Review

A review on interventions to prevent osteoporosis and improve fracture healing in osteoporotic patients

  • Received: 01 July 2020 Accepted: 24 September 2020 Published: 10 October 2020
  • Introduction Proportion of aged people has increased due to improvement in longevity. With advancing age bones loose mass, get weakened, become more prone to osteoporotic fractures. Development of osteoporosis is silent with reduction in total bone mineral content. Bone mineral density is the most important parameter which can measure gravity of OP. The T-score of −2.5 or below indicates OP leading to fractures commonly in bones of spine, hip and distal extremities.
    Objective Purpose of this review is to find out currently available as well as future approaches those could be beneficial in faster recoveries.
    Method Major databases were searched from inception to till January 2020. Relevant main articles and cross references were evaluated. In addition, findings were compared to a previously published review. Pharmacological and bio-molecule interventions decrease bone depletion by reducing bone resorption and enhancing bone formation process, and other external stimuli like strengthening muscular and skeletal tissues are employed to prevent OP and accelerate fracture healing and normal functioning.
    Results A literature review of such interventions showed that bisphosphonates and selective estrogen receptor modulator treatment reduces fracture risks in osteoporosis and increase callus formation during fracture repair but do not reduce total time of fracture healing. Parathyroid hormone (PTH) and its analogues prevent OP by promoting callus formation as well as osteogenesis, enhance coupled remodelling and amount of mineralized tissue. When PTH is combined with bone morphogenetic proteins (BMP) improve mechanical functioning by integrating new bone tissue with old bone tissue. Restraining and supportive therapies like the physical exercises could be beneficial to minimize gravity of osteoporosis. Electromagnetic field therapy and pulsed ultrasonic therapy could be useful after surgical management of fracture or delayed union.
    Conclusion To identify the quantitative effect of these therapies in isolation or in combination, clinical trials under proper experimental settings are especially important. Unlike the therapies for fracture repair in non-osteoporotic patients, the line of treatment and duration of each therapy in isolation or in combination with pharmacological agents, biomolecules, physical stimuli, exercises and lifestyles are necessary.

    Citation: Manishtha Rao, Madhvi Awasthi. A review on interventions to prevent osteoporosis and improve fracture healing in osteoporotic patients[J]. AIMS Medical Science, 2020, 7(4): 243-268. doi: 10.3934/medsci.2020015

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  • Introduction Proportion of aged people has increased due to improvement in longevity. With advancing age bones loose mass, get weakened, become more prone to osteoporotic fractures. Development of osteoporosis is silent with reduction in total bone mineral content. Bone mineral density is the most important parameter which can measure gravity of OP. The T-score of −2.5 or below indicates OP leading to fractures commonly in bones of spine, hip and distal extremities.
    Objective Purpose of this review is to find out currently available as well as future approaches those could be beneficial in faster recoveries.
    Method Major databases were searched from inception to till January 2020. Relevant main articles and cross references were evaluated. In addition, findings were compared to a previously published review. Pharmacological and bio-molecule interventions decrease bone depletion by reducing bone resorption and enhancing bone formation process, and other external stimuli like strengthening muscular and skeletal tissues are employed to prevent OP and accelerate fracture healing and normal functioning.
    Results A literature review of such interventions showed that bisphosphonates and selective estrogen receptor modulator treatment reduces fracture risks in osteoporosis and increase callus formation during fracture repair but do not reduce total time of fracture healing. Parathyroid hormone (PTH) and its analogues prevent OP by promoting callus formation as well as osteogenesis, enhance coupled remodelling and amount of mineralized tissue. When PTH is combined with bone morphogenetic proteins (BMP) improve mechanical functioning by integrating new bone tissue with old bone tissue. Restraining and supportive therapies like the physical exercises could be beneficial to minimize gravity of osteoporosis. Electromagnetic field therapy and pulsed ultrasonic therapy could be useful after surgical management of fracture or delayed union.
    Conclusion To identify the quantitative effect of these therapies in isolation or in combination, clinical trials under proper experimental settings are especially important. Unlike the therapies for fracture repair in non-osteoporotic patients, the line of treatment and duration of each therapy in isolation or in combination with pharmacological agents, biomolecules, physical stimuli, exercises and lifestyles are necessary.


    Rough set theory was initially developed by Pawlak [1] as a new mathematical methodology to deal with the vagueness and uncertainty in information systems. Covering rough set (CRS) theory is a generalization of traditional rough set theory, which is characterized by coverings instead of partitions. Degang Chen et al.[2] proposed belief and plausibility functions to characterize neighborhood-covering rough sets. Essentially, they developed a numerical method for finding reductions using belief functions. Liwen Ma[3] defined the complementary neighborhood of an arbitrary element in the universe and discussed its properties. Based on the concepts of neighborhood and complementary neighborhood, an equivalent definition of a class of CRS is defined or given. Bin Yang and Bao Qing Hu [4] introduced some new definitions of fuzzy-covering approximation spaces and studied the properties of fuzzy-covering approximation spaces and Mas fuzzy covering-based rough set models. On this basis, they proposed three rough set models based on fuzzy coverage as the generalization of Ma model. Yan-Lan Zhang and Mao-Kang Luo[5] studied the relation between relation-based rough sets and covering-based rough sets. In a rough set framework based on relation, they unified five kinds of covering-based rough sets. The equivalence relations of covering-based rough sets and the type of relation-based rough sets were established. Lynn Deer et al. [6] studied 24 such neighborhood operators, which can be derived from a single covering. They also verified the equality between them, reducing the original set to 13 different neighborhood operators. For the latter, they established a partial order showing which operators produce smaller or larger neighborhoods than the others. Li Zhang et al.[7,8] combined the extended rough set theory with the mature MADM problem solving methods and proposed several types of covering-based general multigranulation intuitionistic fuzzy rough set models by using four types of intuitionistic fuzzy neighborhoods. Sang-Eon Han [9,10] set a starting point for establishing a CRS for an LFC-Space and developed the notions of accuracy of rough set approximations. Further, he gave two kinds of rough membership functions and two new rough concepts of digital topological rough set. Qingyuan Xu et al.[11] proposed a rough set method to deal with a class of set covering problem, called unicost set covering problem, which is a well-known problem in binary optimization. Liwen Ma[12] considered some types of neighborhood-related covering rough sets by introducing a new notion of complementary neighborhood. Smarandache[13] proposed the concept of neutrosophic sets in 1999, pointing out that neutrosophic sets is a set composed of the truth-membership, indeterminacy-membership and falsity-membership. Compared with previous models, it can better describe the support, neutrality and opposition of fuzzy concepts. Because of the complexity of practical problems in real life, Wang et al.[14] proposed interval neutrosophic sets(INS) and proved various properties of interval neutrosophic sets, which are connected to operations and relations over interval neutrosophic sets. Nguyen Tho Thong et al.[15] presented a new concept called dynamic interval-valued neutrosophic sets for such the dynamic decision-making applications. Irfan Deli[16] defined the notion of the interval valued neutrosophic soft sets, which is a combination of an interval valued neutrosophic sets and a soft sets. And introduced some definition and properties of interval valued neutrosophic soft sets. Hua Ma et al. [17,18] utilized the INS theory to propose a time-aware trustworthiness ranking prediction approach to selecting the highly trustworthy cloud service meeting the user-specific requirements and a time-aware trustworthy service selection approach with tradeoffs between performance costs and potential risks because of the deficiency of the traditional value prediction approaches. Ye jun[19] defined the Hamming and Euclidean distances between INS and proposed the similarity measures between INS based on the relationship between similarity measures and distances. Hongyu Zhang et al.[20] Defined the operations for INS and put forward a comparison approach based on the related research of interval valued intuitionistic fuzzy sets. Wei Yang et al.[21] developed a new multiple attribute decision-making method based on the INS and linear assignment. Meanwhile he considered the correlation of information by using the Choquet integral. Peide Liu and Guolin Tang[22] combined power average and generalized weighted agammaegation operators to INS, and proposed some agammaegation operators to apply in decision making problem.

    In recent years, many scholars have studied the combined application of rough sets and neutrosophic sets. In order to make a comprehensive overview for neutrosophic fusion of rough set theory Xue Zhan-Ao et al.[23] defifined a new covering rough intuitionistic fuzzy set model in covering approximation space, which is combined by CRS and intuitionistic fuzzy sets. They discussed the properties of lower and upper approximation operators and extended covering rough intuitionistic fuzzy set in rough sets from single-granulation to multi-granulation. Hai-Long Yang et al.[24] proposed single valued neutrosophic rough sets by combining single valued neutrosophic sets and rough sets. They also studied the hybrid model by constructive and axiomatic approaches. Hai-Long Yang et al.[25] combined INS with rough sets and proposed a generalized interval neutrosophic rough sets based on interval neutrosophic relation.They explored the hybrid model through the construction method and the axiomatic method. At the same time, the generalized interval neutrosophic approximation lower and upper approximation operators were defined by the construction method. In this paper we will study the interval neutrosophic covering rough set (INCRS), which is combined by the CRS and INS, and discuss the properties of it. Further we will give the complete proof of them. In order to do so, the remainder of this paper is shown as follows. In Section 2, we briefly review the basic concepts and operational rules of INS and CRS. In Section 3, we propose the definition and the properties of INCRS and give some easy cases to describe it. In Section 4, we discuss some theorems for INCRS and prove them completely. In Section 5, we give a simple application of Interval Neutrosophic Covering Rough Sets. In Section 6, we conclude the paper.

    This section gives a brief overview of concepts and definitions of interval neutrosophic sets, and covering rough sets.

    Definition 2.1.[13] Let X be a space of points (objects), with a class of elements in X denoted by x. A neutrosophic set A in X is summarized by a truth-membership function TA(x), an indeterminacy-membership function IA(x), and a falsity-membership function FA(x).The functions TA(x), IA(x), FA(x) are real standard or non-standard subsets of ]0,1+[. That is TA(x):X]0,1+[IA(x):X]0,1+ and FA(x):X]0,1+[.

    There is restriction on the sum of TA(x),IA(x) and FA(x), so 0supTA(x)+supIA(x)+supFA(x)3+. As mentioned above, it is hard to apply the neutrosophic set to solve some real problems. Hence, Wang et al presented interval neutrosophic set, which is a subclass of the neutrosophic set and mentioned the definition as follows:

    Definition 2.2.[13] Let X be a space of points (objects), with a class of elements in X denoted by x. A single-valued neutrosophic set N in X is summarized by a truth-membership function TN(x), an indeterminacy-membership function IN(x), and a falsity-membership function FN(x). Then an INS A can be denoted as follows:

    A={x,TA(x),IA(x),FA(x)xX} (2.1)

    where TA(x)=[TLA(x),TUA(x)], IA(x)=[ILA(x),IUA(x)], FA(x)=[FLA(x),FUA(x)][0,1] for xX. Meanwhile, the sum of TA(x)IA(x), and FA(x) fulfills the condition 0TA(x)+IA(x)+FA(x)3.

    For convenience, we refer to A=TA,IA,FA=[TLA,TUA],[ILA,IUA],[FLA,FUA] as an interval neutrosophic number (INN), which is a basic unit of INS. In addition, let X=[1,1],[0,0],[0,0] be the biggest interval neutrosophic number, and =[0,0],[1,1],[1,1] be the smallest interval neutrosophic number.

    Definition 2.3.[13] The complement of an INS A=TA,IA,FA=[TLA,TUA],[ILA,IUA],[FLA,FUA] is denoted by AC and which is defined as AC=[FLA,FUA],[1IUA,1ILA],[TLA,TUA]. For any x,yX, an INS 1y and its complement 1X{y} are defined as follows:

    T1y(x)={[1,1],x=y[0,0],xy,I1y(x)=F1y(x)={[0,0],x=y[1,1],xy
    T1x(y)(x)={[0,0],x=y[1,1],xy,I1x1y(x)=F1x(y)(x)={[1,1],x=y[0,0],xy

    Definition 2.4.[16] A={x,TA(x),IA(x),FA(x)} and B={x,TB(x),IB(x),FB(x)} are two interval neutrosophic sets, where TA(x)=[TLA(x),TUA(x)], IA(x)=[ILA(x),IUA(x)], FA(x)=[FLA(x),FUA(x)], and TB(x)=[TLB(x),TUB(x)], IB(x)=[ILB(x),IUB(x)], FB(x)=[FLB(x),FUB(x)], then

    ABTA(x)TB(x),IA(x)IB(x),FA(x)FB(x)ABTA(x)TB(x),IA(x)IB(x),FA(x)FB(x)A=BTA(x)=TB(x),IA(x)=IB(x),FA(x)=FB(x)

    And it satisfies that:

    TA(x)TB(x)TLA(x)TLB(x),TUA(x)TUB(x)TA(x)TB(x)TLA(x)TLB(x),TUA(x)TUB(x)TA(x)=TB(x)TLA(x)=TLB(x),TUA(x)=TUB(x)

    If A and B do not satisfy the above relationship, then they are said to be incompatible.

