\left(t\right)$ overlaps with another existing cell center ${\tilde{a}}^{\mathbf{x}}\left(t\right)$, $\theta$ is resampled until a valid non-overlapping position is found.

(vi)

Inheritance

Upon division, daughter cells $a_1$ and $a_2$ inherit half of the mother cell a’s damage and drug load:

$$\begin{array}{c}\hfill a_i^{dam}\left(t\right)={\displaystyle \frac{1}{2}}{a}^{dam}\left(t\right),a_j^d\left(t\right)={\displaystyle \frac{1}{2}}{a}^d\left(t\right),i,j\in \{1,2\},\end{array}$$

and reset their age to 0. The local oxygen concentration at its position determines the oxygen level of each daughter cell. Daughter cells inherit their mother’s death threshold, proliferation rate, and oxygen consumption rate, with the possibility of mutation in these traits. This mutational mechanism enables initially identical cells to evolve heterogeneous phenotypes over time.

We initialize the zeroth-generation cells with the state vector:

$$\begin{array}{c}\hfill a=\left\{\left(k\right),a^{\mathbf{x}}\left(t\right),o\left(a^{\mathbf{x}}\left(t\right),t\right),0,0,T_k,N_k,M_k\right\}.\end{array}$$

We initialize the cell cycle duration as

$$\begin{array}{c}\hfill M_k\sim \mathrm{Uniform}[3.24\times {10}^4,3.96\times {10}^4]\mathrm{s}.\end{array}$$

Converting to nondimensional units with $\tau =5.76\times {10}^4$ s yields ${\wp }_{age}=M_k\in [0.56,0.69]$. For consistency with Table 3, we use ${\wp }_{age}$ to denote the distribution of cell-cycle durations. To represent biologically realistic heterogeneity in cell cycle progression, we sample initial cell ages $N_k$ from the uniform distribution $N_k\sim \mathrm{Uniform}[0,M_k]$. At last, $T_k$ is the death threshold. We consider two resistance scenarios.

(i)

Preexisting resistance. At initialization, $1\%$ of cells are specified as resistant, with $T_k=a_R^{\mathrm{death}}=2.5$, while the remaining $99\%$ are sensitive, with $T_k=a_S^{\mathrm{death}}=0.5$.

(ii)

Spontaneous mutation. All cells are initialized as sensitive, with $T_k=a_S^{\mathrm{death}}=0.5$.

(vii)

Mutation

Existing mutation models include [38]: (i) random mutation, in which one of the predefined $N>1$ phenotypes is selected with equal probability $p={\displaystyle \frac{1}{N}}$ during mutation, and (ii) linear mutation, where phenotypes evolve deterministically along a predefined path of increasing resistance and aggressiveness.

Although linear mutation avoids abrupt phenotypic jumps, it enforces a deterministic progression toward aggressive phenotypes. It also disregards microenvironmental selection pressures. To address these limitations, we introduce a neutral non-directional mutation algorithm that prevents abrupt trait shifts and enables unbiased phenotypic evolution. Mutations follow a Poisson process with intensity $\mu >0$ per cell per time step:

$$\begin{array}{c}\hfill \mathbb{P}\left(\mathrm{mutation}\mathrm{in}[t,t+\mathrm{\Delta }t]\right)=1-e^{-\mu \mathrm{\Delta }t}\approx \mu \mathrm{\Delta }t\mathrm{for}\mu \mathrm{\Delta }t\ll 1\end{array}$$

On a mutation event, we draw independent multipliers $r_i\sim \mathrm{Uniform}[0.7,1.7]$ for each mutable trait $x_i$ (e.g., ${\alpha }_n$, $a^{death}$, oxygen consumption), not a single multiplier applied to all traits. The new trait value is

$$\begin{array}{c}\hfill x_{i,\mathrm{new}}=r_i\times x_{i,\mathrm{current}}.\end{array}$$

We enforce biologically plausible bounds immediately to ensure traits remain within the specified phenotype envelope:

$$\begin{array}{c}\hfill 0.5x_{i,\mathrm{baseline}}\le x_{i,\mathrm{new}}\le 4x_{i,\mathrm{baseline}}.\end{array}$$

(12)

This design: (i) prevents a single mutation from producing unrealistically extreme trait values, (ii) allows independent evolution of different traits, and (iii) preserves the neutral (non-directional) character of the scheme.

The biological rationale and caveats for the chosen multiplier range $r_i\in [0.7,1.7]$ are that it represents moderate per-event effect sizes. A single mutation can increase a trait by up to $70\%$ or reduce it by up to $30\%$. For example, under a single mutation, the probability that a trait increases by at least $50\%$ is

$$\begin{array}{c}\hfill \mathbb{P}(r_i\ge 1.5)={\displaystyle \frac{1.7-1.5}{1.7-0.7}}=0.2,\end{array}$$

and the probability it increases by at least $40\%$ is 0.30.

Because the scheme is multiplicative, larger phenotype changes (e.g., a $3\times$ increase in the death threshold) require several successive mutations. Using the mean multiplier $\mathbb{E}\left(r_i\right)=1.2$, the expected number of mutation events n such that $1.2^n\ge 3$ is approximately $n\approx 6$. These simple calculations demonstrate the following: (i) single mutations are moderate in size, and (ii) high-level resistance accumulates over multiple events, consistent with gradual phenotype evolution rather than abrupt jumps.

Classical mutation models explicitly tie resistance acquisition to cell division events [38,40]. In contrast, we implement mutation as a time-continuous Poisson process with intensity $\mu$, independent of cell division. At each event, every mutable trait (including $T{h}_{\mathrm{multi}}$) is independently rescaled by $r_i\sim \mathrm{Uniform}[0.7, 1.7]$, with immediate enforcement of the biologically plausible bounds $0.5\le T{h}_{\mathrm{multi}}\le 4$.

The independence assumption reflects biological evidence that mutations can arise from internal cell mechanisms and environmental stresses, rather than being strictly division-dependent [44]. The choice of a Poisson process models the memoryless property of mutation events, decoupling mutation timing in a way that is both analytically tractable and reproducible, regardless of a cell’s division history.

For a comprehensive introduction to stochastic mutation processes, we refer the reader to Ewens [45]. Future iterations of the model could refine this approach by incorporating environment-dependent mutation rates or kinetic signaling modules.

Remark 1.

(Limitation: Neutral Mutation Assumption). In our current framework, resistance mutations are implemented as neutral, meaning they impose a direct fitness cost. This abstraction isolates the effects of microenvironmental heterogeneity without confounding selective pressures. Biologically, however, the acquisition of resistance can impose metabolic burdens or reduce proliferative capacity. Evidence from microbial systems demonstrates that antibiotic resistance acquisition is often associated with a fitness cost for resistant organisms [46]. In cancers, the presence of drug-tolerant persisters (DTPs) (i.e., cells with intrinsically slow growth kinetics that can survive initial drug exposure) further supports the notion that resistance may carry a fitness penalty [47]. Such costs can reshape clonal competition and potentially delay or suppress the expansion of resistant cells. Future extensions should therefore relax the neutrality assumption and incorporate non-neutral mutations. One possible strategy is to link the resistance threshold $a^{death}$ to a reduced proliferation rate ${\alpha }_n$. This modification would enable systematic exploration of trade-offs between therapeutic survival and baseline fitness, thereby aligning the model more closely with experimental observations of resistance evolution.

We implement the tumor cell rules described above within the hybrid solver. To ensure clarity and reproducibility, we summarize their integration with the PDE fields in the global time-stepping flowchart (Figure 4). The computational update steps appear explicitly in the PDE-ABM solver (Algorithm A1) and the ABM update operator (Algorithm A2) in Appendix D.

2.2.2. ABM Angiogenesis Rules

We model endothelial tip cells and vessel cells as discrete agents, denoted $b\in T_t$ and $v\in V_t$, respectively. Chemotaxis, branching, and anastomosis drive angiogenesis. The set $T_t$ contains all tip cells at time t.

(i)

Agent trait

Each endothelial tip cell $b\in T_t$ tracks its spatial position $b^{\mathbf{x}}\left(t\right)$ (unit: m), age $b^{age}$ (unit: s) since last branching, and a lineage identifier ${\mathrm{id}}_b$ (dimensionlessm) that records ancestry:

$$\begin{array}{c}\hfill b=\{{\mathrm{id}}_b,b^{\mathbf{x}}\left(t\right),b^{age}\left(t\right)\}.\end{array}$$

This lineage scheme parallels the tumor-cell identifiers and ensures consistent reconstruction of vascular branching trees. Full definitions and update rules are provided in Appendix C.

The tip cell age evolves as follows:

$$\begin{array}{c}\hfill b^{age}(t+\mathrm{\Delta }t)=b^{age}\left(t\right)+\mathrm{\Delta }t.\end{array}$$

Tip migration uses a lattice of Von Neumann neighbors according to chemotaxis-diffusion dynamics. Directional movement probabilities $P_0$ (stationary), $P_1$ (left), $P_2$ (right), $P_3$ (down), and $P_4$ (up) are derived from the TAF gradient (Equation (17)). The movement probabilities link discrete tip motion to the continuum TAF field; Section 3.2 gives the explicit derivations and numerical implementation.

(ii)

Anastomosis

Anastomosis occurs when a migrating tip enters a lattice site occupied by another tip or vessel agent in the Von Neumann neighborhood. Upon anastomosis, the invading tip ceases migration and branching, converting into a vessel segment and contributing to a closed-loop vascular network. This rule not only yields dynamically evolving vascular structures with loops, consistent with physiological neovascularization, but also prevents tip collisions, ensuring biologically realistic vascular connectivity. Classical angiogenesis models have incorporated similar angiogenesis formulations [25].

For a schematic illustration of the branching of the tip and the formation of vascular loops through anastomosis, see Figure 9 in [25]. Figure 5 shows the anastomosis process. Here, a migrating tip enters an already occupied site. At this point, it stops further migration and branching, converting into a vessel segment. Therefore, this event prevents tip collisions and stabilizes the emerging vascular architecture. It can even form a closed loop in the network.

