This review addresses the topic of biofilms, including their development and the interaction between different counterparts. There is evidence that various diseases, such as cystic fibrosis, otitis media, diabetic foot wound infections, and certain cancers, are promoted and aggravated by the presence of polymicrobial biofilms. Biofilms are composed by heterogeneous communities of microorganisms protected by a matrix of polysaccharides. The different types of interactions between microorganisms gives rise to an increased resistance to antimicrobials and to the host's defense mechanisms, with the consequent worsening of disease symptoms. Therefore, infections caused by polymicrobial biofilms affecting different human organs and systems will be discussed, as well as the role of the interactions between the gram-negative bacteria Pseudomonas aeruginosa, which is at the base of major polymicrobial infections, and other bacteria, fungi, and viruses in the establishment of human infections and diseases. Considering that polymicrobial biofilms are key to bacterial pathogenicity, it is fundamental to evaluate which microbes are involved in a certain disease to convey an appropriate and efficacious antimicrobial therapy.
Citation: Manuela Oliveira, Eva Cunha, Luís Tavares, Isa Serrano. P. aeruginosa interactions with other microbes in biofilms during co-infection[J]. AIMS Microbiology, 2023, 9(4): 612-646. doi: 10.3934/microbiol.2023032
[1] | Tin Phan, Changhan He, Alejandro Martinez, Yang Kuang . Dynamics and implications of models for intermittent androgen suppression therapy. Mathematical Biosciences and Engineering, 2019, 16(1): 187-204. doi: 10.3934/mbe.2019010 |
[2] | Avner Friedman, Harsh Vardhan Jain . A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences and Engineering, 2013, 10(3): 591-608. doi: 10.3934/mbe.2013.10.591 |
[3] | Zhimin Wu, Tin Phan, Javier Baez, Yang Kuang, Eric J. Kostelich . Predictability and identifiability assessment of models for prostate cancer under androgen suppression therapy. Mathematical Biosciences and Engineering, 2019, 16(5): 3512-3536. doi: 10.3934/mbe.2019176 |
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[5] | Heyrim Cho, Allison L. Lewis, Kathleen M. Storey, Anna C. Zittle . An adaptive information-theoretic experimental design procedure for high-to-low fidelity calibration of prostate cancer models. Mathematical Biosciences and Engineering, 2023, 20(10): 17986-18017. doi: 10.3934/mbe.2023799 |
[6] | Leonardo Schultz, Antonio Gondim, Shigui Ruan . Gompertz models with periodical treatment and applications to prostate cancer. Mathematical Biosciences and Engineering, 2024, 21(3): 4104-4116. doi: 10.3934/mbe.2024181 |
[7] | Samantha L Elliott, Emek Kose, Allison L Lewis, Anna E Steinfeld, Elizabeth A Zollinger . Modeling the stem cell hypothesis: Investigating the effects of cancer stem cells and TGF−β on tumor growth. Mathematical Biosciences and Engineering, 2019, 16(6): 7177-7194. doi: 10.3934/mbe.2019360 |
[8] | Leo Turner, Andrew Burbanks, Marianna Cerasuolo . PCa dynamics with neuroendocrine differentiation and distributed delay. Mathematical Biosciences and Engineering, 2021, 18(6): 8577-8602. doi: 10.3934/mbe.2021425 |
[9] | Hao Shen, Xiao-Dong Weng, Du Yang, Lei Wang, Xiu-Heng Liu . Long noncoding RNA MIR22HG is down-regulated in prostate cancer. Mathematical Biosciences and Engineering, 2020, 17(2): 1776-1786. doi: 10.3934/mbe.2020093 |
[10] | Peng Gu, Dongrong Yang, Jin Zhu, Minhao Zhang, Xiaoliang He . Bioinformatics analysis identified hub genes in prostate cancer tumorigenesis and metastasis. Mathematical Biosciences and Engineering, 2021, 18(4): 3180-3196. doi: 10.3934/mbe.2021158 |
This review addresses the topic of biofilms, including their development and the interaction between different counterparts. There is evidence that various diseases, such as cystic fibrosis, otitis media, diabetic foot wound infections, and certain cancers, are promoted and aggravated by the presence of polymicrobial biofilms. Biofilms are composed by heterogeneous communities of microorganisms protected by a matrix of polysaccharides. The different types of interactions between microorganisms gives rise to an increased resistance to antimicrobials and to the host's defense mechanisms, with the consequent worsening of disease symptoms. Therefore, infections caused by polymicrobial biofilms affecting different human organs and systems will be discussed, as well as the role of the interactions between the gram-negative bacteria Pseudomonas aeruginosa, which is at the base of major polymicrobial infections, and other bacteria, fungi, and viruses in the establishment of human infections and diseases. Considering that polymicrobial biofilms are key to bacterial pathogenicity, it is fundamental to evaluate which microbes are involved in a certain disease to convey an appropriate and efficacious antimicrobial therapy.
