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Research article Special Issues

Investigation of the protective and therapeutic effects of Lactobacillus casei and Saccharomyces cerevisiae in a breast cancer mouse model

  • Received: 13 December 2021 Revised: 22 April 2022 Accepted: 27 April 2022 Published: 16 May 2022
  • Introduction 

    The development of novel strategies for cancer therapy is crucial to improve standard treatment protocols.

    Aim 

    This study aimed to determine the protective and therapeutic effects of heat-killed preparations of Lactobacillus casei and Saccharomyces cerevisiae in a breast cancer mouse model.

    Methods 

    Forty-two female BALB/c mice (7–8 weeks old) were divided into six groups (seven mice per group). Four groups were injected with 107 Ehrlich ascites tumor (EAT) cells suspended in phosphate-buffered saline (PBS) subcutaneously into the left side of the mammary fat pad. Tumor growth was monitored weekly until all animals developed a palpable tumor. The tumor-bearing mice in the experimental groups received heat-killed L. casei or S. cerevisiae three times per week for 35 days. The mice in the control group received PBS. The remaining two groups received heated L. casei or S. cerevisiae and then were injected with EAT cells. After 35 days, all mice were sacrificed to determine the immune response.

    Results 

    Animals that received heated S. cerevisiae exhibited the lowest rate of tumor growth compared with the other groups. TGF-β and IL-4 secretion was increased in all mice, whereas the secretion of INF-γ and IL-10 was decreased in breast tissues. Moreover, at the histopathological level, the volume of viable tumor in the control group was higher than in the treated groups.

    Conclusion 

    Supplementary treatment with S. cerevisiae resulted in the best outcome in the breast cancer model compared with other treated and vaccinated groups.

    Citation: Kholoud Baraka, Rania Abozahra, Maged Wasfy Helmy, Nada Salah El Dine El Meniawy, Sarah M Abdelhamid. Investigation of the protective and therapeutic effects of Lactobacillus casei and Saccharomyces cerevisiae in a breast cancer mouse model[J]. AIMS Microbiology, 2022, 8(2): 193-207. doi: 10.3934/microbiol.2022016

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  • Introduction 

    The development of novel strategies for cancer therapy is crucial to improve standard treatment protocols.

    Aim 

    This study aimed to determine the protective and therapeutic effects of heat-killed preparations of Lactobacillus casei and Saccharomyces cerevisiae in a breast cancer mouse model.

    Methods 

    Forty-two female BALB/c mice (7–8 weeks old) were divided into six groups (seven mice per group). Four groups were injected with 107 Ehrlich ascites tumor (EAT) cells suspended in phosphate-buffered saline (PBS) subcutaneously into the left side of the mammary fat pad. Tumor growth was monitored weekly until all animals developed a palpable tumor. The tumor-bearing mice in the experimental groups received heat-killed L. casei or S. cerevisiae three times per week for 35 days. The mice in the control group received PBS. The remaining two groups received heated L. casei or S. cerevisiae and then were injected with EAT cells. After 35 days, all mice were sacrificed to determine the immune response.

    Results 

    Animals that received heated S. cerevisiae exhibited the lowest rate of tumor growth compared with the other groups. TGF-β and IL-4 secretion was increased in all mice, whereas the secretion of INF-γ and IL-10 was decreased in breast tissues. Moreover, at the histopathological level, the volume of viable tumor in the control group was higher than in the treated groups.

    Conclusion 

    Supplementary treatment with S. cerevisiae resulted in the best outcome in the breast cancer model compared with other treated and vaccinated groups.



    Mechanical metamaterials are engineered systems designed to create exotic materials, often prepared in laboratories using two or more components or phases. These materials exhibit unconventional mechanical properties not found in conventional materials [1]. Some notable examples include bandgap metamaterials composed of phononic crystals [2,3], which can be designed to exhibit adjustable frequency intervals where the transmission of mechanical waves is not permitted [4,5]. Tensegrity bandgap metamaterials are formed by alternating lumped masses with tensegrity units of different types, allowing for the creation of mass–spring systems that exhibit tunable frequency bandgaps. These bandgaps can be adjusted by internal and external prestress, as well as by contrasts in mass and spring properties between their elements (refer, e.g., to [6] and references tharein).

