Anatomy-DT: A Cross-Diffusion Digital Twin for Anatomical Evolution

## Abstract

Abstract Accurately modeling the spatiotemporal evolution of tumor morphology from baseline imaging is a pre-requisite for developing digital twin frameworks that can simulate disease progression and treatment response. Most existing approaches primarily characterize tumor growth while neglecting the concomitant alterations in adjacent anatomical structures. In reality, tumor evolution is highly non-linear and heterogeneous, shaped not only by therapeutic interventions but also by its spatial context and interaction with neighboring tissues. Therefore, it is critical to model tumor progression in conjunction with surrounding anatomy to obtain a comprehensive and clinically relevant understanding of disease dynamics. We introduce a mathematically grounded framework that unites mechanistic partial differential equations (PDEs) with differentiable deep learning. Anatomy is represented as a multi-class probability field on the simplex and evolved by a cross-diffusion reaction–diffusion system that enforces inter-class competition and exclusivity. A differentiable implicit–explicit (IMEX) scheme treats stiff diffusion implicitly while handling nonlinear reaction and event terms explicitly, followed by projection back to the simplex. To further enhance global plausibility, we introduce a topology regularizer that simultaneously enforces centerline preservation and penalizes region overlaps. The approach is validated on synthetic datasets (Voronoi, Vessel) and a clinical dataset (UCSF-ALPTDG brain glioma). On synthetic benchmarks, our method achieves state-of-the-art accuracy (e.g., Voronoi-DSC: 95.70 ± 0.30 95.70\pm 0.30 and Vessel-DSC: 71.14 ± 0.25 71.14\pm 0.25 ) while preserving topology, and also demonstrates superior performance on the clinical dataset (UCSF-DSC: 65.37 ± 0.35 65.37\pm 0.35 ). By integrating PDE dynamics, topology-aware regularization, and differentiable solvers, this work establishes a principled path toward anatomy-to-anatomy generation for digital twins that are visually realistic, anatomically exclusive, and topologically consistent. Code will be made available upon acceptance.

## 1 Introduction

Modeling the temporal evolution of anatomical structures is a central challenge in computational oncology and medical image analysis (Ren et al., 2022; Lachinov et al., 2023). Clinical imaging protocols routinely capture patient scans before and after treatment, or during disease progression, yet predictive models that generate plausible future anatomy from a baseline scan remain limited. The main reasons being scarce training datasets and lack in the existing mechanisms to enforce temporal consistency and anatomical plausibility in biological growth dynamics.
There is a need to develop computational surrogates for individual patients that simulate trajectories of disease growth and therapeutic response, thereby informing personalized treatment planning and clinical decision support (Katsoulakis et al., 2024; Kuang et al., 2024).

A potential solution to predicting growth trajectories is through generative AI approaches. Most existing generative modeling techniques in medical imaging, such as conditional generative adversarial networks (GANs) and diffusion models (Ho et al., 2020; Dhariwal & Nichol, 2021; Armanious et al., 2020), operate primarily
generate images from gaussian noise without considering any treatment paradigms as input.
While these methods produce visually realistic outputs, they often fail to guarantee structural
plausibility. To ensure structure correctness, several methods have been proposed that use conditioning mechanisms to generate images (Zhang et al., 2023; Zhao et al., 2023). Using anatomies as control has been shown to improve medical image generation (Bhattacharya et al., 2024; 2025a).
Recent methods can generate anatomically accurate post-treatment images from pre-treatment scans (Bhattacharya et al., 2025b; Liu et al., 2025) when conditioned with patient demographics, genomic markers, etc. The major limitations of these methods are that they are static in nature and cannot handle heterogenous inputs i.e., pre-treatment, pre-operative, post-operative scans, varying treatment schemes, genomic mutations.

Figure: Figure 1: Existing baselines model only tumor growth and do not consider adjacent anatomy changes. Our proposed method, Anatomy-DT, models both anatomy and tumor growth conditioned on different treatment paradigms.
Refer to caption: x1.png

In contrast, mechanistic models of tumor growth and tissue dynamics, often formulated as reaction–diffusion partial differential equations (PDEs) (Martens et al., 2022; Yin et al., 2019; Metzcar et al., 2024), provide interpretable dynamics grounded in biology. These models capture important features such as infiltration along
tissue interfaces or mass effect on surrounding tissue. However, they are typically restricted to single-class tumor representations, lack explicit coupling with surrounding anatomy, and cannot be easily coupled with deep learning frameworks
for end-to-end training. Moreover, they rarely account for global topological invariants, such as the connectivity of ventricles or the tree structure of vessels, which are critical to ensure clinically valid anatomical trajectories. Recently proposed works ((Bhattacharya et al., 2025a) and Gupta et al. (2024)) take topological constraints into account to ensure topological consistency in the generated images(Bhattacharya et al., 2025a; Gupta et al., 2024). With the emergence of digital twins, mechanistic models conditioned on diverse treatment modalities have re-emerged as an important paradigm. Clinically, however, these treatments exert influence not only on the tumor itself but also on adjacent anatomical structures. Capturing the coupled evolution of multiple anatomies is particularly challenging, as it requires maintaining both spatial coherence and topological consistency across interacting tissues. Topology-constrained mechanistic models provide a principled framework to address this challenge, ensuring preservation of anatomical topology while enabling clinically faithful simulations.

In this work, we introduce a novel framework, Anatomy-DT, an anatomy digital twin
that bridges the gap between mechanistic PDEs and data-driven generative models. Our approach represents anatomy as a multi-class probability field $u(x,t)\in\Delta^{K-1}$, where $K$ denotes the number of tissue classes and $\Delta^{K-1}$ is the probability simplex. The temporal evolution of this field is governed by a *cross-diffusion reaction–diffusion system*, which naturally enforces competition between classes and exclusivity of tissue regions. Concretely, the governing PDE is formulated as

$$ $\partial_{t}u_{i}(x,t)\;=\;\nabla\cdot\Bigg(\sum_{j=1}^{K}D_{ij}(u)\,\nabla u_{j}(x,t)\Bigg)\;+\;R_{i}(u)$ $$

where $D_{ij}(u)$ encodes both self-diffusion ($i=j$) and cross-diffusion ($i\neq j$) coefficients, and $R_{i}(u)$ represents local reaction or growth terms (e.g., proliferation, atrophy, or event-driven changes). Such systems generalize classical reaction–diffusion equations Murray (2001); Turing (1990); Yankeelov et al. (2015); Jarrett et al. (2018) and have been studied extensively in multi-species dynamics Shigesada et al. (1979); Chen et al. (2021); Burger et al. (2020). In the anatomical setting, this structure enforces sharp, mutually exclusive tissue boundaries without resorting to ad-hoc normalization.

