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doi: 10.3934/dcdss.2022055
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Mathematical model for simulation of morphological changes associated to crypt fission in the colon

1. 

Departamento de Matemática Aplicada, Instituto de Matemática Estatistica e Computação Científica (IMECC), Universidade Estadual de Campinas (Unicamp), 13083-859 Campinas, SP, Brazil

2. 

Dyrecta Lab, Istituto di Ricerca, I-70014 Conversano, Italy, Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, I-70126 Bari, Italy

* Corresponding author: Giuseppina Settanni

Received  September 2021 Revised  January 2022 Early access March 2022

Morphological changes due to colorectal cancer propagation by an abnormal crypt fission is an interesting application arising in medicine and biology, that we try to analyse by a differential equations model coupled with a discrete crypt fission model. A colonic crypt can slowly change its shape in three steps: growth, bifurcation and fission. Fission is a rare event in a normal tissue, however if transit cells are unable to differentiate, due to an activation of Wnt signaling, then it may become fast and uncontrolled and cause a formation of aberrant crypt foci (ACF), defined as clusters of aberrant crypts. The differential equation system is composed by a convective diffusive equation for transit cell density and an elliptic equation for cell pressure, both defined on a manifold. The discrete crypt fission model acts in a set of adjacent crypts in order to investigate the ACF dynamics, due to a differentiation block. By using a Galerkin finite element method we solve numerically the differential equation model and show some interesting results about an ACF formation caused by a deformation and fission of abnormal crypts.

Citation: Giuseppe Romanazzi, Giuseppina Settanni. Mathematical model for simulation of morphological changes associated to crypt fission in the colon. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022055
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show all references

References:
[1]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1965.

[2]

A. A. AlmetB. D. HughesK. A. LandmanI. S. Näthke and J. M. Osborne, A multicellular model of intestinal crypt buckling and fission, Bull. Math. Biol., 80 (2018), 335-359.  doi: 10.1007/s11538-017-0377-z.

[3]

A. A. AlmetP. K. MainiD. E. Moulton and H. M. Byrne, Modeling perspectives on the intestinal crypt, a canonical system for growth, mechanics, and remodeling, Curr. Opin. Biomed. Eng., 15 (2020), 32-39.  doi: 10.1016/j.cobme.2019.12.012.

[4]

S. P. BachA. G. Renehan and C. S. Potten, Stem cells: The intestinal stem cell as a paradigm, Carcinogenesis, 21 (2000), 469-476.  doi: 10.1093/carcin/21.3.469.

[5]

A. M. BakerB. CereserS. MeltoA. G. FletcherM. Rodriguez-JustoP. J. TadrousA. HumphriesG. EliaS. A. McDonaldN. A. WrightB. D. SimonsM. Jansen and T. A. Graham, Quantification of crypt and stem cell evolution in the normal and neoplastic human colon, Cell. Rep., 8 (2014), 940-947. 

[6]

N. BarkerR. A. RidgwayJ. H. van EsM. van de WeteringH. BegthelM. van den BornE. DanenbergA. R. ClarkeO. J. Sansom and H. Clevers, Crypt stem cells as the cells-of-origin of intestinal cancer, Nature, 457 (2009), 608-611.  doi: 10.1038/nature07602.

[7]

M. Bjerknes, Expansion of mutant stem cell populations in the human colon, J. Theor. Biol., 178 (1996), 381-385.  doi: 10.1006/jtbi.1996.0034.

[8]

B. M. Boman and J. Z. Fields, An APC:WNT counter-current-like mechanism regulates cell division along the human colonic crypt axis: A mechanism that explains how APC mutations induce proliferative abnormalities that drive colon cancer Development, Front. Oncol., 3 (2013), 244.  doi: 10.3389/fonc.2013.00244.

[9]

B. M. BomanJ. Z. FieldsO. Bonham-Carter and O. A. Runquist, Computer modeling implicates stem cell overproduction in colon cancer initiation, Cancer. Res., 61 (2001), 8408-8411. 

[10]

L. BruensS. I. J. EllenbroekJ. van Rheenen and H. J. Snippert, In vivo imaging reveals existence of crypt fission and fusion in adult mouse intestine, Gastroenterology, 153 (2017), 674-677.  doi: 10.1053/j.gastro.2017.05.019.

