Multilevel thresholding using a modified ant lion optimizer with opposition-based learning for color image segmentation

Shikai Wang; Kangjian Sun; Wanying Zhang; Heming Jia; Shikai Wang; Kangjian Sun; Wanying Zhang; Heming Jia

doi:10.3934/mbe.2021155

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 4: 3092-3143. doi: 10.3934/mbe.2021155

Previous Article Next Article

Research article Special Issues

Multilevel thresholding using a modified ant lion optimizer with opposition-based learning for color image segmentation

1.
School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China
2.
College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China
3.
College of Information Engineering, Sanming University, Sanming 365004, China

Received: 06 February 2021 Accepted: 25 March 2021 Published: 02 April 2021

Multilevel thresholding has important research value in image segmentation and can effectively solve region analysis problems of complex images. In this paper, Otsu and Kapur's entropy are adopted among thresholding segmentation methods. They are used as the objective functions. When the number of threshold increases, the time complexity increases exponentially. In order to overcome this drawback, a modified ant lion optimizer algorithm based on opposition-based learning (MALO) is proposed to determine the optimum threshold values by the maximization of the objective functions. By introducing the opposition-based learning strategy, the search accuracy and convergence performance are increased. In addition to IEEE CEC 2017 benchmark functions validation, 11 state-of-the-art algorithms are selected for comparison. A series of experiments are conducted to evaluate the segmentation performance of the algorithm. The evaluation metrics include: fitness value, peak signal-to-noise ratio, structural similarity index, feature similarity index, and computational time. The experimental data are analyzed and discussed in details. The experimental results significantly demonstrate that the proposed method is superior over others, which can be considered as a powerful and efficient thresholding technique.

Keywords:

Citation: Shikai Wang, Kangjian Sun, Wanying Zhang, Heming Jia. Multilevel thresholding using a modified ant lion optimizer with opposition-based learning for color image segmentation[J]. Mathematical Biosciences and Engineering, 2021, 18(4): 3092-3143. doi: 10.3934/mbe.2021155

Related Papers:

[1]	Pragati Gautam, Vishnu Narayan Mishra, Rifaqat Ali, Swapnil Verma . Interpolative Chatterjea and cyclic Chatterjea contraction on quasi-partial $b$ -metric space. AIMS Mathematics, 2021, 6(2): 1727-1742. doi: 10.3934/math.2021103
[2]	Afrah Ahmad Noman Abdou . Chatterjea type theorems for complex valued extended $b$ -metric spaces with applications. AIMS Mathematics, 2023, 8(8): 19142-19160. doi: 10.3934/math.2023977
[3]	Jamshaid Ahmad, Abdullah Eqal Al-Mazrooei, Hassen Aydi, Manuel De La Sen . Rational contractions on complex-valued extended $b$ -metric spaces and an application. AIMS Mathematics, 2023, 8(2): 3338-3352. doi: 10.3934/math.2023172
[4]	Naeem Saleem, Salman Furqan, Mujahid Abbas, Fahd Jarad . Extended rectangular fuzzy $b$ -metric space with application. AIMS Mathematics, 2022, 7(9): 16208-16230. doi: 10.3934/math.2022885
[5]	Naeem Saleem, Hüseyin Işık, Sana Khaleeq, Choonkil Park . Interpolative Ćirić-Reich-Rus-type best proximity point results with applications. AIMS Mathematics, 2022, 7(6): 9731-9747. doi: 10.3934/math.2022542
[6]	Afrah A. N. Abdou, Maryam F. S. Alasmari . Fixed point theorems for generalized $\alpha$ - $\psi$ -contractive mappings in extended $b$ -metric spaces with applications. AIMS Mathematics, 2021, 6(6): 5465-5478. doi: 10.3934/math.2021323
[7]	Wasfi Shatanawi, Taqi A. M. Shatnawi . Some fixed point results based on contractions of new types for extended $b$ -metric spaces. AIMS Mathematics, 2023, 8(5): 10929-10946. doi: 10.3934/math.2023554
[8]	Menaha Dhanraj, Arul Joseph Gnanaprakasam, Gunaseelan Mani, Rajagopalan Ramaswamy, Khizar Hyatt Khan, Ola Ashour A. Abdelnaby, Stojan Radenović . Fixed point theorem on an orthogonal extended interpolative $\psi\mathcal{F}$ -contraction. AIMS Mathematics, 2023, 8(7): 16151-16164. doi: 10.3934/math.2023825
[9]	Yan Han, Shaoyuan Xu, Jin Chen, Huijuan Yang . Fixed point theorems for $b$ -generalized contractive mappings with weak continuity conditions. AIMS Mathematics, 2024, 9(6): 15024-15039. doi: 10.3934/math.2024728
[10]	Hongyan Guan, Jinze Gou, Yan Hao . On some weak contractive mappings of integral type and fixed point results in $b$ -metric spaces. AIMS Mathematics, 2024, 9(2): 4729-4748. doi: 10.3934/math.2024228

Abstract

1. Introduction

Breast cancer is the second most commonly diagnosed cancer in women worldwide. Approximately 75% of breast cancers are estrogen receptor positive (ER+) ^[1]. Estrogen receptors are activated by estradiol binding leading to tumor growth. Standard treatment for ER+ breast cancer involves the use of endocrine therapy after surgery or chemotherapy ^[2,3] to reduce the risk of breast cancer relapse and lead to prolonged life. There are two types of Endocrine therapy for breast cancer including drugs that lower estrogen levels and drugs that block ERs on breast cancer cells. The aromatase inhibitors (AIs) inhibit estrogen biosynthesis leading to reduction of estradiol (E2) levels. The selective ER modulators (SERMs) block ERs and reduce ER activity. The best known SERM is tamoxifen which binds to the ER and modulates its function leading to reduction of tumor cell division rate and slows tumor growth. Tamoxifen has been proven to be effective but most patients eventually develop resistance resulting in tumor recurrence ^[2,3,4,5,6]. It has been reported that tamoxifen may switch the mode of action from antagonism to agonism promoting turmor growth ^[7].

