
In this study, a one-dimensional chemotaxis-haptotaxis model of cancer cell invasion of tissue was numerically and statistically investigated. In the numerical part, the time dependent, nonlinear, triplet governing dimensionless equations consisting of cancer cell (CC) density, extracellular matrix (ECM) density, and urokinase plasminogen activator (uPA) density were solved by the radial basis function (RBF) collocation method both in time and space discretization. In the statistical part, mean CC density, mean ECM density, and mean uPA density were modeled by two different machine learning approaches. The datasets for modeling were originated from the numerical results. The numerical method was performed in a set of parameter combinations by parallel computing and the data in case of convergent combinations were stored. In this data, inputs consisted of selected time values up to a maximum time value and converged parameter values, and outputs were mean CC, mean ECM, and mean uPA. The whole data was divided randomly into train and test data. Trilayer neural network (TNN) and multilayer adaptive regression splines (Mars) model the train data. Then, the models were tested on test data. TNN modeling resulting in terms of mean squared error metric was better than Mars results.
Citation: Bengisen Pekmen, Ummuhan Yirmili. Numerical and statistical approach on chemotaxis-haptotaxis model for cancer cell invasion of tissue[J]. Mathematical Modelling and Control, 2024, 4(2): 195-207. doi: 10.3934/mmc.2024017
[1] | Ricai Luo, Khadija Dawood, Muhammad Kamran Jamil, Muhammad Azeem . Some new results on the face index of certain polycyclic chemical networks. Mathematical Biosciences and Engineering, 2023, 20(5): 8031-8048. doi: 10.3934/mbe.2023348 |
[2] | Bing Wang, Fu Tan, Jia Zhu, Daijun Wei . A new structure entropy of complex networks based on nonextensive statistical mechanics and similarity of nodes. Mathematical Biosciences and Engineering, 2021, 18(4): 3718-3732. doi: 10.3934/mbe.2021187 |
[3] | Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor . Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences and Engineering, 2009, 6(1): 1-25. doi: 10.3934/mbe.2009.6.1 |
[4] | Fu Tan, Bing Wang, Daijun Wei . A new structural entropy measurement of networks based on the nonextensive statistical mechanics and hub repulsion. Mathematical Biosciences and Engineering, 2021, 18(6): 9253-9263. doi: 10.3934/mbe.2021455 |
[5] | Bei Liu, Hongzi Bai, Wei Chen, Huaquan Chen, Zhen Zhang . Automatic detection method of epileptic seizures based on IRCMDE and PSO-SVM. Mathematical Biosciences and Engineering, 2023, 20(5): 9349-9363. doi: 10.3934/mbe.2023410 |
[6] | David Cuesta-Frau, Pau Miró-Martínez, Sandra Oltra-Crespo, Antonio Molina-Picó, Pradeepa H. Dakappa, Chakrapani Mahabala, Borja Vargas, Paula González . Classification of fever patterns using a single extracted entropy feature: A feasibility study based on Sample Entropy. Mathematical Biosciences and Engineering, 2020, 17(1): 235-249. doi: 10.3934/mbe.2020013 |
[7] | Xiaoye Zhao, Yinlan Gong, Lihua Xu, Ling Xia, Jucheng Zhang, Dingchang Zheng, Zongbi Yao, Xinjie Zhang, Haicheng Wei, Jun Jiang, Haipeng Liu, Jiandong Mao . Entropy-based reliable non-invasive detection of coronary microvascular dysfunction using machine learning algorithm. Mathematical Biosciences and Engineering, 2023, 20(7): 13061-13085. doi: 10.3934/mbe.2023582 |
[8] | Enas Abdulhay, Maha Alafeef, Hikmat Hadoush, V. Venkataraman, N. Arunkumar . EMD-based analysis of complexity with dissociated EEG amplitude and frequency information: a data-driven robust tool -for Autism diagnosis- compared to multi-scale entropy approach. Mathematical Biosciences and Engineering, 2022, 19(5): 5031-5054. doi: 10.3934/mbe.2022235 |
[9] | Dawei Li, Enzhun Zhang, Ming Lei, Chunxiao Song . Zero trust in edge computing environment: a blockchain based practical scheme. Mathematical Biosciences and Engineering, 2022, 19(4): 4196-4216. doi: 10.3934/mbe.2022194 |
[10] | Ziqi Peng, Seiroh Okaneya, Hongzi Bai, Chuangxing Wu, Bei Liu, Tatsuo Shiina . Proposal of dental demineralization diagnosis with OCT echo based on multiscale entropy analysis. Mathematical Biosciences and Engineering, 2024, 21(3): 4421-4439. doi: 10.3934/mbe.2024195 |
In this study, a one-dimensional chemotaxis-haptotaxis model of cancer cell invasion of tissue was numerically and statistically investigated. In the numerical part, the time dependent, nonlinear, triplet governing dimensionless equations consisting of cancer cell (CC) density, extracellular matrix (ECM) density, and urokinase plasminogen activator (uPA) density were solved by the radial basis function (RBF) collocation method both in time and space discretization. In the statistical part, mean CC density, mean ECM density, and mean uPA density were modeled by two different machine learning approaches. The datasets for modeling were originated from the numerical results. The numerical method was performed in a set of parameter combinations by parallel computing and the data in case of convergent combinations were stored. In this data, inputs consisted of selected time values up to a maximum time value and converged parameter values, and outputs were mean CC, mean ECM, and mean uPA. The whole data was divided randomly into train and test data. Trilayer neural network (TNN) and multilayer adaptive regression splines (Mars) model the train data. Then, the models were tested on test data. TNN modeling resulting in terms of mean squared error metric was better than Mars results.