(iii)

Branching

We model endothelial tip branching as a Poisson process, with the following intensity:

$$\begin{array}{c}\hfill {\lambda }_{br}(b,t)=c_{br}{\displaystyle \frac{c(b^{\mathbf{x}}\left(t\right),t)}{{\parallel c(·,t)\parallel }_{\infty }}}H(b^{age}\left(t\right)-\psi ).\end{array}$$

Here, $c_{br}$ denotes the baseline branching rate (nondimensional value, corresponding to a dimensional unit of $1/\mathrm{s}$). The term $c(b^{\mathbf{x}}\left(t\right),t)$ represents the local TAF concentration at the tip position $b^{\mathbf{x}}$. Finally, ${\parallel c(·,t)\parallel }_{\infty }$ denotes the domain-wide maximum TAF concentration at time t. The Heaviside function H enforces a minimum branching age: no branching occurs until $b^{age}\left(t\right)\ge \psi$, with $\psi =1.125$ (nondimensional time units).

Normalizing $c(b^{\mathbf{x}},t)$ by ${\parallel c(·,t)\parallel }_{\infty }$ bounds the branching intensity by $c_{br}$. This normalization provides a convenient scaling of the stochastic intensity. However, it introduces a nonlocal dependence: each tip implicitly senses the maximum TAF concentration across the tissue domain. Biologically, this global sensing assumption may not be mechanistically realistic, since endothelial tip cells are known to respond to local gradients of VEGF/TAF rather than global maxima. Nevertheless, this choice stabilizes the numerical range of ${\lambda }_{br}$.

The nonlocality of ${\parallel c(·,t)\parallel }_{\infty }$ represents a modeling simplification. In more physiologically grounded descriptions, we can express the branching intensity as a saturating function of the local TAF concentration. For example, the Michaelis–Menten type is

$$\begin{array}{c}\hfill {\lambda }_{br}(b,t)=c_{br}{\displaystyle \frac{c(b^{\mathbf{x}}\left(t\right),t)}{K_c+c(b^{\mathbf{x}}\left(t\right),t)}},\end{array}$$

where $K_c$ is the half-saturation concentration. Incorporating such nonlinear local response functions is a promising extension for future refinements of the model.

A branching event occurs if $b^{age}>\psi$, at least one Moore neighbor is vacant, and

$$\begin{array}{c}\hfill \mathrm{Uniform}[0,1]<1-e^{-{\lambda }_{br}(b,t)\mathrm{\Delta }t}\approx {\lambda }_{br}(b,t)\mathrm{\Delta }t.\end{array}$$

Upon branching, one daughter cell remains at the original location, while the other occupies a randomly selected vacant site in the Moore neighborhood, resetting their age to 0. Tip motions are restricted to the four orthogonal directions (Von Neumann neighborhood), whereas branching may place a daughter on a diagonal (Moore) site if vacant. Figure 6 illustrates the branching mechanism. When the stochastic branching criterion holds, the parent tip produces two daughter tips. One remains at the original site, while the other occupies a randomly chosen vacant Moore neighbor, which may include diagonal sites. This rule allows the expansion of vascular network, increasing the likelihood of vessel interconnections.

(iv)

Proliferation

Endothelial tip cell proliferation follows a fixed doubling time ${\tau }_{\mathrm{tip}}=1.125$ ($6.48\times {10}^4$ s), consistent with classical hybrid angiogenesis models [25]. Each division elongates the sprout by one cell length. During division, one daughter tip continues migration along the sprout direction, while the other becomes a new stalk segment at the original tip location.

Both tip and stalk (non-tip) endothelial cell proliferation contribute to sprout elongation, but experimental and computational evidence indicate that stalk-cell proliferation is mechanically induced by tension generated at the advancing tip. As noted by Santos-Oliveira et al. [48], “the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts.” Thus, tip-driven stalk proliferation results in elongation by approximately one cell length per cycle, functionally equivalent to the elongation produced by tip proliferation. To simplify the model while preserving this biological effect, we therefore represent proliferation solely through tip cell proliferation, which serves as an effective proxy for the combined contributions of both tip and stalk endothelial cells. This formulation is consistent with classical angiogenesis models, e.g., [25] and Figure 16 therein.

We implement the tip proliferation rule as follows:

$$\begin{array}{c}\hfill \mathrm{Tip}\mathrm{cell}\mathrm{at}\mathrm{position}{\mathit{x}}_{\mathrm{tip}}\underset{\mathrm{division}\mathrm{at}t+{\tau }_{\mathrm{tip}}}{\to }\left\{\begin{array}{c}\mathrm{Daughter}\mathrm{tip}\mathrm{at}{\mathit{x}}_{\mathrm{tip}}+\mathrm{\Delta }\mathbf{x},\hfill \\ \mathrm{Stalk}\mathrm{segment}\mathrm{at}{\mathit{x}}_{\mathrm{tip}}.\hfill \end{array}\right.\end{array}$$

Here, $\mathrm{\Delta }\mathit{x}$ represents the tip migration along the sprout direction. Each tip division adds one new stalk segment behind the migrating tip without generating an additional tip cell at the same location, avoiding redundancy with the branching rule. Figure 7 illustrates the tip proliferation schematically. The red and green nodes represent the migrating tip and the newly created stationary stalk segment, respectively, and the arrow indicates the tip movement along the sprout. At each fixed doubling time, the active tip divides asymmetrically: one daughter advances the sprout by migrating forward, while the other differentiates into a stalk cell at the former tip location. This mechanism ensures that sprout elongation is driven solely by tip proliferation while maintaining exactly one active tip per sprout.

Our current model does not include vessel pruning or regression. Once a site becomes a vessel, it remains one permanently, and regression cannot occur. Such modeling extensions incorporating vessel pruning and regression remain for future work.

(v)

Angiogenic network

The motion of an individual endothelial cell at the capillary sprout tip governs the entire sprout’s movement because the remaining endothelial cells lining the sprout wall are contiguous [25]. Thus, the cumulative paths of tip cells define the angiogenic network:

$$\begin{array}{c}\hfill A_t=\bigcup _{b\in T_t}\left\{b^{\mathbf{x}}\left(s\right):0\le s\le t\right\}.\end{array}$$

Each lattice site intersecting $A_t$ becomes a vessel agent $v=\left\{v^{\mathbf{x}}\left(t\right)\right\}\in V_t$, represented geometrically as a circle of radius $R_c$ inscribed within the site grid. We encode this geometry by the normalized indicator function ${\varphi }_v$ for $v\in V_t$. Vessel agents serve as sources of oxygen and drug delivery in the PDEs, thereby coupling the continuum microenvironmental fields with the ABM.

The present formulation omits explicit mechanical interactions (e.g., Lennard–Jones-type potentials) between endothelial and tumor cells. As a result, cells may overlap (up to $F_{max}$). This simplification follows many prior hybrid PDE-ABM models, which also neglect cell–cell forces for tractability [25,38,40]. For instance, Spill et al. [49] incorporated only random motion and chemotaxis as tip cell motility mechanisms, excluding mechanical interactions.

While this choice isolates the role of chemotaxis and proliferation in angiogenic dynamics, it does not prevent cell overlap. Incorporating simple repulsive-force interactions could enhance biological realism in future work. Established force-based models offer a natural way to integrate such effects [18,19,50]. Their exclusion here allows us to focus specifically on tumor evolution under microenvironmental regulation.

For clarity and reproducibility, the pseudocode in Algorithm A2 summarizes the ABM module per time step. The global flowchart (Figure 4) illustrates its coupling with the PDE fields, including sampling of local concentrations and updating agent-driven sources, and Algorithm A1 details the computational steps of the hybrid PDE-ABM framework.

We present broader limitations and suggested future work in the Discussion. As a prelude to the numerical implementation, we now provide a consolidated summary of the parameter values employed. This bridge is important: the selection and calibration of dimensional quantities directly constrain the behavior of both PDE fields and agent-based dynamics, and hence determine the realism and reproducibility of the simulations.

3. Numerical Implementation

3.1. Parameterization and Nondimensionalization

To establish biologically relevant scales while ensuring numerical stability, we nondimensionalize the model using characteristic parameters:

$$\begin{array}{c}\hfill \tilde{n}={\displaystyle \frac{n}{n_0}},\tilde{c}={\displaystyle \frac{c}{c_0}},\tilde{t}={\displaystyle \frac{t}{\tau }}.\end{array}$$

We choose the length scale $L=5\times {10}^{-3}$ m and set time scale $\tau =L^2/D=5.76\times {10}^4$ s, where D denotes a representative diffusion rate. We set the reference cell density to $n_0=6.4\times {10}^{13}\mathrm{cell}/{\mathrm{m}}^3$, representative of a densely packed tissue rather than a sparse culture. Reported tumor cell diameters vary between 10 and $100\mu \mathrm{m}$ depending on tumor type [38]. Here, we adopt $25\mu \mathrm{m}$ as a representative diameter such that $2R_c=\mathrm{\Delta }x$, i.e., each lattice grid cell accommodates exactly one tumor cell. The reference concentration is set to $c_0=1.0\times {10}^{-7}\mathrm{mol}/{\mathrm{m}}^3$, following Anderson et al. [25]. This value corresponds to a typical TAF concentration scale used in prior experimental studies to test the endothelial chemotactic coefficient ${\chi }_0$. See Appendix E for the detailed parameter estimation.

It is important to note that $n_0$ also serves as the characteristic scale for the agent-based source and sink terms $\sum _{a\in {\mathrm{\Lambda }}_t}{\varphi }_a$ and $\sum _{v\in V_t}{\varphi }_v$. Since each normalized indicator function ${\varphi }_a$ and ${\varphi }_v$ has units of $\mathrm{cell}/{\mathrm{m}}^3$, these sums naturally represent local cell densities (In the three-dimensional case, the normalized indicator functions ${\varphi }_a,{\varphi }_v$ have units of $\mathrm{cell}/{\mathrm{m}}^3$, interpreted as per-volume cell counts (i.e., local cell density). Because most experimental and modeling parameter values reported in the literature are in volumetric (3D) units, we adopt a three-dimensional reference frame for nondimensionalization.). Therefore, $n_0$ provides the appropriate volumetric reference scale for converting per cell production or uptake coefficients ($\eta ,\lambda ,{\rho }_d,S_d,{\rho }_o,S_o$) into terms with correct physical dimensions. This interpretation ensures consistency between the literature-reported per cell rates (which we list directly in Table 3) and our nondimensionalization procedure. We refer readers for the details of the relevant unit conversions to Appendix E.

