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Prostate cancer is commonly treated by a form of hormone therapy called androgen suppression. This form of treatment, while successful at reducing the cancercell population, adversely affects quality of life and typically leads to a recurrence of the cancer in an androgen-independent form. Intermittent androgen suppression aims to alleviate some of these adverse affects by cycling the patient on and off treatment. Clinical studies have suggested that intermittent therapy is capable of maintaining androgen dependence over multiple treatment cycles while increasing quality of life during off-treatment periods. This paper presents a mathematical model of prostate cancer to study the dynamics of androgen suppression therapy and the production of prostate-specific antigen (PSA), a clinical marker for prostate cancer. Preliminary models were based on the assumption of an androgen-independent (AI) cell population with constant net growth rate. These models gave poor accuracy when fitting clinical data during simulation. The final model presented hypothesizes an AI population with increased sensitivity to low levels of androgen. It also hypothesizes that PSA production is heavily dependent on androgen. The high level of accuracy in fitting clinical data with this model appears to confirm these hypotheses, which are also consistent with biological evidence.


I. INTRODUCTION
Prostate cancer is the most common type of non-skin cancer in American men and the second leading cause of cancer mortality. 1Beginning as early as the second decade of life, the development of prostate cancer can require over 50 years to reach a detectable state. 2 Due to the slow growth rate of prostate cancer, chemotherapy has a limited effect on the disease.Instead, treatment focuses on surgery and radiotherapy for localized disease and hormone therapy for metastatic cancer.
The normal prostate is dependent on androgens, specifically testosterone and 5αdihydrotestosterone (DHT), for development and maintenance. 3Prostate cells respond to these androgens by way of the androgen receptor (AR), and in most cases, prostate cancer cells are also dependent on androgen.Since stimulation by androgen is required for the prostate cells and their malignant counterparts to survive and proliferate, prostate cancer can be targeted by a form of hormone therapy called androgen suppression.As the name suggests, androgen suppression lowers the serum androgen levels.The low androgen levels inhibit the growth of cancer cells and induce apoptosis, or natural cell death. 2 While androgen suppression is initially successful at shrinking a Travis Portz is currently at University of Wisconsin-Madison.Electronic mail: portz@wisc.edub Electronic mail: kuang@asu.edu2158-3226/2012/2(1)/011002/14 C Author(s) 2012 2, 011002-1 the prostate cancer in most patients, almost all patients with metastatic disease experience a relapse within several years.At this hormone refractory stage, the androgen-dependent (AD) cells have been replaced by what are commonly referred to as androgen-independent (AI) cells.These cells are able to sustain growth in low-androgen environments and may also be resistant to the apoptotic effects of such an environment. 4ndrogen 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 psychological impact of surgery. 5These drugs include luteinizing hormone-releasing hormone (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. 5,6 ven 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. 6,7 his therapy schedule aims to maintain the apoptotic effect of androgen suppression on prostate cancer cells by ending treatment before a large AI cell population can develop.[9][10] Results of these studies suggest that the overall time of progression to androgen independence is not significantly changd by intermittent therapy. 6However, intermittent therapy still provides better quality of life over continuous therapy. 11creening for prostate cancer has proven effective at reducing the mortality rate by detecting the disease in its earlier stages. 12Screening is done through prostate-specific antigen (PSA) tests and digital rectal examination (DRE).PSA is an enzyme secreted by prostate cells, and elevated or rising levels of PSA in the blood can be an indication of both localized and metastatic prostate cancer.However, 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 cancer.

II. RELATED WORK
Androgen suppression therapy has been studied with a mathematical model to investigate the mechanism for androgen-independent relapse. 13This model assumed 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 this model and predicted that relapse can only be prevented by IAS if normal androgen levels have a negative effect on the growth rate of AI cells. 14This is biologically unlikely since AI cells typically have androgen receptors with increased sensitivity. 15Using biologically likely hypotheses for AI cell growth rates, the model shows that continuous therapy results in a longer time to androgen-independent relapse than intermittent therapy.
A predictive model has been developed which separates the androgen-independent cells into two populations, those that have undergone a reversible change (and can change back to androgendependent cells) and those that have undergone irreversible changes. 16This model is 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 model. 17he evolution of prostate cancer from a local stage to a systemic, androgen-independent stage has been studied with a cell kinetics model. 18This model includes 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.
The role of the androgen receptor and androgen levels in the evolution of prostate cancer have also been studied with a mathematical model. 19The results of this model suggest that low androgen levels during the development stages of prostate cancer may delay the onset of malignancy but result in a more aggressive phenotype if cancer does develop.While our models do not consider the androgen receptor directly, understanding its role in the androgen dependence and mutation of prostate cancer cells with the help of this model is important.
The dynamics of serum PSA levels in the presence of prostate cancer have also been studied with a mathematical model by Swanson et al. 20 This model assumes that healthy prostate cells and prostate cancer cells produce PSA at a constant rate.With the serum PSA being cleared at a constant rate, the model predicts a linear relationship between tumor volume and serum PSA concentration along with some delay.