The entropy of a probability measures the uncertainty of a system. This concept is strongly based on statistical methodology. It is mainly used for chemical structures and their corresponding graphs. It also provides information about graph structure and chemical topologies. It was used as a notion for the first time in 1955. In many scientific and technical fields, entropy has applications. Intrinsic and extrinsic entries are determined in this way. The idea of degree power is used to investigate networks as information functional. The authors put forward the idea of entropy for different topological indices. The entropy of probability distributions is the foundation as described in. This parameter imparts a lot of information about structures, graphs, and chemical topology of networks in network theory, by which a parameter is known as degree powers. It is a base for graph theory in applied mathematics to investigate networks as information functional. Authors, put forward the idea of entropy for a variety of networks, like Shannon used for the entropy of probability distribution.
In modeling and designing any chemical network can be converted into a graph and this theory plays a very important role. By using the graph theoretical transformation of a chemical network or structure, we can determine many properties like physicochemical properties, thermodynamical properties, and biological activities. Among them, by using the topological indies certain properties can be characterized. Edge-weighted base entropy is also from the family of topological indices.
A simple molecular finite graph in chemical graph theory denoted both atom and chemical bond in terms of vertices and edges, respectively. The numeric value of the topological index shows the physical, chemical, and topological properties of a graph. With the help of chemical graph theory, we can model the mathematical phenomenon of chemical networks. It has a relation to the nontrivial application of graph theory for solving molecular problems. This theory plays a vital role in the field of chemical science. Chem-informatics is the combination of chemistry, mathematics, and information science. We can predict bio-activities and physical-chemical properties of the chemical compound by using QSAR and QSPR, which are examined by chem-informative [1].
The authors of [2], computed some topological indies and corresponded to their entropies. They discussed the relationship between numerical and graphical on the basis of topological indies and their entropies. Furthermore, they also discussed the topological indies which give fruitful results for the structural properties of g−C3N4, and some related applications are also available. According to [3], the authors find the entropy value by applying different parameters like the total number of vertices, edges, and degree of any graph. Then they compared the results with different graphs. They resulted that increasing the vertices and edges of graphs, affects increasing the entropy of such graphs. Some crystal structures based on non-kekulean benzenoid sub-structures are discussed in terms of entropies by using the degree-based topological indices [4]. They also computed the relationship between degree-based topological indices and degree-based entropy. They provided some applications in different topics like chemical, biological, and physical reactivity processes. Given in the article [5], topological indices of si2c3−I and si2c3−II and based on the results entropy measures are discussed. Here in [6], the researchers contributed towards a tool for molecular graphs which is based on the weight of edges, known as entropy. They relate the entropy measure with polynomial functions. Particularly, they computed the entropy measure of magnesium iodide structure and find different entropies like Zagreb and atom bond entropies.
In [7], the authors discussed the benzenoid hydrocarbon chemical structure. Shannon applied the benzenoid entropy in the transmission rate of telephonic channels, optical communication, and wireless. The impacts of high features in a different system. In this paper, the author discusses the characteristics and effects of graph entropies in different topological indies like coronoid polycyclic aromatic hydrocarbons. They computed the entropies through topological indies on degree terminal vertices. In [8], researchers find the effective roll of metal-insulator transition superlattices (GST-Sw) with different topological properties. Particularly, they discussed the atom-bond degree-based topological indices which are applied in the heat formation of single crystal and monolayered structure of Ge-sb-te. The relation of this structure is also available. Researchers of [9], find different entropies of crystallography chemical networks. The author particularly computed hyper and augmented Zagreb, forgotten, and Balaban entropies for the crystallography structures of cup-rite cu2o and titanium difluoride tif2 by using different topological tools. First, second, modified and augmented Zagreb indies, symmetric division, harmonic, inverse sum index, and forgotten indies are computed in [10]. They computed these indices for two chemical structures crystal cubic carbon and carbon graphite networks.
In [11], the author derived some degree-based temperature descriptors. He also computed temperature entropies of molecular structure and related to degree-based temperature with the help of specific information functions. He also compared graphically with the calculated function. All the computed results from the given information provided more effective drug and covid-19 vaccines. Authors of [12], focused on the carbon nanotubes which are useful in tissue engineering and investigated the various entropies. They computed some entropies for different variants of carbon nanotubes. They also investigated and compared the results of armchair carbon nanotubes with other forms of structure. The researchers of [13], investigated the relation between the natural polymer of cellulose network and pharmacological applicantion which provided a good result in a different structure. They computed different K-Banhatti indices and their entropies. The nature polymer of cellulose networks and their entropies are effective in various topics of chemical structural theory. Structures of Polycyclic aromatic hydrocarbons are discussed in [14], and authors elaborated intriguing properties based on the graph's theoretical parameters. They discussed entropy measures and their numerical values of given structures. Different entropies of the molecular structure of HCQ, by using degrees of vertices and edges, are discussed in [15]. Particularly, they computed degree-based topological characteristics of hydroxyethyl starch conjugated structures with hydroxychloroquine. They also presented some relations of different entropies with other structures. In the end, they compared the proven results in terms of numerical and graphical form.
Authors of [16], provide the main purpose of this article to study the properties of a graph and then discuss the structure like hyaluronic acid (HA) curcumin conjugates. Moreover, the author computed the entropies by using the degree-based topological indices with the help of the information function. Indices are linked with total π-electron property and measured some entropies of trans-PD-(NH2)s lattice and metal-organic super-lattice structures, by using the different graph parameters, in [17]. Structures of dendrimers based on cyclotriphosphazene (N3P3) which are applied to the topics of balanced and computed the EPR temperature spectrum, are discussed in [18]. First of all the author computed eccentricity-based indices and their entropies. After the above indices result the author presented it numerically and graphically. Entropies of some variants of Y-shaped nano-junctions are discussed in [19]. By using topological indices, knowledge discovery and representations of molecular structures are considered in [20]. Particularly, authors computed topological indies like atom-bond connectivity (ABC), the fourth version of ABC, geometric arithmetic (GA), and the fifth version of GA. Finally, the authors computed the above indices for octa-nano sheets, equilateral triangular, tetra sheets, rectangular sheets, and rectangular tetra sheets.
