## 1 Introduction  
Ever since the discovery of androgen dependency of prostate cells, androgen deprivation therapy (ADT) has played a vital role in the treatment of metastatic and locally advanced prostate cancer [1; 2; 3]. However, controversy remains regarding its best application. Although this treatment will regress tumors in over 90% of patients [4], after prolonged androgen depletion, patients will eventually develop castration-resistant prostate cancer (CRPC) [5]. The development of CRPC can take from a few months to more than ten years [3; 6], after which there is a very limited number of effective treatments and patients suffer high mortality [7]. ADT is expensive and its side effects include sexual dysfunction, hot flashes, and fatigue [8]. Based on some preclinical studies, intermittent androgen suppression (IAS) is suggested as a sensible alternative to ADT [9]. During off-treatment periods, patients enjoy a \"vacation\" from the severe side effects of ADT [8], and studies have suggested that IAS may not negatively affect the time to resistance progression or survival in comparison to ADT [10]. Consequently, IAS is selected by some patients to improve the quality of life and also hopefully to delay the progression to CRPC [4]. Many mathematical models have studied the dynamics of prostate cancer during ADT or IAS [11; 12; 13; 14; 15; 16; 17; 18]. A detailed review of some of these models are presented in the recent book of [PERSON] et al. [19]. [PERSON] et al. are pioneers of mathematically modeling and analyzing the dynamics of IAS [12]. They formulated a system of ordinary differential equations to study the mechanics of ADT and IAS. They considered castrate-resistant (CR) and castrate-sensitive (CS) cell populations as well as androgen levels. Their model included mutations from CS to CR cells, and their focus was on comparing continuous and intermittent therapy and the development of resistance. [PERSON] et al. [14] introduced a piece-wise linear model of three cancer cell populations. Their model included CS cells, CR cells that may mutate into CS cells, and CR cells that will not mutate. Several investigators using [PERSON] et al. 's model [14] have studied estimation of parameters [20; 21], optimal switching times and control in IAS [20; 22; 23], and forecasting CRPC progression [24; 25]. Built on the works of [PERSON] et al. [12] and [PERSON] [26], [PERSON], [PERSON], and [PERSON] (PKN) [13] developed a novel mathematical model to study the dynamics of IAS by using the cell quota model [27] from mathematical ecology, which relates growth rate to an intracellular nutrient, to modeling the growth of both the CS and CR cell populations. The cell quota in [13] is defined as the intracellular androgen concentrations for each cell population. This model is carefully fitted with clinical prostate-specific antigen (PSA) data, where androgen data was used to model the cell quota and other growth parameters. [PERSON] et al. [28] compared the models of [PERSON] et al. [14] and [PERSON] [13] regarding their accuracy of fitting clinical data and predicting future PSA levels. They concluded that while a biologically-based model is important to reveal the underlying processes and my present more robust and better predictions, a simpler model such as that of [PERSON] et al. might also be practical for fitting clinical data and predicting future PSA outcomes of individual patients. In this paper, we present a simplified model to the final model in PKN [13]. Several key terms in our model will be mechanistically formulated. This model is concise and amenable to systematical mathematical analysis of its dynamics. For simplicity, we shall use serum androgen concentration to approximate intracellular androgen. This is reasonable since androgen passively and quickly diffuses through the prostate membrane via concentration gradient [29]. This approach is practical for a typical clinical setting, where the data collected can be applied directly to the model.