Research article Special Issues

Analysis of single-cell RNA-sequencing data identifies a hypoxic tumor subpopulation associated with poor prognosis in triple-negative breast cancer


  • Received: 18 January 2022 Revised: 22 March 2022 Accepted: 28 March 2022 Published: 06 April 2022
  • Triple-negative breast cancer (TNBC) is an aggressive subtype of mammary carcinoma characterized by low expression levels of estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2). Along with the rapid development of the single-cell RNA-sequencing (scRNA-seq) technology, the heterogeneity within the tumor microenvironment (TME) could be studied at a higher resolution level, facilitating an exploration of the mechanisms leading to poor prognosis during tumor progression. In previous studies, hypoxia was considered as an intrinsic characteristic of TME in solid tumors, which would activate downstream signaling pathways associated with angiogenesis and metastasis. Moreover, hypoxia-related genes (HRGs) based risk score models demonstrated nice performance in predicting the prognosis of TNBC patients. However, it is essential to further investigate the heterogeneity within hypoxic TME, such as intercellular communications. In the present study, utilizing single-sample Gene Set Enrichment Analysis (ssGSEA) and cell-cell communication analysis on the scRNA-seq data retrieved from Gene Expression Omnibus (GEO) database with accession number GSM4476488, we identified four tumor subpopulations with diverse functions, particularly a hypoxia-related one. Furthermore, results of cell-cell communication analysis revealed the dominant role of the hypoxic tumor subpopulation in angiogenesis- and metastasis-related signaling pathways as a signal sender. Consequently, regard the TNBC cohorts acquired from The Cancer Genome Atlas (TCGA) and GEO as train set and test set respectively, we constructed a risk score model with reliable capacity for the prediction of overall survival (OS), where ARTN and L1CAM were identified as risk factors promoting angiogenesis and metastasis of tumors. The expression of ARTN and L1CAM were further analyzed through tumor immune estimation resource (TIMER) platform. In conclusion, these two marker genes of the hypoxic tumor subpopulation played vital roles in tumor development, indicating poor prognosis in TNBC patients.

    Citation: Yi Shi, Xiaoqian Huang, Zhaolan Du, Jianjun Tan. Analysis of single-cell RNA-sequencing data identifies a hypoxic tumor subpopulation associated with poor prognosis in triple-negative breast cancer[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 5793-5812. doi: 10.3934/mbe.2022271

    Related Papers:

    [1] Rajab Ali Borzooei, Hee Sik Kim, Young Bae Jun, Sun Shin Ahn . MBJ-neutrosophic subalgebras and filters in BE-algebras. AIMS Mathematics, 2022, 7(4): 6016-6033. doi: 10.3934/math.2022335
    [2] M. Mohseni Takallo, Rajab Ali Borzooei, Seok-Zun Song, Young Bae Jun . Implicative ideals of BCK-algebras based on MBJ-neutrosophic sets. AIMS Mathematics, 2021, 6(10): 11029-11045. doi: 10.3934/math.2021640
    [3] Abdelaziz Alsubie, Anas Al-Masarwah . MBJ-neutrosophic hyper BCK-ideals in hyper BCK-algebras. AIMS Mathematics, 2021, 6(6): 6107-6121. doi: 10.3934/math.2021358
    [4] Dongsheng Xu, Huaxiang Xian, Xiewen Lu . Interval neutrosophic covering rough sets based on neighborhoods. AIMS Mathematics, 2021, 6(4): 3772-3787. doi: 10.3934/math.2021224
    [5] Sumyyah Al-Hijjawi, Abd Ghafur Ahmad, Shawkat Alkhazaleh . A generalized effective neurosophic soft set and its applications. AIMS Mathematics, 2023, 18(12): 29628-29666. doi: 10.3934/math.20231517
    [6] Ning Liu, Zengtai Gong . Derivatives and indefinite integrals of single valued neutrosophic functions. AIMS Mathematics, 2024, 9(1): 391-411. doi: 10.3934/math.2024022
    [7] D. Ajay, P. Chellamani, G. Rajchakit, N. Boonsatit, P. Hammachukiattikul . Regularity of Pythagorean neutrosophic graphs with an illustration in MCDM. AIMS Mathematics, 2022, 7(5): 9424-9442. doi: 10.3934/math.2022523
    [8] Amr Elrawy, Mohamed Abdalla . Results on a neutrosophic sub-rings. AIMS Mathematics, 2023, 8(9): 21393-21405. doi: 10.3934/math.20231090
    [9] D. Jeni Seles Martina, G. Deepa . Some algebraic properties on rough neutrosophic matrix and its application to multi-criteria decision-making. AIMS Mathematics, 2023, 8(10): 24132-24152. doi: 10.3934/math.20231230
    [10] Dongsheng Xu, Xiangxiang Cui, Lijuan Peng, Huaxiang Xian . Distance measures between interval complex neutrosophic sets and their applications in multi-criteria group decision making. AIMS Mathematics, 2020, 5(6): 5700-5715. doi: 10.3934/math.2020365
  • Triple-negative breast cancer (TNBC) is an aggressive subtype of mammary carcinoma characterized by low expression levels of estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2). Along with the rapid development of the single-cell RNA-sequencing (scRNA-seq) technology, the heterogeneity within the tumor microenvironment (TME) could be studied at a higher resolution level, facilitating an exploration of the mechanisms leading to poor prognosis during tumor progression. In previous studies, hypoxia was considered as an intrinsic characteristic of TME in solid tumors, which would activate downstream signaling pathways associated with angiogenesis and metastasis. Moreover, hypoxia-related genes (HRGs) based risk score models demonstrated nice performance in predicting the prognosis of TNBC patients. However, it is essential to further investigate the heterogeneity within hypoxic TME, such as intercellular communications. In the present study, utilizing single-sample Gene Set Enrichment Analysis (ssGSEA) and cell-cell communication analysis on the scRNA-seq data retrieved from Gene Expression Omnibus (GEO) database with accession number GSM4476488, we identified four tumor subpopulations with diverse functions, particularly a hypoxia-related one. Furthermore, results of cell-cell communication analysis revealed the dominant role of the hypoxic tumor subpopulation in angiogenesis- and metastasis-related signaling pathways as a signal sender. Consequently, regard the TNBC cohorts acquired from The Cancer Genome Atlas (TCGA) and GEO as train set and test set respectively, we constructed a risk score model with reliable capacity for the prediction of overall survival (OS), where ARTN and L1CAM were identified as risk factors promoting angiogenesis and metastasis of tumors. The expression of ARTN and L1CAM were further analyzed through tumor immune estimation resource (TIMER) platform. In conclusion, these two marker genes of the hypoxic tumor subpopulation played vital roles in tumor development, indicating poor prognosis in TNBC patients.



