Team:NEFU China/Model
Tumor Prediction Model:
Currently, we tested our case conceptually using in vitro models. In the next phase, we will evaluate our case in mice. Prior to the in vivo study, we wanted to predict the outcomes using mathematical modeling that is based on a radial tumor growth. Beyond that, the model elucidates the effects of vascularization and immune recruitment on tumor growth [1].
A mathematic model that can evaluate the tumor size before and after the treatment was developed in this study. Taken the radial tumor growth and immune recruitment dynamics together, a prediction model of tumor growth is established as follows:
The proposed model is based on the following major assumptions:
(1) The temporal evolution of the average tumor radius is considered, but invasive and diffusive tumor mechanisms are not taken into account.
(2) The death rate pA of tumor cells reflects the lump effect of apoptotic and necrotic processes.
(3) Innate immunity or basal immune surveillance is represented as a minimum presence of active effector cells at any time, even in the absence of cancer cells.
(4) Effector cells are recruited depending on the density of tumor cells according to the Michaelis–Menten dynamics.
(5) The efficacy of immune killing depends on the ability of effector cells to penetrate the tumor bulk via functional tumor-associated vasculature[2][3]. With improved vascularization, the effectors destroy tumor cells not only on the surface of the tumor, but also further inside.
(6) Effector cells die at a constant rate and are inactivated in accordance with their antitumor activity.
Equations:
Based on the assumption of radial symmetry, the model of the system variables is the average tumor radius R(t) and the effector cell density E(t) in tumor vicinity is given by:
Where the time coordinate t has been omitted for the sake of notation simplicity.
L, α, pM, pA, c, λ, K, D1, D0 and δ are positive constants.
L is an intrinsic length scale to represent nutrient content on average, namely consumption, diffusion, and supply.
α is a dimensionless parameter represents the extent of how the tumor-associated vasculature acts on the tumor; therefore, the value of α varies between 0 and 1.
pA is assumed to be a constant for the cell mortality rate, and the rate of cell division can be characterized by pM
c represents the killing rate of tumor cells by the effectors, λ is the immune recruitment rate, and K is the tumor volume at which the recruitment rate is half-maximal.
D1 and D0 are the inactivation and death rates of effector cells, respectively.
δ is the background rate of immune effector recruitment.
Further details about derivation, parametrization and theoretical analysis of the tumor-effector cell recruitment model can be found in Reference #1.
Following the conclusions of paragraphs above, the constant in the equation can be determined as:
Model parameters. Parameter values unavailable in the literature were estimated from in vivo experimental data of tumor growth in immunocompromised Rag1−/− and wild-type (WT) BALB/c mice.
Based on our literature search[11], the constant of the procedure during the treatment can be determined as:
Based on a previous study, xenograft tumors in mice will become palpable when they reach a volume of about 100 mm3 [12], which will be the time to start treatment. The predicted changes of tumor size during the treatment is shown in Fig. 1.
Fig. 1. Predicted tumor growth during the treatment of the engineered bacteria in our case.
References
[1] Hatzikirou, H., et al., Cancer therapeutic potential of combinatorial immuno- and vasomodulatory interventions. Journal of the Royal Society Interface, 2015. 12(112).: pii: 20150439.
[2] Huang, Y.H., et al., Vascular normalizing doses of antiangiogenic treatment reprogram the immunosuppressive tumor microenvironment and enhance immunotherapy. Proceedings of the National Academy of Sciences of the United States of America, 2012. 109(43): p. 17561-17566.
[3] Huang, Y.H., et al., Vascular normalization as an emerging strategy to enhance cancer immunotherapy. Cancer Research, 2013. 73(8).
[4] d'Onofrio, A., A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences. Physica D-Nonlinear Phenomena, 2005. 208(3-4): p. 220-235.
[5] Kuznetsov, V.A., et al., Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bulletin of Mathematical Biology, 1994. 56(8): p. 295-321.
[6] de Pillis, L.G., A.E. Radunskaya, and C.L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Research, 2005. 65(17): p. 7950-7958.
[7] Su, B., et al., Mathematical modelling of immune response in tissues. Computational and Mathematical Methods in Medicine, 2009. 10(1): p. 9-38.
[8] Onofrio, A., Ledzewicz, U., Scha¨ttler, H., On the dynamics of tumor– immune system interactions and combined chemo-and immunotherapy. In new challenges for cancer systems biomedicine, 2012. 1(1): p. 249-266.
[9] Alessandri, K., et al., Cellular capsules as a tool for multicellular spheroid production and for investigating the mechanics of tumor progression in vitro. Proceedings of the National Academy of Sciences of the United States of America, 2013. 110(37): p. 14843-14848.
[10] Delarue, M., et al., Mechanical Control of Cell flow in Multicellular Spheroids. Physical Review Letters, 2013. 110(13).
[11] Hatzikirou, H., et al., Therapeutic Potential of Bacteria against Solid Tumors. Cancer Research, 2017. 77(7): p. 1553-1563.
[12] Choudhury, K.R., I. Kasman, and G.D. Plowman, Analysis of multi-arm tumor growth trials in xenograft animals using phase change adaptive piecewise quadratic models. Statistics in Medicine, 2010. 29(23): p. 2399-2409.
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