Graph entropies of porous graphene chemical networks are found in [21], by using some topological indices. They also provide some application of chosen indices-based entropies, particularly for the porous graphene structure. Different topological indies-based entropies of armchair carbon nanotubes are found in [22]. They provide some applications of variants of carbon nanotubes, like capped uncapped, and semi-capped which are purely used in memory devices and tissue engineering. Authors of [23] introduced the self-powered vertex degree topological indices and computed the graph entropy measures for tessellations networks, such as the isentropic later which is known as a non-isomorphic tessellation of kekulenes. Researchers of [24], computed the entropies of the structures of fractal and Cayley tree-type dendrimers by using the different topological indies and their degree-based results. The chosen structures are types of dendrimers denoted by Fr and Cm,n.
The Mostar, Padmakar-Ivan (PI), and Szeged indies of the molecular graph of two kinds of dendrimers, in which one is Phthalocyanines and 2nd is Porphyrine are discussed in [25]. The author also computed the entropies of these structures and relates these with other topological indies and their entropies.
Three classes of reticular metal-organic frameworks, graph entropies, enumeration of circuits, walks and topological properties are computed and analyzed in [26]. They also used different properties like eccentricities, radius, diameter, vertex, and degree to find the thermochemistry for the reticular network. In the [27] article, the authors describe and explain the characterization of two-dimensional coronene fractals in terms of different topological indies. The author also computed the entropy measure and compared it with other topological indies. they used machine learning techniques for robust computation of enthalpies. Moreover, they used NMR and ESR spectroscopic patterns of coronene fractals. In [28], the authors determine the data difficulty handling by using the entropy measure. For this, they collected the data in a different field from science, for this research work they faced a problem, the data is not accurate, and for this, they used the entropy measure to get the non-redundant, non-correlated data. The authors computed a good solution by using good algorithms for research and entropy measures. There is 25 entropy measure is taken for the classification procedure and compares its result. The crystal structure of the polyphenylene network for photocatalysis is discussed in [29], and find some topological indies. By using these indices they work for the thermodynamic properties namely entropy and heat which are taken fruitful results for the crystal structure. They concluded that effective application due to the chosen topological indies and their corresponding entropies occurred.
A biochemistry network namely t-level hypertrees of corona product of hypertrees with path are produced in [30]. The author also works with different topological indies like eccentricity-based indies and their entropies for the chosen biochemical network. Crystal structure of titanium difluoride TiF2 and the crystallographic structure of CU2O are discussed in the paper given by [31]. The author also computed the different entropies of these structures and relates them to different topological indies like first, second, and third redefined Zagreb indices, fourth atom bond connectivity index, fifth geometric arithmetic, and Sanskruti index and their entropies are discussed. For further results related to the topic is found in [32,33,34]. Moreover, closely related work can be found in [35,36,37,38,39,40,41,42,43,44,45].
Let E(G) be the edge and V(G) be the vertex of a graph. The degree of a vertex v in a graph determine by the attached edges with a vertex in the graph, and it is denoted by dv. A distance d(a,b) between a and b vertices are the shortest length path of the graph. The order and size of the graph are |V(G)| and |E(G)|, respectively. Following are some formulations of the topological indices we used to develop the entropies.
● First redefined Zagreb index
RZ1(G)=∑ab∈E(G)da+dbdadb. | (2.1) |
● 2nd redefined Zagreb index
RZ2(G)=∑ab∈E(G)dadbda+db. | (2.2) |
● Third redefined Zagreb index
RZ3(G)=∑ab∈E(G)dadb(da+db). | (2.3) |
● Augmented Zagreb index
AZ(G)=∑ab∈E(G)(dadbda+db−2)3. | (2.4) |
● Albertson index
A(G)=∑ab∈E(G)|da−db|. | (2.5) |
● Irregularity measure index
IM(G)=∑ab∈E(G)(da−db)2. | (2.6) |
● Reformulated Zagreb index
RZ(G)=∑ab∈E(G)(da−db−2)2. | (2.7) |
● Forgotten index
F(G)=∑a∈V(G)d3a. | (2.8) |
● First gourava index
GO1(G)=∑ab∈E(G)[da+db+dadb]. | (2.9) |
● First hyper gourava index
HGO1(G)=∑ab∈E(G)[da+db+dadb]2. | (2.10) |
● Second gourava index
GO2(G)=∑ab∈E(G)[(da+db).(dadb)]. | (2.11) |
● Second hyper gourava index
HGO2(G)=∑ab∈E(G)[(da+db).(dadb)]2. | (2.12) |
Given above is the first part of our methodology, corresponding to Eqs (2.1)–(2.12), the given below are edge-weighted based entropies. Some of these entropies are developed already, but most of them are new and we are developing and computing for the first time. The notation of entropy is used by the letter λRα(G) for the topological index Rα for the graph G.