Prostate cancer is the most frequently diagnosed cancer and second leading cause of death from cancer in men. Growth of this cancer is stimulated by androgens, or male sexual hormones. These androgens circulate in the blood and diffuse into the tissue where they stimulate the prostate tumor to grow. One treatment for advanced or metastatic prostate cancer is androgen deprivation therapy (ADT). Androgen deprivation may be achieved by hormone therapy, which inhibits the production of androgens in the testicles [11]. Leuprolide acetate and goserelin are drugs currently used for this treatment that can be delivered continuously over extended time periods using depot injections [14]. The influence of the remaining androgens produced by other sources, such as the adrenal glands, can be eliminated by anti-androgens such as flutamide, bicalutamide, enzalutamide, and nilutamide [11]. A combination of anti-androgens with chemical castration via ADT is known as the maximal androgen blockade (MAB). Both ADT and MAB facilitate apoptosis, the programmed death of androgen-dependent (AD) cancer cells, and quickly induce temporal regression of tumors.
Currently, many patients receive continuous androgen suppression (CAS) therapy of ADT and MAB. However, many of these patients undergo a relapse with an increase of the PSA level within three years after the initiation of ADT [9]. Androgen-independent (AI) cells are thought to be responsible for this recurrent tumor growth. These cells are not sensitive to androgen suppression, but rather they replicate even in an androgen depleted environment. The AI cells are generated by mutation from AD tumor cells. Once the tumor acquires these AI cells, androgen deprivation is unable to inhibit the cancer growth and a relapse becomes inevitable. In one study, 5 of the 61 patients demonstrated this progression within 4 years [13]. It is important to prevent a relapse or at least delay the time to relapse as long as possible. At the same time, it is also important to reduce economic costs and alleviate adverse side effects of prolonged androgen suppression such as osteoporosis, cardiovascular disease, anemia, and metabolic disorders [1].
A possible strategy to delay the progression from the AD state to the AI state is intermittent androgen suppression (IAS), which is a form of androgen ablative therapy delivered intermittently with off-treatment periods. On-treatment periods last for several months until PSA levels fall below a prescribed threshold and then, to avoid emergence of AI cells, the IAS therapy introduces off-treatment periods that serve to maintain the androgen-deprivation sensitivity of the cancer cells and restore their apoptotic potential, which can be induced by androgen deprivation [10]. Fourteen studies of nineteen models published have confirmed improvement in the quality of life during the off-treatment periods and alleviation of adverse side effects such as sexual dysfunction, hot flushes, and fatigue [1]. However, it is unknown how to optimally plan the IAS therapy.
Ideta, et al. [9], introduce a mathematical model commonly referred to as the ITTA model that describes the growth of a prostate tumor under IAS therapy based on monitoring of the serum prostate-antigen. By treating the tumor growth as a mixed assembly of androgen-dependent and androgen-independent cells, they investigate the difference between CAS and IAS in order to understand the factors that result in AI relapse. The ITTA model is known as a hybrid dynamical system because tumor growth is continuous in time whereas the on-and off-treatment protocol is discrete in time. In [9] and [18] bifurcation analysis of the ITTA model is used to characterize parameter regions that distinguish a prevention of relapse (cancer free state) from the occurrence of relapse.
