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Article

Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System

1
Institute of Management and Digital Technologies, Russian University of Transport, 127055 Moscow, Russia
2
Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia
3
School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
4
Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences (INM RAS), 119333 Moscow, Russia
5
Moscow Center for Fundamental and Applied Mathematics at INM RAS, 119333 Moscow, Russia
6
World-Class Research Center “Digital Biodesign and Personalized Healthcare”, Sechenov First Moscow State Medical University, 119048 Moscow, Russia
7
Institute of Computer Science and Mathematical Modeling, Sechenov First Moscow State Medical University, 119048 Moscow, Russia
8
Institute for Sound and Vibration Research, University of Southampton, Highfield, Southampton SO17 1BJ, UK
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(19), 3557; https://doi.org/10.3390/math10193557
Submission received: 30 August 2022 / Revised: 22 September 2022 / Accepted: 26 September 2022 / Published: 29 September 2022
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)

Abstract

:
Pancreatic cancer represents one of the difficult problems of contemporary medicine. The development of the illness evolves very slowly, happens in a specific place (stroma), and manifests clinically close to a final stage. Another feature of this pathology is a coexistence (symbiotic) effect between cancer cells and normal cells inside stroma. All these aspects make it difficult to understand the pathogenesis of pancreatic cancer and develop a proper therapy. The emergence of pancreatic pre-cancer and cancer cells represents a branching stochastic process engaging populations of 64 cells differing in the number of acquired mutations. In this study, we formulate and calibrate the mathematical model of pancreatic cancer using the quasispecies framework. The mathematical model incorporates the mutation matrix, fineness landscape matrix, and the death rates. Each element of the mutation matrix presents the probability of appearing as a specific mutation in the branching sequence of cells representing the accumulation of mutations. The model incorporates the cancer cell elimination by effect CD8 T cells (CTL). The down-regulation of the effector function of CTLs and exhaustion are parameterized. The symbiotic effect of coexistence of normal and cancer cells is considered. The computational predictions obtained with the model are consistent with empirical data. The modeling approach can be used to investigate other types of cancers and examine various treatment procedures.

1. Introduction

Pancreatic cancer has the worst prognosis of all major cancer types, with very poor treatment options. Less than 10% of patients will live for more than five years after diagnosis [1,2]. The main reasons for this are as follows.
  • The process of formation of precancerous and cancerous cells represents a successive chain of mutations over a relatively long period of time (10 to 20 years). During this time, the disease often causes very limited noticeable symptoms, and the cancer has often already spread at the time of diagnosis, with the most common sites of metastasis being the liver, lung, and peritoneum [3].
  • Pancreatic tumors comprise cancer cells surrounded by other cell types and extracellular matrix, collectively known as the tumor stroma. The dense stroma of pancreatic cancer makes both diagnosis and treatment more difficult [4,5].
  • A favorable microenvironment for tumor progression exists inside the stroma, built in large part upon symbiotic interactions between non-cancerous pancreatic cells and precancerous and cancerous cells [4,6,7,8,9].
  • The genetic changes that drive pancreatic cancer do not reveal any clear vulnerabilities to available therapies, and the success with existing treatments is very limited [1,10].
Surgery before the disease has spread remains the only treatment likely to be curative; however, <15% of patients are selected for surgery, and there are no broadly used methods for identifying patients at this early, potentially curable, stage. The most common treatment for metastatic or locally advanced pancreatic cancer are aggressive chemotherapies that modestly extend survival and, to date, targeted therapies have provided little or no benefit [1]. Approximately 85% of pancreatic cancers are defined as pancreatic ductal adenocarcinoma (PDAC), and many, perhaps most, of these tumors are believed to develop from precursor lesions termed pancreatic intraepithelial neoplasia (PanIN) [11]. However, the sequences of genetic changes that lead to PDAC and the consequences of these evolutionary processes on the responses to therapy remain unclear [12,13]. Initial models proposed sequential events in key drivers over many years: activating mutation of a copy of KRAS, followed by bi-allelic loss of CDKN2A, TP53, and SMAD4 [14]. However, more recent data imply that multiple loss of function events may occur simultaneously [15].
Therefore, to deepen our understanding of these processes and to derive testable predictions, we elected to develop a mathematical model of PDAC progression. This describes the dynamics of species communities, taking mutations into account: M. Eigen’s replicator system of quasispecies [16].