[11]

P. Buske, J. Galle, N. Barker, G. Aust, H. Clevers and M. Loeffler, A comprehensive model of the spatiotemporal stem cell and tissue organisation in the intestinal crypt, PLoS Comput. Biol., 7 (2011), e1001,045.

[12]

P. BuskeJ. PrzybillaM. LoefflerN. SachsT. SatoH. Clevers and J. Galle, On the biomechanics of stem cell niche formation in the gut-modelling growing organoids, FEBS J., 279 (2012), 3475-3487.  doi: 10.1111/j.1742-4658.2012.08646.x.

[13]

A. J. CarulliL. C. Samuelson and S. Schnell, Unraveling intestinal stem cell behavior with models of crypt dynamics, Integr. Biol. (Camb), 6 (2014), 243-257.  doi: 10.1039/c3ib40163d.

[14]

H. Clevers and R. Nusse, Wnt $\beta$-catenin signaling and disease, Cell, 149 (2012), 1192-1205. 

[15]

G. De MatteisA. Graudenzi and M. Antoniotti, A review of spatial computational models for multi-cellular systems, with regard to intestinal crypts and colorectal cancer development, J. Math. Biol., 66 (2013), 1409-1462.  doi: 10.1007/s00285-012-0539-4.

[16]

A. Di GarboM. D. JohnstonS. J. Chapman and P. K. Maini, Variable renewal rate and growth properties of cell populations in colon crypts, Phys. Rev. E, 81 (2010), 061909.  doi: 10.1103/PhysRevE.81.061909.

[17]

A. D'Onofrio and I. P. M. Tomlinson, A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt, J. Theoret. Biol., 244 (2007), 367-374.  doi: 10.1016/j.jtbi.2006.08.022.

[18]

D. Drasdo and M. Loeffler, Individual-based models to growth and folding in one-layered tissues: Intestinal crypts and early development, Nonlinear Anal., 47 (2001), 245-256.  doi: 10.1016/S0362-546X(01)00173-0.

[19]

C. M. Edwards and S. J. Chapman, Biomechanical modelling of colorectal crypt budding and fission, Bull. Math. Biol., 69 (2007), 1927-1942.  doi: 10.1007/s11538-007-9199-8.

[20]

B. EmerickG. Schleiniger and B. M. Boman, A kinetic model to study the regulation of $\beta$-catenin, APC, and Axin in the human colonic crypt, J. Math. Biol., 75 (2017), 1171-1202.  doi: 10.1007/s00285-017-1112-y.

[21]

B. EmerickG. Schleiniger and B. M. Boman, Multi-scale modeling of APC and $\beta$-catenin regulation in the human colonic crypt, J. Math. Biol., 76 (2018), 1797-1830.  doi: 10.1007/s00285-017-1204-8.

[22]

I. N. FigueiredoC. LealG. Romanazzi and B. Engquist, Homogenization model for aberrant crypt foci, SIAM J. Appl. Math., 76 (2016), 1152-1177.  doi: 10.1137/140967660.

[23]

I. N. Figueiredo, C. Leal, G. Romanazzi and B. Engquist, Biomathematical model for simulating abnormal orifice patterns in colonic crypts, Math. Biosci., 315 (2019), 108221, 11 pp. doi: 10.1016/j.mbs.2019.108221.

[24]

I. N. FigueiredoC. LealG. RomanazziB. Engquist and P. N. Figueiredo, A convection-diffusion-shape model for aberrant colonic crypt morphogenesis, Comput. Vis. Sci., 14 (2011), 157-166.  doi: 10.1007/s00791-012-0170-3.

[25]

I. N. FigueiredoG. RomanazziC. Leal and B. Engquist, A multiscale model for aberrant crypt foci, Procedia Comput. Sci., 18 (2013), 1026-1035.  doi: 10.1016/j.procs.2013.05.268.

