Pith. sign in

REVIEW 3 major objections 6 minor 66 references

Patient-specific prediction of glioblastoma growth via reduced order modeling and neural networks

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A neural network predicts glioblastoma growth 150x faster from two MRIs.

desk verdict Useful pipeline, but the inverse network's validation split is leaky, so the reported accuracy numbers overstate patient-specific generalization. read the letter →

arxiv 2412.05330 v2 pith:ISQC4LU7 submitted 2024-12-04 eess.IV cs.LGcs.NAmath.NAphysics.bio-phq-bio.TO

D. Cerrone , D. Riccobelli , S. Gazzoni , P. Vitullo , F. Ballarin , J. Falco , F. Acerbi , A. Manzoni
show 2 more authors
P. Zunino P. Ciarletta
This is my paper · ORCID
classification eess.IVcs.LGcs.NAmath.NAphysics.bio-phq-bio.TO MSC 92C5065M6068T07
keywords glioblastomadiffuse-interfacemodelproperorthogonaldecompositionneuralnetworksurrogatepatient-specificparameteridentificationtumorgrowthpredictionsensitivityanalysisneuroimaging
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that a patient-specific forecast of glioblastoma growth can be made fast enough for clinical use by combining a diffuse-interface PDE model of tumor invasion with reduced-order surrogates. Its central claim is that two tumor segmentations taken 20 days apart are enough to recover the six biophysical parameters $\nu$, $M_0$, $\kappa$, $\delta$, $\delta_n$, $S_n$ that drive the model, and that a POD-neural-network surrogate then reproduces the full-order simulation about 150 times faster while tracking tumor volume with roughly 96% accuracy. The proof-of-concept is carried out on a realistic brain-shaped mesh reconstructed from one patient's MRI and diffusion tensor imaging, with synthetic tumor evolutions used to train and test the networks. The authors present this as the methodological foundation for a glioblastoma digital twin, not yet as a clinically validated tool.

What carries the argument

The load-bearing mechanism is the POD-NN surrogate pipeline built on the diffuse-interface model. Proper orthogonal decomposition (POD) compresses snapshots of the full-order finite element solutions for $\phi$, $\mu$, and $\hat{n}$ into a basis of 20 modes retaining about 95% of the solution energy; two neural networks then replace the expensive reduced-order projections. The forward network $NN_\phi$ maps $[\nu, M_0, \kappa, \delta, \delta_n, S_n, t]$ to the reduced coefficients of the tumor field, and the inverse network $NN_\mathrm{inv}$ maps the reduced coefficients of two tumor distributions separated by 20 days to the six parameters. The mechanism works because the reduced basis makes both directions low-dimensional: the inverse problem becomes a regression on $2N_\mathrm{POD}$ inputs rather than an optimization over the full PDE, and a forward evaluation takes seconds instead of the roughly 780 s cost of the full-order model. Global Morris and local Monte Carlo sensitivity analyses identify which parameters the data can actually constrain.

What would settle it

Run the trained pipeline on real longitudinal MRI and DTI from a glioblastoma patient not used in training: segment the tumor at two time points 20 days apart, estimate the six parameters, simulate the tumor distribution at a third time point, and compare with the actual segmentation. The central claim predicts volume accuracy near 96% and a morphology that follows the white-matter structure; a substantially larger volume error or a clearly wrong invasion pattern would falsify the transfer claim. A purely synthetic falsifier is to evaluate the surrogate on held-out parameter sets near the boundary of the Table 1 ranges: if the reduced basis learned from the 750 training sets cannot reproduce the full-order solution within the reported error, the surrogate claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that the inverse problem of patient-specific parameter identification for glioblastoma growth is tractable in real time once the forward model is compressed by proper orthogonal decomposition and learned by neural networks. Starting from the diffuse-interface system in Eq. (9) for the tumor phase field $\phi$, the oxygen concentration $\hat{n}$, and the chemical potential $\mu$, the authors build a reduced basis of $N_\mathrm{POD}=20$ modes that retains about 95% of the solution energy, train a forward network mapping the six parameters and time to reduced coefficients, and train an inverse network mapping the reduced coefficients of two tumor distributions at $t_0$ and $t_0+20$ days back to the parameters. On a synthetic test case built on a patient's anatomy, the recovered parameters are close to the ground truth (for example $\nu$ goes from 0.356 to 0.366 d$^{-1}$), and the forecast matches the full-order tumor volume to about 96% accuracy. The surrogate runs about 150 times faster than the full-order model, cutting the simulation time from roughly 780 seconds to about 5 seconds, and the parameter estimation itself takes seconds.

