Screening for prostatehas proven effective at reducing the mortality rate by detecting the disease in its earlier stages.Screening is done through prostate-specific antigen (PSA) tests and digital rectal examination (DRE). PSA is an enzyme secreted by prostateand elevated or rising levels of PSA in the blood can be an indication of both localized and metastatic prostateHowever, the use of PSA levels for screening is controversial because many factors can contribute to increases in PSA concentration such as benign prostate hyperplasia (BPH). False positives and negatives are not uncommon when screening for prostate

Given the adverse effects of androgen suppression therapy, a method to prevent or delay the development of androgen independence while increasing quality of life during treatment is desired. One proposed method is intermittent androgen suppression (IAS). With IAS, androgen suppression drugs are administered during an on-treatment period until a remission of the disease is detected. The drugs are then withheld in an off-treatment period until the disease progresses back to a certain level.This therapy schedule aims to maintain the apoptotic effect of androgen suppression on prostateby ending treatment before a large AIpopulation can develop. Clinical studies have shown that patients are responsive to multiple cycles of thetherapy.Results of these studies suggest that the overall time of progression to androgen independence is not significantly changd by intermittent therapy.However, intermittent therapy still provides better quality of life over continuous therapy.

Androgen suppression can be done by surgical castration or through the use of a combination of drugs. Surgical castration is a simple and inexpensive procedure, but most patients find it difficult to accept given its permanent nature. The use of androgen suppression drugs is more common given the psychologicalof surgery.These drugs include luteinizing hormone-releasing(LHRH) agonists and analogs which lower the amount of testosterone produced by the testicles and antiandrogens which block androgen from activating the androgen receptor. Both methods of androgen suppression have adverse effects in addition to the development of androgen independence. Short-term effects include hot flashes, loss of libido, erectile dysfunction, and fatigue. Long-term effects may include muscle loss, osteoporosis, anemia, and loss of cognitive function.

The normal prostate is dependent on androgens, specifically testosterone and 5α-dihydrotestosterone (DHT), for development and maintenance.Prostaterespond to these androgens by way of the androgen receptor (AR), and in most cases, prostateare also dependent on androgen. Since stimulation by androgen is required for the prostateand their malignant counterparts to survive and proliferate, prostatecan be targeted by a form oftherapy called androgen suppression. As the name suggests, androgen suppression lowers the serum androgen levels. The low androgen levels inhibit theofand induce apoptosis, or naturaldeath.While androgen suppression is initially successful at shrinking the prostatein most patients, almost all patients with metastatic disease experience a relapse within several years. At thisrefractory stage, the androgen-dependent (AD)have been replaced by what are commonly referred to as androgen-independent (AI)Theseare able to sustainin low-androgen environments and may also be resistant to the apoptotic effects of such an environment.

Prostateis the most common type of non-skinin American men and the second leading cause ofmortality.Beginning as early as the second decade of life, the development of prostatecan require over 50 years to reach a detectable state.Due to the slowrate of prostatehas a limited effect on the disease. Instead, treatment focuses on surgery andfor localized disease andtherapy for metastatic

The role of the androgen receptor and androgen levels in the evolution of prostatehave also been studied with a mathematicalThe results of thissuggest that low androgen levels during the development stages of prostatemay delay the onset ofbut result in a more aggressive phenotype ifdoes develop. While ourdo not consider the androgen receptor directly, understanding its role in the androgen dependence and mutation of prostatewith the help of thisis important.

The evolution of prostatefrom a local stage to a systemic, androgen-independent stage has been studied with akineticsThisincludes variables for both local and systemic androgen-dependent populations. The rate of the disease's advancement is scaled by the PSA doubling-time, which is considered to be a more robust metric with respect to the variability of normal PSA levels among men.

A predictivehas been developed which separates the androgen-independentinto two populations, those that have undergone a reversible change (and can change back to androgen-dependentand those that have undergone irreversible changes.Thisis piecewise-linear with constraints on parameters to make predicting a relapse possible. An optimal scheduling method for intermittent therapy has also been developed using this

Androgen suppression therapy has been studied with a mathematicalto investigate the mechanism for androgen-independent relapse.Thisassumed continuous administration of androgen suppression therapy and predicted that the treatment is only successful for a small range of biological parameters. Intermittent androgen suppression was applied to thisand predicted that relapse can only be prevented by IAS if normal androgen levels have a negative effect on therate of AIThis is biologically unlikely since AItypically have androgen receptors with increased sensitivity.Using biologically likely hypotheses for AIrates, theshows that continuous therapy results in a longer time to androgen-independent relapse than intermittent therapy.