To achieve stable numerical integration within deep learning pipelines, we develop a differentiable implicit–explicit (IMEX) scheme Ascher et al. (1995); Hundsdorfer & Verwer (2013) that treats the stiff diffusion operator implicitly, while handling the nonlinear reaction and event terms explicitly. After each update, the solution is projected back onto the simplex, ensuring tissue probability conservation and anatomical plausibility. This design yields a numerically stable PDE layer that is fully compatible with backpropagation and scalable to medical images.

To further guarantee anatomical plausibility beyond local PDE dynamics, we incorporate global topological priors. Anatomical structures are inherently complex, and their organization becomes further disrupted during tumor growth owing to processes such as angiogenesis, stromal remodeling, extracellular matrix degradation, edema formation, and mass effect, which collectively alter both local microarchitecture and global tissue geometry. In this work, we focus on modeling how anatomical structures deform in response to tumor growth. We propose a formulation that enforces topological correctness by combining centerline consistency for complex anatomical structures with a no-overlap constraint across different anatomies.
Topological regularization has been successfully applied to image analysis and generative models Clough et al. (2020); Hofer et al. (2019), but to our knowledge, this is the first integration with a cross-diffusion PDE generative layer. Taken together, the combination of cross-diffusion PDE dynamics, differentiable IMEX integration, and topology-aware regularization constitutes a principled mathematical formulation for generative anatomy-to-anatomy modeling.

We
experiment
on two synthetic datasets, demonstrating Anatomy-DT’s ability to reproduce both local tissue proliferation
and global topology preservation. Beyond synthetic validation, the framework can be directly applied to clinical datasets such as brain MRIs, where baseline masks are evolved into follow-up masks conditioned on treatment variables. By unifying mechanistic priors with differentiable generative modeling, our method lays the foundation for digital twins that are not only visually realistic, but also anatomically and topologically consistent.

Our contributions are as follows: a) We propose Anatomy-DT, a cross-diffusion reaction–diffusion PDE on the simplex for anatomy-to-anatomy generation, introducing inter-class proliferation as a novel generative mechanism, b) We design a differentiable IMEX solver that implicitly handles stiff diffusion, explicitly treats nonlinear terms, and projects onto the simplex to ensure stability, probability conservation, and anatomical plausibility within end-to-end training, c)We introduce topology-preserving regularization by enforcing centerline consistency and inter-anatomy no-overlap, providing structural guarantees absent in standard generative models and d) We demonstrate the effectiveness of the proposed method on a clinical dataset for post-treatment tumor and multiple-anatomy evolution prediction.

Together, these contributions lead to a tumor and multi-anatomy growth modeling paradigm constrained on different treatment types.To the best of our knowledge, this is the first approach that integrates PDE dynamics and topological constraints into digital twin frameworks.

## 2 Methods

Figure: Figure 2: Anatomy-DT architecture. Our proposed method has three primary components: a) A Growth CNN that learns the residual patterns of anatomy growth, b) the cross-diffusion PDE which models multi-anatomy evolution and c) a topology loss function preserving the anatomical structures.
Refer to caption: x2.png

### 2.1 Background

Diffusion-based PDEs are widely used to model spatio-temporal processes in biology particularly in tumor growth, spread of signaling molecules, etc. In biological application, these diffusion PDEs are used in conjunction with reaction representation such as proliferation, chemotherapy kill, etc. (Yin et al., 2019). These systems are termed as Reaction Diffusion PDEs. However, such methods typically focus on modeling the growth of a single class, such as a tumor, while neglecting the concurrent structural and functional changes in surrounding anatomies. Here, we provide a brief background of different diffusion methods for multi-class setting ($k$).
Self-diffusion. For class $k$, the term $\nabla\cdot(D_{k}(x)\nabla p_{k})$ models independent spreading, such as edema expansion along white-matter tracts, where $D_{k}(x)$ denotes a spatially varying diffusivity and $p_{k}$ the probability or density of class $k$. This method smooths local concentrations of different classes but does not enforce interactions between them (Swanson et al., 2000).
Cross-diffusion. The term $-\nabla\cdot(\chi_{kj}(x)\,p_{k}\nabla p_{j})$ encodes inter-class interactions in which the spatial gradients of class $i$ influence the diffusion of class $j$, where ${i,j}\subset k$. This method enforces anatomical exclusivity by avoiding inter-class diffusion (Vanag & Epstein, 2009).
Treatment terms. In tumor growth modeling, in addition to the diffusion terms, there are also reaction terms like proliferation represented as $r_{k}$ and discrete interventions such as surgery or radiotherapy represented as $s_{k}$. The reaction diffusion model can be represented as

$$ $\frac{\partial p_{k}}{\partial t}\;=\;\nabla\cdot\big(D_{k}(x)\nabla p_{k}\big)\;+\;r_{k}(p_{k})\;+\;s_{k}.$ (1) $$

where $p_{k}$ is the probability of each class.

### 2.2 State Representation

In this work, we propose a system that models tumor and anatomy growth. We first define how anatomical structures ($\mathbb{A}_{k}$) are represented over time $t\in[0,T]$. Instead of using discrete segmentation masks, which are non-differentiable and cannot capture uncertainty, we adopt a probabilistic formulation where each pixel/voxel $x$ is described as a distribution over anatomical classes. This formulation is differentiable and captures both relatively stable anatomical structures and evolving pathologies like tumors.

###### Definition 1 (Anatomical State) .

We define the anatomical state at time $t$ as a multi-class probability field

$$ $p:\Omega\times[0,T]\to\Delta^{K-1},\qquad p(x,t)=(p_{1}(x,t),\ldots,p_{K}(x,t)),$ $$

where $\Omega\subset\mathbb{R}^{D}$ denotes the image domain and $\Delta^{K-1}$ is the $(K-1)$-simplex.

