×

Advection-diffusion models for solid tumour evolution in vivo and related free boundary problem. (English) Zbl 1008.92017

Summary: This paper proposes a multicell model to describe the evolution of tumour growth from the avascular stage to the vascular one through the angiogenic process. The model is able to predict the formation of necrotic regions, the control of mitosis by the presence of an inhibitory factor, the angiogenesis process with proliferation of capillaries just outside the tumour surface with penetration of capillary sprouts inside the tumour, the regression of the capillary network induced by the tumour when angiogenesis is controlled or inhibited, say as an effect of angiostatins, and finally the regression of the tumour size.
The three-dimensional model is deduced both in a continuum mechanics framework and by a lattice scheme in order to put in evidence the relation between microscopic phenomena and macroscopic parameters. The evolution problem can be written as a free-boundary problem of mixed hyperbolic-parabolic type coupled with an initial-boundary value problem in a fixed domain.

MSC:

92C50 Medical applications (general)
35R35 Free boundary problems for PDEs
65C20 Probabilistic models, generic numerical methods in probability and statistics
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1142/S0218202599000324 · Zbl 0932.92017 · doi:10.1142/S0218202599000324
[2] DOI: 10.1007/BF02460635 · Zbl 0812.92011 · doi:10.1007/BF02460635
[3] DOI: 10.1016/S0895-7177(96)00174-4 · Zbl 0883.92014 · doi:10.1016/S0895-7177(96)00174-4
[4] DOI: 10.1080/10273669808833021 · Zbl 0917.92013 · doi:10.1080/10273669808833021
[5] DOI: 10.1017/S0956792597003264 · Zbl 0906.92016 · doi:10.1017/S0956792597003264
[6] DOI: 10.1142/S0218202598000664 · Zbl 0947.92014 · doi:10.1142/S0218202598000664
[7] DOI: 10.1016/S0895-7177(97)00143-X · Zbl 0898.92018 · doi:10.1016/S0895-7177(97)00143-X
[8] DOI: 10.1016/S0025-5564(97)00023-0 · Zbl 0904.92023 · doi:10.1016/S0025-5564(97)00023-0
[9] DOI: 10.1142/S0218202599000282 · Zbl 0932.92018 · doi:10.1142/S0218202599000282
[10] Chaplain M. A., Invasion Metastasis 16 pp 222– (1996)
[11] DOI: 10.1007/BF00184647 · Zbl 0830.92013 · doi:10.1007/BF00184647
[12] Chaplain Ed. M. A., Math. Models Methods Appl. Sci. pp 9– (1999)
[13] Conger A. D., Cancer Res. 43 pp 558– (1983)
[14] DOI: 10.1142/S0218202599000269 · doi:10.1142/S0218202599000269
[15] DOI: 10.1084/jem.138.4.745 · doi:10.1084/jem.138.4.745
[16] DOI: 10.1002/jcp.1041380222 · doi:10.1002/jcp.1041380222
[17] Freyer J. P., Cancer Res. 46 pp 3504– (1986)
[18] DOI: 10.1016/0360-3016(95)02065-9 · doi:10.1016/0360-3016(95)02065-9
[19] Haji-Karim M., Cancer Res. 38 pp 1457– (1978)
[20] Inch W. R., Growth 34 pp 271– (1970)
[21] Ed., Invasion Metastasis 16 pp 169– (1996)
[22] DOI: 10.1016/0895-7177(96)00053-2 · Zbl 0852.92010 · doi:10.1016/0895-7177(96)00053-2
[23] DOI: 10.1093/imammb/14.3.189 · doi:10.1093/imammb/14.3.189
[24] DOI: 10.1142/S0218202599000270 · Zbl 0932.92019 · doi:10.1142/S0218202599000270
[25] DOI: 10.1093/imammb/15.2.165 · doi:10.1093/imammb/15.2.165
[26] DOI: 10.1016/S0893-9659(98)00038-X · Zbl 0937.92018 · doi:10.1016/S0893-9659(98)00038-X
[27] DOI: 10.1142/S0218202599000294 · Zbl 0933.92021 · doi:10.1142/S0218202599000294
[28] DOI: 10.1126/science.2451290 · doi:10.1126/science.2451290
[29] DOI: 10.1093/imammb/14.1.39 · doi:10.1093/imammb/14.1.39
[30] DOI: 10.1093/imammb/15.1.1 · doi:10.1093/imammb/15.1.1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.