With this finala newfor the serum PSA concentration is introduced:We assume that PSA is produced by both AD and AIat a baseline rate σplus an additional androgen-dependent rate. Experimental evidence supports the assumption that PSA production is dependent on androgen levels.Hill functions are use for the androgen-dependent rate so that it increases with thequota toward a maximum rate σ. The androgen-dependent rate is split into two terms, one for eachpopulation, because we have already assumed that the two populations respond differently to androgen. We also assume that PSA is cleared from the blood at a constant rate.

Thequotas for androgen are modeled by the same equation as the preliminaryHowever, there are now twoquota variables,) for the AD population and) for the AI population. All of thequota parameters except the minimumquotasandare assumed to be the same for both populations.

The relevance of thequotafollows from the nature of androgen's action and how its signal is transduced. The AR is intracellular, so only intracellular androgen can be sensed, anddepends on AR:androgen binding. Hence androgen is clearly a resource. For the AIandrogen receptors are typically either consitutively activated in an androgen-independent way, are amplified, or are otherwise activated by mutated regulators.In any of these cases, the androgen receptor is still active and some AR:androgen response remains. However, far less androgen is required to achieve the same level ofwhich the Droop formalism captures very nicely with the constraint

Thequota for androgen within the ADis modeled byAs the serum androgen concentration increases, the uptake rate of androgen into theapproaches its maximum. The maximum uptake rate saturates based on the currentquota) and a maximumquota. Androgen within theis used forup to the minimumquota at a rate μand is also assumed to degrade at a constant rate. The formulation of thequota uptake rate is based on aby Packer

Thisincludes mutation between bothpopulations. This change is made under the hypothesis that androgen dependence can be regained by AIin an androgen-rich environment with some sort of switching behavior. The mutation and/or switching rates are given by hill equations,The AD to AI mutation rate,), is low for normal androgen levels and high for low androgen levels. In contrast, the AI to AD mutation rate,), is high for normal androgen levels and low for low androgen levels.

In our preliminarytherate of the ADpopulation is given by Droop'squotaThequotaintroduces a new variable,), which is thequota for androgen. The AD and AIpopulations are modeled byTherate of the ADpopulation is zero when) is at the minimumquota. As) increases, therate approaches its maximum, μ. The apoptosis rate of the ADpopulation and the netrate of the AI population excluding mutation are constant.

Ourare based on the works of Jacksonand IdetaThe Idetaincludes an androgen-dependent (AD)population and an androgen independent (AI)population with mutation from the AD population to the AI population at a rate based on the androgen concentration. We use the version of theirwith a constant AIrate for comparison with ourSee the supplementary material for a formulation of the

The models developed here aim to produce results which accurately match the results of clinical trials. With an accurate model, we hope to gain a greater understanding of the processes at work in prostate cancer and androgen suppression therapy. A model which can be used to predict the course of an individual case of prostate cancer would also be useful in developing a treatment schedule in a clinical setting.

The parameters of theare initially fit by hand to provide good qualitative fits with the clinical PSA data. Once a reasonably close fit has been obtained, a simplex search method is used to minimize the mean square error between the clinical PSA data and the simulated PSA concentrations. This parameter fitting is performed for each of the seven cases on all threeEstimated parameter ranges for the finalare shown in table I . See the supplementary material for the otherNote that the maximum mutation rates in the preliminaryare considerably higher than in the Idetaand the finalThis change is made to accommodate the hypothesis of a switching mechanism in the preliminary

The serum androgen data from the clinical cases is used directly as) for fitting rather thanthe androgen concentration. However, the coarse androgen data must bebefore it can be used as an input to the numerical simulations. Using linearresults in the peaks of the PSA concentrations being significantly delayed. These delays are caused by the slow linear decline in androgen concentration between off-treatment and on-treatment periods in the simulations when the actual androgen concentrations drop very quickly. To obtain more accurate results, an exponential fit is used between the last off-treatment data points and the first on-treatment data points,whereis the time of off-treatment data point,is the time of the on-treatment data point, and γ is the serum androgen clearance rate. The remaining segments of the androgen data areusing piecewise cubic hermite splines to give smoother responses in the simulations. We also generate future androgen levels using a rectangular function based on the average androgen levels during on-treatment and off-treatment periods. The androgen data for the first case are shown in figure. See the supplementary material for cases 2 – 7.

In a clinical study,seven men with stage C and stage D prostatewere treated with intermittent androgen suppression therapy. Androgen withdrawal was maintained for 6 or more months and then interrupted for 2 to 11 months. Serum androgen and PSA concentrations were measured on a monthly basis. When serum PSA concentrations exceeded a threshold of about 20 ng/mL, androgen withdrawal was resumed. This treatment cycle was continued over periods of 21 to 47 months. We use the androgen and PSAdata from the seven cases to simulate and fit theUsing clinical data from an intermittent androgen suppression trial provides a better assessment of the dynamics of prostateas responses to both the initiation and withdrawal of treatment can be observed as opposed to continuous androgen suppression where only a single on-treatment period is observed.

