Nonlinear Analysis: Real World Applications 13 (2012) 78–84
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Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Experimental versus numerical data for breast cancer progression C.L. Jorcyk a , M. Kolev b , K. Tawara a , Barbara Zubik-Kowal c,∗ a
Department of Biology, Boise State University, Boise, ID 83725, USA
b
Department of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Zolnierska 14, 10-561, Poland
c
Department of Mathematics, Boise State University, 1910 University Drive, Boise, ID 83725, USA
article
info
Article history: Received 15 April 2011 Accepted 16 July 2011 Keywords: Cell lines Mammary Tumor progression Animal model Integro-differential equations Numerical simulations
abstract This paper deals with a mouse model of breast cancer based on two mammary adenocarcinoma cell lines derived from a spontaneous tumor of the mammary gland in a female BALB/c mouse. We investigate both animal and mathematical models of tumor progression, and demonstrate a correspondence between the experimental and predicted data. The mathematical model is solved numerically and the laboratory data are utilized in order to find unknown parameters for the model equations. The results of the numerical experiments illustrate that the mathematical model has a potential to describe the growth of cancer cells in vivo. Published by Elsevier Ltd
1. Introduction Breast cancer is the most common type of malignant disease that occurs in women. In 2009, approximately 192,000 women were diagnosed with breast cancer in the United States [1]. Over 40,000 women diagnosed with the disease die yearly due to inadequate detection and treatment options, making breast cancer the second most common cause of cancerrelated death for women in the US. Cells progress from normal, to abnormal, to precancerous, and finally to invasive cancer as they accumulate mutations that inactivate their growth regulating mechanisms. Left undetected, cancer cells will spread from the primary tumor in the breast through the bloodstream or lymphatic system to secondary organs, in a process known as metastasis. Most breast cancer patients die due to the complications that arise from the growth of metastatic lesions in vital organs such as lung, liver, brain, and bone [2,3]. Developing effective therapeutics for breast cancer patients by studying the behavior and pathogenesis of cancer cells is a continuing area of research focus and funding. The use of various animal models allows cancer to be studied in a controlled environment to elucidate cellular mechanisms that could be targeted therapeutically. Mus musculus (mouse) is the primary animal model used by researchers to study the biology and progression of metastatic disease in vivo. In addition to transgenic mice models of cancer, many researchers use a variety of methods to implant breast cancer cells into mice in order to study the different stages of tumor progression. Tail vein, subcutaneous, intracardiac, intraperitoneal, intratibial, or orthotopic injections are some of the techniques available for transplanting cancer cells into mice [4,5]. Each method allows the investigation of a different part of the metastatic cascade. The use of tail vein or intracardiac injections introduces cells directly into the circulation, thereby bypassing the primary tumor growth stage and allowing for maximum dispersal of the cancer cells and resultant metastases. On the other hand, an orthotopic injection refers to an injection of
∗
Corresponding author. Tel.: +1 208 426 2802; fax: +1 208 426 1356. E-mail addresses:
[email protected] (C.L. Jorcyk),
[email protected] (M. Kolev),
[email protected] (K. Tawara),
[email protected] (B. Zubik-Kowal). 1468-1218/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.nonrwa.2011.07.014
C.L. Jorcyk et al. / Nonlinear Analysis: Real World Applications 13 (2012) 78–84
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Table 1 Individual mice tumor average sizes (mm3 ). Days
4T1.2 cells
Days
66c14 cells
– 14 17 21 25 – 31 –
– 17.2 40.81 108.21 218.5 – 643.7 –
10 14 17 21 24 28 31 34
16.68 39.52 101.45 139.86 239.98 413.08 409.08 478.03