In contrast, Portz, et al. [12], and Everett, et al. [6], utilize models that include serum PSA as a compartment that evolves dynamically in response to a combination of the AD cells, AI cells, intracellular androgen, and an independent clearance rate. Moreover, they include bidirectional mutation rates between the AD and AI compartments. Although these models include more details regarding the interaction of the compartments within the ITTA model, the ultimate conclusion in [6] is that a simpler model may provide more accurate information for predictive use.
Hirata, et al. [8], extend the ITTA model so that the androgen independent cells may exist in either a reversible or an irreversible state. During an on-treatment cycle, an androgen dependent cell may change to an androgen independent cell of either type and a reversible AI cell may change to an irreversible AI cell. During an off-treatment cycle, an AI cell that is in the reversible state may change to an AD cell. They show that their model provides a better description of the dynamics of prostate cancer under IAS than a model with only reversible or irreversible cell types. Suzuki, et al. [17], simplify the ITTA model and assume that the androgen levels are constant (i.e., at steady-state) during both on-and off-treatment cycles. This yields two linear, autonomous dynamical systems: one for the on-treatment phase and one for the off-treatment phase. They propose a region-dividing method which exploits the phase plane dynamics and saddle-point nature of the disease free equilibrium to control the switching between on-and off-treatment cycles and guarantee a cancer free state. They show that the region-dividing method effectively controls both the ITTA model and the Hirata model.
In this paper, we modify the ITTA model in order to create and analyze mathematical models for continuous, intermittent and periodic androgen suppression (CAS, IAS, and PAS respectively). Our models for IAS and PAS retain the qualitative behavior of the system under the feedback control defined in [9], yet are simple enough so that sensitivity analysis may be conducted to determine both the dynamic impact of parameters during treatment and the relative importance of parameter values on relapse time. Each of the studies [9], [18], and [17] include models that describe the prevention of relapse (i.e., the cancer free state), but we consider only those parameter values that allow for disease relapse as we are not aware of any clinical data that support prevention of relapse by IAS therapy. Although sensitivity to initial conditions can provide valuable information within a parameter estimation process, here we are focused on the relative importance of parameters under three different treatment options (CAS, IAS, and PAS) and we note the importance of collecting data at various times within a clinical trial that includes these options in order to improve estimates for those parameters.
We now summarize the basic structure and parameters of the ITTA model. First, it is assumed the administration is alternatively either present during on-treatment periods
˙a(t)=−γ[a(t)−a0]−γa0u(t). | (1) |
The tumor growth is described with changes in the numbers of AD and non-reversible AI cells, and we assume the proliferation and apoptosis rates of AD and AI cells are dependent on the androgen concentration
˙x1(t)=[α1p1(a(t))−β1q1(a(t))−m(a(t))]x1(t) | (2) |
˙x2(t)=m(a(t))x1(t)+[α2p2(a(t))−β2]x2(t) | (3) |
where
p1(a)=a(a+k2)−1, | (4) |
q1(a)=k3+(1−k3)a(a+k4)−1, | (5) |
m(a)=m1(1−a/a0), and | (6) |