###### Remark 1 .

The simplex constraint,

$$ $p_{k}(x,t)\geq 0,\quad\sum_{k=1}^{K}p_{k}(x,t)=1,$ $$

ensures mutual exclusivity of tissue classes. This means that even though the model operates in a relaxed probability space, every pixel is assigned to exactly one dominant class at inference.
Such a design allows us to reason jointly about multiple tissues while maintaining anatomical plausibility.

###### Definition 1 (Anatomical State) .

###### Remark 1 .

### 2.3 Anatomy Digital Twin

Anatomy-DT models anatomy and tumor growth across different timepoints given that the patient is subjected to certain treatments. Having established the representation in the previous sub-section, we now specify the model governing temporal evolution of the different anatomies and tumors.
We build on cross-diffusion reaction–diffusion systems, which are widely used in biology and ecology to capture competition between different interacting categories. In this particular context, these models are useful as they not only capture the independent proliferation of each class, but also captures spatial competition and exclusivity between different classes. To enhance expressive power, we further integrate a Growth CNN, which learns residual corrections to the PDE dynamics and better aligns simulated trajectories with observed imaging data.
In conjunction with cross-diffusion and proliferation terms, we propose to incorporate clinical interventions, such as surgical resections, radiotherapy, or chemotherapy (in any combination). To this end, we augment the cross–diffusion reaction–diffusion system with the per-class treatment term $s_{k}$. These terms allow the framework to capture both the natural dynamics of tissue competition and the discontinuities introduced by medical interventions, resulting in a more faithful representation of patient-specific tumor–anatomy evolution.

###### Definition 2 (Cross-Diffusion PDE model) .

The dynamics of $p(x,t)$ are governed by a multi-class cross-diffusion reaction–diffusion system. For each class $k$, the PDE is represented as:

$$ $\partial_{t}p_{k}(x,t)=\nabla\cdot\Big(D_{k}(x)\nabla p_{k}(x,t)-\sum_{j\neq k}\chi_{kj}(x)\,p_{k}(x,t)\nabla p_{j}(x,t)\Big)+r_{k}(p;C,U,x,t)+s_{k}(x,t).$ (2) $$

###### Remark 2 .

Here, $D_{k}(x)$ encodes tissue-specific anisotropic diffusion, $\chi_{kj}(x)$ regulates how classes compete for space, $r_{k}$ models growth or atrophy, and $s_{k}$ captures interventions such as surgery or radiation.
This general formulation integrates continuous biophysical dynamics with discrete medical events, making it well suited for constructing patient-specific digital twins.

For the tumor class, $r_{\text{tumor}}=\alpha(x)\,p_{\text{tumor}}\Big(1-\tfrac{p_{\text{tumor}}}{\kappa(x)}\Big)$, where $\alpha(x)$ is the local growth rate and $\kappa(x)$ is the carrying capacity. This classical logistic growth term captures exponential proliferation at low density and saturation at high density, reflecting biological growth limits.

###### Definition 2 (Cross-Diffusion PDE model) .

###### Remark 2 .

### 2.4 Numerical Integration via IMEX Scheme

Directly integrating equation [2](https://arxiv.org/html/2509.25280v1#S2.E2) is numerically unstable due to stiffness in the diffusion terms.
To address this, we design a differentiable solver based on an implicit–explicit (IMEX) scheme.
This splitting stabilizes integration while preserving sharp intervention effects such as resections. We first decompose the PDE into stiff and non-stiff parts $\partial_{t}p=F_{\text{stiff}}(p)+F_{\text{nonstiff}}(p)$,
with diffusion and cross-diffusion in $F_{\text{stiff}}$, and reaction and intervention terms in $F_{\text{nonstiff}}$. Then we perform IMEX Time Discretization. The scheme is given by $\frac{p^{n+1}-p^{n}}{\Delta t}=F_{\text{stiff}}(p^{n+1})+F_{\text{nonstiff}}(p^{n})$. Implicit updates require solving $(I-\Delta tD_{k}\nabla^{2})p_{k}^{n+1}=\mathrm{rhs}$,
which we approximate with $m$ Jacobi iterations. Unrolling these iterations yields a solver compatible with backpropagation, enabling gradients to flow through temporal dynamics. We observe that the IMEX scheme provides unconditional stability with respect to step size for the diffusion terms, while explicit updates preserve the discontinuities induced by clinical interventions. Projection back to the simplex ensures feasibility of the anatomical state throughout training.

### 2.5 Topology-Preserving Regularization

While PDE dynamics ensure plausible growth and diffusion, they do not guarantee preservation of
global anatomical topology. To prevent unrealistic predictions, such as fragmented white matter
tracts or disconnected cortical gray matter, we introduce a topology-aware regularization based on centerline dice loss Shi et al. (2024) and no-overlap clause Nandanwar & Murty (2018).

###### Definition 3 (Anatomy Structure Regularizer) .

For selected classes $k\in\mathbb{A}_{k}$, we preserve topological consistency via the cl-Dice loss between the predicted soft mask $p_{k}(x,t)$ and ground-truth $q_{k}(x)$. Let $S(\cdot)$ denote a soft skeletonization operator. The predicted ($\mathbb{X}$) and ground-truth ($\mathbb{Y}$) soft skeletons are represented as:

$$ $\mathrm{\mathbb{X}}(p_{k},q_{k})=\frac{\langle S(p_{k}),\,q_{k}\rangle}{\langle S(p_{k}),\,\mathbf{1}\rangle+\varepsilon},\qquad\mathrm{\mathbb{Y}}(p_{k},q_{k})=\frac{\langle S(q_{k}),\,p_{k}\rangle}{\langle S(q_{k}),\,\mathbf{1}\rangle+\varepsilon},$ $$

and the clDice loss

$$ $\mathcal{L}_{\mathrm{clDice}}(p_{k},q_{k})=1-\frac{2\,\mathrm{\mathbb{X}}(p_{k},q_{k})\,\mathrm{\mathbb{Y}}(p_{k},q_{k})}{\mathrm{\mathbb{X}}(p_{k},q_{k})+\mathrm{\mathbb{Y}}(p_{k},q_{k})+\varepsilon}.$ $$

###### Definition 4 (Anatomy Overlap Regularizer) .

To encourage mutual exclusivity among classes, we penalize pairwise overlaps of the soft probabilities:

$$ $\mathcal{L}_{\mathrm{overlap}}(p)=\frac{2}{\mathbb{A}(\mathbb{A}-1)}\sum_{1\leq i<j\leq K}\,\mathbb{E}_{x}\big[p_{i}(x,t)\,p_{j}(x,t)\big].$ $$

where $\mathbb{A}$ is the total number of anatomical classes.

