br Introduction br The prostate cancer is one
The prostate cancer is one of the most common tumors (for men) and one of the hottest research topics in oncology . The prostate Lycopene secrete proteins into the bloodstream, one of which is known as prostate specific antigen (PSA), and a high level of the PSA in serum might indicate the presence of the cancer (malignant tumors) [32,36]. The primary androgen (main male sex hormone) is testosterone which is mainly produced by testes but is also secreted by the adrenal gland. Ninety percent of free testosterone that enters a prostate cell is converted to dihydrotestosterone (DHT) by enzyme 5α-reductase (SRD5A2) . Both testosterone and DHT induce proliferation and inhibit apoptosis of prostate tumor cells. This is why androgen deprivation therapy (ADT) has been the main treatment for prostate cancer to reduce the level of androgen in the serum, decrease the rate of proliferation, and increase the mortality rate of tumor cells . ADT can be accomplished by surgical means (removal of testicles) or by chemical castration which involves the use of luteinizing hormone-releasing hormone agonist (LHRH) to reduce serum testosterone level by ninety five percent, and DHT by an average of sixty percent , combining LHRH agonist with antiandrogens option to achieve maximal androgen blockade (MAB) [6,24]. Typically, ADT succeeds at the beginning of treatment because the dominant tumor cells are those that depend on androgen for growth (androgen-dependent cells) but in most cases ADT cannot prevent a relapse. This is because after several years androgen-dependent (AD) tumor cells become resistant to treatment as these cells mutate into androgen-independent (AI) cells which can proliferate even in an androgen depleted environment [5,11,38]. In some cases, intermittent androgen suppression (IAS) (the treatment is administered intermittently) has been used instead of the continuous androgen suppression (CAS) in the hopes of delaying a relapse of prostate cancer and improving the quality of life [7,25]. IAS therapy schedule depends on
Supported in part by the National Science Fund of China (11571284). ∗ Corresponding author.
E-mail address: [email protected] (W. Wang).
the level of PSA. The treatment is activated when the PSA level is above a certain value L1 and it turns off (patient takes a vacation period) when the PSA is below a threshold value L2 < L1. Intermittent treatment could be effective if the off-treatment period begins before the androgen-dependent cells become resistant to the therapy. Therefore, tumor cells may take longer time to adapt and mutate because of the periodic changes in the environment. These changes could lead to prolong the survival of AD cells (because in off-treatment period AD cells still can proliferate) and delay the dominance of resistance cells (relapse). Nevertheless, there is no definite conclusion that IAS therapy is superior than CAS therapy in all cases in the medical community .
Mathematical modeling and analysis have greatly contributed to the understanding of mechanisms of tumor cell pro-gression under CAS, IAS and immunotherapy response when it is combined with androgen deprivation therapy (see, e.g, [1,8,9,13–15,17,20,21,35,37,41–44,47]). Ideta et al.  proposed a mathematical model to compare CAS and IAS, and discuss the benefit of IAS. Rutter and Kuang  formulated a population model for androgen deprivation therapy with immunother-apy. In their model, the growth rates of AD cells and AI cells are described by logistic functions and AI cells might convert back to AD cells in an androgen-rich environment. A similar model was proposed and studied in Portz and Kuang .
The models in the references [17,35,37] are deterministic. But tumor growth is sensitive to certain fluctuations such as temperature, radiation and chemical products, oxygen supply and nutrients [27–29,41,42], and these environmental pertur-bations are inevitable. Hence, the model of Ideta et al.  was extended to a stochastic model by Tanaka et al.  where stochastic white noises are introduced to the ODE model to represent the intrinsic fluctuations of tumor dynamics. Based on the IAS protocol (on and off-treatment periods schedule) which was applied in clinical trials by Bruchovsky et al. , the authors used numerical simulations to show that their stochastic model is able to describe some features and characteristics observed in clinical trials. Moreover, the noises could be responsible for the variability of responses to the therapy from one patient to another.
Competition between AD cells and AI cells may alter the fate of tumor cells. Their competitive effects have been consid-ered in [35,37,39] with identical competition coe cient. However, the recent oncology researches [16,48] have shown that AD cells and AI cells have distinct shapes and functions. More specifically, AD cells appear as a triangle or spindle form, whereas AI cells look rounder and flatter. In addition, AD cells and AI cells also have different gene levels so that AD cells and AI cells have different functions such as cell proliferation, apoptosis, adhesion and cell motility, which plays an impor-tant role in cell migration and cancer metastasis . Therefore, AD cells and AI cells may compete for nutrient and oxygen with different intensities.