Fulvastrant is the first selective ER degrader (SERD), a pure antagonist which blocks and damages ERs, approved by FDA in 2002 ^[8]. It has been shown that fulvastrant has higher affinity for ERs than tamoxifen ^[3] and has been proven to be effective in patients with ER+ breast cancer after progression on other endocrine therapies such as tamoxifen or AIs ^[4,9]. The maximum feasible dose for fulvestrant is 500 mg given by monthly intramuscular injections ^[7,10,11]. However, fulvestrant gives low bioavailability and a long period of time (3-6 months) to reach steady state plasma concentration ^[5,10]. These disadvantages of fulvestrant result in limitations in its clinical benefit. Therefore, there has been a compelling clinical need to develop oral SERDs to overcome these limitations ^[4,9].

AZD9496 is a new oral SERD and is a potent antagonist and degrader of ERs with high oral bioavailability, rapid absorption ^[5,9,10]. Cyclin-dependent kinase 4 and 6 (CDK4/6) inhibitors in combination with endocrine therapy have shown increased progression-free survival in ER+ breast cancer compared with endocrine therapy alone ^[5,12,13]. Experimental study by Weir et al ^[5] has shown that AZD9496 gives greater tumor growth inhibition compared with fulvestrant. Further study has shown tumor regression in combination therapy using AZD9496 and CDK4/6 ^[5].

A mathematical model of tumor growth in MCF-7 breast cancer cell line with interaction among the cancer cells and immune cells has been developed by Wei ^[14]. The model was based on several experimental results using MCF-7 breast cancer cells. Multistability which resembles the 3 E's of cancer immunoediting (elimination, equilibrium, and escape) was observed in the mathematical model. In this study, treatment terms are to be incorporated into the mathematical model for the study of outcome of treatment using combination therapy with AZD9496 and CDK4/6. To provide details, the mathematical model is presented in Section 2. Numerical simulations and discussion are given in Section 3. Finally, a brief conclusion is made in Section 4.

2. Mathematical model

2.1. Formulation of AZD9496 dynamics

A first-in-human study of AZD9496 has reported a mean terminal half-life ( $t_{1/2}$ ) of 1.4 to 5.7 hours and a rapid absorption with a median $t_{max}$ , the time to reach the peak plasma concentration, of 1.33-3.00 hours ^{[9, 10]}. Let $U$ in $mg$ be the amount of AZD9496 in the body but has not entered the circulation, $Z$ in $mg$ the amount of AZD9496 in the circulation, $\alpha_7$ the absorption rate constant, and $\beta_7$ the elimination rate constant. The dynamics of AZD9496 are described by the following equations:

$\begin{eqnarray} \frac{dU}{dt} & = & -\alpha_7U, \end{eqnarray}$

(2.1)

$\begin{eqnarray} \frac{dZ}{dt} & = & \alpha_7U-\beta_7Z. \end{eqnarray}$

(2.2)

The value for $\beta_7$ is determined by $\ln2/t_{1/2}$ . If $t_{1/2} = 3.5$ , then $\beta_7 = \ln2/(3.5/24) = 4.7541$ . The value for $\alpha_7$ is obtained by solving Eqs. (2.1) and (2.2) followed by setting $Z^{\prime}(t_{max}) = 0$ to find $\alpha_7$ . This gives $\alpha_7 = 24.3659$ .

2.2. Formulation of palbociclib dynamics

Palbociclib, which has been shown to arrest tumor growth, is a CDK4/6 inhibitor administered orally and absorbed slowly from intestine in 6-12 hours with a median $t_{max} = 5.5$ ranging from 2.0 to 9.8, a terminal terminal half life of $25.9\pm 7.5$ (mean $\pm$ SD) hours, and a bioavailability of 46% ^[15]. Let $P$ in $mg$ be the amount of palbociclib in the body but has not entered the circulation, $Q$ in $mg$ the amount of palbociclib in the circulation, $\alpha_8$ the absorption rate constant, and $\beta_8$ the elimination rate constant. The dynamics of palbociclib are described by the following equations:

$\begin{eqnarray} \frac{dP}{dt} & = & -\alpha_8P, \end{eqnarray}$

(2.3)

$\begin{eqnarray} \frac{dQ}{dt} & = & k_1\alpha_8P-\beta_8Q. \end{eqnarray}$

(2.4)

The parameter values $\alpha_8 = 14.1512$ and $\beta_8 = \ln2/1.08 = 0.64$ are obtained in a similar way used in Section 2.1. Because the bioavailability of palbociclib is 46%, about half amount of the drug enters the circulation and $k_1 = 0.5$ is used throughout the paper.