p2(a)=1−κa/a0. | (7) |
Here we use the parameter
(i)κ=0,(ii)κ=1−β2/α2. | (8) |
The coefficients
The functional forms and parameters in (1)-(8) were defined in [9] for both qualitative and quantitative accuracy and, where possible, based on experimental data. For example, since AD cells do not multiply in the absence of androgen and the proliferation rate is known to be approximated by
Parameter | Variable | Baseline Value | Range |
Normal androgen level | 30 nmol/l | 26.25-33.75 nmol/l | |
Androgen concentration rate | 0.08 days |
0.0425-0.1175 days |
|
Androgen dependent proliferation rate | 0.0204 days |
0.0129-0.0279 days |
|
Androgen dependent apoptosis rate | 0.0076 days |
0.00685-0.00835 days |
|
Androgen independent proliferation rate | 0.0242 days |
0.0216-0.0268 days |
|
Androgen independent apoptosis rate | 0.0168 days |
0.0130-0.0206 days |
|
Maximum mutation rate | 0.00005 days |
1.25-8.75 |
|
AD proliferation half-saturation level | 2 nmol/l | 1.25-2.75 nmol/l | |
AD androgen free apoptosis constant | 8 | 7.25-8.75 | |
AD apoptosis rate half-saturation level | 0.5 nmol/l | 0.275-0.725 nmol/l | |
Minimum PSA concentration | 10 ng/ml | 6.5-13.5 ng/ml | |
Maximum PSA concentration | 15 ng/ml | 11.5-18.5 ng/ml | |
Treatment transition rate | 1100 days |
500-1700 days |
a(0)=30,x1(0)=15,x2(0)=0.01. | (9) |
The suggested range of values for each parameter was created to stay within values available in the literature and in the cases where no such information was available, a standard deviation was used to ensure that the sample coefficient of variation remained under 30%. In Section 3 we further discuss the use of these parameter ranges to simulate data for sensitivity analysis.
Recall that the serum PSA is an effective biomarker for the prostate tumor growth. Its concentration is the only observable output of the system and is used as a basis for the intermittent administration in IAS therapy. Since a large amount of serum PSA is secreted by cancer cells, the PSA concentration
y(t)=x1(t)+x2(t). | (10) |
The administration is suspended when the serum PSA concentration falls below
u(t)={0→1wheny(t)=r1and˙y>0,1→0wheny(t)=r0and˙y<0. | (11) |
Based on this model, Ideta, et al., use numerical simulations and bifurcation analysis to show how tumor growth and relapse time are influenced by the net growth rate of the androgen-independent cells and the mutation rate from androgen-dependent cells to androgen-independent cells. In [12] and [6], a direct sensitivity to androgens for both the AD and AI compartments is assumed and their models include a response to intracellular androgen concentrations for both types of cells. However, the coupling of the differential equations (1)-(3) allows for indirect effects of androgen on even the AI compartment through changes in the parameter space. It is important to quantify these effects using sensitivity analysis in order to make a comparison with the outcomes of other models.
The remainder of this paper is organized as follows. In Section 2 we discuss the model under an assumption of continuous androgen suppression that will be used as a baseline for sensitivity analysis that will be extended to the intermittent and periodic models. In Section 3 we construct and analyze a dynamic model for IAS that treats the administration of anti-androgen therapy in (11) as a continuous state variable on the same time scale as the other compartments. In Section 4, we investigate a model for periodic androgen suppression (PAS) where
