## Abstract

The testicular-hypothalamic-pituitary axis regulates male reproductive system functions. Understanding these regulatory mechanisms is important for assessing the reproductive effects of environmental and pharmaceutical androgenic and antiandrogenic compounds. A mathematical model for the dynamics of androgenic synthesis, transport, metabolism, and regulation of the adult rodent ventral prostate was developed on the basis of a model by Barton and Anderson (1997). The model describes the systemic and local kinetics of testosterone (T), 5α-dihydrotestosterone (DHT), and luteinizing hormone (LH), with metabolism of T to DHT by 5α-reductase in liver and prostate. Also included are feedback loops for the positive regulation of T synthesis by LH and negative regulation of LH by T and DHT. The model simulates maintenance of the prostate as a function of hormone concentrations and androgen receptor (AR)-mediated signal transduction. The regulatory processes involved in prostate size and function include cell proliferation, apoptosis, fluid production, and 5α-reductase activity. Each process is controlled through the occupancy of a representative gene by androgen-AR dimers. The model simulates prostate dynamics for intact, castrated, and intravenous T-injected rats. After calibration, the model accurately captures the castration-induced regression of the prostate compared with experimental data that show that the prostate regresses to ∼17 and 5% of its intact weight at 14 and 30 days postcastration, respectively. The model also accurately predicts serum T and AR levels following castration compared with data. This model provides a framework for quantifying the kinetics and effects of environmental and pharmaceutical endocrine active compounds on the prostate.

- rodent ventral prostate
- androgen receptor
- testosterone
- 5α-dihydrotestosterone
- testicular-hypothalamic-pituitary axis

the hypothalamus, pituitary, and testes produce endocrine hormones responsible for regulation of the prostate and other male sexual functions (12, 46). Exogenous endocrine active compounds can disrupt these processes. Some pesticides (e.g., vinclozolin and linuron) are known to have antiandrogenic activity (19, 20). Toxic effects of antiandrogens in male rodents range from developmental effects such as reproductive malformations, retained nipples, and undescended testes to pubertal effects such as delayed puberty and reduced weights of prostate and other reproductive organs. In the pharmaceutical setting, therapeutic drugs such as finasteride, dutasteride, bicalutamide, and flutamide are used to treat benign prostatic hyperplasia and/or prostate cancer by inhibiting androgen-dependent growth processes.

The mechanisms of action of antiandrogens are generally of two forms. The first is the androgen antagonist, which binds to the androgen receptor (AR) but does not stimulate DNA transcription, such as the pharmaceutical compounds flutamide and bicalutamide or the environmental compound vinclozolin. The second is the 5α-reductase inhibitor, which blocks the metabolism of testosterone (T) to 5α-dihydrotestosterone (DHT). The therapeutic drugs dutasteride and finasteride are examples of 5α-reductase inhibitors; no environmental examples are currently known. Both of these mechanisms effectively inhibit AR-mediated DNA transcription, leading to reduction of prostate size and cell numbers and decreased prostatic fluid production. Quantification of the male regulatory processes and their disruptions would aid in the dose-response assessment of environmental endocrine active compounds (4, 5). This requires biologically based quantitative methods describing not only the pharmacokinetics and mode of action of the exogenous compound of interest but also the pharmacokinetics of the endogenous hormones and how they regulate the male reproductive organs.

The objective in this research effort was to understand the normal adult male hormonal regulation of the prostate to set a foundation for quantitatively characterizing the effects of exposure to antiandrogens in male rats. To quantitatively understand the normal adult male hormonal regulation of the prostate, mathematical descriptions of the kinetics of T, DHT, and luteinizing hormone (LH) are required, along with several pharmacodynamic components, such as prostatic AR concentrations, maintenance of prostate size, and regulation of prostatic fluid production.

The regulation of T is crucial for prostate maintenance in that the size and function of the prostate is gene regulated via DNA binding by androgen-bound AR (21–23, 37, 42, 53). T is produced mainly in the testicular Leydig cells in response to LH released from the pituitary (12). It is metabolized to DHT by 5α-reductase in the liver and prostate (39, 50). The enzyme 5α-reductase is upregulated by T and DHT via AR-mediated gene expression (39, 50). LH upregulates the production of T and, hence, DHT, which in turn downregulates LH, creating a negative feedback loop (12). In the blood, T and DHT bind to serum albumin, which appears to play a role in regulating transport and tissue uptake dynamics (15, 33, 49).