Framework for Quasispecies Dynamics

In this study, the quasispecies term implies the set of distinct cell populations (species) that differ with respect to the acquired genetic mutations. The respective mathematical model represents an ODE system describing the evolution of population frequencies, taking into account the set of possible mutations:
d p ( t ) d t = ( Q M ) p ( t ) p ( t ) f ( p )
where p = ( p 1 , p 2 , , p n ) is a vector of frequencies (relative numbers of species). Note that in the classical theory of quasispecies, the total number of species remains constant at t 0 . Matrix Q = ( q i j ) , i , j = 1 , 2 , , n represent the probability of mutations, i.e., q i j sets the probabilities that replicating a species with index j results in a species with number i:
i = 1 n q i j = 1 , j = 1 , 2 , , n
The diagonal matrix M = d i a g ( m 1 , m 2 , , m n ) defines the viral fitness landscape. In other words, each species is assigned a number that characterizes its ability to survive in the system. Usually the fitness landscape is quantified on the basis of species growth rates and other data on the dominant properties of the species. The system of quasispecies dynamics describes the evolution of species communities, taking into account possible mutations, but ignoring the natural mortality rates of species. In order to account for species mortality, a modified (open) quasispecies system [16] was proposed, in which the total number of species is not a constant value. Let u i ( t ) be the abundance of species i, t 0 , M and Q the species adaptation matrix and mutation matrix, respectively.
Let us introduce the following function:
S ( u ) = i = 1 n u i ( t ) = 1 .
The function S ( u ) defines the total number of species at time t 0 . The function Φ ( s ) is characterized by the following properties:
(a)
Φ ( s ) is a smooth function s [ 0 , c ) , c 0 ;
(b)
Φ ( s ) > 0 , s [ 0 , c ) ;
(c)
If 0 < c < + , then lim s c s Φ ( s ) = 0 ;
(d)
Function s Φ ( s ) = 0 has only one maximum at the point s * , 0 < c < c * .
As an example of function Φ ( s ) , we can consider the functions Φ ( s ) = ( c s ) , c > 0 , 0 < s < c , or Φ ( s ) = exp ( γ s ) , γ > 0 , s > 0 .
Consider the diagonal matrix describing species mortality rates D = d i a g ( d 1 , d 2 , , d n ) . Let the elements of the fitness landscape matrix M and the mortality matrix satisfy the following conditions:
0 m m i n m i m m a x 0 d m i n d i d m a x , i = 1 , 2 , , n 0 < d m a x < m m i n .
Here are the model equations describing the population dynamics of species:
d u ( t ) d t = Φ ( s ) ( Q M ) u ( t ) D u ( t ) u ( 0 ) = u 0 0
System (5) is positively invariant for any non-negative initial conditions u ( 0 ) 0 . There exists a unique solution to this problem. When conditions in (4) are satisfied, the total number of species (3) as well as the species abundances are bounded functions on the system trajectories [16]. The average fitness of open system (5) is defined by
f ( u ) = 0 , s ( u ) = 0 i = 1 n m i u i ( t ) i = 1 n d i u i ( t ) = ( Φ ( s ) ) 1 , s ( u ) 0
where the numerator of the expression is equal to the average fitness of quasispecies (1), and the denominator reflects the decline of the species community due to their mortality. Note that the proposed method for describing cell dynamics does not take into account the spatial distribution of cells, which, as a rule, leads to equations of the “reaction-diffusion” type. In the case of the classical Eigen system, it has been proved [17] that the solutions of the “reaction-diffusion” system have the property of diffusion stability with respect to additive additions of diffusion terms. The case of an open replicator system requires a separate study, which is beyond the scope of our research. In this case, to solve the problem, it is possible to use the methods of traveling waves [18,19,20].
In Section 2, the mutational pathways of the normal cells are characterized. In Section 3, the mathematical model of tumor-immune cell dynamics if developed. The results of numerical simulations are presented in Section 4. The results are briefly discussed in the Section 5.