###### Definition 3 (Anatomy Structure Regularizer) .

###### Definition 4 (Anatomy Overlap Regularizer) .

#### Anatomy Topology Loss.

Combining Definitions 4 and 5, we present a combined anatomy topology loss (ATL) function. We use a weighted sum of structure and exclusivity regularizers:

$$ $\mathcal{L}_{\mathrm{ATL}}(p)=\sum_{k\in\mathcal{K}_{\mathrm{cl}}}\lambda_{1}\,\mathcal{L}_{\mathrm{clDice}}(p_{k},q_{k})\;+\;\lambda_{2}\,\mathcal{L}_{\mathrm{overlap}}(p).$ $$

### 2.6 Training

For training, we use a combination of Dice loss and Anatomy Topology Loss (ATL). The Dice loss ensures accurate overlap between predicted and ground-truth anatomical and tumor regions, while ATL enforces topological correctness by preserving the structural integrity of anatomical compartments. The final loss is represented as:

$$ $\min_{\theta}\;\mathcal{L}(\theta)=\underbrace{\sum_{k\in\mathbb{A}}\mathcal{L}_{\text{seg}}(p^{\text{pred}},p^{\text{gt}})}_{\text{Segmentation loss}}+\underbrace{\lambda_{\text{topo}}\sum_{k\in\mathbb{A}}\mathcal{L}_{\text{ATL}}(p_{k})}_{\text{Topology loss}}+\lambda_{\text{reg}}\|\theta\|^{2}$ (3) $$

.

## 3 Experiments and Results

Figure: Figure 3: Datasets. We use two synthetic datasets and a real clinical dataset.
Refer to caption: x3.png

### 3.1 Experimental Setup

Datasets. We first validate our approach on two synthetic benchmarks designed for stress-testing: a Voronoi-based dataset and a vessel-tree dataset (Figure [3](https://arxiv.org/html/2509.25280v1#S3.F3)). These controlled settings allow us to systematically evaluate stability and robustness. Additional implementation and generation details are provided in the Appendix [A](https://arxiv.org/html/2509.25280v1#A1). We further evaluate our framework on a clinical dataset, UCSF-ALPTDG dataset Fields et al. (2024), which provides multi-timepoint imaging of adult brain gliomas.
Baselines. We compare our method against two broad categories of approaches. The first includes deep learning–based models such as U-Net (Ronneberger et al., 2015), ConvLSTM (Shi et al., 2015), and NeuralODE Chen et al. (2018), which represent widely used architectures for image segmentation and temporal modeling. The second comprises PDE-driven formulations, namely Fisher-KPP (Fisher, 1937; Kolmogorov, 1937), and Cross-diffusion (Vanag & Epstein, 2009) models, which provide interpretable mechanistic baselines rooted in tumor growth dynamics. All these models were tuned to take pre-treatment multi-anatomy and tumor labels as input along with the treatment variables and to predict the subsequent changes in both anatomical structures and tumor regions in the post-treatment images. This comparison allows us to evaluate performance relative to both data-driven black-box methods and theory-guided physics-based models.
Evaluation metrics. For evaluation of the generated tumor masks, we use Dice-Sørensen Coefficient (DSC) and Hausdorff Distance-95th percentile (HD95). For all experimentations, we use a 5-fold cross-validation strategy and report the
average performance on 5 folds.

**Table 1: Post-treatment anatomy and tumor mask prediction. We compare our method with different DL and PDE-based baselines. DSC and HD95 Metrics reported as $\mu\pm\sigma$. Best results in bold and second best in underline.**
|  | Voronoi | Vessel | UCSF |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| Method | DSC | HD95 | DSC | HD95 | DSC | HD95 |
| UNet | 66.08 $\pm$ 9.82 | 29.48 $\pm$ 7.67 | 60.31 $\pm$ 2.16 | 36.70 $\pm$ 2.84 | 63.57 $\pm$ 2.29 | 17.87 $\pm$ 6.10 |
| ConvLSTM | 69.86 $\pm$ 3.96 | 26.16 $\pm$ 2.38 | 59.62 $\pm$ 4.09 | 37.11 $\pm$ 3.22 | 64.60 $\pm$ 2.12 | 14.29 $\pm$ 4.32 |
| NeuralODE | 69.01 $\pm$ 11.43 | 28.18 $\pm$ 6.73 | 62.90 $\pm$ 3.40 | 33.98 $\pm$ 4.12 | 58.00 $\pm$ 9.20 | 27.12 $\pm$ 19.88 |
| Fisher-KPP | 71.57 $\pm$ 0.34 | 38.11 $\pm$ 0.30 | 69.04 $\pm$ 0.68 | 27.50 $\pm$ 1.17 | 59.88 $\pm$ 1.36 | 16.35 $\pm$ 3.35 |
| Cross-diffusion | 71.53 $\pm$ 0.35 | 38.11 $\pm$ 0.29 | 66.32 $\pm$ 0.94 | 37.63 $\pm$ 1.65 | 63.22 $\pm$ 1.96 | 10.49 $\pm$ 1.98 |
| Ours | 95.70 $\pm$ 0.30 | 1.56 $\pm$ 0.14 | 71.14 $\pm$ 0.25 | 19.35 $\pm$ 1.12 | 65.37 $\pm$ 0.35 | 10.22 $\pm$ 0.67 |

Figure: Figure 4: Qualitative results. A. We compare the tumor segmentation masks from our generated method with different baselines. We report the DSC scores for each cases. B. We show the ground-truth (in red) and predicted (in green) tumor, white matter tracts, cortical gray matter, and lateral ventricle segmentation contours. We report the DSC score for each structure.
Refer to caption: x4.png