2.3. Mathematical model with treatment

Let $\alpha_9$ and $\alpha_{10}$ be the constants for tumor growth inhibition induced by palbociclib and AZD9496, respectively. The combination treatment using AZD9496 and palbociclib incorporated into the mathematical model proposed by Wei ^[14] is as follows:

$\begin{eqnarray} \frac{dT}{dt} & = & T(ae^{-\alpha_9Q}+\frac{ce^{-\alpha_{10}Z}ET}{1+\alpha_1 E+\beta_1T^2}) (1-T/K)-\frac{p_1TN^2}{1+\alpha_2T+\beta_2N^2} \\ & &-\frac{p_6T^2L}{1+\alpha_6T^2+\beta_6L}. \end{eqnarray}$

(2.5)

$\begin{eqnarray} \frac{dN}{dt} & = & eC-fN-p_2NT+\frac{p_3NT}{1+\alpha_3T+\beta_3N}, \end{eqnarray}$

(2.6)

$\begin{eqnarray} \frac{dC}{dt} & = & \alpha -\beta C, \end{eqnarray}$

(2.7)

$\begin{eqnarray} \frac{dL}{dt} & = & (p_4L_N+\frac{p_5I}{\alpha_4+I}L)(1-L/K_L)\frac{T}{\alpha_5+T}-dL, \end{eqnarray}$

(2.8)

$\begin{eqnarray} \frac{dU}{dt} & = & -\alpha_7U+\sum\limits_{i = 0}^{M_1}s_i\delta(t_{1i}), \quad 0\leq t_{10} \lt t_{11} \lt \cdots \lt t_{1M_1} \end{eqnarray}$

(2.9)

$\begin{eqnarray} \frac{dZ}{dt} & = & \alpha_7U-\beta_7Z, \end{eqnarray}$

(2.10)

$\begin{eqnarray} \frac{dP}{dt} & = & -\alpha_8P+\sum\limits_{i = 0}^{M_2}v_i\delta(t_{2i}), \quad 0\leq t_{20} \lt t_{21} \lt \cdots \lt t_{2M_2} \end{eqnarray}$

(2.11)

$\begin{eqnarray} \frac{dQ}{dt} & = & 0.5\alpha_8P-\beta_8Q, \end{eqnarray}$

(2.12)

$\begin{eqnarray} E(t) & = & \tilde{E}(t-n\tau), \quad t\in[n\tau, (n+1)\tau), \quad n = 0, 1, 2, \cdots, \end{eqnarray}$

(2.13)

where $E(t)$ is a periodic function and $t$ is in days. The variables and parameter values are summarized in . The parameter values $\alpha_9$ and $\alpha_{10}$ are to be determined using experimental data.

Table 1. Parameter values in Eqs. (2.5)-(2.13).

Parameter/Variable	Value	Units	Description
$T$	Variable	Cell	MCF-7 tumor cell population
$N$	Variable	Cell L $^{-1}$	NK cell population
$C$	Variable	Cell L $^{-1}$	WBC population
$L$	Variable	Cell L $^{-1}$	CTL population
$E$	Variable	pmol L $^{-1}$	Estradiol in circulation
$U$	Variable	mg	AZD9496 not in circulation
$Z$	Variable	mg	AZD9496 in circulation
$P$	Variable	mg	Palbociclib not in circulation
$Q$	Variable	mg	Palbociclib in circulation
$a$	0.066	Day $^{-1}$	Tumor growth rate
$c$	$0.00147$	L Cell $^{-1}$ Day $^{-1}$ pmol $^{-1}$	Tumor growth rate induced by E2
$\alpha_1$	0.507	L pmol $^{-1}$	Half saturation constant
$\beta_1$	$7.08\times 10^{-8}$	Cell $^{-2}$	Half saturation constant
$K$	$10^9$	Cell	Tumor cell carrying capacity
$p_1$	$8.7\times 10^{-4}$	L $^2$ Cell $^{-2}$ Day $^{-1}$	NK induced tumor death
$\alpha_2$	$7\times 10^6$	Cell $^{-1}$	Half saturation constant
$\beta_2$	$5.4\times 10^{-5}$	L $^2$ Cell $^{-2}$	Half saturation constant
$\beta$	$6.3\times 10^{-3}$	Day $^{-1}$	WBC death rate
$\alpha$	$5\times 10^7$	Cell L $^{-1}$ Day $^{-1}$	WBC production rate
$e$	0.00486	Day $^{-1}$	Fraction of WBCs becoming NK cells
$f$	0.0693	Day $^{-1}$	NK cell death rate
$p_2$	$3.42\times 10^{-6}$	Cell Day $^{-1}$	NK cell inactivation by tumor cells
$p_3$	$1.87\times 10^{-8}$	Cell $^{-1}$ Day $^{-1}$	NK cell recruitment rate
$\alpha_3$	$1.6\times 10^{-5}$	Cell $^{-1}$	Half saturation constant
$\beta_3$	3.27	L Cell $^{-1}$	Half saturation constant
$p_6$	$2.04\times 10^{-3}$	L Cell $^{-2}$ Day $^{-1}$	CTL induced tumor death
$\alpha_6$	0.268	Cell $^{-2}$	Half saturation constant
$\beta_6$	4343	L Cell $^{-1}$	Half saturation constant
$L_N$	$2.3\times 10^{8}$	Cell L $^{-1}$	Naive CTL population
$K_L$	$8\times 10^{8}$	Cell L $^{-1}$	CTL carrying capacity
$p_4$	$9\times 10^{-5}$	Day $^{-1}$	Fraction of naive CTL activated
$I$	$2.3\times 10^{-11}$	g L $^{-1}$	IL-2 concentration
$\alpha_4$	$2.3\times 10^{-11}$	g L $^{-1}$	Half saturation constant
$p_5$	$4.14$	L Cell $^{-2}$ Day $^{-1}$	CTL growth rate induced by IL-2
$d$	$0.41$	Day $^{-1}$	CTL death rate
$\alpha_5$	1000	Cell	Half saturation constant
$\alpha_7$	24.3659	Day $^{-1}$	Absorption rate of AZD9496
$\beta_7$	4.7541	Day $^{-1}$	Elimination rate of AZD9496
$\alpha_8$	14.1512	Day $^{-1}$	Absorption rate of palbociclib
$\beta_8$	0.64	Day $^{-1}$	Elimination rate of palbociclib
$\alpha_9$	0.01	mg $^{-1}$	Tumor growth inhibition by palbociclib
$\alpha_{10}$	0.2263	mg $^{-1}$	Tumor growth inhibition by AZD9496
$s_i$	Varies	mg	AZD9496 treatment dosage at $t_{1i}$
$v_i$	Varies	mg	Palbociclib treatment dosage at $t_{2i}$