2. Development of the Cancer Mutations

The vast majority (>85%) of pancreatic cancers, defined as pancreatic ductal adenocarcinomas, are driven by activating mutations in a single copy of the KRAS gene (85–95%) and also display very frequent loss of function mutations in TP53 (>75%), CDKN2A (>30–60%), and SMAD4 (30–50%). Here, these mutations have been abbreviated as K, P, C, and S mutations, respectively. Figure 1 represents the chain of successive mutations of growing pancreatic cancer cell populations. Healthy cells ( N ) with the potential for oncogenic transformation to PDAC [21,22] are characterized by a high probability of slow error-free replication p 0 = 0.999999 . Each healthy originating cell can proliferate or die, according to data presented in Table 1, whereas at any given time most of them are non-replicative. Table 1 summarizes the doubling time data for different cell types as well as data on type mortality rates.
With probability of p m 10 7 / 4 per cell per generation, healthy cells will be subject to an activating mutation in a single allele of KRAS. Similarly, we estimate the probability of loss of function mutations in the second copy of CDKN2A (C), TP53 (P), or SMAD4 (S) to be 10 7 / 4 per cell per generation. For simplicity, it is assumed that loss of the first copy of these tumor suppressors has no effect on cell fitness. Therefore, healthy N cells are converted with the probability p leading to K, P, C, or S mutant cells, as shown in Figure 1.
These initiating mutations in key genes, K, C, P, and S, would be expected, in many cases, to lead to oncogene induced senescence. Therefore, the loss from the model of these cells upon mutation is defined by probability p d i . Otherwise, the mutant cells of K type, as demonstrated in Figure 1, can proliferate and die as K-mutants with probabilities shown in Table 1, but can also give a new mutation of P, C, or S type, resulting in K P , K C , or K S double-mutants, correspondingly. At this and consequent stages after adaptation, the new mutants are believed to avoid oncogene induced senescence. These double-mutants can also proliferate, die, or give a rise to triple-mutant cells, i.e., K P S , K P C , K C P , K C S , etc., as shown in Figure 1.
As a result, the overall number of mutated cells is 64. To continue, it is important to enumerate all cell types. The continuous numbering is introduced starting from a single-mutant cells, as illustrated in Table 2 and Figure 1. Overall, the one-letter cells correspond to numbers from 1 to 4, two-letter cells—from 5 to 16, three-letter cells—from 17 to 40, and four-letter cells—from 41 to 64.
Healthy cells mutate into one-letter type with probability q 0 = p = 1 4 10 7 . Given the assumptions stated above, the probabilities of one-letter type mutations are q 1 = p p 1 ( 1 p d ) .
Next, we assume that the probabilities of the mutations converting cells with a single mutation into cells carrying two mutations (i.e., one-letter cell types to the two-letter cell types) and the subsequent mutations of these cells into the three-letter and four-letter cell types are identical:
p m 1 = p 1 = 1 12 , p m 2 = p m 3 = p m 4 = p 2 = 1 24 .
Probabilities of two-letter type mutations are:
q 2 = p p 1 2 ( 1 p d ) 2
Accordingly, three-letter and four-letter:
q 3 = p p 1 3 ( 1 p d ) 3
q 4 = p p 1 4 ( 1 p d ) 4
Unfortunately, data on the probability of instantaneous mortality of p d species are unknown to us. All subsequent calculations will be made using p d = 0.95 .
Fitness rates m i and death rates d i in the stated order of cell type enumeration are presented in Figure 2. Based on the data in Table 1, we construct a matrix M that defines the fitness landscape of species. Let us take 1200 days as a unit of fitness (doubling time of healthy cells). All other indicators of the fitness of cell species are calculated by the ratio of the value of 1200 to the corresponding doubling time of the population of each particular species.