In this article, a model describing the endogenous hormone kinetics of the testicular-pituitary axis and the dynamics of the androgenic regulation of the prostate is presented. The model includes the pharmacokinetics of T, DHT, and LH, as well as the dynamics of AR binding and signaling as they relate to the regulation of the prostate. Development of the model included characterization of critical biological and physiological processes inherent in the hormonal regulation of the male reproductive system, including androgen-LH feedback loops and AR-mediated regulation of the prostate.

The model structure for the hormonal transport, or pharmacokinetic component, is based on standard physiologically based pharmacokinetic (PBPK) modeling (32), whereas the dynamics of the AR-based regulation of the prostate are based on mass action kinetics. These dynamics include binding of T and DHT to the AR, dimerization of androgen-AR complexes, and gene activation by these complexes via DNA binding. The complexes regulate prostatic cellular mass, fluid production, blood flow, AR concentrations, and 5α-reductase activity.

Unknown physiological, kinetic, and binding parameters were estimated using available experimental data from the literature. The resulting calibrated model is capable of quantitatively describing intact and castrated rats at steady state, as well as the dynamics that occur following castration. This model may be used for testing and generating hypotheses, data interpretation, experimental design, and identifying key data gaps in the study of the hormonal regulation of the prostate and provides a framework for elaborations that address the effects of exposure to exogenous, pharmaceutical, or environmental androgens and antiandrogens (5).

## METHODS

### Model Structure: Pharmacokinetic Component

The transport and dispositional kinetics of T and DHT are described using standard PBPK compartmental modeling (32), whereas a single-compartment model is used for LH. Flow-limited compartments are utilized in cases where the uptake rate of a compound into a tissue is limited by the blood flow rate into the tissue rather than the diffusion rate of the compound across cell membranes. Conversely, diffusion-limited compartments are used when the diffusion across cell membranes is slow compared with the blood flow rate into the tissue (32). The pharmacokinetics of T and DHT are modeled with compartments representing the systemic serum, liver, prostate, testes, and rest of body (Fig. 1). LH is modeled with a single systemic compartment that captures androgen-regulated synthesis and the effects of LH on T synthesis in the testes. All terms in the equations are defined in Tables 1, 2, 3, and 4.

#### Systemic serum compartment.

The compartment for systemic serum includes exchange of T and DHT with the tissue compartments, the binding dynamics of T and DHT to albumin, and a basal input rate of T, due to limited synthesis of T outside the testes (29). A simplifying assumption is made that all concentrations are rapidly equilibrated between red blood cells and serum.

The rate equation for the kinetics of T in the systemic serum is given as (1) The first four terms represent T exchange with the rest of body, prostate, liver, and testes compartments, respectively, followed by the basal synthesis of T. Free concentrations CT_{blf} are affected by albumin binding in the blood as in *Eq. 3* below. It is assumed that only free T is available to enter the testes and rest-of-body compartments, whereas both free and albumin-bound T are available for uptake into the metabolically active prostate and liver (33).

The equation for the kinetics of DHT in systemic serum has a similar form, except that basal DHT synthesis is not included, because it is assumed that all significant amounts of DHT are produced in the prostate and liver via metabolism (1). In addition, the rest-of-body compartment for DHT incorporates the testes, which is modeled as a separate compartment for T to capture the dynamics of T synthesis. The equation for DHT transport in the systemic serum is given by (2) We assume standard binding kinetics for T and DHT binding to albumin: (3) (4) with free T, DHT, and albumin concentrations determined using mass balance: (5) (6) (7)

#### Liver compartment.

The kinetics of T and DHT in the liver are described by using a flow-limited compartmental model. Metabolism of T to DHT via 5α-reductase in the liver is modeled with a Michaelis-Menten term, and linear terms are included for the general metabolic clearance of T and DHT in the liver (49). The rate equations for T and DHT in the liver are given by (8) and (9) Free concentrations of T and DHT in the liver are given by the algebraic equations (10) so that free concentrations are proportional to total concentrations.

