Physics-Regularized Multi-Modal Image Assimilation for Brain Tumor Localization
Authors: Michal Balcerak
Tamaz Amiranashvili
Andreas Wagner
Jonas Weidner
Petr Karnakov
Johannes C. Paetzold
Ivan Ezhov
Petros Koumoutsakos
Benedikt Wiestler
bjoern menze
Venue: NeurIPS 2024 poster (2024)
Keywords: Inverse Problems
System Identification
Physics-Informed
Biomechanical Modeling
Tumor Growth
Primary area: machine_learning_for_healthcare
TL;DR: We propose a physics-regularized learning approach on dynamic discrete meshes to address the complex inverse problem of tumor localization.

Abstract: Physical models in the form of partial differential equations serve as important priors for many under-constrained problems. One such application is tumor treatment planning, which relies on accurately estimating the spatial distribution of tumor cells within a patient’s anatomy. While medical imaging can detect the bulk of a tumor, it cannot capture the full extent of its spread, as low-concentration tumor cells often remain undetectable, particularly in glioblastoma, the most common primary brain tumor. Machine learning approaches struggle to estimate the complete tumor cell distribution due to a lack of appropriate training data. Consequently, most existing methods rely on physics-based simulations to generate anatomically and physiologically plausible estimations. However, these approaches face challenges with complex and unknown initial conditions and are constrained by overly rigid physical models. In this work, we introduce a novel method that integrates data-driven and physics-based cost functions, akin to Physics-Informed Neural Networks (PINNs). However, our approach parametrizes the solution directly on a dynamic discrete mesh, allowing for the effective modeling of complex biomechanical behaviors. Specifically, we propose a unique discretization scheme that quantifies how well the learned spatiotemporal distributions of tumor and brain tissues adhere to their respective growth and elasticity equations. This quantification acts as a regularization term, offering greater flexibility and improved integration of patient data compared to existing models. We demonstrate enhanced coverage of tumor recurrence areas using real-world data from a patient cohort, highlighting the potential of our method to improve model-driven treatment planning for glioblastoma in clinical practice.

Physics-Regularized Multi-Modal Image Assimilation for Brain Tumor Localization
Michal Balcerak' Tamaz Amiranashvili Andreas Wagner', Jonas Weidner", Petr Karnakov", .Johannes C. Paetzold', Ivan Ezhov?
Petros Koumoutsakos Benedikt Wiestler2t Bjoern Menze
University of Zurich, TTechnical University of Munich Harvard University, 'Imperial College London
Abstract
Physical models in the form of partial differential equations serve important pri- ors for many coner-constrained problems. One such application tumor treatment planning, which relies accurately estimating the spatial distribution of tumor cells within a patient's anatomy. While medical imaging can detect the bulk of a tumor, it cannot capture the full extent of its spread, as low-concentration tumor cells often remain undetectable, particularly in glioblastoma, the most common pri- mary brain tumor. Machine learning approaches struggle to estimate the complete tumor cell distribution due to lack of appropriate training data. Consequently, most existing methods rely on physics-based based simulations to generate anatomically and physiologically plausible estimations. However, these approaches face chal- lenges with complex and unknown initial conditions and constrained by overly rigid physical models. this work, we introduce a novel method that integrates data-driven and physics-based cost functions, akin Physics-Informed Neural Networks (PINNs). However, our approach parametrizes the solution directly on dynamic discrete mesh, allowing for the effective modeling of complex biome- chanical behaviors. Specifically, we propose unique discretization scheme that quantifies how well the learned spatiotemporal distributions of tumor and brain tis- sues adhere to their respective growth and elasticity equations. This quantification acts as a regularization term, offering greater flexibility and improved integration of patient data compared to existing models. We demonstrate enhanced coverage tumor recurrence areas using real-world data from a patient cohort, highlight- ing the potential of our method to improve model-driven treatment planning for glioblastoma in clinical practice.
Introduction
The management of gliomas, particularly glioblastomas, highly challenging due their infiltration beyond the tumor margins detectable in medical imaging. This infiltrative behavior complicates the precise personalization radiotherapy. Current clinical practice applies uniform radiation 1.5 em margin around the active tumor core visible in Magnetic Resonance Imaging (MRI) scans. However, this approach overlooks critical factors such heterogeneous infiltration patterns, varying brain tissues, and anatomical barriers, failing to account for the highly individualized dynamics of
"Corresponding author: nicchal.alceree @ @uzh.ch 'Contributed equally as senior authors.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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tumor spread in each patient. Personalizing radiotherapy to account for this spread remains an unmet clinical need in neuro-oncology [1].
The computational modeling of tumor growth has become a promising solution to this challenge [2, 3, 4, 5], by formalizing the tumor growth process through mathematical models of varying complexity [6]. However, personalizing these models to the limited clinical observational (image) data of patient remains formidable challenge. To address it, two main computational approaches have emerged:
Hard-Constraiine Physics-Based Models: These models use Partial Differential Equations (PDEs) to constrain the solution space, ensuring that output tumor cell distributions adhere strictly to predefined physical laws [7. 8, 9, 10, 11].
However, since PDEs are approximations of the underlying stochastic processes in complex biological environments, they may overconstrain the system. While these models offer structure and insight, they fail to capture the intricate behaviors inherent biological processes like tumor growth. Additionally, linking image observations to state variables requires careful consideration. As a result, the full integration of data is hindered by these limitations, reducing the effectiveness of physical models in representing empirical data.
Data-Driven Models:
These models [12, 13] offer great flexibility by leveraging access to imaging datasets. They can directly be learned from - and applied to the patients' diagnostic scans. However, they often lack rigorous quality control and fail to integrate crucial biological insights, such as tissue-specific infiltration patterns and brain topology. This makes them less reliable for clinical application, especially when there insufficient data to infer them purely from imaging information.
In Search of a Hybrid Approach: Combining physical insights on spatiotemporal distributions of state variables with increased model flexibilityy could be highly beneficial. One approach achieve Informed
this is by testing the physics residual in selected locations, as demonstrated in Neural Networks (PINNs) [14, 15, 16, 17, 18]. Another way involves projecting the learned solution into a latent space that adheres to physics constraints [19, 20]. A different framework, Optimizing a Discrete Loss (ODIL) [21], constructs residual of PDEs using grid-based discretization. Unlike methods that optimize the weights of Multilayer Perceptrons (MLPs) that implicitly parametrize the unknown fields in the PINNs method, ODIL directly optimizes the discretized unknown fields, which can be faster and more stable due to the local (rather than global, as in MLPs) dependencies of the solution on the learnable weights.
Both ODIL [22] and PINNs [23] have recently been employed to model growth equations conditioned on MRI and metabolic imaging of patients. ODIL advances oatt--of-the-rr radiotherapy planning by relaxing the constraints of the growth equation, thereby enhancing the capture of tumor recurrence. A similar trend is observed in computer vision's novel view synthesis, where neural radiance field MLPs [24] are being replaced by discrete representations [25, 26, 27].
However, none of these hybrid approaches have modeled brain-tumor interactions dealing with constraints imposed by local structural anatomy, such as the relative composition of brain tissues and their microstructure, or by the impact of gross anatomical deformations visible in structural scans of large tumors. While biomechanical model extensions can address the latter, incorporating tissue elasticity remains challenging due to the unknown initial anatomy and the computational complexity of the tumor growth dynamics. So far, no hybrid approach free from hard PDE constraints has successfully linked biological processes, such as tumor growth, with models of the biological environment, like tissue elasticity. Only a few studies have used PINNs for biomechanical modeling in synthetic scenarios, demonstrating that the displacement field of materials be learned from observations [28, 29].
The main contributions of this work are follows:
We construct discrete soysics-based residuals as measures of differences between learned spa- tiotemporal distributions and those predicted by discretized physical equations governing tumor growth and tissue biomechanics. Using a Lagrangian perspective where particles carry tissue intensities, we project these dynamics onto static Fulerian grids to condition the learnable tumor cell and tissue distributions available observations. Unlike data-driven models, hybrid