The nondimensionalization in Equation (5) defines dimensionless parameters as ratios relative to the chosen characteristic scales. For example, ${\tilde{D}}_n$ and ${\tilde{D}}_c$ denote diffusion coefficients normalized by the reference diffusivity D, ${\tilde{\xi }}_c$ and ${\tilde{\xi }}_o$ represent decay rates scaled by the characteristic time $\tau$, and $\tilde{\eta }$, $\tilde{\lambda }$, ${\tilde{\rho }}_d$, and ${\tilde{\rho }}_o$ correspond to per cell production or uptake rates rescaled by the reference cell density $n_0$. The chemotactic sensitivity is characterized by ${\tilde{\chi }}_0$, and $\alpha =c_0/k_1$ quantifies the ratio between the reference TAF concentration and the receptor half-saturation constant. Collectively, these dimensionless quantities measure the relative strength of competing processes on biologically relevant spatiotemporal scales. The resulting dimensionless quantities are summarized in Table 3, which provides the authoritative parameter set used in all simulations. These values are also available in machine-readable format (params.json, params.csv) in the GitHub Repository in the Data Availability Statement Section. Under this scaling, a dimensionless time unit of $t=0.5$ corresponds to $2.88\times {10}^4$ s, consistent with experimental tumor growth measurements [41,42].

With the nondimensional model and parameter set fixed, we now describe the numerical methods used to integrate the PDEs and update the agent rules.

3.2. Numerical Implementation

Building upon the mathematical framework in Section 2.1 and Section 2.2, we now describe the computational methodology used to simulate the coupled tumor–vascular system. The focus is on discretization schemes that ensure numerical stability and accurate coupling between continuum fields and agent-based dynamics.

We employ a hybrid numerical strategy. The reaction–diffusion equations for drug, oxygen, and TAF (Equations (2)–(4)) are solved using the ADI method, which is unconditionally stable for linear diffusion and permits larger time steps than explicit schemes. The endothelial density equation (Equation (1)), which contains a nonlinear chemotactic flux, is advanced by a forward Euler method subject to the Courant–Friedrichs–Lewy (CFL) condition [51]:

$$\begin{array}{c}\hfill \mathrm{\Delta }t\le min\left\{{\displaystyle \frac{\mathrm{\Delta }{x}^2}{4D_n}},{\displaystyle \frac{\mathrm{\Delta }x}{{2\parallel \chi \left(c\right)\nabla c\parallel }_{\infty }}}\right\},\end{array}$$

(13)

where ${\parallel ·\parallel }_{\infty }$ denotes the $L^{\infty }\left(U\right)$-norm. This restriction guarantees stability and prevents spurious oscillations.

The computational domain is discretized on a uniform Cartesian grid with $(N_x+1)\times (N_y+1)=100\times 100$ nodes, spanning a $2.5\times {10}^{-5}{\mathrm{m}}^2$ tissue region with mesh spacing $\mathrm{\Delta }x=\mathrm{\Delta }y=5.0\times {10}^{-5}\mathrm{m}$. To ensure compatibility between the ADI solver and the explicit chemotaxis scheme, the time step $\mathrm{\Delta }t$ is chosen to satisfy the CFL condition (13).

Coupling between PDEs and agents is achieved through a global time-stepping loop. Within each global iteration, PDEs are advanced with step $\mathrm{\Delta }t$ for m substeps until reaching the agent-based update interval $\mathrm{\Delta }{t}^{\prime }=m\mathrm{\Delta }t$. At this point, agents sample local PDE fields and update motility, proliferation, mutation, branching, and anastomosis. Source and sink terms are then passed back to the PDEs, closing the coupling cycle. This multiscale update procedure reflects the faster evolution of biochemical fields relative to discrete cellular events, while maintaining accuracy and consistency across scales.

We discretize the endothelial cell equation using central differences in space and forward Euler in time:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial n}{\partial t}}=D_n\mathrm{\Delta }n-\nabla ·\left(\chi \left(c\right)n\nabla c\right),\end{array}$$

using central differences:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial n}{\partial t}}& \approx {\displaystyle \frac{n_{i,j}^{k+1}-n_{i,j}^k}{\mathrm{\Delta }t}},\hfill \\ \hfill D_n\mathrm{\Delta }n& \approx D_n{\displaystyle \frac{n_{i-1,j}^k+n_{i+1,j}^k+n_{i,j-1}^k+n_{i,j+1}^k-4n_{i,j}^k}{\mathrm{\Delta }{x}^2}}.\hfill \end{array}$$

The nonlinear chemotaxis term $-\nabla ·\left(\chi \right(c)n\nabla c)$ is discretized using a probabilistic finite difference scheme inspired by hybrid discrete–continuum models [25,40,52]. In this approach, the directional bias of cell motion is encoded as movement probabilities within a von Neumann neighborhood, and the divergence is then approximated by finite differences through fluxes at half indices:

$$\begin{array}{cc}\hfill \nabla ·(\chi n\nabla c)& \approx {\displaystyle \frac{F_{i+\frac{1}{2},j}-F_{i-\frac{1}{2},j}}{\mathrm{\Delta }x}}+{\displaystyle \frac{G_{i,j+\frac{1}{2}}-G_{i,j-\frac{1}{2}}}{\mathrm{\Delta }x}},\hfill \end{array}$$

with

$$\begin{array}{cc}\hfill F_{i+{\displaystyle \frac{1}{2}},j}:=& \chi (c_{i+{\displaystyle \frac{1}{2}},j})n_{i+{\displaystyle \frac{1}{2}},j}{\displaystyle \frac{c_{i+1,j}-c_{i,j}}{\mathrm{\Delta }x}}\hfill \\ \hfill F_{i-{\displaystyle \frac{1}{2}},j}:=& \chi (c_{i-{\displaystyle \frac{1}{2}},j})n_{i-{\displaystyle \frac{1}{2}},j}{\displaystyle \frac{c_{i,j}-c_{i-1,j}}{\mathrm{\Delta }x}}\hfill \\ \hfill G_{i,j+{\displaystyle \frac{1}{2}}}:=& \chi (c_{i,j+{\displaystyle \frac{1}{2}}})n_{i,j+{\displaystyle \frac{1}{2}}}{\displaystyle \frac{c_{i,j+1}-c_{i,j}}{\mathrm{\Delta }x}}\hfill \\ \hfill G_{i,j-{\displaystyle \frac{1}{2}}}:=& \chi (c_{i,j-{\displaystyle \frac{1}{2}}})n_{i,j-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{c_{i,j}-c_{i,j-1}}{\mathrm{\Delta }x}}\hfill \end{array}$$

with half-index values approximated by linear interpolation, for example:

$$\begin{array}{c}\hfill \chi (c_{i+{\displaystyle \frac{1}{2}},j})={\displaystyle \frac{\chi \left(c_{i+1,j}\right)+\chi \left(c_{i,j}\right)}{2}},n_{i+{\displaystyle \frac{1}{2}},j}={\displaystyle \frac{n_{i+1,j}+n_{i,j}}{2}}.\end{array}$$

The same approach applies to calculating the values of $\chi (c_{i-{\displaystyle \frac{1}{2}},j}),\chi (c_{i,j+{\displaystyle \frac{1}{2}}}),\chi (c_{i,j-{\displaystyle \frac{1}{2}}})$ and $n_{i-{\displaystyle \frac{1}{2}},j},n_{i,j+{\displaystyle \frac{1}{2}}},n_{i,j-{\displaystyle \frac{1}{2}}}$ in the remaining directions. Substituting the flux expressions yields the discrete update

$$\begin{array}{c}\hfill n_{i,j}^{k+1}=n_{i,j}^k+\mathrm{\Delta }t\left({\displaystyle \frac{D_n}{\mathrm{\Delta }{x}^2}}{\delta }^2{n}_{i,j}^k-{\displaystyle \frac{1}{\mathrm{\Delta }x}}(\delta F_{i,j}+\delta G_{i,j})\right)\end{array}$$

(14)

Prior analysis by Wang et al. [51] establishes consistency, stability, convergence, nonnegativity, and mass conservation for this explicit chemotaxis discretization, which we summarize below.

Theorem 1.

(i) 

Consistency. The local truncation errors of the endothelial chemotaxis and ADI schemes satisfy

$$\begin{array}{c}\hfill {\tau }_{\mathrm{endo}}=\mathcal{O}(\mathrm{\Delta }t+h^2),{\tau }_{\mathrm{ADI}}=\mathcal{O}(\mathrm{\Delta }{t}^2+h^2),\mathrm{for}\mathrm{\Delta }x=\mathrm{\Delta }y=h.\end{array}$$

(ii) 

Stability. The ADI scheme is unconditionally stable for pure linear diffusion. The finite difference scheme for endothelial chemotaxis is conditionally stable under the CFL condition (Equation (13)). Moreover, subject to the additional constraints

$$\begin{array}{cc}\hfill \mathrm{\Delta }x& \le 2{\displaystyle \frac{D_n}{{\parallel \chi \left(c\right)\nabla c\parallel }_{\infty }}},\mathrm{\Delta }t\le {\displaystyle \frac{1}{4{\displaystyle \frac{D_n}{\mathrm{\Delta }{x}^2}}+2{\displaystyle \frac{{\parallel \chi \left(c\right)\nabla c\parallel }_{\infty }}{\mathrm{\Delta }x}}}},\hfill \end{array}$$

(15)

all motility probabilities $P_0,P_1,P_2,P_3,P_4$ remain nonnegative.

(iii) 

Convergence. Let $n(x,y,t)$ be the exact solution of the endothelial chemotaxis equation, and let $n_{i,j}^k$ be its numerical approximation. Under the CFL constraints (Equations (13) and (15)), and assuming sufficient regularity of n and c, the scheme satisfies

$$\begin{array}{c}\hfill \parallel n_{i,j}^k-n(x_i,y_j,t_k)\parallel =\mathcal{O}(\mathrm{\Delta }t+\mathrm{\Delta }{x}^2).\end{array}$$

(iv) 

Nonnegativity. With nonnegative initial data and the CFL conditions (Equations (13) and (15)), the finite difference update preserves nonnegativity of $n_{i,j}^k$ at each time step.