3. Model Calibration for Cancer and Precancerous Cell Dynamics

To proceed with the equations of the model, we note that the abundance of nutrients in the pancreas facilitates favorable conditions for healthy, cancerous, and precancerous cells. Until the very latest stages of the disease, the number of healthy pancreatic cells is in vast excess over the number of cancer cells. Therefore, we assume that the number of healthy cells remains constant and the system of equations stated below describes only growth dynamics for cancerous and precancerous cells.
Let S ^ be the upper limit of all cells in the pancreas, S c ( t ) —number of cancerous and precancerous cells at time t 0 , S c ( 0 ) = 0 .
The supply of cells with nutrients is described by function ψ ( t ) , t 0 , which is the solution to the equation
d ψ ( t ) d t = r ψ ( t ) 1 ψ ( t ) K , ψ ( 0 ) = 10 5 ,
where K = 0.18 (scaled with respect to the maximal value 1), and r = 0.00075 day 1 .
The system of equations governing the population dynamics of the number of cancerous and precancerous cells is given by
d v i ( t ) d t = ( ψ ( t ) S c ( t ) ) S ^ ( Q M v ( t ) ) i + γ i d i v i ( t ) , v i ( 0 ) = 0 , i = 1 , , 64 .
Here v i ( t ) is the frequency of the ith cell type relative to S ^ , M—diagonal matrix of the fitness landscape M = d i a g ( m 1 , m 2 , , m 64 ) , d i —the corresponding death rates. The constants γ i c ( i = 1 , 2 , , 64 ) characterize the effect of symbiosis between healthy cells and cancerous and precancerous cells of type i. The numerical values of constants γ i c are given in Table 3.
Let q i j are elements of matrix Q, then we have
( Q M v ( t ) ) i = j = 1 64 q i j m j v j ( t ) , i = 1 , 2 , , 64 .
Equation (10) implies that the average fitness of the system is given by
f ( t ) = ψ ( t ) S c ( t ) 1
The tumor cells expressing non-self antigens are eliminated by antitumor immune responses mediated by CD8+ T cells (CTL) [23]. However, during cancer progression, CTLs loss their function and acquire an exhaustion state due to immunosuppression within the tumor micro-environment [23,24,25]. The system of equations accounting for interactions between cancerous and stromal cells and the effect of cancer antigen-induced CD8 T cell response T ( t ) leads to the model system of ODEs presented below:
d v i ( t ) d t = ( ψ ( t ) S c ( t ) ) ( S ^ ( Q M v ( t ) ) i + γ i c ) d i v i ( t ) γ i v i ( t ) T ( t ) T ^ K T + β ( S ^ S c ( t ) ) 2 + T ^ · T ( t ) ; v i ( 0 ) = 0 , i = 1 , 2 , , 64 .
The last term in the above equation describes the T-cell mediated elimination of cancer cells depending on the functional state of the tumor-specific T cells.
The equation for cancer-specific CD8 T cells takes into account their homeostatic turnover, the cancer-antigens induction of their clonal expansion, and the functional exhaustion taking place when the number of cancer cells increases above a certain threshold.
d T ( t ) d t = T * T ^ + b S ^ · S c ( t ) T ( t ) ω + S ^ 2 ( S c ( t ) ) 2 d T T ( t ) , T ( 0 ) = 0.3
Hence, it is considered that positive and negative regulations determine the strength of the immune response. For a low or medium abundance of cancer cells, the antiviral immune response is stimulated (the numerator term). However, at high density of cancer cells it gets down-regulated (the denominator term). In particular, for high tumor cell density, the down-regulation of T cell activity takes place (known as functional exhaustion). The description follows the framework previously suggested for virus infections [26,27].
If the we take into account the immune cell-mediated elimination of the cancer cells, then the fitness of the system is defined by
f i m ( t ) = f ( t ) 1 1 + C ( t ) D ( t ) .
Here the functions C ( · ) , D ( · ) are given by
C ( t ) = T ^ · T ( t ) · i = 1 64 γ i v i ( t ) K T + β ( S ^ · S c ( t ) ) 2 + T ^ · T ( t ) ,
D ( t ) = i = 1 64 d i v i ( t ) ,
and f ( t ) is the fitness of the system without the impact of the immune system.
Here, T ^ = 2.5 · 10 5 cells/mL refers to the homeostatic level of tumor-antigen-specific CTLs, K T = 10 3 cells/mL estimates the CTL density for a half-maximal elimination rate of tumor cells, T * = 10 3 cells/mL/day is the homeostatic supply, d T = 0.01 (1/day) is the CTL death rate coefficient, β = 10 11 (cells/mL) 2 , b = 10 day 1 , ω = 10 7 (cells/mL) 2 , stand for the down-regulation of the CTL effector function, the clonal proliferation rate of CTLs, and the threshold for cancer cell abundance inducing functional exhaustion in CTLs. For simulations, we used a rescaled value of CTL with respect to their homeostatic abundance T ^ .
The software implementation of the model is written in Python version 3.8.8 using jupyter-notebook version 6.3.0. To solve the system of differential equations, the odeint() function from the scipy.integrate module was used. The error of this method for each value of the function y i is equal to:
e i = 1.5 · 10 8 · | y i | + 1.5 · 10 8
Graphs were built with a step of 0.1.