#### Testes compartment.

The testes compartment has a unique structure based on the specific biology of the testes that is important for T since the testicular Leydig cells are the major source for T synthesis (12). Here, we give a brief outline of the model; for a comprehensive development of the model structure and equations, the reader is referred to Barton and Anderson (6). As previously mentioned, the kinetics of DHT in the testes are incorporated in the rest-of-body compartment, so that the testes compartment is utilized only for T.

The testes compartment is divided into two subcompartments, representing the seminiferous tubules (ST) and the interstitial fluid (IF). To account for the testicular shunt (30) that directs a portion of the blood flow directly into the spermatic cord (bypassing the testes), the blood flow rate to the testes compartment is split into two components (Fig. 1). One component of the blood flow feeds into the IF subcompartment, and the other component flows into the spermatic cord and feeds directly into the systemic serum. The bias of the split is controlled by the model parameter ts.

Circulating T reenters the IF via the blood and then diffuses between the IF and ST as with a standard diffusion-limited compartment (32). The ST compartment is present to facilitate future elaborations to address spermatogenesis. The diffusion of T between the IF and ST is controlled in the model by the parameter μ. To account for the synthesis of T in the Leydig cells, there is a source of T production in the IF (12). Recalling that LH upregulates T synthesis in the Leydig cells (6), we assume a positive linear feedback mechanism in the model with rate constant *k*_{1T}. Once the testicular venous blood leaves the testes compartment, it immediately reenters the systemic serum via the spermatic cord. The rate equation for T in the IF subcompartment is given by (11) With only diffusion occurring in the ST compartment, the corresponding rate equation is defined as (12) The algebraic equation for the testicular venous blood describes the concentration of T as proportional to the concentration in the interstitial fluid due to nonspecific binding and is given by (13)

#### Prostate compartment.

The prostate is a complex tissue with multiple lobes and multiple cell types, which vary in their androgen responsiveness (41). The model currently is parameterized to describe the ventral lobe of the prostate, which represents 60–70% of the rat prostate (11). The prostate compartment includes transport to and from the blood, the metabolism of T to DHT by 5α-reductase, and the dynamics of AR binding, dimerization, and DNA binding (Fig. 2). The rate equations for the kinetics of T and DHT in the prostate are given (respectively) by (14) and (15) Free prostatic T and DHT are available to bind to AR. The androgen-bound AR complexes T:AR and DHT:AR then form homo- and heterodimers. These dimers bind to DNA and stimulate the regulation of various processes. The equations for T and DHT binding to AR are given by (16) and (17) The homo- and heterodimers (abbreviated as DT, DD, and TT) have similar kinetics: (18) (19) (20) The dimers above are able to bind to DNA sites to form DNA-dimer complexes. In the model, there are four distinct DNA binding sites that are involved in regulating 5α-reductase synthesis, cell proliferation, apoptosis, and prostatic fluid production. Although numerous genes are likely responsible for the regulation of the last three processes, in the model the simplifying assumption was made to represent each process by a single gene.

The four representative genes in the model are available to bind to all three dimer types, yielding 12 equations for the concentrations of the DNA-dimer complexes. One example is given here. The following equation is for the concentration of the DHT:AR-DHT:AR dimer bound to the 5α-reductase synthesis-regulating gene: (21) where parameters and variables for the 12 DNA-dimer complexes are given in Tables 1, 2, and 4. The dissociation rates for the DNA-dimer complexes are determined to reflect relative potencies between T and DHT as well as between the four representative DNA binding sites. Specifically, we assume that the relative potencies for the dissociation of DNA from the dimers T:AR-T:AR, D:AR-D:AR, and D:AR-T:AR are 6, 0.5, and 4, respectively. These potencies account for the fact that DHT is significantly more potent in mediating AR-based effects than T is (55). Moreover, the four representative DNA binding sites have relative potencies for dissociation with DNA-dimer complexes that reflect the differential responses of the four regulatory events (5α-reductase activity, fluid production, cell proliferation, and apoptosis) to AR-mediated signaling. These potencies are 0.14, 0.4, 1.4, and 0.2, respectively, and were estimated as part of the parameter estimation process described below. The model is formulated so that the occupancy of each gene can hold any value between 0 and 1, representing the fraction of total possible occupancy.