    Definition 2.5. A and B are two INNs, we have the following basic properties of INNs.

     (1) AAB,BAB (2) ABA,ABB (3) (AB)c=ACBC; (4) (AC)C=A

    Definition 2.6.[25] Let X be a finite set space of points (objects), and R be an equivalence relation on X. Denote by X/R the family of all equivalence classes induced by R. Obviously X/R gives a partition of X. (X, R) is called an interval neutrosophic approximation space. For xX, the lower and upper approximations of A are defined as below:

    R(A)={xX|[x]RA},R+(A)={xX|[x]RA},

    where

    [x]R={yX|(x,y)R}. It follows that R(A)AR+(A)

    If R(A)R+(A), A is called a rough set.

    Definition 2.7.[3] Let X be a space of points (objects) and C={C1,C2,,Cm} be a family of subsets of X. If none of the elements in C is empty and mi=1Ci=X, then C is called a covering of X, and (X,C) is called a covering approximation space.

    Definition 2.8.[3] Let (X,C) be a covering approximation space. For any xX, the neighborhood of x is defined as mi=1{CiC|xCi}, which is denoted by Nx.

    Definition 2.9.[24] Let (X,C) be a covering approximation space. For any xX, the lower and upper approximations of A are defined as below:

    C(A)={xX|NxA},C+(A)={xX|NxA}

    Based on the definition of neighborhood, the new covering rough models can be obtained.

    We will give the definition of interval neutrosophic covering rough sets in this section, meanwhile we'll also use some examples for the sake of intuition. In addition, we will given some properties and their proofs of INCRS.

    Definition 3.1. Let X be a space of points (objects). For any [s,t][0,1] and C={C1,C2,,Cm}, where Ci={Tc,iIci,Fci} and CiINS(i=1,2,,m). For xX,CkC, then Ck(x)[s,t], where TCk(x)[s,t],ICk(x)[1t,1s], FCk(x)[1t,1s]. Then C is called a interval neutrosophic [s,t] covering of X.

    Definition 3.2. Let C={C1,C2,,Cm} be an interval neutrosophic [s,t] covering of X. If 0[s,t][s,t], C is an interval neutrosophic [s,t]covering of X.

    Proof. C={C1,C2,,Cm} is a interval neutrosophic [s,t] covering of X. ThusCk(x)[s,t], and satisfy TCk(x)[s,t],ICk(x)[1t,1s], FCk(x)[1t,1s]. when 0[s,t][s,t], we can get 0[s,t][s,t]TCk(x) and 0ICk(x)[1s,1t][1s,1t],0FCk(x)[1s,1t][1s,1t]. So C is a interval neutrosophic left[s,t] covering of X.

    Definition 3.3.[26] Suppose C={C1,C2,,Cm} is an interval neutrosophic [s,t] covering of X. Ifs=t=β, then C is called a interval neutrosophic β covering of X.

    Definition 3.4. Suppose C={C1,C2,,Cm} is an interval neutrosophic [s,t] covering of X, where Ci={Tc,iIci,Fci} and CiINS(i=1,2,,m). For xX, the interval neutrosophic [s,t] neighborhood of x is defined as follows:

    N[s,t]x(y)={CiC|TCi(x)[s,t],ICi(x)[1t,1s],FCi(x)[1t,1s]}.

    Definition 3.5.[26] Let C={C1,C2,,Cm} be an interval neutrosophic [s,t] covering of X, where Ci={Tc,iIci,Fci} and CiINS(i=1,2,,m). If s=t=β, then the interval neutrosophic [s,t] neighborhood of x is degraded as the interval neutrosophic β neighborhood of x.

    Theorem 3.6. Let C={C1,C2,,Cm} be an interval neutrosophic [s,t] covering of X, where Ci={Tc,iIci,Fci} and CiINS(i=1,2,,m). x,y,zX, some propositions are shown as follows:

    (1) N[s,t]x(x)[s,t];

    (2) if N[s,t]x(y)[s,t] and N[s,t]y(z)[s,t], then N[s,t]x(z)[s,t];

    (3) CixX{N[s,t]x|Ci(x)[s,t]},i{1,2,,m};

    (4) if [s1,t1][s2,t2][s,t], then N[s1,t1]xN[s2,t2]x.

    Proof. (1)

    N[s,t]x(x)=(TCi(x)[s,t],ICi(x)[1t,1s],FCi(x)[1t,1s])(x)=(Ci(x)[s,t]Ci)(x)=Ci(x)[s,t]Ci(x)[s,t].

    (2)

    If N[s,t]x(y)[s,t], then N[s,t]x(y)=(TCi(x)[s,t],ICi(x)[1t,1s],FCi(x)[1t,1s]Ci)(y) =(Ci(x)[s,t]Ci)(y)=Ci(x)[s,t]Ci(y)[s,t], thus Ci(x)[s,t]Ci(y)[s,t], similarly, it can be obtained that Ci(y)[s,t]Ci(z)[s,t]. So Ci(x)[s,t]Ci(z)[s,t], thus N[s,t]x(z)=(TCi(x)[s,t],ICi(x)[1t,1s],FCi(x)[1t,1s]Ci)(z)=(Ci(x)[s,t]Ci)(z)=Ci(x)[s,t]Ci(z)[s,t]

    (3)

    N[s,t]x={CiC|TCi(x)[s,t],ICi(x)[1t,1s],FCi(x)[1t,1s]}=(Ci(x)[s,t]Ci)Ci, hence for any xX, it can be obtained that CixX{N[s,t]x(x)|Ci(x)[s,t]},(i=1,2,,m)

    (4)

    {CiC|TCi(x)[s1,t1],ICi(x)[1t1,1s1],FCi(x)[1t1,1s1]}={CiC|Ci(x)[s1,t1]}. When [s1,t1][s2,t2], it is obvious that {CiC|Ci(x)[s1,t1]}{CiC|Ci(x)[s2,t2]}, then {CiC|Ci(x)[s1,t1]}{CiC|Ci(x)[s2,t2]},, that is N[s1,t1]xN[s2,t2]x.

    Example 1. Let X be a space of a points(objects), with a class of elements in X denoted by x, C={C1,C2,C3,C4} is a interval neutrosophic covering of X, which is shown in Table 1. Set [s,t]=[0.4,0.5], and it can be gotten that C is a interval neutrosophic [0.4,0.5] covering of X.

    Table 1.  The interval neutrosophic [0.4,0.5] covering of X.
    C1 C2 C3 C4
    x1 [0.4,0.5],[0.2,0.3],[0.3,0.4] [0.4,0.6],[0.1,0.3],[0.2,0.4] [0.7,0.9],[0.2,0.3],[0.4,0.5] [0.4,0.5],[0.3,0.4],[0.5,0.7]
    x2 [0.6,0.7],[0.1,0.2],[0.2,0.3] [0.6,0.7],[0.1,0.2],[0.2,0.3] [0.3,0.6],[0.2,0.3],[0.3,0.4] [0.5,0.7],[0.2,0.3],[0.1,0.3]
    x3 [0.3,0.6],[0.3,0.5],[0.8,0.9] [0.5,0.6],[0.2,0.3],[0.3,0.4] [0.4,0.5],[0.2,0.4],[0.7,0.9] [0.3,0.5],[0.0,0.2],[0.2,0.4]
    x4 [0.7,0.8],[0.0,0.1],[0.1,0.2] [0.6,0.7],[0.1,0.2],[0.1,0.3] [0.6,0.7],[0.3,0.4],[0.8,0.9] [0.4,0.5],[0.5,0.6],[0.3,0.4]

     | Show Table
    DownLoad: CSV

    N[0.4,0.5]x1=C1C2C3,N[0.4,0.5]x2=C1C2C4,N[0.4,0.5]x3=C2C4,N[0.4,0.5]x4=C1C2.

    The interval neutrosophic [0.4,0.5] neighborhood of xi(i=1,2,3,4) is shown in Table 2. Obviously, the interval neutrosophic [0.4,0.5] neighborhood of xi(i=1,2,3,4) is covering of X.

    Table 2.  The interval neutrosophic [0.4,0.5] neighborhood of xi(i=1,2,3,4).
    x1 x2 x3 x4
    N[0.4,0.5]x1 [0.4,0.5],[0.2,0.3],[0.4,0.5] [0.3,0.6],[0.3,0.5],[0.8,0.9] [0.3,0.5],[0.2,0.4],[0.7,0.9] [0.6,0.7],[0.3,0.4],[0.8,0.9]
    N[0.4,0.5]x2 [0.4,0.5],[0.3,0.4],[0.5,0.7] [0.5,0.7],[0.2,0.3],[0.2,0.3] [0.3,0.5],[0.2,0.4],[0.3,0.4] [0.4,0.5],[0.5,0.6],[0.3,0.4]
    N[0.4,0.5]x3 [0.4,0.5],[0.3,0.4],[0.5,0.7] [0.5,0.7],[0.2,0.3],[0.2,0.3] [0.3,0.5],[0.2,0.3],[0.3,0.4] [0.4,0.5],[0.5,0.6],[0.3,0.4]
    N[0.4,0.5]x4 [0.4,0.5],[0.2,0.3],[0.3,0.4] [0.6,0.7],[0.1,0.2],[0.2,0.3] [0.3,0.6],[0.2,0.3],[0.3,0.4] [0.6,0.7],[0.1,0.2],[0.1.0.3]

     | Show Table
    DownLoad: CSV

    The interval neutrosophic [s,t] covering was presented in the previous section. Based on this, the coverage approximation space can be obtained.

    Definition 3.7.[26] Let C={C1,C2,,Cm} be an interval neutrosophic [s,t] covering of X, where Ci={TciIci,Fci} and CiINS(i=1,2,,m). Then (X,C) is called a interval neutrosophic [s,t] covering approximation space.

    Definition 3.8. Let (X,C) be an interval neutrosophic [s,t] covering approximation space, for any A INS, the lower approximation operator C_[s,t](A) and the upper approximation operator ¯C[s,t](A) of interval neutrosophic A are defined as follows: C_[s,t](A)={TC_[s,t](A),IC_[s,t](A),FC_[s,t](A)},¯C[s,t](A)={T¯C[s,t](A),I¯C[s,t](A),F¯C[s,t](A)}, where

    TC_[s,t](A)={TA(y)FN[s,t]x(y)|yX},IC_[s,t](A)={IA(y)([1,1]IN[s,t]x(y))|yX},
    FC_[s,t](A)={FA(y)TN[s,t]x(y)|yX},T¯C[s,t](A)={TA(y)TN[s,t]x(y)|yX},
    I¯C[s,t](A)={IA(y)IN[s,t]x(y))T|yX},F¯C[s,t](A)={FA(y)FN[s,t]x(y)|yX}.

    For any xX, then A is called an interval neutrosophic [s,t] covering rough set, if C_[s,t](A)¯C[s,t](A).

    Example 2. Let A be a interval neutrosophic set, where

    A(x1)=[0.4,0.6],[0.2,0.4],[0.3,0.4],A(x2)=[0.4,0.5],[0.1,0.3],[0.2,0.4],
    A(x3)=[0.4,0.5],[0.2,0.5],[0.3,0.6],A(x4)=[0.3,0.5],[0.2,0.4],[0.4,0.6].

    Then the lower approximation operator C_[0.4,0.5](A) and the upper approximation operator ¯C[0.4,0.5](A) of interval neutrosophic A can be calculated by Definition 3.8.

    C_[0.4,0.5](A)(x1)=[0.4,0.6],[0.2,0.5],[0.4,0.6],C_[0.4,0.5](A)(x2)=[0.3,0.5],[0.2,0.5],[0.4,0.5],
    C_[0.4,0.5](A)(x3)=[0.3,0.5],[0.2,0.5],[0.4,0.5],C_[0.4,0.5](A)(x4)=[0.3,0.5],[0.2,0.5],[0.4,0.6].
    ¯C[0.4,0.5](A)(x1)=[0.4,0.5],[0.2,0.4],[0.4,0.5],¯C[0.4,0.5](A)(x2)=[0.4,0.5],[0.2,0.3],[0.2,0.4],
    ¯C[0.4,0.5](A)(x1)=[0.4,0.5],[0.2,0.3],[0.2,0.4],¯C[0.4,0.5](A)(x2)=[0.4,0.5],[0.1,0.3],[0.2,0.4].

