Mathematical model for the androgenic regulation of the prostate in intact and castrated adult male rats

Modelling the distinct roles of epithelial and stromal androgen receptor in the regulation of prostate epithelial dynamics

The prostate is an androgen-responsive organ, but the complex cellular and molecular interactions that mediate these responses remain incompletely defined. Here, I synthesise the existing literature to derive a simple conceptual framework describing the androgen-dependent regulation of prostate epithelial dynamics. In this framework, epithelial androgen receptor (AR) cell-autonomously controls luminal cell height, whereas stromal AR regulates the synthesis of growth factors that promote luminal cell survival and proliferation. With the additional aid of a reanalysis of single-cell RNA-seq data, I also propose that insulin-like growth factor 1 (IGF1) functions as a key androgen-dependent growth factor coordinating stromal-to-epithelial paracrine communication. A novel mathematical model based on this framework was able to quantitatively fit experimental data describing prostate regression and regeneration. Model analysis demonstrates how the luminal cell population can maintain a stable equilibrium size via competition for and degradation of stroma-derived IGF1 and how this population size can be controlled by androgen levels, without a requirement for distinct luminal cell subsets. Moreover, model simulations were able to qualitatively recapitulate experimental observations in inflammatory and cancerous states, thereby providing insights into potential disease mechanisms. This simple model could therefore serve as a foundation for more comprehensive modelling of both the healthy and diseased prostate.

Modeling the synergistic properties of drugs in hormonal treatment for prostate cancer

Prostate cancer is one of the most prevalent cancers in men, with increasing incidence worldwide. This public health concern has inspired considerable effort to study various aspects of prostate cancer treatment using dynamical models, especially in clinical settings. The standard of care for metastatic prostate cancer is hormonal therapy, which reduces the production of androgen that fuels the growth of prostate tumor cells prior to treatment resistance. Existing population models often use patients' prostate-specific antigen levels as a biomarker for model validation and for finding optimal treatment schedules; however, the synergistic effects of drugs used in hormonal therapy have not been well-examined. This paper describes the first mathematical model that explicitly incorporates the synergistic effects of two drugs used to inhibit androgen production in hormonal therapy. The drugs are cyproterone acetate, representing the drug family of anti-androgens that affect luteinizing hormones, and leuprolide acetate, representing the drug family of gonadotropin-releasing hormone analogs. By fitting the model to clinical data, we show that the proposed model can capture the dynamics of serum androgen levels during intermittent hormonal therapy better than previously published models. Our results highlight the importance of considering the synergistic effects of drugs in cancer treatment, thus suggesting that the dynamics of the drugs should be taken into account in optimal treatment studies, particularly for adaptive therapy. Otherwise, an unrealistic treatment schedule may be prescribed and render the treatment less effective. Furthermore, the drug dynamics allow our model to explain the delay in the relapse of androgen the moment a patient is taken off treatment, which supports that this delay is due to the residual effects of the drugs.

Review: Mathematical Modeling of Prostate Cancer and Clinical Application

We review and synthesize key findings and limitations of mathematical models for prostate cancer, both from theoretical work and data-validated approaches, especially concerning clinical applications. Our focus is on models of prostate cancer dynamics under treatment, particularly with a view toward optimizing hormone-based treatment schedules and estimating the onset of treatment resistance under various assumptions. Population models suggest that intermittent or adaptive therapy is more beneficial to delay cancer relapse as compared to the standard continuous therapy if treatment resistance comes at a competitive cost for cancer cells. Another consensus among existing work is that the standard biomarker for cancer growth, prostate-specific antigen, may not always correlate well with cancer progression. Instead, its doubling rate appears to be a better indicator of tumor growth. Much of the existing work utilizes simple ordinary differential equations due to difficulty in collecting spatial data and due to the early success of using prostate-specific antigen in mathematical modeling. However, a shift toward more complex and realistic models is taking place, which leaves many of the theoretical and mathematical questions unexplored. Furthermore, as adaptive therapy displays better potential than existing treatment protocols, an increasing number of studies incorporate this treatment into modeling efforts. Although existing modeling work has explored and yielded useful insights on the treatment of prostate cancer, the road to clinical application is still elusive. Among the pertinent issues needed to be addressed to bridge the gap from modeling work to clinical application are (1) real-time data validation and model identification, (2) sensitivity analysis and uncertainty quantification for model prediction, and (3) optimal treatment/schedule while considering drug properties, interactions, and toxicity. To address these issues, we suggest in-depth studies on various aspects of the parameters in dynamical models such as the evolution of parameters over time. We hope this review will assist future attempts at studying prostate cancer.