(v) 

Mass Conservation. With one-sided Neumann boundary approximation, the scheme conserves the total mass:

$$\begin{array}{c}\hfill \sum _{i,j}{n}_{i,j}^{k+1}=\sum _{i,j}{n}_{i,j}^k.\end{array}$$

To interpret the update probabilistically, we recast the finite difference scheme as a weighted sum of von Neumann neighborhood contributions:

$$\begin{array}{c}\hfill n_{i,j}^{k+1}=n_{i,j}^k{P}_0+n_{i+1,j}^k{P}_1+n_{i-1,j}^k{P}_2+n_{i,j+1}^k{P}_3+n_{i,j-1}^k{P}_4,\end{array}$$

(16)

where $P_0$ is the probability of remaining stationary and $P_1$–$P_4$ correspond to movement into the four von Neumann neighbors (Figure 8a).

We derive the movement probabilities as

$$\left\{\begin{array}{cc}{P}_0=1-4{\displaystyle \frac{D_n\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}}\hfill & \hfill -{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i+1,j}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i+1,j}-c_{i,j})\\ & \hfill +{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i-1,j}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i,j}-c_{i-1,j})\\ & \hfill -{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i,j+1}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i,j+1}-c_{i,j})\\ & \hfill +{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i,j-1}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i,j}-c_{i,j-1})\\ P_1={\displaystyle \frac{D_n\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}}\hfill & \hfill -{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i+1,j}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i+1,j}-c_{i,j})\\ P_2={\displaystyle \frac{D_n\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}}\hfill & \hfill +{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i-1,j}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i,j}-c_{i-1,j})\\ P_3={\displaystyle \frac{D_n\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}}\hfill & \hfill -{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i,j+1}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i,j+1}-c_{i,j})\\ P_4={\displaystyle \frac{D_n\mathrm{\Delta }t}{\mathrm{\Delta }{x}^2}}\hfill & \hfill +{\displaystyle \frac{\mathrm{\Delta }t{\chi }_0}{4\mathrm{\Delta }{x}^2}}\left({\displaystyle \frac{1}{1+\alpha c_{i,j-1}}}+{\displaystyle \frac{1}{1+\alpha c_{i,j}}}\right)(c_{i,j}-c_{i,j-1})\end{array}\right.$$

(17)

with chemotactic sensitivity modeled as $\chi \left(c\right)={\chi }_0/(1+\alpha c)$. These coefficients combine symmetric diffusion with chemotactic bias from centered differences, producing net movement up TAF gradients.

Raw weights are clipped to remain nonnegative,

$$\begin{array}{c}\hfill P_i=max\{P_i,0\},\end{array}$$

and then normalized so that they sum to 1:

$$\begin{array}{c}\hfill \sum _i{P}_i=1.\end{array}$$

Under CFL conditions (Equations (13) and (15)), positivity holds automatically and only normalization is needed.

Cell movement is sampled stochastically: a random number $r\sim \mathrm{Uniform}[0,1]$ is compared against cumulative probabilities

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill R_0& =[0,P_0],\hfill \\ \hfill R_j& =\left(\sum _{i=0}^{j-1}{P}_i,\sum _{i=0}^j{P}_i\right],j=1,\dots ,4,\hfill \end{array}\end{array}$$

(18)

and the cell moves in the direction associated with the interval containing r [25,40,52]. This construction ensures conservation of total cell number and a complete probabilistic partition of the unit interval.

Finally, to validate the directional behavior of the chemotactic flux ${\mathbf{J}}_{\mathrm{chemo}}=\chi \left(c\right)n\nabla c$ in Equation (1), we visualized the flux field under a representative tumor-derived TAF profile, $c(x,y)=e^{-0.05({(x-1.5)}^2+{(y-1.5)}^2)}$. As shown in Figure 8b, the flux vectors align with the gradient of $c(x,y)$ and consistently point toward the chemotactic source, confirming the correct implementation of the discretized flux term and directional consistency under tumor-induced TAF gradients.

To solve the diffusion-dominated PDEs for molecular fields (Equations (2)–(4)), we employ the ADI method. At each time step, two tridiagonal systems are solved sequentially: first implicit in x and explicit in y, then reversed. Reaction terms are evaluated explicitly.

For a generic reaction–diffusion equation

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u}{\partial t}}=D\mathrm{\Delta }u+f(u,x,y,t),\end{array}$$

the ADI discretization reads

$$\begin{array}{cc}\hfill \left(1-{\displaystyle \frac{D\mathrm{\Delta }t}{2\mathrm{\Delta }{x}^2}}{\delta }_x^2\right)U^{k+1/2}& =\left(1+{\displaystyle \frac{D\mathrm{\Delta }t}{2\mathrm{\Delta }{y}^2}}{\delta }_y^2\right)U^k+{\displaystyle \frac{\mathrm{\Delta }t}{2}}f\left(U^k\right),\hfill \\ \hfill \left(1-{\displaystyle \frac{D\mathrm{\Delta }t}{2\mathrm{\Delta }{y}^2}}{\delta }_y^2\right)U^{k+1}& =\left(1+{\displaystyle \frac{D\mathrm{\Delta }t}{2\mathrm{\Delta }{x}^2}}{\delta }_x^2\right)U^{k+1/2}+{\displaystyle \frac{\mathrm{\Delta }t}{2}}f\left(U^k\right),\hfill \end{array}$$

(19)

where $U^{k+1/2}\approx u\left((k+1/2)\mathrm{\Delta }t\right)$. Here, ${\delta }_x^2$ and ${\delta }_y^2$ denote second-order central difference operators.

This Douglas ADI scheme [24] is unconditionally stable for pure diffusion and computationally efficient for reaction–diffusion systems. In practice, this stability permits large time steps, but to ensure consistency with the explicit chemotaxis update, we restrict $\mathrm{\Delta }t$ to satisfy the CFL conditions (Equations (13) and (15)). This hybrid implicit–explicit strategy balances stability, accuracy, and efficiency in simulating the coupled dynamics.

We couple PDE and ABM updates in a global time-stepping loop. At each global iteration, we advance PDEs for m substeps of size $\mathrm{\Delta }t$ until reaching the ABM step $\mathrm{\Delta }{t}^{\prime }=m\mathrm{\Delta }t$. Agents then sample local fields, update motility, proliferation, mutation, branching, and anastomosis, and update PDE source and sink terms. The global hybrid solver flowchart (Figure 4), PDE update algorithm (Algorithm A1), and ABM update algorithm (Algorithm A2) contain full implementation details.

4. Mathematical Analysis

Recent advances in stochastic population models have established rigorous frameworks for demographic noise [53]. Inspired by such approaches, we extend stability and bifurcation analysis to the tumor angiogenesis context, where spatial heterogeneity and chemotaxis play key roles.

4.1. Unidirectional Coupling

In our baseline model (Equation (6)–(9)), endothelial cells chemotactically migrate along TAF gradients but do not secrete TAF; only hypoxic tumors act as TAF sources. We refer to this setting as unidirectional coupling: endothelial cells chemotactically migrate along TAF gradients but do not secrete TAF. Biologically, this reflects the canonical pathway in which hypoxia induces VEGF secretion via hypoxia-inducible factor-1$\alpha$ (HIF-1$\alpha$) [10], and vessels mitigate hypoxia by restoring oxygen supply. The resulting negative feedback, where endothelial chemotaxis reduces TAF secretion, suggests a stabilizing effect that suppresses spatial heterogeneity.

To formalize this intuition, we consider the tumor-oxygen-endothelial-TAF system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}m=D_m\mathrm{\Delta }m+{\alpha }_{n}m(1-m/m_{max})H(o-o_{\mathrm{hyp}}),\hfill \\ {\partial }_{t}o=D_o\mathrm{\Delta }o-{\xi }_{o}o-{\rho }_{o}m+S_o(1-o)n,\hfill \\ {\partial }_{t}n=D_n\mathrm{\Delta }n-\nabla ·(\chi \left(c\right)n\nabla c),\hfill \\ {\partial }_{t}c=D_c\mathrm{\Delta }c+\eta mH(o_{\mathrm{hyp}}-o)-{\xi }_{c}c-\lambda cn.\hfill \end{array}\right.\end{array}$$

(20)

where m denotes the continuum tumor density field, o oxygen concentration, n endothelial density, and c TAF concentration. Tumor growth follows logistic kinetics with carrying capacity $m_{max}$ and proliferation rate ${\alpha }_n$. The Heaviside functions $H(o-o_{\mathrm{hyp}})$ and $H(o_{\mathrm{hyp}}-o)$ enforce oxygen-threshold-dependent switching of proliferation and TAF secretion, respectively.

On the other hand, in the simplified formulation Equation (20), we approximate tumor density by a continuous logistic field $m(x,t)$. We couple TAF production to hypoxia through a Heaviside switch $H(o_{\mathrm{hyp}}-o)$, which activates secretion when oxygen concentration falls below the hypoxia threshold $o_{\mathrm{hyp}}$. This binary switch simplifies the analysis and facilitates linearization around steady states. We note, however, that in reality, VEGF secretion responds gradually to decreasing oxygen levels rather than switching abruptly [54]. Thus, the Heaviside approximation is a modeling simplification. While a sharp cutoff suffices for our illustrative bifurcation analysis, future extensions could incorporate gradual hypoxia-VEGF coupling. In simulations the full model (Equation (20)) can use smooth kinetics to better reflect biological realism, and the Heaviside is only for analytical tractability.

Theorem 2 (Pattern Suppression under Unidirectional Coupling).

In the unidirectional tumor-oxygen-endothelial-TAF system (20), where only tumor cells secrete TAF and endothelial cells respond chemotactically but do not produce TAF, no spatial pattern emerges.

Proof (Sketch).

Linearizing around both normoxic ($c_0=0$) and hypoxic ($c_0>0$) equilibria yields dispersion relations with eigenvalues which have negative real parts. In both cases, the Routh–Hurwitz conditions are satisfied, so perturbations decay. Full derivations are given in Appendix F.  □

We note that this analysis assumes the linear saturation law for oxygen supply. In Appendix G, we repeat the argument under a Michaelis–Menten supply law and obtain the same conclusion: both the normoxic and hypoxic steady states remain linearly stable, and all nontrivial perturbations decay. Thus, the suppression of spatial patterns under unidirectional coupling is not an artifact of the linear approximation but persists under the more realistic nonlinear kinetics.