| Show Table

DownLoad: CSV

A study conducted by Weir et al. ^[5] used an MCF-7 xenograft model to study in vivo efficacy of AZD9496 which was administered orally once daily (QD) at several selected dosages. In this fitting process, the experimental data are the tumor volumes on day 21 after treatment. Since the mathematical model has been developed for human patients, the dosages in mg/kg are converted into those for a patient of 60 kg throughout the paper. Let $x_i$ , $i = 1, 2, \cdots, 6$ be the dosages of ADZ9496 in and $y_i = T(21)$ is the solution to Eqs. (2.5)-(2.13) with the initial condition $(T(0), N(0), C(0), L(0), U(0), Z(0), P(0), Q(0)) = (8.72\times10^7, 2.5\times10^8, 4.3\times10^9, 6.6 \times10^8, 0, 0, 0, 0)$ corresponding to the dosage $x_i$ . Experimental data have shown WBC counts of $(4.3\pm0.8)\times10^3$ cells/ $\mu L$ and lymphocyte counts of $(3.5\pm0.6)\times10^3$ cells/ $\mu L$ of in mice ^[16]. Berrington et al. ^[17] have reported that 7% (2-13%) and 19% (13-32%) of lymphocytes are NK cells and CTLs, repectively. Therefore, $C(0) = 4.3\times10^9$ , $N(0) = 3.5\times 10^9\times 0.07 = 2.5\times10^8$ and $L(0) = 3.5\times 10^9\times 0.19 = 6.6\times10^8$ are used in this fitting process. The parameter value for $\alpha_{10}$ is to be determined. It has been found in this work that Eqs. (2.5)-(2.13) with parameter values given in ^[14] poorly fit the data set. Therefore, parameters $a$ , $c$ , and $\alpha_{10}$ are used as fitting parameters. The result of the fitting process gives $a = 0.066$ , $c = 0.00147$ , and $\alpha_{10} = 0.2263$ . Figure 1 shows the experimental data and the fitting curve.

Figure 1. Curve fitting for the parameter values

$a$ ,

$c$ , and

$\alpha_{10}$ .

DownLoad: Full-Size Img PowerPoint

Weir et al. ^[5] also conducted an experiment to study the efficacy of combination therapy using AZD9496 and palbociclib. The result has shown that a tumor with an initial volume of 0.42 cm $^3$ (about $10^8$ cells) reduced to that with 0.25 cm $^3$ (about $6\times 10^7$ cells) after 18 days when treated with AZD9496 at 5 mg/kg QD and palbociclib at 50 mg/kg QD. Consider that the maximum tolerated dose (MTD) of palbociclib is 125 mg. The combination doses at 300 mg QD for AZD9496 and 125 mg QD for palbociclib are selected. The parameter value $\alpha_9$ is determined to satisfy $T(18) = 6\times 10^7$ by solving Eqs. (2.5)-(2.13) with $T(0) = 10^8$ . This results in $\alpha_9 = 0.01003$ .

3. Numerical simulation and discussion

3.1. Tumor dynamics without treatment

The system under study has a smaller intrinsic tumor growth rate, $a$ , and a larger factor, $c$ , of tumor growth induced by E2 than those in ^[14]. Recall that the system using parameter values in ^[14] exhibits three stable equilibria, and the immune system is able to eliminate or control a tumor smaller than 2 mm in diameter. The system under study exhibits two stable equilibria without treatment (Figure 2). They are tumor free equilibrium and large tumor equilibrium. and show that the immune system is able to eliminate a tumor of $10^3$ cells ( $\approx$ 0.2 mm in diameter) while a tumor of $10^4$ cells ( $\approx$ 0.43 mm in diameter) grows large with time. Tumor population dynamics depend strongly on tumor cell properties and individual variations.

Figure 2. Tumor dynamics with initial conditions (a)

$(10^3, 4\times10^8, 8\times10^9, 8\times10^8, 0, 0, 0, 0)$ and (b)

$(10^4, 4\times10^8, 8\times10^9, 8\times10^8, 0, 0, 0, 0)$ . Parameter values are shown in Table 1.