To effectively analyze the effects of Intermittent Androgen Suppression (IAS), we first look at a Continuous Androgen Suppression (CAS) therapy model in order to establish a baseline for which sensitivity analysis can be performed in a straightforward manner. Moreover, this treatment regimen is currently the only recommended option as a standard of care outside of Europe [4]. CAS is modelled by letting
Given the coupled system of differential equations (1)-(3), an understanding of the influence that parameters within the model have on solutions is important since changes in the values of those parameters (e.g., the androgen concentration rate,
˙s=s(t)∂f∂x+∂f∂θ, | (12) |
s(0)=0. | (13) |
Each equation in a dynamical system yields a corresponding sensitivity equation for each parameter within the model. Thus, for a model that has
˙saγ=−γsaγ−a,˙sx1γ=(α1dp1da−β1dq1da−dmda)x1saγ+[α1p1(a)−β1q1(a)−m(a)]sx1γ,˙sx2γ=(dmdax1+α2dp2dax2)saγ+m(a)sx1γ+[α2p2(a)−β2]sx2γ, |
where
dp1da=k2(a+k2)2,dq1da=k4(1−k3)(a+k4)2,dmda=−m1a0,dp2da=−κa0. |
We note here that the derivative of
The sensitivity solution for androgen concentration with respect to the parameter
Next, we consider the sensitivity of the androgen dependent cells to changes in the parameter space. In Figure 3 we present the sensitivity functions for the AD population with respect to the parameters
In contrast, sensitivity of the AI population to changes in the parameter space remains relatively low until many months of treatment have elapsed. In Figure 4 we see that sensitivity of this compartment to the androgen concentration rate
In Section 5, we further explore the CAS model in order to examine the effect of our choice for the value of
In order to compare the parameter sensitivity in this scenario with what we computed for CAS treatment, we first reformulate the algorithmic definition as an ordinary differential equation. First, we consider the switch from on-treatment to off-treatment (from
H(y−(r0−ϵ))H(r0+ϵ−y), | (14) |
where
H(y−K)={0if y≤K,1if y>K. | (15) |
Because of numerical inaccuracies,
To create a smooth transition from
f10(u)=u(u−(1+ˆϵ)), | (16) |
where
Finally, the dynamics of
˙u10=H(y−(r0−ϵ))H(r0+ϵ−y)H(−˙y)u(u−(1+ˆϵ)). | (17) |
Similarly, we model the switch from off-treatment to on-treatment (i.e., from
˙u01(t)=H(y−(r1−ϵ))H(r1+ϵ−y)H(˙y)(u+ˆϵ)(1−u). | (18) |
We model the back and forth switching in (11) by adding (17) and (18). Hence, we have the following ordinary differential equation:
˙u=λ[H(y−(r0−ϵ))H(r0+ϵ−y)H(−˙y)u(u−(1+ˆϵ))+H(y−(r1−ϵ))H(r1+ϵ−y)H(˙y)(u+ˆϵ)(1−u)]. | (19) |
Here we use a parameter
To accurately conduct parameter sensitivity computations, we need to ensure that the right side of (19) is differentiable. Thus, we will use the arctangent function to approximate the step functions in (19). In particular, if
G[z(θ)]=1πtan−1[10000z(θ)]+12, | (20) |
so that
ddθG[z(θ)]=10000π(1+[10000z(θ)]2)−1dzdθ, | (21) |
and hence continuity of the derivative of
˙a=f(1)(a)+g(1)(u), | (22) |
˙x1=f(2)(a), | (23) |
˙x2=f(3)(a), | (24) |
˙u=g(2)(u). | (25) |
where
f(1)(a)=−γ(a(t)−a0) | (26) |
g(1)(u)=−γa0u(t), | (27) |
f(2)(a)=[α1p1(a(t))−β1q1(a(t))−m(a(t))]x1(t), | (28) |
f(3)(a)=m(a(t))x1(t)+[α2p2(a(t))−β2q2(a(t))]x2(t), | (29) |
g(2)(u)=λ[G(y−(r0−ϵ))G(r0+ϵ−y)G(−˙y)u(u−(1+ˆϵ))+G(y−(r1−ϵ))G(r1+ϵ−y)G(˙y)(u+ˆϵ)(1−u)]. | (30) |
We use the following system for the sensitivity to each parameter
˙saθ=f(1)a(a)saθ(t)+f(1)θ(a)+g(1)u(u)suθ(t)+g(1)θ(u), | (31) |
˙sx1θ=f(2)x1(a)sx1θ(t)+f(2)θ(a), | (32) |
˙sx2θ=f(3)x2(a)sx2θ(t)+f(3)θ(a), | (33) |
˙suθ=g(2)u(u)suθ(t)+g(2)θ(u). | (34) |