the cancer cells into their area of origin, in this case the mammary tissue. This method promotes primary tumor growth followed by potential, subsequent metastatic events [5]. Since metastasis is a multi-step process, cancer cell implantation techniques such as orthotopic injections accurately mimic human breast cancer progression. 2. The 66c14 and 4T1.2 orthotopic model of mouse mammary cancer In this study, we utilized the 66c14 and 4T1.2 syngeneic mouse model of metastatic breast cancer. Our laboratory data are presented in Table 1. 66c14 and 4T1.2 cells were originally derived from a spontaneous tumor of the mammary gland in an inbred BALB/c mouse [6]. These cell lines were selected for their distinct metastatic profiles upon orthotopic reintroduction into BALB/c mice. 4T1.2 cells aggressively metastasize to lung, brain, liver, and bone mimicking the human situation, while 66c14 cells are mildly metastatic to lung and lymph node [6,7]. This difference in metastatic potential is thought to be due in part to the low caveolin-1 expression in 4T1.2 cells, which has previously been shown to be suppressed in metastatic human breast cancer [8]. In the work presented here, we were interested in studying the growth pattern of 66c14 and 4T1.2 primary tumors in vivo. 66c14 and 4T1.2 cells were maintained in minimum essential media alpha (MEMα) containing 10% fetal bovine serum (FBS), 1 mM sodium pyruvate, and 100 units/ml of penicillin and streptomycin. The cells were incubated at 37 °C with 5% CO2 and 95% humidity. 66c14 or 4T1.2 (1 × 105 ) cells were orthotopically injected into the 4th mammary gland of 4-week old female BALB/c mice (n = 5). One hundred percent of mice injected with the cancer cells developed primary tumors, and tumor size was measured using calipers at set time-points ranging from 10 to 34 days. To calculate tumor volume in mm3 , the following equation was used: tumor volume (mm3 ) = (length × width2 )/2. Our findings demonstrated that orthotopically injected 4T1.2 cells result in tumor growth at a faster rate than 66c14 cells (Table 1). Furthermore, the condition of the mice injected with 4T1.2 cells deteriorated faster, presumably due to a larger metastatic burden. 3. Mathematical model In addition to in vitro and in vivo experiments, which are the major tools in cancer research, methods of mathematical modeling have been shown to be useful for better understanding of the processes of carcinogenesis. Mathematical and computational approaches possess the ability to describe in quantitative terms the complex and highly nonlinear interactions between non-cancerous and cancer cells [9–12]. In particular, over the last two decades, kinetic theory for active particles (KTAP) and partial differential equations have been successfully applied by Bellomo et al. to model the spread of tumors [11,13–15]. Within the framework of KTAP, the phenomena studied are described by integro-differential equations of Boltzmann type, which include a variable u presenting the functional activity of interacting entities called active particles. This methodology has been successfully applied for modeling various processes and phenomena considered by applied and life sciences, for example, complex living systems [16,17], population dynamics [18], politics and social sciences [19,20], psychological relationships [21], investigation of traffic flow [22], etc. Recent development and applications of this theory can be found in the book by Bellomo [9]. Recently, it has been demonstrated in [23,24] that the KTAP has the potential to describe the growth of prostate and breast tumor cells. The theory has been used in [23] to predict prostate tumor progression and a correspondence has been obtained between the numerical [23] and laboratory [25] data. In [24], it has been demonstrated that it is possible to find such parameter values for the model partial differential equations so that their solutions correspond to clinical data, therefore showing the potential to extend the applications of the kinetic theory to breast cancer. In this paper, we have applied the model equations to the 66c14 and 4T1.2 syngeneic mouse model. We have utilized the laboratory data to estimate the parameter values for the kinetic model and the results of our numerical experiments demonstrate that the dynamics of the model equations properly capture the development of the 66c14 and 4T1.2 primary tumors in vivo. For the model equations, we consider the populations of cancer cells (the first population denoted with i = 1), helper T cells (i = 2), cytotoxic T lymphocytes (i = 3), antigen presenting cells (i = 4), antigen-loaded APCs (i = 5), and host environment cells (i = 6). The distribution density of the ith population with activation state u ∈ [0, 1] at time t ≥ 0 is
80
C.L. Jorcyk et al. / Nonlinear Analysis: Real World Applications 13 (2012) 78–84 Table 2 Tumor-immune system dynamics variables. i
Abbr.