DownLoad: Full-Size Img PowerPoint

3.2. Tumor dynamics with monotherapy using AZD9496

Based on an MCF-7 xenograft model in mice treated with AZD9496, it was reported that a dose of 5 mg/kg is the minimum dose required for significant tumor inhibition ^[9]. A dose $s_i = 300$ QD is selected in the simulation and the results are shown in Figures 3(a)-(c). - show that the treatment with AZD9496 at a dose of 300 mg QD is able to eliminate a tumor of $10^6$ cells ( $\approx$ 2 mm in diameter) and control a tumor of $10^7$ cells ( $\approx$ 4 mm in diameter) but the treatment fails to control a tumor of $10^8$ cells ( $\approx$ 9 mm in diameter). Increasing the dose $s_i$ to 3000 mg QD still fails to control a tumor of $10^8$ cells (). Experimental data ^[5] have shown little difference in inhibition of tumor growth between AZD9496 doses of 5 mg/kg ( $s_i = 300$ ) and 50 mg/kg ( $s_i = 3000$ ). Similar results have also been observed by ^[4].

Figure 3. Tumor dynamics of monotherapy using AZD9496 with selected initial tumor cell population levels and dosages. (a)

$T(0) = 10^6$ and

$s_i = 300$ , (b)

$T(0) = 10^7$ and

$s_i = 300 QD$ , (c)

$T(0) = 10^8$ and

$s_i = 300$ QD, and (d)

$T(0) = 10^8$ and

$s_i = 3000$ QD. Parameter values are shown in Table 1.

DownLoad: Full-Size Img PowerPoint

3.3. Tumor dynamics with monotherapy using palbociclib

The maximum dose of palbociclib used in an experimental study by Weir et al. ^[5] is 50 mg/kg ( $v_i = 3000$ ) QD, which is a high dose. Figure 4 shows that monotherapy using palbociclib is ineffective. This agrees with the results in the experimental study by Weir et al. ^[5].

Figure 4. Tumor dynamics of monotherapy using palbociclib with

$T(0) = 10^4$ and

$v_i = 3000$ QD. Parameter values are shown in Table 1.

DownLoad: Full-Size Img PowerPoint

3.4. Tumor dynamics with combination therapy

Combination therapy allows the use of lower dosages of each therapy drug to reduce toxicity and overcome resistance. It can also produce synergistic effect and enhance response. In this simulation, doses $s_i = 300$ (AZD9496) QD and $v_i = 125$ (palbociclib) QD are selected. shows that the combination therapy can produce synergistic effect and control a tumor of $10^8$ cells which would otherwise grow large when treated with monotherapy using AZD9496 or palbociblib alone.

Figure 5. Tumor dynamics of combination therapy using AZD9496 and palbociclib with

$T(0) = 10^8$ ,

$s_i = 300$ QD, and

$v_i = 125$ QD. Parameter values are shown in Table 1.

DownLoad: Full-Size Img PowerPoint

3.5. Variation in treatment schedule

First-in-human studies have been conducted to determine safety and tolerability for AZD9496 ^{[9, 10]}. Patients received AZD9496 in a dose escalation design with 20 mg QD to 600 mg twice daily (BID). Although the maximum tolerated dose was not reached, Hamilton et al. ^[9] has recommended the 250 mg BID dose for subsequent AZD9496 study. shows that monotherapy using AZD9496 alone at 250 mg BID is able to eliminate a tumor of $10^7$ cells. This treatment has a better outcome than that of the treatment at 500 mg QD (). Dividing a single daily dose into two smaller doses and administered twice daily can be more effective. Furthermore, shows that AZD9496 250 mg BID combined with palbociclib 125 mg QD is able to eliminated a tumor of $10^8$ cells.

Figure 6. Tumor dynamics of combination therapy using AZD9496 twice daily and palbociclib once daily with (a)

$T(0) = 10^7$ and

$s_i = 250$ BID, (b)

$T(0) = 10^7$ and

$s_i = 500$ QD, and (c)

$T(0) = 10^8$ and

$s_i = 250$ BID, and

$v_i = 125$ QD. Parameter values are shown in Table 1.

DownLoad: Full-Size Img PowerPoint

3.6. Variation in tumor cell proliferation rate

The doubling time of MCF-7 breast cancer cells ranges between 30 hours and several days ^[18,19]. A doubling time of 30 hours is corresponding to $a = ln2/(30/24) = 0.55$ while that of 10 days is corresponding to $a = ln2/10 = 0.069$ . There is a large variation in the intrinsic tumor growth rate $a$ . Assume a breast cancer with $a = 0.2$ , where the doubling time is about 3.5 days. shows that combination therapy with doses $s_i = 300$ (AZD9496) QD and $v_i = 125$ (palbociclib) QD is not able to control a tumor of $10^8$ cells. Increasing the dose of AZD9496 is not effective (figure not shown). shows that increasing the dose of palbociclib to 250 mg QD is able to control a tumor of $10^8$ cells. However, this dose is beyond the MTD of palbociclib.

Figure 7. Tumor dynamics of combination therapy using AZD9496 and palbociclib with (a)

$T(0) = 10^8$ ,

$s_i = 300$ QD, and

$v_i = 125$ QD, and (b)

$T(0) = 10^8$ ,

$s_i = 300$ QD, and

$v_i = 250$ QD. Parameter values are shown in Table 1 except for

$a = 0.2$ .

DownLoad: Full-Size Img PowerPoint

4. Conclusion

This paper, which is based on a previously developed mathematical model ^[14], studies a combination therapy using AZD9496 as a degrader of ERs and palbociclib as a CDK4/6 inhibitor. Treatment terms are modeled based on clinical and experimental data and information of pharmacokinetic parameters of the above drugs ^[5,9,10,15].