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method remains robust due to physics regularization while being more flexible at adapting data than hurd-constrained numerical simulations,
Our method allows for the estimation of the initial condition of the biomechanical environment. We demonstrate the ability to learn the complex unknown initial condition, which, in our case, is not just the origin of the pathology but also the initial state of the brain tissue anatomy.
We validate our method on the downstream task of radiotherapy planning using the largest publicly available dataset known to us, achieving new state-of-the-al performance in capturing tumor recurrence.
Learned patient's undeformed healthy enstomy
diatribution within deformed by turnor anstory
Learned dyromicss boundary consitione
Dyymaskc
T1Gd
FLAIR
PET-PET
tecretccomm Emenningccs
Prysice-eguitriied learning spatiotemport
Crose Greet Rlodoky FIG
Uebrown tsaues
Or Cupuuaayty
Cansbraptro
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Figure I: Method overview: (a) 3D MRI and PET scans of a glioblastoma patient (b) Preprocessed input includes brain tissue maps, tumor segmentation, and metabolic map from FET-PET. (c) Tumor cell distribution and brain anatomy inferred using a loss function based on assumptions about physical processes, initial conditions. (d) Outcomes: initial healthy anatomy, spatial tumor cell distribution. and system identification parameters.
Figure provides an overview of our method, showing how we condition the learnable full spatial tumor cell distribution on the patient data and regularize using soft assumptions about the physics of tumor growth, tissue elasticity, and initial conditions for both the anatomy and the tumor. In addition to multiparametric MRI scans (T1Gd, T1, FLAIR)), we incorporate Fluoroethyl-L-ttrrosnn PET (FET-PET), specialized form of Positron Emission Tomography (PET), as additional imaging modality that captures the metabolic activity of tumor cells.
Physical Assumptions
Our model integrates physical assumptions ensure that the learned tumor cell distribution adheres to established biological and mechanical principles. These assumptions are incorporated into the loss function regularization terms alongside initial condition assumptions and data terms.
2.1 Physical Model
We consider the problem learning the tumor cell distribution within the domain Q (0, 1]. where (0. represents the assumed time of tumor growth, and the unit cube Ω [0, 1] describes our spatial domain.
The objective is to determine a tumor cell density function e(æ, f), (x, t) E Ω x (0, 1] approximately satisfying the reaction-diifusion-aaveee equation [30]:
dc
De
Sc
(vc)
(1)
Øt
cell migration through diffusion
cell proliferation
tissue displacements through advection
where De = V (D(m) represents the diffusion of the cells with the tissue-dependent diffusion constant D(m), S(c) pc(1 c) describes the cell growth with growth factor p. and is
"Code is available https: http:///htttb...om/aaaaaaaaaaaaa

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velocity field moving the cells in the interior of the domain due to the tissue's elastic properties and the tumor-induced stresses in the tissue.
The brain tissue vector m, which represents the percentage-wise concentration of different tissue [mw f), mam (x, t), (2, f)], where mwm, MGM, and mesF
types, is defined (2; denote the concentrations of white matter, gray matter, and cerebrospinal fluid, respectively.
The brain tissue vector m is governed by:
d,m + div(m @ v) 0,
(2)
where @ denotes the outer product. We assume Neumann boundary conditions for both e and m. For the no-flux condition applies at the borders of diffusive tissues, specifically the combined white and gray matter. For each tissue component in m, the no-flux boundary is assumed to be the constant brain boundary visible in the MRI scans.
The tumor induced stresses lead to displacement field u. such that the velocity given by v = 0,u. We assume quasi-static mechanical equilibrium due to the slow tumor growth rate relative to tissue mechanical responses, leading to:
( +
where Y regulates the impact of the tumor the displacement and is learnable patient-specific parameter. We apply the Neo-Hookean model [31] to represent hyperelastic behavior in biological tissues, where the stress a is related to the deformation gradient F,j = 6ig + Bu with the Lamé parameters À and (material properties in Appendix G), averaged across different tissue types to accommodate the heterogeneity:
atz
FFF - I) Aln(J)I,
(4)
with J= det The averaged Lamé parameters, Ā and p. are computed based on the proportional contributions of constituent tissues, ensuring the stress tensor accurately reflects the composite mechanical properties of the mixed tissues. This methodology facilitates detailed simulations adhering to the elasticity laws relevant to heterogeneous biological environments.
The set of learnable parameters Dynamics describing the tumor dynamics, is thus given by Dynamics (DaM, RG p. 7 where DOM and DwM RDDM are the diffusivities in pure gray and white matter regions, which are averaged to the effective diffusivity with a weighted average based on the tissue proportions.
2.2 Discrete Physics Residuals
We use different approaches to discretize the elasticity equations and the reaction- -iffiinndiifusion- advection equations. For the former, we use a time-dependent grid in space, described by the N₂ points (p)) is)'ss
time ntt < n S N,, which initially form a uniform grid on Ω. For the latter, we further partition the domain f2 at the rith point in time into N,, cells 2 < S Nz. such that Ω N 2z. Note that the nodes defining the cell boundaries are carried by the particles p". whose movement, governed by tissue elasticity, gradually deforms the cells ?2 from their initial rectangular shape.
The displacement field ut is computed based on the initial (t 0) and current = nat) positions of particles p" within the tissue by u." = p" -p". Using central finite differences on the initial uniform reference grid, we can consecutively approximate Vuy. F Vey, V V.J.. and define the tissue residual:
LTisuu Flasticity PDE:
ΣΣ Of
As the particles, representing grid points, move with the tissue's deformation, they inherently accommodate the advection. Thus, the observed displacement of tumor cells attributed to advection absorbed into the changes in particle positions. This leads to simplification the tumor equation when solved in the moving reference frame of the particles.

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We use an implicit Euler scheme combined with a cell-centered finite-volume discretization to discretize the tumor equation. We denote the boundaries between two cells 1 _ i.j S N₂ by 2 T3 ang n 057.. Further, the index set of all neighboring cells of d is given by N; [ : 052, n 8Ω, * BAje # 2} and assumed to stay independent of n, due to the extend of our deformations (see limitations Section 7). For function ex at time nAt at the ith cell, we discretize the diffusion D and reaction S operators by:
P
Σ Di JEN.
and S.c"] = [2]] c.))
(6)
|, x
where D is the harmonic mean of D. the cell boundary, 1 denotes the interface area, and |[2]] denotes the cell volume. We thus obtain the tumor loss by:
Nx
2 At
Trunnrr Growth PDE
-DcC D S(c,5]
(7)
Physics Assumptions
Liisse Elasticity PDE
a
t= 0
t
Initial Assumptions
Imaging Data Reconstruction
L/Halthyy Anatomy
L TTssmmss
hoy brain antiomy
BITer nerooal
leamoed
Time
Tumor Starts from Gaussian
Leure Ledemas Litetabllc Map
turtor disticutioo
Innging Funaion
cogmenzziion und meMabok) map
LTame Growth
Figure 2: Learning process overview: optimization of spatiotemporal distributions tumor cells and tissues. (a) Initial condition penalties enforce symmetric healthy anatomy, with the initial tumor distribution
0 as a small Gaussian blob. (b) Physics penalties regularize dynamics between the initial and the final time. The first row shows gray matter contours with particle positions; the second row shows white matter contours with learned tumor concentrations. (c) Agreement of tumor distribution with anatomical tissues, visible tumor segmentations, and metabolic map after transforming the final tumor distribution through the imaging function.
2.3 Particle-Grid Projections
While Lagrangian frame representation from the point of view of particles beneficial for accurately modeling advection, an Eulerian frame is essential for rendering the advected particle states into the image space. Denoting the Eulerian and Lagrangian frames G and P respectively, an Eulerian peeeee field Jo and Lagrangian field F! are related through:
WifF
F Σ WFFF
F.
(8)
E, Wij
where indexes grid nodes, indexes particles, and wij represents the weight of the trilinear shape function defined on node i and evaluated at the location of particle j.
Initial Assumptions
We model the initial conditions of the tumor and brain tissues by incorporating assumptions into our loss functions, ensuring the model aligns with the expected biological and anatomical state of the brain before and at the onset of tumor growth.
5

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3.1 Initial Tumor Distribution
We assume that the initial tumor distribution can be represented by a small Gaussian blob. To enforce this assumption, we penalize the difference between (u)) and a Gaussian function centered at Tp E Ω. The center of this Gaussian blob, zo. is part of the learnable parameters Omnital (vo, m )
The initial condition loss for the tumor cells is defined as:
N₂
(1i (-(xi x00)2 ?
))
L Tamor stans from a Gaussian
[O) - D1 exp
(9)
D2
where D1 0.5, D2 0.02 are constants and r; is the position of the center of Q?.
3.2 Healthy Anatomy
The deformations are clearly visible in the MRI images and are an integral part of understanding growth dynamics. To obtain a plausible estimate of the unperturbed brain, we need to incorporate an anatomical prior.
healthy brain is roughly symmetric between hemispheres, as justified by anatomical and func- tional observations. Imaging techniques such as MRI reveal symmetrical tissue structures, and electreencephalography (EEG) recordings show symmetrical patterns of electrical activity in the brain [32, 33J. This assumption applies to key tissue types, including white matter, gray matter, and cerebrospinal fluid. We construct loss function, denoted as Heallhy Anatomy, to quantify the asymmetry of the learned F (m"). The implementation details are provided in Appendix E. We do not use the unperturbed anatomy explicitly for inference of the tumor cells, however, we need additional constraints to bound from Eq. that couples tumor cells with tissue dynamics. While symmetry of the healthy brain is a coarse assumption, we soften it by quantifying the asymmetry in lower resolutions.
Estimation white matter tissues and visible tumor segmentation
Learned fult-spatial tumor distribution regularized by the phyaica residual
Learned initial condition for the tissues
Average brain tempiate for reference
MRI 3D scan
b
Ediraa Erhandng
0.2
0.4
06
0.8
Netroiiccom core
Tumpor call caraily
Figure 3: Inference overview: (a) Patient's MRI 3D scans. (b) Estimated tissues through non-rigid registration of the average brain, showing white matter and visible tumor segmentations. (c) Learned tumor cell distribution, regularized by the physies residual and aligned with patient data. (d) Learned initial condition of the tissues representing healthy anatomy. (e) Average brain template for reference, rigidly registered to the MRI scan, Notable anatomical differences include the lack of matter passage between the hemispheres next to the tumor, which could affect tumor cell inference results.
4 Imaging Data Reconstruction
Imaging data refers to information directly observed from the patient's MRI and FET-PET 3D scans [34], including visible tumor parts and surrounding brain tissues. As shown in Figure 1, this data includes estimated brain tissue probability maps, visible tumor segmentation, and a metabolic map derived from FET-PET scans, focusing on the enhancing core and edema. MRT scans provide segmentation maps [35], revealing the visible tumor. Using non-rigid atlas image registration with

