Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics

Jian Guo Liu; Min Tang; Li Wang; Zhennan Zhou

doi:10.3934/dcdsb.2018297

Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics

Jian Guo Liu, Min Tang, Li Wang, Zhennan Zhou

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n ^γ . For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

Original language	English (US)
Pages (from-to)	3011-3035
Number of pages	25
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	24
Issue number	7
DOIs	https://doi.org/10.3934/dcdsb.2018297
State	Published - Jul 2019

Bibliographical note

Funding Information:
J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135.

Funding Information:
2010 Mathematics Subject Classification. 35K55, 35B25, 76D27, 92C50. Key words and phrases. Tumor growth model, free boundary limit, Hele-Shaw flow model. J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135. ∗ Corresponding author: Zhennan Zhou.

Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.

Keywords

Free boundary limit
Hele-shaw flow model
Tumor growth model

Publisher link

10.3934/dcdsb.2018297

Find It

Find It at U of M Libraries

Cite this

@article{9061eb29cb0c43a6bbf7619ba632b408,

title = "Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics",

abstract = " In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n γ . For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.",

keywords = "Free boundary limit, Hele-shaw flow model, Tumor growth model",

author = "Liu, {Jian Guo} and Min Tang and Li Wang and Zhennan Zhou",

note = "Funding Information: J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135. Funding Information: 2010 Mathematics Subject Classification. 35K55, 35B25, 76D27, 92C50. Key words and phrases. Tumor growth model, free boundary limit, Hele-Shaw flow model. J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135. ∗ Corresponding author: Zhennan Zhou. Publisher Copyright: {\textcopyright} 2019 American Institute of Mathematical Sciences. All rights reserved.",

year = "2019",

month = jul,

doi = "10.3934/dcdsb.2018297",

language = "English (US)",

volume = "24",

pages = "3011--3035",

journal = "Discrete and Continuous Dynamical Systems - Series B",

issn = "1531-3492",

publisher = "Southwest Missouri State University",

number = "7",

}

TY - JOUR

T1 - Analysis and computation of some tumor growth models with nutrient

T2 - From cell density models to free boundary dynamics

AU - Liu, Jian Guo

AU - Tang, Min

AU - Wang, Li

AU - Zhou, Zhennan

N1 - Funding Information: J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135. Funding Information: 2010 Mathematics Subject Classification. 35K55, 35B25, 76D27, 92C50. Key words and phrases. Tumor growth model, free boundary limit, Hele-Shaw flow model. J. Liu is partially supported by KI-Net NSF RNMS grant No.11-07444, NSF grant DMS-1812573 and NSF grant DMS-1514826. M. Tang is supported by Science Challenge Project No. TZZT2017-A3-HT003-F and NSFC 91330203. Z. Zhou is partially supported by RNMS11-07444 (KI-Net) and the start up grant from Peking University. L. Wang is partially supported by the start up grant from SUNY Buffalo and NSF grant DMS-1620135. ∗ Corresponding author: Zhennan Zhou. Publisher Copyright: © 2019 American Institute of Mathematical Sciences. All rights reserved.

PY - 2019/7

Y1 - 2019/7

N2 - In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n γ . For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

AB - In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density n is governed by the Darcy law via the pressure p(n) = n γ . For finite γ, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As γ → ∞, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

KW - Free boundary limit

KW - Hele-shaw flow model

KW - Tumor growth model

UR - http://www.scopus.com/inward/record.url?scp=85065477890&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065477890&partnerID=8YFLogxK

U2 - 10.3934/dcdsb.2018297

DO - 10.3934/dcdsb.2018297

M3 - Article

AN - SCOPUS:85065477890

SN - 1531-3492

VL - 24

SP - 3011

EP - 3035

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 7

ER -

Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this