Free concentrations of prostatic T, DHT, AR, and DNA sites are given by (22) (23) (24) and (25) where NT_{p} and ND_{p} are coefficients for nonspecific binding of T and DHT, respectively, in the prostate, and ∑(T) and ∑(D) are all the DNA-bound dimers containing T and D, respectively. The equations for DNA occupancy are given by (26)

#### Rest-of-body compartment.

The rest-of-body compartment represents all tissues and organs not included in the remaining compartments and includes exchange with the systemic serum. Note that the testes are included in the rest-of-body compartment for DHT but not for T since this tissue is modeled separately for T synthesis dynamics. This results in two different blood flow rates, *Q*_{b} and *Q*_{bt} for the rest-of-body compartments that exclude and include the testes. The transport equations for T and DHT in the rest of body are given by (27) and (28)

#### LH pharmacokinetics.

The kinetics of LH are described in the model with a single equation that incorporates the synthesis and elimination of LH in the serum. The synthesis of LH is negatively regulated by T and DHT concentrations, with a small basal amount of synthesis assumed. Similar to the upregulation of T by LH, we assume a piecewise linear feedback mechanism to describe downregulation of LH by T. Elimination of LH is proportional to the amount of LH in the system. The transport equation is given by (29) where α_{T} and α_{D} represent the relative potency for inhibiting LH production, *k*_{LH1} and *k*_{LH2} describe the decline in LH with increasing androgen, *k*_{LH3} denotes the minimum LH synthesis rate, and *k*_{LH4} represents LH degradation. See Fig. 3 for a representative graph of the LH synthesis rate function.

### Model Structure: Pharmacodynamic Component

The model describes the dynamics of AR concentrations, prostatic blood flow, 5α-reductase activity, prostatic ductal lumen mass, and prostate cellular mass. Prostatic blood flow is regulated by the size of the prostatic cellular mass, whereas 5α-reductase activity, cellular mass, and ductal lumen mass are regulated via occupancy of DNA binding sites.

#### AR concentrations.

Although the AR has been shown to be upregulated by androgens (48), here we make the simplifying assumption that prostatic AR is produced at a constant rate and free AR is eliminated at a linearly proportional rate. We further assume that androgen-bound AR is not available for elimination. The equations for total AR amount and concentration, respectively, are as follows: (30) Recall that the concentration of free prostatic AR is given in *Eq. 24*.

#### Prostatic blood flow.

Although it has been shown that prostatic blood flow decreases almost immediately after castration (45), here we make the simplifying assumption that the blood flow to the prostate is proportional to the volume of the prostatic cell mass V_{pc}: (31)

#### 5α-Reductase activity.

The activity of 5α-reductase and, hence, the metabolism of T are regulated by AR binding and signaling (18, 43). Here, we assume that *V*_{max} is directly proportional to the occupancy, DNAo_{5a}, of the representative 5α-reductase gene: (32)

#### Androgen-sensitive prostatic cellular mass.

On the basis of experimental data (11, 42), we assume that the prostate contains a basal cellular mass, VPC_{2}, that is not affected by androgens and an androgen-sensitive cellular mass that is regulated via the occupancy of the representative cell proliferation and apoptosis genes (Fig. 4). The equation for the volume of the androgen-sensitive cellular mass is as follows: (33) This is based on the assumption that the proliferation rate is proportional to the occupancy DNAo_{cp} of the representative cell proliferation gene and that the rate of cell formation increases with prostate cellular mass. Moreover, the death rate is negatively related to the occupancy DNAo_{cd} of the representative apoptosis gene, since in this case DNA occupancy results in protection from apoptosis (36).

#### Androgen-sensitive prostatic ductal lumen mass.

As with the prostate cellular mass, we assume that there is a basal amount VPL_{2} of the ductal lumen that remains unaffected by androgen concentrations and an androgen-sensitive amount that is regulated by the representative fluid production gene in the model. The governing equation is given by (34) The growth rate of the ductal lumen is proportional to the occupancy DNAo_{sec} of the representative fluid production gene and the size of the androgen-sensitive prostate cellular mass relative to the intact, steady-state cellular mass VPC_{1b}. Specifically, when cellular mass decreases following castration or antiandrogen exposure, the rate of fluid production concurrently decreases, since there are fewer cells present to secrete fluid. We also assume that there is linear degradation of ductal lumen as fluid exits the prostate.