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tumor masking [34]. we estimate tissues both outside and hidden by the tumor, including white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF). These observations offer partial information about both the anatomy and the tumor cell distribution.
The imaging function (see Figure 2) [22, 36] bridges simulated tumor cell densities with visible tumor segmentation and brain tisuuss as well as the metabolic map. It depends on parameters for thresholding the tumor cell distribution obtain segmentation, denoted Otmaging which must be learned.
We construct a loss function that minimizes the difference between model predictions and patient observables. Specifically, Leore matches the visible tumor core, LEdema matches the visible edema, and Metabolic Map matches the metabolic map profile. Tissue intensities carried by particles are set to reproduce the patient's is estimated tissues at / = 1. While these intensities are predetermined, particle positions weakly constrained, matching the tissues (WM, GM, CSF) to the observed structures outside the tumor is enforced through Tissess The constructed penalty terms align our spatiotemporal distributions of tumor cells and tissues with patient data characteristics. Implementation details of the data losses and the imaging function are provided in Appendix D.
5 Combined Loss Function
The loss function Lroal constructed balance the contributions of physics-based regularization, assumptions about the initial conditions, and the imaging empirical data. This facilitates com- prehensive approach to infer the full spatial tumor cell distribution at the time of the patient's 's data acquisition (see Figure 3 for an overview of the inference process and Figure 2 for the learning process visualization):
Lotal = a Lumor Growth PDE + 022 Elasticity PDE +
Physical Assumptions
0g LgLTun
Tumor Starts From Gaussion + az Healthy Anatomy
Initial Assumptions
as Ccure Org Lydema
Qr7 Metaholic Map + Org Tissues
(10)
Imaging Data Reconstruction
All a. parameters were determined based on experiments with synthetic data involving both single focal and local multi-focal tumors. Extensive ablation studies are provided in Appendix C, and the details of the synthetic experiment setup can be found in Appendix F.
6 Validation and Ablation Study of Radiotherapy Planning
There is no objective ground truth for pre-operative tumor cell distribution, so methods are evaluated through downstream tasks like defining radiation target volumes. An "accurate" method targets areas that later show progression in follow-up MRI, acknowledging that radiation delays but does not eliminate tumor regrowth. An "inaccurate" method spares these areas or targets regions that remain tumor-free. The current "Standard Plan" defines the target volume using 1.5 cm isosurface around the MRI-segmented tumor core, while our approach generates isosurfaces from learned tumor cell distributions. Figure 4 visualizes the Standard Plan and a plan based on our learned tumor cell distribution. Each model's 's target volume simply a redistribution of the target volume of the Standard Plan.
We assess accuracy by measuring the overlap between the target volumes and regions of later tumor progression, calculating the Recurrence Coverage [%]. This metric represents the percentage of the tumor progression region that falls within the radiotherapy target volume given model, after rigid alignment of both scans.
Additional experiments studying properties of the tumor growth and imaging model synthetic data, ie., without relying on indirect validation via radiotherapy planning, are given in Appendix B.
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Pre-operative MRI 3D scan
Distance from tumor segmentation
Learned tumor cell distribution
cm
1.0
1.5 cm
2.5 cm
0.0
Necrotic core Enhancing core Edema
Recurrence Coverage 62.33% Recurrence Coverage 77.00%
Radiotherapy Contour
Tumor Recurrence
Figure 4: Radiotherapy planning: a) Pre-operative segmentations with white matter concentration as background. b) Distance map from the tumor core segmentation with 1.5 em contour and within diffusive tissue, constituting the Standard Plan. c) Our learned tumor cell distribution with the isosurface contour where the enclosed volume equals the total volume of the Standard Plan.
6.1 Implementation details
Our computational model uses multi-resolution method [37] with additional, coarser grids to accelerate convergence. We employ four levels of grid refinement, resulting in 152,880,044 88,808 unknowns. At the finest grid level of 72 x 72 72 x 96, each grid point contains four unknowns: three for particle positions and one for tumor cell density.
The learning process includes system identification of unknowns: ODynamics Onitial and Omagingg which parameterize tumor growth dynamies, tissue elasticity, initial conditions, and imaging characteristics. Using the Adam optimizer, convergence takes around 3 hours on an NVIDIA RTX A6000 GPU.
6.2 Dataset
We use a publicly available dataset of 58 patients with preoperative MRI scans and preoperative FET- PET imaging, along with follow-up MRI scans at the time of the first visible tumor recurrence [22].
We would like emphasize that this is the largest publicly available dataset of this kind, i.c., with multi-modal imaging data and follow-up MRI. Its size is common for medical studies and sufficient motivate clinical trials (see, for example, [38, 39, 40] on clinical investigations into the of PET for radiotherapy planning). The problem we address a critical challenge in cancer treatment research that calls for MI solutions. However, data acquisition is costly and time-consuming, and clinical datasets, such as ours with only a few dozen complete observations, are insufficient to support purely data-driven ML approaches (as visible from results reported in Table At the same time, it is this inherent limitation that is encouraging us to incorporate physics-based assumptions and thhyics--nfoomee regularizations into an ML model to effectively address the driving clinical problem.
The follow-up up images with visible tumor recurrences not used method during the optimiza- tion or hyper-paramedt search at any point; therefore, the dataset can be considered 100% unseen. The decision not any recurrence data for hyper-parameter tuning is avoid diluting the already small dataset.
6.3 Models and Features
Our analysis includes range of models, each categorized by their distinct capabilities:
Numerical Physics Simulations: Employ methods such as Finite Element Method (FEM) and Finite Difference Method (FDM) to model tumor growth dynamics and account for dynamic tissue behaviors. The dynamic parameters are determined through numerous simulations with varying parameters. The brain tissues' initial conditions are based on average brains.

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Data-Driven Neural Networks (Unconstrained): Use Convolutional Neural Networks (CNNs) to directly predict likely recurrence locations, representing a population-basee data-driven approach.
Data-Driven Neural Networks (Physies-Consttaanee):: Similar the above unconstrained approach but limited to finding parameters for physics simulations, incorporating hard physics constraints.
Static Grid Discretization: A previous state-of-the thrte-t method that uses Optimizing a Discrete Loss (ODIL) and static grids to penalize the learned spatiotemporal tumor cell distribution by quan- tifying the discrepancy with the tumor growth equation, keeping brain anatomy static throughout the entire process.
Standard Plan: Apply uniform safety margins around the tumor core, serving as the baseline for volume determination and reflecting current clinical practice post-tumor resection surgery radiotherapy planning.
Specific implementations within these categories are explained in Appendix
6.4 Results
Table 1, specifically the "Recurrence Coverage [%] (Any)" column, and Figure 5a present results where recurrence is defined by any segmentation visible in the follow-up MRI scan. Our method achieved 74.7% recurrence coverage, surpassing the previous state-of-the-art at 72.9% and the Standard Plan 70.0%. Using a predefined average brain reduced the performance to 73.4%. The mrysics-constrainee data-driven approach slightly outperforms the Standard Plan at 67.1% 1 and significantly outperforms its unconstrained variant 59.0%. This highlights the advantage of adding physics constraints, Figure 5a shows the distance between 'Greater' and 'Less' categories increased from 26% 36%.
Table 1, using the "Recurrence Coverage [%] (Enhancing Core)" column, and Figure 5b show results where recurrence is defined by enhancing core segmentation, per RANO guidelines [41]. Our method leads with 89.9% average recurrence coverage, compared to 89.0% for static grid discretization and 87.3% for the Standard Plan. Numerical physics simulations are more robust at 86.2% compared to 84.3% for the physics-constraine remains the lowest at 66.8%. Results are close because the enhancing core often occurs near pre- operative MRI segmentations, leading to many methods achieving 100% coverage (seen in 'Equal' in Figure 5b). However, Figure 5b shows an increase in the preference from 19% 28%.
data-driven method. The unconstrained data-driven method
In both recurrence definitions, mean estimate uncertainty is substantial, as Recurrence Coverage for individual patients vary significantly. These uncertainties the standard errors of the mean. Nonetheless, Figure 5 shows that even slight improvements in average Recurrence Coverage consistently enhance radiotherapy outcomes compared to the Standard Plan baseline.
Table I: Comparison of recurrence segmentation coverage given equal radiation volume.
Model
Recurrence Coveragel (Any)
Recurrence Coverage! (Fnhancing Core)
Dynamical Tissues
Inferable Healthy Anatomy
Population- Based Data-
Physics-
Constrained
Driven
NN (Unconstrained)
59.0 i 4.3
66.8 4.9
x
x
NN (Physics-Cone (Pyssic-Coostrainee))
70.4
3.7
$4.3 3.3
Hard
Numerical Physics Simulations
67.1 3.8
86.2 3.6
Hard
Standard Plan
70.0 3.8
87.3
3.6
x
x
Static Grid Discretization
72.9
3.5
89.0
Soft
Ours (wlo inferable anatomy)*
73.1 3.2
89.3 4 2.9
X
Soft
3.1
Wee set the initial condition for the brain tissues as hard constraint, representing the average brain.