III. MODEL DEVELOPMENT
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.
Our models are based on the works of Jackson et al. 13 and Ideta et al.. 14 The Ideta model includes an androgen-dependent (AD) cell population and an androgen independent (AI) cell population with mutation from the AD population to the AI population at a rate based on the androgen concentration.We use the version of their model with a constant AI proliferation rate for comparison with our models.See the supplementary material for a formulation of the model. 21

A. Preliminary model
In our preliminary model, the growth rate of the AD cell population is given by Droop's cell quota model. 22The cell quota model introduces a new variable, Q(t), which is the cell quota for androgen.The AD and AI cell populations are modeled by The proliferation rate of the AD cell population is zero when Q(t) is at the minimum cell quota q.
As Q(t) increases, the growth rate approaches its maximum, μ m .The apoptosis rate of the AD cell population and the net growth rate of the AI population excluding mutation are constant.This model includes mutation between both cell populations.This change is made under the hypothesis that androgen dependence can be regained by AI cells in 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, m 1 (Q), is low for normal androgen levels and high for low androgen levels.In contrast, the AI to AD mutation rate, m 2 (Q), is high for normal androgen levels and low for low androgen levels.
The cell quota for androgen within the AD cells is modeled by As the serum androgen concentration increases, the uptake rate of androgen into the cells approaches its maximum.The maximum uptake rate saturates based on the current cell quota Q(t) and a maximum cell quota q m .Androgen within the cells is used for growth up to the minimum cell quota at a rate μ m and is also assumed to degrade at a constant rate b.The formulation of the cell quota uptake rate is based on a model by Packer et al. 23 The serum PSA concentration, P(t), is again modeled as a linear function of the two cancer cell populations:

B. Final model
In the final model, we use the cell quota model for both the AD and AI cell populations: Both cell populations have the same maximum proliferation rate μ m .To give the AI cells greater capacity for proliferation in low androgen environments compared to AD cells, we select a lower minimum cell quota for the AI cells, q 2 < q 1 .As before, the apoptosis rates are constant for both populations.
The relevance of the cell quota model follows from the nature of androgen's action and how its signal is transduced.The AR is intracellular, so only intracellular androgen can be sensed, and proliferation depends on AR:androgen binding.Hence androgen is clearly a resource.For the AI cells, androgen receptors are typically either consitutively activated in an androgen-independent way, are amplified, or are otherwise activated by mutated regulators. 24,25 n 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 of proliferation, which the Droop formalism captures very nicely with the constraint q 2 < q 1 .
The cell quotas for androgen are modeled by the same equation as the preliminary model, However, there are now two cell quota variables, Q 1 (t) for the AD population and Q 2 (t) for the AI population.All of the cell quota parameters except the minimum cell quotas q 1 and q 2 are assumed to be the same for both populations.The mutation rates between the AD and AI cell populations take the same form as in the preliminary model,  8 and interpolated using equation ( 13) at the beginning of each treatment cycle and using cubic hermite splines between other data points.Future androgen levels for making predictions with our model are also shown.The dashed vertical line separates the interpolated clinical data from the prediction.
We assume that PSA is produced by both AD and AI cells at a baseline rate σ 0 plus an additional androgen-dependent rate.Experimental evidence supports the assumption that PSA production is dependent on androgen levels. 26Hill functions are use for the androgen-dependent rate so that it increases with the cell quota toward a maximum rate σ i .The androgen-dependent rate is split into two terms, one for each cell population, 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.