Load-bearing premise

The whole pipeline assumes that the diffuse-interface model in Eq. (9), with its literature parameter ranges and DTI-derived tensors, is an adequate description of real glioblastoma growth, and that two tumor snapshots 20 days apart carry enough information to identify the parameters that actually drive the forecast.

Editorial extensions

If this is right

  • If the framework transfers to real clinical scans, a treating team could estimate a patient's proliferation and oxygen parameters from two routine MRIs and simulate the likely tumor distribution at a future date within a single clinical visit.
  • Because the surrogate separates the fixed brain anatomy (tensors $T$ and $D$, mesh) from the six parameters, forecasts for a new set of parameters on the same anatomy are near-instant; the main remaining cost is retraining for a new patient's anatomy.
  • The sensitivity results imply that clinical calibration should focus on the proliferation rate $\nu$, the oxygen consumption rate $\delta_n$, and the oxygen supply rate $S_n$, while $\delta$ and $\kappa$ are weakly identifiable and can be pinned to literature values with little loss in volume prediction.
  • By learning reduced coefficients directly from simulation data, the pipeline avoids hyper-reduction techniques such as DEIM, so the reported speed-up does not require projecting the nonlinear terms of Eq. (9).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the authors do not pursue is to use the inverse network's output as a proposal generator inside a Bayesian or ensemble framework, converting the reported mean parameter error of about 15% into calibrated uncertainty intervals for surgical planning.
  • The 20-day input spacing is tied to the clinical follow-up protocol; a testable extension would train the same architecture on variable time gaps (10, 30, 60 days) to see how identifiability of $\nu$ and $\delta_n$ degrades as the gap shrinks.
  • Because all validation is synthetic, the 96% volume accuracy measures consistency with the model that generated the data; the clinically meaningful test is against real recurrences, which the authors state is the object of an ongoing study.
  • Since the weakly identifiable parameters $\delta$ and $\kappa$ are also those with wide literature ranges, a pragmatic variant would freeze them at population medians and reduce the inverse problem to four parameters, likely improving stability.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes a computational pipeline for glioblastoma growth prediction and patient-specific parameter identification. A diffuse-interface PDE model (Eq. 9) is discretized with finite elements, reduced via POD, and replaced by a neural network surrogate (POD-NN) for the forward problem. A second neural network (NNinv) maps the POD coefficients of two tumor distributions 20 days apart to six model parameters. All training and validation are performed on synthetic data generated from a single patient's brain anatomy (MRI/DTI). The authors report a ~150x speedup for the forward surrogate, a ~15% parameter estimation error, and "96% accuracy" in tumor volume forecasting, and they support the results with global Morris and local Monte Carlo sensitivity analyses.

Significance. If the inverse-network generalization were properly established, the pipeline would offer a fast, interpretable tool for estimating GBM growth parameters from longitudinal imaging, with potential clinical value. The use of a physics-based forward model with DTI-derived anisotropic tensors and the inclusion of global and local sensitivity analyses are strengths. However, the central claim of patient-specific generalization is not yet supported because the inverse-network test split is not parameter-disjoint and the accuracy metrics are not quantitatively defined. The work is a promising proof-of-concept, but its current validation is insufficient for the stated claims.