    In the case of a forced mass–spring chain, when the length of such a system is finite and/or when the presence of a boundary breaks the periodicity condition, it is well known that the Bloch–Floquet wave form solutions of the dynamic problem do not longer satisfy the equations of motion. Two alternative strategies are available to overcome such an issue. The first one consists of solving the ordinary differential equations of motion of the discrete system with aid of the competent initial conditions, and accounting for any kinematic restrictions [6]. Such a strategy is generally accurate, but has a computational disadvantage if the number of unit cells is very large. The second strategy consists of deriving a continuum model of the mass-spring system equipped with competent Partial Differential Equations (PDEs) and Boundary Conditions (BCs). Such an approach can be accurate, in the presence of a correct mathematical formulation, only when the wave-length is larger than the size ε of the unit cell. However it offers the advantage that the associated computational effort is either negligible in presence of analytical solutions to the equations of motion or dependent on the adopted mesh size in the case of finite element approximations.

    This work develops a continuum model of the tensegrity mass–spring system recently proposed in [6] (see Section 2 for a review of this model). The approach that is followed moves on from the introduction of kinematic descriptors at the continuum level for the chain domain (i.e., 'hosting' and 'resonant' displacement fields) and next introduces appropriate Piola's ansatzes to link such descriptors to those corresponding to the discrete model [7,8,9,10]. The adopted ansatz guarantees that the continuum descriptors correspond to the limits for ε0 of the discrete counterparts. Using the continuum displacement fields and a variational principle for the action functional system, the work derives the PDEs of the Maxwell chain as well as the dispersion relation at the continuum level (Section 3). Analytic solutions of the wave dynamics of the continuum model are presented in Section 4, which illustrate its main features and permit the validation of the dispersion relation presented in the previous section. The paper ends with concluding remarks and directions for future work in Section 5.

    We hereafter recall the main features of the tensegrity Maxwell chain model diffusely presented in [6], which is graphically illustrated in Figure 1. It is composed of a number of N tensegrity θ=1 prisms ('PH prisms') [14] arranged in parallel with 2N minimal regular tensegrity prisms ('PR prisms' or T3 prisms) [16]. The PH prisms are interposed between N+1 hosting masses MH, while two PR prisms are interposed between two consecutive MH masses, being separated by a resonant MR mass. The MH and MR masses exhibit characteristic sizes (rH and rR) and rotational moments of inertia JH=nHMHr2H and JR=nRMRr2R with respect to the axis of the chain, respectively, where nH and nR are two scalar factors depending on the shape of such elements (nH=nR=1/2 in the case of masses consisting of circular discs with radii rH and rR).

    Figure 1.  Graphical illustration of the tensegrity Maxwell chain.

    The model analyzed hereafter describes the response of the chain in the small displacement regime from the reference configuration, by reducing the prism units to linear springs endowed with stiffness coefficients KH (PH prisms) and KR (PH prisms). The latter coincides with the tangent values of the axial stiffness coefficients of the PH and PR prisms in correspondence to the reference configuration and are assumed to be positive. Let uHi denote the axial displacement of the ith hosting mass and let ϑHi denote the twisting rotation that accompanies the axial deformation of the PH prism interposed between the hosting masses i and i+1. Similarly, let us use the symbols uRi and ϑRi to indicate the axial displacement and the twisting rotation of the ith resonant mass [11], respectively (Figure 1). We assume the following compatibility equations between the axial and twisting motions of the hosting and resonant prisms [6]

    uHi=p ϑHi,  i=0,...,N;uRi = p ϑRi,  i=0,...,N1 (2.1)

    Here, p is a coupling parameter to be determined by studying the kinematics of the PH and PR prisms in proximity to the reference configuration. Equation (2.1) permits to reduce the kinematics of the Maxwell chain to the axial displacements uHi and uRi of the hosting and resonant units (see [6] for more details).