    In the realms of economics and game theory, the term "duopoly game" is used to describe a scenario in which only two dominant companies or competitors are active within a particular market or industry [1]. These two companies, with their substantial market share, have a significant impact on market dynamics. In a duopoly game, each firm's conduct is highly dependent on the other, since its strategic choices like pricing, output levels, and marketing tactics directly affect each other's results and market positions [2].

    With the worsening environmental pollution and the advancement of the circular economy concept, remanufacturing has emerged as a compelling subject in both business and academia. Remanufacturing used products can decrease the government's spending on environmental pollution treatment, enhance resource utilization by businesses and society, and simultaneously achieve mutually beneficial economic, environmental, and social outcomes [3,4]. In the electronics manufacturing sector, renowned international companies such as Apple, HP, and Fuji Xerox have integrated remanufacturing into their overall business strategy to achieve the development of both the economy and environment. Xerox, for instance, succeeded in conserving 200 million in raw material costs through remanufacturing over five years [5]. The Global Refurbished and Used Mobile Phones Market reached an estimated value of about USD 57.45 billion in 2022 [6], and the global Automotive Parts Remanufacturing Market reached an estimated value of about USD 66.30 billion in the same year 2022 [7].

    Manufacturers frequently face the decision of whether or not to remanufacture their end-of-life products, and they often avoid remanufacturing due to concerns that remanufactured products may adversely affect sales of higher-margin new products. For example, in 2005, third-party remanufacturer (TPR) represented 54% of the aftermarket for automotive parts in Europe and 66% worldwide [8]. Manufacturers face this dilemma when assessing the benefits of remanufacturing against the potential negative effects on new products. The decision-making process involves a range of considerations, encompassing economic, environmental, and social factors. However, this approach could have negative consequences for manufacturers in industries where their end-of-life products are attractive to third-party remanufacturers. These third-party remanufacturers may significantly undercut the sales of the original manufacturer. This dilemma highlights the complex dynamics of the remanufacturing industry and the potential impact on the sales and market position of the original manufacturer. The attractiveness of end-of-life products to third-party remanufacturers introduces a competitive challenge that manufacturers must carefully consider when deciding whether to pursue remanufacturing.

    Recent research literature has shown considerable interest in exploring remanufacturing problems. Geyer [9] examined the remanufacturing strategies of various manufacturers, including BMW, Kodak, and Xerox, and showed that they achieved significant profitability by realizing cost savings through remanufacturing practices. Guide [10] investigated the competitive dynamics between remanufacturing and manufacturing by developing a novel optimal pricing model. Vorasayan [11] employed mathematical programs to establish the best recycling price and quantity. In [12], the authors developed a two-period model that involves competition between the original equipment manufacturer (OEM) and an independent operator capable of producing remanufactured products in the second period.

    Financial dynamical systems have the potential to exhibit chaotic behavior, where even minor alterations can lead to significantly amplified and unpredictable consequences, contributing to fluctuations and instabilities within the game [2,14,15].

    The complex interaction among investors, geopolitical factors, and economic policies can potentially thrust the market into chaos, resulting in unforeseen fluctuations in asset values, extreme volatility in prices, or sudden market collapses [2,12,18,19,20,21]. Consequently, accurately forecasting long-term trends becomes notably challenging. Hence, such chaotic behavior is deemed unsuitable for economic and financial systems, proving to be harmful. Thus, the manufacturer and the third-party both desire market stability, as it enables them to make decisions more easily and consistently pursue maximum profits. However, the closed-loop supply chain market is intricate; altering decision variables can shift the market from stability to chaos. Therefore, the manufacturer must collaborate with the third-party and implement measures to mitigate or prevent chaos, promoting market development and stability.

    Numerous researchers and scholars have invested significant effort in devising effective methods to manage or eradicate such undesirable behavior [18,23,24,25]. While many of them are devoted to investigating chaos control in duopoly games [26,27,28,29], there is a scarcity of literature specifically addressing chaos control in remanufacturing duopoly games. Furthermore, the Ott, Grebogi, and Yorke (OGY) control method has not been explored in such remanufacturing models to the best of our knowledge.

    This study focuses on the dynamic exploration and chaos control in a remanufacturing duopoly game [12,30]. Within the market, we examine two firms: an OEM that exclusively produces new products, and a TPR that exclusively produces distinct products. The OEM and the TPR engage in repeated output competition, with access to only partial market information. It is understood that customers' willingness to pay for original and remanufactured products differs due to the need to distinguish between new products sold by the OEM and remanufactured products sold by the TPR. The OEM manufactures and sells new products in period "t" with a marginal cost of cn, while the TPR remanufactures these products in period "t+1" with a marginal cost of cr, satisfying the condition cn>cr. The symbol δ represents a consumer's willingness to pay for a remanufactured product, which is a fraction of their willingness to pay for the new products. Consumer willingness to pay is uniformly distributed in the range [0,1] and is heterogeneous. The variables qn and qr denote the demand for remanufactured and new products, respectively [12,30].

    This paper is organized as follows: In the second section, we delve into an examination of the nature of the game under consideration, accompanied by a brief analysis of the stability of the Nash equilibrium. The third section is dedicated to exploring the presence of chaos using Lyapunov exponents (LEs), bifurcation diagrams, and spectral entropy (SE), incorporating a comparative analysis of the results obtained from these three analytical tools. Section 4 introduces and applies the OGY method to stabilize the unstable Nash equilibrium point. In Section 5, numerical simulations are conducted to validate the effectiveness of the OGY method in stabilizing the game, employing the marginal costs cn and cr. The paper is then concluded in the final section.

    We consider a recurrent remanufacturing duopoly game with diverse competition strategies and heterogeneous players, assuming that the original equipment manufacturer is boundedly rational and pursues profit maximization as its business objective, while the third-party remanufacturer is adaptive and aims for share maximization based on achieving a certain profit. The dynamics of the resulting remanufacturer duopoly game are described by the following two-dimensional discrete system [12]:

    {qn(t+1)=qn(t)[1+α(12qn(t)δqr(t)cn)],           qr(t+1)=min{(1β)qr(t)+1+v2β(1qn(t)crδ),qn(t)}. (2.1)

    Here, α, β[0,1] are the relative speed of output adjustment of OEM and TPR. The attitude between the profit and market share for the TPR is given by v[0,1].