major comments (3)
  1. [Section 3.2] The inverse-network data set of 15,000 input-output pairs is split randomly into 11,000 training and 4,000 test pairs. Since all 20 pairs for a given parameter set are generated from the same simulation trajectory, a test pair almost certainly comes from a parameter set that also appears in the training set. Consequently, the reported 15% parameter error (Fig. 4) and the 96% volume accuracy claimed in Section 4 do not demonstrate generalization to a new patient's parameter set; they may reflect memorization of parameter-set-specific signatures. The authors should re-split the data by parameter set (e.g., hold out a subset of the 750 parameter sets entirely) and report test errors on unseen parameter sets. They should also state whether the parameter set in Eq. (10) used for the Fig. 5 proof-of-concept was held out during training.
  2. [Section 4 / Fig. 5] The conclusion states "an accuracy of 96% in forecasting tumor volume" but no metric is defined in the text or computed from the data shown in Fig. 5. The volume fraction curves appear close, but the paper should report a quantitative error measure (e.g., relative L2 error or relative volume error at t=30 days) computed on a properly held-out test set. As written, this claim is unverifiable and inconsistent with the vague "well-tracked" description in Section 3.2.
  3. [Sections 3.1-3.2 and Conclusions] The inverse network is trained and evaluated on synthetic data generated from a single brain anatomy and a single initial tumor condition (the Gaussian φ0 defined in Section 3.2). The proof-of-concept in Fig. 5 uses the same initial condition and anatomy. Thus the claim of "patient-specific parameter identification" is currently limited to the DTI-derived tensors; generalization to different tumor shapes, locations, and brain geometries is untested. Although the Conclusions acknowledge this, the Abstract and Introduction should be tempered to reflect that this is a single-anatomy, single-initial-condition proof-of-concept.
minor comments (6)
  1. [Section 3.2] The paragraph beginning "The computational demand of the POD-Galerkin solution is generally high..." appears twice verbatim; remove one occurrence.
  2. [Section 4] The reported speed-up is quoted as "approximately 150 times" in Section 3.2 and Fig. 5, but the Conclusions say "computational speed-up of approximately 99%"; the latter is undefined and inconsistent, and should be corrected.
  3. [Section 3.2] The elapsed time for parameter estimation is described as "of the order of seconds"; please report the actual inference time (e.g., the mean over the test set).
  4. [Section 2.1] The model in Eq. (9) includes parameters ε (diffuse interface thickness) and r (anisotropy tuning factor) that are set a priori; these should be listed explicitly as fixed parameters, and the sensitivity analysis should state that their effects are not studied.
  5. [Fig. 4] The learning curves show training and test errors for the direct and inverse networks, but the panels are not labeled; please clarify which panel corresponds to which network and whether the plotted error is absolute or relative.
  6. [Appendix A.2] In the normalization of the tensor T, the notation with hats (ˆT) is not clearly distinguished from T; please clarify the relationship between D, T, and ˆT.

Circularity Check

1 steps flagged · score 6.0 of 10

Inverse-network test split is not parameter-disjoint, so the reported parameter recovery and 96% volume accuracy partly reflect memorization rather than patient-specific generalization.

  1. fitted input called prediction [Section 3.2, Proof-of-concept, NNinv training/test data generation]
    "To train the inverse neural network, denoted by NNinv, we extract twenty pairs of tumor distributions, each separated by a distance of twenty days, for each of the 750 parameter sets. This results in a total of Ninv data = 15000 input-output pairs. These are then split into a training set containing Ninv train = 11000 elements and a test set with Ninv test = 4000 elements."

    Because all 20 pairs from one parameter set are draws from the same simulation, a random pair-level split makes test and training share parameter sets with near-certainty: for any test pair, the probability that none of its 19 sibling pairs was selected for the 11000-element training set is (4000/15000)^19, effectively zero. The network can memorize parameter-set-specific signatures instead of learning a patient-independent inverse map. The reported ~15% parameter error and the Fig. 5 recovery (Eq. (10) vs Eq. (11)) therefore do not demonstrate generalization to an unseen patient; the test 'prediction' is statistically forced by training on the same parameter sets. The paper neither reports a parameter-disjoint split nor states that Eq. (10) was held out.

full rationale

The forward model (Eq. (9)) is a mechanistic diffuse-interface PDE with parameter ranges from independent literature (Table 1) and is solved by FEM/POD; the ROM and POD-NN are standard surrogate constructions, so the forward derivation is self-contained and not circular. The central circularity is confined to the inverse-network validation: the train/test split is performed on input-output pairs rather than on parameter sets, so test pairs are drawn from parameter sets already represented in training. This makes the reported parameter-recovery accuracy and the 96% volume-forecast accuracy partly a memorization check, not a patient-specific prediction. The paper honestly labels the study a synthetic proof-of-concept, which limits clinical extrapolation but does not repair the split flaw. Same-group self-citations ([30], [32], [48]) support model and learning-strategy choices, but no uniqueness theorem or external result is smuggled in, so they are not independently load-bearing. Overall: partial circularity in the central validation claim, score 6.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the fidelity of the phase-field tumor model and on the sufficiency of two snapshots for parameter identification. The model uses literature-based parameter ranges and several constitutive choices (double-well potential, oxygen-dependent growth, DTI-derived tensors with a hand-set anisotropy factor). No new physical entities are introduced. The paper's own sensitivity analysis indicates that two of the six parameters are weakly identifiable, which is the main structural risk.