Numerical simulation shows that monotherapy using AZD9496 alone can help control a tumor with a restricted size while using palbociclib alone is ineffective. This has also been observed in the experimental study conducted by Weir ^[5] using an MCF-7 xenograft model in mice. A clinical trial has reported that some patients receiving AZD9496 had stable cancer ^[9]. Combination therapy using AZD9496 and palbociclib can generate synergistic effect and induce tumor regression for a larger tumor, which would otherwise grow large when monotherapy with either AZD9496 or palbociclib is used. This also agrees with the experimental results in ^[5].

A change in dosing schedule may affect the outcome of a treatment. Dividing a large single dose into two smaller doses may be more effective. Finally, an example with a larger tumor cell proliferation rate is studied. Numerical simulation shows that combination therapy can inhibit tumor growth if a lager dose, which is beyond MTD, of palbociclib is used. It is suggested that triple therapy using immunotherapy and the above two drugs may be able to treat the tumor with safety and efficacy.

Acknowledgments

This research work was supported by the Ministry of Science and Technology of Taiwan under the grant MOST107-2115-M-035-006.

Conflict of interest

The author declares no conflicts of interest in this paper.

References

[1]	N. M. Zaitoun, M. J. Aqel, Survey on Image Segmentation Techniques, Procedia Comput. Sci., 65 (2015), 797-806. doi: 10.1016/j.procs.2015.09.027
[2]	M. Sridevi, C. Mala, A Survey on Monochrome Image Segmentation Methods, Procedia Technol., 6 (2012), 548-555. doi: 10.1016/j.protcy.2012.10.066
[3]	A. K. M. Khairuzzaman, S. Chaudhury, Multilevel thresholding using grey wolf optimizer for image segmentation, Expert Syst. Appl., 86 (2017), 64-76. doi: 10.1016/j.eswa.2017.04.029
[4]	J. Tang, Y. Wang, C. Huang, H. Liu, N. Al-Nabhan, Image edge detection based on singular value feature vector and gradient operator, Math. Biosci. Eng., 17 (2020), 3721-3735. doi: 10.3934/mbe.2020209
[5]	X. Song, Y. Wang, Q. Feng, Q. Wang, Improved graph cut model with features of superpixels and neighborhood patches for myocardium segmentation from ultrasound image, Infinite Study, 2019.
[6]	X. Lu, Z. You, M. Sun, J. Wu, Z. Zhang, Breast cancer mitotic cell detection using cascade convolutional neural network with U-Net, Math. Biosci. Eng., 18 (2021), 673-695. doi: 10.3934/mbe.2021036
[7]	H. Jia, K. Sun, W. Song, X. Peng, C. Lang, Y. Li, Multi-Strategy Emperor Penguin Optimizer for RGB Histogram-Based Color Satellite Image Segmentation Using Masi Entropy, IEEE Access, 7 (2019), 134448-134474. doi: 10.1109/ACCESS.2019.2942064
[8]	S. Wang, H. Jia, X. Peng, Modified salp swarm algorithm based multilevel thresholding for color image segmentation, Math. Biosci. Eng., 17 (2019), 700-724.
[9]	A. Dirami, K. Hammouche, M. Diaf, P. Siarry, Fast multilevel thresholding for image segmentation through a multiphase level set method, Signal Process., 93 (2013), 139-153. doi: 10.1016/j.sigpro.2012.07.010
[10]	E. Hamuda, M. Glavin, E. Jones, A survey of image processing techniques for plant extraction and segmentation in the field, Comput. Electron. Agric., 125 (2016), 184-199. doi: 10.1016/j.compag.2016.04.024
[11]	S. Kotte, R. K. Pullakura, S. K. Injeti, Optimal multilevel thresholding selection for brain MRI image segmentation based on adaptive wind driven optimization, Measurement, 130 (2018), 340-361. doi: 10.1016/j.measurement.2018.08.007
[12]	N. Otsu, A threshold selection method from gray-level histograms, IEEE Trans. Syst. Man Cybern., 9 (1979), 62-66. doi: 10.1109/TSMC.1979.4310076
[13]	J. N. Kapur, P. Sahoo, A. K. C. Wong, A new method for gray-level picture thresholding using the entropy of the histogram, Comput. Vis. Graph Image Process., 29 (1985), 273-285. doi: 10.1016/0734-189X(85)90125-2
[14]	A. K.Bhandari, V. K. Singh, A. Kumar, G. K. Singh, Cuckoo search algorithm and wind driven optimization based study of satellite image segmentation for multilevel thresholding using Kapur's entropy, Expert Syst. Appl., 41 (2014), 3538-3560. doi: 10.1016/j.eswa.2013.10.059
[15]	M. A. E. Aziz, A. A. Ewees, A. E. Hassanien, Whale Optimization Algorithm and Moth-Flame Optimization for multilevel thresholding image segmentation, Expert Syst. Appl., 83 (2017), 242-256. doi: 10.1016/j.eswa.2017.04.023
[16]	K. P. Baby Resma, M. S. Nair, Multilevel thresholding for image segmentation using Krill Herd Optimization algorithm, J. King Saud Univ. Comput. Inf. Sci., (2018), forthcoming.
