1. Sumana Ghosh and Sandip Banerjee Department of Mathematics
Mathematical Modeling of Cancer Immune system with Humoral and Cell Mediated Immune Responses
Sumana Ghosh and Sandip Banerjee
Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee , Uttarakhand, India.
BIOMAT 2017

3. Humoral vs. Cell Mediated Response

4. B cells are a type of white blood cell that produce antibodies.
Plasma cells are white blood cells which produce large volumes of antibodies.
Antibodies are gamma globulin proteins that are found in blood or other bodily fluids of vertebrates and are used by the immune system to identify and neutralize foreign objects.
An antigen is a substance/molecule that, when introduced into the body, triggers the production of an antibody by the immune system, which will then kill or neutralize the antigen that is recognized as a foreign and potentially harmful invader.

5. Tumor Modeling by various Authors
Brian Sleeman
V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor
A.S. Perelson, L. Preziosi,
J. Adam and N. Bellomo
D. Kirschner and J.C. Panetta
M. Bodnar and U. Fory’s
M. Kolev
M. Chaplain and Alexander Anderson
Denise Kirschner and Alexei Tsygvintsev
L.G. de Pillis and A.Radunskaya
Seema Nanda, Helen Moore and Suzanne Lenhart
Ram Rup Sarkar and Sandip Banerjee

6. Papers on Humoral mediated Immune System
R.O. Dillman and J.A. Koziol, A Mathematical Model of Monoclonal Antibody Therapy in Leukemia, Mathl. Modeling, 9 (1)(1987),
M. Kolev, Mathematical Modeling of the Competition between Tumors and the Immune Systems Considering the Role of Antibodies, Journal of Mathematical and Computer Modeling, 37(2003),

8. The Journal of Clinical Investigation , Volume 119 Number 8 August 2009
How cancer cells are killed?
When the antibodies bind with cancerous cell, it causes lysosomes (small acid containing sacs) inside the cell to swell and burst rapidly releasing their toxic contents with fatal results for cancerous cells, which is non-apoptotic in nature.
Also, the antibodies are internalized by the cancerous cells and the conjugated toxin kills them.

10. The model
Let B, P, A,T are the no of large B-cells, Plasma cells, Antibody and Tumor cells respectively.
Initial condition : B(0) = B0>0, P(0) = P0>0, A(0) = A0>0 and T(0) = T0>0 where, 0<u<1.

11. The model
Let B, P, A,T are the no of large B-cells, Plasma cells, Antibody and Tumor cells respectively.
Initial condition : B(0) = B0 > 0, P(0) = P0 > 0, A(0) = A0 >0 and T(0) = T0 > 0 where, 0 < u < 1.

12. Basic properties of the model
Positivity and Boundedness of solutions
Theorem 1: Let the initial conditions of the system be positive, then the solutions ( B(t), P(t), A(t), T(t) ) of the system are non-negative for all t > 0.
Theorem 2: The solutions of the system with positive initial conditions are bounded where,
and

13. Equilibrium points:
Trivial equilibrium point
Boundary equilibrium point
Cancerous free equilibrium point
Positive interior equilibrium point
where,

14. Local Stability Analysis:
Trivial equilibrium point is always unstable.
Boundary equilibrium point is unstable if
but in that case and does not exist.
Cancer free equilibrium point is Locally Asymptotically Stable (LAS) if
Positive interior equilibrium point is LAS if it exist.

15. Global Stability Analysis
Theorem 3: The LAS cancer-free equilibrium point is globally asymptotically stable, provided
Theorem 2: The LAS positive interior E* equilibrium point is globally asymptotically stable, if
Note:
(i) For , the cancer free equilibrium point is globally asymptotically stable and the endemic equilibrium point does not exists.
(ii) For , the endemic equilibrium point is globally asymptotically stable and cancer free equilibrium point is unstable.

16. Bifurcation Analysis
Using Sotomayor theorem [30] , it has been shown that the system experiences a transcritical bifurcation at the cancer-free equilibrium point as the parameter passes through the bifurcation parameter value (unstable to stable)

17. Estimation of Parameters
The growth rate a1 of the large B lymphocyte is estimated to be between 0.02 and 0.2 hr−1 .
The life time of plasma cells ranges from few days to few weeks. Thus the natural death rate μ1 of the plasma cells is approximated to vary between to 0.02 hr−1.
K1 = carrying capacity of B cells = 106 cells, u = 0.1.
b1 = rate of conversion of B cells to Plasma cells = 0.01 hr−1
Optimal Strategies in Immunology, B-Cell Differentiation and Proliferation, Perelson, 1976, Journal of Mathematical Biology.

18. Parameter Estimation
To generate synthetic data, we first generate a figure by taking human data, patient 9, as given in Table 2 of [41]. This figure (see FigA) demonstrates the behaviour of the model by de pillis et al. [41], who has used parameters taken from experimental results of patient from Rosenberg's study [110] on metastatic melanoma. In the generated figure, they have investigated a tumor comprising of 106 cells, a size that, in silico, the innate immune system cannot control on its own. The figures shows the effect of immunotherapy alone against the tumor, namely, short doses of IL-2.

19. Best Fit estimate for the model parameters
Parameter Estimation
The data for cancerous cell decay are extracted (say, for 20 time points) by using Data Thief ( and some noise is added to the data, we denote this as observed values, Tobs (say). At the starting of the estimation process, initially we choose (within meaningful biological range) the values of the parameters and arbitrarily, which we want to estimate.
Next, we solve the model equations numerically with these initial values of the parameters and obtain the solution of the model at those time points, where the observed values have been obtained, which is denoted as calculated values, Tcal (say). Now, we use the least squares method to minimize the sum of the residuals, to obtain the estimated values.
Best Fit estimate for the model parameters

20. Parameter values

21. Numerical Results β1> β*=2.967*10-8, (i) A,B is for β1=3.5*10-8,
(i) For , the cancer free equilibrium point is globally asymptotically stable and the endemic equilibrium point does not exists.
(ii) For , the endemic equilibrium point is globally asymptotically stable and cancer free equilibrium point is unstable (saddle).
β1> β*=2.967*10-8, (i) A,B is for β1=3.5*10-8,
β1<β*=2.967* (ii) C is for β1=1.0135*10-9

22. Fig (A, B, C) for, β1<β*=2.967×10-8, figure D is for, β1> β*

23. Conclusions
The dynamic clearly indicate that a significant role is played by β1, the effectiveness of the antibodies to kill the cancerous cells directly.
The stability of the high number of cancerous cells equilibrium point implies that reducing the cancer burden through any effective treatment is not sufficient enough to kill all the cancerous cells. Once the treatment stops, the system, even with an untraceable sign of cancer, will return to high number of cancerous cells state.
However, alteration of system parameters through monoclonal antibody therapy of cancer, may have the ability to change the stability nature of the cancer free equilibrium point and allowing a new treatment protocol to eradicate cancerous cells. Cetuximab (Erbitux), a monoclonal antibody approved to treat colon cancer and head and neck cancers.

24. THANK YOU