IV. SIMULATION
In a clinical study, 8 seven men with stage C and stage D prostate cancer were 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 PSA time series data from the seven cases to simulate and fit the models.Using clinical data from an intermittent androgen suppression trial provides a better assessment of the dynamics of prostate cancer as 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 serum androgen data from the clinical cases is used directly as A(t) for fitting rather than modeling the androgen concentration.However, the coarse androgen data must be interpolated before it can be used as an input to the numerical simulations.Using linear interpolation results 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, where t i is the time of off-treatment data point, t f is the time of the on-treatment data point, and γ is the serum androgen clearance rate.The remaining segments of the androgen data are interpolated using 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 1.
See the supplementary material for cases 2 -7. 21he parameters of the models are 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 three models.Estimated parameter ranges for the final model are shown in table I. See the supplementary material for the other models. 21Note that the maximum mutation rates in the preliminary model are considerably higher than in the Ideta model and the final model.This change is made to accommodate the hypothesis of a switching mechanism in the preliminary model.

A. Ideta model
The simulation results for the Ideta model are shown in figures 2 and 3.The model is capable of fitting the first PSA spike from an off-treatment period as seen in cases 1 -3.However, subsequent PSA spikes are much smaller or non-existent in the simulations.
A closer look at the case 1 results gives some insight as to why the model has 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 simulated proliferation and apoptosis rates for the case are shown in figure 2(b).During the first and second on-treatment periods, there were small jumps in the androgen concentration which resulted in lower net AD cell death 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 figure 2(b) that the net growth rate 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 AD cell population being almost entirely eradicated.Since the AI population has a constant net growth rate, this model cannot 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.

B. Preliminary model
The simulation results for the preliminary model are illustrated in figures 4 and 5.With the addition of AI to AD mutation, this model is capable of producing PSA spikes from multiple treatment cycles.However, the model now has trouble achieving the low PSA levels in the ontreatment periods that we see in the clinical results.
Again, we take a closer look at the case 1 results to gain some insight as to why the model fails to accurately fit the clinical results during simulation.As with the Ideta model, the slight jumps in androgen concentration during the first and second on-treatment periods caused reduced AD cell death rate in the simulation.The use of the cell quota model did 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 AI cell population while the AD cell population recovers significantly faster than in the Ideta model.This is caused by the high AI to AD mutation rates which can be seen in figure 4(b).As a result, the model maintains the androgen-dependence of the tumor and we get PSA spikes from each treatment cycle.However, during the on-treatment periods, the mutation from AD to AI cells and the constant proliferation rate of the AI cells result in an AI cell population that is too large to give us the low PSA concentrations needed to fit the clinical data.

C. Final model
The simulation results for the final model are shown in figures 6 and 7. We can clearly see that this model is better at producing the PSA spikes at the end of off-treatment periods and the low PSA values during on-treatment periods than either model 1 or model 2. Once again taking a closer look at the case 1 results, we see that both the cell quota model for the AI population and the androgen-dependent PSA production contributed to the more accurate fit.In the first two on-treatment periods, the growth rate of the AI population is significantly reduced.In the third and fourth on-treatment periods, the AI population declines.Figure 6(b) shows the cell quotas for both populations.The lower AI cell quota in the third and fourth on-treatment periods explain the decline in the AI population during those periods.Looking at the cell populations alone, the model would have still failed to provide an accurate fit of the clinical data if the PSA equation from models 1 and 2 had been used.The addition of the androgen-dependent PSA production enables the model to fit the sharp spikes and low valleys in the PSA concentration without having unrealistic proliferation, apoptosis or mutation rates.We see spikes in the cell quotas during the first and second on-treatment periods as a result of the small jumps in androgen concentration that caused problems in the first two models.These jumps again affect the model's ability to fit the declining PSA values during those periods, especially the first on-treatment period.Despite this problem, the model is able to fit the clinical data extremely well during the rest of the treatment periods and in the other cases.
Table II shows the mean squared error between the simulated PSA levels and the clinical PSA levels for each case and model.We 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 preliminary models both had seven free parameters, while the final model had 8 free parameters.The results clearly confirm that the final model produces the best fits out of the three.The difference between the first two models can be attributed to the preliminary model's inability to produce the low PSA levels during on-treatment periods.The Ideta model's inability 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.

D. Predictions
The results of running the final model for another treatment cycle beyond the clinical data are shown in figures 8 and 9.We note that the patients in cases 1, 2, 3, and 5 had stage C cancer, while the patients in cases 4, 6, and 7 had stage D (metastatic) cancer. 8Our model predicts uncontrolled growth in 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.The model also predicts a poor response to another treatment cycle for the patient in case 3, who had already undergone two long treatment cycles.