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Comparieon Standard Radioherapy Plan Recureene: any follow up segmentation
Greater
b
Comparison to the Standard Radiotherapy Plan. Recurence: entanstig coretsloo tollowup MRI segmeniations
Greater
Equal
Equal
Less
10%
Dhinl Mimelli
Dheretarter
Moran pa
Sirulttons
Qal
Cart infeatee
DrpaDinee Vo. Der Orweeh
Figure 5: Direct vatient-by-patinn comparisons to the Standard Plan of radiotherapy plans with equal total volumes: "Greater," "Equal," and "Less" refer to the direct numerical comparison of Recurrence Coverage. a) Recurrence is defined as the union of edema, enhancing core, and necrotic core on the follow-up MRI segmentation. b) Recurrence is defined as the enhancing segmentation.
the follow-up MRI
As shown in Figure 5, the gap between the "Greater" and "Less" categories represents the number of patients who benefit from our method compared to the standard plan. We observe consistent improvement with our method, demonstrating superior performance beyond just higher mean recur- rence coverage, which could be skewed by outliers. The results are statistically significant, as shown in Table in Appendix B.
7 Conclusion
In this work, we present the first successful approach to a soft-constrained physics system- identification problem that combines the complex biological process of tumor growth with the biomechanical environment such as elastic brain tissues. Our method integrates physics-based con- straints with multi-modal imaging data enhance tumor treatment planning for glioblastomas. The method balances adherence to observed patient data and physics- based penalties through a unique discretization scheme, serving flexible spatiotemporal regularization term.
Additionally, our method provides estimates of the initial condition, i.e., tumor-free, pre-deformation anatomy. Such healthy brain anatomy can be utilized in various studies, e.g., for aligning with post-surgical scans to assess the extent of tumor removal and detect complications [42, 43, 44].
Overall. our results show that this approach outperforms previous state-of-the-un techniques in covering tumor recurrence areas, with improved performance on real-world patient data. Finally, we want to point out that relaxing physical model constraints has broad applicability beyond glioblastoma localization. This approach can extend to other real-worke problems where rigid physics- based models limited but still relevant.
The convergence speed of the method depends on multiresolution grid, assuming that problems can be represented at lower resolutions. However, this approach may not be directly applicable PDEs with narrow sharee interfaces. Additionally, while large deformations are allowed, we assume they are not large enough to cause material rupture, which may not be optimally handled the Lagrangian perspective without modifications the presented scheme.
Limitations:
Acknowledgments and Disclosure of Funding
We extend our gratitude to John Lowengrub and Ray Zirui Zhang from the University of California, Irvine, for their invaluable insights and discussions. This research was supported by the Helmut Horten Foundation and the DCoMEX project.
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A Baselines Implementation
Here we provide details regarding the specific baseline implementations:
Numerical Physics Simulations: We employ the numerical forward simulation scheme from [45] which was implemented on GPU [13]. The simulations follows the equation 1. We initialize the brain tissues with a rigidly registered average brain. For Dyyamicss E Iinnial, and Oimaging we use the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) method [10, 46).
Data-Driven Neural Networks (Unconstrained): We use nnU-Net [47] which takes brain tissue distribution, FET-PET map, and pre-operative tumor segmentations to directly predict core tumor recurrences. After predicting the recurrences, we construct distance distribution from the predicted recurrences. This distribution is then thresholded, similar to other methods, to create a binary radiotherapy treatment map of the same volume the Standard Plan. The network is trained using an 80%/20% train-test split.
Data-Driven Neural Networks (Physiss-Cnnstrainee):: We employ a recently published method [13] that uses CNNs to map pre-operative tumor segmentation to Dynamics and runs numerical simulation following equation 1. This method uses an average brain as the initial anatomy. After inference, the tumor cell distribution non-rigidly registered to the patient's brain anatomy.
This method, implemented in [22], uses estimated brain tissues (4) as the initial condition for the anatomy and later keeps the tissues static, effectively setting Dynamics in Equation to O.
Static Grid Discretization:
Standard Plan: This approach uses 1.5 em uniform safety margin around the resection cavity and/or remaining tumor MRI imaging, aligning with both North American and European guidelines [48, 49].
B Synthetic Results
To show how the method performs tasks where there is ground truth, method and GliODIL [16] (previous SOTA, Static Grid) are additionally evaluated on synthetic cases generated by numerical physics solvers under various conditions. See Table for the results and Table for statistical tests.
RMSE Focal Origin) [%]
RMSE (3 Focal Origins) [%]
Input Type
Ours
Static Grid [16]
Statie Grid [16]
Tumor Core
10.2 + 0.4
12.2 + 0.6
13.3 + 0.5
13.0 0.7
Tumor Core, Edema, and Metaholic Map
2.5 + 0.3
5.5 + 0.8
4.0 + 0.5
6.9 + 0.5
Tumor Core, Edema, and Metabolic Map,
0.3 0.1
3.3 0.2
0.4 0.1
0.3
Oranis Dynamics
Table 2:
Synthetic results comparing RMSE for 30 patients with different input types for one and three tumor origins, using method and Static Grid Discretization (GliODIL [16]). The test dataset was not used for parameter tuning. Parameter ranges in Appendix D and y E [0, 1.5] from Eq. 3 in the manuscript. Ours wins because GliODIL cannot deform brain tissues through its static grid limitations.
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Recurrence (Any)
Recurrence (Enhancing Core)
Comparison
p-value
p-value
Ours vs Standard Plan (1 (1.5cm)
0.014
0,00082
Ours vs Static Grid Discretization (GHODDIL]16)
0.043
0.039
Table 3: Statistical significance tests comparing recurrence coverage (any and enhancing core) among different models using the Wilcoxon signed-rank test. The data is not normally distributed which tested using Shapiro-Win Wilk test resulting in p-values below 1E-12.
C Extended Ablation Study
Additional information regarding the loss coefficients:
02 is essential for grid stability. Without it, the grid would be unstable, causing folding and rendering the results meaningless. set to zero, 0g can be replaced by a Laplacian- regularization, visualized in Figure 6b and Figure 6c, though this clearly results in less biologically plausible deformations.
04 regularizes the initial anatomy to bound from Eq 3. It should be kept low due to the natural asymmetry in brains. More details are provided in Appendix
Cg. and Og are crucial for grounding PDE solutions in reality (most important, all other losses can be seen simply regularisation of these components). Imaging data reconstruction loss cannot be calibrated using synthetic studies as all the other alphas. Instead of using recurrence data to calibrate them, we set them such that they provide balanced contribution to the total loss
ag should be kept low since the initial tumor shape is unknown. However, decreasing it too much causes local minimum with tumor.
On serves as the main regularization term. It should not >> oc to avoid solutions not flexible enough to properly accommodate patient data (see Numerical Physics Simulations results in Table I; this setup performs poorly), nor should be - to prevent overfitting imaging data alone, effectively collapsing to standard plan level performance.
Removing elements of the imaging data reconstruction loss introduces ill-posedness and degradation of synthetic results (see Table 4). Removing tumor growth physics regularizations effectively reduces our solution performance close the standard plan. Removing particle dynamics (static grid) reduces our model to [16] Static Grid Discretization ("GliODIL").
a)
b)
C)
Liou O3y T-Tnnor Goowth FOE Giy L-tisse Hasticyy FDE
Plykel Asorpption:
Og Tanor Stats For Gussia + 04 Eggeatt Anaromy
Asumptons
Or/ocos + Gg + Br Etteboti Map + og Taass
braging
02
07
Figure 6: Deformation through particle movement: Total loss of our model for reference. b) Particle position with tissue elasticity. c) Particle position without tissue elasticity. These particle positions are learned for the pre-operative time. For grid stability in e), additional regularization is required. For > 0 we observe one large localized deformation caused by the tumor, which is more biologically plausible than multiple smaller deformations visible for a2 0.
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Model Variation
Recurrence (Any)
Recurrence (Enhancing Core)
Coverage [%]
IoU+
Coverage [%)
IoU+
Ours, as (gg
0. (xy 0,
0.15
83.5
3.7
0.05 0.01
Ours, og
63.5 + 4.0
0.15 + 0.03
86.5 + 3.6
0.06 + 0.02
Ours, 0
70.0 + 3.8
0.16 + 0.02
87.4 + 3.4
0.06 + 0.02
Standard Plun (1.5cm)
70.0 + 3.8
0.16 - 0.02
87.3 3.6
0.04 + 0.01
Ours, 01
70.9
- 0.02
3.4
0.05 + 0.02
Ours, 02
88.1
0.02
Ours, 04
3.9
0.16
89.0 3.4
0.06 + 0.02
Static Grid Discretization (GliODIL, (16)
72.9 + 3.5
0,16 + 0.02
89.0 3.3
0.06 + 0.02
Ours
74.7 + 3.1
0.19 = 0.02
89.9 = 2.7
0.08 0.02
Table 4: Ablation study showing the impact of different model variations on recurrence coverage and IoU. Large deviations account for various sizes of tumors and recurrences, yet the results remain consistent and statistically significant (Table 3). *loU heavily penalizes the fact that the radiotherapy volumes are equal between baselines and much larger than typical recurrences. Equal volume is to make comparisons with the Standard Plan without volume/recall trade-offs.
D Imaging Model
The core of the method's alignment with the data is encapsulated by the loss function components, which associate the simulated outputs with key imaging traits such as the tumor core, surrounding edema, metabolic activity detected through FET-PET imaging, and visible brain tissues. This loss function depends on two learnable parameters Omaging = {Adown: pup and is expressed as:
Imaging Data Reconstruction
Lcree (c, Oup) + Ledemal (c, O Oup) + Metabolic Map (c) + rissue (m.) (11)
The loss function is composed of individual terms corresponding to distinct anatomical features:
Loore relates tumor cell concentrations above the threshold Oup to the tumor core region.
LEdema delincates the edema area surrounding the tumor, regulated by the lower and upper thresholds Oawnn and Oup-
Metaabllic Mup assesses the metabolic activity indicated by FET-PET signals within the edema and core regions, using simple correlation metric, Trissuus connects visible brain tissues with those inferred by the method.
These adaptive parameters {dowwn up enable the model to accommodate variations in MRI/FET-PET imaging contrasts and noise levels.