    The stability has been thoroughly examined in [12]. Let ˉδ denote the critical quantity

    ˉδ=1v(3+v)cn+(1+v)cr+(1v(3+v)cn+(1+v)cr)2+8(1+v)2cr2(1+v),

    and we then distinguish two cases.

    In this case, system (2.1) has only one fixed point E(qn,qr), called the Nash equilibrium, provided that

    (1+v)(crδ)2cn+2>0and(1+v)(δ+δcn2cr)>0.

    Here,

    qn=(1+v)(crδ)2cn+24(1+v)δ,

    and

    qr=(1+v)(δ+δcn2cr)δ(4(1+v)δ).

    Proposition 2.1. The Nash equilibrium E is locally asymptotically stable, provided that

    cn>1+(crδ)(1+ν2)+(β2)(4δ(1+ν))α(4β(2δ1+ν2))=˜cn. (2.2)

    Or, equivalently,

    cr<δ+(cn1(β2)(4δ(1+ν))α(4β(2δ1+ν2)))21+ν=˜cr. (2.3)

    In this case, the game has two equilibrium points: a boundary equilibrium E0=(0,0) and a conditional equilibrium point E=(1cn2+δ,1cn2+δ), for cn<1.

    Proposition 2.2. The boundary equilibrium E0 is unstable (saddle point), and the conditional equilibrium E is locally asymptotically stable, provided that:

    cn>max(cNSn,cFn)=˜cn, (2.4)

    where cNSn=12+δδα, and cFn=12(2+δ)α(2δ).

    Example 2.1. Considering the marginal cost cn]0,1] as a control parameter:

    a) Taking the set of parameter values α=5.8, b=v=0.5, δ=0.3353, and cr=0.2, we have δ=ˉδ=0.3353 for cn=ˉcn0.5962, and thus

    for cn<ˉcn, we have ˉδ>δ, (in this case ˜cn0.9727, but cn<ˉcn<˜cn, thus E is unstable. This aligns with the observation from Figure 1(a), where |λ1|>1 and |λ2|<1 signify the instability of the Nash equilibrium E, characterized as a saddle point).

    Figure 1.  Evolution of the eigenvalues versus cn for: (a) α=5.8,b=0.5,v=0.5,δ=0.3353, and cr=0.2, (b) α=6.9,β=δ=v=0.5, and cr=0.2.

    for cn>ˉcn we have ˉδ<δ, (in this case ˜cn0.5163, so cn>˜cn, thus E is locally asymptotically stable. This aligns with the observation from Figure 1(a), where |λ1,2|<1).

    b) Taking the set of parameter values α=5.8, β=v=cn=0.5, and δ=0.3353, we have δ=ˉδ=0.3353 for cr=ˉcr0.16779, and thus

    for cr<ˉcr, we have δ>ˉδ, (in this case ˜cn0.5163, but cn=0.5<˜cn, thus E is unstable. This corresponds to the observation from Figure 1(b), where |λ1|>1 and |λ2|<1 indicate the instability of the Nash equilibrium E, identified as a saddle point).

    for cr>ˉcr, we have δ<ˉδ, (in this case ˜cr0.0544, but cr>ˉcr>˜cr, thus E is unstable. This aligns with the observation from Figure 1(b), where |λ1|>1 and |λ2|<1 signify the instability of the Nash equilibrium E, characterized as a saddle point).

    There are various tools for the investigation of the complexity and chaos in dynamical systems, such as Lyapunov exponent, bifurcation diagram, 01 test, phase portrait SE, and others [19,20,21]. First, we highlight the complexity of the model by using the SE test.

    By employing the correlation algorithm, when we refer to the complexity of chaotic systems, we describe how much a chaotic sequence resembles a random one [1]. The more a model is close to the random sequence, the higher its complexity.

    We employ the SE algorithm to quantify the complexity of the understudy duopoly game (2.1).

    Here is an outline of the SE complexity algorithm, which can generate an energy distribution via the Fourier transform and determine the corresponding spectrum entropy using the Shannon entropy [19].

    Step 1: Let ZN(n),n=0,1,,N1 be a chaotic pseudo-randomness sequence of length N. With the aim of the spectrum being able to reflect the signal's energy information more effectively, we should first remove the DC (direct current) part of ZN using the subsequent formula:

    z(n)=z(n)ˉμ; where ˉμ=1NN1n=0z(n). (3.1)

    Step 2: We apply a discrete Fourier transform to the sequence z(n).

    Z(k)=N1n=0z(n)ej2πNnk=N1n=0z(n)WnkN, where k=0,1,2,,N1. (3.2)

    Step 3: In this step, we calculate the relative power spectrum. The first half of the sequence for converted Z(k) is taken and calculated using Parseval's theorem. At one of its frequency points, the power spectrum's value is

    p(k)=1N|Z(K)|2, where k=0,1,2,,N/21. (3.3)

    Then, the relative power spectrum of the sequence is given by

    Pk=p(k)ptot =1N|Z(k)|21NN/21k=0|Z(k)|2=|Z(k)|2N/21k=0|Z(k)|2, we have N/21k=0Pk=1. (3.4)

    Step 4: The SE se is then obtained by utilizing the Shannon entropy and the relative power spectral density Pk.

    se=N/21k=0PklnPk. (3.5)

    If Pk=0, define PklnPk=0. It is demonstrable that the value of the SE converges to ln(N/2). For comparative analysis, the SE SE is normalized as

    SE(N)=seln(N/2). (3.6)

    The presence of a non-uniform distribution in the sequence power spectrum leads to a simpler structure in the sequence spectrum and a distinct oscillation pattern in the signal. In such cases, the SE measure is smaller, indicating a lower complexity. Conversely, when the distribution is more uniform, the complexity is higher. Figure 2 depicts the relationship between SE and the parameters cn and cr for the understudy model (2.1).

    Figure 2.  Representation of the spectral entropy (using the sequence qn(t)) of the game (2.1) with the initial conditions qn(0)=0.22, qr(0)=0.21 for the parameter values α=5.8, β=v=0.5, and δ=0.3353, versus: (a) cn[0.4987,0.65], with cr=0.2, (b) cr[0,0.25], with cn=0.5.

    Figure 2(a) illustrates the SE versus the marginal cost cn of the OEM for cn[0.49,0.65]. We can see that the SE increases when the marginal cost cn decreases, indicating a possible appearance of the chaotic behavior in the windows cn[0.4893,0.547].