4. Computational Experiments

Figure 3 shows the results of numerical calculations of the system without taking into account the interaction with the cells of the immune system. It predicts the evolution of the abundance of the mutants determined by the differences in their fitness values. Cells bearing one mutation appear earlier and have larger population size than that of the two-, three-, and four-mutation cells by a factor of ten.
Figure 4 shows the results of experiments taking into account the interaction of cancer cells with the effector CD8+ T cells of the immune system. The following graphs show the change in the total number of cancer cells under the influence of cells of the immune system. The CTL-mediated elimination of cancer cells slows down the emergence of the single- and double-mutation cells. However, the CTL response fails to control the emergence and abundance of tumor cells with three and four mutations.
The dynamics of the CTL response to tumor antigens is depicted in Figure 5. Initially, the appearance of non-self antigens expressed by tumor cells induces the clinal expansion of CTLs. However, when the density of cancer cells reaches a certain threshold, the immune response is down-regulated (days 650), and the CTLs acquire an exhaustion phenotype.
Finally, the model can be used to predict the evolution of the mean cancer call population fitness determining the rate of cancer progression. Figure 6 shows the dynamics of the mean fitness corresponding to uncontrolled growth of cancer cells and the effect of immune control, which delays the emergence of more aggressive cancer quasispecies.

5. Conclusions

In this work, we developed a quantitative mathematical model of the evolutionary dynamics of pancreatic cancer cells. The model describes the emergence of cancer cells bearing different numbers of mutations as evolving quasispecies systems consisting of 64 subpopulations bearing one, two, three, and four mutations in the most commonly mutated genes in pancreatic cancer, KRAS, CDKN2A, TP53, and SMAD4. In addition, the antitumor CTL response is taken into account and the model consistently reproduces the sequential evolution of cancer cell populations.
In the future, improvements on the model can expand the scope of parameters that contribute to the performance of the model, bringing it closer to the reality of the disease. Further model development should consider the relationship between the number of mutations and the antigenicity of the respective cancer cells, and potentially additional mutations beyond the four considered here. Furthermore, the incorporation of further existing and future data should improve the validity of estimates of the different cell doubling times and mortality rates for specific mutated cell types that are used to calibrate the model dynamics. Finally, there would be benefits to applying stochastic modeling approaches with the potential to reproduce the unpredictable heterogeneous course of the disease. In summary, our current model and future improvements will provide an analytical tool to examine approaches to the treatment of pancreatic ductal adenocarcinoma, including the chemotherapy, surgery, reinvigoration of exhausted CTLs, and the CAR-T cell therapies [24,25,28,29].

Author Contributions

Conceptualization, A.S.B., N.L., D.Y. and G.B.; methodology, A.S.B., N.L., D.Y. and G.B.; software, M.C.; validation, A.S.B., N.L. and D.Y.; formal analysis, A.S.B. and M.C.; investigation, A.S.B., N.L., M.C., D.Y., D.G., R.S. and G.B.; data curation, N.L.; writing—A.S.B., N.L., M.C., D.Y., D.G., R.S. and G.B.; writing—review and editing, A.S.B., N.L., M.C., D.Y., D.G., R.S. and G.B.; visualization, M.C., D.G. and R.S.; supervision, A.S.B.; funding acquisition, A.S.B. and D.Y. All authors have read and agreed to the published version of the manuscript.

Funding

The reported study was funded by RFBR and the Royal Society of London (RS), project number 21-51-10006. A.B. was supported by the Moscow Center for Fundamental and Applied Mathematics at Lomonosov Moscow State University (agreement with the Ministry of Education and Sciences of the Russian Federation No. 075-219-1621) and D.G., R.S. and G.B. were partly supported by the Moscow Center for Fundamental and Applied Mathematics at INM RAS (agreement with the Ministry of Education and Sciences of the Russian Federation No. 075-15-2022-286).

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Abbreviations

The following abbreviations are used in this manuscript:
PDACPancreatic ductal adenocarcinoma
CTLeffector CD8+ T lymphocyte
ODEordinary differential equation