The rate constants for cell proliferation and apoptosis in *Eq. 33* were chosen so that both terms of the equation are equal when the model reaches intact steady-state values, thus yielding equilibrium and no change in prostate cellular mass. The rate constants for fluid secretion and fluid flow in *Eq. 34* were chosen similarly so that the entire ventral prostate is at equilibrium when steady-state values are reached.

Note that in *Eqs. 33* and *34*, the ratio VPC_{1}(*t*)/VPC_{1b} can become unbounded under conditions of supernormal hormone levels and prostate growth. However, in the scope of these modeling efforts, supernormal hormone levels will not be simulated, and therefore supernormal prostate volumes are not expected. If one were to use the model under conditions of supernormal hormone levels, it would be necessary to address this limitation, likely using experimental data from androgen implant experiments in intact and castrated rats to define an appropriate function.

#### Algebraic equation for prostate volume.

Adding the basal and androgen-sensitive volumes of the prostate cellular mass and ductal lumen mass in the model, we are able to compute the total volume of the prostate as a function of time: (35)

### Model Implementation and Calibration

The model was implemented computationally in Matlab (The Mathworks, Natick, MA). Parameters for the model were chosen on the basis of values from the literature where possible. Other parameter values were estimated by comparison with experimental data, and a selected set was reestimated using standard least squares parameter estimation techniques (51).

One parameter estimation problem was formulated to estimate pharmacokinetic parameters such as tissue/blood partition coefficients and the feedback parameters for LH and T synthesis. These parameters were estimated using a submodel that contains only the pharmacokinetics of T, DHT, and LH without any of the pharmacodynamic phenomena. The estimation problem involved comparing submodel simulations to intact and castrate steady-state values and data as in (6).

A second parameter estimation problem was formulated with the full model to estimate pharmacodynamic parameters, such as binding constants, prostatic fluid production rate, 5α-reductase *V*_{max}, and prostatic cell proliferation and death rates. These parameters were estimated in comparison with intact and castrate steady-state behaviors and the dynamics of prostate regression following castration.

The third and final estimation problem was formulated with the full model and included all parameters estimated in the first two estimation problems, where the initial value for each unknown parameter was the optimal value obtained from the respective previous estimation problem. This allowed for a more accurate fit to the overall model, since a submodel was used in the first problem to estimate reasonable starting points for the pharmacokinetic parameters. Model fits using the final parameter values are presented in results. A complete list of model parameters with values and references is given in Tables 2–4.

## RESULTS

The model was used to predict the dynamics of androgen levels and prostate regulation under various conditions, including the intact steady state, exogenous dosing of T via intravenous injections, and castration. The simulations reported in this section vary in time duration from 8 h to 30 days. Although rats grow significantly in the time span of 30 days and the prostate grows at a similar rate to the whole body (8, 37), we assume a constant body mass in the model and report all simulations and experimental data as fractions of intact values.

### Predicting Steady-State Concentrations

Table 5 reports steady-state concentrations of T, LH, and DHT in serum and various tissues for intact and castrate conditions. These concentrations were obtained by simulating an unperturbed intact or castrate state until equilibrium was obtained for all model species. The results given in Table 5 suggest that the model is capable of predicting both intact and castrate steady-state concentrations of T, DHT, and LH compared with the data.

### Simulating an Intravenous Dose of T

Results of a simulated intravenous injection of 0.02 nmol T are shown in Fig. 5, which depicts time courses for T and LH concentrations in blood and T concentrations in the testes. The plots illustrate the ability of the model to simulate T and LH regulation in serum and the testes after a perturbation of the intact steady-state system. Note the immediate spike in serum T concentrations from the steady-state concentration of 7.6 nM, and the resulting acute decrease in serum LH concentrations, followed by decaying oscillatory effects due to the stabilizing ability of the feedback system. After ∼4 h from the time of injection, the system is essentially stable.