    Figure 2(b) illustrates the evolution of the SE versus the marginal cost cr, of the TPR firm, within the range cr[0;0.25]. The SE shows an increase as the marginal cost cr rises, with the possible appearance of the chaotic behavior in the windows cn[0.16,0.25].

    The Lyapunov exponents tool aims to assess the divergence or convergence of two trajectories initiated from nearby points. These exponents provide a means to discern various dynamics within the underlying game, such as regular, periodic, chaotic, and hyperchaotic behavior. The presence of at least one positive Lyapunov exponent signifies the chaotic nature of the game.

    Investigating now the dynamics of the duopoly game (2.1) by employing bifurcation diagrams and the Largest Lyapunov Exponent (LLE) calculated using the Wolf Swift algorithm [20].

    Figure 3 illustrates the LLE and bifurcation diagram versus the marginal cost cn for the set of parameter values: α=5.8, β=ν=0.5, δ=0.3353, and cr=0.2. It is evident that LLE is positive for cn[0.4897,0.5466] confirming the chaotic behavior previously predicted by the SE. Additionally, we can infer that the game exhibits periodic behavior (period-8 points, period-4 points, and period-2 points) within the interval cn[0.5466,0.5961]. We observe the stationary behavior for cn[0.5962,0.65].

    Figure 3.  (a) Evolution of the Largest Lyapunov Exponent versus cn (b) bifurcation diagram of qr(t) versus cn, indicating a transition to chaos via inverse period-doubling scenario.

    Figure 4 depicts the LLE and the bifurcation of qr(t) with respect to the marginal cost cr[0,0.25], for the parameter values α=5.8, β=cn=ν=0.5, and δ=0.3353. Obviously, LLE is positive for almost cr [0.1533,0.2366] confirming the chaotic behavior predicted by the SE. Furthermore, the game exhibits periodic behavior in the interval [0,0.1533]. It is apparent that the largest Lyapunov exponents align well with the bifurcation diagram.

    Figure 4.  (a) Evolution of the Largest Lyapunov Exponent versus the marginal cost cr, (b) bifurcation diagram of qr(t) with respect to the marginal cost cr.

    The duopoly game (2.1) exhibits strange attractors for various parameter values, some of them are illustrated in Figures 5 and 6, in blue, accompanied by the corresponding unstable Nash equilibrium depicted in red.

    Figure 5.  Two chaotic attractors of the duopoly game (2.1) with the initial conditions qn(0)=0.22,qr(0)=0.21: (a) cr=0.2,cn=0.5,v=0.5,β=0.5,α=5.8, and δ=0.3353, (b) cr=0.25,cn=0.5,v=0.5,β=0.5,α=5.8, and δ=0.3353.
    Figure 6.  Two chaotic attractors of the duopoly game (2.1) with the initial conditions qn(0)=0.22,qr(0)=0.21: (a) cr=0.2,cn=0.5,v=0.5,β=0.5,α=6.9, and δ=0.5, (b) cr=0.2027,cn=0.5,v=0.5,β=0.5,α=6.9, and δ=0.5.

    For instance, considering the parameter values α=5.8, β=v=cn=12, δ=0.3353, and cr=15, we obtain the following Lyapunov exponents: λ1=0.4476,λ2=1.0816. As λ1>0, it indicates that the duopoly game (2.1) features a chaotic attractor. The Kaplan–Yorke dimension is calculated as d=1.5177, revealing a fractal nature.

    It is not advisable to select parameter values for cn and cr within the chaotic zone, as this would result in the duopoly game (2.1) exhibiting chaotic behavior in the market. Given that both firms lack knowledge of each other's strategies, we propose implementing control measures in the game to prevent the occurrence of chaotic market conditions. The objective is to assist the firms in the duopoly game (2.1) in making informed decisions by selecting appropriate values for their marginal costs cn and cr.

    To evaluate the system's performance, particularly when it displays chaos, we use aggregate market profits as a measure. Let MT(α,cr) denote the aggregate market profits during the time period T, then

    MT(α,cr)=Tt=0(Πn(qn(t),qr(t))+Πr(qn(t),qr(t))), (3.7)

    where, Πn and Πr represent the profit functions of OEM and TPR respectively given by Eqs (4) and (5) in [12]. We consider the adjustment speed α and the TPR cost cr as a variable parameters and adopt the same parameter configurations as illustrated in Figure 2, and T=100. The diagram in Figure 7 depicts the aggregate profits versus α and cr where we observe that, when the adjustment speed of the OEM or TPR cost cr falls within the chaotic range, the market's aggregate profits experience a significant decrease in comparison to the equilibrium state case, meaning that the chaotic output dynamics is harmful to the market.

    Figure 7.  Aggregate profits of the duopoly game (2.1) with the initial conditions qn(0)=0.22,qr(0)=0.21, and the parameter values cn=0.5,v=0.5,β=0.5, and δ=0.3353 (a) versus the adjustment speed α with cr=0.2, (b) versus the TPR cost cr with α=5.8.

    Chaotic behavior in economics is unwelcome, as it adversely affects investor confidence, economic growth, and operational challenges. It also heightens volatility and financial risks. The manifestation of chaos in duopoly games leads to unpredictability in the output decisions of both OEM and TPR players due to its sensitivity to infinitesimal errors. Consequently, there is a desire to discover methods to control chaos within economic systems. Various approaches have been suggested, including feedback control [23], adaptive control [24], and OGY method [18,25], among others. In this study we opt for the OGY method due to its non-invasive application (very small parameter adjustment during a very small time period), ensuring that the game's evolution between TPR and OEM will be stabilized without great effort.

    The fundamental concept behind the OGY method involves making small time-dependent perturbations to an accessible control parameter of the system to guide the system state toward its unstable equilibrium, utilizing the stable manifold as a vehicle to achieve the desired outcome as illustrated in Figure 8.

    Figure 8.  Schematic of OGY control method.

    Consider the map

    xn+1=F(xn,p), (4.1)

    where xnR2, F:R2×RR2, and pR is an externally accessible control parameter.

    The map (4.1) can be linearized around the fixed point xF(ˉp) as follows:

    xn+1xF(ˉp)=A(xnxF(ˉp))+B(pˉp),

    where A=Fx|x=xF(ˉp) and B=Fp|p=ˉp. Then,

    Δp=pˉp=KT(xnxF(ˉp)),

    ere, K is a gain vector that will be determined later. The controlled system can be written as

    xn+1xF(ˉp)=A(xnxF(ˉp))BKT(xnxF(ˉp))=(ABKT)(xnxF(ˉp)).