    In this section we'll give you some theorems about INCRS and a complete proof of them.

    Theorem 1. (1) C_[s,t](X)=X,¯C[s,t]()=;

    (2) C_[s,t](AC)=(¯C[s,t](A))C,¯C[s.t](AC)=(C_[s,t](A))C;

    (3) C_[s,t](AB)=C_[s,t](A)C_[s,t](B),¯C[s,t](AB)=¯C[s,t](A)¯C[s,t](B);

    (4) If AB, then C_[s,t](A)C_[s,t](B),¯C[s,t](A)¯C[s,t](B);

    (5) C_[s,t](AB)C_[s,t](A)C_[s,t](B),¯C[s,t](AB)¯C[s,t](A)¯C[s,t](B);

    (6) If 0[s,t][s,t], then C_[s,t](A)C_[s,t](A),¯C[s,t](A)¯C[s,t](A).

    proof. (1) TC_[s,t](X)={TX(y)FN[s,t]x(y)|yX}=[1,1],

    IC_[s,t](X)={IX(y)([1,1]IN[s,t]x(y))|yX}=[0,0],

    FC_[s,t](X)={FX(y)TN[s,t]x(y)|yX}=[0,0],

    C_[s,t](X)=TC_[s,t](X),IC_[s,t](X),FC_[s,t](X)=[1,1],[0,0],[0,0]=X;

    T¯C[s,t]()={T(y)TN[s,t]x(y)|yX}=[0,0],

    I¯C[s,t]()={I(y)IN[s,t]x(y)|yX}=[1,1],

    F¯C[s,t]()={F(y)FN[s,t]x(y)|yX}=[1,1],

    ¯C[s,t]()=T¯C[s,t](),I¯C[s,t](),F¯C[s,t]()=[0,0],[1,1],[1,1]=.

    (2) AC=FA,[1,1]IA,TA,

    TC_[s,t](AC)={FA(y)FN[s,t]x(y)|yX}=F¯C[s,t](A),

    IC_[s,t](AC)={([1,1]IA(y))([1,1]IN[s,t]x(y))|yX}

    =[1,1]{IA(y)IN[s,t]x(y)|yX}=[1,1]I¯C[s,t](A)

    FC_[s,t](AC)={TA(y)TN[s,t]x(y)|yX}=T¯C[s,t](A),

    C_[s,t](AC)={TC_[s,t](AC),IC_[s,t](AC),FC_[s,t](AC)}={F¯C[s,t](A),[1,1]I¯C[s,t](A),T¯C[s,t](A)}

    (¯C[s,t](A))C={F¯C[s,t](A),[1,1]I¯C[s,t](A),T¯C[s,t](A)}=C_[s,t](AC).

    Similarly, it can be gotten that ¯C[s.t](AC)=(C_[s,t](A))C

    (3) AB={TATB,IAIB,FAFB},

    TC_[s,t](AB)={(TA(y)TB(y))FN[s,t]x(y)|yX}

    ={(TA(y)FN[s,t]x(y))(TB(y)FN[s,t]x(y))|yX}=TC_[s,t](A)TC_[s,t](B),

    IC_[s,t](AB)={(IA(y)IB(y))([1,1]IN[s,t]x(y))|yX}

    ={(IA(y)([1,1]IN[s,t]x(y)))(IB(y)([1,1]IN[s,t]x(y)))|yX} =IC_[s,t](A)IC_[s,t](A),

    FC_[s,t](AB)={(FA(y)FB(y))TN[s,t]x(y)|yX}

    ={(FA(y)(TN[s,t]x(y)))(FB(y)(TN[s,t]x(y)))|yX}=FC_[s,t](A)FC_[s,t](A),

    C_[s,t](AB)={TC_[s,t](AB),IC_[s,t](AB),FC_[s,t](AB)}

    ={TC_[s,t](A)TC_[s,t](B),IC_[s,t](A)IC_[s,t](A),FC_[s,t](A)FC_[s,t](A)}=C_[s,t](A)C_[s,t](B).

    Similarly, it can be gotten that ¯C[s,t](AB)=¯C[s,t](A)¯C[s,t](B)

    (4) If AB, then TATB,IAIB,FAFB.

    When TATB, then {TA(y)FN[s,t]x(y)|yX}{TB(y)FN[s,t]x(y)|yX},

    thus {TA(y)FN[s,t]x(y)|yX} {TB(y)FN[s,t]x(y)|yX},

    hence {TB(y)FN[s,t]x(y)|yX}, {TB(y)FN[s,t]x(y)|yX}, that is TC_[s,t](A)TC_[s,t](B).

    When IAIB, then {IA(y)(1IN[s,t]x(y))|yX}{IB(y)([1,1]IN[s,t]x(y))|yX}.

    Thus {IA(y)([1,1]IN[s,t]x(y))|yX}{IB(y)([1,1]IN[s,t]x(y))|yX},

    hence IC_[s,t](A)IC_[s,t](A).

    When FAFB, then {FA(y)TN[s,t]x(y)|yX}{FB(y)TN[s,t]x(y)|yX},

    thus {FA(y)TN[s,t]x(y)|yX}{FB(y)TN[s,t]x(y)|yX}, so FC_[s,t](A)FC_[s,t](A),C_[s,t](A)C_[s,t](B).

    Similarly, it can be gotten that ¯C[s,t](A)¯C[s,t](B).

    (5) It is obvious that AAB,BAB, ABA,ABB.

    So C_[s,t](A)C_[s,t](AB),C_[s,t](B)C_[s,t](AB), ¯C[s,t](AB)¯C[s,t](A), ¯C[s,t](AB)¯C[s,t](B).

    Hence C_[s,t](A)C_[s,t](B)C[s,t](AB), ¯C[s,t](AB)¯C[s,t](A)¯C[s,t](B).

    (6) If 0[s,t][s,t], then N[s,t]xN[s,t]x. Thus TN[s,t]xTN[s,t]x, IN[s,t]xIN[s,t]x, FN[s,t]xFN[s,t]x, hence {TA(y)FN[s,t]x(y)|yX}{TA(y)FN[s,t]x(y)|yX},

    {IA(y)([1,1]IN[s,t]x(y))|yX}{IA(y)([1,1]IN[s,t]x(y))|yX},

    {FA(y)TN[s,t]x(y)|yX}{FA(y)TN[s,t]x(y)|yX}.

    That is C_[s,t](A)C_[s,t](A). Similarly, it can be gotten that ¯C[s,t](A)¯C[s,t](A).

    Theorem 2. Let (X,C) be an interval neutrosophic [s,t] covering approximation space, then the following statements are equivalent:

    (1) C_[s,t]()=;

    (2) ¯C[s,t](X)=X;

    (3) For anyxX, {yX|CiC((Ci(x)[s,t])(Ci(y)=X))}.

    Proof. {yX|CiC((Ci(x)[s,t])(Ci(y)=X))} means for each xX and Ci(x)[s,t],yX such that Ci(y)=X, satisfying N[s,t]x(y)=X.

    (1)(3) If C_[s,t]()=, then

    C_[s,t]()={FN[s,t]x(y),([1,1]IN[s,t]x(y)),TN[s,t]x(y)|yX}= yX,

    FN[s,t]x(y)=[0,0],IN[s,t]x(y)=[0,0],TN[s,t]x(y)=[1,1], that is N[s,t]x(y)=X.

    (3)(2) If N[s,t]x(y)=X, then

    ¯C[s,t](X)={TN[s,t]x(y),IN[s,t]x(y),FN[s,t]x(y)|yX}={[1,1],[0,0],[0,0]}=X.

    (2)(1) It is proved by the rotation of C_ and ¯C. So they are equivalent.

    Theorem 3. Let (X,C) be an interval neutrosophic [s,t] covering approximation space. Ais an INS and B is an constant interval neutrosophic set, where B=[α,α+],[β,β+],[γ,γ+]. It satisfies that for any xX, [α,α+],[β,β+],[γ,γ+](x)=[α,α+],[β,β+],[γ,γ+].

    If {yX|CiC((Ci(x)[s,t])(Ci(y)=X))}, then

    (1) C_[s,t](B)=B,¯C[s,t](B)=B;

    (2) C_[s,t](AB)=C_[s,t](A)B,¯C[s,t](AB)=¯C[s,t](A)B.

    Proof. (1) {yX|CiC((Ci(x)[s,t])(Ci(y)=X))} means for each xX and Ci(x)[s,t], yX, such that Ci(y)=X, then N[s,t]x(y)=X.

    TB_[s,t]={[α,α+]FN[s,t]x(y)|yX}=[α,α+],

    IB_[s,t]={[β,β+]([1,1]IN[s,t]x(y))|yX}=[β,β+],

    FB_[s,t]={[γ,γ+]TN[s,t]x(y)|yX}=[γ,γ+].

    So that C_[s,t](B)=B. Similarly, it can be gotten that ¯C[s,t](B)=B.

    (2) TC_[s,t](AB)={(TA(y)[α,α+])FN[s,t]x(y)|yX}={TA(y)FN[s,t]x(y)|yX}[α,α+],

    IC_[s,t](AB)={(IA(y)[β,β+])([1,1]IN[s,t]x(y))|yX}

    ={IA(y)([1,1]IN[s,t]x(y))|yX}[β,β+],

    FC_[s,t](AB)={(FA(y)[γ,γ+])TN[s,t]x(y)|yX}={FA(y)TN[s,t]x(y)|yX}[γ,γ+],

    Thus C_[s,t](AB)=C_[s,t](A)B. Similarly, it can be proofed that ¯C[s,t](AB)=¯C[s,t](A)B.

    Corollary. When α=α+=α,β=β+=β,γ=γ+=γ,B=α,β,γ It can be gotten that

    (1) C_[s,t]α,β,γ=α,β,γ,¯C[s,t]α,β,γ=α,β,γ;

    (2) C_[s,t](Aα,β,γ)=C_[s,t](A)α,β,γ,¯C[s,t](Aα,β,γ=¯C[s,t](A)α,β,γ.

    The proof is omitted.

    Theorem 4. Let (X,C) be an interval neutrosophic [s,t] covering approximation space. Ais an INS and AX, for any xX, there are

    (1) ¯C[s,t](1y)(x)=N[s,t]x(y);

    (2) C_[s,t](1X{y})(x)=(N[s,t]x(y))C.

    Proof. T¯C[s,t](1y)(x)={T1y(z)TN[s,t]x(z)|zX}

    =(T1y(y)TN[s,t]x(y))(zX{y}(T1y(z)TN[s,t]x(z)))

    =([1,1]TN[s,t]x(y))([0,0]TN[s,t]x(z))=TN[s,t]x(y),

    I¯C[s,t](1y)(x)={I1y(z)IN[s,t]x(z)|zX}

    =(I1y(y)IN[s,t]x(y))(zX{y}(I1y(z)IN[s,t]x(z)))

    =([0,0]IN[s,t]x(y))([1,1]IN[s,t]x(z))=IN[s,t]x(y),

    F¯C[s,t](1y)(x)={F1y(z)FN[s,t]x(z)|zX}

    =(F1y(y)FN[s,t]x(y))(zX{y}(F1y(z)FN[s,t]x(z)))

    =([0,0]FN[s,t]x(y))([1,1]FN[s,t]x(z))=FN[s,t]x(y).

    So ¯C[s,t](1y)(x)=N[s,t]x(y).

    Similarly, it can be gotten that C_[s,t](1X{y})(x)=(N[s,t]x(y))C, and the proof process is omitted.

    Theorem 5. Let (X,C) be an interval neutrosophic [s,t] covering approximation space. Ais an INS and AX, for any xX, if (N[s,t]x)CAN[s,t]x, then C_[s,t](C_[s,t](A))C_[s,t](A)A¯C[s,t](A)¯C[s,t](¯C[s,t](A)).

    Proof. (N[s,t]x)C=FN[s,t]x,([1,1]IN[s,t]x),TN[s,t]x.