### Simulating Serum T Levels After Castration

Systemic T concentrations drop rapidly following castration (9, 26). Data show that, within the first 30 min, serum T levels drop to ∼15% of intact values. After 6 h, serum T levels are reported to be less than 5% of intact values. Figure 6 depicts these data and the corresponding model prediction. Serum T in castrated rats is often close to assay detection limits and is quite variable, as evidenced by values ranging from 0.4 to 5% of intact values in Kyprianou and Isaacs (26). Therefore, an extratesticular synthesis rate of T included in the systemic serum compartmental model was calibrated to give 5% of intact levels, which yielded a good overall fit with all of the experimental data used to calibrate the model.

### Simulating Regression of the Prostate Postcastration

Suzuki et al. (48) report substantial decreases in prostatic AR levels in response to castration. Figure 7 depicts the model prediction vs. the experimental data. As can be seen, the predicted AR levels at each time point are within the given error bars for the data.

Several studies have demonstrated a sigmoidal decrease in prostate weight in time after castration (27, 28, 40, 42). Figure 8 includes model predictions of postcastration prostate weight vs. experimental data. The model predictions match well with a majority of the data across the different experiments. Moreover, the simulations suggest that the prostate has lost all androgen-sensitive cellular mass and ductal lumen mass by about the 11th day after castration. The remaining androgen-insensitive prostate mass is ∼16.4% of the total intact mass.

In addition to predicting the time course of the entire prostate volume after castration, model simulations capture the regression of the ductal lumen component. Figure 9 depicts model predictions of the ductal lumen mass after castration compared with experimental data (42), showing a good match between the data and the model.

As illustrated by the experimental data presented in Figs. 6–9, there is a clear sequence of regulatory events that occurs after castration, leading to decreased prostate weight. Within 1 h, there is a sharp decrease in serum androgen concentrations followed by a drop in prostatic AR levels over a couple of days. As AR concentrations decrease, the prostate responds more slowly with decreased fluid production, so that the ductal lumen mass shrinks to less than 10% of the intact mass within a week. Finally, the prostatic cellular mass regresses at a slower pace, reaching its minimum ∼2 wk after castration. These dynamics, which are shown together with corresponding model simulations in Fig. 10, are consistent with a gene-regulatory delay in the prostatic response to androgen removal.

### Model Validation

The predictive accuracy of the model was tested by comparing simulations with experimental data that were not used in the calibration process. These include prostatic blood flow rate measurements after castration as well as additional data for postcastration regression of the prostate. Figure 11 depicts model simulations compared with the additional prostate weight and blood flow data. The model accurately predicts the prostate weight data, although it overpredicts the weight at 72 h postcastration. The data from these two studies at this particular time point happen to be significantly lower than data at the same point from the studies used to calibrate the model (see Fig. 8), which explains why the model is overpredicting the new data.

Moreover, the model predictions for prostatic blood flow rate after castration are reasonably accurate compared with the data (Fig. 11). Note that no prostatic blood flow data were used in the model calibration process, so the accurate prediction of prostatic blood flow helps to further validate the model as a plausible representation of the biology. Because the current state of the model approximates the relationship between blood flow and tissue volume without describing the complex causal relationship (24, 36a), these results may further suggest that the simplified description provides a reasonable approximation of prostatic blood flow behavior.

## DISCUSSION

The objective of this research effort was to understand the normal adult male hormonal regulation of the prostate as a foundation for quantitatively characterizing the effects of environmental or pharmaceutical antiandrogen exposure in male rats. Because it is impossible to capture all physiological and biological details in a mathematical model, a typical approach is to describe the critical determinants of biological behavior while reasonably simplifying many other details. The result is a model that is a mix of mechanistic and empirical elements. The goal is then to obtain a parameter set that produces biologically plausible behavior and that matches well with available experimental data.

The model captures the kinetics of the endogenous androgens and their regulatory effects on the size and function of the prostate in the adult male rat. It also is capable of simulating intact steady-state conditions and accurately predicting the experimentally observed sequence of events that occurs after castration, including the immediate drop in serum androgen concentrations.