VI. DISCUSSION
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  8. Prediction of the final model, case 1.The dashed vertical line separates the clinical fit and the prediction.The PSA concentration and AI population increase significantly with another off-treatment period.However, the subsequent on-treatment period remains effective in stopping further growth.
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.
The final 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.][26] 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. 26 This hypothesis differs from the Jackson 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 ince our final 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 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. 13,14 ince the addition of androgen-dependent PSA production to the final 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.
Overall, the high level of accuracy at which the final model fit the clinical is surprising.While the model does have a fair amount of complexity, only eight parameters (the maximum growth rate, the minimum cell quotas, the apoptosis rates, and the PSA production rates) were varied between the individual cases to produce the fits.These eight parameters are all central to the dynamics of the model, have significant biological meaning, and can be expected to vary between individuals.Future work could look into simplifying the model where possible without sacrificing its accuracy.Another future problem is using the model to predict the progression of prostate cancer without having all of the data immediately available for fitting.Our simulations used a single set of parameter values for each case to fit the full clinical time series data, but there is no reason that the parameter values cannot be updated over time as more data points become available.
A clinical data validated mathematical model of prostate cancer growth under intermittent androgen suppression therapy

Travis Portz Yang Kuang Arizona State University
August 2011 1 Supplementary Material

Ideta Model
The Ideta et al. model [1] includes two separate cancer cell populations, androgen-dependent (AD) cells, represented by the variable X 1 (t), and androgen independent (AI) cells, represented by the variable X 2 (t).These population variables are modeled by The proliferation and apoptosis rates of the AI cells are constant, whereas the proliferation and apoptosis rates of the AD cells are functions of the serum androgen concentration, A(t).The proliferation rate of the AD population is α 1 p(A) where p(A) is Thus, as the androgen concentration increases, the proliferation rate of the AD population approaches its maximum α 1 .In a completely androgen-free environment, the AD cells cease to proliferate altogether.The apoptosis rate of the AD population is β 1 q(A) where q(A) is This gives us an apoptosis rate that ranges from β 1 in an androgen-rich environment to β 1 k 2 in an androgen-free environment.Thus we choose k 2 > 1 to have a lower net growth rate in low androgen environments.
The AD cells mutate to become AI cells at a rate Thus, as the androgen level decreases, the mutation rate approaches m 1 .At the normal androgen level a 0 , the mutation rate is zero.AI cells do not mutate to become AD cells under normal conditions in this model.However, if the androgen level exceeds a 0 , then the AD to AI mutation rate is negative.This negative mutation rate could be interpreted as AI to AD mutation.

FIG. 1 .
FIG.1.Androgen data interpolation, case 1.The data are taken from a clinical trial 8 and interpolated using equation (13) at the beginning of each treatment cycle and using cubic hermite splines between other data points.Future androgen levels for making predictions with our model are also shown.The dashed vertical line separates the interpolated clinical data from the prediction.
FIG.2.Ideta model results, case 1.The clinical PSA measurements are illustrated by red circles.The AD and AI cell populations are overlaid as dashed black and gray lines to show the source of the simulated PSA concentrations.The proliferation rate, α 1 p(A), and the apoptosis rate, β 1 q(A), for the AD population are shown in (b).The AI population has constant proliferation and apoptosis rates.This model does not match the second and third PSA peaks well.

7 FIG. 3 .
FIG. 3. Ideta model results, cases 2 -7.Again, we see that this model has difficulty matching PSA peaks beyond the first treatment cycle.

FIG. 4 .
FIG. 4. Preliminary model results, case 1. Cell quota dependent mutation rates are shown in (b).This model has difficulty matching the low PSA levels during on-treatment periods.

7 FIG. 5 .
FIG. 5. Preliminary model results, cases 2 -7.This model has difficulty matching the sharp declines in PSA levels as seen in cases 2 and 3.

FIG. 6 .
FIG. 6. Final model results, case 1. Cell quotas are shown in (b) with the minimum quotas indicated by dashed lines.In general, the PSA data is matched very accurately.

7 FIG. 7 .
FIG. 7. Final model results, cases 2 -7.Again, this model matches the clinical data very accurately compared to the first two models.

7 FIG. 9 .
FIG. 9. Predictions of the final model, cases 2 -7.The additional cycle is effective for cases 2 and 7. Cases 4, 6, and 7 experience rapid growth of the AI population during the extra off-treatment period.

TABLE I .
Final model parameter ranges.

TABLE II .
Comparison of model fits, including mean squared error (MSE) and Schwarz Bayesian Criterion (SBC).Lower values indicate better fits for both MSE and SBC.