● Entropy of first redefined Zagreb index
λRZ1(G)=−1RZ1(G)log[∏ab∈E(G)(da+dbdadb)da+dbdadb]+log(RZ1(G)). | (2.13) |
● Entropy the second redefined Zagreb index
λRZ2(G)=−1RZ2(G)log[∏ab∈E(G)(dadbda+db)dadbda+db]+log(RZ2(G)). | (2.14) |
● Entropy the third redefined Zagreb index
λRZ3(G)=−1RZ3(G)log∏ab∈E(G)[dadb(da+db)][dadb(da+db)]+log(RZ3(G)). | (2.15) |
● Entropy the augmented Zagreb index
λAZ(G)=−1AZI(G)log[∏ab∈E(G)[(dadbdadb−2)3](dadbdadb−2)3]+log(AZ(G)). | (2.16) |
● Entropy of Albertson index
λA(G)=−1AI(G)log[∏ab∈E(G)(|da−db|)(|da−db|)]+log(A(G)). | (2.17) |
● Entropy irregularity measure
λIM(G)=−1IM(G)log[∏ab∈E(G)[(da−db)2](da−db)2]+log(IM(G)). | (2.18) |
● Entropy reformulated Zagreb index
λRZ(G)=−1RZ(G)log[∏ab∈E(G)[(da−db−2)2](da−db−2)2]+log(RZ(G)). | (2.19) |
● Entropy of forgotten index
λF(G)=−1F(G)log[∏a∈V(G)[d3a]d3a]+log(F(G)). | (2.20) |
● Entropy of first gourava index
λGO1(G)=−1GO1(G)log[∏ab∈E(G)(da+db+dadb)(da+db+dadb)]+log(GO1(G)). | (2.21) |
● Entropy of first hyper gourava index
λHGO1(G)=−1HGO1(G)log[∏ab∈E(G)(da+db+dadb)(da+db+dadb)]+log(HGO1(G)). | (2.22) |
● Entropy of second gourava index
λGO2(G)=−1GO2(G)log[∏ab∈E(G)[(da+db)dadb][(da+db)dadb]]+log(GO2(G)). | (2.23) |
● Entropy of second hyper gourava index
λHGO2(G)=−1HGO2(G)log[∏ab∈E(G)[(da+db)dadb][(da+db)dadb]]+log(HGO2(G)). | (2.24) |
Clay mineral tetrahedral sheets are represented by the notation TSCMβ,γ. Various edge degree-based entropy measurements are applied to tetrahedral sheets in the aforementioned paper. One of the platonic graphs, the tetrahedral graph, has four vertices and six edges. It can be seen as a solid. The only planer construction of its isomorphic graphs, such as the complete graph K4 and wheel graph W4, is the tetrahedral graph. Chemical graph theory uses tetrahedral sheets made by tetrahedral graph polymerization to reflect silicone and other clay minerals. On tetrahedral sheets, interdisciplinary topics can be accommodated via combinatorial properties such as labeling, coloring, enumeration, and indexing as well as algorithmic operations such as shortest path and spanning tree. Figure 1 depicts the tetrahedral sheets of clay minerals TSCMβ,γ for β=2,γ=2. There are topological indices available in [46] for Eqs (4.1)–(4.12).
The graph TSCMβ,γ contains 10βγ+7β+γ are vertices and 24βγ+12β total count of edges. In TSCMβ,γ there are two types of vertices having degrees 3 or 6. The vertex set of TSCMβ,γ can be distributed according to their degrees.
Let
Vi={a∈V(TSCMβ,γ):da=i}. |
This indicates that the vertices of degrees i are present in the set Vi. According to their degree, the set of vertices is as follows:
V3={a∈V(TSCMβ,γ):da=3}V6={a∈V(TSCMβ,γ):da=6} |
Since |V3|=4βγ+6β+2γ and |V3|=6βγ+β−γ, we can also subdivide the edges of TSCMβ,γ into following three subsets according to the degree of its end vertices.
E3,3={ab∈TSCMβ,γ:da=3,db=3}E3,6={ab∈TSCMβ,γ:da=3,db=6}E6,6={ab∈TSCMβ,γ:da=6,db=6} |
Note that E(TSCMβ,γ)=E3,3∪E3,6∪Eβ,γ. The number of edges incident to one vertex of degree 3 and other vertices are 3,6 are 4β+2γ, 12βγ+10β+2γ respectively, so |E3,3|=4β+2γ, |E3,3|=12βγ+10β+2γ. The edges that are incident to two vertices of degree 6 are now the remaining number of edges, which are |E6,6|=12βγ−2β−4γ.
In this section, we will present our main results and some computational work in terms of topological indices and corresponding entropies.
RZ1I(G)=60βγ+20β−5γ. | (4.1) |
RZ2I(G)(G)=77128βγ+972β−1296γ. | (4.2) |
RZ3I(G)(G)=3275251242875βγ+83883843686000β−1781362531372000γ. | (4.3) |
AZ(G)=144βγ+60β−6γ. | (4.4) |
A(G)=36βγ+30β+6γ. | (4.5) |
IM(G)=108βγ+90β+18γ. | (4.6) |
RZ(G)=1392βγ+345β−270γ. | (4.7) |
F(G)=1404βγ+378β−162γ. | (4.8) |
GO1(G)=603βγ+234β−108γ. | (4.9) |
HGO1(G)=36396βγ+3582β−7308γ. | (4.10) |
GO2(G)=7128βγ+972β−1296γ. | (4.11) |
HGO2(G)=2554416βγ−9944β−688176γ. | (4.12) |
Now discuss the following properties of general topological invariant (IG) based on degree of vertices in a graphs (G).
● Entropy of first redefined Zagreb index
λRZ1(G)=−1RZ1(G)log[∏ab∈E(G)da+dbdadbda+dbdadb]+log(RZ1(G)). |
Computing the First Redefined Zagreb index by using Eq (2.1), we will get the result given in Eq (4.1). Now using Eq (4.1) in the formula of entropy of First Redefined Zagreb index, which is given in the Eq (2.13). After simplification, we will get the result given in the Eq (4.13).putting the values
λRZ1(G)=−110βγ+7β+γlog[[(23)23]4β+2γ×[(12)12]12βγ+10β+2γ×[(13)13]12βγ−2β−4γ]. |
λRZ1(G)=−110βγ+7β+γlog[(0.7631)(0.7071)(0.6934)]+log[10βγ+7β+γ]. | (4.13) |
● Entropy the second redefined Zagreb index
λRZ2(G)=−1RZ2(G)log[∏ab∈E(G)[dadbda+db][dadbda+db]]+log(RZ2(G)). |
Computing the Second Redefined Zagreb index by using Eq (2.1), we will get the result given in Eq (4.1). Now using Eq (4.1) in the formula of entropy of Second Redefined Zagreb index, which is given in the Eq (2.13). After simplification, we will get the result given in the Eq (4.14). Putting the value
λRZ2(G)=−160βγ+20β−5γlog[[2323]4β+2γ×[12]12βγ+10β+2γ×[13][13]12βγ−2β−4γ]+log[60βγ+20β−5γ]. |
λRZ2(G)=−160βγ+20β−5γlog[1.8371427]+log[60βγ+20β−5γ]. | (4.14) |
● Entropy the third redefined Zagreb index
λRZ3(G)=−1RZ3(G)log[∏ab∈E(G)[dadb(da+db)]dadb(da+db)]+log(RZ3(G)). |
Computing the Third Redefined Zagreb index by using Eq (2.2), we will get the result given in Eq (4.2). Now using Eq (4.2) in the formula of entropy of Third Redefined Zagreb index, which is given in the Eq (2.14). After simplification, we will get the result given in the Eq (4.15). Putting the values
λRZ3(G)=−17128βγ+972β−1296γlog[[3232]4β+2γ×[(2)2]12βγ+10β+2γ×[(3)3]12βγ−2β−4γ]+log[7128βγ+972β−1296γ]. |
λRZ3(G)=−17128βγ+972β−1296γlog[[(54)(54)(162)(162)](432)(432)]+log[7128βγ+972β−1296γ]. | (4.15) |
● Entropy the augmented Zagreb index {AZ(G)}
λAZ(G)=−1AZ(G)log[∏ab∈E(G)[(dadbdadb−2)3](dadbdadb−2)3]+log(AZ(G)),=−13275251242875βγ+83883843686000β−1781362531372000γ. |
Computing the augmented Zagreb AZ(G) index by using Eq (2.3), we will get the result given in Eq (4.3). Now using Eq (4.3) in the formula of entropy of augmented Zagreb AZ(G) index, which is given in the Eq (2.15). After simplification, we will get the result given in the Eq (4.16).
