Article Contents

2024, Volume 18, Issue 5: 1223-1242. Doi: 10.3934/ipi.2024012

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On forward and inverse problems for a DCIS model with free boundaries in mathematical biology

Hongyu Liu^1, and
Keji Liu^2, ,

1.
Department of Mathematics, City University of HongKong, Kowloon, HongKong, China
2.
School of Mathematics, Dishui Lake Advanced Finance Institute, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China

^*Corresponding author: Keji Liu
^*Corresponding author: Keji Liu

Received: October 2023

Revised: March 2024

Early access: March 2024

Published: October 2024

Keji Liu is supported by [the NNSF of China under grant No. 12071275] and Hongyu Liu is supported by [the NSFC/RGC Joint Research Scheme, N_CityU101/21, ANR/RGC Joint Research Scheme, A-CityU203/19, and the Hong Kong RGC General Research Funds (projects 11311122, 12301420 and 11300821)]

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Abstract

We are concerned with the mathematical study of a DCIS model which arises in characterizing the biological development of breast cancer. A salient feature of a DCIS model is the presence of free boundaries for describing the tumor growth which is not known in advance. In this paper, we are particularly interested in the case with general free boundaries which may be asymmetric. We first propose an iterative finite difference method for the forward problem and show that the method is of 2nd order in both space and time. Then we propose to study an inverse problem of recovering the nutrient consumption rate by the incisional biopsy data. We establish the unique identifiability of the inverse problem and develop a novel reconstruction scheme based on a certain integral formulation. Extensive numerical experiments are conducted to corroborate the theoretical findings. Our study opens up a new direction of research in mathematical biology with many potential extensions and developments.

Keywords:

Mathematics Subject Classification: Primary: 35R30, 35R35, 65N21.

Citation:

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Figure 1. The demonstrations of (a) the normal duct, (b) the atypical hyperplasia and (c) DCIS [16]

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Figure 2. The schematic illustration of the DCIS model with asymmetric free boundaries

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Figure 3. A schematic illustration of the pathological section at $ t = T $

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Figure 4. The direct problem of example 1: the approximations of asymmetric free boundaries $ \varphi_i(t) (i = 1,2) $ for (a) $ t\in[0,0.5] $ and (b) $ t\in[0,1] $

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Figure 5. The inverse problem of example 1: the reconstructions of consumption rate $ \lambda(x) $ at (a) $ t = 0.5 $ and (b) $ t = 1 $; the forecasts of the moving boundaries $ \varphi_i(t) (i = 1,2) $ for (c) $ t\in[0.5,1.5] $ and (d) $ t\in[1,1.5] $

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Figure 6. The direct problem of example 2: the approximations of asymmetric free boundaries $ \varphi_i(t) (i = 1,2) $ for (a) $ t\in[0,0.5] $ and (b) $ t\in[0,1] $

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Figure 7. The inverse problem of example 2: the reconstructions of consumption rate $ \lambda(x) $ at (a) $ t = 0.5 $ and (b) $ t = 1 $; the forecasts of the moving boundaries $ \varphi_i(t) (i = 1,2) $ for (c) $ t\in[0.5,1.5] $ and (d) $ t\in[1,1.5] $

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Figure 8. The direct problem of example 3: the approximations of asymmetric free boundaries $ \varphi_i(t) (i = 1,2) $ for (a) $ t\in[0,0.25] $ and (b) $ t\in[0,0.5] $

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Figure 9. The inverse problem of example 3: the reconstructions of consumption rate $ \lambda(x) $ at (a) $ t = 0.25 $ and (b) $ t = 0.5 $; the forecasts of the moving boundaries $ \varphi_i(t) (i = 1,2) $ for (c) $ t\in[0.25,0.8] $ and (d) $ t\in[0.5,0.8] $

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Figure 10. The direct problem of example 4: the approximations of asymmetric free boundaries $ \varphi_i(t) (i = 1,2) $ for (a) $ t\in[0,0.5] $ and (b) $ t\in[0,1] $

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Figure 11. The inverse problem of example 4: the reconstructions of consumption rate $ \lambda(x) $ at (a) $ t = 0.5 $ and (b) $ t = 1 $; the forecasts of the moving boundaries $ \varphi_i(t) (i = 1,2) $ for (c) $ t\in[0.5,1.5] $ and (d) $ t\in[1,1.5] $

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Table 1. The maximum relative errors $ \delta_\infty^u $ and $ \delta_\infty^\varphi $, and the order of accuracy $ Ord_\infty $ in experiment 1

$ \tau=h $	$ \delta_\infty^u $	$ \delta_\infty^\varphi $	$ Ord_\infty $
1/20	2.915E$ - $03	1.330E$ - $03	*
1/40	8.554E$ - $04	3.680E$ - $04	1.775
1/80	2.304E$ - $04	9.660E$ - $05	1.887
1/160	5.973E$ - $05	2.473E$ - $05	1.947

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Table 2. The maximum relative errors $ \delta_\infty^u $ and $ \delta_\infty^\varphi $, and the order of accuracy $ Ord_\infty $ in experiment 2

$ \tau=h $	$ \delta_\infty^u $	$ \delta_\infty^\varphi $	$ Ord_\infty $
1/20	3.937E$ - $04	3.761E$ - $03	*
1/40	1.289E$ - $04	9.382E$ - $04	1.750
1/80	3.937E$ - $05	2.342E$ - $04	1.872
1/160	1.136E$ - $05	5.854E$ - $05	1.925

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Table 3. The maximum relative errors $ \delta_\infty^u $ and $ \delta_\infty^\varphi $, and the order of accuracy $ Ord_\infty $ in experiment 3

$ \tau=h $	$ \delta_\infty^u $	$ \delta_\infty^\varphi $	$ Ord_\infty $
1/20	5.442E$ - $02	8.278E$ - $03	*
1/40	8.715E$ - $03	2.420E$ - $03	1.743
1/80	2.587E$ - $03	6.527E$ - $04	1.857
1/160	7.335E$ - $04	1.692E$ - $04	1.923

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Table 4. The maximum relative errors $ \delta_\infty^u $ and $ \delta_\infty^\varphi $, and the order of accuracy $ Ord_\infty $ in experiment 4

$ \tau=h $	$ \delta_\infty^u $	$ \delta_\infty^\varphi $	$ Ord_\infty $
1/20	1.678E$ - $03	1.207E$ - $03	*
1/40	4.540E$ - $04	3.287E$ - $04	1.886
1/80	8.168E$ - $05	7.040E$ - $05	1.958
1/160	2.936E$ - $05	2.644E$ - $05	1.992

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References

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