free parameters (2)
  • r (anisotropy tuning factor) = 3
    Set by hand in Appendix A.2 to modulate anisotropy of the DTI-derived tensor T; for r=1 the tensor is just a rescaling of D, but r=3 is chosen without fitting or justification.
  • epsilon (diffuse interface thickness) = not specified
    Section 2.3 fixes epsilon a priori and does not report its value; it controls interface width in the Cahn-Hilliard energy and affects tumor morphology.
assumptions (5)
  • domain assumption The brain is modeled as a two-phase mixture with saturation condition phi_c + phi_l = 1 and equal densities close to water.
    Section 2.1, Eq. (1). This is the foundation of the phase-field model.
  • domain assumption The mass source terms satisfy Gamma_c = -Gamma_l, enforcing mixture incompressibility (grad v = 0).
    Section 2.1, right after Eq. (1).
  • domain assumption Fickian constitutive law J = -(1/M0) T grad mu with Landau free energy Eq. (6) and double-well potential Eq. (7).
    Section 2.1, Eqs. (6)-(8).
  • domain assumption Tumor growth rate depends linearly on oxygen availability as Gamma = nu gamma (n/ns - delta) h(phi).
    Section 2.1, Eq. (8).
  • domain assumption Two snapshots of the tumor 20 days apart contain enough information to identify the six patient-specific parameters.
    Section 3.1 defines the inverse map; Section 3.4 shows weak identifiability of delta and kappa, so this premise is partially contradicted by the paper's own sensitivity analysis.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Patient-specific prediction of glioblastoma growth via reduced order modeling and neural networks." pith.science (2026). https://pith.science/paper/ISQC4LU7

@misc{pith2026241205330,
  author       = {Pith},
  title        = {Pith review of: Patient-specific prediction of glioblastoma growth via reduced order modeling and neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISQC4LU7}},
  note         = {Machine review of arXiv:2412.05330}
}
read the original abstract

Glioblastoma is among the most aggressive brain tumors in adults, characterized by patient-specific invasion patterns driven by the underlying brain microstructure. In this work, we present a proof-of-concept for a mathematical model of GBL growth, enabling real-time prediction and patient-specific parameter identification from longitudinal neuroimaging data. The framework exploits a diffuse-interface mathematical model to describe the tumor evolution and a reduced-order modeling strategy, relying on proper orthogonal decomposition, trained on synthetic data derived from patient-specific brain anatomies reconstructed from magnetic resonance imaging and diffusion tensor imaging. A neural network surrogate learns the inverse mapping from tumor evolution to model parameters, achieving significant computational speed-up while preserving high accuracy. To ensure robustness and interpretability, we perform both global and local sensitivity analyses, identifying the key biophysical parameters governing tumor dynamics and assessing the stability of the inverse problem solution. These results establish a methodological foundation for future clinical deployment of patient-specific digital twins in neuro-oncology.

Figures

Figures reproduced from arXiv: 2412.05330 by the authors.