This analysis crucially relies on the assumption of unidirectional coupling. Endothelial cells chemotactically migrate along TAF gradients but do not secrete TAF. This kind of unidirectional chemotaxis model (cells responding to fixed signals without self-production) is known to be stable [55]. In contrast, if endothelial cells were allowed to secrete or amplify TAF (bidirectional coupling), the feedback structure would change. Such bidirectional signaling may give rise to chemotactic instabilities (see Section 4.2 and related discussion). This distinction is essential for interpreting both our results and comparisons with the literature.

4.2. Bidirectional Coupling

The preceding analysis established that unidirectional coupling does not produce Turing instabilities. In contrast, clinical observations of the tumor microenvironment often reveal the coexistence of vascularized and hypoxic regions. To reconcile this discrepancy, we introduce a reduced endothelial-chemoattractant (n-c) subsystem that incorporates bidirectional coupling. For clarity, we define bidirectional coupling as the case in which endothelial cells chemotactically migrate along TAF gradients and secrete TAF simultaneously.

Tumor-associated endothelial cells secrete TAF such as angiopoietin-1 (Ang-1), which acts synergistically with VEGFs to amplify angiogenesis [56,57,58]. In addition, autocrine VEGF production by endothelial cells has been documented in human placental endothelium, where VEGF release strongly correlates with sprouting activity in vitro [59]. Additional evidence of autocrine VEGF production can be found therein. Furthermore, tumor-derived microvesicles carrying activated oncogenic receptors can upregulate endothelial VEGF expression, shifting them toward an autocrine angiogenic phenotype within the tumor microenvironment [60]. Collectively, these findings indicate that endothelial cells are not passive responders to tumor-secreted cues. Instead, they can actively sustain and amplify angiogenesis through both paracrine and autocrine pathways.

To capture this feedback, we extend the endothelial–TAF pair in Equation (20) by allowing endothelial cells to produce chemoattractant at a rate ${\eta }_n\ge 0$. The resulting system is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_{t}n=D_n\mathrm{\Delta }n-\nabla ·(\chi \left(c\right)n\nabla c),\hfill \\ {\partial }_{t}c=D_c\mathrm{\Delta }c+{\eta }_{n}n-{\xi }_{c}c,\hfill \end{array}\right.\end{array}$$

(21)

with homogeneous Neumann boundary conditions. Here, $D_n,D_c>0$ are diffusion coefficients, ${\xi }_c>0$ is the clearance rate of c, and $\chi \left(c\right)={\chi }_0/(1+\alpha c)$ is the chemotactic sensitivity. We use ${\eta }_n$ to denote the TAF production rate by endothelial cells, and distinguish it from $\eta$, which denotes the TAF production rate by tumor cells. Setting ${\eta }_n=0$ recovers the unidirectional coupling mechanism. The bidirectional coupling scenario contains a self-reinforcing feedback loop:

$$\begin{array}{cc}\hfill \cdots \stackrel{\chi \left(c\right)\nabla c}{⟶}& n\mathrm{aggregation}\stackrel{{\eta }_{n}c}{⟶}c\sec \mathrm{retion}\stackrel{\chi \left(c\right)\nabla c}{⟶}\hfill \\ \hfill & \mathrm{enhanced}n\mathrm{migration}\stackrel{{\eta }_{n}c}{⟶}\mathrm{enhanced}c\sec \mathrm{retion}\stackrel{\chi \left(c\right)\nabla c}{⟶}\cdots \hfill \end{array}$$

In this section, we assume uniform baseline densities $n_0$, $c_0$ set by tumor production and vessel uptake, and focus on perturbations of $n,c$. We emphasize that the analysis of the reduced n-c subsystem captures the essence of the patterning mechanism, and acknowledge that the other fields (oxygen, drug, tumor) remain near uniform.

Theorem 3 (Pattern Formation under Bidirectional Coupling).

The system Equation (21) admits a spatially homogeneous steady state $(n_0,c_0)$ with

$$\begin{array}{c}\hfill {\eta }_n{n}_0-{\xi }_c{c}_0=0.\end{array}$$

Linearizing around this steady state $(n,c)=(n_0+\tilde{n},c_0+\tilde{c})$ yields the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\partial }_t\tilde{n}=D_n\mathrm{\Delta }\tilde{n}-{\chi }_{\mathrm{eff}}{n}_0\mathrm{\Delta }\tilde{c},\hfill \\ {\partial }_t\tilde{c}=D_c\mathrm{\Delta }\tilde{c}+{\eta }_n\tilde{n}-{\xi }_c\tilde{c}.\hfill \end{array}\right.\end{array}$$

Here,

$$\begin{array}{c}\hfill {\chi }_{\mathrm{eff}}{\chi }_0/(1+\alpha c_0)\end{array}$$

is the effective chemotaxis sensitivity parameter at the steady state $(n_0,c_0)$. We expand the small perturbations with respect to each spectral pair of the Neumann Laplacian:

$$\begin{array}{c}\hfill \mathrm{\Delta }{\varphi }_j+{\lambda }_j{\varphi }_j=0\mathrm{on}U,{\nabla }_{\overrightarrow{n}}{\varphi }_j=0\mathrm{on}\partial U,\end{array}$$

where $\overrightarrow{n}$ is the unit outward normal vector on $\partial U$. This expansion yields the following dispersion relation:

$$\begin{array}{c}\hfill {\sigma }^2+a_1\left({\lambda }_j\right)\sigma +a_0\left({\lambda }_j\right)=0,\end{array}$$

with

$$\begin{array}{c}\hfill a_1\left({\lambda }_j\right)=(D_n+D_c){\lambda }_j+{\xi }_c,a_0\left({\lambda }_j\right)=D_n{D}_c{\lambda }_j^2+(D_n{\xi }_c-{\eta }_n{\chi }_{\mathrm{eff}}{n}_0){\lambda }_j.\end{array}$$

Turing instability requires

$$\begin{array}{c}\hfill D_n{D}_c{\lambda }_j^2+(D_n{\xi }_c-{\eta }_n{\chi }_{\mathrm{eff}}{n}_0){\lambda }_j<0\end{array}$$

for some ${\lambda }_j$. Only when

$$\begin{array}{c}\hfill D_n{\xi }_c<{\eta }_n{\chi }_{\mathrm{eff}}{n}_0,\end{array}$$

linear instability arises if the squared wavenumber ${\lambda }_j$ satisfies the following:

$$\begin{array}{c}\hfill {\underline{\lambda }}_j=0<{\lambda }_j<{\displaystyle \frac{{\eta }_n{\chi }_{\mathrm{eff}}{n}_0-D_n{\xi }_c}{D_n{D}_c}}{\overline{\lambda }}_j.\end{array}$$

In a finite-domain U, the admissible wavenumbers are discrete, with the maximum constrained by ${\overline{\lambda }}_j={\displaystyle \frac{{\eta }_n{\chi }_{\mathrm{eff}}{n}_0-D_n{\xi }_c}{D_n{D}_c}}$.

Furthermore, $a_1\left({\lambda }_j\right)>0$, precluding a Hopf bifurcation. More explicitly, since $a_1>0$ and $a_0<0$ at instability, the eigenvalues are real and one becomes positive, so patterns are stationary (no oscillatory eigenvalues). Complex conjugate roots, when present, are strictly damped. Hence, the only linear route to pattern formation is the stationary Turing band $a_0\left({\lambda }_j\right)<0$.

Proof (Sketch).

Linearization of the bidirectional system about the homogeneous equilibrium $(n_0,c_0)$ leads to a quadratic dispersion relation. Instability occurs exactly when $D_n{\xi }_c<{\eta }_n{\chi }_{\mathrm{eff}}{n}_0$, in which case a band of unstable wavenumbers exists and stationary Turing patterns form. Otherwise all perturbations decay. Full algebraic details are provided in Appendix F.  □

Remark 2.

(i)

All results in Section 4.1 and Section 4.2 are derived from linearization around homogeneous steady states. They determine conditions for the onset of instability but do not address nonlinear dynamics (e.g., pattern selection), which remain open questions for future work.

(ii)

Our analysis is in a two-dimensional cross-section, which necessarily omits true three-dimensional features such as vascular tortuosity and branching geometry. By Weyl’s law [61], the eigenvalue count satisfies

$$\begin{array}{c}\hfill N\left(\lambda \right)\propto \mathrm{vol}(\Omega ){\lambda }^{d/2},\end{array}$$

the number of Laplacian eigenmodes below a threshold grows faster in 3D ($d=3$) than in 2D ($d=2$). Thus, three-dimensional geometries generally admit more unstable modes within a given band, and one may expect similar instabilities, potentially with even broader unstable ranges.

(iii)

An important corollary of Theorem 3 is that the instability band arises only through a Turing instability: since $a_1\left(\lambda \right)>0$, complex-conjugate eigenvalues are strictly damped and no Hopf bifurcation occurs. Thus, the model predicts the formation of stable spatial patterns rather than sustained oscillations. This outcome is consistent with classical results on chemotaxis-driven patterning, as reviewed in the Keller–Segel framework [62].

The feedback loop formalized in (21) is strongly grounded in experimental evidence. Endothelial cells not only respond to exogenous VEGF and other TAFs but also secrete angiogenic factors such as Ang-1 and VEGF themselves, both constitutively and in response to oncogenic signals. This autocrine secretion establishes a self-reinforcing loop: local accumulation of endothelial cells enhances TAF release, which in turn sharpens chemotactic gradients and drives further aggregation. Tumor-derived microvesicles further enhance this positive feedback loop by upregulating endothelial VEGF expression, effectively converting endothelial cells from passive responders into active drivers of angiogenesis. In the model, this dynamic is encapsulated by the parameter ${\eta }_n$, which closes the feedback loop between endothelial density and chemoattractant concentration. From a systems perspective, bidirectional coupling increases the likelihood of runaway positive feedback and lowers the threshold for pattern-forming instabilities. From a biological perspective, it provides a parsimonious explanation for how endothelial cells can collectively sustain and amplify vascular sprouting within the tumor microenvironment. Thus, the inclusion of ${\eta }_n>0$ in our model is not only mathematically consequential but also biologically justified by emerging experimental findings on autocrine endothelial signaling.