[26]

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Figure 1.  Colon crypt structure: stem cells (in red) at the bottom, transit cells (in cyan) in the mid-crypt region and differentiated cells (in blue) in the last third of the crypt. (Figure created in BioRender.com)
Figure 2.  Colonic epithelium region: (A) a distribution of seven adjacent crypts (circle) in the colonic epithelium, restrained in the inter-crypts region (hexagon), reproduces the histological image (B) of a colonic epithelium region from M. A. Hill "Embryology Gastrointestinal Tract - Colon Histology" [34]
Figure 3.  Crypt fission in 3-D: (A) Normal crypt at step 1. (B) Deformed crypt with a bifurcation starting from the bottom up to $ 2/3 \,h $ and an orifice enlargement at step 2. (C) Complete fission with formation of two new crypts at step 3
Figure 4.  Crypt fission steps, top view: (A) normal crypt at step 1; (B) deformed crypt at step 2; (C) two new crypts generated by a complete fission at step 3
Figure 5.  Deformed crypt at step 2, sections: (A) from the orifice to the height $ 2/3\, h $; (B) plan region at $ 2/3\, h $ splitting the two half spheroids; (C) two half spheroids from $ 2/3\, h $ to the bottom
Figure 6.  Crypt distribution at step 2: only the central crypt is deformed with an enlargement of the orifice following three possible directions (A) "Left-up/Right-down", (B) "Up/Down", (C) "Right-up/Left-down"
Figure 7.  Evolution of a crypt distribution at each step: (A) all crypts are normal at step 1. (B) The central crypt is deformed in the direction "Up/Down" at step 2. (C) The final complete fission generates, at step 3, two new crypts in the same direction "Up/Down" and tissue regions around
Figure 8.  Other possible evolution of crypt distribution at step 3, due to a complete fission of the central crypt in the direction: (A) "Left-up/Right-down"; (B) "Right-up/Left-down". Two new crypts are generated in the respectively directions, while tissue regions appear around
Figure 9.  Geometrical configuration of $ S = S_c\cup S_{ext} $, where $ S_c $ (light grey) is a half prolate spheroid of height $ h $ centred in $ (x_{1,C},x_{2,C},h) $ and $ S_{ext} $ (dark grey) is the hexagonal inter-crypts region
Figure 10.  Domain decomposition of a crypt in step 1 and possible directions for a crypt deformation in step 2
Figure 11.  (A) In step 1 an abnormal transit cell distribution with $ T>0.65 $ is located close to the orifice (black circle) in the "Up" region of the central crypt; (B) in step 2 the central crypt deforms in the "Up-Down" direction; (C) in step 3 the deformed crypt undergoes fission generating two new crypts, called "Up Crypt" and "Down Crypt"; (D) after step 3 the "Up Crypt" and "Down Crypt", having a maximum density $ T>0.65 $ respectively in the "Down" and "Up" region, see Figure 12, both deform again in the "Up/Down" direction
Figure 12.  The histogram represents the maximun transit cell density $ T $ distributed in each one of the six regions of the two new crypts, the "Up Crypt" (yellow/light) and "Down Crypt" (cyan/dark), generated in step 3 after a fission in the direction "Up-Down"
Figure 13.  (A) In step 1 an abnormal transit cell distribution with $ T>0.65 $ is located close to the orifice (black circle) in the "Left-up" region of the central crypt; (B) in step 2 the central crypt deforms in the "Left-up/Right-down" direction; (C) in step 3 the deformed crypt undergoes fission generating two new crypts, called "Up Crypt" and "Down Crypt"; (D) after step 3 the 'Up Crypt" and "Down Crypt", having maximum density $ T>0.65 $ respectively in the "Right-down" and "Left-up" region, see Figure 14, both deform again in the "Left-up/Right-down" direction
Figure 14.  The histogram represents the maximun transit cell density $ T $ distributed in each one of the six regions of the two new crypts, the "Up Crypt" (yellow/light) and "Down Crypt" (cyan/dark), generated in step 3 after a fission in the direction "Left-up-Right-down"
Figure 15.  (A) In step 1 an abnormal cell distribution with $ T>0.65 $ is widely allocated close to the orifice (black circle) between the region "Up" and "Right-up" of the central crypt; (B) in step 2 the central crypt deforms in the "Up/Down" direction; (C) in step 3 the deformed crypt undergoes fission in the same direction, generating two new crypts, called "Up Crypt" and "Down Crypt"; (D) after step 3 the "Up Crypt" and "Down Crypt", having a maximum density $ T>0.65 $ respectively in the "Down" and "Right-up" region, as shown in Figure 16, deform respectively along the direction "Up/Down" and "Right-up/Left-down"
Figure 16.  The histogram represents the maximun transit cell density $ T $ distributed in each one of the six regions of the two new crypts, the "Up Crypt" (yellow/light) and "Down Crypt" (cyan/dark), generated in step 3 after a fission in direction "Up/Down"
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