Population
Activation state u ∈ [0, 1]
1 2 3 4 5 6
CC Th CTL APC AgAPC HE
Cancer cells Helper T cells Cytotoxic T lymphocytes Antigen presenting cells Antigen-loaded APCs Host environment cells
Recognition of CC by APC Cytokines produced by Th cells Destruction of CC Not considered Not considered Not considered and f6 constant
denoted by fi (t , u),
fi : [0, ∞) × [0, 1] → R+ , i = 1, . . . , 6.
Table 2 shows the population variables and their abbreviations for i = 1, . . . , 6. Moreover, we denote by ni ( t ) =
1
∫
fi (t , u)du,
ni : [0, ∞) → R+ , i = 1, . . . , 6,
(3.1)
0
the number of the ith cells per unit volume at time t ≥ 0. Our goal is to determine n1 (t ) by solving the following model equations
∂ f1 (t , u) = p(161) ∂t
1
∫
f1 (t , v)dv − d12 f1 (t , u)
1
∫
0
(1)
v f2 (t , v)dv + t16 2u 0
− d13 f1 (t , u)
1
∫
f1 (t , v)dv − u f1 (t , u) 2
u
1
∫
v f3 (t , v)dv
(3.2)
0
∫ 1 ∂ f2 (2) (2) (t , u) = p16 (1 − u) + p25 (1 − u)n5 (t ) v f2 (t , v)dv ∂t 0 ∫ u (2) 2 + t25 n5 (t ) 2 (u − v)f2 (t , v)dv − (1 − u) f2 (t , u) 0
− d21 f2 (t , u)
1
∫
f1 (t , v)dv − d26 f2 (t , u) + S2 (t , u)
(3.3)
0
∫ ∫ 1 u ∂ f3 (3) (3) (3) 2 (t , u) = p16 (1 − u) + p25 (1 − u)n5 (t ) v f2 (t , v)dv + t23 2 (u − v)f3 (t , v)dv − (1 − u) f3 (t , u) ∂t 0 0 ∫ 1 ∫ 1 × f2 (t , v)dv − d31 f3 (t , u) f1 (t , v)dv − d36 f3 (t , u) + S3 (t , u) (3.4) 0
d dt d dt
(4)
0
(4)
n4 (t ) = p16 + p25 n5 (t )
1
∫
(5)
v f2 (t , v)dv − b14 n4 (t ) 0
(5)
n5 (t ) = b14 n4 (t )
1
∫
v f1 (t , v)dv − d51 n5 (t ) 0
1
∫
v f1 (t , v)dv − d46 n4 (t )
(3.5)
f1 (t , v)dv − d56 n5 (t ) + S5 (t ).
(3.6)
0 1
∫ 0
The meanings of the parameters and corresponding gain, loss and conservative terms participating in the model Eqs. (3.2)– (3.6) are described in Table 3. The functions Si (t , u),
i = 2, 3
describe the possible inlet of Th cells and CTLs, respectively, and S5 (t ) is a source term describing the possible influx of antigen-loaded APCs. A general theory for systems of the type (3.2)–(3.6) is presented in the papers by Arlotti et al. [26,27] and in the book by Arlotti et al. [28]. The entire model (3.2)–(3.6) is a system of partial integro-differential equations, which are not complete and have to be supplemented by initial conditions. For the conditions, we apply the experimental data of Table 1. The parameter values of the model can also be found by using the experimental data of Table 1 and comparing them with numerical approximations to the solutions of (3.2)–(3.6). In Section 4, we solve (3.2)–(3.6) numerically and determine parameter values for the 66c14 and 4T1.2 orthotopic model of mouse mammary cancer.
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Table 3 Model parameters and variables. Par.