Towards a non-animal risk assessment for anti-androgenic effects in humans

Toxicology testing is undergoing a transformation from a system based on high-dose studies in laboratory animals to one founded primarily on in vitro methods that evaluate changes in normal cellular signalling pathways using human-relevant cells or tissues. We review the tools and approaches that could be used to develop a non-animal safety assessment for anti-androgenic effects in humans, with a focus on the molecular initiating events (MIEs) that human disorders indicate critical for normal functioning of the hypothalamus-pituitary-testicular (HPT) axis. In vitro test systems exist which can be used to characterize the effects of test chemicals on some MIEs such as androgen receptor antagonism, inhibition of steroidogenic enzymes or 5α-reductase inhibition. When used alongside information describing the pharmacokinetics of a specific chemical exposure, these could be used to inform a pathways-based safety assessment. However, some parts of the HPT axis such as events occurring in the hypothalamus or pituitary are not well represented by accepted in vitro methods. In vitro tools to characterize perturbations in these events need to be developed before a fully integrated model of the HPT axis can be described. Knowledge gaps also exist which prevent us from using in vitro data to predict the type and severity of in vivo effect(s) that could arise from a given level of in vitro anti-androgenic activity. This means that more work is needed to reliably link an MIE with an adverse outcome. However, especially for chemicals with low anti-androgenic activity, human exposure data can be used to put in vitro mode of action data into context for risk-based safety decision-making.

A partial differential equation model of metastasized prostatic cancer

Biochemically failing metastatic prostate cancer is typically treated with androgen ablation. However, due to the emergence of castration-resistant cells that can survive in low androgen concentrations, such therapy eventually fails. Here, we develop a partial differential equation model of the growth and response to treatment of prostate cancer that has metastasized to the bone. Existence and uniqueness results are derived for the resulting free boundary problem. In particular, existence and uniqueness of solutions for all time are proven for the radially symmetric case. Finally, numerical simulations of a tumor growing in 2-dimensions with radial symmetry are carried in order to evaluate the therapeutic potential of different treatment strategies. These simulations are able to reproduce a variety of clinically observed responses to treatment, and suggest treatment strategies that may result in tumor remission, underscoring our model's potential to make a significant contribution in the field of prostate cancer therapeutics.

A multiscale, mechanism-driven, dynamic model for the effects of 5α-reductase inhibition on prostate maintenance

A systems-level mathematical model is presented that describes the effects of inhibiting the enzyme 5α-reductase (5aR) on the ventral prostate of the adult male rat under chronic administration of the 5aR inhibitor, finasteride. 5aR is essential for androgen regulation in males, both in normal conditions and disease states. The hormone kinetics and downstream effects on reproductive organs associated with perturbing androgen regulation are complex and not necessarily intuitive. Inhibition of 5aR decreases the metabolism of testosterone (T) to the potent androgen 5α-dihydrotestosterone (DHT). This results in decreased cell proliferation, fluid production and 5aR expression as well as increased apoptosis in the ventral prostate. These regulatory changes collectively result in decreased prostate size and function, which can be beneficial to men suffering from benign prostatic hyperplasia (BPH) and could play a role in prostate cancer. There are two distinct isoforms of 5aR in male humans and rats, and thus developing a 5aR inhibitor is a challenging pursuit. Several inhibitors are on the market for treatment of BPH, including finasteride and dutasteride. In this effort, comparisons of simulated vs. experimental T and DHT levels and prostate size are depicted, demonstrating the model accurately described an approximate 77% decrease in prostate size and nearly complete depletion of prostatic DHT following 21 days of daily finasteride dosing in rats. This implies T alone is not capable of maintaining a normal prostate size. Further model analysis suggests the possibility of alternative dosing strategies resulting in similar or greater effects on prostate size, due to complex kinetics between T, DHT and gene occupancy. With appropriate scaling and parameterization for humans, this model provides a multiscale modeling platform for drug discovery teams to test and generate hypotheses about drugging strategies for indications like BPH and prostate cancer, such as compound binding properties, dosing regimens, and target validation.

Effects of Eurycoma longifolia on Testosterone Level and Bone Structure in an Aged Orchidectomised Rat Model

Testosterone replacement is the choice of treatment in androgen-deficient osteoporosis. However, long-term use of testosterone is potentially carcinogenic. Eurycoma longifolia (EL) has been reported to enhance testosterone level and prevent bone calcium loss but there is a paucity of research regarding its effect on the bone structural parameters. This study was conducted to explore the bone structural changes following EL treatment in normal and androgen-deficient osteoporosis rat model. Thirty-six male Sprague-Dawley rats aged 12 months were divided into normal control, normal rat supplemented with EL, sham-operated, orchidectomised-control, orchidectomised with testosterone replacement, and orchidectomised with EL supplementation groups. Testosterone serum was measured both before and after the completion of the treatment. After 6 weeks of the treatment, the femora were processed for bone histomorphometry. Testosterone replacement was able to raise the testosterone level and restore the bone volume of orchidectomised rats. EL supplementation failed to emulate both these testosterone actions. The inability of EL to do so may be related to the absence of testes in the androgen deficient osteoporosis model for EL to stimulate testosterone production.