with initial conditions
saθ(0)=sx1θ(0)=sx2θ(0)=suθ(0)=0. | (35) |
The sensitivity solutions for the parameter
In contrast, the sensitivities for the mutation parameter
In Figure 10 we see the sensitivities to the parameter
Thus, we generally expect the same behavior of our sensitivity functions during periods of on-treatment as are observed in the CAS model, but an opposite behavior is generally observed during periods of off-treatment. These swings in sensitivity generally spike at points in time where treatment is turned either on or off. For
We also examine the relative sensitivity of our model to each parameter in order to determine which parameters have the biggest impact on the long-term behavior of our solution components. Using the range of parameter values presented in Table 1 and an assumption that each parameter was normally distributed in its range, independent of the other parameter values, 100 randomized parameter sets were generated and used to generate simulated data sets
J(q)=∑k||z(tk;q0)−ˆz(tk;q)||22, |
representing a squared error in the Euclidean norm of the difference between our solutions generated using the baseline parameters (represented by
∂J∂q(q0)=2∑k(z(tk;q0)−ˆz(tk;q))T∂z∂q(tk;q0). |
For each randomized parameter set, we calculate the relative sensitivity to a given parameter
ΥJθ:=∂J∂θ×θJ. |
In Figure 11, we present a summary of these values for the case
In Ideta, et al., Figures 8 and 9 show simulations of the IAS model for
TR={t>0|y(t)=¯y}, | (36) |
where
The IAS model (1)-(10) controls the treatment indicator function
Ton=∫TR0u(τ)dτ. | (37) |
It can be seen in Figure 13 that in general
The periodic nature of
ˆu(t)={1when0≤t<T/2,0whenT/2≤t<T. | (38) |
The PAS model consists of equations (1)-(8) with treatment control (38). After substituting (38) into (1), the androgen dynamics are
ˆa(t)={a0e−γt,when0≤t<T/2,a0e−γt+a0[1−e−γ(t−T/2)],whenT/2≤t<T, | (39) |
with
A(t)=[α1p1(a)−β1q1(a)−m(a)0m(a)α2p2(a)−β2]. | (40) |
Denoting initial conditions as
Φ(t)=[exp(∫t0[f1(τ)−m(τ)]dτ)0ψ(t)exp(∫t0f2(τ)dτ)], | (41) |
where
f1(τ)=α1p1(a(τ))−β1q1(a(τ)), | (42) |
f2(τ)=α2p2(a(τ))−β2, | (43) |
ψ(t)=e∫t0f2(τ)dτ∫t0m(a)e∫t0(f1(τ)−m(τ)−f2(τ))dτdτ. | (44) |
Figure (14) depicts the serum PSA
Simulation of the PAS model reveals that for
limt→∞x1(t)=0,limt→∞x2(t)=∞. | (45) |
The following theorem details conditions under which (45) holds for the PAS model.
Theorem 4.1. If the parameters are chosen so that
∫T0[f1(τ)−m(τ)]dτ<0andκ<ρ−1(1−β2/α2), | (46) |
where
Proof. The equation for
x1(t)=x1(0)exp(∫t0[f1(τ)−m(τ)]dτ)=p1(t)exp(ζ1t) | (47) |
where
ζ1=T−1∫T0[f1(τ)−m(τ)]dτ, | (48) |
and
We next determine
x2(t)=x2(0)exp(∫t0f2(τ)dτ)+ψ(t) | (49) |
with
xh2(t)=x2(0)exp(∫t0f2(τ)dτ)=p2(t)exp(b2t) | (50) |
where
b2=T−1∫T0f2(τ)dτ=α2(1−κˉa/a0)−β2, | (51) |
where
ˉa=T−1∫T0a(τ)dτ=a0γT(e−γT/2−e−γT)+a02. | (52) |
Using the bound
In order to determine a parameter set such that the inequalities (46) hold, we first compute
ζ1=γ−1T−1[−α1v2−α1w12+a0α1c−12s21−k3β1γT+(1−k3)β1v4+β1(1−k3)w13−a0β1(1−k3)c−13s31−m1γT/2−m1(e−γT/2−1)+m1(eγT/2−1)(e−γT−e−γT/2)], | (53) |
where
vi=ln(a0e−γT/2+kia0+ki),wij=ln(cie−γT+cjcie−γT/2+cj),sij=ln(cieγT+cjcieγT/2+cj), |
and
c1=a0(1−eγT/2),c2=a0+k2,c3=a0+k4. |
Numerically, we find a critical value
We next use the results in Theorem 4.1 to derive a closed form expression that approximates the relapse time
ˆx(1)1(t)=x1(0)et/τ1,0≤t<γ−1, and | (54) |
ˆx(2)1(t)=ˆx(1)1(γ−1)e−|ζ1|(t−γ−1),t≥γ−1, | (55) |
where
τ−11=α1p1(a0)−β1q1(a0)=α1a0(a0+k2)−1−β1[k3+(1−k3)a0(a0+k4)−1]. | (56) |
We next approximate