Figure 1
Figure 1. Representation of the computational pipeline. The geometry and the distribution of the tumor is known at for t = t0. From this datum, we perform the POD and get the reduced order solution estimation for the direct problem (POD-NN procedure). Given the distribution of the grown tumor at t = t1 = t0 + 20 days, we train a neural network to solve the inverse problem estimating the patient-specific parameters. Xeon® Proc… view at source ↗
Figure 2
Figure 2. A representation of the computational domain extrac￾ted from the MRI data (a) with superposed color map of a component of the tensor D extracted from DTI data in mm2 d −1 (b); reconstructed domain of the brain cortex (c) and corresponding sections for grey matter (blue), white matter (light blue), and tumour (red) in the labeled mesh (d). guage for declaration of finite element discretization of variational forms. T… view at source ↗
Figure 3
Figure 3. Plot of the solution ϕ within a fixed sagittal plane intersecting the tumor centroid at t= 0 (left), 15 (center), 30 days (right). Solid lines indicate the FOM solution (black), the POD-Galerkin solution (orange), the POD-NN solution (red), and the FOM solution obtained using the parameter obtained in the inverse problem (blue) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Absolute (top) and relative (bottom) mean squared error e over the epochs in the training of the direct (top) and the inverse (bottom) neural networks. The solid lines indicate the errors over the training set (blue) and over the test sets (orange). is required, it may…
Figure 5
Figure 5. Figure 5: FOM, POD-Galerkin, POD-NN, and patient-specific FOM solutions of a GBL concentration ϕ at t= 0, 15, 30 days (top) and corresponding computational times (bottom,left). Volume fraction of tumor over time (bottom, right). The parameters used in the FOM and POD-Galerkin mo…
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Local sensitivity analysis results: probability distribu￾tions for each of the parameter estimated as an output of the inverse problem. For each distribution, the cor￾responding mean and standard deviation values are also displayed. values. This observation suggests th…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    P . Y. Wen et al. ‘Glioblastoma in adults: a Soci- ety for Neuro-Oncology (SNO) and European Society of Neuro-Oncology (EANO) consensus review on current management and future dir- ections’. Neuro-Oncology 22.8 (2020), pp. 1073– 1113

    work page 2020

  2. [2]

    H. L. Smith, N. Wadhwani and C. Horbinski. ‘Major features of the 2021 WHO classification of CNS tumors’. Neurotherapeutics 19.6 (2022), pp. 1691–1704

    work page 2022

  3. [3]

    D. N. Louis et al. ‘The 2021 WHO Classific- ation of Tumors of the Central Nervous Sys- tem: a summary’. Neuro-Oncology 23.8 (2021), pp. 1231–1251

    work page 2021

  4. [4]

    B. T. Whitfield and J. T. Huse. ‘Classification of adult-type diffuse gliomas: impact of the World Health Organization 2021 update’. Brain pathology 32.4 (2022), e13062

    work page 2022

  5. [5]

    Weller et al

    M. Weller et al. ‘EANO guidelines on the dia- gnosis and treatment of diffuse gliomas of adulthood’. Nature reviews Clinical oncology 18.3 (2021), pp. 170–186

    work page 2021

  6. [6]

    T. S. Deisboeck et al. ‘Pattern of self- organization in tumour systems: complex growth dynamics in a novel brain tumour spheroid model’. Cell Proliferation 34.2 (2001), pp. 115–134

    work page 2001

  7. [7]

    Stupp et al

    R. Stupp et al. ‘Radiotherapy plus Concomitant and Adjuvant Temozolomide for Glioblastoma’. New England Journal of Medicine 352.10 (2005), pp. 987–996

    work page 2005

  8. [8]

    Omuro and L

    A. Omuro and L. M. DeAngelis. ‘Glioblastoma and Other Malignant Gliomas: A Clinical Re- view’. JAMA 310.17 (2013), p. 1842

    work page 2013

Show all 66 references
  1. [9]

    T. J. Brown et al. ‘Association of the Extent of Resection With Survival in Glioblastoma: A Systematic Review and Meta-analysis’. JAMA Oncology 2.11 (2016), p. 1460

    2016

  2. [10]

    Rivera et al

    M. Rivera et al. ‘Updates on Surgical Manage- ment and Advances for Brain Tumors’. Current Oncology Reports 23.3 (2021)

    2021

  3. [11]

    Cavallo et al

    C. Cavallo et al. ‘The utilization of fluorescein in brain tumor surgery: a systematic review’. Journal of Neurosurgical Sciences 62.6 (2018)

    2018

  4. [12]

    S. L. Hervey-Jumper and M. S. Berger. ‘Max- imizing safe resection of low- and high-grade glioma’. Journal of Neuro-Oncology 130.2 (2016), pp. 269–282

    2016

  5. [13]

    Petrecca et al

    K. Petrecca et al. ‘Failure pattern following complete resection plus radiotherapy and temo- zolomide is at the resection margin in patients with glioblastoma’. Journal of Neuro-Oncology 111.1 (2012), pp. 19–23

    2012

  6. [14]

    Chirica et al

    C. Chirica et al. ‘One Step Forward—The Cur- rent Role of Artificial Intelligence in Glio- blastoma Imaging’. Life 13.7 (2023), p. 1561