Description
u t
Activation state Time
p16
Generation of new tumor cells
t16 d12 d13
Steady progress of the cancer cells toward decreasing their activation states Destruction of cancer cells due to the activity of Th cells Destruction of cancer cells due to the response of CTLs
p16
Generation of new Th cells by HE
p16
Production of new CTLs by HE
p25
Generation of new Th cells due to the interactions between Th cells and antigen-loaded APCs
p25
Production of new CTLs due to the interactions between Th cells and antigen-loaded APCs
t25
Steady progress of the Th cells toward increasing their activation states due to the interactions between Th cells and antigen-loaded APCs
t23 d21 d31 d26 d36
Steady progress of the CTLs toward increasing their activation states due to the interactions between Th cells and CTLs Destruction of Th cells due to their interactions with cancer cells Destruction of CTLs due to their interactions with cancer cells Natural death of Th cells Natural death of CTLs
p16
Constant production of new APCs by HE
p25
Generation of new APCs due to the interactions between Th cells and antigen-loaded APCs
b14 d46 d51 d56
Appearance of new antigen-loaded APCs due to the interactions between APCs and CCs Natural death of APCs Destruction of antigen-loaded APCs due to their interactions with cancer cells Natural death of antigen-loaded APCs
(1 )
(1)
(2 ) (3 ) (2 ) (3 )
(2) (3)
(4) (4)
(5)
4. Numerical approximations for the 66c14 and 4T1.2 orthotopic model of mouse mammary cancer The series of mammary carcinoma cell lines developed from spontaneous tumors of the mammary gland in a female BALB/c mouse provide a resource to apply differential equations and model the cellular immune response against cancer and complicated interactions between the six main populations ni (t ), i = 1, . . . , 6, used in the kinetic theory for active particles. The goal of this section is to apply the five model Eqs. (3.2)–(3.6) to compute numerical approximations to the 4T1.2 and 66c14 cell lines. The proliferation essay described in Table 1 is utilized to find the parameter values needed in (3.2)–(3.6) for each cell line. Since it is not possible to solve the Eqs. (3.2)–(3.6) exactly, we solve this system numerically. In the first step, we transform the integro-differential equations (3.2)–(3.6) to a larger system composed of ordinary differential equations. For this transformation, we discretize the Eqs. (3.2)–(3.6) with respect to the activation state by applying the equidistant gridpoints ui = i1u,
i = 0, 1, 2, . . . , N ,
for the activation state u ∈ [0, 1]. The step size 1u is defined by 1u = 1/N, where N corresponds to the number of the grid-points. The larger the number N, the more accurate numerical approximations fj,i (t ) ≈ fj (t , ui )
(4.7)
are computed within machine precision (and further depression of N is not needed). Here, j indicates the population for the distribution density function fj and i indicates the localization of the activation state u ∈ [0, 1]. The grid-points ui ∈ [0, 1] are utilized not only for (4.7) but also to compute the approximations for the integrals 1
∫
f1 (t , v)dv ≈ Q0N f1 (t , v) , 0 1
∫
v fj (t , v)dv ≈ Q0N v fj (t , v) ,
j = 1, 2, 3,
0
∫
1
∫
ui ui
(4.8) f1 (t , v)dv ≈
QiN
f1 (t , v) ,
(ui − v)fj (t , v)dv ≈ Q0i (ui − v)fj (t , v) ,
j = 2, 3.
0
Here, Q denotes an arbitrary numerical quadrature based on equidistant nodes. The lower and upper indices for Q indicate the interval of integration, and the corresponding integrand is included between the square brackets.
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C.L. Jorcyk et al. / Nonlinear Analysis: Real World Applications 13 (2012) 78–84 Table 4 Parameter values for growth of primary tumors in female mice. Param. (1 )
4T1.2
66c14
p16
2.5 · 10
p16
2.5 · 10−4
(2 ) (3)
−1
Param.
