On a subdiffusive tumour growth model with fractional time derivative

Fritz, Marvin; Kuttler, Christina; Rajendran, Mabel Lizzy; Wohlmuth, Barbara; Scarabosio, Laura

doi:10.1093/imamat/hxab009

Cited by 22 publications

(14 citation statements)

References 96 publications

(64 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…E.g., for α = 0.1 the energy drops from E = 0.5 to E ≈ 0.16 at t ≈ 0.16, whereas for α = 0.9 it goes from E = 0.5 to E ≈ 0.3 at t ≈ 0.24. This corresponds to the instantaneous process of time-fractional PDEs, see also the numerical studies in [51]. Furthermore, we observe that the energy reaches its asymptotic regime at E ≈ 0.12 at different times for each α, e.g., for α = 0.1 at t ≈ 1.25 versus t ≈ 2.4 for α = 0.7.…”

Section: Simulation Resultssupporting

confidence: 61%

See 1 more Smart Citation

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

Fritz¹,

Khristenko²,

Wohlmuth³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Time-fractional partial differential equations are nonlocal in time and show an innate memory effect. In this work, we propose an augmented energy functional which includes the history of the solution. Further, we prove the equivalence of a time-fractional gradient flow problem to an integer-order one based on our new energy. This equivalence guarantees the dissipating character of the augmented energy. The state function of the integerorder gradient flow acts on an extended domain similar to the Caffarelli-Silvestre extension for the fractional Laplacian. Additionally, we apply a numerical scheme for solving time-fractional gradient flows, which is based on kernel compressing methods. We illustrate the behavior of the original and augmented energy in the case of the Ginzburg-Landau energy functional.

show abstract

Section: Simulation Resultssupporting

confidence: 61%

“…We pass to the limit k → ∞ and apply compactness methods to return to the variational form of the time-fractional gradient flow. Recently, the Faedo-Galerkin method has been applied to various time-fractional PDEs, see, e.g., [21,41,[51][52][53].…”

Section: Well-posedness Of Time-fractional Gradient Flowsmentioning

confidence: 99%

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

Fritz¹,

Khristenko²,

Wohlmuth³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…T dx, then one obtains the time-fractional reaction-diffusion equation as studied in a tumor growth setting in Fritz et al (2021c).…”

Section: Nonlocal-in-space: Cell-to-cell and Cell-to-matrix Adhesionmentioning

confidence: 99%

“…As the tumor grows, the surrounding host tissues generate mechanical stress, restricting the tumor's growth. In the papers (Faghihi et al, 2020;Lima et al, 2016Lima et al, , 2017, mechanical deformation in a tumor development model was first mentioned, and in terms of analysis, it was first examined in Fritz et al (2021c) in a diffusion-type tumor model and subsequently Garcke et al (2021) (HPC) library written in C++ and therefore, yields higher potential for code optimization and saving run times than in FEniCS. We refer to our GitHub https://github.com/CancerModeling/Angiogenesis3D1D where our code is freely accessible.…”

Section: Mechanical Deformationmentioning

confidence: 99%

Tumor evolution models of phase-field type with nonlocal effects and angiogenesis

Fritz¹

2023

Preprint

View full text Add to dashboard Cite

In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on diffusive interface models and follow the phase-field approach that describes the tumor as a collection of cells. Such systems are based on a multiphase approach that employs constitutive laws and balance laws for individual constituents. In mathematical oncology, numerous biological phenomena are involved, including temporal and spatial nonlocal effects, complex nonlinearities, stochasticity, and mixeddimensional couplings. Using the models, for instance, we can express angiogenesis and cell-to-matrix adhesion effects. Finally, we offer some methods for numerically approximating the models and show simulations of the tumor's evolution in response to various biological effects.

show abstract

“…For prostate cancer, a combination of chemotherapy with antiangiogenic therapy and the optimization of the treatment are given in [49]. Chemotherapy is also included in the multispecies phase-field model approach of [50].…”

Section: Introductionmentioning

confidence: 99%

A phase-field model for non-small cell lung cancer under the effects of immunotherapy

Wagner

Schlicke

Fritz

et al. 2023

Preprint

Self Cite

View full text Add to dashboard Cite

Formulating tumor models that predict growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our model consists of a Cahn-Hilliard-based phase-field model describing the evolution of proliferative and necrotic tumor cells. These are coupled to a simplified nutrient model that drives the growth of the proliferative cells and their decay into necrotic cells. The applied immunotherapy decreases the proliferative cell concentration. Here, we model the immunotherapeutic agent concentration in the entire lung over time by an ordinary differential equation (ODE). Finally, reaction terms provide a coupling between all these equations. By assuming spherical, symmetric tumor growth and constant nutrient inflow, we simplify this full 3D cancer simulation model to a reduced 1D model. We can then resort to patient data gathered from computed tomography (CT) scans over several years to calibrate our model. For the reduced 1D model, we show that our model can qualitatively describe observations during immunotherapy by fitting our model parameters to existing patient data. Our model covers cases in which the immunotherapy is successful and limits the tumor size, as well as cases predicting a sudden relapse, leading to exponential tumor growth. Finally, we move from the reduced model back to the full 3D cancer simulation in the lung tissue. Thereby, we show the predictive benefits a more detailed patient-specific simulation including spatial information could yield in the future.

show abstract

On a subdiffusive tumour growth model with fractional time derivative

Cited by 22 publications

References 96 publications

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy

Tumor evolution models of phase-field type with nonlocal effects and angiogenesis

A phase-field model for non-small cell lung cancer under the effects of immunotherapy

Contact Info

Product

Resources

About