## 1 Introduction
### Prostate Cancer: Physiology and Treatments  
In this section, we aim to give a summary of the biological and clinical details of prostate cancer and its treatments. The prostate is a walnut-sized gland that is part of the endocrine system and the male reproductive system. Its main job is to secrete seminal fluids that are used to nourish and transport sperm. The function, proliferation, and atrophy of healthy and cancerous prostate cells depend highly on the circulating level of androgen, particularly testosterone [7]. Androgens are steroid hormones that regulate the development and maintenance of male characteristics by binding to androgen receptors  
Figure 1: Statistics of prostate cancer in the US—data obtained from [4]. (AR), which is a cytoplasmic receptor that is activated by binding to androgens, which induce the AR to migrate to the nucleus and bind to the Androgen Response Elements (ARE) within the DNA [8]. The testicular-hypothalamic-pituitary axis, also known as the hypothalamus-pituitary-gonadal axis--where gonads indicate the reproductive system--produces androgen in a negative feedback loop. This means that the higher the androgen level in the body, the lower its production rate becomes [9]. Approximately 90%-95% of the body's androgen is produced by the testes and the remainder by the adrenal gland [10]. Once androgens, or testosterone, enter the prostate, about 90% is converted to the more potent form 5\(\alpha\)-Dihydrotestosterone (DHT) by the 5\(\alpha\)-reductase enzyme [11]. DHT has been observed to be a better agonist to the androgen receptors, with about 2-3 and 15-30 times higher affinity than testosterone and adrenal androgens, respectively [12; 13]. Additionally, the unbinding rate of DHT and AR is roughly five times slower than that of testosterone and AR [14]. Once androgens bind to the AR, they are transported to the nucleus to bind to the ARE. This process results in the transcription of androgen-regulated genes, which signals the proliferation and inhibition of apoptosis in both healthy and cancerous prostate cells. The proliferation of prostate cells produces a prostate-specific antigen (PSA). Most of this byproduct of prostate cell growth is normally contained within the prostate wall; however, in the presence of prostate cancer cells, the cell wall is disrupted, which allows for an increased leakage of PSA into the bloodstream. For this reason, a simple blood test can be used to measure the level of PSA, which acts as a biomarker for prostate cancer and is often used as a proxy for tumor volume in mathematical models [15]. A simplified schematic of prostate cancer dynamics is described in Figure 2.  
Figure 2: A simplified schematic of the dynamics of androgen and prostate-specific antigen in prostate cancer-adapted from [16]. (1) Serum androgen flows into the tissue of prostatic epithelial cells. (2) It is then converted to Dihydrotestosterone (DHT) by 5-\(alpha\) reductase. (3) Both testosterone and DHT bind to androgen receptors. (4) This further activates the proliferative and survival signals. (5) Cancer activity results in the production of prostate-specific antigen (PSA), which is leaked out of the cells (6). Note that the gray compartments indicate measurable concentrations that are relatively easier to obtain. The diagnosis of prostate cancer usually starts with a blood test to measure the PSA level. To verify the presence of a prostate tumor, a digital rectal exam, a transrectal ultrasound, and/or a prostate biopsy may be carried out. If cancer is confirmed, clinicians also assign the patient a Gleason score, which measures the resemblance of biopsy cancer tissues to healthy tissues. The Gleason score is often used as part of the clinical process to determine which treatment would be best for the patient [17], but it is not often utilized in mathematical models. For localized prostate cancer, the main options are: watchful waiting and active surveillance, surgical removal of the prostate, and radiation.