### 3.2 Clinical Applications.

Quantitative analysis. In Table [1](https://arxiv.org/html/2509.25280v1#S3.T1), we report DSC and HD95 scores for anatomy and tumor mask segmentation tasks. U-Net and ConvLSTM reached DSCs of $63.57\pm 2.29$ and $64.60\pm 2.12$ with HD95 of $17.87\pm 6.10$ and $14.29\pm 4.32$, respectively, while NeuralODE performed worse ($58.00\pm 9.20$, $27.12\pm 19.88$). Fisher–KPP achieved a DSC $\approx 59.88\pm 21.36$ with HD95 $16.35\pm 3.35$, and cross-diffusion acheived ($63.22\pm 1.96$, $10.49\pm 1.98$). Our method surpassed all baselines with the highest DSC ($65.37\pm 0.35$) and lowest HD95 ($10.22\pm 0.67$), demonstrating both accuracy and stability. Clinically, improved boundary precision reduces uncertainty in treatment planning, while the lower variance indicates robustness that is crucial for consistent deployment across heterogeneous patient populations. Additionally, in Figure [5](https://arxiv.org/html/2509.25280v1#S3.F5).A, we compare the DSC of the proposed method with different baselines for different anatomies. In Figure [5](https://arxiv.org/html/2509.25280v1#S3.F5).B, we report cl-Dice for computing the topological accuracy of the generated anatomical structures.
We observe that Anatomy-DT achieved higher DSC and cl-Dice for both anatomies compared to the baselines. In summary, we conclude that our proposed method achieves the best performance on both combined segmentation and individual anatomy segmentation tasks.

Figure: Figure 5: Results on different structures. We show box plots for DSC and cl-Dice scores of different structures (WMT and CGM).
Refer to caption: x5.png

Qualitative analysis. In Figure [4](https://arxiv.org/html/2509.25280v1#S3.F4).B, we show the ground truth and predicted tumor and anatomy masks for four different patients overlaid on top of the FLAIR sequence. We observe that our proposed method consistently achieved high DSC across different tumor and anatomies. In Figure [4](https://arxiv.org/html/2509.25280v1#S3.F4).A, we compare the generated tumor and different anatomies from our method with different baselines like UNet, ConvLSTM, and NeuralODE. We provide zoomed-in views of the critial anatomy regions. We observe that our method accurately captures more granular patterns compared to the baselines.

### 3.3 Toy datasets

Quantitative analysis. Table [1](https://arxiv.org/html/2509.25280v1#S3.T1) reports segmentation performance on the toy Voronoi and Vessel datasets. On the Voronoi dataset, our method achieves a DSC of 0.9570$\pm$0.0030 and HD95 of 1.56$\pm$0.14, a $\sim$34% gain in overlap and $\sim$94% reduction in boundary error compared to the best baseline. PDE-only models (Fisher, Cross-diffusion) show consistent DSC of around 0.715 but very poor HD95 ($\sim$38), indicating coarse front propagation without fine boundary fidelity. On Vessel dataset, which emphasizes thin and filamentary structures, our method attains DSC of 0.7114$\pm$0.0025 and HD95 of 19.35$\pm$1.12. This marks a $\sim$7.3% DSC gain over Cross-diffusion and a $\sim$43% HD95 reduction versus NeuralODE. The gains are most evident in HD95, reflecting improved adherence to elongated vascular boundaries.
Overall, the toy results show that PDE-only models capture broad dynamics but miss fine structures, while learning-only baselines underperform on topology-sensitive geometries. Our method consistently improves both overlap and boundary metrics with low variance across folds.
Stability and Sensitivity Analyses.
We conducted a series of controlled ablations to evaluate the stability and robustness of our cross-diffusion PDE model.
Varying the integration step size $\Delta t$ while fixing the rollout horizon $T\approx 1.0$ revealed a clear trade-off: smaller steps achieved high Dice ($0.9584$ at $\Delta t=0.1$) and low HD95, whereas large steps led to numerical collapse (Dice $0.7851$ at $\Delta t=0.3$).
Analysis of the Jacobi solver confirmed diminishing returns beyond 4 iterations, with optimal accuracy obtained at 1–2 iterations and only marginal degradation at higher counts; sweeping the relaxation factor $\omega$ around the default ($\omega=0.8$–1.0) showed minimal sensitivity.
Resolution scaling highlighted strong performance up to $128\times 128$, but severe degradation at larger grids, consistent with increased stiffness of the PDE operator.
Sweeps over cross-diffusion strength $\chi$, tumor carrying capacity $\hat{C}$, and TV regularization $\lambda_{tv}$ all produced stable, near-constant Dice around $0.953$, with slight improvements at moderate regularization.
Finally, constraining the spatial growth rate ($k_{\max}$) demonstrated a sharp stability boundary: performance peaked at $k_{\max}=2.0$ (Dice $0.9540$), but collapsed when overly permissive ($k_{\max}=10.0$, Dice $0.61$).
Collectively, these results delineate the regime where the proposed PDE model yields robust, high-fidelity forecasts while exposing failure modes outside stable parameter ranges.
Ablation Analysis. To disentangle the contributions of spatial modeling and topology constraints, we conducted a controlled ablation study (Table [2](https://arxiv.org/html/2509.25280v1#S3.T2)). Removing spatial features led to a substantial performance drop: DSC decreased to $64.82\pm 4.99$ and HD95 increased to $40.27\pm 2.22$. Introducing spatial dynamics (without Growth CNN) yielded the largest gain, improving Dice to $70.53\pm 0.45$ and reducing HD95 by nearly half ($20.23\pm 0.29$), showcasing the importance of spatially aware growth modeling. Adding the topology loss maintained competitive Dice ($70.50\pm 7.64$) while preserving boundary coherence, as reflected in a moderate HD95 ($20.23\pm 0.30$).
We observe that combining Growth CNN along with the spatial terms and topology loss achieved the best performance (DSC: 95.7$\pm$0.3 and HD95: 1.56$\pm$0.14).