ˆx(1)2(t)=(x2(0)−θ1)eb1t+θ1et/τ1,0≤t<γ−1, and | (57) |
ˆx(2)2(t)=(ˆx(1)2(γ−1)−θ2)eb2(t−γ−1)+θ2e−|ζ1|(t−γ−1),γ−1≤t<T/2, | (58) |
where the constants are defined as
b1=α2p2(a0)−β2=α2(1−κ)−β2,b2=α2p2(0)−β2=α2−β2, | (59) |
θ1=m1x1(0)[τ−11−b1]−1,θ2=m1ˆx(1)1(γ−1)[|ζ1|−b2]−1. | (60) |
For
TR≈1α2−β2ln(¯yˆx(1)2(γ−1)−θ2)+γ−1. | (61) |
For
˜x(1)2(t)=ˆx(2)2(T)exp{(α2−β2)(t−n+12T)},nT≤t<(n+12)T, and | (62) |
˜x(2)2(t)=ˆx(2)2(T)exp{(α2−β2)12nT},(n+12)T≤t<(n+1)T, | (63) |
where
TR≈1α2−β2ln(¯yˆx(2)2(T))+Nc+12T, | (64) |
where
Nc=⌈2T(α2−β2)ln(¯yˆx(2)2(T))⌉. | (65) |
Figure 15 shows comparisons of the actual and approximated relapse times from (61) and (64) as
To approximate the relapse time for the continuous model, where we can solve (1) analytically, we have
limT→∞|ζ1|=β1(k3+12c−13a0(1−k3))−12c−12α1−12m1. | (66) |
Figure 16 shows the relapse time for
For a generic parameter
ΥTRθ:=∂TR∂θ×θTR. | (67) |
We use the sensitivity index to investigate the sensitivity of the relapse time to the parameters in the model and summarize these results in Figures 17 and 18 and Table 3. For the periodic model we compute
Parameter | Periodic (ⅰ) | Periodic (ⅱ) | Continuous |
For the continuous model we compute
Table 3 displays the mean values of the sensitivity indices. In all cases, the mean and median values were very close. These values inform us as to how changes in each parameter influence relapse time. For example, if
In this paper we have investigated mathematical models that describe the treatment of advanced prostate cancer by androgen suppression. Our dynamical system models, derived from the hybrid dynamical system presented in [9], allow us to simulate continuous androgen suppression (CAS), intermittent androgen suppression (IAS), and periodic androgen suppression (PAS). These models describe the behavior of the androgen dynamics as well as the dynamics of both androgen dependent (AD) and non-reversible androgen independent (AI) cancer cells. We alter feedback mechanism for the treatment control function
Our dynamic sensitivity analysis for the CAS model produced an expected result -compartments in the model were highly sensitive to parameters influencing their growth and decay, but only during times when the solution variables for those compartments had a large enough magnitude that we could expect their presence to be measurable in a clinical experiment. When considering the dynamic sensitivities for the IAS model, this pattern continues in an expected way as we switch between on-treatment and off-treatment periods of time. However, we learned that the spikes in the sensitivity of androgen (
Unlike intermittent androgen suppression where the on and off treatment is governed by certain threshold levels of total PSA, treatment with periodic androgen suppression requires that the patient switch between on and off treatment at certain intervals defined by the period
In this paper we derived our models using only the foundational model of Ideta, et al., from [9]. In this case, the AI cells are not reversible and the AD cells are independent of the AI cells. This allowed us to approximate relapse time in a relatively straightforward manner as the model equations are only forward coupled. Future work may include investigations of IAS and PAS models where the AI cell types are both reversible and irreversible. In this case, the AD cell proliferation will also depend on the AI cells during the off-treatment cycle as described in [8], and a more detailed analysis of the relapse time is necessary as the model equations are backward coupled.