    The stability of the controlled system is determined by the eigenvalues of (ABKT). The system is controllable if the controllability matrix

    P=[B,AB],

    has full rank 2.

    Assume the matrix A has two eigenvalues λs and λu, with two left eigenvectors vs and vu, and two right eigenvectors ws and wu. We adjust the parameter p such that wTu(xn+1xF(ˉp))=0. We get

    KT=λuwTuwTuB  and  wTuB0.

    Then, the control law can be written as

    Δp={KT(xnxF(ˉp))if  |KT(xnxF(ˉp))|<ξ,0elsewhere. (4.2)

    ● If 0<δ<ˉδ. Then,

    A=FX=[12αqnδαqn1+v2β1β]   and  Bcr=Fcr=[0(1+v)βδ].

    The cr controllability matrix is

    Pcr=[Bcr,ABcr]=[0αβqn(v+1)(1+v)βδβδ(β1)(v+1)],

    and its determinant is

    det(Pcr)=1δαβ2qn(v2+2v+1)0.

    Thus, we conclude that TPR can stabilize the Nash equilibrium E by adjusting its marginal cost cr.

    ● If ˉδ<δ<1, then

    A=FX=[12αqnδαqn10] and Bcn=Fcn=[αqn0].

    The cn controllability matrix is

    Pcn=[Bcn,ABcn]=[αqnαqn(2αqn1)0αqn],

    and its determinant is

    det(Pcn)=α2(qn)20.

    Then, we conclude that the OEM firm can stabilize the game around the Nash equilibrium E by adjusting its marginal cost cn.

    We conduct numerical simulations to confirm the theoretical findings. First, we use the OEM marginal cost cn as the control parameter, and then we use the TPR marginal cost cr as the control parameter.

    Let us consider cn as a control parameter and set the other parameters as follows: α=5.8, β=δ=v=0.5, and cr=0.2. For cn=cn=0.5 we have E=(0.2279,0.1317), and the Jacobian matrix

    A=[1.64390.44320.3750.5],  with Bcn=[1.3220]

    and the control gain KT=(1.3,0.26).

    Figure 9 illustrates the response of the controlled duopoly game along with the applied control effort. The control is activated when the system state approaches the unstable equilibrium E at t=54 and the marginal cost cn is adjusted by a small perturbation of order 103 during the short time period t[54,56]. Subsequently, the control is established at t=57, stabilizing the duopoly game to its Nash equilibrium.

    Figure 9.  Response of the controlled duopoly game (2.1) using the OEM marginal cost cn with the parameter values α=5.8, β=v=0.5, δ=0.3353, and cr=0.2, for cn=cn=0.5.

    Let us consider cr as a control parameter and set the other parameters as follows:

    α=5.8, β=v=cn=0.5, and δ=0.3353. For, cr=cr=0.2, we get the Nash equilibrium E(0.2279,0.1317), the Jacobian matrix

    A=[1.64390.44320.37500.5], with Bcr=[01.1184],

    and the control gain KT=(7.6930,1.5368).

    Figure 10 shows the response of the controlled duopoly game along with the control effort. The control was activated when the system state is sufficiently close to the unstable equilibrium E at t=96 and the TPR marginal cost cr is slightly adjusted by a perturbation of order 104 during the short time period t[96,102]. Subsequently, the control is established at t=103, stabilizing the duopoly game to its Nash equilibrium.

    Figure 10.  Response of the controlled duopoly game (2.1) using the TPR marginal cost cr with the parameter values α=5.8, β=v=cn=0.5, and δ=0.3353 for cr=cr=0.2.

    In contrast to other control methods, the application of the OGY method to stabilize markets offers the advantage of achieving control with minimal effort of a single firm within a brief time frame. This means that the method can lead the market towards stability in a smooth manner without compromising the firms in the game.

    For example, in the present study, the control was realized through adjusting the OEM cost cn by |Δcn|<6×103 over only two units of time. Additionally, it was accomplished when adjusting the TPR cost cr by a perturbation |Δcr|<1.1×103 over only 6 units of time. In the counter party, Kopel's model was controlled in [26] using a state feedback method through adjusting state of the first firm with a perturbation |u(t)|<0.4 over about 60 units of time. In [26], a Cournot model was stabilized using a delay feedback control (Pyragas method) by adjusting the state of the first firm during approximately 50 units of time. In [27], a Cournot model was stabilized using the state variable feedback and parameter variation method during approximately 40 units of time.

    We examined a duopoly game model involving distinct competition strategies and heterogeneous players. The findings reveal significant impacts of the marginal cost cn for the original equipment manufacturer (OEM) and the marginal cost cr for the third-party remanufacturer (TPR) on their competitive dynamics within the game. Under specific parameter configurations, a decrease in cn induces a transition in game behavior from chaos to regular, while an increase in cr results in a shift from regular to chaotic behavior. By applying a control law derived from the OGY method, with either the marginal cost cn of the OEM or the marginal cost cr of the TPR as a control parameter, the chaotic behavior is effectively eliminated, leading to the stabilization of the duopoly game at its Nash equilibrium point E.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This research has been funded by Scientific Research Deanship at University of Ha'il–Saudi Arabia through project number (RG-23 045).

    All authors declare no conflicts of interest in this paper.