We adopt sigmoid functions to portray the gradational transitions observed at tumor region margins. The sigmoid, (zx), is specified as:
(
(12)
1te
Here, 3 modulates the steepness of the transition and is set to 3 50. For the tumor core:
N.
Lcore Oup)
Σ
(0up
c. a),
(13)
1=0
for the edema:
No a (0awn - + (O ch + +a)),
Teaterna (c, Oaown, Oup)
Σ
(14)
1=0
where offsets the thresholds and is set to 0.05.
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The metabolic activity within the tumor is evaluated by the loss term LPET, which measures the correlation between the simulated metabolic signal and actual FET-PET scan observations:
(c) pp ppr))
(15)
Here, corr(c ,p represents the Pearson correlation coefficient between the predicted tumor cell densities C, and the restricted FET-PET signal " across all voxels within within the edema and enhancing core regions, providing direct measure of how well the model predictions align with the observed metabolic profiles.
E
Assumption of Brain Symmetry at Initial Time Without the Tumor
Our model employs a multi-resolution analysis to assess brain symmetry, particularly useful since the brain's 's symmetry is more evident at coarser resolutions. This approach is applied to key tissue types: white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF), each analyzed at full, 2x, and 4x downsampled levels to accommodate both the brain's structural nuances and computational efficiency.
The process for each tissue type involves converting tissue data into particles, then calculating symmetry loss at each resolution. This calculation entails:
1. Downsample the tissue representation, using average pooling
2. Divide the tissue's downsampled representation along its height, reflecting the brain's hemispherical division.
3. Mirror one hemisphere and compare it to the other, quantifying symmetry by calculating the mean absolute difference between them.
The total symmetry loss accumulates across all tissues and resolutions to measure deviation from an ideal symmetric state:
2 Σ
Litaatyy Brain (m)
Σ1
(oss scale factor 2*).
(16)
The multi-resolution approach not only addresses the limitations of assuming brain symmetry at detailed levels but also streamlines computational efforts. The smoother landscape at downsampled resolutions especially favorable for particle movement through gradient descent.
F
Synthetic Experiments
To calibrate the loss function weights, we generated a synthetic dataset for single focal and localized multi-focal tumors by solving PDEs using finite difference method numerical solver [13]. An average brain model was used to represent the spatial distribution of brain tissues. Tumor growth model parameters, elasticity, imaging model parameters, and 1 or 3 focal locations were varied using uniform random distributions.
The metric used was the RMSE between the learned and ground truth tumor cell distributions. In total, 100 synthetic patients were generated. Table 5 provides the details the parameter ranges used for generating the synthetic single focal and multi-focal tumor datasets.
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Table 5: Parameter ranges for generating synthetic single focal and multi-focal tumor datasets
Shared parameters
Parameter
Min
Max
D w
0.035
0.2
7
1.5
P
0.035
0.2
R
10
30
necro
0.70
0.85
Oup
0,45
0.60
down Tim
0.15
0.35
100
Single focal tumor center (mm)
(20., y0. 20)
57.6
96
Multi-focal tumor centers (mm)
Tumor center (z)ga yo. 2g)
57.6
96
Tumor center (w yū. 20)
(x yd. 2))
9.6
Tumor center (x)) Vo 20
(x Vo- 26 9.6
G Material Parameters for Biological Tissues
The table below summarizes the Young's modulus and Poisson's ratio for different tissue types used in the hyperelastic modeling of biological tissues.
Table 6: Young's modulus and Poisson's 's ratio for biological tissues.
Tissue Type
Young's 's Modulus (Pa)
Poisson's Ratio
Gray Matter (GM)
2100
0.4
White Matter (WM)
2100
0.4
Cerebrospinal Fluid (CSF)
100
0.1
Tumor
8000
0.45
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NeurIPS Paper Checklist
1. Claims
Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification:
The claims are justified. The physics loss is clearly formulated and the results are visualized.
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The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
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It is fine to include aspirational goals as motivation as long as is clear that these goals attained by the paper.
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Question:
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Answer: [Yes]
Justification: Yes, we have addressed the multi-resolution grid, which operates under the assumption that the problem hand can reliably approximated lower-than-targer resolutions.
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Question: For each theoretical result, does the paper provide the full of assumptions and complete (and correct) proof?
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Answer: [NA]
Justification: There no theoretical results a striet sense. It easy to see that At + 0, Az + 0 the RHS of the discrete residual converges to the original PDEs.
Guidelines:
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Reviews:
[{"review_id": "oWTFndBKbl", "reviewer": "Reviewer_smuQ", "summary": "This paper presents a method for brain tumor modeling through a joint data-driven and physics-based approach. The proposed approach leverages a multi-task loss that incorporates physical assumptions and prior healthy anatomy to more faithfully model the tumor cell distribution. This method outperforms existing computational approaches and the clinical standard with respect to recurrence coverage", "questions": "- According to line 207, recurrence coverage is the “percentage of the tumor segmentation… that is encompassed within the radiotherapy target volume defined by each model.” If this is true, then wouldn’t a predicted radiotherapy target volume consisting of the entire scan obtain 100% recurrence coverage? Unless I am misunderstanding, an existing overlap metric (which penalizes such “overprediction”) might be more appropriate.\n- Line 5: “Deep-learning based” -> “Deep learning-based”\n- Line 234: Missing parenthetical “(Physics-Constrained:”", "limitations": "Limitations are briefly addressed in Section 7.", "rating": 5, "confidence": 1, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The presentation quality is extremely high. Care was taken to logically organize sections of the paper with helpful illustrations and definitions throughout.\n- The treatment of related work is thorough, and domain-specific details are effectively communicated throughout.\n- Each component of the proposed method is well-motivated and specifically catered to the application domain.\n- Experiments appear to be sound, and the proposed method outperforms relevant baselines.", "weaknesses": "- I am slightly confused by the recurrence coverage metric. Has this metric been used in prior work? Why not a well-established overlap metric such as intersection over union, for example?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your thoughtful feedback regarding the recurrence coverage. Please note our global response comment with additional results and explanations. Below, we address your specific questions.**\n\n\n**Q1**: *According to line 207, recurrence coverage is the “percentage of the tumor segmentation… that is encompassed within the radiotherapy target volume defined by each model.” If this is true, then wouldn’t a predicted radiotherapy target volume consisting of the entire scan obtain 100% recurrence coverage? Unless I am misunderstanding, an existing overlap metric (which penalises such “overprediction”) might be more appropriate.* **Weakness**: *Why not a well-established overlap metric such as intersection over union, for example?*\n\n**A1**: We apologize for not communicating it properly in the manuscript. The radiotherapy target volume is fixed and equal to the standard plan total volume. The standard plan was generated according to the current clinical guidelines [n6] (1.5cm around the visible preoperative tumor). The radiation volume is bounded by the standard plan volume and is taken equal among baselines, typically ranging from 10% to 25% of the brain volume depending on the extent of the preoperative tumor.\nBoth metrics (IoU and recurrence coverage) reward overlap and penalize overprediction but in different ways. Recurrence coverage can be seen as recall and is calculated as $\\frac{|R \\cap V|}{|R|}$, where R is the set of recurrence voxels and V is the model-selected radiotherapy volume. The volume |V| is always equal across all baselines for a given patient, defined by the total volume of the standard plan. IoU heavily penalizes a solution for being larger than the eventual recurrence. The small IoU values across all baselines may be misleading without the context, as well as biased towards a few large recurrences (|R| comparable with |V|), skewing the metric. However, we absolutely agree that reporting more metrics might help to contextualize our results. Therefore, based on the reviewer's suggestion, we have included the IoU in Table 1 of the one-page PDF as well as the RMSE for synthetic results in Table 2 of the same document.\n\nIn addition, it would not be easy to compare models with different recalls and different volumes, as it is unclear how to handle the trade-offs in the clinical setup.\n\n**Q2,Q3**: *Line 5: “Deep-learning based” -> “Deep learning-based”, Line 234: Missing parenthetical “(Physics-Constrained:”.*\n\n**A2,A3**: Thank you for pointing this out.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2gY27ETx49", "reviewer": "Reviewer_YNED", "summary": "This paper presents a new approach to integrate physics-based tumor growth constraint with multi-modal imaging data to predict tumor cell distribution and thus enhance tumor treatment planning for glioblastomas. The core of the methods includes a discrete physics residual and initial assumptions encoding initial tumor distribution and symmetrical pattern between hemispheres. Experiments were performed on a small real dataset, and the success was measured by the “recurrence coverage” defined as the percentage of tumor (based on follow-up MRI) within the model-predicted volume. Experimental results demonstrate improved recurrence coverage of the presented method compared to fully data-driven or fully physics-based methods.", "questions": "Clarifications on the data split used to derive the reported results will be appreciated. \n\nThe statistical significance of the reported performance (Table 1) will be appreciated. \n\nClarifications on the results related to Fig 5 (see questions above) will be appreciated. \n\nHow sensitive are the presented methods to the different terms in the loss functions, and which are the most important terms?