[17]	A. Ibrahim, A. Ahmed, S. Hussein, A. E. Hassanien, Fish image segmentation using Salp Swarm Algorithm, in The International Conference on Advanced Machine Learning Technologies and Applications (AMLTA2018), Springer, (2018), 42-51.
[18]	S. Ouadfel, A. Taleb-Ahmed, Social spiders optimization and flower pollination algorithm for multilevel image thresholding: A performance study, Expert Syst. Appl., 55 (2016), 566-584. doi: 10.1016/j.eswa.2016.02.024
[19]	M. Díaz-Cortés, N. Ortega-Sánchez, S. Hinojosa, D. Oliva, E. Cuevas, R. Rojas, et al., A multi-level thresholding method for breast thermograms analysis using Dragonfly algorithm, Infrared Phys. Technol., 93 (2018), 346-361. doi: 10.1016/j.infrared.2018.08.007
[20]	S. C. Satapathy, N. Sri Madhava Raja, V. Rajinikanth, A. S. Ashour, N. Dey, Multi-level image thresholding using Otsu and chaotic bat algorithm, Neural Comput. Appl., 29 (2018), 1285-1307. doi: 10.1007/s00521-016-2645-5
[21]	M. Salvi, F. Molinari, Multi-tissue and multi-scale approach for nuclei segmentation in H & E stained images, BioMed. Eng. OnLine., 17 (2018), 89. doi: 10.1186/s12938-018-0518-0
[22]	Y. Feng, H. Zhao, X. Li, X. Zhang, H. Li, A multi-scale 3D Otsu thresholding algorithm for medical image segmentation, Digital Signal Process., 60 (2017), 186-199. doi: 10.1016/j.dsp.2016.08.003
[23]	D. Zhao, L. Liu, F. Yu, A. A. Heidari, M. Wang, G. Liang, et al., Chaotic random spare ant colony optimization for multi-threshold image segmentation of 2D Kapur entropy, Knowl. Based Syst., 216 (2021), 106510. doi: 10.1016/j.knosys.2020.106510
[24]	D. Zhao, L. Liu, F. Yu, A. A. Heidari, M. Wang, D. Oliva, et al., Ant colony optimization with horizontal and vertical crossover search: Fundamental visions for multi-threshold image segmentation, Expert Syst. Appl., 167 (2021), 114122. doi: 10.1016/j.eswa.2020.114122
[25]	L. He, S. Huang, An efficient krill herd algorithm for color image multilevel thresholding segmentation problem, Appl. Soft Comput., 89 (2020), 106063. doi: 10.1016/j.asoc.2020.106063
[26]	I. Hilali-Jaghdam, A. B. Ishak, S. Abdel-Khalek, A. Jamal, Quantum and classical genetic algorithms for multilevel segmentation of medical images: A comparative study, Comput. Commun., 162 (2020), 83-93. doi: 10.1016/j.comcom.2020.08.010
[27]	B. Wu, J. Zhou, X. Ji, Y. Yin, X. Shen, An ameliorated teaching-learning-based optimization algorithm based study of image segmentation for multilevel thresholding using Kapur's entropy and Otsu's between class variance, Inf. Sci., 533 (2020), 72-107. doi: 10.1016/j.ins.2020.05.033
[28]	S. Mirjalili, The Ant Lion Optimizer, Adv. Eng. Software, 83 (2015), 80-98. doi: 10.1016/j.advengsoft.2015.01.010
[29]	M. J. Hadidian-Moghaddam, S. Arabi-Nowdeh, M. Bigdeli, D. Azizian, A multi-objective optimal sizing and siting of distributed generation using ant lion optimization technique, Ain Shams Eng. J., 9 (2018), 2101-2109. doi: 10.1016/j.asej.2017.03.001
[30]	M. Raju, L. C. Saikia, N. Sinha, Automatic generation control of a multi-area system using ant lion optimizer algorithm based PID plus second order derivative controller, Int. J. Electr. Power Energy Syst., 80 (2016), 52-63. doi: 10.1016/j.ijepes.2016.01.037
[31]	P. Saxena, A. Kothari, Ant Lion Optimization algorithm to control side lobe level and null depths in linear antenna arrays, Int. J. Electron. Commun., 70 (2016), 1339-1349. doi: 10.1016/j.aeue.2016.07.008
[32]	E. Umamaheswari, S. Ganesan, M. Abirami, S. Subramanian, Cost Effective Integrated Maintenance Scheduling in Power Systems using Ant Lion Optimizer, Energy Procedia, 117 (2017), 501-508. doi: 10.1016/j.egypro.2017.05.176
[33]	P. D. P. Reddy, V. C. V. Reddy, T. G. Manohar, Ant Lion optimization algorithm for optimal sizing of renewable energy resources for loss reduction in distribution systems, J. Electr. Syst. Inf. Technol., 5 (2018), 663-680. doi: 10.1016/j.jesit.2017.06.001
[34]	D. Oliva, S. Hinojosa, M. A. Elaziz, N. Ortega-Sánchez, Context based image segmentation using antlion optimization and sine cosine algorithm, Multimedia Tools Appl., 77 (2018), 25761-25797. doi: 10.1007/s11042-018-5815-x
[35]	C. Jin, Z. Ye, L. Yan, Y. Cao, A. Zhang, L. Ma, et al., Image Segmentation Using Fuzzy C-means Optimized by Ant Lion Optimization, in 2019 10th IEEE International Conference on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications (IDAACS), IEEE, (2019), 388-393.
[36]	X. Yue, H. Zhang, A Novel Industrial Image Contrast Enhancement Technique Based on an Improved Ant Lion Optimizer, Arab J. Sci. Eng., 46 (2021), 3235-3246. doi: 10.1007/s13369-020-05148-4