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1. | Tin Phan, Sharon M. Crook, Alan H. Bryce, Carlo C. Maley, Eric J. Kostelich, Yang Kuang, Review: Mathematical Modeling of Prostate Cancer and Clinical Application, 2020, 10, 2076-3417, 2721, 10.3390/app10082721 | |
2. | Yixuan Zou, Fei Tang, Jeffery C. Talbert, Chee M. Ng, Chih-Pin Chuu, Using medical claims database to develop a population disease progression model for leuprorelin-treated subjects with hormone-sensitive prostate cancer, 2020, 15, 1932-6203, e0230571, 10.1371/journal.pone.0230571 | |
3. | Huan Yang, Yuanshun Tan, Dynamic behavior of prostate cancer cells under antitumor immunity and pulse vaccination in a random environment, 2021, 105, 0924-090X, 2645, 10.1007/s11071-021-06614-w |
Parameter | Variable | Baseline Value | Range |
Normal androgen level | 30 nmol/l | 26.25-33.75 nmol/l | |
Androgen concentration rate | 0.08 days |
0.0425-0.1175 days |
|
Androgen dependent proliferation rate | 0.0204 days |
0.0129-0.0279 days |
|
Androgen dependent apoptosis rate | 0.0076 days |
0.00685-0.00835 days |
|
Androgen independent proliferation rate | 0.0242 days |
0.0216-0.0268 days |
|
Androgen independent apoptosis rate | 0.0168 days |
0.0130-0.0206 days |
|
Maximum mutation rate | 0.00005 days |
1.25-8.75 |
|
AD proliferation half-saturation level | 2 nmol/l | 1.25-2.75 nmol/l | |
AD androgen free apoptosis constant | 8 | 7.25-8.75 | |
AD apoptosis rate half-saturation level | 0.5 nmol/l | 0.275-0.725 nmol/l | |
Minimum PSA concentration | 10 ng/ml | 6.5-13.5 ng/ml | |
Maximum PSA concentration | 15 ng/ml | 11.5-18.5 ng/ml | |
Treatment transition rate | 1100 days |
500-1700 days |
Parameter | Periodic (ⅰ) | Periodic (ⅱ) | Continuous |
Parameter | Variable | Baseline Value | Range |
Normal androgen level | 30 nmol/l | 26.25-33.75 nmol/l | |
Androgen concentration rate | 0.08 days |
0.0425-0.1175 days |
|
Androgen dependent proliferation rate | 0.0204 days |
0.0129-0.0279 days |
|
Androgen dependent apoptosis rate | 0.0076 days |
0.00685-0.00835 days |
|
Androgen independent proliferation rate | 0.0242 days |
0.0216-0.0268 days |
|
Androgen independent apoptosis rate | 0.0168 days |
0.0130-0.0206 days |
|
Maximum mutation rate | 0.00005 days |
1.25-8.75 |
|
AD proliferation half-saturation level | 2 nmol/l | 1.25-2.75 nmol/l | |
AD androgen free apoptosis constant | 8 | 7.25-8.75 | |
AD apoptosis rate half-saturation level | 0.5 nmol/l | 0.275-0.725 nmol/l | |
Minimum PSA concentration | 10 ng/ml | 6.5-13.5 ng/ml | |
Maximum PSA concentration | 15 ng/ml | 11.5-18.5 ng/ml | |
Treatment transition rate | 1100 days |
500-1700 days |
Parameter | Periodic (ⅰ) | Periodic (ⅱ) | Continuous |