λAZ(G)=−13275251242875βγ+83883843686000β−1781362531372000γlog[[(54)54][4β+2γ]×[(162)(162)][12βγ+10β+2γ]×[(432)(432)][12βγ−2β−4γ]]+log[13275251242875βγ+83883843686000β−1781362531372000γ]. |
log[(64)64(187)187(3610)3610]+log[3275251242875βγ+83883843686000β−1781362531372000γ]. | (4.16) |
● Entropy for variation of Randić index R′(G)
λR′(G)=−1R′(G)log[∏ab∈E(G)[1max(dadb)][1max(dadb)]]+log(R′(G)) |
Computing the Variation of Randić index R′(G) by using Eq (2.4), we will get the result given in Eq (4.4). Now using Eq (4.4) in the formula of entropy of Variation of Randić index R′(G), which is given in the Eq (2.16). After simplification, we will get the result given in the Eq (4.17). Putting the values
λR′(G)=−1144βγ+60β−6γlog[((729343)729343)4β+2γ×((729512)729512)12βγ+10β+2γ×((58324913)58324913)12βγ−2β−4γ]+1144βγ+60β−6γ. |
λR′(G)=−1144βγ+60β−6γlog[1max31max61max6]+[144βγ+60β−6γ]. | (4.17) |
● Entropy Albertson index
λA(G)=−1A(G)log[∏ab∈E(G)(|da−db|)(|da−db|)]+log(A(G)). |
Computing the Albertson index by using Eq (2.5), we will get the result given in Eq (4.5). Now using Eq (4.5) in the formula of entropy of Albertson index, which is given in the Eq (2.17). After simplification, we will get the result given in the Eq (4.18). Putting the values
λA(G)=−136βγ+30β+6γlog[(3)12βγ+10β+2γ]+log[36βγ+30β+6γ]. | (4.18) |
● Entropy irregularity measure
λIM(G)=−1IM(G)log[∏ab∈E(G)[(da−db)2](da−db)2]+log(IM(G)). |
Computing the Irregularity measure by using Eq (2.6), we will get the result given in Eq (4.6). Now using Eq (4.6) in the formula of entropy of Irregularity measure, which is given in the Eq (2.18). After simplification, we will get the result given in the Eq (4.18). Putting the values
λIM(G)=−1108βγ+90β+18γlog[((9)9)12βγ+10β+2γ]+log[108βγ+90β+18γ]. | (4.19) |
● Enrtopy Reformulated Zagreb index
λRZ(G)=−1RZ(G)log[∏ab∈E(G)[(da−db−2)2](da−db−2)2]+log(RZ(G)). |
Computing the reformulated Zagreb index by using Eq (2.7), we will get the result given in Eq (4.7). Now using Eq (4.7) in the formula of entropy of Reformulated Zagreb index, which is given in the Eq (2.19). After simplification, we will get the result given in the Eq (4.20). Putting the values
λRZ(G)=−11392βγ+345β−270γlog[[(4)4]4β+2γ×[(25)25]12βγ+10β+2β×[(4)4]12βγ−2β−4γ]+log[1392βγ+345β−270γ]. | (4.20) |
● Entropy of forgotten index
λF(G)=−1F(G)log[∏a∈V(G)[d3a]d3a]+log(F(G)). |
Computing the forgotten Zagreb index by using Eq (2.8), we will get the result given in Eq (4.8). Now using Eq (4.8) in the formula of entropy of Forgotten Zagreb index, which is given in the Eq (2.20). After simplification, we will get the result given in the Eq (4.21). Puttting the values
λF(G)=−11404βγ+378β−162γlog[[((3)3)27]4βγ+6β+2γ×[((6)3)216]6βγ+β−γ]+log[1404βγ+378β−162γ]. | (4.21) |
● Entropy of first gourava index
λGO1(G)=−1GO1(G)log[∏ab∈E(G)(da+db+dadb)(da+db+dadb)]+log(GOI1(G)). |
Computing the first gourava index by using Eq (2.9), we will get the result given in Eq (4.9). Now using Eq (4.9) in the formula of entropy of First Gourava index, which is given in the Eq (2.21). After simplification, we will get the result given in the Eq (4.22). Putting the values.