[37]	Z. Wu, D. Yu, X. Kang, Parameter identification of photovoltaic cell model based on improved ant lion optimizer, Energy Convers. Manage., 151 (2017), 107-115. doi: 10.1016/j.enconman.2017.08.088
[38]	K. R. Subhashini, J. K. Satapathy, Development of an Enhanced Ant Lion Optimization Algorithm and its Application in Antenna Array Synthesis, Appl. Soft Comput., 59 (2017), 153-173. doi: 10.1016/j.asoc.2017.05.007
[39]	S. K. Majhi, S. Biswal, Optimal cluster analysis using hybrid K-Means and Ant Lion Optimizer, Karbala Int. J. Mod. Sci., 4 (2018), 347-360. doi: 10.1016/j.kijoms.2018.09.001
[40]	R. Sarkhel, N. Das, A. K. Saha, M. Nasipuri, An improved Harmony Search Algorithm embedded with a novel piecewise opposition based learning algorithm, Eng. Appl. Artif. Intell., 67 (2018), 317-330. doi: 10.1016/j.engappai.2017.09.020
[41]	A. A. Ewees, M. A. Elaziz, E. H. Houssein, Improved grasshopper optimization algorithm using opposition-based learning, Expert Syst. Appl., 112 (2018), 156-172. doi: 10.1016/j.eswa.2018.06.023
[42]	M. A. Ahandani, Opposition-based learning in the shuffled bidirectional differential evolution algorithm, Swarm Evol. Comput., 26 (2016), 64-85. doi: 10.1016/j.swevo.2015.08.002
[43]	N. H. Awad, M. Z. Ali, P. N. Suganthan, Ensemble sinusoidal differential covariance matrix adaptation with Euclidean neighborhood for solving CEC2017 benchmark problems, in 2017 IEEE Congress on Evolutionary Computation (CEC), IEEE, (2017), 372-379.
[44]	R. Roy, S. Laha, Optimization of Stego image retaining secret information using genetic algorithm with 8-connected PSNR, Procedia Comput. Sci., 60 (2015), 468-477. doi: 10.1016/j.procs.2015.08.168
[45]	A. Tanchenko, Visual-PSNR measure of image quality, J. Visual Commun. Image Represent., 25 (2014), 874-878. doi: 10.1016/j.jvcir.2014.01.008
[46]	Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861
[47]	V. Bruni, D. Vitulano, An entropy based approach for SSIM speed up, Signal Process., 135 (2017), 198-209. doi: 10.1016/j.sigpro.2017.01.007
[48]	C. Li, A. C. Bovik, Content-partitioned structural similarity index for image quality assessment, Signal Process. Image Commun., 25 (2010), 517-526. doi: 10.1016/j.image.2010.03.004
[49]	L. Zhang, L. Zhang, X. Mou, D. Zhang, FSIM: A feature similarity index for image quality assessment, IEEE Trans. Image Process., 20 (2011), 2378-2386. doi: 10.1109/TIP.2011.2109730
[50]	J. John, M. S. Nair, P. R. A. Kumar, M. Wilscy, A novel approach for detection and delineation of cell nuclei using feature similarity index measure, Biocybern. Biomed. Eng., 36 (2016), 76-88. doi: 10.1016/j.bbe.2015.11.002
[51]	S. K. Dinkar, K. Deep, Opposition based Laplacian Ant Lion Optimizer, J. Comput. Sci., 23 (2017), 71-90. doi: 10.1016/j.jocs.2017.10.007
[52]	M. Wang, X. Zhao, A. A. Heidari, H. Chen, Evaluation of constraint in photovoltaic models by exploiting an enhanced ant lion optimizer, Sol. Energy, 211 (2020), 503-521. doi: 10.1016/j.solener.2020.09.080
[53]	The Berkeley Segmentation Dataset and Benchmark. Available from: https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/.
[54]	H. Jia, X. Peng, W. Song, C. Lang, Z. Xing, K. Sun, Multiverse Optimization Algorithm Based on Lévy Flight Improvement for Multithreshold Color Image Segmentation, IEEE Access, 7 (2019), 32805-32844. doi: 10.1109/ACCESS.2019.2903345
[55]	A. K. M. Khairuzzaman, S. Chaudhury, Masi entropy based multilevel thresholding for image segmentation, Multimed. Tools Appl., 78 (2019), 33573-33591. doi: 10.1007/s11042-019-08117-8
[56]	A. K. Bhandari, A. Kumar, G. K. Singh, Modified artificial bee colony based computationally efficient multilevel thresholding for satellite image segmentation using Kapur's, Otsu and Tsallis functions, Expert Syst. Appl., 42 (2015), 1573-1601. doi: 10.1016/j.eswa.2014.09.049
[57]	S. Kotte, P. R. Kumar, S. K. Injeti, An efficient approach for optimal multilevel thresholding selection for gray scale images based on improved differential search algorithm, Ain Shams Eng. J., 9 (2018), 1043-1067. doi: 10.1016/j.asej.2016.06.007
[58]	V. K. Bohat, K. V. Arya, A new heuristic for multilevel thresholding of images, Expert Syst. Appl., 117 (2019), 176-203. doi: 10.1016/j.eswa.2018.08.045
[59]	A. K. Bhandari, A novel beta differential evolution algorithm-based fast multilevel thresholding for color image segmentation, Neural Comput. Appl., 32 (2020), 4583-4613. doi: 10.1007/s00521-018-3771-z
[60]	D. H. Wolpert, W. G. Macready, No free lunch theorems for optimization, IEEE Trans. Evol. Comput., 1 (1997), 67-82. doi: 10.1109/4235.585893

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)