    [1] F. Bray, J. Ferlay, I. Soerjomataram, R. L. Siegel, L. A. Torre, A. Jemal, Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA Cancer J. Clin., 68 (2018), 394–424. https://doi.org/10.3322/caac.21492 doi: 10.3322/caac.21492
    [2] R. Dent, M. Trudeau, K. I. Pritchard, W. M. Hanna, H. K. Kahn, C. A. Sawka, et al., Triple-negative breast cancer: clinical features and patterns of recurrence, Clin. Cancer Res., 13 (2007), 4429–4434. https://doi.org/10.1158/1078-0432.CCR-06-3045 doi: 10.1158/1078-0432.CCR-06-3045
    [3] A. Marra, G. Viale, G. Curigliano, Recent advances in triple negative breast cancer: the immunotherapy era, BMC Medicine, 17 (2019), 90. https://doi.org/10.1186/s12916-019-1326-5 doi: 10.1186/s12916-019-1326-5
    [4] G. Zarrilli, G. Businello, M. V. Dieci, S. Paccagnella, V. Carraro, R. Cappellesso, et al., The tumor microenvironment of primitive and metastatic breast cancer: implications for novel therapeutic strategies, Int. J. Mol. Sci., 21 (2020), 8102. https://doi.org/10.3390/ijms21218102 doi: 10.3390/ijms21218102
    [5] S. D. Soysal, A. Tzankov, S. E. Muenst, Role of the tumor microenvironment in breast cancer, Pathobiology, 82 (2015), 142–152. https://doi.org/10.1159/000430499 doi: 10.1159/000430499
    [6] P. V. Loo, T. Voet, Single cell analysis of cancer genomes, Curr. Opin. Genet. Dev., 24 (2014), 82–91. https://doi.org/10.1016/j.gde.2013.12.004 doi: 10.1016/j.gde.2013.12.004
    [7] A. A. Pollen, T. J. Nowakowski, J. Shuga, X. Wang, A. A. Leyrat, J. H. Lui, et al., Low-coverage single-cell mRNA sequencing reveals cellular heterogeneity and activated signaling pathways in developing cerebral cortex, Nat. Biotechnol., 32 (2014), 1053–1058. https://doi.org/10.1038/nbt.2967 doi: 10.1038/nbt.2967
    [8] B. Treutlein, D. G. Brownfield, A. R. Wu, N. F. Neff, G. L. Mantalas, F. H. Espinoza, et al., Reconstructing lineage hierarchies of the distal lung epithelium using single-cell RNA-seq, Nature, 509 (2014), 371–375. https://doi.org/10.1038/nature13173 doi: 10.1038/nature13173
    [9] Q. H. Nguyen, N. Pervolarakis, K. Blake, D. Ma, R. T. Davis, N. James, et al., Profiling human breast epithelial cells using single cell RNA sequencing identifies cell diversity, Nat. Commun., 9 (2018), 2028. https://doi.org/10.1038/s41467-018-04334-1 doi: 10.1038/s41467-018-04334-1
    [10] M. Bartoschek, N. Oskolkov, M. Bocci, J. Lövrot, C. Larsson, M. Sommarin, et al., Spatially and functionally distinct subclasses of breast cancer-associated fibroblasts revealed by single cell RNA sequencing, Nat. Commun., 9 (2018), 5150. https://doi.org/10.1038/s41467-018-07582-3 doi: 10.1038/s41467-018-07582-3
    [11] D. Hanahan, R. A. Weinberg, Hallmarks of cancer: the next generation, Cell, 144 (2011), 646–674. https://doi.org/10.1016/j.cell.2011.02.013 doi: 10.1016/j.cell.2011.02.013
    [12] B. Muz, P. de la Puente, F. Azab, A. K. Azab, The role of hypoxia in cancer progression, angiogenesis, metastasis, and resistance to therapy, Hypoxia (Auckl), 3 (2015), 83–92. https://doi.org/10.2147/HP.S93413 doi: 10.2147/HP.S93413
    [13] X. Jing, F. Yang, C. Shao, K. Wei, M. Xie, H. Shen, et al., Role of hypoxia in cancer therapy by regulating the tumor microenvironment, Mol. Cancer, 18 (2019), 157. https://doi.org/10.1186/s12943-019-1089-9 doi: 10.1186/s12943-019-1089-9
    [14] X. Sun, H. Luo, C. Han, Y. Zhang, C. Yan, Identification of a hypoxia-related molecular classification and hypoxic tumor microenvironment signature for predicting the prognosis of patients with triple-negative breast cancer, Front. Oncol., 11 (2021), 700062. https://doi.org/10.3389/fonc.2021.700062 doi: 10.3389/fonc.2021.700062
    [15] X. Yang, X. Weng, Y. Yang, M. Zhang, Y. Xiu, W. Peng, et al., A combined hypoxia and immune gene signature for predicting survival and risk stratification in triple-negative breast cancer, Aging (Albany NY), 13 (2021), 19486–19509. https://doi.org/10.18632/aging.203360 doi: 10.18632/aging.203360
    [16] R. Gao, S. Bai, Y. C. Henderson, Y. Lin, A. Schalck, Y. Yan, et al., Delineating copy number and clonal substructure in human tumors from single-cell transcriptomes, Nat. Biotechnol., 39 (2021), 599–608. https://doi.org/10.1038/s41587-020-00795-2 doi: 10.1038/s41587-020-00795-2
    [17] M. J. Goldman, B. Craft, M. Hastie, K. Repečka, F. McDade, A. Kamath, et al., Visualizing and interpreting cancer genomics data via the Xena platform, Nat. Biotechnol., 38 (2020), 675–678. https://doi.org/10.1038/s41587-020-0546-8 doi: 10.1038/s41587-020-0546-8
    [18] D. M. Gendoo, N. Ratanasirigulchai, M. S. Schroder, L. Paré, J. S. Parker, A. Prat, et al., Genefu: an R/Bioconductor package for computation of gene expression-based signatures in breast cancer, Bioinformatics, 32 (2016), 1097–1099. https://doi.org/10.1093/bioinformatics/btv693 doi: 10.1093/bioinformatics/btv693
    [19] P. Jezequel, D. Loussouarn, C. Guerin-Charbonnel, L. Campion, A. Vanier, W. Gouraud, et al., Gene-expression molecular subtyping of triple-negative breast cancer tumours: importance of immune response, Breast Cancer Res., 17 (2015), 43. https://doi.org/10.1186/s13058-015-0550-y doi: 10.1186/s13058-015-0550-y
    [20] Y. Hao, S. Hao, E. Andersen-Nissen, W. M. Mauck III, S. Zheng, A. Butler, et al., Integrated analysis of multimodal single-cell data, Cell, 184 (2021), 3573–3587. https://doi.org/10.1016/j.cell.2021.04.048 doi: 10.1016/j.cell.2021.04.048