λGO1(G)=−1603βγ+234β−108γlog[[(15)15]4β+2γ×[(27)27]12βγ+10β+2γ×[(48)48]12βγ−2β−4γ]+log[603βγ+234β−108γ]. | (4.22) |
● Entropy of first hyper gourava index
λHGO1(G)=−1HGO1(G)log[∏ab∈E(G)(da+db+dadb)(da+db+dadb)]+log(HGO1(G)). |
Computing the first hyper gourava index by using Eq (2.10), we will get the result given in Eq (4.10). Now using Eq (4.10) in the formula of entropy of First Hyper Gourava index, which is given in the Eq (2.22). After simplification, we will get the result given in the Eq (4.23). Putting the values
λHGO1(G)=−136396βγ+3582β−7308γlog[[(15)15]4β+2γ×[(27)27]12βγ+10β+2γ×[(48)48]12βγ−2β−4γ]+log[36396βγ+3582β−7308γ]. | (4.23) |
● Entropy of second gourava index
λGO2(G)=−1GO2(G)log[∏ab∈E(G)[(da+db)dadb][(da+db)dadb]]+log(GO2(G)). |
Computing the second gourava index by using Eq (2.11), we will get the result given in Eq (4.11). Now using Eq (4.11) in the formula of entropy of Second Gourava index, which is given in the Eq (2.23). After simplification, we will get the result given in the Eq (4.24). Putting the values.
λGO2(G)=−17128βγ+972β−1296γlog[[(54)54]4β+2γ×[(162)162]12βγ+10β+2γ×[(432)432]12βγ−2β−4γ]+log[7128βγ+972β−1296γ]. | (4.24) |
● Entropy of second hyper gourava index
λHGO2(G)=−1HGO2(G)log[∏ab∈E(G)[(da+db)dadb][(da+db)dadb]]+log(HGO2(G)). |
Computing the scond Hyper Gourava index by using Eq (2.12), we will get the result given in Eq (4.12). Now using Eq (4.12) in the formula of entropy of Second Hyper Gourava index, which is given in the Eq (2.24). After simplification, we will get the result given in the Eq (4.25). Putting the values
λHGOI2(G)=−12554416βγ−99144β−68817γlog[((54)54)4β+4γ×((162)162)12βγ+10β+2γ]+log((432)432)12βγ−2β−4γ+log[2554416βγ−99144β−688176γ]. | (4.25) |
A network or structure's edge weight-based entropy offers structural details and in-depth content in the form of mathematical equations. This study contained the edge valency-based entropies of tetrahedral sheets of clay minerals. It highlights the molecular attributes in the form of a logarithmic function and offers the structural information of chemical networks or their related build-up graphs. These entropies are newly developed in this work and can be considered as an application or computational work for other networks or structures as a future direction for the researchers.
This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).
The author declare that they have no conflict of interest.
[1] |
T. L. Jackson, S. R. Lubkin, N. O. Siemers, D. E. Kerri, P. D. Senter, J. D. Murray, Mathematical and experimental analysis of localization of anti-tumor antibody-enzyme conjugates, Brit. J. Cancer, 80 (1999), 1747–1753. https://doi.org/10.1038/sj.bjc.6690592 doi: 10.1038/sj.bjc.6690592
![]() |
[2] |
A. R. A. Anderson, M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bull. Math. Biol., 60 (1998), 857–900. https://doi.org/10.1006/bulm.1998.0042 doi: 10.1006/bulm.1998.0042
![]() |
[3] | A. R. A. Anderson, M. A. J. Chaplain, E. L. New Man, R. J. C. Steele, A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med., 2 (2000), 129–154. |
[4] |
J. A. Sherratt, M. A. J. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291–312. https://doi.org/10.1007/s002850100088 doi: 10.1007/s002850100088
![]() |
[5] |
A. Matzavinos, M. A. J. Chaplain, Travelling-wave analysis of a model of the immune response to cancer, Comput. R. Biol., 327 (2004), 995–1008. https://doi.org/10.1016/j.crvi.2004.07.016 doi: 10.1016/j.crvi.2004.07.016
![]() |
[6] |
M. A. J. Chaplain, G. Lolas, Mathematical modelling of cencer cell invasion of tissue: the role of the urokinase plasminogen activation system, Math. Models Methods Appl. Sci., 11 (2005), 1685–1734. https://doi.org/10.1142/S0218202505000947 doi: 10.1142/S0218202505000947
![]() |
[7] |
M. A. J. Chaplain, G. Lolas, Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity, Netw. Heterog. Media, 1 (2006), 399–439. https://doi.org/10.3934/nhm.2006.1.399 doi: 10.3934/nhm.2006.1.399
![]() |
[8] |
A. Gerisch, M. A. J. Chaplain, Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion, J. Theor. Biol., 250 (2008), 684–704. https://doi.org/10.1016/j.jtbi.2007.10.026 doi: 10.1016/j.jtbi.2007.10.026
![]() |
[9] |
N. Bellomo, A. Bellouquid, E. de Angelis, The modelling of the immune competition by generalized kinetic (Boltzmann) models: review and research perspectives, Math. Comput. Model., 37 (2003), 1131–1142. https://doi.org/10.1016/S0895-7177(03)80007-9 doi: 10.1016/S0895-7177(03)80007-9
![]() |
[10] |
H. Enderling, A. R. A. Anderson, M. A. J. Chaplain, A. J. Munro, J. S. Vaidya, Mathematical modelling of radiotherapy strategies for early breast cancer, J. Theor. Biol., 241 (2006), 158–171. https://doi.org/10.1016/j.jtbi.2005.11.015 doi: 10.1016/j.jtbi.2005.11.015
![]() |
[11] |