5. Results

Our analysis predicts that any Fourier mode with squared wavenumber lies in the unstable band $(0,{\overline{\lambda }}_j)$ evolves into spatial patterns. However, we also note that even with the inequality $0<{\lambda }_j<{\overline{\lambda }}_j$ satisfied, a finite-domain may prevent patterning. The continuous unstable band $(0,{\overline{\lambda }}_j)$ must contain at least one admissible discrete Neumann eigenvalue of the domain. We demonstrate this constraint below.

5.1. Finite-Domain Constraints

Consider the Neumann spectral problem on the rectangle $U=[0,L_a]\times [0,L_b]$. The Neumann eigenpairs take the form

$$\begin{array}{cc}\hfill {\varphi }_{p,q}& =c_{p,q}cos\left({\displaystyle \frac{p\pi x}{L_a}}\right)cos\left({\displaystyle \frac{q\pi y}{L_b}}\right),\hfill \\ \hfill {\lambda }_{p,q}& ={\left({\displaystyle \frac{p\pi }{L_a}}\right)}^2+{\left({\displaystyle \frac{q\pi }{L_b}}\right)}^2,\hfill \end{array}$$

for $(p,q)\in {\mathbb{Z}}^2$. The dispersion relation reads

$$\begin{array}{c}\hfill {\sigma }^2+a_1\left({\lambda }_{p,q}\right)\sigma +a_0\left({\lambda }_{p,q}\right)=0,\end{array}$$

with $a_1\left({\lambda }_{p,q}\right)=(D_n+D_c){\lambda }_{p,q}+{\xi }_c$ and $a_0\left({\lambda }_{p,q}\right)=D_n{D}_c{\lambda }_{p,q}^2+(D_n{\xi }_c-{\eta }_n{\chi }_{\mathrm{eff}}{n}_0){\lambda }_{p,q}$. The continuous unstable band reads $(0,{\overline{\lambda }}_{p,q})$ where ${\overline{\lambda }}_{p,q}={\displaystyle \frac{{\eta }_n{\chi }_{\mathrm{eff}}{n}_0-D_n{\xi }_c}{D_n{D}_c}}$. Assume $L_a\le L_b$. Patterns occur if and only if at least one admissible ${\lambda }_{p,q}$ lies in $(0,{\overline{\lambda }}_{p,q})$. The first nontrivial Neumann mode has squared wavenumber ${\pi }^2/L_b^2$, so the necessary and sufficient condition for patterning reads

$$\begin{array}{c}\hfill {\displaystyle \frac{{\pi }^2}{L_b^2}}<{\overline{\lambda }}_{p,q}\Rightarrow L_b>{\displaystyle \frac{\pi }{\sqrt{{\overline{\lambda }}_{p,q}}}}.\end{array}$$

Note that ${\displaystyle L_b>{\displaystyle \frac{\pi }{\sqrt{{\overline{\lambda }}_{p,q}}}}}$ only guarantees that the first nontrivial Neumann mode (with $p=0,q=1$) lies in the unstable band. The domain must exceed a critical linear size to admit any unstable Neumann mode.

Next, we evaluate the critical length using the parameter set in Table 3. We vary the endothelial-chemoattractant production rate ${\eta }_n$ across orders of magnitude and set $n_0=1$, $D_n=4.61\times {10}^{-4}$, $D_c=0.12$, $\alpha =0.6$, ${\xi }_c=0.002$, ${\chi }_0=0.0599$, and $L_a=L_b=5$. The effective chemotactic sensitivity equals ${\chi }_{\mathrm{eff}}={\displaystyle \frac{{\chi }_0}{1+\alpha c_0}}={\displaystyle \frac{{\chi }_0}{1+\alpha {\eta }_n{n}_0/{\xi }_c}}$.

Table 6. Critical domain length $L_{\mathrm{crit}}$ for instability under Neumann boundary condition. We report values in dimensionless units, with physical length obtained by multiplying by the characteristic scale $L=5\mathrm{mm}$. All parameter values are taken from the authoritative set in Table 3; see also the machine-readable files params.json and params.csv in the GitHub Repository in the Data Availability Statement Section.

${\mathit{\eta }}_{\mathit{n}}$ (Production)	Computed ${\mathit{\chi }}_{\mathbf{eff}}$	${\overline{\mathit{\lambda }}}_{\mathit{p},\mathit{q}}$ Upper Bound	Critical Domain Length ${\mathit{L}}_{\mathbf{crit}}=\frac{\mathit{\pi }}{\sqrt{{\overline{\mathit{\lambda }}}_{\mathit{p},\mathit{q}}}}$ (Dimensionless Units, 1 Unit = 5 mm)
0.001	0.04608	0.81625	3.47727
0.01	0.01498	2.69031	1.91535
0.05	0.00374	3.36706	1.71208
0.1	0.00193	3.47621	1.68499
0.5	0.00040	3.56873	1.66300
1	0.00020	3.58065	1.66023

lists the computed values of ${\overline{\lambda }}_{p,q}$ and the critical length $L_{\mathrm{crit}}$ for ${\eta }_n\in \{0.001,0.01,0.05,0.1,0.5,1\}$. Increasing ${\eta }_n$ expands the unstable band $(0,{\overline{\lambda }}_{p,q})$ and reduces the minimal domain $L_{\mathrm{crit}}$ that supports patterning.

Table 7. First nine Neumann Laplacian eigenmodes on the unit square and instability membership for representative ${\eta }_n$ values used in the manuscript. A checkmark (✓) indicates the mode satisfies ${\lambda }_{p,q}<{\overline{\lambda }}_{p,q}$ and therefore lies inside the unstable band for that ${\eta }_n$.

Index	Mode $(\mathit{p},\mathit{q})$	${\mathit{\lambda }}_{\mathit{p},\mathit{q}}$	${\mathit{\eta }}_{\mathit{n}}=0.001$	${\mathit{\eta }}_{\mathit{n}}=0.01$	${\mathit{\eta }}_{\mathit{n}}=0.05$	${\mathit{\eta }}_{\mathit{n}}=0.1$	${\mathit{\eta }}_{\mathit{n}}=0.5$	${\mathit{\eta }}_{\mathit{n}}=1$
1	(0, 0)	0	–	–	–	–	–	–
2	(0, 1)	${\displaystyle \frac{{\pi }^2}{25}}\approx 0.3948$	✓	✓	✓	✓	✓	✓
3	(1, 0)	${\displaystyle \frac{{\pi }^2}{25}}\approx 0.3948$	✓	✓	✓	✓	✓	✓
4	(1, 1)	${\displaystyle \frac{2{\pi }^2}{25}}\approx 0.7896$	✓	✓	✓	✓	✓	✓
5	(0, 2)	${\displaystyle \frac{4{\pi }^2}{25}}\approx 1.5791$		✓	✓	✓	✓	✓
6	(2, 0)	${\displaystyle \frac{4{\pi }^2}{25}}\approx 1.5791$		✓	✓	✓	✓	✓
7	(1, 2)	${\displaystyle \frac{5{\pi }^2}{25}}\approx 1.9739$		✓	✓	✓	✓	✓
8	(2, 1)	${\displaystyle \frac{5{\pi }^2}{25}}\approx 1.9739$		✓	✓	✓	✓	✓
9	(2, 2)	${\displaystyle \frac{8{\pi }^2}{25}}\approx 3.1583$			✓	✓	✓	✓

reports the first nine Neumann modes on ${[0,5]}^2$ and marks which modes satisfy ${\lambda }_{p,q}<\overline{\lambda }$ for the ${\eta }_n$ values above. The trivial uniform mode $(0,0)$ with ${\lambda }_{0,0}=0$ does not participate in Turing instability.

Using the instability condition $D_n{\xi }_c<{\eta }_n{\chi }_{\mathrm{eff}}{n}_0$ together with ${\chi }_{\mathrm{eff}}={\displaystyle \frac{{\chi }_0}{1+\alpha {\eta }_n{n}_0/{\xi }_c}}$, we obtain the explicit lower bound

$$\begin{array}{c}\hfill {\eta }_n>{\displaystyle \frac{D_n{\xi }_c}{({\chi }_0-D_n\alpha )n_0}}≕{\eta }_n^0.\end{array}$$

With the parameter value above and $n_0=1$ we compute ${\eta }_n^0=2.4281\times {10}^{-6}$. Therefore, any net endothelial cell to TAF production comparable to or larger than $2.43\times {10}^{-6}$ crosses the linear threshold that permits a Turing chemotactic band, though a finite-domain must still contain an admissible mode.

Throughout, we assumed homogeneous Neumann (no-flux) boundaries, consistent with modeling tumor tissue bounded by impermeable surroundings. For completeness and comparison, we also consider periodic boundaries, which are widely used to emulate a toroidal domain and thereby approximate an effectively infinite tissue without edge effects. The smallest nonzero periodic mode has squared wavenumber ${(2\pi /L_b)}^2$. Hence, periodic boundaries alter the minimal admissible wavelength and change the finite-domain threshold to

$$\begin{array}{c}\hfill {\displaystyle \frac{4{\pi }^2}{L_b^2}}<{\overline{\lambda }}_{p,q}\Rightarrow L_b>{\displaystyle \frac{2\pi }{\sqrt{{\overline{\lambda }}_{p,q}}}},\end{array}$$

which explains the factor-of-two difference in critical linear dimension relative to Neumann boundaries. Appendix H gives further details.

Remark 3 (Boundary Conditions).

The choice of boundary condition alters the smallest admissible eigenvalue: under Neumann BCs it is ${\pi }^2/L^2$, while under periodic BCs it is ${\left(2\pi \right)}^2/L^2$. Consequently, the critical domain size differs by a factor-of-two,

$$\begin{array}{c}\hfill L_{\mathrm{crit}}^{\mathrm{Neumann}}={\displaystyle \frac{\pi }{\sqrt{\overline{\lambda }}}},L_{\mathrm{crit}}^{\mathrm{Periodic}}={\displaystyle \frac{2\pi }{\sqrt{\overline{\lambda }}}}.\end{array}$$

We explicitly note this distinction to clarify that the instability threshold depends not only on system parameters but also on boundary conditions.