    [21] D. Aran, A. P. Looney, L. Liu, E. Wu, V. Fong, A. Hsu, et al., Reference-based analysis of lung single-cell sequencing reveals a transitional profibrotic macrophage, Nat. Immunol., 20 (2019), 163–172. https://doi.org/10.1038/s41590-018-0276-y doi: 10.1038/s41590-018-0276-y
    [22] A. Liberzon, C. Birger, H. Thorvaldsdottir, M. Ghandi, J. P. Mesirov, P. Tamayo, The Molecular Signatures Database (MSigDB) hallmark gene set collection, Cell Syst., 1 (2015), 417–425. https://doi.org/10.1016/j.cels.2015.12.004 doi: 10.1016/j.cels.2015.12.004
    [23] S. Jin, C. F. Guerrero-Juarez, L. Zhang, I. Chang, R. Ramos, C. H. Kuan, et al., Inference and analysis of cell-cell communication using CellChat, Nat. Commun., 12 (2021), 1088. https://doi.org/10.1038/s41467-021-21246-9 doi: 10.1038/s41467-021-21246-9
    [24] J. F. Prud'homme, F. Fridlansky, M. Le Cunff, M. Atger, C. Mercier-Bodart, M. F. Pichon, et al., Cloning of a gene expressed in human breast cancer and regulated by estrogen in MCF-7 cells, DNA, 4 (1985), 11–21. https://doi.org/10.1089/dna.1985.4.11 doi: 10.1089/dna.1985.4.11
    [25] M. C. Rio, J. P. Bellocq, J. Y. Daniel, C. Tomasetto, R. Lathe, M. P. Chenard, et al., Breast cancer-associated pS2 protein: synthesis and secretion by normal stomach mucosa, Science, 241 (1988), 705–708. https://doi.org/10.1126/science.3041593 doi: 10.1126/science.3041593
    [26] T. Vogl, A. Stratis, V. Wixler, T. Völler, S. Thurainayagam, S. K. Jorch, et al., Autoinhibitory regulation of S100A8/S100A9 alarmin activity locally restricts sterile inflammation, J. Clin. Invest., 128 (2018), 1852–1866. https://doi.org/10.1172/JCI89867 doi: 10.1172/JCI89867
    [27] Q. Fang, S. Yao, G. Luo, X. Zhang, Identification of differentially expressed genes in human breast cancer cells induced by 4-hydroxyltamoxifen and elucidation of their pathophysiological relevance and mechanisms, Oncotarget, 9 (2018), 2475–2501. https://doi.org/10.18632/oncotarget.23504 doi: 10.18632/oncotarget.23504
    [28] N. O'Brien, T. M. Maguire, N. O'Donovan, N. Lynch, A. D. Hill, E. McDermott, et al., Mammaglobin a: a promising marker for breast cancer, Clin. Chem., 48 (2002), 1362–1364. https://doi.org/10.1093/clinchem/48.8.1362 doi: 10.1093/clinchem/48.8.1362
    [29] M. Zafrakas, B. Petschke, A. Donner, F. Fritzsche, G. Kristiansen, R. Knüchel, et al., Expression analysis of mammaglobin A (SCGB2A2) and lipophilin B (SCGB1D2) in more than 300 human tumors and matching normal tissues reveals their co-expression in gynecologic malignancies, BMC Cancer, 6 (2006), 88. https://doi.org/10.1186/1471-2407-6-88 doi: 10.1186/1471-2407-6-88
    [30] D. Carter, J. F. Douglass, C. D. Cornellison, M. W. Retter, J. C. Johnson, A. A. Bennington, et al., Purification and characterization of the mammaglobin/lipophilin B complex, a promising diagnostic marker for breast cancer, Biochemistry, 41 (2002), 6714–6722. https://doi.org/10.1021/bi0159884 doi: 10.1021/bi0159884
    [31] T. L. Colpitts, P. Billing-Medel, P. Friedman, E. N. Granados, M. Hayden, S. Hodges, et al., Mammaglobin is found in breast tissue as a complex with BU101, Biochemistry, 40 (2001), 11048–11059. https://doi.org/10.1021/bi010284f doi: 10.1021/bi010284f
    [32] S. Robson, S. Pelengaris, M. Khan, c-Myc and downstream targets in the pathogenesis and treatment of cancer, Recent Pat. Anticancer Drug Discov., 1 (2006), 305–326. https://doi.org/10.2174/157489206778776934 doi: 10.2174/157489206778776934
    [33] P. C. Fernandez, S. R. Frank, L. Wang, M. Schroeder, S. Liu, J. Greene, et al., Genomic targets of the human c-Myc protein, Genes Dev., 17 (2003), 1115–1129. https://doi.org/10.1101/gad.1067003 doi: 10.1101/gad.1067003
    [34] J. H. Patel, A. P. Loboda, M. K. Showe, L. C. Showe, S. B. McMahon, Analysis of genomic targets reveals complex functions of MYC, Nat. Rev. Cancer, 4 (2004), 562–568. https://doi.org/10.1038/nrc1393 doi: 10.1038/nrc1393
    [35] C. Attwooll, E. L. Denchi, K. Helin, The E2F family: specific functions and overlapping interests, EMBO J., 23 (2004), 4709–4716. https://doi.org/10.1038/sj.emboj.7600481 doi: 10.1038/sj.emboj.7600481
    [36] T. Yu, L. Liang, X. Zhao, Y. Yin, Structural and biochemical studies of the extracellular domain of Myelin protein zero-like protein 1, Biochem. Biophys. Res. Commun., 506 (2018), 883–890. https://doi.org/10.1016/j.bbrc.2018.10.161 doi: 10.1016/j.bbrc.2018.10.161
    [37] K. M. McCarthy, I. B. Skare, M. C. Stankewich, M. Furuse, S. Tsukita, R. A. Rogers, et al., Occludin is a functional component of the tight junction, J. Cell Sci., 109 (1996), 2287–2298. https://doi.org/10.1242/jcs.109.9.2287 doi: 10.1242/jcs.109.9.2287
    [38] J. Sakata, T. Shimokubo, K. Kitamura, S. Nakamura, K. Kangawa, H. Matsuo, et al., Molecular cloning and biological activities of rat adrenomedullin, a hypotensive peptide, Biochem. Biophys. Res. Commun., 195 (1993), 921–927. https://doi.org/10.1006/bbrc.1993.2132 doi: 10.1006/bbrc.1993.2132
    [39] K. Miyashita, H. Itoh, N. Sawada, Y. Fukunaga, M. Sone, K. Yamahara, et al., Adrenomedullin promotes proliferation and migration of cultured endothelial cells, Hypertens. Res., 26 Suppl (2003), S93–98. https://doi.org/10.1291/hypres.26.S93 doi: 10.1291/hypres.26.S93
    [40] N. Ferrara, H. P. Gerber, J. LeCouter, The biology of VEGF and its receptors, Nat. Med., 9 (2003), 669–676. https://doi.org/10.1038/nm0603-669 doi: 10.1038/nm0603-669