    When (N[s,t]x)CA,thus FN[s,t]xTA, [1,1]IN[s,t]xIA,TN[s,t]xFA,

    so TC_[s,t](A)={TA(y)FN[s,t]x(y)|yX}={TA(y)|yX}TA,

    IC_[s,t](A)={IA(y)([1,1]IN[s,t]x(y))|yX}={IA(y)|yX}IA,

    FC_[s,t](A)={FA(y)TN[s,t]x(y)|yX}={FA(y)|yX}FA.

    That is C_[s,t](A)A. Similarly, A¯C[s,t](A).

    According to theorem 1(4), C_[s,t](C_[s,t](A))C_[s,t](A)A¯C[s,t](A)¯C[s,t](¯C[s,t](A)).

    Theorem 5 gives a sufficient condition for C_[s,t](A)A¯C[s,t](A), and then theorem 6 will give the necessary condition.

    Theorem 6. Let (X,C) be an interval neutrosophic [s,t] covering approximation space. AX, if xX,Ci(x)[s,t]Ci(x)=X(i={1,2,m}), and then

    C_[s,t](A)AˉC[s,t](A).

    Proof. xX,Ci(x)[s,t]Ci(x)=X(i={1,2,m}), which means xX,N[s,t]x=X=[1,1],[0,0],[0,0].

    TC_[s,t](A)={TA(y)FN[s,t]x(y)|yX}={TA(y)[0,0]|yX}={TA(y)|yX}TA,

    IC_[s,t](A)={IA(y)([1,1]IN[s,t]x(y))|yX}={IA(y)[1,1]|yX}={IA(y)|yX}IA,

    FC_[s,t](A)={FA(y)TN[s,t]x(y)|yX}={FA(y)[1,1]|yX}={FA(y)|yX}FA.

    So C_[s,t](A)A.

    T¯C[s,t](A)={TA(y)TN[s,t]x(y)|yX}={TA(y)|yX}TA,

    I¯C[s,t](A)={IA(y)IN[s,t]x(y)|yX}={IA(y)|yX}IA,

    F¯C[s,t](A)={FA(y)FN[s,t]x(y)|yX}={FA(y)|yX}FA.

    So A¯C[s,t](A).

    Hence C_[s,t](A)A¯C[s,t](A).

    Theorem 7. Let C={C1,C2,,Cm} be an interval neutrosophic [s,t] covering of X.AINS,¯C and C_ are the upper and lower approximation operator, which are defined in defination 3.8. Then we can get that:

     (1)  C is serial C_[s,t]α,β,λ=α,β,λ,α,β,λ[0,1],C_[s,t]()=,ˉC[s,t]α,β,λ=α,β,λ,α,β,λ[0,1],ˉC[s,t](X)=X;

     (2) C is reflexive C_[s,t](A)A,AˉC[s,t](A);

     (3) C is symmetric C_[s,t](1X(y}))(x)=C_[s,t](1X{x})(y),x,yX,ˉC[s,t](1y)(x)=ˉC[s,t](1x)(y),x,yX;

     (4) C is transitive C_[s,t](A)C_[s,t](C_[s,t](A)),ˉC[s,t](ˉC[s,t](A))ˉC[s,t](A).

    Proof. (1) When C is serial, then it satisfies yX and N[s,t]x(y)=X. So it can be proved by Theorem 3, Theorem 4 and Deduction.

    (2) When C is reflexive, then N[s,t]x(x)=X=[1,1],[0,0],[0,0]

    TC_[s,t](A)(x)={TA(y)FN[s,t]x(y)|yX}TA(x)FN[s,t]x(x)=TA(x),

    IC_[s,t](A)(x)={IA(y)([1,1]IN[s,t]x(y))|yX}IA(x)[1,1]=IA(x),

    FC_[s,t](A)(x)={FA(y)TN[s,t]x(y)|yX}FA(x)[1,1]=FA(x).

    That is C_[s,t](A)A.

    If C_[s,t](A)A, let A=1X(x), and x,yX, then

    TN[s,t]x(x)=(TN[s,t]x(x)[1,1])[0,0]

    =(TN[s,t]x(x)F(1X{x})(x))(yX{x}(TN[s,t]x(y)F(1X{x})(y)))

    ={TN[s,t]x(y)F(1X{x})(y)|yX}

    =FC_[s,t](1X{x})(x)F(1X{x})(x)=[1,1],

    [1,1]IN[s,t]x(x)={([1,1]IN[s,t]x(x))[1,1]}[0,0]

    ={([1,1]IN[s,t]x(x))I(1X{x})(x)}{yX{x}(([1,1]IN[s,t]x(y))I(1X{x})(y))}

    ={I(1X{x})(x)([1,1]IN[s,t]x(y))|yX}

    =IC_[s,t](1X{x})(x)I(1X{x})(x)=[1,1],

    so IN[s,t]x(x)=[0,0].

    FN[s,t]x(x)={FN[s,t]x(x))[0,0]}[1,1]

    ={FN[s,t]x(x)T(1X{x})(x)}{yX{x}(FN[s,t]x(y)T(1X{x})(y))}

    ={T(1X{x})(x)FN[s,t]x(y)|yX}

    =TC_[s,t](1X{x})(x)T(1X{x})(x)=[0,0].

    That is N[s,t]x(x)=[1,1],[0,0],[0,0]=X. So C is reflexive. Meanwhile, it is easy to prove the other part by the same way.

    (3) TC_[s,t](1X{x})(y)={T(1X{x})(z)FN[s,t]y(z)|zX}

    ={FN[s,t]y(x)T(1X{x})(x)}{zX{x}(FN[s,t]y(z)T(1X{x})(z))}

    ={FN[s,t]y(x)[0,0]}[1,1] =FN[s,t]y(x), TC_[s,t](1X{y})(x)={T(1X{y})(z)FN[s,t]x(z)|zX}

    ={FN[s,t]x(y)T(1X{y})(y)}{zX{y}(FN[s,t]x(z)T(1X{y})(z))}

    ={FN[s,t]x(y)[0,0]}[1,1]

    =FN[s,t]x(y),

    IC_[s,t](1X{x})(y)={I(1X{x})(z)([1,1]IN[s,t]y(z))|zX}

    ={([1,1]IN[s,t]y(x))I(1X{x})(x)}{zX{x}(([1,1]IN[s,t]y(z))I(1X{x})(z))}

    ={([1,1]IN[s,t]y(x))[1,1]}[0,0]

    =[1,1]IN[s,t]y(x),

    IC_[s,t](1X{y})(x)={I(1X{y})(z)([1,1]IN[s,t]x(z))|zX}

    ={([1,1]IN[s,t]x(y))I(1X{y})(y)}{zX{y}(([1,1]IN[s,t]x(z))I(1X{y})(z))}

    ={([1,1]IN[s,t]x(y))[1,1]}[0,0]

    =[1,1]IN[s,t]x(y),

    FC_[s,t](1X{x})(y)={F(1X{x})(z)TN[s,t]y(z)|zX}

    ={TN[s,t]y(x)F(1X{x})(x)}{zX{x}(TN[s,t]y(z)F(1X{x})(z))}

    ={TN[s,t]y(x)[1,1]}[0,0]

    =TN[s,t]y(x),

    FC_[s,t](1X{y})(x)={F(1X{y})(z)TN[s,t]x(z)|zX}

    ={TN[s,t]x(y)F(1X{y})(y)}{zX{y}(TN[s,t]x(z)F(1X{y})(z))}

    ={TN[s,t]x(y)[1,1]}[0,0]

    =TN[s,t]x(y).

    So when is symmetric, it satisfies TN[s,t]x(y)=TN[s,t]y(x),IN[s,t]x(y)=IN[s,t]y(x), FN[s,t]x(y)=FN[s,t]y(x), that is N[s,t]x(y)=N[s,t]y(x), then

    TC_[s,t](1X{x})(y)=TC_[s,t](1X{y})(x),

    IC_[s,t](1X{x})(y)=IC_[s,t](1X{y})(x),FC_[s,t](1X{x})(y)=FC_[s,t](1X{y})(x)

    That is C_[s,t](1X{x})(y)=C_[s,t](1X{y})(x).

    It is similar to get ¯C[s,t](1y)(x)=¯C[s,t](1x)(y), and the proof is omitted.

    (4) If C is transitive, then {TN[s,t]x(y)TN[s,t]y(z)|yX}TN[s,t]x(z),

    {IN[s,t]x(y)IN[s,t]y(z)|yX}IN[s,t]x(z), {FN[s,t]x(y)FN[s,t]y(z)|yX}FN[s,t]x(z).

    TC_[s,t](C_[s,t](A))(x)={TC_[s,t](A)(y)FN[s,t]x(y)|yX}={{TA(z)FN[s,t]y(z)|zX}FN[s,t]x(y)|yX}

    =yXzX(TA(z)FN[s,t]x(z)FN[s,t]x(y))=zX(yX(FN[s,t]y(z)FN[s,t]x(y))TA(z))

    zX(FN[s,t]x(z)TA(z))=TC_[s,t](A)(x),

    IC_[s,t](C_[s,t](A))(x)={IC_[s,t](A)(z)(1IN[s,t]x(y))|yX}

    ={{IA(z)(1IN[s,t]y(z))|zX}(1IN[s,t]x(y))|yX}

    =yXzX(IA(z)(1IN[s,t]y(z))(1IN[s,t]x(y))=zX((1yX(IN[s,t]y(z)IN[s,t]x(y))IA(z)

    zX(1IN[s,t]x(z))IA(z)=IC_[s,t](A)(x),

    FC_[s,t](C_[s,t](A))(x)={FC_[s,t](A)(y)TN[s,t]x(y)|yX}={{FA(z)TN[s,t]y(z)|zX}TN[s,t]x(y)|yX}

    =yXzX(FA(z)TN[s,t]y(z)TN[s,t]x(y))=zX(yX(TN[s,t]y(z)TN[s,t]x(y)))FA(z)

    zX(TN[s,t]x(z)FA(z))=FC_[s,t](A)(x),

    so C_[s,t](A)C_[s,t](C_[s,t](A)).

    Similarly, it can be gotten that ¯C[s,t](¯C[s,t](A))¯C[s,t](A). If C_[s,t](A)C_[s,t](C_[s,t](A)), let A=1X{x} and x,y,zX, \\ from the proving process of (3), we have

    TN[s,t]x(z)=FC_[s,t](1X{z})(x)FC_[s,t](C_[s,t](1X{z})(x) ={FC_[s,t](1X{z})(y)TN[s,t]x(y)|yX}

    ={TN[s,t]y(z)TN[s,t]x(y)|yX},

    [1,1]IN[s,t]x(z)=IC_[s,t](1X{z})(x)IC_[s,t](C_[s,t](1X{z})(x) {IC_[s,t](1X{z})(y)([1,1]IN[s,t]x(y))|yX}

    ={([1,1]IN[s,t]y(z)([1,1]IN[s,t]x(y)|yX},

    Thus IN[s,t]x(z){IN[s,t]y(z)IN[s,t]x(y)|yX}.

    FN[s,t]x(z)=TC_[s,t](1X{z})(x)TC_[s,t](C_[s,t](1X{z})(x) ={TC_[s,t](1X{z})(y)FN[s,t]x(y)|yX}

    ={FN[s,t]y(z)FN[s,t]x(y)|yX},

    Therefore C is transitive. When ¯C[s,t](¯C[s,t](A))¯C[s,t](A), it can be proved C is transitive by the same way.

    In medicine, a combination of drugs is usually used to cure a disease. Suppose, X={xj,j=1,2,,n} is a collection of n drugs, V={yi,i=1,2,,m} are m important symptom (such as fever, cough, fatigue, phlegm, etc.) of diseases (such as: 2019-NCOV, etc.), and Ci(xj) represents the effective value of medication for the treatment of symptoms.

    Let [s,t] be the evaluation range. For each drugxjX, if there is at least one symptom yiV that causes the effective value of drug xj for the treatment of symptom yi to be in the [s,t] interval, thenC={Ci:i=1,2,,m} is the interval neutrosophic [s,t] covering on X. Thus, for each drug xj, we consider the set of symptoms {yi:Ci(xj)[s,t]}.

    The interval neutrosophic [s,t] neighborhood of xj is N[s,t]xj={CiC|TCi(xj)[s,t],ICi(xj)[1t,1s],FCi(xj)[1t,1s]}(xk)=(Ci(x)[s,t]Ci)Ci(xk,k=1,2,,n. This represents the effective value interval for each drugxk for all symptoms in the symptom set {yi:Ci(xj)[s,t]}. We consider as the upper and lower thresholds of effective values of s and t. If they are lower than the lower threshold, there will be no therapeutic effect; if they are higher than the upper threshold, the therapeutic effect will be too strong, and it is easy to cause other side effects to the body during the treatment (regardless of the situation of reducing the usage). Let an interval neutrosophic set of A represent the therapeutic ability of all drugs in X that can cure disease X. Since Ais imprecise, we consider the approximation of A, that is, the lower approximation and the upper approximation of interval neutrosophic covering rough.