The model also captures the cascade of events that occur following castration and rapid elimination of circulating T (except for continuing low levels of adrenal production). The hypothesis is incorporated that the binding affinities of T and DHT for AR are similar to what is reported in Ref. 53, with equilibrium dissociation constants of 0.49 and 0.34 nM, respectively, but rapid degradation of free AR is prevented by the two androgens to different extents due to the approximately threefold faster on and off rates for T than for DHT (53, 54). Thus the greater potency of DHT for gene activation in this model in part reflects greater stability of the receptor ligand complex. This stability initially mitigates the rapid decline in serum T (occurring in hours), so that prostatic AR declines over days. The consequences of declining AR show up first in diminished gene occupancy related to 5α-reductase activity, production of prostatic fluid, and finally in decreased cell proliferation and increased apoptosis (40). This sequence of events is accurately captured by the model simulations of postcastration dynamics.

Although cell proliferation, apoptosis, and fluid secretion are each regulated by multiple genes, characterizing each process with a single gene proved to be a reasonable simplification. These simplified empirical descriptions of known complex biological processes can be augmented if necessary to include greater mechanistic details. Such details may then be incorporated into the model as needed, based on the particular problem to be solved.

This model has a description of the central axis feedback between LH and T that revises that used in Barton and Andersen (6). The new description returns to the same steady state for serum concentrations of T in intact animals after perturbations such as an intravenous dose of T or an implant releasing constant levels of T at physiological or lower concentrations (simulations not shown). The previous model would establish a new different steady state for serum T concentrations. Experimental data are not entirely consistent on this issue, although return to control steady-state T levels in intact animals with implants giving physiological or lower serum T concentrations is reported by Darras et al. (11), Perheentupa et al. (38), and Zirkin et al. (56), supporting the new empirical description of the feedback mechanism.

In a case where limited data are available to qualitatively or quantitatively capture the underlying biology of the central axis feedback, the model can serve as a tool to test different hypotheses about biological mechanisms. Current efforts are underway to explore the dynamics of androgens and prostate mass in response to external androgen implants. Model simulations may be able to motivate new experiments to help elucidate the regulatory mechanisms of androgen levels and AR-mediated prostate maintenance under different hormonal conditions.

This model has 25 unknown parameters that needed to be estimated compared with experimental data. This must be addressed carefully to minimize problems of mathematical ill-posedness and to maximize the biological plausibility of the model. Parameter estimation problems often are subject to ill-posedness, which means that there may not be a solution to the problem, there may be more than one solution, or the problem may be highly sensitive to perturbations in data or parameters (25).

In this case, the unknown parameters first were separated into two subsets based on function and estimated using two separate estimation problems. As described in methods, the pharmacokinetic parameters were estimated by comparing a pharmacokinetic submodel to measurements of steady-state androgen concentrations as in Ref. 6, while the pharmacodynamic parameters were estimated using the full model compared with steady-state concentrations and measurements of postcastration dynamics.

This approach helped address ill-posedness by splitting the estimation into two problems, effectively reducing the size of the unknown parameter space for each problem. Moreover, the parameters to be estimated were grouped with experimental data that were more directly related to the relevant biology for each parameter subset (pharmacokinetic vs. pharmacodynamic). Incorporating a variety of experimental data also can help reduce ill-posedness by requiring the model to reproduce multiple behaviors. In particular, the postcastration dynamics data represent measurements of several different model variables in time, so that the resulting model simulations capture the biology over several time scales and complexity scales ranging from the receptor level up to the tissue and systemic levels.

The final step involved solving a third estimation problem for all parameters from the first two problems using the full model. Because the pharmacokinetic parameters were estimated using a submodel, it was important to reestimate all parameters together. The pharmacokinetic submodel did not contain androgen binding to AR, which can significantly alter tissue concentrations by sequestering T and DHT in the prostate, for example. The third estimation problem allowed for recalibration of the pharmacokinetic (as well as pharmacodynamic) parameters by taking the full model into account. The resulting final parameter values did not significantly change the model fits to the data for postcastration dynamics but did improve the predictions of steady-state androgen tissue concentrations. The parameters that changed the most after the third estimation problem were related to LH synthesis, albumin-androgen binding rates, tissue partition coefficients, and the *V*_{max} for 5α-reductase in the liver. Overall, the final parameter set provides a good fit to the wide variety of data used in the calibration process.

