## 1 Introduction  
Prostate cancer (PCa) develops within the prostate gland of men, and is the second most common cancer in men, and the second leading cause of cancer deaths in men in the US [1]. Like the cells it develops from, PCa is dependent on the male sex hormone testosterone for growth and survival. Due to this dependency, advanced PCa is treated with androgen deprivation therapy (ADT), a form of chemical castration wherein the bioavailability of testosterone is inhibited [2; 3]. PCa patients respond well to ADT initially, with response rates as high as 80 to 90 percent. Unfortunately, a majority of these patients progress to a castration-resistant state, necessitating alternative interventions, such as chemotherapy and immunotherapy [4].  
In particular, one such treatment is sipuleucel-T (Provenge), an autologous cellular immunotherapy approved by the FDA in 2010 for treating patients with metastatic castration-resistant PCa (mCRPC) [5]. Treatment with Provenge begins with harvesting the patients' own peripheral blood mononuclear cells, which are then cultured in the presence of prostatic acid phosphatase and granulocyte-macrophage colony-stimulating factor (GM-CSF). This activates the mononuclear cells transforming them into antigen presenting cells, which are then re-injected into the patient, spread over three doses typically given every two weeks [6]. The injected antigen presenting cells train the patients' own immune system to recognize and kill tumor cells. However, Provenge has exhibited only limited success clinically, and has therefore not been widely adopted [7].  
Given that ADT is known to have a profound effect on the immune system [8], researchers are now investigating the therapeutic potential of combining ADT and immunotherapy treatments such as Provenge [9]. However, when two or more treatments are combined, arriving at the optimal dosing and scheduling regimen can be difficult. Indeed, even for Provenge administered as a monotherapy, optimal inter-dose intervals have not been established clinically [10]. Arriving at desired optimal protocols purely experimentally would be prohibitively expensive in terms of both time and resources. Furthermore, it is impossible to test all possible combinations.  
A powerful and practical way to predict the therapeutic potential of combination therapy in cancer, is to use mechanistic modeling. Multiscale mathematical models have already been used to optimize PCa treatment [11; 12]. We have recently published a mechanistic model of the immune response to PCa elicited by Provenge [10], providing explanations for the limited success that this vaccine has exhibited clinically, and why an optimal vaccination schedule has not been established. Here, we adapt this model to better understand how various cells and cytokines of the immune system respond to PCa treatment with Provenge, and use this knowledge to predict optimal dosing and scheduling regimens when Provenge is given in combination with ADT. Our models of PCa and immune system interactions build on the tumor-immune interaction models of [PERSON] et al. [13] and [PERSON] et al. [14], and on [PERSON]'s model of PCa treatment with ADT and immunotherapy [15].  
The remainder of the paper is organized as follows. In the Materials and Methods section we present our model of the immune and tumor response after treatment with ADT or Provenge. In the Results section we explore the potential of combining ADT and Provenge. We conclude with a discussion on the significance of our findings.