V. Andasari, A. Gerisch, G. Lolas, A. P. South, M. A. J. Chaplain, Mathematical modeling of cancer cell invasion of tissue: biological insight from mathematical analysis and computational simulation, J. Math. Biol., 63 (2011), 141–171. https://doi.org/10.1007/s00285-010-0369-1 doi: 10.1007/s00285-010-0369-1
![]() |
[12] |
M. Dehghan, V. Mohammadi, Comparison between two meshless methods based on collocation technique for the numerical solution of four-species tumor growth model, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 204–219. https://doi.org/10.1016/j.cnsns.2016.07.024 doi: 10.1016/j.cnsns.2016.07.024
![]() |
[13] |
M. Dehghan, N. Narimani, An element-free Galerkin meshless method for simulating the behavior of cancer cell invasion of surrounding tissue, Appl. Math. Model., 59 (2018), 500–513. https://doi.org/10.1016/j.apm.2018.01.034 doi: 10.1016/j.apm.2018.01.034
![]() |
[14] |
G. Meral, I. C. Yamanlar, Mathematical analysis and numerical simulations for the cancer tissue invasion model, Commun. Fac. Sci. Univ. Ank. Ser., 68 (2019), 371–391. https://doi.org/10.31801/cfsuasmas.421546 doi: 10.31801/cfsuasmas.421546
![]() |
[15] |
G. Meral, DRBEM-FDM solution of a chemotaxis-haptotaxis model for cancer invasion, J. Comput. Appl. Math., 354 (2019), 299–309. https://doi.org/10.1016/j.cam.2018.04.047 doi: 10.1016/j.cam.2018.04.047
![]() |
[16] |
P. R. Nyarko, M. Anokye, Mathematical modeling and numerical simulation of a multiscale cancer invasion of host tissue, AIMS Math., 54 (2019), 3111–3124. https://doi.org/10.3934/math.2020200 doi: 10.3934/math.2020200
![]() |
[17] |
L. C. Franssen, T. Lorenzi, A. E. F. Burgess, M. A. J. Chaplain, A mathematical framework for modelling the metastatic spread of cancer, Bull. Math. Biol., 81 (2019), 1965–2010. https://doi.org/10.1007/s11538-019-00597-x doi: 10.1007/s11538-019-00597-x
![]() |
[18] |
F. Hatami, M. B. Ghasemi, Numerical solution of model of cancer invasion with tissue, Appl. Math., 4 (2013), 1050–1058. https://doi.org/ 10.4236/am.2013.47143 doi: 10.4236/am.2013.47143
![]() |
[19] |
Y. Tao, C. Cui, A density-dependent chemotaxis-haptotaxis system modeling cancer invasion, J. Math. Anal. Appl., 367 (2010), 612–624. https://doi.org/10.1016/j.jmaa.2010.02.015 doi: 10.1016/j.jmaa.2010.02.015
![]() |
[20] |
Y. Tao, M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differ. Equations, 257 (2014), 784–815. https://doi.org/10.1016/j.jde.2014.04.014 doi: 10.1016/j.jde.2014.04.014
![]() |
[21] |
A. Amoddeo, Moving mesh partial differential equations modelling to describe oxygen induced effects on avascular tumour growth, Cogent Phys., 2 (2015), 1050080. https://doi.org/10.1080/23311940.2015.1050080 doi: 10.1080/23311940.2015.1050080
![]() |
[22] |
A. Amoddeo, A moving mesh study for diffusion induced effects in avascular tumour growth, Comput. Math. Appl., 75 (2018), 2508–2519. https://doi.org/10.1016/j.camwa.2017.12.024 doi: 10.1016/j.camwa.2017.12.024
![]() |
[23] |
A. Amoddeo, Indirect contributions to tumor dynamics in the first stage of the avascular phase, Symmetry, 12 (2020), 1546. https://doi.org/10.3390/sym12091546 doi: 10.3390/sym12091546
![]() |
[24] |
A. Amoddeo, Mathematical model and numerical simulation for electric field induced cancer cell migration, Math. Comput. Appl., 26 (2021), 4. https://doi.org/10.3390/mca26010004 doi: 10.3390/mca26010004
![]() |
[25] |
S. Ganesan, S. Lingeshwaran, Galerkin finite element method for cancer invasion mathematical model, Comput. Math. Appl., 73 (2017), 2603–2617. https://doi.org/10.1016/j.camwa.2017.04.006 doi: 10.1016/j.camwa.2017.04.006
![]() |
[26] |
S. Ganesan, S. Lingeshwaran, A biophysical model of tumor invasion, Commun. Nonlinear Sci. Numer. Simul., 46 (2017), 135–152. https://doi.org/10.1016/j.cnsns.2016.10.013 doi: 10.1016/j.cnsns.2016.10.013
![]() |
[27] |
G. Meral, C. Surulescu, Mathematical modelling, analysis and numerical simulations for the influence of heat shock proteins on tumour invasion, J. Math. Anal. Appl., 408 (2013), 597–614. https://doi.org/10.1016/j.jmaa.2013.06.017 doi: 10.1016/j.jmaa.2013.06.017
![]() |
[28] |
E. Barbera, G. Valenti, Wave features of a hyperbolic reaction-diffusion model for chemotaxis, Wave Motion, 78 (2018), 116–131. https://doi.org/10.1016/j.wavemoti.2018.02.004 doi: 10.1016/j.wavemoti.2018.02.004
![]() |
[29] |
N. Sfakianakis, A. Madzvamuse, M. A. J. Chaplain, A hybrid multiscale model for cancer invasion of the extracellular matrix, Multiscale Model. Simul., 18 (2018), 824–850. https://doi.org/10.1137/18M11890 doi: 10.1137/18M11890
![]() |
[30] |
J. Urdal, J. O. Waldeland, S. Evje, Enhanced cancer cell invasion caused by fibroblasts when fluid flow is present, Biomech. Model. Mechanobiol., 18 (2019), 1047–1078. https://doi.org/10.1007/s10237-019-01128-2 doi: 10.1007/s10237-019-01128-2
![]() |
[31] |
M. Los, A. Klusek, M. A. Hassaan, K. Pingali, W. Dwinel, M. Paszynski, Parallel fast isogeometric L2 projection solver with GALOIS system for 3D tumor growth simulations, Comput. Methods Appl. Meth. Eng., 343 (2001), 291–312. https://doi.org/10.1016/j.cma.2018.08.036 doi: 10.1016/j.cma.2018.08.036