Mathematical modeling of prostate cancer progression in response to androgen ablation therapy

Prostate cancer progression depends in part on the complex interactions between testosterone, its active metabolite DHT, and androgen receptors. In a metastatic setting, the first line of treatment is the elimination of testosterone. However, such interventions are not curative because cancer cells evolve via multiple mechanisms to a castrate-resistant state, allowing progression to a lethal outcome. It is hypothesized that administration of antiandrogen therapy in an intermittent, as opposed to continuous, manner may bestow improved disease control with fewer treatment-related toxicities. The present study develops a biochemically motivated mathematical model of antiandrogen therapy that can be tested prospectively as a predictive tool. The model includes "personalized" parameters, which address the heterogeneity in the predicted course of the disease under various androgen-deprivation schedules. Model simulations are able to capture a variety of clinically observed outcomes for "average" patient data under different intermittent schedules. The model predicts that in the absence of a competitive advantage of androgen-dependent cancer cells over castration-resistant cancer cells, intermittent scheduling can lead to more rapid treatment failure as compared to continuous treatment. However, increasing a competitive advantage for hormone-sensitive cells swings the balance in favor of intermittent scheduling, delaying the acquisition of genetic or epigenetic alterations empowering androgen resistance. Given the near universal prevalence of antiandrogen treatment failure in the absence of competing mortality, such modeling has the potential of developing into a useful tool for incorporation into clinical research trials and ultimately as a prognostic tool for individual patients.

The Role of Non-Aromatizable Testosterone Metabolite in Metabolic Pathways

Dihydrotestosterone (DHT) originates via irreversible reduction of testosterone by catalytic activity of 5α-reductase enzyme and it is demonstratively the most effective androgen. Androgens influence adipose tissue in men either directly by stimulation of the androgen receptor or indirectly, after aromatization, by acting at the estrogen receptor. DHT as a non-aromatizable androgen could be responsible for a male type fat distribution. The theory of non-aromatizable androgens as a potential cause of a male type obesity development has been studied intensively. However, physiological levels of DHT inhibit growth of mature adipocytes. In animal models, substitution of DHT in males after gonadectomy has a positive effect on body composition as a testosterone therapy. Thus, DHT within physiological range positively influences body composition. However, there are pathological conditions with an abundance of DHT, e.g. androgenic alopecia and benign prostatic hyperplasia. These diseases are considered as risk factors for development of metabolic syndrome or atherosclerosis. In obese people, DHT metabolism in adipose tissue is altered. Local abundance of non-aromatizable androgen has a negative effect on adipose tissue and it could be involved in pathogenesis of metabolic and cardiovascular diseases. Increased DHT levels, compared to physiological levels, have negative effect on development of cardiovascular diseases. Difference between the effect of physiological and increased level brings about certain paradox.

The evolutionary impact of androgen levels on prostate cancer in a multi-scale mathematical model

Background

Androgens bind to the androgen receptor (AR) in prostate cells and are essential survival factors for healthy prostate epithelium. Most untreated prostate cancers retain some dependence upon the AR and respond, at least transiently, to androgen ablation therapy. However, the relationship between endogenous androgen levels and cancer etiology is unclear. High levels of androgens have traditionally been viewed as driving abnormal proliferation leading to cancer, but it has also been suggested that low levels of androgen could induce selective pressure for abnormal cells. We formulate a mathematical model of androgen regulated prostate growth to study the effects of abnormal androgen levels on selection for pre-malignant phenotypes in early prostate cancer development.

Results

We find that cell turnover rate increases with decreasing androgen levels, which may increase the rate of mutation and malignant evolution. We model the evolution of a heterogeneous prostate cell population using a continuous state-transition model. Using this model we study selection for AR expression under different androgen levels and find that low androgen environments, caused either by low serum testosterone or by reduced 5α-reductase activity, select more strongly for elevated AR expression than do normal environments. High androgen actually slightly reduces selective pressure for AR upregulation. Moreover, our results suggest that an aberrant androgen environment may delay progression to a malignant phenotype, but result in a more dangerous cancer should one arise.