4T1.2
−1
3.4 · 10−4
2.4 · 10
66c14
d21
−4
2.5 · 10
3.4 · 10−4
d26
2.3 · 10−1
1.2 · 10−1
p16
2.5 · 10−4
3.4 · 10−4
d31
2.5 · 10−1
8.5 · 10−2
p16
2.5 · 10−4
3.4 · 10−4
d36
2.3 · 10−1
1.2 · 10−1
(4)
(2)
p25
2.5 · 10−1
2.4 · 10−1
d46
2.3 · 10−1
1.2 · 10−1
p25
1.4 · 10−2
2.1 · 10−1
d51
2.5 · 10−4
3.4 · 10−4
(3)
(4)
p25
1.6 · 10−2
7.3 · 10−3
d56
2.3 · 10−1
1.2 · 10−1
t16
1.6 · 10−2
7.3 · 10−3
a2
8.8 · 10−12
4.2 · 10−6
(1)
(2)
t25
1.6 · 10−2
7.3 · 10−3
b2
9.4 · 10−2
2.7 · 10−1
t25
1.6 · 10−2
7.3 · 10−3
a3
1.3 · 10−1
4.4 · 10−2
b14 d12 d13
1.9 · 10−1 1.6 · 10−2 2.5 · 10−4
3.0 · 10−2 7.3 · 10−3 3.4 · 10−4
b3 a5 b5
6.1 · 10−6 8.3 · 10−2 2.4 · 10−2
9.1 · 10−6 2.0 · 10−2 1.4 · 10−1
(3) (5 )
We apply (4.7) and (4.8) to the partial integro-differential equations (3.2)–(3.6) and obtain the following system
(1) N 2 (t ) = f1 (t , v) + t16 2ui Qi f1 (t , v) − ui f1,i (t ) dt − d12 f1,i (t )Q0N v f2 (t , v) − d13 f1,i (t )Q0N v f3 (t , v)
df1,i
(1)
p16 Q0N
(4.9)
df2,i
(t ) = p(162) (1 − ui ) + p(252) (1 − ui )n5 (t )Q0N v f2 (t , v) dt (2) − d21 f2,i (t )Q0N f1 (t , v) − d26 f2,i (t ) + t25 n5 (t ) + Wi (f2 ) + S2 (t )
(4.10)
df3,i (t ) = p(163) (1 − ui ) + p(253) (1 − ui )n5 (t )Q0N v f2 (t , v) dt (3) − d31 f3,i (t )Q0N f1 (t , v) t23 + Wi (f3 )QiN f2 (t , v) − d36 f3,i (t ) + S3 (t )
(4.11)
dn4 (t ) = p(164) + p(254) n5 (t )Q0N v f2 (t , v) − b(145) n4 (t )Q0N v f1 (t , v) − d46 n4 (t )
(4.12)
(t ) = b(145) n4 (t )Q0N v f1 (t , v) − d51 n5 (t )Q0N f1 (t , v) − d56 n5 (t ) + S5 (t ).
(4.13)
dt dn5 dt
The components Wi (f2 ) and Wi (f3 ) in (4.9)–(4.13) are defined by
Wi (fj ) = 2Q0i (ui − v)fj (t , v) − (1 − ui )2 fj,i (t ),
j = 2, 3,
and we consider the source terms in the linear form Si (t ) = ai t + bi , i = 2, 3, 5. The system (4.9)–(4.13) includes 3N + 5 ordinary differential equations. This means that the more grid-points ui are applied for the activation state, the more equations have to be solved. (1) (1) The parameter values p16 , t16 , d12 , d13 , . . . , which are utilized in the model (3.2)–(3.6) and the numerical scheme (4.9)– (4.13) are unknown and have to be determined according to the experimental data from Table 1. We have computed the approximations fj,i by solving (4.9)–(4.13) with different parameter values. For the computations, we have applied the code ode15s from the Matlab ODE suite [29]. For each set of parameters we have calculated the functions n1 (t ) from the first quadrature in (4.8) and (3.1) with i = 1. We have applied the composite trapezoidal rule to (4.8) and (3.1). We have compared the model outputs n1 (t ) to the experimental data and by minimizing the sums of squared errors we have chosen these parameter values for which the function n1 (t ) is the closest to the experimental data. From this procedure, we have found parameter estimations, which are listed in Table 4. Results of numerical experiments with the parameter values from Table 4 are presented in Fig. 1. Fig. 1 shows that the model (4.9)–(4.13) expresses the essential features of the growth patterns of 66c14 and 4T1.2 (1) primary tumors in vivo. From the estimated parameter values listed in Table 4, we observe that p16 , which characterizes the generation of new tumor cells, is larger for the more aggressive 4T1.2 tumor than for the less metastatic 66c14 tumor (2) by 10−2 . Furthermore, the parameter p25 , which is used to describe the generation of new Th cells due to the interactions (5)
between Th cells and AgAPCs, is larger for 4T1.2 than for 66c14 also by 10−2 and the parameter b14 , which characterizes the appearance of new antigen-loaded APCs due to the interactions between APCs and CCs, is larger for 4T1.2 than for 66c14
C.L. Jorcyk et al. / Nonlinear Analysis: Real World Applications 13 (2012) 78–84
83
Fig. 1. Experimental in vivo data (grid-points) versus predicted data (solid). The in vivo data are indicated by ◃ and ◦ for the 66c14 and 4T1.2 cells, respectively.