**Table 2: Ablation study on spatial regularizer, growth CNN, and topology loss.**
| Spatial | Growth CNN | Topology | DSC ($\mu\pm\sigma$) | HD95 ($\mu\pm\sigma$) |
| --- | --- | --- | --- | --- |
| ✗ | ✓ | ✗ | 64.82 $\pm$ 4.99 | 40.27 $\pm$ 2.22 |
| ✓ | ✗ | ✗ | 70.53 $\pm$ 0.45 | 20.23 $\pm$ 0.29 |
| ✓ | ✗ | ✓ | 70.50 $\pm$ 0.44 | 20.23 $\pm$ 0.30 |
| ✓ | ✓ | ✓ | 95.70 $\pm$ 0.30 | 1.56 $\pm$ 0.14 |

### 3.4 Discussions

Our findings show that integrating physics-informed modeling with data-driven learning consistently improves both segmentation accuracy and stability. On synthetic benchmarks, our method outperformed PDE-only baselines, which captured coarse dynamics but failed at fine boundaries, and deep learning models, which struggled on topology-sensitive structures. Sensitivity analyses further confirmed that our PDE backbone remains robust across solver configurations, with clear stability limits at extreme parameter ranges. On the clinical UCSF dataset, our approach achieved the highest DSC and lowest HD95 with reduced variance, corroborating its robustness and clinical applicability. Clinically, improved boundary fidelity can directly reduce uncertainty in radiotherapy margin definition and surgical planning, thereby enhancing treatment precision. The findings demonstrate that our framework exhibits strong robustness and holds significant promise for translation into clinical practice.

## 4 Related Work

Generative Modeling in Medical Imaging.
Deep generative models such as GANs, VAEs, and diffusion processes have become standard tools for medical image synthesis, translation, and augmentation (Isola et al., 2017; Kingma et al., 2016; Ho et al., 2020; Dhariwal & Nichol, 2021). Several works extend these methods to clinical contexts, including modality transfer and reconstruction (Chartsias et al., 2017; Armanious et al., 2020), and more recently, diffusion-based synthesis (Özbey et al., 2023; Kim & Park, 2024). Longitudinal prediction—generating a follow-up exam from a baseline—has been pursued with conditional generative models or deformation fields derived from registration networks (Jie et al., 2016; Qin et al., 2019). However, these approaches typically emphasize pixel fidelity rather than anatomical validity; they may produce overlapping tissues, disconnected lobes, or topologically implausible vessels. A few recent works use topological constraints for guiding diffusion models (Bhattacharya et al., 2025a; Gupta et al., 2024; Xu et al., 2025). Our work differs by embedding biological priors directly in the generative process: tissues are constrained to evolve within a cross-diffusion PDE on the probability simplex, with topology regularized explicitly via persistent homology.
Digital Twins in Oncology. DT technology in oncology has gained prominence for its potential to model tumor growth Enderling & AJ Chaplain (2014); Oden et al. (2010) and predict treatment responses Lal et al. (2020); Chaudhuri et al. (2023); Sun et al. (2023). Early models utilized cellular automata Mallet & De Pillis (2006); Moreira & Deutsch (2002) and reaction-diffusion equations Weis et al. (2015); Gatenby & Gawlinski (1996); Konukoglu et al. (2009) to simulate tumor proliferation and response to therapies. Recent advancements have integrated these mechanistic models with machine learning techniques, enabling more accurate predictions by incorporating longitudinal imaging and multi-omics data. For instance, the TumorTwin Kapteyn et al. (2025) framework offers a modular approach to constructing patient-specific DTs, facilitating the simulation of tumor dynamics and treatment outcomes. Additionally, predictive DTs have been developed to optimize radiotherapy planning by accounting for spatially varying tumor characteristics and treatment uncertainties Chaudhuri et al. (2023). Despite these advancements, challenges remain in low data resources and complex treatment modeling pipelines,
which are critical for the clinical application of DTs in oncology.

## 5 Conclusion

We proposed a principled framework for anatomy-to-anatomy evolution that unites cross-diffusion PDE dynamics, differentiable IMEX solvers, and topology-preserving regularization within a deep learning setting. By formulating tissue evolution as a probability field constrained to the simplex, our approach enforces exclusivity and inter-class competition, while persistent homology ensures global structural validity. Theoretical guarantees of weak-solution existence, unconditional stability, and simplex invariance distinguish our method from conventional generative models. Anatomy-DT’s PDE–deep learning formulation offers a mathematically grounded path toward clinically meaningful digital twins that simulate disease trajectories in a stable, interpretable, and topologically consistent manner.

Acknowledgements. This research was partially supported by National Institutes of Health (NIH) and National Cancer Institute (NCI) grants 1R21CA258493-01A1, 1R01CA297843-01, 3R21CA258493-02S1, 1R03DE033489-01A1, and National Science Foundation (NSF) grant 2442053. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

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## Appendix

Here, we provide additional details on datasets, additional results and theoretical proofs.

## Appendix A Datasets

For quantitative comparisons and stress-testing, we synthesize two datasets namely Voronoi and Vessel. Details of these datasets are provided here:
Voronoi. The Voronoi dataset generates multi-class smooth, organ-like regions using soft Voronoi partitions, where each class is formed by placing random sites and assigning pixels based on distance (shown in Figure [6](https://arxiv.org/html/2509.25280v1#A1.F6).A). This creates natural-looking multi-organ structures with smooth boundaries. An additional tumor class is seeded near the center and modeled as a Gaussian blob, which expands at the second timepoint while slightly displacing its neighboring tissue. The dataset is well suited for testing models on smooth anatomical boundaries and tumor–organ interactions.
Vessel. The Vessel dataset focuses on vascular structures. It first builds a branching random-walk skeleton to mimic a vessel tree, then dilates it to obtain thickened vessels (shown in Figure [6](https://arxiv.org/html/2509.25280v1#A1.F6).B). Surrounding regions are filled with soft Voronoi partitions to represent lobes, and a tumor is seeded near a vessel, reflecting biological tendencies of tumor growth along vasculature. Between the two timepoints, vessels thicken and the tumor grows preferentially along vessel proximity, creating realistic dynamics. Together, Voronoi and VesselTree datasets provide complementary challenges: one emphasizes smooth organ partitions, while the other emphasizes branching topology and vessel-guided tumor growth.

Figure: Figure 6: Additional examples of the toy datasets: Voronoi (A) and Vessel (B).
Refer to caption: x6.png

## Appendix B Additional results

Sensitivity analysis. The DSC and HD95 scores are reported for different Anatomy-DT parameters in Figure [7](https://arxiv.org/html/2509.25280v1#A2.F7).
Qualitative results. Additional results are shown Figure [8](https://arxiv.org/html/2509.25280v1#A2.F8).