    2023

  7. [15]

    Lemée, A

    J.-M. Lemée, A. Clavreul and P . Menei. ‘Intrat- umoral heterogeneity in glioblastoma: don’t forget the peritumoral brain zone’. Neuro- Oncology 17.10 (2015), pp. 1322–1332

    2015

  8. [16]

    Chirizzi et al

    C. Chirizzi et al. ‘Next-generation agents for fluorescence-guided glioblastoma surgery’. Bioengineering & Translational Medicine (2023)

    2023

  9. [17]

    N. F. Marko et al. ‘Extent of Resection of Glioblastoma Revisited: Personalized Survival Modeling Facilitates More Accurate Survival Prediction and Supports a Maximum-Safe- Resection Approach to Surgery’. Journal of Clin- ical Oncology 32.8 (2014), pp. 774–782

    2014

  10. [18]

    K. R. Swanson, E. C. Alvord and J. D. Murray. ‘A mathematical modelling tool for predicting survival of individual patients following re- section of glioblastoma: a proof of principle’. British Journal of Cancer 98.1 (2008), pp. 113–119

    2008

  11. [19]

    Gallaher et al

    J. Gallaher et al. ‘From cells to tissue: How cell scale heterogeneity impacts glioblastoma growth and treatment response’. PLoS Compu- tational Biology 16.2 (2020), e1007672

    2020

  12. [20]

    Storey et al

    B. Storey et al. ‘Modeling oncolytic viral ther- apy, immune checkpoint inhibition, and the complex dynamics of innate and adaptive im- munity in glioblastoma treatment’. Frontiers in Physiology 11 (2020), p. 151. Submitted — Under review 13 Patient-specific prediction of glio...

    2020

  13. [21]

    Surendran et al

    A. Surendran et al. ‘Agent-based modelling re- veals the role of the tumor microenvironment on the short-term success of combination te- mozolomide/immune checkpoint blockade to treat glioblastomas’. Journal of Pharmacology and Experimental Therapeutics 387.1 (2023), pp. 66– 77

    2023

  14. [22]

    Mongeon et al

    S. Mongeon et al. ‘Spatial computational mod- elling illuminates the role of the tumour mi- croenvironment for treating glioblastoma with immunotherapies’. npj Systems Biology and Ap- plications 10 (2024), p. 91

    2024

  15. [23]

    Falco et al

    J. Falco et al. ‘In Silico Mathematical Model- ling for Glioblastoma: A Critical Review and a Patient-Specific Case’. Journal of Clinical Medi- cine 10.10 (2021), p. 2169

    2021

  16. [24]

    R. M. Bowen. ‘Theory of Mixtures’. In: Con- tinuum Physics. Elsevier, 1976, pp. 1–127

    1976

  17. [25]

    D. M. Anderson, G. B. McFadden and A. A. Wheeler. ‘Diffuse-Interface Methods in Fluid Mechanics’. Annual Review of Fluid Mechanics 30 (1998), pp. 139–165

    1998

  18. [26]

    S. M. Wise et al. ‘Three-dimensional multis- pecies nonlinear tumor growth—I: Model and numerical method’. Journal of Theoretical Biology 253.3 (2008), pp. 524–543

    2008

  19. [27]

    M. C. Colombo et al. ‘Towards the Personalized Treatment of Glioblastoma: Integrating Patient- Specific Clinical Data in a Continuous Mechan- ical Model’. PLOS ONE 10.7 (2015), e0132887

    2015

  20. [28]

    G. A. Ateshian. ‘On the theory of reactive mix- tures for modeling biological growth’. Biomech- anics and Modeling in Mechanobiology 6 (2007), pp. 423–445

    2007

  21. [29]

    Byrne and L

    H. Byrne and L. Preziosi. ‘Modelling solid tu- mour growth using the theory of mixtures’. Mathematical Medicine and Biology 20.4 (2003), pp. 341–366

    2003

  22. [30]

    Agosti et al

    A. Agosti et al. ‘A computational frame- work for the personalized clinical treatment of glioblastoma multiforme’. Zeitschrift für An- gewandte Mathematik und Mechanik 98.12 (2018), pp. 2307–2327

    2018

  23. [31]