The results of running the finalfor another treatment cycle beyond the clinical data are shown in figures. We note that the patients in cases 1, 2, 3, and 5 had stage Cwhile the patients in cases 4, 6, and 7 had stage D (metastatic)Ourpredicts uncontrolledin the AI population for the stage D cases even though the PSA concentrations do respond to the final on-treatment period in cases 6 and 7. Thealso predicts a poor response to another treatment cycle for the patient in case 3, who had already undergone two long treatment cycles.

Table II shows the mean squared error between the simulated PSA levels and the clinical PSA levels for each case andWe also show the Schwarz Bayesian Criterion, which includes an adjustment for the number of free parameters. In this case, the fits for the Ideta and preliminaryboth had seven free parameters, while the finalhad 8 free parameters. The results clearly confirm that the finalproduces the best fits out of the three. The difference between the first twocan be attributed to the preliminaryinability to produce the low PSA levels during on-treatment periods. The Idetainability to fit multiple spikes in the PSA data does not affect the mean square error as much because there are fewer data points in the off-treatment periods than in the on-treatment periods.

Once again taking a closer look at the case 1 results, we see that both thequotafor the AI population and the androgen-dependent PSA production contributed to the more accurate fit. In the first two on-treatment periods, therate of the AI population is significantly reduced. In the third and fourth on-treatment periods, the AI population declines. Figureshows thequotas for both populations. The lower AIquota in the third and fourth on-treatment periods explain the decline in the AI population during those periods. Looking at thepopulations alone, thewould have still failed to provide an accurate fit of the clinical data if the PSA equation from1 and 2 had been used. The addition of the androgen-dependent PSA production enables theto fit the sharp spikes and low valleys in the PSA concentration without having unrealisticapoptosis or mutation rates. We see spikes in thequotas during the first and second on-treatment periods as a result of the small jumps in androgen concentration that caused problems in the first twoThese jumps again affect theability to fit the declining PSA values during those periods, especially the first on-treatment period. Despite this problem, theis able to fit the clinical data extremely well during the rest of the treatment periods and in the other cases.

Again, we take a closer look at the case 1 results to gain some insight as to why thefails to accurately fit the clinical results during simulation. As with the Idetathe slight jumps in androgen concentration during the first and second on-treatment periods caused reduced ADdeath rate in the simulation. The use of thequotadid not appear to attenuate the effect of these small androgen concentration jumps by any considerable amount. During the off-treatment periods, we see declines in the AIpopulation while the ADpopulation recovers significantly faster than in the IdetaThis is caused by the high AI to AD mutation rates which can be seen in figure. As a result, themaintains the androgen-dependence of theand we get PSA spikes from each treatment cycle. However, during the on-treatment periods, the mutation from AD to AIand the constantrate of the AIresult in an AIpopulation that is too large to give us the low PSA concentrations needed to fit the clinical data.

A closer look at the case 1 results gives some insight as to why thehas trouble fitting the clinical results. In this case, the patient's therapy schedule consisted of four on-treatment periods and three off-treatment periods. The simulatedand apoptosis rates for the case are shown in figure. During the first and second on-treatment periods, there were small jumps in the androgen concentration which resulted in lower net ADdeath rates in the simulation. This prevented the simulated PSA levels from declining as quickly as the actual measured PSA levels. We can also see from figurethat the netrate during the off-treatment period is significantly smaller than the net death rate during the on-treatment period. Combine this with the decreasing duration of the off-treatment periods, and the result is an ADpopulation being almost entirely eradicated. Since the AI population has a constant netrate, thiscannot produce oscillations in the PSA levels without the AD population. Instead, all of the cases show PSA levels approaching the exponential curve of the AI population once it has overtaken the AD population.

VI. DISCUSSION Section: Choose Top of page ABSTRACT I.INTRODUCTION II.RELATED WORK III.MODEL DEVELOPMENT IV.SIMULATION V.RESULTS VI.DISCUSSION << REFERENCES CITING ARTICLES

The models developed here focus on the androgen-dependent dynamics of prostate cancer. All proposed models utilize an AD cell population and an AI cell population. The two preliminary models assume that proliferation and apotosis rates of the AI cells are truly independent of androgen levels. This gives these models trouble fitting the oscillations in PSA levels that are seen in clinical results. The preliminary model attempts to compensate with a substantial mutation rate and/or switching behavior from the AI population to the AD population, but the model then has trouble fitting the low PSA levels in the on-treatment periods. Furthermore, there is no biological evidence suggesting the existence of a mechanism for AI to AD mutation or switching.