Figure: Figure 7: Sensitivity analysis.
Refer to caption: x7.png

Figure: Figure 8: Additional qualitative results.
Refer to caption: x8.png

## Appendix C Theoretical Guarantees

###### Theorem 1 (Existence and Uniqueness) .

Under standard assumptions on diffusion ($D_{k}$ coercive), cross-diffusion ($\chi_{kj}$ bounded), and Lipschitz reaction terms $r_{k}$, the PDE system in equation [2](https://arxiv.org/html/2509.25280v1#S2.E2) admits a unique weak solution $p(x,t)$ on $[0,T]$.

###### Theorem 2 (Stability of IMEX Scheme) .

The implicit treatment of diffusion and cross-diffusion ensures unconditional stability with respect to step size $\Delta t$. The explicit treatment of reaction and intervention terms preserves discontinuities introduced by clinical events.

###### Lemma 1 (Feasibility and Conservation) .

Projection to the simplex $\Delta^{K-1}$ guarantees $p_{k}(x,t)\geq 0$ and $\sum_{k=1}^{K}p_{k}(x,t)=1$ for all $t$. Thus, the solution remains a valid probabilistic anatomical state throughout integration.

###### Theorem 3 (Topology Preservation via ATL) .

Let $\mathcal{L}_{\mathrm{ATL}}(p)=\lambda_{1}\sum_{k\in\mathcal{K}_{\mathrm{cl}}}\mathcal{L}_{\mathrm{clDice}}(p_{k},q_{k})+\lambda_{2}\,\mathcal{L}_{\mathrm{overlap}}(p)$ with $\lambda_{1},\lambda_{2}>0$.
Assume $S(\cdot)$ is the soft skeletonization used in clDice and satisfies the standard properties in Shi et al. (2024): continuity, idempotence on 1-pixel wide skeleta, and morphological thinning consistency.
Then any minimizer $p^{\star}$ of $\mathcal{L}_{\mathrm{ATL}}$ over feasible probability fields satisfies:
(i) for each $k\in\mathcal{K}_{\mathrm{cl}}$, the predicted and reference skeleta coincide (connectivity/homotopy preserved), and
(ii) $p_{i}^{\star}(x)\,p_{j}^{\star}(x)=0$ a.e. for all $i\neq j$ (mutual exclusivity).

###### Proof.

(A) clDice term enforces skeleton (connectivity) agreement.
For a fixed class $k$, clDice is

$$ $\mathcal{L}_{\mathrm{clDice}}(p_{k},q_{k})=1-\frac{2\,\langle S(p_{k}),q_{k}\rangle}{\langle S(p_{k}),\mathbf{1}\rangle+\langle S(q_{k}),\mathbf{1}\rangle+\varepsilon}\,,$ $$

which equals 0 iff the two directional skeleton precisions/recalls are $1$, i.e., $S(p_{k})\subseteq q_{k}$ and $S(q_{k})\subseteq p_{k}$ up to null sets. Under the assumptions on $S(\cdot)$, this yields equality of skeleta and, by the clDice analysis, preservation of connectivity up to homotopy for binary segmentations in 2D/3D. Thus, for any $\lambda_{1}>0$, the ATL minimizer must satisfy $S(p_{k})=S(q_{k})$ for all $k\in\mathcal{K}_{\mathrm{cl}}$. ∎

(B) Overlap term enforces exclusivity.
The overlap penalty

$$ $\mathcal{L}_{\mathrm{overlap}}(p)=\frac{2}{K(K-1)}\sum_{1\leq i<j\leq K}\mathbb{E}_{x}\big[p_{i}(x)\,p_{j}(x)\big]$ $$

is nonnegative and equals 0 iff $p_{i}(x)\,p_{j}(x)=0$ a.e. for all $i\neq j$. Hence at any minimizer with $\lambda_{2}>0$ we have pairwise exclusivity almost everywhere. This is consistent with standard exclusion/non-overlap losses used to enforce disjoint organs in multi-class medical segmentation.

(C) Combined ATL.
Since both terms are nonnegative, the minimum of $\mathcal{L}_{\mathrm{ATL}}$ is achieved only when each term attains its minimum. Therefore a minimizer $p^{\star}$ simultaneously satisfies (A) and (B): skeleton (connectivity) agreement for selected classes and mutual exclusivity across all classes. ∎

###### Theorem 1 (Existence and Uniqueness) .

###### Theorem 2 (Stability of IMEX Scheme) .

###### Lemma 1 (Feasibility and Conservation) .

###### Theorem 3 (Topology Preservation via ATL) .

###### Proof.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: N/A
soundness: 2
strengths: 1. The paper focuses on a well-motivated problem to model the anatomy-toantomy evaluation for longitudinal medical imaging analysis.
2. The paper includes both synthetic and clinical datasets with both quantitative and qualitative results.
3. The paper includes the ablation study to verify the effectiveness of each key designs including spatial modeling, topology constraints and growth CNN.
summary: The paper presents Anatomy-DT that integrates cross-diffusion PDEs, differentiable IMEX solvers, and topology-aware regularization to model the spatiotemporal change of anatomical structures and tumors.
weaknesses: 1. Although the paper is well motivated, it only evaluates on one real dataset and the performance improvement seems to be trivial (e.g., less than 1% compared to runner-up). Therefore, the generalizability in the real-world application remains a concern.
2. Some of the experiment details are missing, for example, what is the testbed of ablation study. Also, it is quite confusing how the baselines are implemented. Were the topology and regularization terms also applied to the baseline method? More details should be described in the paper.
3. How’s the ablation performance on the other two datasets?
4. The related work is placed in an uncommon position.