    Pozzi et al

    G. Pozzi et al. ‘T cell therapy against can- cer: A predictive diffuse-interface mathematical model informed by pre-clinical studies’. Journal of Theoretical Biology 547 (2022), p. 111172

    2022

  24. [32]

    Agosti et al

    A. Agosti et al. ‘A personalized mathematical tool for neuro-oncology: A clinical case study’. International Journal of Non-Linear Mechanics 107 (2018), pp. 170–181

    2018

  25. [33]

    Gorin et al

    F. Gorin et al. ‘Perinecrotic glioma prolifera- tion and metabolic profile within an intracereb- ral tumor xenograft’. Acta Neuropathologica 107 (2004), pp. 235–244

    2004

  26. [34]

    Garcke et al

    H. Garcke et al. ‘A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport’. Mathematical Models and Methods in Applied Sciences 26.06 (2016), pp. 1095–1148

    2016

  27. [35]

    E. A. Swabb, J. Wei and P . M. Gullino. ‘Diffu- sion and convection in normal and neoplastic tissues’. Cancer Research 34.10 (1974), pp. 2814– 2822

    1974

  28. [36]

    K. R. Swanson, E. C. Alvord Jr and J. Murray. ‘A quantitative model for differential motility of gliomas in grey and white matter’. Cell Pro- liferation 33.5 (2000), pp. 317–329

    2000

  29. [37]

    Martínez-González et al

    A. Martínez-González et al. ‘Hypoxic cell waves around necrotic cores in glioblastoma: a biomathematical model and its therapeutic implications’. Bulletin of Mathematical Biology 74 (2012), pp. 2875–2896

    2012

  30. [38]

    Chatelain et al

    C. Chatelain et al. ‘Emergence of microstruc- tural patterns in skin cancer: a phase separa- tion analysis in a binary mixture’. New Journal of Physics 13.11 (2011), p. 115013

    2011

  31. [39]

    Clatz et al

    O. Clatz et al. ‘Realistic simulation of the 3-D growth of brain tumors in MR images coup- ling diffusion with biomechanical deforma- tion’. IEEE Transactions on Medical Imaging24.10 (2005), pp. 1334–1346

    2005

  32. [40]

    Bedogni and M

    B. Bedogni and M. B. Powell. ‘Hypoxia, melano- cytes and melanoma–survival and tumor devel- opment in the permissive microenvironment of the skin’. Pigment Cell & Melanoma Research 22.2 (2009), pp. 166–174

    2009

  33. [41]

    Tierra and F

    G. Tierra and F. Guillén-González. ‘Numerical methods for solving the Cahn–Hilliard equa- tion and its applicability to related energy- based models’. Archives of Computational Meth- ods in Engineering 22 (2015), pp. 269–289

    2015

  34. [42]

    J. S. Hesthaven, G. Rozza and B. Stamm. Cer- tified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer International Publish- ing, 2015

    2015

  35. [43]

    Quarteroni, A

    A. Quarteroni, A. Manzoni and F. Negri. Re- duced Basis Methods for Partial Differential Equa- tions. Springer International Publishing, 2016. 14 Submitted — Under review Patient-specific prediction of glioblastoma growth

    2016

  36. [44]

    Audouze, F

    C. Audouze, F. De Vuyst and P . B. Nair. ‘Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis’. International Journal for Numerical Methods in Engineering 80.8 (2009), pp. 1025–1057

    2009

  37. [45]

    Rapún and J

    M.-L. Rapún and J. M. Vega. ‘Reduced order models based on local POD plus Galerkin pro- jection’. Journal of Computational Physics 229.8 (2010), pp. 3046–3063

    2010

  38. [46]

    Himpe, T

    C. Himpe, T. Leibner and S. Rave. ‘Hierarch- ical approximate proper orthogonal decompos- ition’. SIAM Journal on Scientific Computing 40.5 (2018), A3267–A3292

    2018

  39. [47]

    Q. Wang, J. Hesthaven and D. Ray. ‘Non- intrusive reduced order modeling of unsteady flows using artificial neural networks with ap- plication to a combustion problem’. Journal of Computational Physics 384 (2019), pp. 289–307

    2019

  40. [48]

    Agosti et al

    A. Agosti et al. ‘Learning patient-specific para- meters for a diffuse interface glioblastoma model from neuroimaging data’. Mathematical Methods in the Applied Sciences 43.15 (2020), pp. 8945–8979