    Example 3. LetX be a space of a points (objects), with a class of elements in X denoted by x, being a interval neutrosophic covering of X, which is shown in Table 3. Set [s,t]=[0.4,0.5], and it can be gotten that C is a interval neutrosophic [s,t] covering of X. N[0.4,0.5]x1=C1C2C3,N[0.4,0.5]x2=C1C4,N[0.4,0.5]x3=C2C4,N[0.4,0.5]x4=C2C3. The interval neutrosophic [s,t]neighborhood of xi(i=1,2,3,4) is shown in Table 4. Obviously, the interval neutrosophic[s,t]neighborhood of xi(i=1,2,3,4) is covering of X.

    Table 3.  The interval neutrosophic [0.4,0.5] neighborhood of xi(i=1,2,3,4).
    C1 C2 C3 C4
    x1 [0.4,0.5],[0.2,0.3],[0.4,0.5] [0.4,0.6],[0.1,0.3],[0.3,0.5] [0.7,0.9],[0.2,0.3],[0.4,0.5] [0.4,0.5],[0.3,0.4],[0.6,0.7]
    x2 [0.6,0.7],[0.1,0.2],[0.2,0.3] [0.2,0.4],[0.1,0.2],[0.2,0.3] [0.3,0.6],[0.3,0.5],[0.8,0.9] [0.5,0.7],[0.2,0.3],[0.4,0.6]
    x3 [0.3,0.5],[0.2,0.3],[0.4,0.5] [0.5,0.6],[0.2,0.3],[0.3,0.4] [0.3,0.5],[0.2,0.4],[0.3,0.4] [0.5,0.6],[0.0,0.2],[0.3,0.4]
    x4 [0.7,0.8],[0.6,0.7],[0.1,0.2] [0.6,0.7],[0.1,0.2],[0.1,0.3] [0.6,0.7],[0.3,0.4],[0.3,0.5] [0.3,0.5],[0.5,0.6],[0.6,0.7]

     | Show Table
    DownLoad: CSV
    Table 4.  The interval neutrosophic [0.4,0.5] covering of X.
    C1 C2 C3 C4
    x1 [0.4,0.5],[0.2,0.3],[0.4,0.5] [0.2,0.4],[0.3,0.5],[0.8,0.9] [0.3,0.5],[0.2,0.4],[0.4,0.5] [0.6,0.7],[0.6,0.7],[0.3,0.5]
    x2 [0.4,0.5],[0.3,0.4],[0.6,0.7] [0.5,0.7],[0.2,0.3],[0.4,0.6] [0.3,0.5],[0.2,0.3],[0.4,0.5] [0.3,0.5],[0.6,0.7],[0.6,0.7]
    x3 [0.4,0.5],[0.3,0.4],[0.6,0.7] [0.2,0.4],[0.2,0.3],[0.4,0.6] [0.5,0.6],[0.2,0.3],[0.3,0.5] [0.3,0.5],[0.5,0.6],[0.6,0.7]
    x4 [0.4,0.6],[0.2,0.3],[0.4,0.5] [0.2,0.4],[0.3,0.5],[0.8,0.9] [0.3,0.5],[0.2,0.4],[0.3,0.5] [0.6,0.7],[0.3,0.4],[0.3,0.5]

     | Show Table
    DownLoad: CSV

    Let A be an interval neutrosophic set, and

    A(x1)=[0.2,0.4],[0.2,0.4],[0.3,0.4],A(x2)=[0.5,0.7],[0.1,0.3],[0.2,0.4],A(x3)=[0.3,0.4],[0.2,0.5],[0.3,0.5],A(x4)=[0.5,0.6],[0.2,0.4],[0.4,0.6].

    The lower approximation operator C_[0.4,0.5](A) and the upper approximation operator ¯C[0.4,0.5](A) of the intelligent set A in the interval can be obtained by definition 3.9.

    C_[0.4,0.5](A)(x1)=[0.4,0.5],[0.2,0.5],[0.4,0.6],C_[0.4,0.5](A)(x2)=[0.4,0.5],[0.2,0.5],[0.3,0.5],C_[0.4,0.5](A)(x3)=[0.3,0.5],[0.2,0.5],[0.3,0.5],C_[0.4,0.5](A)(x4)=[0.3,0.5],[0.2,0.5],[0.4,0.6].¯C[0.4,0.5](A)(x1)=[0.5,0.6],[0.2,0.4],[0.4,0.5],¯C[0.4,0.5](A)(x2)=[0.5,0.7],[0.2,0.3],[0.4,0.5],¯C[0.4,0.5](A)(x3)=[0.3,0.5],[0.2,0.3],[0.3,0.5],¯C[0.4,0.5](A)(x4)=[0.5,0.6],[0.2,0.4],[0.3,0.5].

    Then A is the interval neutrosophic [s,t] covering of X.

    And we can get that

    (1) A(x2)[0.4.0.5],C_[0.4,0.5](A)(x2)[0.4.0.5],¯C[0.4,0.5](A)(x2)[0.4.0.5]. Therefore, drug x2plays an important role in the treatment of diseaseA.

    (2) A(x3)<[0.4.0.5],C_[0.4,0.5](A)(x3)<[0.4.0.5],¯C[0.4,0.5](A)(x3)<[0.4.0.5]. So drug x3 has no effect on the treatment of diseaseA.

    (3) A(x1)<[0.4.0.5],C_[0.4,0.5](A)(x1)[0.4.0.5],¯C[0.4,0.5](A)(x1)[0.4.0.5]. Therefore, drug x1 has less effect on the treatment of disease A than drug x2 and drug x4.

    In this paper, we propose the interval neutrosophic covering rough sets by combining the CRS and INS. Firstly, the paper introduces the definition of interval neutrosophic sets and covering rough sets, where the covering rough set is defined by neighborhood. Secondly, Some basic properties and operation rules of interval neutrosophic sets and covering rough sets are discussed. Thirdly, the definition of interval neutrosophic covering rough sets are proposed. Then, this paper put forward some theorems and give their proofs of interval neutrosophic covering rough sets. Lastly, we give the numerical example to apply the interval neutrosophic covering rough sets in the real life.

    The authors wish to thank the editors and referees for their valuable guidance and support in improving the quality of this paper. This research was funded by the Humanities and Social Sciences Foundation of Ministry of Education of the Peoples Republic of China (17YJA630115).

    The authors declare that there is no conflict of interest.


    Acknowledgments



    There were no sources of funding of this study. This review report was not published in a repository.

    Conflict of interest



    Both the authors declare no conflicts of interest in this paper.