When an unstable mode $e^{\sigma \left({\lambda }_j\right)t}{\varphi }_j$ exists, the dominate pattern follows the mode with maximal growth rate, i.e., $e^{\sigma \left({\lambda }_{j_0}\right)t}{\varphi }_{j_0}$, where $\Re \left(\sigma \left({\lambda }_{j_0}\right)\right)$ attains its maximum. We identify the dominant squared wavenumber ${\lambda }_{j_0}$ via

$$\begin{array}{c}\hfill j_0=arg\underset{j}{min}|{\lambda }_j-{\lambda }^{\ast }{|,\mathrm{where}{\displaystyle \frac{d\sigma \left(\lambda \right)}{d\lambda }}|}_{{\lambda }^{\ast }}=0.\end{array}$$

The wavelength for each wavenumber $\sqrt{{\lambda }_j}$ is ${\displaystyle \frac{2\pi }{\sqrt{{\lambda }_j}}}$. Therefore, the $n-c$ subsystem Equation (21) predicts spatially periodic patterns with spacing ${\displaystyle \frac{2\pi }{\sqrt{{\lambda }_{j_0}}}}$.

Thus, the model predicts that beyond a very small threshold of autocrine TAF production by endothelial cells, spatial patterns of vessels will emerge on domains with length exceeding $L_{\mathrm{crit}}$. Using our nondimensional parameters, $L_{\mathrm{crit}}=1.66$, which corresponds to about $\approx 8.3$ mm in physical units.

5.2. Bifurcation Diagram

We fix the square domain $U={[0,5]}^2$ and plot dispersion curves $\Re \left({\sigma }_{+}\left(\lambda \right)\right)$ versus $\lambda$ for ${\eta }_n\in \{0.001,0.01,0.05,0.1,0.5,1\}$ in Figure 9. For each ${\eta }_n$, the dominant eigenvalue $\Re \left({\sigma }_{+}\left(\lambda \right)\right)$ rises to a local maximum and then falls below zero as $\lambda$ increases. For fixed $\lambda$, larger values of ${\eta }_n$ shift the dispersion curve upward, leading to higher values of $\Re \left({\sigma }_{+}\left(\lambda \right)\right)$ and thereby reflecting the destabilizing effect of TAF production. The horizontal green solid line marks the zero-eigenvalue threshold $\Re \left(\sigma \right)=0$. Its intersection with each curve determines the bifurcation value of $\lambda$, beyond which spatial patterns disappear. As ${\eta }_n$ increases, this bifurcation value increases, indicating that stronger TAF production enlarges the unstable band $(0,\overline{\lambda })$ and thus promotes a wider range of unstable modes.

Figure 10 presents the bifurcation diagram of $max\Re \left({\sigma }_{+}\left({\eta }_n\right)\right)$ as a function of ${\eta }_n$. The black arrow highlights the critical bifurcation value of ${\eta }_n$, above which spatial patterns appear. Beyond this threshold, increasing ${\eta }_n$ further amplifies $max\Re \left(\sigma \right)$. Simulation yields a critical value ${\eta }_n^{\ast }=0.0004315$, which differs from the theoretical small parameter threshold $2.43\times {10}^{-6}$. This discrepancy arises because very small ${\eta }_n$ produce an unstable band that remains too narrow to contain any admissible discrete wavenumber on the chosen domain. In such cases, the maximum ${max}_{\lambda }\Re \left({\sigma }_{+}\left({\eta }_n\right)\right)$ is evaluated over an empty set, and we assign it the value zero whenever ${\eta }_n<{\eta }_n^{\ast }$.

Solving the discrete mode condition $D_n{\xi }_c={\eta }_n{\chi }_{\mathrm{eff}}{n}_0$ for the smallest nonzero Neumann eigenvalue yields the computational threshold

$$\begin{array}{c}\hfill {\eta }_n^{\ast }={\displaystyle \frac{A{\xi }_c}{{\chi }_0{n}_0{\xi }_c-A\alpha n_0}},A=D_n{\xi }_c+{\displaystyle \frac{{\pi }^2{D}_n{D}_c}{max{\{L_a,L_b\}}^2}}.\end{array}$$

(22)

Substituting the parameter values yields ${\eta }_n^{\ast }=4.2888\times {10}^{-4}$, in close agreement with the simulation threshold reported in Figure 10. Appendix I contains the derivation.

The factor ${\pi }^2/max{\{L_a,L_b\}}^2$ originates from the smallest nonzero Neumann eigenvalue on the rectangle $U=[0,L_a]\times [0,L_b]$. The Neumann eigenpairs are

$$\begin{array}{c}\hfill {\varphi }_{p,q}(x,y)\propto cos\left({\displaystyle \frac{p\pi x}{L_a}}\right)cos\left({\displaystyle \frac{q\pi y}{L_b}}\right),{\lambda }_{p,q}={\left({\displaystyle \frac{p\pi }{L_a}}\right)}^2+{\left({\displaystyle \frac{q\pi }{L_b}}\right)}^2,\end{array}$$

for integers $p,q\ge 0$. The trivial spatially uniform mode corresponds to $(p,q)=(0,0)$ with ${\lambda }_{0,0}=0$. The first nontrivial modes are $(1,0)$ and $(0,1)$, whose squared wavenumbers are ${\pi }^2/L_a^2$ and ${\pi }^2/L_b^2$. Thus, the smallest nonzero eigenvalue is

$$\begin{array}{c}\hfill {\lambda }_{min}={\displaystyle \frac{{\pi }^2}{max{\{L_a,L_b\}}^2}},\end{array}$$

which is the value we substitute into the instability inequality ${\eta }_n{\chi }_{\mathrm{eff}}{n}_0>D_n{\xi }_c+D_n{D}_c\lambda$ to obtain the finite-domain threshold. Equivalently, the discrete eigenindex $(p,q)$ directly encodes the number of half-waves in each spatial direction and therefore links the admissible modes to physical domain lengths $L_a,L_b$. For periodic boundaries, the corresponding minimal nonzero mode is $(p,q)=(1,0)$ or $(0,1)$ but with eigenvalue ${(2\pi /L)}^2$, which explains the factor-of-two difference in the critical linear dimension (Neumann vs. periodic) discussed elsewhere in the text.

We give some practical notes for the reader. Because ${\chi }_{\mathrm{eff}}={\chi }_0/(1+\alpha c_0)$ depends implicitly on ${\eta }_n$ through $c_0={\eta }_n{n}_0/{\xi }_c$, the inequality that defines ${\eta }_n^{\ast }$ is nonlinear in ${\eta }_n$. The form (22) makes the finite-domain dependence explicit and is valid precisely when ${\chi }_0{\xi }_c-A\alpha >0$; we therefore recommend reporting whether this positivity condition holds for the parameter regimes used in any numerical experiment (as done above).

5.3. Turing Instability

We validate the theoretical predictions under Neumann boundary conditions by selecting

$$\begin{array}{cc}\hfill & n_0=1,D_n=4.61\times {10}^{-4},D_c=0.5\times 0.12,\alpha =0.6,\hfill \\ \hfill & {\xi }_c=5\times 0.002,{\chi }_0=20\times 0.0599,L_a=4,L_b=1.\hfill \end{array}$$

These parameters yield the critical domain length $L=0.1179$. Figure 11 shows Turing patterns in the coupled n–c subsystem. Endothelial density first organizes into elongated stripes and then splits into dot-like clusters as spatial heterogeneity amplifies. In contrast, the TAF field evolves from a nearly homogeneous state to stripes, demonstrating the interplay between the two fields during pattern selection.

We next examine four representative parameter regimes under Neumann boundary conditions. In scenario I, we set

$$\begin{array}{cc}\hfill & {\eta }_n=1,n_0=1,D_n=4.61\times {10}^{-4},D_c=0.12,\alpha =0.6,\hfill \\ \hfill & {\xi }_c=0.002,{\chi }_0=0.0599,L_a=1,L_b=1,\hfill \end{array}$$

which gives the critical domain length $L=1.6602$. Since $L>L_a$, the system admits no unstable modes, and patterns do not form (Figure 12a–c and Figure 13a–c). In scenario II, we set

$$\begin{array}{cc}\hfill & {\eta }_n=1,n_0=1,D_n=4.61\times {10}^{-4},D_c=0.5\times 0.12,\alpha =0.6,\hfill \\ \hfill & {\xi }_c=5\times 0.002,{\chi }_0=20\times 0.0599,L_a=1,L_b=1,\hfill \end{array}$$

which decreases the critical length to $L=0.1179$. The reduction in $D_c$ and the amplification of $({\xi }_c,{\chi }_0)$ ensure $L<L_a$ and produce discrete unstable modes that generate Turing patterns (Figure 12d–f and Figure 13d–f). In scenario III, we impose unidirectional coupling by choosing

$$\begin{array}{cc}\hfill & {\eta }_n=0,n_0=1,D_n=4.61\times {10}^{-4},D_c=0.5\times 0.12,\alpha =0.6,\hfill \\ \hfill & {\xi }_c=5\times 0.002,{\chi }_0=20\times 0.0599,L_a=1,L_b=1.\hfill \end{array}$$

This choice removes endothelial feedback. The system does not sustain Turing patterns, and c decays as shown in Figure 12g–i and Figure 13g–i. In Scenario IV, we increase diffusion with

$$\begin{array}{cc}\hfill & {\eta }_n=1,n_0=1,D_n=100\times 4.61\times {10}^{-4},D_c=0.5\times 0.12,\alpha =0.6,\hfill \\ \hfill & {\xi }_c=5\times 0.002,{\chi }_0=20\times 0.0599,L_a=1,L_b=1,\hfill \end{array}$$

which yields $L=1.1931$. Because $L>L_a$, unstable modes are not available and patterns do not form (Figure 12j–l and Figure 13j–l).

These comparisons show three principles. First, $D_c$, ${\xi }_c$, and ${\chi }_0$ jointly control pattern onset, as Scenarios I and II illustrate. Second, bidirectional coupling is necessary and sufficient for instability, as Scenarios II and III demonstrate. Third, large endothelial diffusion coefficient $D_n$ suppresses patterns by enlarging the critical domain length, as Scenarios II and IV confirm. The maximal unstable mode

$$\begin{array}{c}\hfill \overline{\lambda }={\displaystyle \frac{{\eta }_n\frac{{\chi }_0}{1+\alpha {\eta }_n{n}_0/{\xi }_c}{n}_0-D_n{\xi }_c}{D_n{D}_c}}={\xi }_c{\displaystyle \frac{\frac{{\eta }_n{\chi }_0{n}_0}{{\xi }_c+\alpha {\eta }_n{n}_0}-D_n}{D_n{D}_c}}.\end{array}$$

does not vary monotonically with ${\xi }_c$. Scenarios I and II show that increasing ${\xi }_c$ can still permit patterns when concurrent changes in $D_c$ and ${\chi }_0$ enhance the net instability.