\n\nHow is asymmetry measured as an optimization objective?", "limitations": "The authors presented some discussion about the limitations of the current work. Adding discussion about limitations related to the limited sample size for the experiments would be appreciated.", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Ability to integrate physics-based constraints and multi-modal imaging data is important and a difficult problem to address. The presented work is thus of important potential in bringing physics-informed learning into biomedical tasks.\n\nThe dataset and task considered are quite complex involving multi-modal images and pre-operative and follow-up comparisons. I applaud the authors for the efforts devoted in this type of challenging tasks.", "weaknesses": "The experimental evaluation of the presented work was performed on a very small dataset with unclear training-test split. This raises some question about the general conclusion that can be drawn from the results obtained on limited samples.\n\nGiven the reported standard deviation (eg. Table 1), the obtained margin of improvement appears to be marginal without statistical significance. \n\nThe writing of the paper lacks clarify at places. For instance, the results presented in Figure 5 was unclear. What are the “greater”, “less”, and “equal” categories, and why a bigger gap between the “greater” and “less” categories indicate a favorable performance?\n\nThe objective of the presented model includes a large number of terms. The sensitivity of the model performance to the include/exclusion of these different terms and their hyperparameters deserves substantial analyses that are missing in the current paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your thoughtful feedback regarding the dataset size and lack of statistical tests, despite relatively high uncertainties. We appreciate acknowledging the challenges of the task.  Please note our global response comment with additional results and explanations.**\n\n**Q1:** *Clarifications on the data split used to derive the reported results will be appreciated. Additional weakness:  performed on a very small dataset.*\n\n**A1:** Please let us mention this is the largest publicly available dataset of this kind, including metabolic PET imaging alongside MRI, and follow-up with recurrence. For example, in [n1], [n2], and [n3] PET-based radiotherapy planning has been evaluated in comparably sized datasets. However, from a machine learning perspective, the size of the dataset is what effectively forces us to use physics-based assumptions and regularization. \nOur hyperparameters were selected in synthetic studies with the methodology discussed in Appendix D. Our method does not use recurrence images at any stage other than for final testing, and the synthetic patients have fixed seeds for reproducibility purposes. The clinical dataset is, therefore, a fully unseen test set for our results. Only the NN (Unconstrained) baseline was trained on recurrences since it is purely data-driven and with 80/20 train/test split over 5x cross-folding, and we report only the results from the unseen folds. \n\n**Q2:** *The statistical significance of the reported performance (Table 1) will be appreciated.* \n\n**A2:** Please see Table 3 in the one-page PDF with the tests. Our results are statistically significant and will be included in the final manuscript thanks to the reviewer suggestion. We understand the skepticism drawn from Table 1 in the main manuscript alone. \n\n**Q3:** *Clarifications on the results related to Fig 5 (see questions above) will be appreciated.* \n\n**A3:** We appreciate your comment and recognize that our explanation could have been clearer. In Figure 5, “Greater,” “Less,” and “Equal” refer to the quality of patient outcomes given radiotherapy from various methods (denoted below each histogram) compared to the standard plan in terms of recurrence coverage. “Greater” quantifies how many patients would have received a better plan, “Less” - how many would have received a worse plan. The gap between “Greater” and “Less” shows how many patients would benefit from our method compared to the standard plan. We observe a consistent improvement with our method, illustrating superior performance beyond just higher mean recurrence coverage, which could be skewed by outliers. This was meant to complement Table 1, but in hindsight, statistical test results with low p-values would be more straightforward (as in Table 3 of the one-page PDF).\n\n**Q4:** *How sensitive are the presented methods to the different terms in the loss functions, and which are the most important terms?*\n\n**A4:** We agree that an extension of the ablation studies is valuable, and we have provided such studies in Figure 1 and Table 1 of the one-page PDF. Additional information regarding the loss coefficients (see Figure 1a of the one-page PDF for reference regarding alphas):\n\n$\\alpha_2$ is essential for grid stability. Without it, the grid would be unstable, causing folding and rendering the results meaningless. If set to zero, $\\alpha_2$ can be replaced by a Laplacian-based regularization, as visualized in Figure 1b and 1c, though this clearly results in less biologically plausible deformations. \n\n$\\alpha_4$ regularizes the initial anatomy to bound $\\gamma$ from Eq 3. It should be kept low due to the natural asymmetry in brains. More details are provided in Appendix C.\n\n$\\alpha_6$, $\\alpha_7$, $\\alpha_8$, and $\\alpha_9$ are crucial for grounding PDE solutions in reality (most important, all other losses can be seen simply as regularisation of these components). Visible data loss cannot be calibrated using synthetic studies as all the other alphas. Instead of using recurrence data to calibrate them, we set them such that they provide a balanced contribution to the total loss\n\n $\\alpha_3$ should be kept low since the initial tumor shape is unknown. However, decreasing it too much causes a local minimum with no tumor. \n\n$\\alpha_1$ serves as the main regularization term. It should not be $\\rightarrow \\infty$ to avoid solutions not flexible enough to properly accommodate patient data (see Numerical Physics Simulations results in Table 1 of the manuscript; this setup performs poorly), nor should it be $\\rightarrow 0$ to prevent overfitting to visible data alone, effectively collapsing to standard plan level performance.\n\nRemoving elements of the visible data loss introduces ill-posedness and degradation of synthetic results (see Table 2 of the one-page PDF). Removing tumor growth physics regularizations effectively reduces our solution to performance close to the standard plan. Removing particle dynamics (static grid) reduces our model to [16] Static Grid Discretization (“GliODIL”). \n\n**Q5:** *How is asymmetry measured as an optimization objective?*\n\n**A5:** Asymmetry is measured by calculating the total symmetry loss across different tissue types (white matter, gray matter, cerebrospinal fluid) and resolutions (full, 2x, 4x downsampled). \n\nThe steps are:\n* Downsample the tissue data.\n* Divide the downsampled data into two halves, corresponding to the brain's hemispheres.\n* Mirror one hemisphere and compare it to the other by calculating the mean absolute difference.\n\nPlease see Appendix C for a formal definition of the symmetry ('Healthy Brain') loss.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FUR8SCLuHT", "reviewer": "Reviewer_ZDBA", "summary": "The authors propose a method using soft physics-based regularization to predict the distribution of brain tumor. The main contribution is a novel discretization scheme of physics equations to model brain tumor.", "questions": "1. Is there any bias in the evaluation metrics since there are no ground truth labels and authors use recurrence coverage?", "limitations": "1. The authors should discuss the limitations of their methods in more details, especially related to the lack of ground truth labels to train the model", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This work is an improvement for physics-based approaches for medical imaging \n\n2. The proposed method of discretizing the physics residuals is novel and potentially applicable to related problems and of interest to the research community \n\n3. The paper is clearly written and the technical details are sound and sufficient", "weaknesses": "1. The main weakness is the lack of ground truth data to train and evaluate the model performance. The evaluation is performed not on the original task of brain tumor detection but rather on a downstream task of “recurrence coverage”, which measures the percentage of the tumor detected in follow-up MRIs (rather than the original). \n\n2. Related to above, the authors only use one metric to evaluate their model, which may be insufficient, especially since there is no ground truth labels", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your thoughtful feedback regarding the ground truth labels and limited amount of available data. Please note our global response comment with additional results and explanations. Below we address your specific questions.**\n\n**Weakness**: *Lack of ground truth data to train and evaluate the model performance. The evaluation is performed not on the original task of brain tumor detection but rather on a downstream task.*\n\n**Answer**: We acknowledge that this aspect of the problem was not adequately communicated in the manuscript. We thank the reviewer for raising this issue, which has enabled us to refine our message in the discussion.\n\nPersonalizing radiotherapy is a critical but currently unmet need in neuro-oncology. This problem is clearly complicated by the fact that there is indeed no ground truth for the spread of tumor cells into the brain. To solve this problem, we developed a physics-informed framework that estimates this tumor cell spread from pre-operative imaging, with the clear aim of informing personalized radiotherapy for glioblastoma patients despite the lack of reliable, vast amounts of training data for purely data-driven approaches. To evaluate the quality of our predictions and compare it to several baselines, we consequently evaluated all models against the relevant clinical task for which they were developed. Estimating the recurrence coverage and comparing it to standard clinical radiotherapy plans is a well-established metric in this field [n4,n5]. We, therefore, disagree with the notion of this being a weakness of our work, but rather want to emphasize that our method - which is the first physics-informed framework that tackles deformations and the underlying biological processes in any field - excels at solving the highly challenging and clinically relevant task of personalized radiotherapy planning. We outperform all evaluated methods with statistically significant results (Table 3, one-page global PDF). \n\n**Question**: *Is there any bias in the evaluation metrics since there are no ground truth labels and authors use recurrence coverage?