    [1] Kilbanski A, Adams-Campbell L, Bassford T, et al. (2001) Osteoporosis prevention, diagnosis and therapy. JAMA 285: 785-795. doi: 10.1001/jama.285.6.785
    [2]  U.S. National Library of Medicine Medline Plus. Osteoporosis–overview. [Accessed September 22, 2017]. Updated September 2017. Available from: http://bit.ly/2sQEYYg.
    [3] Gomez-Cabello A, Ara I, Gonzalez-Agüero A, et al. (2012) Effects of training on bone mass in older adults: A systematic review. Sports Med 42: 301-325. doi: 10.2165/11597670-000000000-00000
    [4] Nguyen ND, Ahlborg HG, Center JR, et al. (2007) Residual lifetime risk of fracture in women and men. J Bone Miner Res 22: 781-788. doi: 10.1359/jbmr.070315
    [5] Melton LJ, Chrischilles EA, Cooper C, et al. (1992) Perspective. How many women have osteoporosis? J Bone Miner Res 7: 1005-1010. doi: 10.1002/jbmr.5650070902
    [6] Center JR, Bliuc D, Nguyen TV, et al. (2007) Risk of subsequent fracture after low trauma fracture in men and women. JAMA 297: 387-394. doi: 10.1001/jama.297.4.387
    [7] Bolander ME (1992) Regulation of fracture repair by growth factors. Proc Soc Exp Biol Med 200: 165-170. doi: 10.3181/00379727-200-43410A
    [8] Einhorn TA (1998) The cell and molecular biology of fracture healing. Clin Orthop Relat Res 355: S7-S21. doi: 10.1097/00003086-199810001-00003
    [9] Ferguson C, Alpern E, Miclau T, et al. (1999) Does adult fracture repair recapitulate embryonic skeletal formation? Mech Dev 87: 57-66. doi: 10.1016/S0925-4773(99)00142-2
    [10] Gerstenfeld LC, Cullinane DM, Barnes GL, et al. (2003) Fracture healing as a post-natal developmental process: Molecular, spatial, and temporal aspects of its regulation. J Cell Biochem 88: 873-884. doi: 10.1002/jcb.10435
    [11] Vortkamp A, Pathi S, Peretti GM, et al. (1998) Recapitulation of signals regulating embryonic bone formation during postnatal growth and in fracture repair. Mech Dev 71: 65-76. doi: 10.1016/S0925-4773(97)00203-7
    [12] Sambrook P, Cooper C (2006) Osteoporosis. Lancet 367: 2010-2018. doi: 10.1016/S0140-6736(06)68891-0
    [13] Dennison E, Medley J, Cooper C (2006) Who is at risk of osteoporosis? Women's Health Med 3: 152-154. doi: 10.1383/wohm.2006.3.4.152
    [14] Bonnick SL (1998)  Bone Densitometry in Clinical Practice New Jersey: Humana Press Inc..
    [15] Binkley N, Adler R, Bilezikian JP, et al. (2014) Osteoporosi diagnosis in men: The T-score controversy revisited. Curr Osteoporor Rep 12: 403-409. doi: 10.1007/s11914-014-0242-z
    [16] Kanis JA, Johansson H, Harvey NC, et al. (2018) A brief history of FRAX. Arch Osteoporos 13: 118. doi: 10.1007/s11657-018-0510-0
    [17] Kanis JA, Johnell O, Oden A, et al. (2008) FRAX and the assessment of fracture probability in men and women from the UK. Osteoporos Int 19: 385-397. doi: 10.1007/s00198-007-0543-5
    [18] Lorentzon M, Cummings SR (2015) Osteoporosis: The evolution of a diagnosis. J Intern Med 277: 650-661. doi: 10.1111/joim.12369
    [19] Deloumeau A, Molto A, Roux C, et al. (2017) Determinants of short-term fracture risk in patients with a recent history of low-trauma non-vertebral fracture. Bone 105: 287-291. doi: 10.1016/j.bone.2017.08.018
    [20] Ferrari SL, Abrahamsen B, Napoli N, et al. (2018) Diagnosis and management of bone fragility in diabetes: AN emerging challenge. Osteoporos Int 29: 2585-2596. doi: 10.1007/s00198-018-4650-2
    [21] Cummings SR, Black DM, Rubin SM (1989) Lifetime risks of hip, Colles', or vertebral fracture and coronary heart disease among white postmenopausal women. Arch Intern Med 149: 2445-2448. doi: 10.1001/archinte.1989.00390110045010
    [22] Butcher JL, MacKenzie EJ, Cushing B, et al. (1996) Long term outcomes after low extremity trauma. J Trauma 41: 4-9. doi: 10.1097/00005373-199607000-00002
    [23] MacKenzie EJ, Bose MJ, Pollak AN, et al. (2005) Long term persistence of disability following severe lower limb trauma. Results of a seven year follow up. J Bone Joint Surg Am 87: 1801-1809.
    [24] Qaseem A, Snow V, Shekelle P, et al. (2008) Pharmacologic treatment of low bone density or osteoporosis to prevent fractures: A clinical practice guideline from the American College of Physicians. Ann Intern Med 149: 404-415. doi: 10.7326/0003-4819-149-6-200809160-00007
    [25] Russell RG (2011) Bisphosphonates: The first 40 years. Bone 49: 2-19. doi: 10.1016/j.bone.2011.04.022
    [26] Odvina CV, Zerwekh JE, Rao DS, et al. (2005) Severely suppressed bone turnover: A potential complication of alendronate therapy. J Clin Endocrinol Metab 90: 1294-1301. doi: 10.1210/jc.2004-0952
    [27] Yates J (2013) A meta-analysis characterizing the dose-response relationships for three oral nitrogen-containing bisphosphonates in postmenopausal women. Osteoporos Int 24: 253-262. doi: 10.1007/s00198-012-2179-3
    [28] Zhang J, Wang R, Zhao YL, et al. (2012) Efficacy of intravenous zoledronic acid in the prevention and treatment of osteoporosis: A meta-analysis. Asian Pac J Trop Med 5: 743-748. doi: 10.1016/S1995-7645(12)60118-7
    [29] Crandall CJ, Newberry SJ, Diamant A, et al. (2014) Comparative effectiveness of pharmacologic treatments to prevent fractures: An updated systematic review. Ann Intern Med 161: 711-723. doi: 10.7326/M14-0317
    [30] Barrionuevo P, Kapoor E, Asi N, et al. (2019) Efficacy of pharmacological therapies for the prevention of fractures in postmenopausal women: A network meta-analysis. J Clin Endocrinol Metab 104: 1623-1630. doi: 10.1210/jc.2019-00192
    [31] Freemantle N, Cooper C, Diez-Perez A, et al. (2013) Results of indirect and mixed treatment comparison of fracture efficacy for osteoporosis treatments: A meta-analysis. Osteoporos Int 24: 209-217. doi: 10.1007/s00198-012-2068-9
    [32] Hak DJ, Fitzpatrick D, Bishop JA, et al. (2014) Delayed union and nonunions: Epidemiology, clinical issues, and financial aspects. Injury 45: S3-S7. doi: 10.1016/j.injury.2014.04.002
    [33] Hegde V, Jo JE, Andreopoulou P, et al. (2016) Effect of osteoporosis medications on fracture healing. Osteoporos Int 27: 861-871. doi: 10.1007/s00198-015-3331-7
    [34] Saito T, Sterbenz JM, Malay S, et al. (2017) Effectiveness of anti-osteoporotic drugs to prevent secondary fragility fractures: Systematic review and meta-analysis. Osteoporos Int 28: 3289-3300. doi: 10.1007/s00198-017-4175-0
    [35] Duckworth AD, McQueen MM, Tuck CE, et al. (2019) Effect of alendronic acid on fracture healing: A multicenter randomized placebo-controlled trial. J Bone Miner Res 34: 1025-1032. doi: 10.1002/jbmr.3679
    [36] Lim EJ, Kim JT, Kim CH, et al. (2019) Effect of preoperative bisphosphonate treatment on fracture healing after internal fixation treatment of intertrochanteric femoral fractures. Hip Pelvis 31: 75-81. doi: 10.5371/hp.2019.31.2.75
    [37] Goodship AE, Walker PC, McNally D, et al. (1994) Use of a bisphosphonate (pamidronate) to modulate fracture repair in ovine bone. Ann Oncol 5: S53-S55. doi: 10.1093/annonc/5.suppl_2.S53
    [38] Peter CP, Cook WO, Nunamaker DM, et al. (1996) Effect of alendronate on fracture healing and bone remodeling in dogs. J Orthop Res 14: 74-79. doi: 10.1002/jor.1100140113
    [39] Miettinen SS, Jaatinen J, Pelttari A, et al. (2009) Effect of locally administered zoledronic acid on injury-induced intramembranous bone regeneration and osseointegration of a titanium implant in rats. J Orthop Sci 14: 431-436. doi: 10.1007/s00776-009-1352-9
    [40] Skripitz R, Johansson HR, Ulrich SD, et al. (2009) Effect of alendronate and intermittent parathyroid hormone on implant fixation in ovariectomized rats. J Orthop Sci 14: 138-143. doi: 10.1007/s00776-008-1311-x
    [41] Goldhahn J, Feron JM, Kanis J, et al. (2012) Implications for fracture healing of current and new osteoporosis treatments: An ESCEO consensus paper. Calcif Tissue Int 90: 343-353. doi: 10.1007/s00223-012-9587-4
    [42] Gerstenfeld LC, Sacks DJ, Pelis M, et al. (2009) Comparison of effects of the bisphosphonate alendronate versus the RANKL inhibitor denosumab on murine fracture healing. J Bone Miner Res 24: 196-208. doi: 10.1359/jbmr.081113
    [43] Adami S, Libanati C, Boonen S, et al. (2012) Denosumab treatment in postmenopausal women with osteoporosis does not interfere with fracture-healing: Results from the freedom trial. J Bone Jt Surg Am 94: 2113-2119. doi: 10.2106/JBJS.K.00774
    [44] Hanley DA, Adachi JD, Bell A, et al. (2012) Denosumab: Mechanism of action and clinicaloutcomes. Int J Clin Pract 66: 1139-1146. doi: 10.1111/ijcp.12022
    [45] Bone HG, Bolognese MA, Yuen CK, et al. (2011) Effects of denosumab treatment and discontinuation on bone mineral density and bone turnover markers in post menopausal women with low bone mass. J Clin Endocrinol Metab 96: 972-980. doi: 10.1210/jc.2010-1502
    [46] Papaionnou A, Morin S, Cheung AM, et al. (2010) 2010 clinical practice guidelines for the diagnosis and management of osteoporosis in Canada: Summary. CMAJ 182: 1864-1873. doi: 10.1503/cmaj.100771
    [47] Russell RG, Rogers MJ (1999) Bisphophonates: From the laboratory to clinic and back again. Bone 25: 97-106. doi: 10.1016/S8756-3282(99)00116-7
    [48] Papapoulos S, Chapurlat R, Libanati C, et al. (2012) Five years of denosumab exposure in women with postmenopausal osteoporosis: Results from the first two years of the FREEDOM extension. J Bone Miner Res 27: 694-701. doi: 10.1002/jbmr.1479
    [49] Seeman E, Delmas PD, Henley DA, et al. (2010) Microarchitectural deterioration of cortical and trabecular bone: Differing effects of denosumab and alendronate. J Bone Miner Res 25: 1886-1894. doi: 10.1002/jbmr.81
    [50] Lv F, Cai X, Yang W, et al. (2020) Denosumab or romosozumab therapy and risk of cardiovascular events in patients with primary osteoporosis: Systemic review and meta-analysis. Bone 130: 115121. doi: 10.1016/j.bone.2019.115121
    [51] Riggs BL, Khosla S, Melton LJ (2002) Sex steroids and the construction and conservation of the adult skeleton. Endo Rev 23: 279-302. doi: 10.1210/edrv.23.3.0465
    [52] Sahiner T, Aktan E, Kaleli B, et al. (1998) The effects of postmenopausal hormone replacement therapy on sympathetic skin response. Maturitas 30: 85-88. doi: 10.1016/S0378-5122(98)00049-8
    [53] Cho CH, Nuttall ME (2001) Therapeutic potential of oestrogen receptor ligands in development for osteoporosis. Expert Opin Emerg Drugs 6: 137-154.
    [54] Riggs BL, Hartmann LC (2003) Selective estrogen-receptor modulators–mechanisms of action and application to clinical practice. N Engl J Med 348: 618-629. doi: 10.1056/NEJMra022219
    [55] Nilsson S, Koehler KF (2005) Oestrogen receptors and selective oestrogen receptor modulators: Molecular and cellular pharmacology. Basic Clin Pharmacol Toxicol 96: 15-25. doi: 10.1111/j.1742-7843.2005.pto960103.x
    [56] Gennari L, Merlotti D, Valleggi F, et al. (2007) Selective estrogen receptor modulators for postmenopausal osteoporosis: Current state of development. Drugs Aging 24: 361-379. doi: 10.2165/00002512-200724050-00002
    [57] Gennari L, Merlotti D, De Paola V, et al. (2008) Bazedoxifene for the prevention of postmenopausal osteoporosis. Ther Clin Risk Manag 4: 1229-1242. doi: 10.2147/TCRM.S3476
    [58] Delmas PD, Bjarnason NH, Mitlak BH, et al. (1997) Effects of raloxifene on bone mineral density, serum cholesterol concentrations, and uterine endometrium in postmenopausal women. N Engl J Med 337: 1641-1647. doi: 10.1056/NEJM199712043372301
    [59] Lufkin EG, Whitaker MD, Nickelsen T, et al. (1998) Treatment of established postmenopausal osteoporosis with raloxifene: A randomized trial. J Bone Miner Res 13: 1747-1754. doi: 10.1359/jbmr.1998.13.11.1747
    [60] Ettinger B, Black DM, Mitlak BH, et al. (1999) Reduction of vertebral fracture risk in postmenopausal women with osteoporosis treated with raloxifene: Results from a 3-year randomized clinical trial. Multiple outcomes of raloxifene evaluation (MORE) Investigators. JAMA 282: 637-645. doi: 10.1001/jama.282.7.637
    [61] Maricic M, Adachi JD, Sarkar S, et al. (2002) Early effects of raloxifene on clinical vertebral fractures at 12 months in postmenopausal women with osteoporosis. Arch Intern Med 162: 1140-1143. doi: 10.1001/archinte.162.10.1140