by 1.6 · 10−1 . Additionally, the parameter d56 , which corresponds with the natural death of AgAPCs, is larger for 4T1.2 than for 66c14 by 1.1 · 10−1 . Moreover, the parameter d26 , which characterizes the natural death of Th cells, is also larger for the more aggressive 4T1.2 tumor than for 66c14 by 1.1 · 10−1 and the parameter d31 , which is used to model the destruction of CTLs due to their interactions with CCs, is larger for 4T1.2 than for 66c14 by 1.7 · 10−1 . The parameters d36 and d46 , which characterize the deaths of CTLs and APCs, respectively, are larger for 4T1.2 than for 66c14 by 1.1 · 10−1 . We also observe that since the parameter values of a2 are not significant for both tumors, the numerical experiments suggest that the source function S2 (t ), which describes the possible inlet of Th cells, does not depend on time. Additionally, S2 (t ) is smaller for the more aggressive 4T1.2 tumor than for the milder 66c14 tumor by 1.8 · 10−1 . From Fig. 1, we observe that the curve evolution for the more aggressive 4T1.2 tumor is concave upward over the whole time interval of the laboratory experiment. On the other hand, the curve for the less metastatic 66c14 tumor evolves differently. It is concave upward but only in the first half of the time interval of the laboratory measurements and then it is concave downward in the second half, turning from concave upward to downward by about the 21st or 22nd day. The curve evolution corresponding to the less aggressive 66c14 tumor shows that, in the case of 66c14 cells, the immune system protects the organism better against cancer, which is in agreement with the fact that 66c14 cells are less metastatic than the aggressive 4T1.2 cells. 5. Concluding remarks and future directions We have analyzed both animal and mathematical models for the investigation of mammary carcinoma progression. The development and characterization of a progressive series of two adenocarcinoma cell lines from a spontaneous tumor of the mammary gland in a female BALB/c mouse have been described by means of integro-differential equations of Boltzmann type. The numerical approximations to the solutions of the proposed mathematical model have shown good agreement with the laboratory data of the tumor growth rates. As the development and utilization of experimental and computational methods can improve the understanding of the carcinogenic processes, our future work will address expanded mathematical models and new mammary adenocarcinoma cell lines derived from mouse models. References [1] A. Jemal, R. Siegel, E. Ward, Y. Hao, J. Xu, M.J. Thun, Cancer statistics, Cancer J. Clin. 59 (2009) 225–249. [2] R.E. Coleman, R.D. Rubens, The clinical course of bone metastases from breast cancer, Br. J. Cancer 55 (1987) 61–66. [3] E.F. Solomayer, I.J. Diel, G.C. Meyberg, C. Gollan, G. Bastert, Metastatic breast cancer: clinical course, prognosis and therapy related to the first site of metastasis, Breast Cancer Res. Treat. 59 (2000) 271–278. [4] I.S. Kim, S.H. Baek, Mouse models for breast cancer metastasis, Biochem. Biophys. Res. Commun. 394 (2010) 443–447. [5] A. Fantozzi, G. Christofori, Mouse models of breast cancer metastasis, Breast Cancer Res. 8 (2006) 212. [6] M. Lelekakis, J.M. Moseley, T.J. Martin, et al., A novel orthotopic model of breast cancer metastasis to bone, Clin. Exp. Metastasis 17 (1999) 163–170. [7] E.K. Sloan, K.L. Stanley, R.L. Anderson, Caveolin-1 inhibits breast cancer growth and metastasis, Oncogene 23 (2004) 7893–7897. [8] A.F. Hurlstone, G. Reid, J.R. Reeves, et al., Analysis of the CAVEOLIN-1 gene at human chromosome 7q31.1 in primary tumours and tumour-derived cell lines, Oncogene 18 (1999) 1881–1890. [9] N. Bellomo, Modelling Complex Living Systems, Birkhäuser, Boston, 2008. [10] A. Bellouquid, M. Delitala, Modelling Complex Biological Systems—A Kinetic Theory Approach, Birkhäuser, Boston, 2006. [11] N. Bellomo, N.K. Li, P.K. Maini, On the foundations of cancer modelling: selected topics, speculations, and perspectives, Math. Models Methods Appl. Sci. 18 (2008) 593–646. [12] N. Bellomo, B. Carbonaro, Toward a mathematical theory of living systems focusing on developmental biology and evolution: a review and perspectives, Phys. Life Rev. 8 (2011) 1–18. [13] N. Bellomo, G. Forni, Complex multicellular systems and immune competition: new paradigms looking for a mathematical theory, Curr. Top. Dev. Biol. 81 (2008) 485–502.