    It is an easy task to obtain the expressions of the kinetic energy K and the internal elastic energy Uint of the discrete Maxwell chain model as follows:

    K=12MeqH(˙uHN)2+N1i=012[MeqH(˙uHi)2+MeqR(˙uRi)2] (2.2)
    Uint=N1i=012[KR((uHiuRi)2+(uHi+1uRi)2) + KH(uHi+1uHi)2], (2.3)

    where we have employed the dot notation for time derivatives, and we have introduced the equivalent masses

    MeqH= MH(1+nHr2Hp2),    MeqR = MR(1+nRr2Rp2) (2.4)

    The external energy of the Maxwell chain is given by

    Uext=Fext,eqH,0uH0+N1i=0[Fext,eqH,iuHi(1δ0i)+Fext,eqR,iuRi]+Fext,eqH,NuHN, (2.5)

    where the following equivalent axial forces have been introduced:

    Fext,eqH,i= FextH,i+MextH,ip,   Fext,eqR,i = FextR,i+MextR,ip (2.6)

    FextH,i and FextR,i denote the external forces and the external twisting moments MextH,i and MextR,i applied to the chain.

    It is worth to note that we decompose the external energy (2.5) into three addends. The first and third addends are the contributions from the external world, respectively, to the extreme left and to the extreme right-hand sides of the chain. The second addend contains the rest of the external energies. As a matter of fact, this last second addend is decomposed into two parts. The first is the contribution of the external energies due to the external forces applied to the hosting masses with i=1,...,N1. The second is the contribution of the external energies due to the external forces applied to the resonance masses with i=0,...,N1.

    We now introduce a one-dimensional continuum domain of length L, characterized by two displacement fields uH(X,t) and uR(X,t) (see Figure 1) as kinematic descriptors, where X[0,L] is a spatial variable (cf. Figure 1) and t[t1,t2] is the time variable. This domain represents the support of a continuum model for the Maxwell chain, which consists of a mixture of two solids: the first solid describes the hosting masses MH of the discrete model, while the second one describes the resonance masses MR. Both solids are defined not only for every instant of time t[t1,t2] but also for every point X[0,L] of the reference configuration. In addition, each of the two solids has its own dynamics, one being defined by the displacement field uH(X,t) and the other by the displacement field uR(X,t).

    The following Piola's ansatz

    f(Xi)=fi,i=0,...,N,εLNi=0fiε=L0f(X)dX, (3.1)

    are assumed to link the continuum kinematic descriptors to the discrete ones introduced in the previous section, where f is either uH or uR, and fi is either uHi or uRi. Such ansatzes let us identify the realizations of the continuum descriptors uH(X,t) and uR(X,t) at the points of application of the ith discrete masses MH and MR with the discrete descriptors uHi(t) and uRi(t). Making use of Eq (3.1), we are led to rewrite the kinetic energy (2.2) of the discrete Maxwell chain as follows:

    K=Ni=0ε12[MeqHε(˙uHi)2+MeqRε(˙uRi)2]=L012[ϱH˙u2H+ϱR˙u2R]dX, (3.2)

    where the linear mass densities ϱH and ϱR are given by

    ϱH=MeqHε,ϱR=MeqRε. (3.3)

    Let us now look at the continuum model as the limit for ε0 of a sequence of discrete chain models. In correspondence to such a limit, one easily realizes that Eq (3.3) do not degenerate only if the values MH and MR (see also Eq (2.4)) rescale with the size ε of the unit cell of the discrete chain. We therefore assume that the values of the masses MH and MR tend to zero for ε0, in such a way that the linear mass densities ϱH and ϱR remain finite.

