Direction-dependent turning leads to anisotropic diffusion and persistence

Loy, Nadia; Hillen, Thomas; Painter, Kevin J.

doi:10.1017/s0956792521000206

Cited by 7 publications

(15 citation statements)

References 53 publications

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“…This model (25,26) is the standard go-or-grow model, which has been studied in several publications [3,34,15,40].…”

Section: The Isotropic Casementioning

confidence: 99%

“…We use parameter estimation in Section 4.1 to estimate ϕ. Now consider the time derivative of the sum of equations (25,26) and obtain…”

Section: A Single Equation For the Bulkmentioning

confidence: 99%

“…Here we present the results of numerical simulations in two dimensions for the full microtube model (1-5), along with the anisotropic diffusion approximations (25)(26) and (28). Regarding the numerical methods, the equations that involve transport terms in (1-5) are integrated with the methods developed in [26,25]. The remaining equations are either of reaction-diffusion type or ODEs, and they are integrated with an Explicit Euler time integration scheme.…”

Section: Numerical Test Casesmentioning

confidence: 99%

“…A tumour is initialised as bulk at the edge of a portion of tissue of length 1 mm. We note that the fibres are considered to be isotropic; hence, model (22)(23) is identical to (25)(26), and model ( 28) is identical to (27). The invasion process for the full model (1-5) has previously been shown in Figure 4.…”

Section: Test 1: Invasion Wavesmentioning

confidence: 99%

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Modelling Microtube Driven Invasion of Glioma

Hillen,

Loy,

Painter

et al. 2023

Preprint

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Malignant gliomas are notoriously invasive, a major impediment against their successful treatment. This invasive growth has motivated the use of predictive partial differential equation models, formulated at varying levels of detail, and including (i) "proliferation-infiltration'' models, (ii) "go-or-grow'' models, and (iii) anisotropic diffusion models. Often, these models use macroscopic observations of a diffuse tumour interface to motivate a phenomenological description of invasion, rather than performing a detailed and mechanistic modelling of glioma cell invasion processes. Here we close this gap. Based on experiments that support an important role played by long cellular protrusions, termed tumour microtubes, we formulate a new model for microtube-driven glioma invasion. In particular, we model a population of tumour cells that extend tissue-infiltrating microtubes. Mitosis leads to new nuclei that migrate along the microtubes and settle elsewhere. A combination of steady state analysis and numerical simulation is employed to show that the model can predict an expanding tumour, with travelling wave solutions led by microtube dynamics. A sequence of scaling arguments allows us reduce the detailed model into simpler formulations, including models falling into each of the general classes (i), (ii), and (iii) above. This analysis allows us to clearly identify the assumptions under which these various models can be a posteriori justified in the context of microtube-driven glioma invasion. Numerical simulations are used to compare the various model classes and we discuss their advantages and disadvantages.

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“…This model (25,26) is the standard go-or-grow model, which has been studied in several publications [3,34,15,40].…”

Section: The Isotropic Casementioning

confidence: 99%

“…We use parameter estimation in Section 4.1 to estimate ϕ. Now consider the time derivative of the sum of equations (25,26) and obtain…”

Section: A Single Equation For the Bulkmentioning

confidence: 99%

Section: Numerical Test Casesmentioning

confidence: 99%

Section: Test 1: Invasion Wavesmentioning

confidence: 99%

See 2 more Smart Citations

Modelling Microtube Driven Invasion of Glioma

Hillen,

Loy,

Painter

et al. 2023

Preprint

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show abstract

“…The spatially dependent parameter µ(x) is the turning rate, while 1/µ(x) is the mean run time at location x. The traditional literature has focused on the case of constant turning rate [12,23], while direction-dependent turning rates have recently been studied in Reference [24]. The kernel T(x, v, v ) denotes the probability density of switching velocity from v to v, given that a turn occurs at location x.…”

Section: Transport Equationsmentioning

confidence: 99%

Anisotropic Network Patterns in Kinetic and Diffusive Chemotaxis Models

Thiessen

Hillen

2021

Mathematics

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For this paper, we are interested in network formation of endothelial cells. Randomly distributed endothelial cells converge together to create a vascular system. To develop a mathematical model, we make assumptions on individual cell movement, leading to a velocity jump model with chemotaxis. We use scaling arguments to derive an anisotropic chemotaxis model on the population level. For this macroscopic model, we develop a new numerical solver and investigate network-type pattern formation. Our model is able to reproduce experiments on network formation by Serini et al. Moreover, to our surprise, we found new spatial criss-cross patterns due to competing cues, one direction given by tissue anisotropy versus a different direction due to chemotaxis. A full analysis of these new patterns is left for future work.

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Modelling microtube driven invasion of glioma

Hillen,

Loy,

Painter

et al. 2023

J. Math. Biol.

View full text Add to dashboard Cite

Malignant gliomas are notoriously invasive, a major impediment against their successful treatment. This invasive growth has motivated the use of predictive partial differential equation models, formulated at varying levels of detail, and including (i) “proliferation-infiltration” models, (ii) “go-or-grow” models, and (iii) anisotropic diffusion models. Often, these models use macroscopic observations of a diffuse tumour interface to motivate a phenomenological description of invasion, rather than performing a detailed and mechanistic modelling of glioma cell invasion processes. Here we close this gap. Based on experiments that support an important role played by long cellular protrusions, termed tumour microtubes, we formulate a new model for microtube-driven glioma invasion. In particular, we model a population of tumour cells that extend tissue-infiltrating microtubes. Mitosis leads to new nuclei that migrate along the microtubes and settle elsewhere. A combination of steady state analysis and numerical simulation is employed to show that the model can predict an expanding tumour, with travelling wave solutions led by microtube dynamics. A sequence of scaling arguments allows us reduce the detailed model into simpler formulations, including models falling into each of the general classes (i), (ii), and (iii) above. This analysis allows us to clearly identify the assumptions under which these various models can be a posteriori justified in the context of microtube-driven glioma invasion. Numerical simulations are used to compare the various model classes and we discuss their advantages and disadvantages.

show abstract

Direction-dependent turning leads to anisotropic diffusion and persistence

Cited by 7 publications

References 53 publications

Modelling Microtube Driven Invasion of Glioma

Modelling Microtube Driven Invasion of Glioma

Anisotropic Network Patterns in Kinetic and Diffusive Chemotaxis Models

Modelling microtube driven invasion of glioma

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