---

confidence: 5
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Classical PDE models use a single scalar tumor field. How does Anatomy-DT initialize and evolve multi-class anatomies—do all tissues obey the same PDE or class-specific diffusion matrices?
2. Which topological invariants (e.g., Betti numbers, Euler characteristic) are actually preserved by the clDice + overlap loss? Is any persistent-homology-based measure computed, or is topology regularization purely heuristic?
3. How does the model manage patients with variable or irregular imaging intervals?
4. What exact operator is used for projection, and how are gradients propagated through it?
5. Have you compared against ImageFlowNet, which also predicts anatomy-to-anatomy transformations (seems highly relevant to your topic )?
soundness: 3
strengths: 1. The paper addresses a very valuable problem. It bridges mechanistic tumor PDEs and deep learning for a digital twin application, utilizing the neighboring anatomy jointly. This is a very valuable design (clinically). 
2. The paper provides a very reasonable mathematical structure. Cross-diffusion on the simplex elegantly captures tissue competition and exclusivity; the IMEX solver offers a stable and differentiable temporal evolution layer. Incorporating clDice and overlap penalties meaningfully constrains geometric plausibility, addressing a known issue in medical image generation. 
3. The presentation is very clear overall.
summary: This paper introduces Anatomy-DT, a hybrid framework combining cross-diffusion PDEs, differentiable implicit–explicit (IMEX) integration, and topology-preserving regularization for simulating tumor and anatomy evolution from baseline imaging. The authors define the anatomy representation as a probability field on the simplex, enforcing tissue exclusivity, while a small Growth CNN learns residual corrections to the PDE dynamics. The approach is tested on synthetic datasets (Voronoi, Vessel) and a clinical UCSF glioma dataset, outperforming U-Net, ConvLSTM, NeuralODE, and classical PDE baselines in Dice and HD95 metrics. According to the results shown in the paper, Anatomy-DT achieves the best performance.
weaknesses: 1. The baselines (U-Net, ConvLSTM, NeuralODE) are not designed for deformation or flow-based modeling. **Comparisons should include ImageFlowNet (https://arxiv.org/abs/2406.14794) or other neural image-registration / diffeomorphic flow methods that explicitly predict voxel-wise motion fields.** Without these, the evaluation does not fully test the advantage of Anatomy-DT over strong, domain-relevant baselines.
2. It is unclear whether the CNN predicts residuals for the reaction term, diffusion coefficients, or updates directly on the state variable.
3. The “projection to the simplex” step ensures feasibility, but its implementation (exact projection vs softmax normalization) is unspecified; differentiability through this step is questionable.
4. The phrase “diffusion-based PDE” may be misinterpreted as diffusion models used in generative AI. The paper should clarify that the diffusion here refers to physical heat diffusion, not stochastic diffusion modeling.
5. All experiments involve only two timepoints (baseline to follow-up). Handling irregular or continuous time series is unclear; the PDE formalism could, in principle, allow this, but it is never demonstrated or validated.

---

confidence: 4
contribution: 3
initial_rating: 4
presentation: 2
questions: 1) An ablation showing PDE performance vs. CNN or combined is missing. Could you show how relevant the PDE really is ?  

2) Are the parameters in the PDE learnt or set in advance? How can these learnt parameters be guaranteed to reflect the biological principles behind the tumour and its anatomy? 

3) Could you provide more detailed information about the number of patients, the train/validation/test splits ?
soundness: 3
strengths: 1. Clinical relevance. This work represents a novel method for joint tumor-anatomy modeling. Past research has focused mainly in modeling tumor growth while in this paper, the authors propose a coupled evolution of tumors and adjacent anatomical structures, which sounds more relevant for clinical applications. Anatomy-DT uses cross-diffusion PDEs on the probability simplex is clearly defined. 

2. Contributions. There are two interesting contributions in this work. First, the differentiable IMEX solver which treats stiff diffusion implicitly while handling nonlinear terms can be generalised to other deep learning pipelines. The Topology-aware regularization is effective at reinforcing structural information unlike generative models that do not necessarily preserve the structure.

3. Evaluation: The model is evaluated on both synthetic (Voronoi, Vessel) and clinical (UCSF-ALPTDG) datasets. The performance improvements over baselines are promising.
summary: This paper introduces Anatomy-DT, a mathematically grounded framework that unites mechanistic partial
differential equations (PDEs) with differentiable deep learning to model tumor progression in conjunction with surrounding anatomy.
Anatomy is resented as a multi-class probability field on the simplex evolved by a cross-diffusion reaction–diffusion system, employing a differentiable implicit–explicit (IMEX) scheme and integrating a topology regularizer. The approach is validated on synthetic datasets (Voronoi, Vessel) and a clinical dataset (UCSF-ALPTDG brain glioma).
weaknesses: 1. Concerns on generalisation. Only one clinical dataset is used, the UCSF-ALPTDG brain glioma dataset. Generalization to other cancer types and other imaging modalities is not shown.

2. Methodology. There is no sufficient information that explains how the Growth CNN operates. It is unclear how much of the performance can be attributed to CNN vs PDE. 

3. Parameter choices. There is no reasoning behind the selection of the cross-diffusion coefficients, growth rates, and carrying capacities.

4. Baseline comparison missing. The current method is not compared to any diffusion models conditioned on anatomy.

5. Dataset Limitations. The synthetic datasets (Voronoi, Vessel) may not fully represent the amount of anatomical evolution.

6. Reproducibility. Some implementation details are missing for the choice of hyperparameters, training setup, and architectures.

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confidence: 4
contribution: 2
initial_rating: 2
presentation: 2
questions: N/A
soundness: 2
strengths: N/A
summary: This paper presents a mathematically grounded framework for modeling the spatiotemporal evolution of tumor morphology and surrounding anatomy to support digital twin simulations of disease progression and treatment response. Unlike previous methods that focus only on tumor growth, the proposed approach jointly models tumor and anatomical changes using a cross-diffusion reaction–diffusion system combined with differentiable deep learning. It employs an implicit–explicit (IMEX) scheme for stable numerical integration and introduces a topology regularizer to preserve anatomical structures and prevent overlaps. Validated on synthetic (Voronoi, Vessel) and clinical (UCSF-ALPTDG glioma) datasets, the method achieves state-of-the-art accuracy and topological consistency, offering a principled foundation for anatomically realistic and clinically meaningful digital twin models.
weaknesses: I noticed that Fisher-KPP (Fisher, 1937; Kolmogorov, 1937) and Cross-Diffusion (Vanag & Epstein, 2009) are used as comparison methods in this paper. How can such outdated methods be considered appropriate for comparison?
Additionally, UNet and ConvLSTM are also quite old methods.


Decision: Reject