![]() |
[32] |
J. J. Benito, A. Garcia, L. Gavete, M. Negreanu, F. Urena, A. M. Vargas, Solving a chemotaxis-haptotaxis system in 2D using generalized finite difference method, Comput. Math. Appl., 80 (2020), 762–777. https://doi.org/10.1016/j.camwa.2020.05.008 doi: 10.1016/j.camwa.2020.05.008
![]() |
[33] |
J. J. Benito, A. Garcia, L. Gavete, M. Negreanu, F. Urena, A. M. Vargas, Convergence and numerical solution of a model for tumor growth, Mathematics, 9 (2021), 1355. https://doi.org/10.3390/math9121355 doi: 10.3390/math9121355
![]() |
[34] |
H. Y. Hin, T. Xiang, Negligibility of haptotaxis effect in a chemotaxis-haptotaxis model, Math. Models Methods Appl. Sci., 31 (2021), 1373–1417. https://doi.org/10.1142/S0218202521500287 doi: 10.1142/S0218202521500287
![]() |
[35] |
H. Shen, X. Wei, A parabolic-hyperbolic system modeling the tumor growth with angiogenesis, Nonlinear Anal. Real World Appl., 64 (2022), 103456. https://doi.org/10.1016/j.nonrwa.2021.103456 doi: 10.1016/j.nonrwa.2021.103456
![]() |
[36] |
Y. He, X. Liu, Z. Chen, J. Zhu, Y. Ziong, K. Li, et al., Interaction between cancer cells and stromal fibroblasts is required for activation of the uPAR-uPA-MMP-2 cascade in pancreatic cancer metastasis, Clin. Cancer Res., 13 (2007), 11. https://doi.org/10.1158/1078-0432.CCR-06-2088 doi: 10.1158/1078-0432.CCR-06-2088
![]() |
[37] |
C. Melzer, J. Ohe, H. Otterbein, H. Ungefroren, R. Hass, Changes in uPA, PAI-1, and TGF-β production during breast cancer cell interaction with human mesenchymal stroma/stem-like cells (MSC), Int. J. Mol. Sci., 20 (2019), 2630. https://doi.org/10.3390/ijms20112630 doi: 10.3390/ijms20112630
![]() |
[38] |
J. Huang, L. Zhang, D. Wan, L. Zhou, S. Zheng, S. Lin, et al., Extracellular matrix and its therapeutic potential for cancer treatment, Signal Transduct. Target. Ther., 6 (2021), 153. https://doi.org/10.1038/s41392-021-00544-0 doi: 10.1038/s41392-021-00544-0
![]() |
[39] |
E. Henke, R. Nandigama, S. Ergun, Extracellular matrix in the tumor microenvironment and its impact on cancer therapy, Front. Mol. Biosci., 6 (2020), 160. https://doi.org/10.3389/fmolb.2019.00160 doi: 10.3389/fmolb.2019.00160
![]() |
[40] |
A. A. Shimpi, C. Fischbach, Engineered ECM models: opportunities to advance understanding of tumor heterogeneity, Curr. Opin Cell Biol., 72 (2021), 1–9. https://doi.org/10.1016/j.ceb.2021.04.001 doi: 10.1016/j.ceb.2021.04.001
![]() |
[41] |
K. Dass, A. Ahmad, A. S. Azmi, S. H. Sarkar, F. H. Sarkar, Evolving role of uPA/uPAR system in human cancers, Cancer Treat. Rev., 34 (2008), 122–136. https://doi.org/10.1016/j.ctrv.2007.10.005 doi: 10.1016/j.ctrv.2007.10.005
![]() |
[42] |
P. Pakneshan, M. Szyf, R. Farias-Eisner, S. A. Rabbani, Reversal of the hypomethylation status of urokinase (uPA) promoter blocks breast cancer growth and metastasis, J. Biol. Chem., 279 (2004), 31735–31744. https://doi.org/10.1074/jbc.M401669200 doi: 10.1074/jbc.M401669200
![]() |
[43] |
M. W. Pickup, J. K. Mouw, V. M. Weaver, The extracellular matrix modulates the hallmarks of cancer, EMBO Rep., 15 (2014), 1243–1253. https://doi.org/10.15252/embr.201439246 doi: 10.15252/embr.201439246
![]() |
[44] |
A. W. Holle, J. L. Young, J. P. Spatz, In vitro cancer cell-ECM interactions inform in vivo cancer treatment, Adv. Drug Deliv. Rev., 97 (2016), 270–279. https://doi.org/10.1016/j.addr.2015.10.007 doi: 10.1016/j.addr.2015.10.007
![]() |
[45] |
E. J. Kansa, Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl., 19 (1990), 147–161. https://doi.org/10.1016/0898-1221(90)90271-K doi: 10.1016/0898-1221(90)90271-K
![]() |
[46] | G. E. Fasshauer, Meshfree approximation methods with matlab, World Scientific Publications, 2007. |
[47] | G. E. Fasshauer, M. McCourt, Kernel-based approximation methods using MATLAB, World Scientific Publications, 2015. |
[48] | L. N. Trefethen, Spectral methods in matlab, Oxford University Press, 2000. |
[49] | T. W. Chow, S. Y. Cho, Neural networks and computing, Imperial College Press, 2007. |
[50] |
J. J. Friedman, Multivariate adaptive regression splines, Ann. Stat., 19 (1991), 11–67. https://doi.org/10.1214/aos/1176347963 doi: 10.1214/aos/1176347963
![]() |
[51] |
B. P. Geridonmez, Machine learning approach to the temperature gradient in the case of discontinuous temperature boundary conditions in a triangular cavity, J. Phys., 2514 (2023), 012010. https://doi.org/10.1088/1742-6596/2514/1/012010 doi: 10.1088/1742-6596/2514/1/012010
![]() |
1. | Amal S. Alali, Shahbaz Ali, Muhammad Adnan, Delfim F. M. Torres, On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs, 2023, 15, 2073-8994, 1911, 10.3390/sym15101911 | |
2. | Muhammad Farhan Hanif, Hasan Mahmood, Shahbaz Ahmad, On degree-based entropy measure for zero-divisor graphs, 2024, 16, 1793-8309, 10.1142/S1793830923501045 |