    [62] Langdahl BL, Silverman S, Fujiwara S, et al. (2018) Real-world effectiveness of teriparatide on fracture reduction in patients with osteoporosis and comorbidities or risk factors for fractures: Integrated analysis of 4 prospective observational studies. Bone 116: 58-66. doi: 10.1016/j.bone.2018.07.013
    [63] Lou S, Lv H, Wang G, et al. (2016) The effect of teriparatide on fracture healing of osteoporotic patients: A meta-analysis of randomized controlled trials. Biomed Res Int .
    [64] Kim SM, Kang KC, Kim JW, et al. (2017) Current role and application of teriparatide in fracture healing of osteoporotic patients: A systematic review. J Bone Metab 24: 65-73. doi: 10.11005/jbm.2017.24.1.65
    [65] Johansson T (2016) PTH 1-34 (teriparatide) may not improve healing in proximal humerus fractures. A randomized, controlled study of 40 patients. Acta Orthop 87: 79-82. doi: 10.3109/17453674.2015.1073050
    [66] Huang TW, Chuang PY, Lin SJ, et al. (2016) Teriparatide improves fracture healing and early functional recovery in treatment of osteoporotic intertrochanteric fractures. Medicine (Baltimore) 95: e3626. doi: 10.1097/MD.0000000000003626
    [67] Kim SJ, Park HS, Lee DW, et al. (2019) Short-term daily teriparatide improve postoperative functional outcome and fracture healing in unstable intertrochanteric fractures. Injury 50: 1364-1370. doi: 10.1016/j.injury.2019.06.002
    [68] Bernhardsson M, Aspenberg P (2018) Abaloparatide versus teriparatide: A head to head comparison of effects on fracture healing in mouse models. Acta Orthop 89: 674-677. doi: 10.1080/17453674.2018.1523771
    [69] Miller PD, Hattersley G, Riis BJ, et al. (2016) Effect of abaloparatide vs placebo on new vertebral fractures in postmenopausal women with osteoporosis: A randomized clinical trial. JAMA 316: 722-733. doi: 10.1001/jama.2016.11136
    [70] Miller PD, Hattersley G, Lau E, et al. (2018) Bone mineral density response rates are greater in patients treated with abaloparatide compared with those treated with placebo or teriparatide: Results from the ACTIVE phase 3 trial. Bone 120: 137-140. doi: 10.1016/j.bone.2018.10.015
    [71] Langdahl BL, Silverman S, Fujiwara S, et al. (2018) Real-world effectiveness of teriparatide on fracture reduction in patients with osteoporosis and comorbidities or risk factors for fractures: Integrated analysis of 4 prospective observational studies. Bone 116: 58-66. doi: 10.1016/j.bone.2018.07.013
    [72] Wojda SJ, Donahue SW (2018) Parathyroid hormone for bone regeneration. J Orthop Res 36: 2586-2594. doi: 10.1002/jor.24075
    [73] Cheng ZY, Ye T, Ling QY, et al. (2018) Parathyroid hormone promotes osteoblastic differentiation of endothelial cells via the extracellular signal-regulated protein kinase 1/2 and nuclear factor-kappaB signaling pathways. Exp Ther Med 15: 1754-1760.
    [74] Swarthout JT, D'Alonzo RC, Selvamurugan N, et al. (2002) Parathyroid hormone-dependent signaling pathways regulating genes in bone cells. Gene 282: 1-17. doi: 10.1016/S0378-1119(01)00798-3
    [75] Krishnan V, Bryant HU, Macdougald OA (2006) Regulation of bone mass by wnt signaling. J Clin Investig 116: 1202-1209. doi: 10.1172/JCI28551
    [76] Sims NA, Ng KW (2014) Implications of osteoblast-osteoclast interactions in the management of osteoporosis by antiresorptive agents denosumab and odanacatib. Curr Osteoporos Rep 12: 98-106. doi: 10.1007/s11914-014-0196-1
    [77] Wan M, Yang C, Li J, et al. (2008) Parathyroid hormone signaling through low-density lipoprotein-related protein 6. Genes Dev 22: 2968-2979. doi: 10.1101/gad.1702708
    [78] Li X, Zhang Y, Kang H, et al. (2005) Sclerostin binds to LRP5/6 and antagonizes canonical Wnt signaling. J Biol Chem 280: 19883-19887. doi: 10.1074/jbc.M413274200
    [79] Koide M, Kobayashi Y (2018) Regulatory mechanisms of sclerostin expression during bone remodeling. J Bone Miner Metab 37: 9-17. doi: 10.1007/s00774-018-0971-7
    [80] Keller H, Kneissel M (2005) SOST is a target gene for PTH in bone. Bone 37: 148-158. doi: 10.1016/j.bone.2005.03.018
    [81] Pazianas M (2015) Anabolic effects of PTH and the “anabolic window”. Trends Endocrinol Metab 26: 111-113. doi: 10.1016/j.tem.2015.01.004
    [82] Chandler H, Lanske B, Varela A, et al. (2018) Abaloparatide, a novel osteoanabolic PTHrP analog, increases cortical and trabecular bone mass and architecture in orchiectomized rats by increasing bone formation without increasing bone resorption. Bone 120: 148-155. doi: 10.1016/j.bone.2018.10.012
    [83] Kakar S, Einhorn TA, Vora S, et al. (2007) Enhanced chondrogenesis and Wnt signaling in PTH-treated fractures. J Bone Miner Res 22: 1903-1912. doi: 10.1359/jbmr.070724
    [84] Andreassen TT, Ejersted C, Oxlund H (1999) Intermittent parathyroid hormone (1-34) treatment increases callus formation and mechanical strength of healing rat fractures. J Bone Miner Res 14: 960-968. doi: 10.1359/jbmr.1999.14.6.960
    [85] Yu M, D'Amelio P, Tyagi AM, et al. (2018) Regulatory T cells are expanded by teriparatide treatment in humans and mediate intermittent PTH-induced bone anabolism in mice. EMBO Rep 19: 156-171. doi: 10.15252/embr.201744421
    [86] Liu Y, Wang L, Kikuiri T, et al. (2011) Mesenchymal stem cell-based tissue regeneration is governed by recipient T lymphocytes via IFN-gamma and TNF-alpha. Nat Med 17: 1594-1601. doi: 10.1038/nm.2542
    [87] Subbiah V, Madsen VS, Raymond AK, et al. (2010) Of mice and men: Divergent risks of teriparatide-induced osteosarcoma. Osteoporos Int 21: 1041-1045. doi: 10.1007/s00198-009-1004-0
    [88] Lou S, Lv H, Li Z, et al. (2018) Parathyroid hormone analogues for fracture healing: Protocol for a systematic review and meta-analysis of randomised controlled trials. BMJ Open 8: e019291. doi: 10.1136/bmjopen-2017-019291
    [89] Ozturan KE, Demir B, Yucel I, et al. (2011) Effect of strontium ranelate on fracture healing in the osteoporotic rats. J Orthop Res 24: 1651-1661.
    [90] Li YF, Luo E, Feng G, et al. (2011) Systemic treatment with strontium ralenate promotes tibial fracture healing in the osteoporotic rats. J Orthop Res 29: 138-142. doi: 10.1002/jor.21204
    [91] Tarantino U, Celi M, Saturnino L, et al. (2010) Strontium ralenate and bone healing: Report of two cases. Clin Cases Miner Bone Metab 7: 65-68.
    [92] Alegre DN, Ribeiro C, Sousa C, et al. (2012) Possible benefits of strontium ralenate in complicated long bone fractures. Rheumatol Int 32: 439-443. doi: 10.1007/s00296-010-1687-8
    [93] Scaglione M, Fabbri L, Casella F, et al. (2016) Strontium ranelate as an adjuvant for fracture healing: Clinical, radiological, and ultrasound findings in a randomized controlled study on wrist fractures. Osteoporos Int 27: 211-218. doi: 10.1007/s00198-015-3266-z
    [94] Murray SS, Murray BEJ, Wang JC, et al. (2016) The history and histology of bone morphogenetic protein. Histol Histopathol 31: 721-732.
    [95] Canalis E, Economides AN, Gazzerro E (2003) Bone morphogenetic proteins, their antagonists, and the skeleton. Endocr Rev 24: 218-235. doi: 10.1210/er.2002-0023
    [96] Onishi T, Ishidou Y, Nagamine T, et al. (1998) Distinct and overlapping patterns of localization of bone morphogenetic protein (BMP) family members and a BMP type II receptor during fracture healing in rats. Bone 22: 605-612. doi: 10.1016/S8756-3282(98)00056-8
    [97] Pluhar GE, Turner AS, Pierce AR, et al. (2006) A comparison of two biomaterial carriers for osteogenic protein-1 (BMP-7) in an ovine critical defect model. J Bone Jt Surg Br 88: 960-966. doi: 10.1302/0301-620X.88B7.17056
    [98] Sawyer AA, Song SJ, Susanto E, et al. (2009) The stimulation of healing within a rat calvarial defect by mPCL-TCP/collagen scaffolds loaded with rhBMP-2. Biomaterials 30: 2479-2488. doi: 10.1016/j.biomaterials.2008.12.055
    [99] Cipitria A, Reichert JC, Epari DR, et al. (2013) Polycaprolactone scaffold and reduced rhBMP-7 dose for the regeneration of critical-sized defects in sheep tibiae. Biomaterials 34: 9960-9968. doi: 10.1016/j.biomaterials.2013.09.011
    [100] Guzman JZ, Merrill RK, Kim JS, et al. (2017) Bone morphogenetic protein use in spine surgery in the United States: How have we responded to the warnings? Spine J 17: 1247-1254. doi: 10.1016/j.spinee.2017.04.030
    [101] Carragee EJ, Hurwitz EL, Weiner BK (2011) A critical review of recombinant human bone morphogenetic protein-2 trials in spinal surgery: Emerging safety concerns and lessons learned. Spine J 11: 471-491. doi: 10.1016/j.spinee.2011.04.023
    [102] James AW, LaChaud G, Shen J, et al. (2016) A Review of the clinical side effects of bone morphogenetic protein 2. Tissue Eng Part B Rev 22: 284-297. doi: 10.1089/ten.teb.2015.0357
    [103] Glaeser JD, Salehi K, Kanim LEA, et al. (2018) Anti-inflammatory peptide attenuates edema and promotes bmp-2-induced bone formation in spine fusion. Tissue Eng Part A 24: 1641-1651. doi: 10.1089/ten.tea.2017.0512
    [104] Bara JJ, Dresing I, Zeiter S, et al. (2018) A doxycycline inducible, adenoviral bone morphogenetic protein-2 gene delivery system to bone. J Tissue Eng Regen Med 12: e106-e118. doi: 10.1002/term.2393
    [105] Kolk A, Tischer T, Koch C, et al. (2016) A novel nonviral gene delivery tool of BMP-2 for the reconstitution of critical-size bone defects in rats. J Biomed Mater Res A 104: 2441-2455. doi: 10.1002/jbm.a.35773
    [106] Wang M, Park S, Nam Y, et al. (2018) Bone-fracture-targeted dasatinib-oligoaspartic acid conjugate potently accelerates fracture repair. Bioconjug Chem 29: 3800-3809. doi: 10.1021/acs.bioconjchem.8b00660
    [107] Li X, Ominsky MS, Niu QT, et al. (2008) Targeted deletion of the sclerostin gene in mice results in increased bone formation and bone strength. J Bone Miner Res 223: 860-869. doi: 10.1359/jbmr.080216
    [108] Van Lierop AH, Appelman-Dijkstra NM, Papapoulos SE (2017) Sclerostin deficiency in humans. Bone 96: 51-62. doi: 10.1016/j.bone.2016.10.010
    [109] Koide M, Kobayashi Y (2019) Regulatory mechanisms of sclerostin expression during bone remodeling. J Bone Miner Metab 37: 9-17. doi: 10.1007/s00774-018-0971-7
    [110] Regard JB, Zhong Z, Williams BO, et al. (2012) Wnt signaling in bone development and disease: Making stronger bone with Wnts. Cold Spring Harb Perspect Biol 4: a007997. doi: 10.1101/cshperspect.a007997
    [111] Shi C, Li J, Wang W, et al. (2011) Antagonists of LRP6 regulate PTH-induced cAMP generation. Ann N Y Acad Sci 1237: 39-46. doi: 10.1111/j.1749-6632.2011.06226.x
    [112] Wijenayaka AR, Kogawa M, Lim HP, et al. (2011) Sclerostin stimulates osteocyte support of osteoclast activity by a RANKL-dependent pathway. PLoS One 6: e25900. doi: 10.1371/journal.pone.0025900
    [113] Alaee F, Virk MS, Tang H, et al. (2014) Evaluation of the effects of systemic treatment with a sclerostin neutralizing antibody on bone repair in a rat femoral defect model. J Orthop Res 32: 197-203. doi: 10.1002/jor.22498
    [114] Ominsky MS, Brown DL, Van G, et al. (2015) Differential temporal effects of sclerostin antibody and parathyroid hormone on cancellous and cortical bone and quantitative differences in effects on the osteoblast lineage in young intact rats. Bone 81: 380-391. doi: 10.1016/j.bone.2015.08.007
    [115] Cosman F, Crittenden DB, Ferrari S, et al. (2018) Frame Study: The foundation effect of building bone with 1 year of romosozumab leads to continued lower fracture risk after transition to denosumab. J Bone Miner Res 33: 1219-1226. doi: 10.1002/jbmr.3427
    [116] Graeff C, Campbell GM, Pena J, et al. (2015) Administration of romosozumab improves vertebral trabecular and cortical bone as assessed with quantitative computed tomography and finite element analysis. Bone 81: 364-369. doi: 10.1016/j.bone.2015.07.036
    [117] Saag KG, Petersen J, Brandi ML, et al. (2017) Romosozumab or alendronate for fracture prevention in women with osteoporosis. N Engl J Med 377: 1417-1427. doi: 10.1056/NEJMoa1708322
    [118] Weske S, Vaidya M, Reese A, et al. (2018) Targeting sphingosine-1-phosphate lyase as an anabolic therapy for bone loss. Nat Med 24: 667-678. doi: 10.1038/s41591-018-0005-y