    [41] M. I. Lin, W. C. Sessa, Vascular endothelial growth factor signaling to endothelial nitric oxide synthase: more than a FLeeTing moment, Circ. Res., 99 (2006), 666–668. https://doi.org/10.1161/01.RES.0000245430.24075.a4 doi: 10.1161/01.RES.0000245430.24075.a4
    [42] S. Sugo, N. Minamino, K. Kangawa, K. Miyamoto, K. Kitamura, J. Sakata, et al., Endothelial cells actively synthesize and secrete adrenomedullin, Biochem. Biophys. Res. Commun., 201 (1994), 1160–1166. https://doi.org/10.1006/bbrc.1994.1827 doi: 10.1006/bbrc.1994.1827
    [43] M. J. Miller, A. Martinez, E. J. Unsworth, C. J. Thiele, T. W. Moody, T. Elsasser, et al., Adrenomedullin expression in human tumor cell lines. Its potential role as an autocrine growth factor, J. Biol. Chem., 271 (1996), 23345–23351. https://doi.org/10.1074/jbc.271.38.23345 doi: 10.1074/jbc.271.38.23345
    [44] K. Dawas, M. Loizidou, A. Shankar, H. Ali, I. Taylor, Angiogenesis in cancer: the role of endothelin-1, Ann. R. Coll. Surg. Engl., 81 (1999), 306–310.
    [45] N. Zhu, L. Gu, J. Jia, X. Wang, L. Wang, M. Yang, W. Yuan, Endothelin-1 triggers human peritoneal mesothelial cells' proliferation via ERK1/2-Ets-1 signaling pathway and contributes to endothelial cell angiogenesis, J. Cell. Biochem., 120 (2019), 3539–3546. https://doi.org/10.1002/jcb.27631 doi: 10.1002/jcb.27631
    [46] Y. Katanasaka, Y. Kodera, Y. Kitamura, T. Morimoto, T. Tamura, F. Koizumi, Epidermal growth factor receptor variant type III markedly accelerates angiogenesis and tumor growth via inducing c-myc mediated angiopoietin-like 4 expression in malignant glioma, Mol. Cancer, 12 (2013), 31. https://doi.org/10.1186/1476-4598-12-31 doi: 10.1186/1476-4598-12-31
    [47] R. Kolb, P. Kluz, Z. W. Tan, N. Borcherding, N. Bormann, A. Vishwakarma, et al., Obesity-associated inflammation promotes angiogenesis and breast cancer via angiopoietin-like 4, Oncogene, 38 (2019), 2351–2363. https://doi.org/10.1038/s41388-018-0592-6 doi: 10.1038/s41388-018-0592-6
    [48] M. S. Airaksinen, M. Saarma, The GDNF family: signalling, biological functions and therapeutic value, Nat. Rev. Neurosci., 3 (2002), 383–394. https://doi.org/10.1038/nrn812 doi: 10.1038/nrn812
    [49] J. Kang, P. X. Qian, V. Pandey, J. K. Perry, L. D. Miller, E. T. Liu, et al., Artemin is estrogen regulated and mediates antiestrogen resistance in mammary carcinoma, Oncogene, 29 (2010), 3228–3240. https://doi.org/10.1038/onc.2010.71 doi: 10.1038/onc.2010.71
    [50] J. Kang, J. K. Perry, V. Pandey, G. C. Fielder, B. Mei, P. X. Qian, et al., Artemin is oncogenic for human mammary carcinoma cells, Oncogene, 28 (2009), 2034–2045. https://doi.org/10.1038/onc.2009.66 doi: 10.1038/onc.2009.66
    [51] A. Banerjee, Z. S. Wu, P. Qian, J. Kang, V. Pandey, D. X. Liu, et al., ARTEMIN synergizes with TWIST1 to promote metastasis and poor survival outcome in patients with ER negative mammary carcinoma, Breast Cancer Res., 13 (2011), R112. https://doi.org/10.1186/bcr3054 doi: 10.1186/bcr3054
    [52] A. Banerjee, P. Qian, Z. S. Wu, X. Ren, M. Steiner, N. M. Bougen, et al., Artemin stimulates radio- and chemo-resistance by promoting TWIST1-BCL-2-dependent cancer stem cell-like behavior in mammary carcinoma cells, J. Biol. Chem., 287 (2012), 42502–42515. https://doi.org/10.1074/jbc.M112.365163 doi: 10.1074/jbc.M112.365163
    [53] H. Zhang, C. C. Wong, H. Wei, D. M. Gilkes, P. Korangath, P. Chaturvedi, et al., HIF-1-dependent expression of angiopoietin-like 4 and L1CAM mediates vascular metastasis of hypoxic breast cancer cells to the lungs, Oncogene, 31 (2012), 1757–1770. https://doi.org/10.1038/onc.2011.365 doi: 10.1038/onc.2011.365
    [54] A. Banerjee, Z. S. Wu, P. X. Qian, J. Kang, D. X. Liu, T. Zhu, et al., ARTEMIN promotes de novo angiogenesis in ER negative mammary carcinoma through activation of TWIST1-VEGF-A signaling, PLoS One, 7 (2012), e50098. https://doi.org/10.1371/journal.pone.0050098 doi: 10.1371/journal.pone.0050098
    [55] A. Friedli, E. Fischer, Novak-Hofer I, S. Cohrs, K. Ballmer-Hofer, P. A. Schubiger, et al., The soluble form of the cancer-associated L1 cell adhesion molecule is a pro-angiogenic factor, Int. J. Biochem. Cell Biol., 41 (2009), 1572–1580. https://doi.org/10.1016/j.biocel.2009.01.006 doi: 10.1016/j.biocel.2009.01.006
    [56] H. Hall, J. A. Hubbell, Matrix-bound sixth Ig-like domain of cell adhesion molecule L1 acts as an angiogenic factor by ligating alphavbeta3-integrin and activating VEGF-R2, Microvasc. Res., 68 (2004), 169–178. https://doi.org/10.1016/j.mvr.2004.07.001 doi: 10.1016/j.mvr.2004.07.001
    [57] H. Hall, V. Djonov, M. Ehrbar, M. Hoechli, J. A. Hubbell, Heterophilic interactions between cell adhesion molecule L1 and alphavbeta3-integrin induce HUVEC process extension in vitro and angiogenesis in vivo, Angiogenesis, 7 (2004), 213–223. https://doi.org/10.1007/s10456-004-1328-5 doi: 10.1007/s10456-004-1328-5
    [58] M. Zhang, W. Zhang, Z. Wu, S. Liu, L. Sun, Y. Zhong, et al., Artemin is hypoxia responsive and promotes oncogenicity and increased tumor initiating capacity in hepatocellular carcinoma, Oncotarget, 7 (2016), 3267–3282. https://doi.org/10.18632/oncotarget.6572 doi: 10.18632/oncotarget.6572
  • Reader Comments
  • © 2022 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5011) PDF downloads(320) Cited by(14)

Figures and Tables

Figures(10)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog