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Global dynamics of a model of joint hormone treatment with dendritic cell vaccine for prostate cancer

  • * The corresponding author

    * The corresponding author 
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  • Advanced prostate cancer is often treated by androgen deprivation therapy, which is initially effective but gives rise to fatal treatment-resistant cancer. Intermittent androgen deprivation therapy improves the quality of life of patients and may delay resistance towards treatment. Immunotherapy alters the bodies immune system to help fight cancer and has proven effective in certain types of cancer. We propose a model incorporating androgen deprivation therapy (intermittent and continual) in conjunction with dendritic cell vaccine immunotherapy. Simulations are run to determine the sensitivity of cancer growth to dendritic cell vaccine therapy administration schedule. We consider the limiting case where dendritic cells are administered continuously and perform analysis on the full model and the limiting cases of the model to determine necessary conditions for global stability of cancer eradication.

    Mathematics Subject Classification: Primary:92C50, 34D20;Secondary:34D23.


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  • Figure 1.  PSA serum concentration, androgen dependent (AD), and androgen independent (AI) cell concentrations for various dendritic cell vaccine injection times with an $e_1=e_2$ of 0.75. More frequent injections result delay of the rise of fatal androgen independent cancer

    Figure 2.  Limit cycle solutions for androgen dependent $X_1$, and androgen independent $X_2$ cancer cells. The minimal value of $e_1$ required to produce limit cycle behavior is noted above each solution. As vaccine timing decreases, minimal $e_1$ necessary to have stable disease state decreases

    Figure 3.  PSA serum level, AD cell density, and AI cell density for continual dendritic cell vaccinations, with an injection rate 0.04 billion cells for various values of $e_1$. It is apparent that in this continuous case, a wider range of $e_1$ is able to suppress the growth of cancer and elongate the cycles of IAD

    Figure 4.  Birfurcation digram for parameter $e_1$, a measure of cytotoxicity of T-cells. Maximum PSA level in black, minimum PSA level in green. Carrying capacity is stable only for $e_1=0$. Immediately after, we have a Hopf bifurcaton and limit cycles, until we reach $e_1\approx 0.66$, after which the disease-free steady state is stable

    Figure 5.  Comparisons of the full system and quasi-steady state system for various values of $e_1$: 0, 0.15, 0.25, and 0.65, assuming androgen deprivation therapy is constantly on. These values of $e_1$ show many differing dynamics. The quasi-steady state system closely approximates the full system in every case, but shows slight differences in the case of $e_1=0.15$

    Table 1.  Values of parameters (P), explanations, and cited sources of every parameter used in this mathematical model

    P Biological Meaning Value Source
    $r_1$ AD cell proliferation rate 0.025/day [ 1 ]
    $d_1$ AD cell death rate 0.064/day [ 1 ]
    $K$ cancer cell carrying capacity 11 billion
    $k_4$ AI to AD mutation half-saturation 1.7
    $r_2$ AI net cell growth rate 0.006/day [ 1 ]
    $m_1$ maximum mutation rate from AD to AI 0.00005/day [ 19 ]
    $m_2$ maximum mutation rate from AI to AD 0.00015/day [ 37 ]
    $a_0$ base level androgen concentration 30 ng/ml [ 19 ]
    $\gamma$ androgen clearance and production rate 0.08/day [ 19 ]
    $\omega$ cytokine clearance rate 10/day [ 38 ]
    $\mu$ T cell death rate 0.03//day [ 24 ]
    $c$ dendritic cell death rate 0.14/day [ 31 ]
    $e_1$ max rate T cells kill AD cancer cells 0-1/day [ 24 ]
    $g_1$ AD cancer cell saturation level for T cell kill rate 10 x $10^9$ cells [ 24 ]
    $e_2$ max rate T cells kill AI cancer cells 0-1/day [ 24 ]
    $g_2$ AI cancer cell saturation levelfor T cell kill rate 10 x $10^9$ cells [ 24 ]
    $e$ T cell max activation rate 20 x $10^6$ cells/day [ 24 ]
    $g$ DC saturation level for T cell activation 400 x $10^6$ cells [ 40 ]
    $e_3$ max clonal expansion rate 0.1245/day [ 24 ]
    $g_3$ IL-2 saturation level for T cell clonal expansion 1000 ng/ml [ 24 ]
    $e_4$ max rate T cells produce IL-2 5 x $10^{-6}$ ng/ml/cell/day [ 24 ]
    $g_4$ cancer cell saturation level for T cell stimulation 10 x $10^9$ cells [ 24 ]
    $D_1$ DC vaccine dosage 300 x $10^6$ cells [ 40 ]
    $c_1$ AD cell PSA level correlation 1 x $10^{-9}$ ng/ml/cell [ 19 ]
    $c_2$ AI cell PSA level correlation 1 x $10^{-9} $ ng/ml/cell [ 19 ]
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