    [119] Xu R, Yallowitz A, Qin A, et al. (2018) Targeting skeletal endothelium to ameliorate bone loss. Nat Med 24: 823-833. doi: 10.1038/s41591-018-0020-z
    [120] Baltzer AWA, Whalen JD, Wooley P, et al. (2001) Gene therapy for osteoporosis: Evaluation in a murine ovariectomy model. Gene Ther 8: 1770-1776. doi: 10.1038/sj.gt.3301594
    [121] Feng Q, Zheng S, Zheng J (2018) The emerging role of micro RNAs in bone remodeling and its therapeutic implications for osteoporosis. Biosci Rep 38: BSR20180453. doi: 10.1042/BSR20180453
    [122] Hemigou P, Poignard A, Beaujean F, et al. (2005) Percutaneous autologous bone marrow grafting for non-unions. Influence of the number and concentration of progenitor cells. J Bone Joint Surg Am 87: 1430-1437.
    [123] Kawaguchi H, Oka H, Jingushi S, et al. (2010) A local application of recombinant human fibroblast growth factor 2 for tibial shaft fractures: A randomized placebo-controlled trial. J Bone Miner Res 12: 2735-2743. doi: 10.1002/jbmr.146
    [124] DiGiovanni CW, Lin SS, Baumhauer JF, et al. (2013) Recombinant human platelet derived growth factor-BB and beta-tricalcium phosphate (rhPDGF-BB/beta-TCP): An alternative to autologous bone graft. J Bone Joint Surg Am 95: 1184-1192. doi: 10.2106/JBJS.K.01422
    [125] Brighton CT, Black J, Friedenberg ZB, et al. (1981) A multicenter study of the treatment of non-unions with constant direct current. J Bone Joint Surg Am 63: 847-851. doi: 10.2106/00004623-198163050-00030
    [126] Brighton CT (1981) Treatment of non-unions of the Tibia with constant direct current (1980 Fitts Lecture, AAST). J Trauma 21: 189-195. doi: 10.1097/00005373-198103000-00001
    [127] Scott G, King JB (1994) A prospective, double blind trial of electrical capacitive coupling in the treatment of non-union of long bones. J Bone Joint Surg Am 76: 820-826. doi: 10.2106/00004623-199406000-00005
    [128] Sharrard WJ (1990) A double blind trial of pulsed electromagnetic fields for delayed union of tibial fractures. J Bone Joint Surg Br 72: 347-355. doi: 10.1302/0301-620X.72B3.2187877
    [129] Mollen B, De Silva V, Busse JW, et al. (2008) Electrical stimulation for long bone fracture healing: A meta-analysis of randomosed control trials. J Bone Joint Surg Am 90: 2322-2330. doi: 10.2106/JBJS.H.00111
    [130] Li S, Jiang H, Wang B, et al. (2018) Magnetic resonance spectroscopy for evaluating effect of pulsed electromagnetic fields on marrow adiposityin postmenopausal women with osteopenia. J Comput Assist Tomography 42: 792-797. doi: 10.1097/RCT.0000000000000757
    [131] Catalano A, Loddo S, Bellone F, et al. (2018) Pulsed electromagnetic fields modulate bpne metabolism via RANKL/OPG and Wnt/beta-catenin pathways in women with postmenopausal osteoporosis: A pilot study. Bone 116: 42-46. doi: 10.1016/j.bone.2018.07.010
    [132] Ziegler P, Nussler AK, Wilbrand B, et al. (2019) Pulsed electromagnetic field therapy improves osseous consolidation after high tibial osteotomy in elderly patients–A randomized placebo-controlled, double blind trial. J Clin Med 8: 2008. doi: 10.3390/jcm8112008
    [133] Leung KS, Lee WS, Tsui HF, et al. (2004) Complex tibial fracture outcomes following treatment with low-intensity pulsed ultrasound. Ultrasound Med Biol 30: 389-395. doi: 10.1016/j.ultrasmedbio.2003.11.008
    [134] Siska PA, Gruen GS, Pape HC (2008) External adjuncts to enhance fracture healing: What is the role of ultrasound? Injury 39: 1095-105. doi: 10.1016/j.injury.2008.01.015
    [135] Schortinghuis J, Bronckers AL, Stegenga B, et al. (2005) Ultrasound to stimulate early bone formation in a distraction gap: A double blind randomised clinical pilot trial in the edentulous mandible. Arch Oral Biol 50: 411-420. doi: 10.1016/j.archoralbio.2004.09.005
    [136] El-Bialy TH, Elgazzar RF, Megahed EE, et al. (2008) Effects of ultrasound modes on mandibular osteodistraction. J Dent Res 87: 953-957. doi: 10.1177/154405910808701018
    [137] Leung KS, Cheung WH, Zhang C, et al. (2004) Low intensity pulsed ultrasound stimulates osteogenic activity of human periosteal cells. Clin Orthop Relat Res 253-259. doi: 10.1097/00003086-200401000-00044
    [138] Pilla AA, Mont MA, Nasser PR, et al. (1990) Non-invasive low-intensity pulsed ultrasound accelerates bone healing in the rabbit. J Orthop Trauma 4: 246-253. doi: 10.1097/00005131-199004030-00002
    [139] Busse JW, Bhandari M, Kulkarni AV, et al. (2002) The effect of low-intensity pulsed ultrasound therapy on time to fracture healing: A meta-analysis. CMAJ 166: 437-441.
    [140] Heckman JD, Sarasohn-Kahn J (1997) The economics of treating tibia fractures. The cost of delayed unions. Bull Hosp Joint Dis 56: 63-72.
    [141] Romano CL, Zavaterelli A, Meani E (2006) Biophysical treatment of septic non-unions. Archivio di Ortopedia e Reumatologia 117: 12-13.
    [142] Mundi R, Petis S, Kaloty R, et al. (2009) Low-intensity pulsed ultrasound: Fracture healing. Indian J Orthop 43: 132-140. doi: 10.4103/0019-5413.50847
    [143] Schandelmaier S, Kaushal A, Lytvyn L, et al. (2017) Low intensity pulsed ultrasound for bone healing: A systematic review of randomised controlled trials. BMJ 356: j656.
    [144] Yadollahpour A, Rashidi S (2017) Therapeutic applications of low-intensity pulsed ultrasound in osteoporosis. Asian J Pharm 11: S1-S6.
    [145] Farmer ME, Harris T, Madans JH, et al. (1989) Anthropometric indicators and hip fracture. The NHANES I epidemiologic follow-up study. J Am Geriatr Soc 37: 9-16. doi: 10.1111/j.1532-5415.1989.tb01562.x
    [146] Cosman F, Lindsay R, LeBoff MS, et al. (2014) Clinician's guide to prevention and treatment of osteoporosis. Osteoporos Int 25: 2359-2381. doi: 10.1007/s00198-014-2794-2
    [147] National Osteoporosis Foundation Osteoporosis exercise for strong bones. Available from: https://www.nof.org/patients/fracturesfall-prevention/exercisesafe-movement/osteoporosis-exercise-for-strong-bones/.
    [148] Pfeifer M, Minne H (2005) Bone loading exercise recommendations for prevention and treament of osteoporosis. Int Osteoporosis Foundation .
    [149] (2020)  NIH Osteoporosis and Related Bone Disease National Resource Center. Health Topics: Osteoporosis. Available from: www.bones.nih.gov.
    [150] Carneiro MB, Alves DPL, Mercadante MT (2013) Physical therapy in the postoperative of proximal femur fracture in elderly. Literature review. Acta Ortop Bras 21: 175-178. doi: 10.1590/S1413-78522013000300010
    [151] Meys G, Kalmet PHS, Sanduleau S, et al. (2019) A protocol for permissive weight-bearing during allied health therapy in surgically treated fractures of the pelvis and lower extremities. J Rehabil Med 51: 290-297. doi: 10.2340/16501977-2532
    [152] Baer M, Neuhaus V, Pape HC, et al. (2019) Influence of mobilization and weight bearing on in-hospital outcome in geriatric patients with hip fractures. SICOT J 5: 4. doi: 10.1051/sicotj/2019005
    [153] Senderovich H, Kosmopoulos A (2018) An insight into the effect of exercises on the prevention of osteoporosis and associated fractures in high-risk individuals. Rambam Maimonides Med J 9: e0005. doi: 10.5041/RMMJ.10325
    [154] Benedetti MG, Furlini G, Zati A, et al. (2018) The effectiveness of physical exercise on bone density in osteoporotic patients. BioMed Res Int .
    [155] Erhan B, Ataker Y (2020) Rehabilitation of patients with osteoporotic fractures. J Clin Densitom In Press.
    [156] Atkins GJ, Welldon KJ, Wijenayaka AR, et al. (2009) Vitamin K promotes mineralization, osteoclast to osteocyte transition, and an anticatabolic phenotype by gamma-carboxylation-dependent and -independent mechanisms. Am J Physio Cell Physiol 297: C1358-1367. doi: 10.1152/ajpcell.00216.2009
    [157] Lee NK, Sowa H, Hinoi E, et al. (2007) Endocrine regulation of energy metabolism by the skeleton. Cell 130: 456-469. doi: 10.1016/j.cell.2007.05.047
    [158] Yamaguchi M, Weitzmann MN (2011) Vitamin K2 stimulates osteoblastogenesis and suppresses osteoclastogenesis by suppressing nF-kappa B activation. Int J Mol Med 27: 3-14.
    [159] Palermo A, Tuccinardi D, D'Onofrio L, et al. (2017) Vitamin K and osteoporosis: Myth or reality? Metabolism 70: 57-71. doi: 10.1016/j.metabol.2017.01.032
    [160] Rossini M, Bianchi G, Di Munno O, et al. (2006) Treatment of osteoporosis in clinical practice (TOP) study group. Determinants of adherence to osteoporosis treatment in clinical practice. Osteopros Int 17: 914-921. doi: 10.1007/s00198-006-0073-6
    [161] Bischoff-Ferrari B, Giovannucci E, Willett WC, et al. (2006) Estimation of optimal serum concentration of 25-hydroxy vitamin D for multiple health outcomes. Am J Clin Nutr 84: 18-28. doi: 10.1093/ajcn/84.1.18
    [162] Holick MF, Siris ES, Binkley N, et al. (2005) Prevalence of vitamin D inadequacy among post menopausal North American women receiving osteoporosis therapy. J Clin Endocrinol Metab 90: 3215-3224. doi: 10.1210/jc.2004-2364
    [163] Adani S, Giannini S, Bianchi G, et al. (2009) Vitamin D status and response to treatment in postmenopausal osteoporosis. Osteoporos Int 20: 239-244. doi: 10.1007/s00198-008-0650-y
    [164] Merskey HE (1986) Classification of chronic pain: Description of chronic pain syndromes and definition of pain terms. Pain .
    [165] Catalano A, Martino G, Morabito N, et al. (2017) Pain in osteoporosis: From pathophysiology to therapeutic approach. Drugs Aging 34: 755-765. doi: 10.1007/s40266-017-0492-4
    [166] Edwards MH, Dennison EM, Sayer AA, et al. (2015) Osteoporosis and sarcopenia in older age. Bone 80: 126-130. doi: 10.1016/j.bone.2015.04.016
    [167] Lange U, Teichmann J, Uhlemann C (2005) Current knowledge about physiotherapeutic strategies in osteoporosis prevention and treatment. Rheumatol Int 26: 99-106. doi: 10.1007/s00296-004-0528-z
    [168] Ehde DM, Dillworth TM, Turner JA (2014) Cognitive-behavioural therapy for individuals with chronic pain: efficacy, innovations, and direction for research. Am Psycho 69: 153-166. doi: 10.1037/a0035747
    [169] Iwamoto J, Takeda T, Sato Y, et al. (2005) Effect of whole-body vibration exercise on lumbar bone mineral density, bone turnover, and chronic back pain in postmenopausal women treated with alendronate. Aging Clin Exp Res 17: 157-163. doi: 10.1007/BF03324589
    [170] O'Connor JP, Lysz T (2008) Celecoxib, NSAIDs and the skeleton. Drugs Today (Barc) 44: 693-709. doi: 10.1358/dot.2008.44.9.1251573
    [171] Vellucci R, Consalvo M, Celidoni L, et al. (2016) Implications of analgesics use in osteoporotic-related pain treatment: focus on opioids. Clin Cases Miner Bone Metab 13: 89-92.
    [172] Adolphson P, Abbaszadegan H, Jonsson U, et al. (1993) No effects of piroxicam on osteopenia and recovery after Colles' fracture. A randomized, double-blind, placebo-controlled, prospective trial. Arch Orthop Trauma Surg 112: 127-130. doi: 10.1007/BF00449987
    [173] Davis TR, Ackroyd CE (1998) Non-steroidal anti-inflammatory agents in management of Colles' fractures. Br J Clin Prct 42: 184-189.
    [174] Bauer DC, Orwell ES, Fox KM, et al. (1996) Aspirin and NSAID use in older women: Effect on bone mineral density and fracture risk. Study of osteoporotic fractures research group. J Bone Miner Res 11: 29-35. doi: 10.1002/jbmr.5650110106
    [175] Alkhiary YM, Gerstenfeld LC, Elizabeth K, et al. (2005) Enhancement of experimental fracture-healing by systemic administration of recombinant human parathyroid hormone (PTH 1–34). J Bone Joint Surg Am 87: 731-741.
    [176] Morgan EF, ZD Mason, Bishop G, et al. (2008) Combined effects of recombinant human BMP-7 (rhBMP-7) and parathyroid hormone (1–34) in metaphyseal bone healing. Bone 43: 1031-1038. doi: 10.1016/j.bone.2008.07.251
    [177] Jorgensen NR, Schwarz P (2011) Effects of anti-osteoporosis medications on fracture healing. Curr Osteoporos Rep 9: 149-145. doi: 10.1007/s11914-011-0065-0
    [178] Sarahrudi K, Thomas A, Albrecht C, et al. (2012) Strongly enhanced levels of sclerostin during human fracture healing. J Orthop Res 30: 1549-1155. doi: 10.1002/jor.22129
    [179] Kamiya N (2012) The role of BMPs in bone anabolism and their potential targets SOST and DKK1. Curr Mol Pharmacol 5: 153-163. doi: 10.2174/1874467211205020153
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