*\n\n**Answer**: We now understand that communicating results only through a domain specific metric is limiting our message. In addition, there are scenarios where major parts of our method could be applicable to problems that have ground truth labels/distributions. Following the reviewer's suggestion, our method and the previous state-of-the-art [16] have been additionally evaluated on synthetic cases generated by numerical physics solvers under various conditions (both ill-posed and well-posed, violating modeling assumptions about the initial condition of the tumor and adhering to them). This is highlighted in Table 2 of the attached one-page PDF. \n\nThe Recurrence coverage metric we employed is a clinically relevant metric that directly evaluates the models’ performance on the clinical task they were developed for. By aiming to encompass areas of high tumor cell density (which are not visible on standard MRI, but are prone to give rise to recurrence during the disease course) into the radiotherapy plan while keeping the total radiation volume constant with respect to the standard plan, recurrence coverage immediately quantifies any improvements in radiotherapy planning. However, we agree that it is important to judge models through multiple metrics’ lenses and have, therefore, now added also IoU alongside the recurrence coverage.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tsptZWPUqT", "reviewer": "Reviewer_S9Sg", "summary": "The authors proposed a discrete physic-based loss function which can be minimized to infer the spatial distribution of tumor cell distribution, which can not be assessed by imaging data only. The loss maps imaging data to a tumor tissue biomechanical model to solve inverse problems and to better understand individual tumor progression processes. One specific promise of the model is that it can learn the unknown initial condition of the brain tissue before the pathology. Their model is validated by achieving results comparable to state-of-the-art results on the task of tumor recurrence prediction from preoperative brain MRI.", "questions": "- Could the authors highlight the differences between [16] and their paper? Does addition of the tissue deformation in the physical modeling really improve the results?\n- Could the authors better explain the interest of the initial condition reconstruction?\n- Is the data-driven neural network trained on only 20% of the 58 data from radiotherapy planning dataset? Is it enough to train such a model? Could we obtain better results by increasing the training dataset?\n- How are inferred the parameters theta_up and theta_down? \n- In the appendix, in section B, it seems that c_i should be written instead of u_i (equation 14).\n- While it seems that visually the brain tissues exhibit symmetry, it should not be completely symmetric as we can observe on Figure 3 on the average brain template. It seems that “Healthy anatomy” term could lead to overly symmetric structures (see Figure 3 on the learned initial condition between the right and left hemisphere). It appears to me that this could be a limitation of the reconstruction method. Could the authors elaborate on this? The value of the paper could be increased with some analysis/discussion of this reconstruction.", "limitations": "The authors incorporated a “limitations” paragraph to address the convergence seed of their method and their tissue deformation model. However, the paper lacks some discussion in regard to their contribution compared to reference [16]. To which extent does the tissue deformation model improve the results? How can we assert the correctness of the initial condition reconstruction?", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The proposed method incorporates brain tissue deformation seen on medical images to include a tissue displacement term in their biomechanical to more accurately simulate individual tumor growth. Thanks to this contribution, the model can predict an estimation of the brain tissue before the apparition of the tumor thanks to a loss term ensuring symmetry of brain tissue structures to simulate a healthy anatomy. The explanation about the different loss terms are clear and well written.", "weaknesses": "In the reference [16], presented as the previous state-of-the-art method for recurrence coverage prediction, it seems that they obtained better results than the results displayed in table 1. More precisely, they obtained a recurrence coverage on all tumor areas of 76.09% which is better than the method presented in the submitted paper. It appears to me that the models were validated on the same dataset, but perhaps I missed something. Could the authors clarify this point?\n\nFollowing the previous point, it seems the paper lacks discussion about the use of the “Healthy anatomy” loss term. Compared to the results displayed in [16], it seems that there are no significant improvements using this new loss. How can we assert the correctness of the initial healthy brain tissue reconstruction? The authors claim that this initial state could be used to better identify post-surgical complications, but no clear method is described.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Thank you for your thoughtful feedback and diligent comment regarding the numbers in [16].  Please note our global response comment with additional results and explanations.**\n\n**Q1**: *Could the authors highlight the differences between [16] and their paper? Does addition of the tissue deformation in the physical modelling really improve the results?  [16] obtained a recurrence coverage on all tumor areas of 76.09% which is better than the method presented in the submitted paper*\n\n**A1**: We confirm that the original authors of [16] reported different (higher) numbers for both their method and the standard plan for the Recurrence (Any) experiment. We reproduced their numbers (76.09%) using a 2cm margin instead of the 1.5cm margin they claimed. After contacting them, they acknowledged the error and will correct it. The 2cm margin used to be standard but was later changed to 1.5cm, likely causing their mistake. Larger margins increase radiotherapy volumes, naturally raising recall for all baselines. Results for 2cm and 1.5cm cannot be directly compared. Comparing correct 1.5cm margin numbers, we outperform their method (Standard Grid Discretization, \"GliODIL\", cf. Table 1 in the manuscript).\n\nOur method is a generalization of [16], where authors assume $\\gamma$ in our Eq 3 to be 0, which makes the tissues static throughout the growth process. This is empirically not true since the deformations are clearly visible on the MRI. While our method does not require more input data, using the deformation allows us to increase the recurrence coverage while also providing a more physically plausible solution.\n\nTo the best of our knowledge, this is the first learnable approach using physics-informed losses that applies to dynamical domains with underlying physical processes on a large or any scale.\n\n**Q2**: *Could the authors better explain the interest of the initial condition reconstruction? The authors claim that this initial state could be used to better identify post-surgical complications, but no clear method is described.*\n\n**A2**: Effectively, initial condition assumptions further constrain the solution space, as just the equations alone are not enough because the equations themselves have many unknowns ( $\\theta_\\text{Dynamics}$ ).\n\nWe need another time-point (other than preoperative) to further constrain the solution. While the initial condition in itself (position and shape of the initial tumor, initial anatomy) are not directly used to estimate the tumor cells at the preoperative timepoint, they are essential to receive a plausible unique solution. The importance of the initial reconstruction is presented in Table 1 of the one-page PDF, where setting this loss $\\alpha_4$ to 0 clearly reduces performance.\n\nBesides improving tumor cell density estimation, inferring the initial brain anatomy might have further clinical uses: e.g., knowing the “healthy” anatomy prior to tumor development eases atlas-based registration approaches to perform voxel-based lesion mapping (such as in [n7]). However, such uses are beyond the scope of our evaluation.\n\n**Q3**: *Is the data-driven neural network trained on only 20% of the 58 data from radiotherapy planning dataset? Is it enough to train such a model? Could we obtain better results by increasing the training dataset?*\n\n**A3**: The split for the NN (Unconstrained) is 80/20 for train/test, meaning 80% of data is used for training and 20% for testing. We employ 5-fold cross-validation to ensure that all cases are inferred by the model without overfitting. If the 80/20 split produces very low results, it is unlikely that changing the split would significantly alter the outcome.\n\n**Q4**: *How are the parameters theta_up and theta_down inferred?*\n\n**A4**: They are inferred in the same optimization loop as the tumor cell/particle solution and dynamics, using the Adam optimizer. Their true values are practically unknown and depend on factors such as the MRI instrument's field strength, contrast, and normalization. Given the same segmentations, a large gap between theta_up and theta_down values would result in very localized, steep tumors, while a small gap would lead to diffusive tumors spreading across the brain. The knowledge about the steepness comes from the FET-PET modality, whose metabolic map signal directly correlates with the cells.\n\n**Q5**: *While it seems that visually the brain tissues exhibit symmetry, it should not be completely symmetric as we can observe on Figure 3 on the average brain template. It seems that “Healthy anatomy” term could lead to overly symmetric structures (see Figure 3 on the learned initial condition between the right and left hemisphere). It appears to me that this could be a limitation of the reconstruction method. Could the authors elaborate on this? The value of the paper could be increased with some analysis/discussion of this reconstruction.*\n\n**A5**: The symmetry of the brain hemispheres is only an approximation of reality. However, an assumption on the initial anatomy is necessary for the uniqueness of the solution (namely, the parameter $\\gamma$ from Eq. 3 which couples growth of tumor cells with tissue dynamics needs at least two constraints at different time-points to be bounded). We soften the symmetry assumption by calculating it in lower resolutions (fine, downsampled 2x and 4x, cf. Appendix C). This puts more emphasis on larger structures than individual voxel tissue intensity, such as the orientation of the ventricles and rough tissue symmetries. In Fig. 3 of the main manuscript, the learned initial anatomy shows some asymmetries, i.e. in the top left and top right corners. An interesting future direction is to find even better penalties for the healthy brain assumption, such as using a neural network to identify and quantify discrepancies in reconstructed healthy brain tissues. We appreciate the suggestion to expand the discussion on this topic and will incorporate the points above in the final version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear Reviewers,