The results presented here show the ability of the model to reproduce a variety of different biological behaviors, including the intact and castrate steady-state conditions, as well as the dynamics of androgens and the regression of the prostate following castration. Model simulations match well with multiple types of experimental measurements on both the systemic and tissue-specific levels, including hormone and AR concentrations, prostatic blood flow, enzymatic activity of 5α-reductase, and AR-mediated regulation of prostate size and function.

Based on these results, the model and the estimated parameter set seem to provide a plausible representation of the biology. This is further supported by simulations showing that the model matches well with postcastration prostatic blood flow data that were not used in the model calibration process. This model validation can be further enhanced as additional experimental data become available.

By using a systems approach to the gene-regulated processes involved in prostate maintenance, this model may be used as a tool for data analysis, generating and testing hypotheses, and experimental design in the research of the prostate. Moreover, the design of this model can serve as a template for developing models for many other gene-regulated biological processes. The PBPK backbone efficiently describes the transport of relevant compounds through the tissues, while the tissue-specific pharmacodynamic submodel captures key biological activity related to protein synthesis and signal transduction.

The versatility of the model and its capability of capturing the dynamics of prostate behavior under varying conditions make it useful in both toxicological and pharmaceutical settings. Because some pesticides have androgen-antagonistic properties (5), augmentation of the model to simulate the kinetics and effects of androgen antagonist exposure would make it a suitable tool for risk assessment. Furthermore, with enhancements, the model is a potential drug development tool to predict a response to administration of 5α-reductase inhibitors and androgen antagonists. Both classes of antiandrogens are frequently used in therapies to treat diseases such as benign prostatic hyperplasia and prostate cancer. By calibrating the model with experimental data for a given antiandrogen, the model then can be used to predict the aggregate effect on the prostate of similar compounds by adjusting compound-specific parameters such as binding constants. This type of analysis can yield crucial decision-making estimates of EC_{50}, E_{max}, and efficacious dose ranges and frequencies, as well as the ability to quantitatively link biomarker data to clinical drug efficacy.

The ability of the model to be expanded allows for the potential to address a broad range of issues associated with male reproductive function, especially prostate function and maintenance. One major objective is to characterize the effects of prepubertal exposure to antiandrogens, such as vinclozolin, in male rats, because this has been proposed as a bioassay for detecting compounds with endocrine disrupting activity (47). This will involve adapting the model to describe the normal pubertal maturation process as well as the effects on that process that result from prepubertal vinclozolin exposure. Such an elaboration will require modeling prepubertal regulation dominated by androstanediol and the transition to adult androgen regulation dominated by T and DHT. These modeling efforts will provide a quantitative means for understanding the dynamic changes in the male reproductive endocrine system that occur during pubertal development, such as tissue growth, androgen production in the testes, and concentrations of androgens and binding proteins (2, 3, 13, 14, 34, 44, 47). The pubertal model will also aid in quantitatively describing how antiandrogens interfere with these pubertal changes, enhance understanding of the dose-response relationships, and improve accuracy in the risk assessment of these chemicals (5).

## GRANTS

L. K. Potter and M. G. Zager were funded by the Environmental Protection Agency (EPA)/University of North Carolina Toxicology Research Program, Training Agreement TA-829472, with the Curriculum in Toxicology, University of North Carolina, Chapel Hill, NC. This work was reviewed by the EPA and approved for publication but does not necessarily reflect official Agency policy. Mention of trade names or commercial products does not constitute endorsement or recommendation by the EPA for use.

## Acknowledgments

We thank Dr. Roger Rittmaster (GlaxoSmithKline, Research Triangle Park, NC) and colleagues at Dalhousie University (Halifax, NS, Canada) for providing us with experimental data.

L. K. Potter's current address: Scientific Computing & Mathematical Modeling, GlaxoSmithKline, Research Triangle Park, NC 27709.

M. G. Zager's current address: Pharmacokinetics, Dynamics and Metabolism, La Jolla Laboratories, Pfizer, Inc., San Diego, CA 92121.

## Footnotes

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- Copyright © 2006 by American Physiological Society