84
C.L. Jorcyk et al. / Nonlinear Analysis: Real World Applications 13 (2012) 78–84
[14] S. De Lillo, M.C. Salvatori, N. Bellomo, Mathematical tools of the kinetic theory of active particles with some reasoning on the modelling progression and heterogeneity, Math. Comput. Modelling 45 (2007) 564–578. [15] M. Kolev, A mathematical model of cellular immune response to leukemia, Math. Comput. Modelling 41 (2005) 1071–1081. [16] N. Bellomo, M. Delitala, From the mathematical kinetic, and stochastic game theory to modelling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev. 5 (2008) 183–206. [17] N. Bellomo, C. Bianca, M. Delitala, Complexity analysis and mathematical tools towards the modelling of living systems, Phys. Life Rev. 6 (2009) 144–175. [18] L. Arlotti, N. Bellomo, K. Latrach, From the Jäger and segel model to kinetic population dynamics: nonlinear evolution problems and applications, Math. Comput. Modelling 30 (1999) 15–40. [19] M.L. Bertotti, M. Delitala, Conservation laws and asymptotic behavior of a model of social dynamics, Nonlinear Anal. RWA 9 (2008) 183–196. [20] M. Lo Schiavo, The modelling of political dynamics by generalized kinetic (Boltzmann) models, Math. Comput. Modelling 37 (2003) 261–281. [21] N. Bellomo, B. Carbonaro, On the complexity of multiple interactions with additional reasoning about Kate, Jules and Jim, Math. Comput. Modelling 47 (2008) 168–177. [22] M. Delitala, A. Tosin, Mathematical modelling of vehicular traffic: a discrete kinetic theory approach, Math. Models Methods Appl. Sci. 17 (2007) 901–932. [23] Z. Jackiewicz, C.L. Jorcyk, M. Kolev, B. Zubik-Kowal, Correlation between animal and mathematical models for prostate cancer progression, Comput. Math. Methods Med. 10 (2009) 241–252. [24] K. Drucis, M. Kolev, W. Majda, B. Zubik-Kowal, Nonlinear modeling with mammographic evidence of carcinoma, Nonlinear Anal. RWA 11 (2010) 4326–4334. [25] A. Calvo, N. Xiao, J. Kang, C.J.M. Best, I. Leiva, M.R. Emmert-Buck, C. Jorcyk, J.E. Green, Alterations in gene expression profiles during prostate cancer progression: functional correletions to tumorigenicity and down-regulation of selenoprotein-P in mouse and human tumors, Cancer Res. 62 (2002) 5325–5335. [26] L. Arlotti, A. Gamba, M. Lachowicz, A kinetic model of tumour/immune system cellular interactions, J. Theor. Med. 4 (2002) 39–50. [27] L. Arlotti, N. Bellomo, E. De Angelis, Generalized kinetic (Boltzmann) models: mathematical structures and applications, Math. Models Methods Appl. Sci. 12 (2002) 567–591. [28] L. Arlotti, N. Bellomo, E. De Angelis, M. Lachowicz, Generalized Kinetic Models in Applied Sciences, World Sci., New Jersey, 2003. [29] L. Shampine, M.W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput. 18 (1997) 1–22.