    2020

  41. [49]

    Chaturantabut and D

    S. Chaturantabut and D. C. Sorensen. ‘Nonlin- ear Model Reduction via Discrete Empirical Interpolation’. SIAM Journal on Scientific Com- puting 32.5 (2010), pp. 2737–2764

    2010

  42. [50]

    Farhat et al

    C. Farhat et al. ‘Computational bottlenecks for PROMs: precomputation and hyperreduction’. In: Volume 2: Snapshot-Based Methods and Al- gorithms. Ed. by P . Benner et al. Berlin, Boston: De Gruyter, 2020, pp. 181–244

    2020

  43. [51]

    Hesthaven and S

    J. Hesthaven and S. Ubbiali. ‘Non-intrusive re- duced order modeling of nonlinear problems using neural networks’. Journal of Computational Physics 363 (2018)

    2018

  44. [52]

    D. C. Liu and J. Nocedal. ‘On the limited memory BFGS method for large scale optim- ization’. Mathematical Programming 45.1 (1989), pp. 503–528

    1989

  45. [53]

    M. D. Morris. ‘Factorial sampling plans for preliminary computational experiments’. Tech- nometrics 33.2 (1991), pp. 161–174

    1991

  46. [54]

    Saltelli et al

    A. Saltelli et al. Global sensitivity analysis: the primer. John Wiley, 2008

    2008

  47. [55]

    Possenti et al

    L. Possenti et al. ‘A global sensitivity ana- lysis approach applied to a multiscale model of microvascular flow’. Computer Methods in Biomechanics and Biomedical Engineering (2020), pp. 3356–3373

    2020

  48. [56]

    Vitullo et al

    P . Vitullo et al. ‘Sensitivity analysis of a multi- physics model for the vascular microenviron- ment’. International journal for numerical methods in biomedical engineering 39.11 (2023), e3752

    2023

  49. [57]

    G. L. Celora et al. ‘Phenotypic variation mod- ulates the growth dynamics and response to radiotherapy of solid tumours under normoxia and hypoxia’. Journal of Theoretical Biology 527 (2021), p. 110792

    2021

  50. [58]

    Ambrosi and F

    D. Ambrosi and F. Mollica. ‘On the mechan- ics of a growing tumor’. International journal of engineering science 40.12 (2002), pp. 1297–1316

    2002

  51. [59]

    R. K. Jain, J. D. Martin and T. Stylianopoulos. ‘The role of mechanical forces in tumor growth and therapy’. Annual review of biomedical engin- eering 16.1 (2014), pp. 321–346

    2014

  52. [60]

    Riccobelli

    D. Riccobelli. ‘Surface tension-driven boundary growth in tumour spheroids’. Interface Focus in press (2025)

    2025

  53. [61]

    Lassila and G

    T. Lassila and G. Rozza. ‘Parametric free- form shape design with PDE models and re- duced basis method’. Computer Methods in Ap- plied Mechanics and Engineering 199.23-24 (2010), pp. 1583–1592

    2010

  54. [62]

    Manzoni, A

    A. Manzoni, A. Quarteroni and G. Rozza. ‘Model reduction techniques for fast blood flow simulation in parametrized geometries’. Inter- national Journal for Numerical Methods in Biomed- ical Engineering 28.6-7 (2012), pp. 604–625

    2012

  55. [63]

    Zhang et al

    L. Zhang et al. ‘Shape-informed surrogate models based on signed distance function do- main encoding’. arXiv preprint arXiv:2409.12400 (2024)

    2024 arXiv

  56. [64]

    Acerbi et al

    F. Acerbi et al. ‘Mechano-Biological Features in a Patient-Specific Computational Model of Glioblastoma’. In: Neuromethods. Springer US, 2020, pp. 265–287

    2020

  57. [65]

    Sanai and M

    N. Sanai and M. S. Berger. ‘Operative Tech- niques for Gliomas and the Value of Extent of Resection’. Neurotherapeutics 6.3 (2009), pp. 478– 486

    2009

  58. [66]

    Weller et al

    M. Weller et al. ‘European Association for Neuro-Oncology (EANO) guideline on the dia- gnosis and treatment of adult astrocytic and oligodendroglial gliomas’. The Lancet Oncology 18.6 (2017), e315–e329. Submitted — Under review 15

    2017

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.