model is based on two key hypotheses: that the so-called “androgen-independent” population has a higher sensitivity to low androgen concentrations and that PSA production is also dependent on androgen levels to a substantial degree. The first hypothesis is consistent with evidence of a hypersensitive pathway to androgen independence. 15,24–26 93, 1687 (2001). 15. M. Grossmann, H. Huang, and D. Tindall, J. Nat. Cancer Inst., 1687 (2001). https://doi.org/10.1093/jnci/93.22.1687 91, 483 (2004). 24. M.-E. Taplin and S. Balk, J. Cellular Biochem., 483 (2004). https://doi.org/10.1002/jcb.10653 25, 897 (2011). 25. K. Lamont and D. Tindall, Mol. Endocrinol., 897 (2011). https://doi.org/10.1210/me.2010-0469 1, 34 (2001). 26. B. Feldman and D. Feldman, Nat. Rev. Cancer, 34 (2001). https://doi.org/10.1038/35094009 26 1, 34 (2001). 26. B. Feldman and D. Feldman, Nat. Rev. Cancer, 34 (2001). https://doi.org/10.1038/35094009 et al. and Ideta et al. models which assume that androgen levels have either no effect or a negative effect on the net growth rate of the AI population. 13,14 4, 187 (2004). 13. T. Jackson, Discrete Cont. Dyn. Syst. Ser. B, 187 (2004). https://doi.org/10.3934/dcdsb.2004.4.187 18, 593 (2008). 14. A. Ideta, G. Tanaka, T. Takeuchi, and K. Aihara, J. Nonlinear Sci., 593 (2008). https://doi.org/10.1007/s00332-008-9031-0 model produced significantly more accurate results than the Ideta model, we believe that the hypothesis of a hypersensitive AI population is more likely to be true. The finalis based on two key hypotheses: that the so-called “androgen-independent” population has a higher sensitivity to low androgen concentrations and that PSA production is also dependent on androgen levels to a substantial degree. The first hypothesis is consistent with evidence of a hypersensitive pathway to androgen independence.Some possible mechanisms for hypersensitivity to androgen include amplification of the androgen receptor, increased stability and nuclear localization of the androgen receptor, and increased rates of conversion of androgens from testosterone to DHT.This hypothesis differs from the Jacksonand Idetawhich assume that androgen levels have either no effect or a negative effect on the netrate of the AI population.Since our finalproduced significantly more accurate results than the Idetawe believe that the hypothesis of a hypersensitive AI population is more likely to be true.

13,14 4, 187 (2004). 13. T. Jackson, Discrete Cont. Dyn. Syst. Ser. B, 187 (2004). https://doi.org/10.3934/dcdsb.2004.4.187 18, 593 (2008). 14. A. Ideta, G. Tanaka, T. Takeuchi, and K. Aihara, J. Nonlinear Sci., 593 (2008). https://doi.org/10.1007/s00332-008-9031-0 model was so important in providing an accurate fit of clinical data, we believe that androgen levels should be taken into consideration when serum PSA concentrations are used to screen for prostate cancer and to track treatment progress. This is especially important in the context of intermittent androgen suppression where PSA levels are being used to determine when treatment periods should begin and end. A low PSA level may not imply a small cancer cell population if the androgen levels are also low, and stopping androgen suppression when there is still a large cancer cell population may be detrimental to the treatment of a patient. Determining the relative androgen dependence of a tumor is also not possible using PSA measurements alone. Instead, a mathematical model such as the final model proposed here can be used to monitor AD and AI cell populations. Based on the predicted populations, more intelligent decisions regarding treatment scheduling can be made. For example, the predictions of our final model above suggest that cases 3, 4, 6, and 7 should not undergo another off-treatment period since the AI population would grow rapidly and not respond well to further treatment. The second hypothesis, that PSA production is dependent on androgen, is consistent with biological evidence suggesting that the PSA gene is regulated by the androgen receptor.Since the addition of androgen-dependent PSA production to the finalwas so important in providing an accurate fit of clinical data, we believe that androgen levels should be taken into consideration when serum PSA concentrations are used to screen for prostateand to track treatment progress. This is especially important in the context of intermittent androgen suppression where PSA levels are being used to determine when treatment periods should begin and end. A low PSA level may not imply a smallpopulation if the androgen levels are also low, and stopping androgen suppression when there is still a largepopulation may be detrimental to the treatment of a patient. Determining the relative androgen dependence of ais also not possible using PSA measurements alone. Instead, a mathematicalsuch as the finalproposed here can be used to monitor AD and AIpopulations. Based on the predicted populations, more intelligent decisions regarding treatment scheduling can be made. For example, the predictions of our finalabove suggest that cases 3, 4, 6, and 7 should not undergo another off-treatment period since the AI population wouldrapidly and not respond well to further treatment.