Conclusions

The model represents a useful initial framework for understanding the role of androgens in prostate cancer etiology, and it suggests that low androgen levels can increase selection for phenotypes resistant to hormonal therapy that may also be more aggressive. Moreover, clinical treatment with 5α-reductase inhibitors such as finasteride may increase the incidence of therapy resistant cancers.

Reviewers

This article was reviewed by Ariosto S. Silva (nominated by Marek Kimmel) and Marek Kimmel.

Perspectives on Mathematical Modeling for Receptor-Mediated Processes

Mathematical modeling is a potent in silico tool that can help investigate, interpret, and predict the behavior of biological systems. The first step is to develop a working hypothesis of the biology. Then by "translating" the biological phenomena into equations, models can harness the power of mathematical analysis techniques to explore the dynamics and interactions of the biological components. Models can be used together with traditional experimental models to help design new experiments, test hypotheses, identify mechanisms, and predict outcomes. This article reviews the process of building, calibrating, and using mathematical models in the context of the kinetics of receptor and signal transduction biology. An example model related to the androgen receptor-mediated regulation of the prostate is presented to illustrate the steps in the modeling process and to highlight the potential for mathematical modeling in this area.

Modeling systems-level regulation of host immune responses

Many pathogens are able to manipulate the signaling pathways responsible for the generation of host immune responses. Here we examine and model a respiratory infection system in which disruption of host immune functions or of bacterial factors changes the dynamics of the infection. We synthesize the network of interactions between host immune components and two closely related bacteria in the genus Bordetellae. We incorporate existing experimental information on the timing of immune regulatory events into a discrete dynamic model, and verify the model by comparing the effects of simulated disruptions to the experimental outcome of knockout mutations. Our model indicates that the infection time course of both Bordetellae can be separated into three distinct phases based on the most active immune processes. We compare and discuss the effect of the species-specific virulence factors on disrupting the immune response during their infection of naive, antibody-treated, diseased, or convalescent hosts. Our model offers predictions regarding cytokine regulation, key immune components, and clearance of secondary infections; we experimentally validate two of these predictions. This type of modeling provides new insights into the virulence, pathogenesis, and host adaptation of disease-causing microorganisms and allows systems-level analysis that is not always possible using traditional methods.

The immune response is a complex network of processes activated in a host upon infection. Pathogens seek to disrupt or evade these processes to ensure their own survival and proliferation. This article provides a systems-level analysis of the immune response against two related bacterial species in the Bordetella genus, B. bronchiseptica and B. pertussis. B. pertussis, the causative agent of whooping cough, has lost many of the virulence factors of its B. bronchiseptica–like progenitor, and is using different strategies for the modulation of the immune system. We have synthesized two separate network models for the interaction of these pathogens with their hosts. Each network is then translated into a predictive dynamic model and is validated by comparison with available experimental data. The model offers predictions regarding cytokine regulation and the effects of perturbations of the immune system, as well as the time course of infections in hosts that had previously encountered the pathogens. We experimentally validate the prediction that convalescent hosts can rapidly clear both pathogens, while antibody transfer cannot substantially reduce the duration of a B. pertussis infection. This type of modeling provides new insights into the virulence, pathogenesis, and host adaptation of disease-causing microorganisms and can be readily extended to other pathogens.

Effect of Daily Treatment with Thai Herb, Kaempferia parviflora, in Hershberger Assay Using Castrated Immature Rats

The aim of the present study was to investigate the testosterone-like effect of Kaempferia parviflora (KP). Castrated immature rats were randomized and divided into two groups (control and KP-treatment groups). The rats (n=7-8) were treated daily for 5 days by oral route with water in the control group and 1,000 mg/kg of KP in the treatment group. All rats were decapitated 24 h after their last dose and then blood samples were collected for assay of serum FSH, LH, testosterone, progesterone and corticosterone levels. The seminal vesicles plus coagulating glands, ventral prostate, levator ani muscle plus bulbocavernosus muscle, glans penis, kidneys and the adrenal glands were collected and weighed for organ wet weight. Body weight and weight of food intake were recorded throughout the study period. The results show that relative body weight gain in the KP-treatment group was significantly increased 24 and 48 h after the first dose (P<0.05) and then was indistinguishable from the control group. There were no significant differences in the relative reproductive and non-reproductive organ weights between the groups, although all organ weights, except for the glans penis, tended to increase in the KP-treatment group. The serum testosterone levels were significantly increased in the KP-treatment group. There were no significant differences in the serum FSH, LH, progesterone, or corticosterone levels between the groups, even though the serum progesterone level tended to increase and serum LH level tended to decrease in the KP-treatment group. The present study indicates that KP has no testosterone-like effect on reproduction in male rats.