While our hybrid PDE-ABM framework can, in principle, generate explicit spatial simulations (e.g., vessel networks, resistant niches), the present work is methodological and does not attempt to construct a general-purpose simulation platform. Instead, we present dispersion relations, bifurcation diagrams, and Turing patterns under bidirectional coupling both as representative outputs of the framework and as validation of stability analysis. These results demonstrate that the model is capable of producing conditions under which heterogeneous vascular clusters and hypoxic niches may emerge, although we do not visualize full-scale simulations here.

In summary, our mathematical analysis shows that when endothelial cells respond only to TAF, the coupled tumor-oxygen-endothelial-TAF system is linearly stable. Under these conditions, it cannot generate Turing instability. This result highlights a stabilizing effect of unidirectional endothelial-TAF coupling, where vessels relieve hypoxia and thereby suppress further TAF production. However, introducing bidirectional endothelial–chemoattractant coupling, motivated by tumor-associated endothelial cell biology, restores the possibility of chemotactic instability. In this reduced n-c subsystem, instability arises when endothelial-driven TAF production is sufficiently strong relative to endothelial motility and TAF clearance. Under these conditions, the system develops spatially periodic vessel aggregation. The analysis further reveals that domain size and boundary conditions constrain the admissible unstable modes. Transient oscillations may emerge from damped dynamics, but not from Hopf bifurcations. Together, these results provide a mechanistic explanation for the coexistence of vascularized and hypoxic regions in tumors. Spatial self-organization arises not from canonical tumor-to-vessel signaling alone, but from bidirectional endothelial–TAF coupling. This mechanism is consistent with the universality of chemotactic pattern formation [55]: unidirectional coupling suppresses patterns, while bidirectional coupling restores them.

Biological Interpretations: The instability condition

$$\begin{array}{c}\hfill D_n{\xi }_c<{\eta }_n{\chi }_{\mathrm{eff}}{n}_0\end{array}$$

captures the competition between chemotactic aggregation and diffusive or decay damping. High TAF production ${\eta }_n$ and strong chemotactic sensitivity ${\chi }_{\mathrm{eff}}$ drives aggregation, while large endothelial diffusion $D_n$ or rapid TAF clearance ${\xi }_c$ counteract it. When the condition holds, the n–c subsystem predicts vascular patterns with characteristic spacing

$$\begin{array}{c}\hfill {\displaystyle \frac{2\pi }{\sqrt{{\lambda }_{j_0}}}},\end{array}$$

set by diffusivities.

Biologically, this implies that even with uniform initial conditions, endothelial cells self-organize into clusters separated by hypoxic regions. Our model therefore predicts the clinically observed high spatial heterogeneity of the tumor microenvironment, where vascularized zones coexist with poorly perfused, hypoxic regions (Figure 1 in [63]). Such heterogeneity yields uneven oxygenation and drug penetration, creating protected niches for resistance persistence [47,64,65]. We further find no sustained oscillations (no Hopf bifurcation), consistent with angiogenic oscillations being transient phenomena [66].

In summary, the analysis predicts that chemotaxis-driven self-organization produces vascular and hypoxic heterogeneity, explaining why tumors often harbor resistant niches. Therapeutically, strategies that tuning TAF clearance ${\xi }_c$, reduce chemotactic sensitivity ${\chi }_0$, or increase random motility $D_n$ may suppress such pattern formation and improve perfusion homogeneity.

6. Discussion

We demonstrated that bidirectional endothelial–TAF coupling is both necessary and sufficient to generate Turing instability, whereas unidirectional models remain stable. Allowing endothelial cells to secrete chemoattractant transforms a homogeneous vasculature into clusters separated by hypoxic, poorly perfused regions. This mechanism provides a direct link between angiogenic signaling and tumor microenvironmental heterogeneity. By deriving an analytic threshold ${\eta }_n>{\eta }_n^{\ast }$ and corroborating it numerically, we identify endothelial-derived TAF production as the principal driver of instability.

Our results provide a new theoretical perspective within both classical and modern frameworks of pattern formation. The analytic instability criterion $D_n{\xi }_c<{\eta }_n{\chi }_{\mathrm{eff}}{n}_0$ links tumor angiogenesis directly to Turing morphogenesis theory [23]. This connection has implications for tissue engineering and regenerative medicine, where predictable vascularization is required to sustain cell viability and ensure homogeneous delivery of oxygen and therapeutics. Our results highlight tunable parameter levers, such as TAF clearance, chemotactic sensitivity, and endothelial motility, that can be adjusted to promote or suppress vascular patterning.

Previous hybrid PDE-ABM studies emphasized coupling discrete and continuum representations [17] but lacked analytic instability criteria. Our analysis fills the gap by showing how bidirectional endothelial-TAF coupling alone suffices to create vascular patterning and protected niches that can foster drug resistance. This finding extends prior computational models with predictive criteria derived from stability theory. This approach also resonates with recent work on chemotaxis and pattern formation, where coupling reaction–diffusion dynamics and chemotaxis enlarges instability regions and drives morphogenetic patterning in development [67]. Here, we adapt that perspective to pathological angiogenesis, showing that endothelial autocrine production can convert vessels from passive responders to active drives of spatial self-organization.

The biological and translational implications arise directly from the instability condition and the numerical simulations. Within tumors, endothelial-driven hypoxic refugia create protected niches that favor resistant clones and limit drug penetration. Although these features are well-known in tumor microenvironments, prior work has not connected them to bidirectional endothelial-TAF coupling. Our simulations (Figure 11, Figure 12 and Figure 13) confirm the theoretical analysis by showing that spatial Turing patterns appear only under bidirectional coupling when reduced TAF diffusion coefficient and elevated chemotactic sensitivity drive the system above the instability threshold. Increasing the endothelial diffusion coefficient or removing bidirectional feedback suppresses pattern formation. The parameters ${\eta }_n$, $D_c$, $D_n$, ${\xi }_c$, and ${\chi }_0$ therefore act together to regulate vascular heterogeneity. These findings suggest that feedback-induced endothelial patterning can create hypoxic refugia and that targeted modulation of chemotaxis, TAF clearance, or vascular motility may reduce such resistance niches, informing therapeutic design and vascular engineering strategies.

Although linear stability analysis predicts the onset and dominant wavelength of vascular patterns, nonlinear effects determine long-term morphology and amplitude. Branching, anastomosis, and other discrete ABM dynamics are likely to influence pattern selection. A systematic bifurcation analysis, supported by ensemble simulations across mutation rates, repair dynamics, and therapy schedules, would clarify how nonlinear interactions shape perfusion heterogeneity and resistance emergence. Such extensions are necessary to translate mathematical predictions into robust therapeutic strategies.

Despite its contributions, the present study should be viewed as a mechanistic proof of principle. Several limitations warrant discussion:

(i)

Dimensionality. The model is restricted to a two-dimensional tissue slice with homogeneous Neumann boundaries, whereas real tumors and engineered tissues are three-dimensional with irregular geometries and mixed boundary conditions. Spectral theory indicates that the number of Laplacian eigenvalues below a given threshold $\lambda$ scales as $N\left(\lambda \right)\propto \mathrm{vol}(\Omega ){\lambda }^{d/2}$, so more unstable modes arise in 3D ($d=3$) than in 2D ($d=2$). Consequently, our 2D simulations provide a conservative estimate: real 3D tissues with tortuous vasculature are expected to exhibit broader unstable bands, richer spatial patterning, and potentially lower thresholds for instability. Extending the framework to three dimensions will therefore be essential for quantitative calibration against in vivo data.

(ii)

Biochemical kinetics and transport. Linear kinetic laws for oxygen and drugs capture low-concentration dynamics but neglect Michaelis–Menten saturation and active efflux. Blood flow is treated as idealized and constant, though in vivo perfusion is dynamic, rerouted, and occasionally reversed within single vessels [68]. Incorporating spatiotemporally variable flow fields and nonlinear kinetics would refine predictions of hypoxia, perfusion heterogeneity, and resistance niches.

(iii)

Microenvironmental and therapeutic detail. Immune and stromal cells, which profoundly affect tumor progression and therapy response, are not explicitly represented. Drug dynamics are simplified as diffusion, uptake, and decay, without accounting for pharmacokinetics/pharmacodynamics or combination therapies. These omissions limit realism in resistance evolution and therapeutic predictions; coupling with immune ABMs and pharmacological models will be needed.

• We discuss additional caveats, including mechanics, motility variants, alternative resistance schemes, and applications beyond oncology, in Appendix J.

We treat this study as a mechanistic proof-of-principle rather than a simulation atlas. We intentionally focus on deriving analytic instability thresholds and hybrid coupling principles, leaving full-scale PDE-ABM visualizations and biological validation for future work. To ensure reproducibility, we provide the scripts used to generate the analytic figures (dispersion plots, bifurcation plots, and Turing patterns; Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13). Extending this methodological framework into a general-purpose computational platform capable of visualizing vessel morphologies and resistance niches will be an important next step.

Together, our framework links reaction–diffusion theory to vascular clustering and perfusion heterogeneity, which in turn foster drug-resistant niches. Bidirectional endothelial-TAF coupling constitutes a robust mechanism for chemotaxis-driven Turing instabilities in tumor angiogenesis. The analytic threshold and the instability condition $D_n{\xi }_c<{\eta }_n{\chi }_{\mathrm{eff}}{n}_0$ identify concrete intervention points targeting feedback disruption to suppress vascular patterning and resistance evolution. Addressing the dimensional, kinetic, and microenvironmental simplifications highlighted above will be essential to enhance the model’s predictive power and translational relevance, broadening its applicability to both oncology and regenerative medicine. More broadly, this study demonstrates how hybrid mathematical models can yield actionable biological insights, serving as a bridge between theoretical analysis, experimental validation, therapeutic design, and regenerative medicine.