**Please see the one-page PDF with additional ablation studies, synthetic data results (with ground truth), and statistical tests for the recurrence coverage results.** 

We thank all reviewers for their constructive feedback that helped us improve the paper. We are encouraged by their comments:

* "I applaud the authors for the efforts devoted to this type of challenging task."

* "The proposed method of discretizing the physics residuals is novel and potentially applicable to related problems and of interest to the research community."

* "The explanation about the different loss terms is clear and well written."

* "The presentation quality is extremely high."

We have been working diligently to address your critiques: specifically, the use of a single metric in our experiments, the absence of an extended ablation study, and resolving some communication issues related to the presentation of our results, the train/test split, and addressing the size of our dataset compared to other domains.

**Train/test split of our model, size of the dataset**

We would like to emphasize that this is the largest publicly available dataset of this kind (with the follow-up scans). This size of data is standard for medical studies and sufficient to motivate clinical trials (see, for example, [n1,n2,n3] on clinical investigations into the use of PET for radiotherapy planning). The problem we address is an example of a vital (cancer research) problem with very costly and time-consuming data gathering but because it is so important to the community and to improve patient outcomes, researchers need to develop new ML methods with these limitations in mind. We have now added this explanation to the discussion.
The size of the dataset of 58 patients is relatively small compared to other ML domains, which effectively forces to use physics-based assumptions and physics-informed regularizations to approach the problem, which is not solvable otherwise (as shown by the weak results of purely data-driven methods in Table 1 of the manuscript). Despite the limited data size, our improvements are statistically significant (see Table 3 in the one-page PDF).

The follow-up images with visible tumor recurrences are not used by our method during the optimization or hyper-parameter search at any point; therefore, the dataset can be considered 100% unseen. The decision not to use any recurrence data for hyper-parameter tuning is to avoid diluting the already small dataset. 

**Ground truth labels and generalization**

Personalizing radiotherapy is a challenging task because there is no ground truth for the true spread of tumor cells into the brain. To meet this urgent clinical need, we developed our framework precisely to handle this scenario. Thus, our framework was tuned using well-defined synthetic results in various scenarios (Appendix D) and only validated in the clinical situation of radiotherapy planning against the later recurrence. We would like to highlight that our framework is the first physics-informed framework that estimates deformations and underlying biological processes successfully on a large or any scale. 

To show how the method performs on tasks where there is ground truth, thanks to the reviewer ZDBA suggestions, our method and the previous SOTA are additionally evaluated on synthetic cases generated by numerical physics solvers under various conditions (cf. Tab.2 in the PDF and extended discussion in the answers to ZDBA).

**Recurrence Coverage metric**

Recurrence coverage is an established metric [n4, n5], motivated by clinical translation. Given the potential side effects of radiotherapy, it is crucial to balance recurrence coverage against radiotherapy volume. By keeping the volumes constant, we eliminate the need to consider trade-offs related to volume. Recurrence Coverage is defined as recall $\frac{|R \cap V|}{|R|}$ where $R$ is the set of recurrence voxels and $V$ is the predicted radiotherapy volume. Importantly, $|V|$ is always equal across all baselines for a given patient and is defined by the total volume of the standard plan (1.5cm volume around the preoperative tumor core).
In the PDF, we have included (thanks to the suggestion from reviewer smuQ) results for IoU: $\frac{|R \cap V|}{|R \cup V|}$. The values are relatively small across all methods since $|R| \ll |V|$, leading to $|R \cap V| \ll |R \cup V|$, but can bring new perspective. 

**Parameter importance**
 
The tumor cell localization task requires many assumptions to constrain the problem. Please refer to Figure 1a in the one-page PDF for information on the different loss terms (expanded in answers to reviewer YNED). Extensive ablation experiments are presented in Table 1 of the PDF,  thanks to the reviewer YNED suggestion. 

**Symmetric healthy brain assumption**

We do not necessarily need to infer the initial anatomy for inference of the tumor cells, however, we need additional constraints to bound $\gamma$ from Eq. 3 that couples tumor cells with tissue dynamics. While symmetry of the healthy brain is a coarse assumption (rightly pointed out by reviewer S9Sg), we soften it by quantifying the asymmetry in lower resolutions (see Appendix C). Previous SOTA [16] assumes static tissues, which is an even coarser assumption, since the deformations are clearly visible in the MRI images.

[n1] Rieken et al., Radiother. Oncol., 2013. doi:10.1016/j.radonc.2013.06.043

[n2] Munck Af Rosenschold et al., Neuro-Oncol., 2015. doi:10.1093/neuonc/nou316

[n3] Grosu et al., Int. J. Radiat. Oncol. Biol. Phys., 2005. doi:10.1016/j.ijrobp.2005.02.059

[n4] Lipkova et al., IEEE Trans. Med. Imaging, 2019. doi:10.1109/TMI.2019.2902044

[n5] Metz et al., Neuro-Oncol. Adv., 2024. doi:10.1093/noajnl/vdad171

[n6] Niyazi M, et al., Radiother Oncol. 2023;184:109663. doi: 10.1016/j.radonc.2023.109663

[n7] Lai A, et al., J Clin Oncol. 2011;29(34):4482-90. doi: 10.1200/JCO.2010.33.8715

Meta-review: This work presents a new approach to integrate physics-based tumor growth constraint with multi-modal imaging data to predict tumor cell distribution and thus enhance tumor treatment planning for glioblastomas. The core of the methods includes a discrete physics residual and initial assumptions encoding initial tumor distribution and symmetrical pattern between hemispheres. Experiments were performed on a small real dataset, and the success was measured by the recurrence coverage defined as the percentage of tumor (based on follow-up MRI) within the model-predicted volume. Experimental results demonstrate improved recurrence coverage of the presented method compared to fully data-driven or fully physics-based methods.

The presented work is of important potential in bringing physics-informed learning into biomedical tasks and demonstrates the ability to integrate physics-based constraints and multi-modal imaging data. The proposed method of discretizing the physics residuals is novel and potentially applicable to related problems and of interest to the research community. The paper is clearly written and the technical details are sound and sufficient.

The reviewers have some concerns: the experimental evaluation of the presented work was performed on a very small dataset with unclear training-test split. This raises some question about the general conclusion that can be drawn from the results obtained on limited samples. Given the reported standard deviation, the obtained margin of improvement appears to be marginal without statistical significance. The objective of the presented model includes a large number of terms. The sensitivity of the model performance to the include/exclusion of these different terms and their hyperparameters deserves substantial analyses that are missing in the current paper. Further concerns are about the evaluation: the main weakness is the lack of ground truth data to train and evaluate the model performance. The evaluation is performed not on the original task of brain tumor detection but rather on a downstream task of “recurrence coverage”, which measures the percentage of the tumor detected in follow-up MRIs (rather than the original). Related to above, the authors only use one metric to evaluate their model, which may be insufficient, especially since there is no ground truth labels. 

The rebuttal addressed these concerns, and the paper has been improve accordingly. Most reviewers are convinced by the rebuttal.  The work addresses an important research problem and shows great promise, it has the potential to be highly impactful.
Decision: Accept (poster)