    Before we move on to identify the internal energy from Eq (2.3) of the continuum model, we need to write the displacement uHi+1 of the cell i+1 in terms of that uHi of the cell i, with the use of both Piola's ansatz (3.1), adapted for the field uH(X,t), and the following Taylor's series expansion

    uHi+1(t)=uH(Xi+1,t)=uH(Xi+ε,t)uH(Xi,t)+εuH(Xi,t)+12ε2uH(Xi,t) (3.4)

    Here, we have employed the prime notation for the derivatives with respect to the spatial variable X. Eq (3.4) allows us to write from Eq (2.3) the internal energy of the continuum model into the following form:

    Uint=Ni=1ε2{[KRε(uH(Xi,t)uR(Xi,t))2] (3.5)
    +[KRε(uH(Xi,t)+εuH(Xi,t)uR(Xi,t))2] (3.6)
    +[KHε(εuH(Xi,t))2]}. (3.7)

    which, making use of Eq (3.1), leads us finally to

    Uint=L0[12κR(uHuR)2+12κH(uH)2]dX (3.8)

    The continuum stiffness constants κH and κR, which appear in Eq (3.8), are related to the discrete counterparts KH and KR through

    κH=KHε,κR=2KRε. (3.9)

    We now go back to the identification of the continuum model as the limit for ε0 of a sequence of discrete models. In such a limit, Eq (3.9) do not degenerate only if the values κH and κR rescale with ε according to (3.9). This implies that it must result in KH for ε0, in such a way that κH remains finite. Similarly, it must result in KR0 for ε0, so that κR remains finite. It is worth noting that inclusion of the last term in (3.5) would include higher-order gradient terms in the final PDEs that are here neglected. Besides, quadratic terms in (2.3) have implied linear PDEs. However, for large deformation, nonlinear PDEs would be obtained by considering exponents in (2.3) higher than 2. In this general case, the difference between the formulations in the reference or in the present configurations should be considered via the so-called Piola's transformation [15]

    We finally identify the external energy (2.5), making use of Piola's ansatz (3.1), for both the fields uH(X,t) and uR(X,t). We obtain the following expression of Uext for the continuum model:

    Uext=Fext,eqH,0(t)uH(0,t)+Fext,eqH,N(t)uH(L,t),+N1i=0ε[Fext,eqH,i(t)εuH(Xi,t)(1δ0i)+Fext,eqR,i(t)εuR(Xi,t)] (3.10)

    from which, making use of Eq (3.1), we get to

    Uext=Fext,eqH,0(t)uH(0,t)+Fext,eqH,N(t)uH(L,t)+L0[bH(X,t)uH(X,t)+bR(X,t)uR(X,t)]dX. (3.11)

    The distributed external forces bH(X,t) and bR(X,t), which appear in Eq (3.11), are related to the discrete counterparts Fext,eqH,i(t) and Fext,eqR,i(t) through

    bH(Xi,t)=Fext,eqH,i(t)εi=1,...,N1,bR(Xi,t)=Fext,eqR,i(t)ε,i=0,...,N1. (3.12)

    Considering again the limit for ε0 of a sequence of discrete models, we realize that Eq (3.12) do not degenerate in correspondence to such a limit case only if the values bH and bR rescale with ε according to (3.12). This implies that it must result in both Fext,eqH,i0 and Fext,eqR,i0 for ε0, in such a way that bH and bR remain finite for ε0.

    The action functional of the continuum model can be formulated into the following standard form (see, e.g., [17])

    A=t2t1[KUint+Uext], (3.13)

    where t1 and t2 are two times at which we prescribe that the displacement fields uH and uR are equal to known values uαH and uαR (α=1,2), through

    uH(X,t1)=u1H,uH(X,t2)=u2H,uR(X,t1)=u1R,uR(X,t2)=u2R. (3.14)

    We now introduce the following action principle [17],

    δA=0,δuHCH,δuRCR, (3.15)

    for any admissible variations CH of uH and CR of uR. It is an easy task to obtain from Eqs (3.2), (3.8), (3.11), and (3.13) the following result:

    δA=t2t1[L0[δuH(κR(uHuR)+κHuHϱH¨uH+bH)]dX]dt+t2t1[L0[δuR(κR(uHuR)ϱR¨uR+bR)]dX]dt+t2t1[δuH(0,t)(Fext,eqH,0(t)+κHuH(0,t))]dt+t2t1[δuH(L,t)(Fext,eqH,L(t)κHuH(L,t))]dt. (3.16)

    Invoking the arbitrariness of the variations in (3.15), we can finally derive the following PDEs ruling the dynamic problem of the continuum model:

    κR(uHuR)κHuH+ϱH¨uHbH=0 (3.17)
    κR(uHuR)+ϱR¨uRbR=0 (3.18)

    which are accompanied by the BCs

    uH(0,t)=uH0(t)orκHuH(0,t)=Fext,eqH,0(t) (3.19)
    uH(L,t)=uHL(t)orκHuH(L,t)=Fext,eqH,L(t). (3.20)

    Let us now introduce plane wave solutions for the displacement fields uH(X,t) and uR(X,t), through

    uH=Re{uH0exp[I(ωtkwX)]},uR=Re{uR0exp[I(ωtkwX)]} (3.21)

    where Re indicates the real part operator; I is the imaginary unit; uH0, uR0, ω, and kw, respectively, denote the complex amplitudes, the angular frequency, and the wave number of the traveling waves. Making use of Eq (3.21), we easily obtain (neglecting all the external forces and the external energy terms) the following dispersion relation of the continuum model

    ωMoc,Mac(kw)=fM(kw)±ΔM(kw)2ϱHϱR (3.22)

    where the function fM(kw) and the discriminant ΔM(kw) have the expressions given below

    fM(kw)=(ϱH+ϱR)κR+ϱRκHk2w (3.23)
    ΔM(kw)=[fM(kw)]24ϱHϱRκRκHk2w). (3.24)

    In the limit ε0, Eq (3.22) reduce to the form

    ωMackwκHϱH+ϱR, (3.25)
    ωMocκR(ϱH+ϱR)ϱHϱR. (3.26)

    Making use of the positions (3.3) and (3.9), it is easily verified that both the Eqs (3.25) and (3.26) reduce to Eq (25), in the reference [6], of the discrete model, which has been presented to provide the value of the circular frequency of the optic branch for kw=0.

    It is useful to apply the above formulas to the micro-scale physical models of the Maxwell chain studied in [6]. Such models employ PR prisms with a 5.5 mm height in the reference configuration, equilateral triangular bases with an 8.7 mm edge, 0.28 mm Spectra strings (5.48 GPa Young modulus), and 0.8 mm Ti6Al4V bars (120 GPa Young modulus). In addition, they use PH prisms showing 11 mm height in the reference configuration, equilateral triangular bases with a 6.11 mm edge, Spectra strings and Ti6Al4V bars identical to those of the PR prisms (we refer the reader to [6] for more detailed information about such units). The masses forming the above models are circular discs, such that MH=16.03 grams and MR=4 grams (nH=nR=1/2). The study presented in [6] estimates KH=61.04 N/mm, KR=13.70 N/mm, and p2.33 mm/rad.

    Graphical illustrations of the dispersion relations exhibited by discrete and continuum models of the chain under examination are provided in Figure 2, by distinguishing two different cases: discrete and continuum systems in which the longitudinal and twisting motions of the PH and PR prisms are coupled, according to Eq (2.1), as well as discrete and continuum systems in which the same prisms are free to tangentially slide against the lumped masses, so as that the twisting rotations of the tensegrity units are not transferred to the masses (uncoupled system). In the first case, we have MeqH=349.51 grams and MeqR=24.85 grams, while in the second case, it results in MeqH=MH and MeqR=MR. Figure 2 shows that the acoustic and optic branches forming the dispersion relations of the continuum systems correctly reduce to the analogous branches of the discrete systems in the long wavelength regime (kw0). In the coupled system we predict ωMoc(0)=173 Hz, while in the uncoupled system, we predict ωMoc(0)=465 Hz.

    Figure 2.  A comparison between the dispersion relations of the discrete and continuous models for the physical model of the Maxwell chain with coupled (a) and uncoupled (b) longitudinal and twisting motions.