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Figure 1. Scheme illustrating the evolution of possible mutations of healthy cell N. The branch following the initial mutation K is shown in detail. For clarity, the evolutionary schemes evolving from the alternative initial P(2), C(3), and S(4) mutations are not included in the diagram.
Figure 1. Scheme illustrating the evolution of possible mutations of healthy cell N. The branch following the initial mutation K is shown in detail. For clarity, the evolutionary schemes evolving from the alternative initial P(2), C(3), and S(4) mutations are not included in the diagram.
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Figure 2. Fitness rates m i and death rates d i of mutated cells. Colors correspond to one-letter (blue), two-letter (orange), three-letter (green), and four-letter (magenta) mutations.
Figure 2. Fitness rates m i and death rates d i of mutated cells. Colors correspond to one-letter (blue), two-letter (orange), three-letter (green), and four-letter (magenta) mutations.
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Figure 3. Dynamics of the populations of cancerous and precancerous cells without taking into account the effects of immune cells. The x-axis is the time in days. The y-axis is the number of (quasispecies) mutated pancreatic cells. There are: (a) single mutation cells, (b) two mutation cells, (c) three mutation cells, (d) KPCS cells bearing four mutations, (e) dynamics of nutrients intake.
Figure 3. Dynamics of the populations of cancerous and precancerous cells without taking into account the effects of immune cells. The x-axis is the time in days. The y-axis is the number of (quasispecies) mutated pancreatic cells. There are: (a) single mutation cells, (b) two mutation cells, (c) three mutation cells, (d) KPCS cells bearing four mutations, (e) dynamics of nutrients intake.
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Figure 4. Dynamics of the populations of cancerous and precancerous cells, taking into account the impact of effector CTLs of the immune system. (a) One-mutation cells, (b) two-mutation cells, (c) three-mutation cells, (d) KPCS cells.
Figure 4. Dynamics of the populations of cancerous and precancerous cells, taking into account the impact of effector CTLs of the immune system. (a) One-mutation cells, (b) two-mutation cells, (c) three-mutation cells, (d) KPCS cells.
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Figure 5. Dynamics of the anti-tumor CTL response. The x-axis is time in days. The y-axis is the relative number of cells on a scale of 10 4 .
Figure 5. Dynamics of the anti-tumor CTL response. The x-axis is time in days. The y-axis is the relative number of cells on a scale of 10 4 .
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Figure 6. Mean system fitness growth of the cancer cell population without the effect of immune system (yellow line at the top) and taking into account the elimination of cancer cells by effector CTLs (red line at the bottom).
Figure 6. Mean system fitness growth of the cancer cell population without the effect of immune system (yellow line at the top) and taking into account the elimination of cancer cells by effector CTLs (red line at the bottom).
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Table 1. Data on doubling times and mortality rates for cell types bearing specific mutations.
Table 1. Data on doubling times and mortality rates for cell types bearing specific mutations.
Mutation TypeDoubling Time (Days)Mortality
K (KRAS)200 3.22 %
P (TP53)800 0.86 %
C (CDKN2A)800 1 %
S (SMAD4)900 0.9 %
K C 140 15 %
K S 180 20 %
K P 180 8 %
P S 750 5 %
C P 600 5 %
C S 600 5 %
K C S 120 20 %
K P C 120 10 %
K P S 120 10 %
C P S 500 5 %
K P C S 100 8 %
Table 2. An illustration of the enumeration rule for the one-letter and two-letter designated types of mutated cells.
Table 2. An illustration of the enumeration rule for the one-letter and two-letter designated types of mutated cells.
Cell typeKPCS K P K C K S P K
Number12345678
Cell Type P C P S C K C P C S S K S P S C
Number910111213141516
Table 3. Estimates of the effect of symbiosis between healthy cells and cancerous and precancerous cells of type i.
Table 3. Estimates of the effect of symbiosis between healthy cells and cancerous and precancerous cells of type i.
Mutation type, iKPCS K C K S K P
Symbiosis, γ i c ( 10 5 ) 4112202424
Mutation type, i P S C P C S K C S K P C K P S C P S K P C S
Symbiosis, γ i c ( 10 5 ) 3332322020203620
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Bratus, A.S.; Leslie, N.; Chamo, M.; Grebennikov, D.; Savinkov, R.; Bocharov, G.; Yurchenko, D. Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System. Mathematics 2022, 10, 3557. https://doi.org/10.3390/math10193557

AMA Style

Bratus AS, Leslie N, Chamo M, Grebennikov D, Savinkov R, Bocharov G, Yurchenko D. Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System. Mathematics. 2022; 10(19):3557. https://doi.org/10.3390/math10193557

Chicago/Turabian Style

Bratus, Alexander S., Nicholas Leslie, Michail Chamo, Dmitry Grebennikov, Rostislav Savinkov, Gennady Bocharov, and Daniil Yurchenko. 2022. "Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System" Mathematics 10, no. 19: 3557. https://doi.org/10.3390/math10193557

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