    Let us derive the wave equation of the continuum model of the Maxwell chain, by making use of the results derived in the previous section. We start by rewriting Eq (3.21) into the following form:

    uH=Re{ˉuH0(X)exp[I(ωt)]}+uHS(X),uR=Re{ˉuR0(X)exp[I(ωt)]}+uRS(X), (4.1)

    where now ˉuH0(X) and ˉuR0(X) are unknown functions of the spatial coordinate X, and uHS(X) and uRS(X) are those displacement functions produced by a static load at the boundary and assuming zero external distributed forces (bH=bR=0). Thus, Eqs (3.17) and (3.18) yield,

    κR(uHSuRS)κHuHS=0 (4.2)
    κR(uHSuRS)=0 (4.3)

    and uHS(X) and uRS(X) are solved once a particular set of boundary conditions are considered. Keeping this in mind, one can write Eqs (3.17) and (3.18) making use of Eq (4.1), to obtain

    κR(ˉuH0ˉuR0)κHˉu0H+ϱHω2ˉuH0=0 (4.4)
    κR(ˉuH0ˉuR0)+ϱRω2ˉuR0=0. (4.5)

    The insertion of Eq (4.5) into Eq (4.4) leads us to the second-order linear homogeneous differential equation with respect to the spatial coordinate X

    κHˉu0H+ω2ˉϱHˉuH0=0 (4.6)

    where we have introduced the equivalent mass density

    ˉϱH=κR(ϱR+ϱH)ϱRϱHω2κRϱRω2 (4.7)

    Eq (4.6) admits the following general solution:

    ˉuH0=Re{c1exp[α1X]+c2exp[α2X]} (4.8)

    where we introduced the integration constants c1 and c2, to be determined through the competent boundary conditions, and we set

    α1,2=± ωˉϱHκH. (4.9)

    Moving on to determine analytic solutions for ˉuR0(X), we now insert Eq (4.8) into Eq (4.5), obtaining

    ˉuR0=κRϱR ω2+κR[Re{c1exp[α1X]+c2exp[α2X]}] (4.10)

    Let us now determine the constants c1 and c2 in the particular case of a chain lying vertically in space, which is subject to a mass M applied to the top end (X=L) and a sinusoidal displacement excitation at the bottom end (X=0). The base excitation has an amplitude of u0 and can be written as

    uH(0,t)=u0cos(ωt) (4.11)

    It is useful to separate the dynamic and static parts of Eq (4.1). For what concerns the dynamic part, we analyze the boundary conditions

    ˉuH0(0)=u0ˉuH0(L)=0. (4.12)

    which, once inserted into Eq (4.8), leads us to compute the integration constants c1 and c2, and to obtain the dynamic amplitude of the hosting displacement field

    ˉuH0=u0 Re{sec[α1L]cos[α1(LX)]} (4.13)

    and from Eq (4.10) that of the resonant one,

    ˉuR0=κR u0ϱR ω2+κRRe{sec[α1L]cos[α1(LX)]} (4.14)

    For what concerns the static terms uHS and uRS, we observe that Eq (4.3) implies the following result in static equilibrium conditions:

    κR(uHSuRS)=0. (4.15)

    which is accompanied by the static boundary condition

    uHS(0)=0 (4.16)

    On the other hand, in the example under consideration, it is a simple task to obtain the following differential equation for uHS [20]

    uHS(L)=MgκH (4.17)

    where g is the gravity acceleration. Making use of Eqs (4.16) and (4.17) in Eq (4.15), we finally obtain

    uHS(X)=uRS(X)=MgκHX (4.18)

    We are now able to cast Eq (4.1) into the following form:

    uH(X,t)=u0 Re{sec[α1L]cos[α1(LX)]exp[I(ωt)]}MgκHX (4.19)
    uR(X,t)=κR u0ϱR ω2+κR Re{sec[α1L]cos[α1(LX)]exp[I(ωt)]}MgκHX (4.20)

    Since we are assuming κH>0 (cf. Section 2), Eqs (4.19) and (4.20), in association with the positions (4.9), let us conclude that one obtains periodic harmonic solutions for uH and uR only when α1 and α2 are imaginary (ˉϱH>0). Oppositely, we obtain decaying solutions when α1 and α2 are real (ˉϱH<0). The frequency bandgap region obviously corresponds to ˉϱH<0, since in this case the system supports wave solutions that exponentially decay when X approaches L [18,19]. It is easily verified that the values of ω>0 such that ˉϱH<0 are contained in the interval comprised between zero and the frequency ωMoc provided by Eq (3.26). Such a result provides validation of the dispersion relation derived in the previous section, and highlights that the analyzed chain can be effectively employed as an isolation device for vertical vibrations of the top mass M. It is not difficult to generalize the results obtained in the present section to the case in which the examined system is under the action of a not-negligible self-weight. For such an example, indeed, the dynamic parts of uH and uR remains unchanged as compared to the present case, while the static parts of these functions will assume a quadratic expression with respect to X [20].

    We have derived a continuum model for a Maxwell-type mass–spring chain with tensegrity architecture [12,13], which exhibits θ=1 tensegrity prisms [14] arranged in parallel with T3 prisms [16] and lumped masses. When compared to the discrete model presented in [19], the theory formulated in the present work is useful to obtain analytic solutions for the wave propagation problem of the analyzed system in the high wave-length regime, where structural applications are relevant [21,22]. Such a feature has been demonstrated through the analysis of a physical example, which refers to the vibration isolation problem of a mass, e.g., a device to be protected against mechanical vibrations in a hospital or another essential building. It is worth noting that the mass–spring chain studied in this work exhibits internal resonance properties that can be employed to widen the frequency bandgap width of the system, as compared to the case of a diatomic tensegrity mass–spring chain, where such a width can only be tuned by playing with mass and stiffness contrasts between the elements of the unit cell [23].

    The research presented in this work paves the way for the design of novel tensegrity metamaterials serving as next-generation vibration isolation devices. Such systems will exhibit properties that can be easily adjusted to the structure to be protected due to the tunability of the frequency bandgap region [19]. The presence of geometrical and mechanical nonlinearities in such devices will be studied through future research, accounting, e.g., for the presence of superelastic cables made of shape memory alloys [24] and higher order terms in the internal an kinetic energies [25,26]. Their use in forming anisotropic and/or porous metamaterials [27,28], energy harvesters [29], and sensors and actuators for structural health monitoring [30] will also be investigated in future studies.

    L.P. formulated the initial idea for the continuum model, while F.F., J.d.C.M., and R.Z.C. conceived the tensegrity modeling. L.P. and F.F. led the conceptualization, idealization, and supervision phases of the project. F.F. provided funding. J.d.C.M. led the development of the analytical results, while R.Z.C. led the numerical analysis of the physical models. All authors contributed to the first draft and the revised version of the manuscript.

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

    This research has been funded under the National Recovery and Resilience Plan (NRRP), by the European Union–NextGenerationEU, within the project with grant number P2022CR8AJ (FF PI). This research has also been funded by the NextGenerationEU PRIN2022 research projects with grant numbers 20224LBXMZ (FF PI) and 2022P5R22A ("Ricerca finanziata dall'Unione Europea–Next Generation EU"). FF and JdCM also acknowledge the support by the Italian Ministry of Foreign Affairs and International Cooperation within the Italy-USA Science and Technology Cooperation Program 2023–2025, Project "Next-generation green structures for natural disaster-proof buildings" (grant No. US23GR15).

    The authors declare that there is no conflict of interest.



    Author contributions



    Sarah Abdelhamid and Rania Abozahra conceived and designed the study. Maged Wasfy and Nada Elmeniawy conducted the experiments. Sarah Abdelhamid and Kholoud Baraka analyzed the data. Kholoud Baraka wrote the manuscript. All authors read and approved the manuscript.

    Conflict of interest



    We declare that we have no conflict of interest.

    Ethical approval



    All applicable international, national, and/or institutional guidelines for the care and use of animals were followed. This research adhered to the accepted principles of ethical conduct according to the approval reference number (618PM5) by the Research Ethics Committee of the Faculty of Pharmacy, Damanhour University. Informed consent from the parents of young patients was obtained prior to undertaking testing and molecular investigation of their specimens.

    Data availability



    All data generated or analysed during this study are included in this published article.

    Limitations of the study



    Although EAT model is widely used for breast cancer assays, xenograft animal model of mammary glands using human cancer cell lines is still highly recommended to increase the reliability of results. In addition, development of more convenient vehicular methods for bacterial delivery is still highly recommended for future clinical applications.

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