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Research article Special Issues

Biogenic synthesis, characterization and effects of Mn-CuO composite nanocatalysts on Methylene blue photodegradation and Human erythrocytes

  • Each year more than 150, 000 tons of dyes are released in effluents by industries. These chemicals entities non-biodegradable and toxic can be removed from effluent by metallic nanomaterials. The aqueous extract of Manotes expansa leaves is used as reducing and stabilizing agent in the biogenic synthesis of Mn-CuO nanocomposites. The nanoparticles obtained were characterized using UV-visible spectroscopy, X-ray Diffraction (XRD), X-ray Fluorescence, Dynamic Light Scattering (DSL), and Scanning Electron Microscopy (SEM). The hemotoxicity of biosynthesized nanomaterials was assessed by evaluating their hemolytic activity using erythrocytes as a model system. The photocatalytic activity of Mn-CuO was carried out by photocatalytic degradation of Methylene Blue dye as a model. The results obtained by UV-vis spectroscopy showed a Plasmonic Surface Resonance band at 408 nm. XRD and X-ray fluorescence made it possible to identify the presence of particles of formula Mn0.53Cu0.21O having crystallized in a Hexagonal system (a = 3.1080 Å and c = 5.2020 Å). Spherical morphology and average height 49.34 ± 6.71 nm were determined by SEM and DSL, respectively. The hemolytic activity of biosynthesized nanomaterials revealed that they are not hemotoxic in vitro (% hemolysis 3.2%) and 98.3% of Methylene Blue dye was removed after 120 min under irradiation with solar light in the presence of Mn-CuO nanocomposites.

    Citation: Carlos N. Kabengele, Giresse N. Kasiama, Etienne M. Ngoyi, Clement L. Inkoto, Juvenal M. Bete, Philippe B. Babady, Damien S. T. Tshibangu, Dorothée D. Tshilanda, Hercule M. Kalele, Pius T. Mpiana, Koto-Te-Nyiwa Ngbolua. Biogenic synthesis, characterization and effects of Mn-CuO composite nanocatalysts on Methylene blue photodegradation and Human erythrocytes[J]. AIMS Materials Science, 2023, 10(2): 356-369. doi: 10.3934/matersci.2023019

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  • Each year more than 150, 000 tons of dyes are released in effluents by industries. These chemicals entities non-biodegradable and toxic can be removed from effluent by metallic nanomaterials. The aqueous extract of Manotes expansa leaves is used as reducing and stabilizing agent in the biogenic synthesis of Mn-CuO nanocomposites. The nanoparticles obtained were characterized using UV-visible spectroscopy, X-ray Diffraction (XRD), X-ray Fluorescence, Dynamic Light Scattering (DSL), and Scanning Electron Microscopy (SEM). The hemotoxicity of biosynthesized nanomaterials was assessed by evaluating their hemolytic activity using erythrocytes as a model system. The photocatalytic activity of Mn-CuO was carried out by photocatalytic degradation of Methylene Blue dye as a model. The results obtained by UV-vis spectroscopy showed a Plasmonic Surface Resonance band at 408 nm. XRD and X-ray fluorescence made it possible to identify the presence of particles of formula Mn0.53Cu0.21O having crystallized in a Hexagonal system (a = 3.1080 Å and c = 5.2020 Å). Spherical morphology and average height 49.34 ± 6.71 nm were determined by SEM and DSL, respectively. The hemolytic activity of biosynthesized nanomaterials revealed that they are not hemotoxic in vitro (% hemolysis 3.2%) and 98.3% of Methylene Blue dye was removed after 120 min under irradiation with solar light in the presence of Mn-CuO nanocomposites.



    Neuronal activities generate the electrical current in the brain, and further result in the potential changes over the scalp. Electroencephalography (EEG) is a technique used to record the potential changes on the scalp. Even though fMRI, PET, MEG and other brain-imaging tools are widely used in brain research, they are limited by low spatial/temporal resolution, cost, mobility and suitability for long-term monitoring. For example, fMRI has the advantage of providing spatially-resolved data, but suffers from an ill-posed temporal inverse problem, i.e., a map with regional activations does not contain information about when and in which order these activations have occurred [1]. In contrast, EEG signals have been successfully used to obtain useful diagnostic information (neural oscillations and response times) in clinical contexts. Further, they present the advantage to be highly portable, inexpensive, and can be acquired at the bedside or in real-life environments with a high temporal resolution. Because of the lack of significant patient risks, EEG is additionally suited for long-term monitoring.

    EEG offers the possibility of measuring the electrical activity of neuronal cell assemblies on the sub-millisecond time scale [2,3,4]. EEG source imaging further identifies the positions or distributions of electric fields based on EEG signals collected on the scalp [5]. This new tool is widely used in cognitive neuroscience research, and has also found important applications in clinical neuroscience such as neurology, psychiatry and psychopharmacology [6,7]. In cognitive neuroscience, the majority of the studies investigate the temporal aspects of information processing by analyzing event related potentials (ERP). In neurology, the study of sensory or motor evoked potentials is of increasing interest, but the main clinical application concerns with the localization of epileptic foci. In psychiatry and psychopharmacology, a major focus of interest is the localization of sources of certain EEG frequency bands. Localizing the activity sources of a given scalp EEG measurement is achieved by solving the so-called inverse problem [8]. These kinds of inverse problems are usually ill-posed and their solutions are non-unique [9,10].

    Leahy et al. [11] investigated the accuracy of forward and inverse techniques for EEG and MEG dipole localization using a human skull phantom. El Badia and Ha-Duong [12] established an algebraic method to identify the number, locations and moments of electrostatic dipoles in 2D or 3D domain from the Cauchy data on the boundary. Chafik et al. [13] further provided an error estimate without proof. Nara and Ando [14] provided a new projective method for 3D source reconstruction by projecting the sources onto a Riemann sphere. Kang and Lee [15] proposed an algorithm for solving the inverse source problem of a meromorphic function and apply their method to an electrical impedance tomography (EIT) problem. El Badia [16] established a uniqueness result and a local Lipschitz stability estimate for an anisotropic elliptic equation, assuming that the sources are a linear combination of a finite number of monopoles and dipoles. The author also proposed a global Lipschitz stability estimate for dipolar sources. Baratchart et al. [17] solved the inverse source problem by locating the singularities of a meromorphic function from the 2D boundary measurements using best rational or meromorphic approximations.

    Chung and Chung [18] proposed an algorithm for detecting the combination of monopolar and multipolar point sources for elliptic equations in the 2D domain from the Neumann and Dirichlet boundary data. Kandasmamy et al. [19] proposed a novel technique, called "analytic sensing", to estimate the positions and intensities of point sources in 2D for a Poisson's equation. Analytic sensing also used the reciprocity gap principle, but with a novel design of an analytic function which behaved like a sensor. The authors evaluated their estimation accuracy by Cramér-Rao lower bound. Nara and Ando [20] proposed an algebraic method to localize the positions of multiple poles in meromorphic function field from an incomplete boundary. They investigated the accuracy of the algorithm for the open arc or the closed arc, and for the arc enclosing the poles or not enclosing the poles. El Badia and Nara [21] established the uniqueness and local stability result for the inverse source problem of the Helmholtz equation in an interior domain, assuming the source is composed of multiple point sources.

    Clerc et al. [22] applied best rational approximation techniques in the complex plane to EEG source localization and offered stability estimates. Mdimagh and Ben Saad [23] identified the point sources in a scalar problem modeled by Helmholtz equation, using reciprocity gap principle and assuming the sources are harmonic in time. They proved local Lipschitz stability by two methods: one was derived from the Gâteaux differentiability, and the other used particular test functions in the reciprocity gap functional. Vorwerk et al. [24] studies the important role of head tissue conductivity in EEG dipole reconstruction. Rubega et al. [25] estimated EEG source dipole orientation based on singular-value decomposition. Michel and Brunet provided a thorough review on EEG source imaging. There exist several reconstruction methods, such as minimum norm estimates (MNE) [26], low resolution electrical tomography (LORETA) [27,28] or multiple-signal classification algorithm (MUSIC) [29,30], etc.

    Recently, Muñoz-Gutiérrez et al. [31] managed to improve the accuracy of EEG source reconstruction by decomposing the EEG signals into frequency bands with different methods, such as empirical mode decomposition (EMD) and wavelet transform (WT). Kaur et al. [32] presented a new method of EEG source localization using variational mode decomposition (VMD) and standardized the low resolution brain electromagnetic tomography (sLORETA) inverse model. Their VMD-sLORETA model could locate EEG sources in the brain in a very accurate way. Oikonomou and Kompatsiaris [33] developed a novel Bayesian approach for EEG source localization. They incorporated a new sparse prior for the localization of EEG sources with the variational Bayesian (VB) framework and obtained more accurate localization of EEG sources than state-of-the-art approaches.

    In our study we need new methods to detect small changes in EEG source for which dipole methods have advantage. We followed the analytic dipole method by El Badia and Ha-Duong [12] and derived a new error estimate for this source localization method. We provided a mathematical proof of this estimate. We then use simulated data to validate the method. The simulation results support our error estimation, which has a different distance power than a similar error estimate in [22].

    We organize the rest of the paper as follows. In section 2, we introduce the method and its formulation. In section 3, we provide the error estimate of an inverse EEG source localization problem in a bounded domain and its mathematical proof. In section 4, we use simulated data to valid the method and error estimate. A brief conclusion and discussion is in Section 5.

    The electric field E is the negative gradient of the potential u.

    E=u. (2.1)

    The quasi-static approximation means all time derivatives in the equation are set to zero. By quasi-static approximation of Maxwell equation ×HDt=J, we have

    ×H=J

    where H is the magnetizing field, J is the total current density, and D is the displacement field.

    Since the divergence of a curl is always zero, we have

    (×H)=J=0.

    EEG problem can be modeled by a Poisson equation.

    (σu)=(σE)=(JJp)=J=0Jp=Jp=F,

    where σ is the conductivity, Jp is the primary current density, and F is the source term.

    If we assume the source is composed of a finite number of point charges, then by linear combination, we have

    F=mk=1qkδ(rrk), (2.2)

    where m is the number of point charges, qk are values of charges, and rk are the locations of the point charges.

    If we assume the source is composed of a finite number of dipoles, we have

    F=mk=1pkδ(rrk),

    where m is the number of dipoles, pk are the moments (or strengths) of the dipoles, and rk are the centers of dipoles.

    The dipolar source reconstruction problem can be viewed as a Poisson problem.

    Δu=mk=1pkδ(rrk) in Ω, (2.3)
    u=f on Γ, (2.4)
    uν=φ on Γ, (2.5)

    where f and φ are known, and ν is the outer unit normal vector.

    We will use the concept of reciprocity gap functional [34]:

    R(v)=uν,vH1/2(Γ),H1/2(Γ)u,vνH1/2(Γ),H1/2(Γ)=φ,vH1/2(Γ),H1/2(Γ)f,vνH1/2(Γ),H1/2(Γ), (2.6)

    where v is a harmonic function in Ω:

    vH(Ω)={wH1(Ω)Δw=0}. (2.7)

    By Green's formula, we have

    R(v)=mk=1pkv(rrk),vH(Ω). (2.8)

    Let m be the number of dipoles in the brain. Assume mM in our problem, i.e., there is an upper bound for the number of dipoles.

    Let us consider the harmonic polynomials

    vj(x,y)=(x+iy)j,jN.

    Then, in 2D case

    R(vj)=mk=1pkvj(rk)=mk=1[pk1pk2](xk+iyk)j=mk=1[pk1pk2][x(x+iy)jy(x+iy)j]x=xk,y=yk=mk=1[pk1pk2][j(xk+iyk)j11j(xk+iyk)j1i]=mk=1[pk1pk2][1i]j(xk+iyk)j1=jmk=1(pk1+ipk2)(xk+iyk)j1.

    We define

    βj:=R(vj)j=Mk=1(pk1+ipk2)(xk+iyk)j1,j=1,2,...,2M1. (2.9)

    Let

    ηj=[βjβj+1βj+M1]CM,1jM, (2.10)

    and

    Zi=[ηi,ηi+1,...,ηi+M1]=[βiβi+1βi+M1βi+1βi+2βi+Mβi+M1βi+Mβi+2M2],iN.

    Then,

    Z1=[η1,η2,...,ηM]=[β1β2βMβ2β3βM+1βMβM+1β2M1].

    The number m of dipoles is estimated as the rank of Z1.

    Now we can reduce the size of the matrix by recalculating βj and ηj with M replaced by m. Then, the m vectors η1,...,ηm are independent.

    To get the estimates of the positions we need to construct an m×m matrix T such that ηj+1=Tηj,j=1,...,m. Then,

    [η2,...,ηm+1]=T[η1,...,ηm].

    So,

    T=[η2,...,ηm+1][η1,...,ηm]1=[β2β3βm+1β3β4βm+2βm+1βm+2β2m][β1β2βmβ2β3βm+1βmβm+1β2m1]1=Z2Z11.

    The positions of dipoles are estimated as the eigenvalues of T.

    We now show that the eigenvalues of T are the positions of dipoles. Let us first look at an example η2=Tη1.

    Tη1=T[β1β2βm]=T[p1+p2++pmp1S1+p2S2++pmSmp1Sm11+p2Sm12++pmSm1m]=p1T[1S1Sm11]+p2T[1S2Sm12]++pmT[1SmSm1m],

    where pk=pk1+ipk2,k=1,2,...,m is the moment and Sk=xk+iyk,k=1,2,...,m is the position.

    η2=[β2β3βm+1]=[p1S1+p2S2++pmSmp1S21+p2S22++pmS2mp1Sm1+p2Sm2++pmSmm]=p1S1[1S1Sm11]+p2S2[1S2Sm12]++pmSm[1SmSm1m],

    where pk=pk1+ipk2,k=1,2,...,m is the moment and Sk=xk+iyk,k=1,2,...,m is the position.

    Since [1S1Sm11],[1S2Sm12],...,[1SmSm1m] are independent and the results are similar for ηj+1=Tηj,j=1,2,...,m, we know S1,S2,...,Sm are just the eigenvalues of T.

    Now the question is how to get T. Only η1 and η2 are not enough to determine T because vectors have no inverse. So, we use the redundant information to construct the matrices Z1 and Z2 such that T=Z2Z11, where Z1 is invertible because η1,...,ηm are independent.

    To estimate the moments of dipoles we will write Eq (2.9) in matrix form. Notice that now we use m instead of M.

    [β1β2βm]=[S01S02S0mS11S12S1mSm11Sm12Sm1m][p1p2pm], (2.11)

    where pk=pk1+ipk2,k=1,2,...,m is the moment and Sk=xk+iyk,k=1,2,...,m is the position.

    We can write Eq (2.11) in matrix form

    b=Sp, (2.12)

    where b=[β1β2βm],S=[S01S02S0mS11S12S1mSm11Sm12Sm1m], and p=[p1p2pm]. Then, the moments of dipoles in 2D are estimated as

    p=S1b. (2.13)

    Equation (2.13) works in the ideal case of no noise. In reality, due to the noise in the measurements and in the sources, we need find a linear operator L to estimate the moments, i.e.,

    ˜p=Lb (2.14)

    where ˜p represents the estimates of the moments, and b represents the quantities obtained from the measurements.

    Considering the noise accompanied in the measurements, we rewrite Eq (2.12) as

    b=Sp+n,

    where n is a random vector of mean 0. Let N be the covariance matrix of n. Also, assume that ˜p is normally distributed with mean p and its covariance matrix is P.

    Using multiple measurements and the statistical estimation theory we can find the linear operator L which minimizes the expected difference ErrL between the estimated moments ˜p and the exact moments p.

    ErrL=˜pp2=Lbp2=L(Sp+n)p2=(LSI)p+Ln2=Mp+Ln2(where M=LSI)=Mp2+Ln2(by independence of p and n)=Tr(MPMT)+Tr(LNLT).

    Setting the gradient of ErrL to 0 and solving for L, we get the optimal linear operator

    L=PST(SPST+N)1. (2.15)

    Then, by Eq (2.14) we get the best estimates of the moments.

    Theorem 2.1 (Uniqueness of solutions). Let ui,i=1,2 be the solutions of the problems

    (σui)=mik=1pk(i)δS(i)k in Ω,
    uiν=φ on Γ,

    such that

    u1=u2 on Γ,

    then

    m1=m2=m,
    pk(1)=pk(2),k=1,2,...,m,
    S(1)k=S(2)k,k=1,2,...,m.

    The solution of Poisson equation is the convolution of the fundamental solution of Laplace equation and the source function.

    w(x)=12π[m2k=1pk(xSk)|xS(2)k|2m1k=1pk(xSk)|xS(1)k|2],n=2.
    w(x)=14π[m2k=1pk(xSk)|xS(2)k|3m1k=1pk(xSk)|xS(1)k|3],n=3.

    As EEG imaging data are typically noisy, especially determining the rank of a near singular matrix is very unstable, the error of the numerical reconstruction method needs to be studied. Chafik et al. [12,13] proposed that when the norms of the perturbations (g=˜ff,h=˜φφ) are small in H1/2×H1/2, there exist a>0 and b>0 such that k=1,2,...,m,

    \begin{eqnarray} \|\tilde S_k-S_k\|_2&\le&\frac{m(1-R^m)}{d^{m-1}(1-R)}\max\left\{\binom{m-1}jR^j, 0\le j\le m-1\right\}\cdot \\ &&\cdot\left(a\|g\|_{H^{1/2}(\Gamma)}+b\|h\|_{H^{-1/2}(\Gamma)}\right), \end{eqnarray} (3.1)

    where S_k = x_k+iy_k is the exact position of the k th dipole, \tilde S_k = \tilde x_k+i\tilde y_k is the estimated position of the k th dipole, d is the minimal distance between S_k and \tilde S_k , and R\neq1 is a real number bigger than the norm of any point on \Gamma . However, the analysis is not given by Chafik et al.

    Here we present a new error estimate and provide a proof.

    Theorem 3.1. Suppose m dipoles are enclosed in a circular boundary of radius R . The potential f on the boundary and the gradient of the potential \varphi perpendicular to the boundary are known. If T is the measurements without noise, and \tilde T is the measurements with noise, then the error estimate is given by

    \begin{eqnarray} &&\|T-\tilde T\|_\infty\qquad\\ &\le&2m\left(\|\varphi\|_2R^{2m}\sqrt{2\pi R}+\|f\|_2R^{2m}\sqrt{2\pi R}\right)\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)\qquad\\ &&+2m^2\left(\|\varphi\|_2R^{2m}\sqrt{2\pi R}+\|f\|_2R^{2m}\sqrt{2\pi R}\right)^2\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)^2, \qquad \end{eqnarray} (3.2)

    where p is the moment of dipoles and d is the smallest distance between any two dipoles.

    Proof. We define

    \begin{equation} \nonumber Z_i = \left[ {\begin{array}{*{20}{c}}\beta_i&\beta_{i+1}&\cdots&\beta_{i+m-1}\\ \beta_{i+1}&\beta_{i+2}&\cdots&\beta_{i+m}\\ \vdots\\ \beta_{i+m-1}&\beta_{i+m}&\cdots&\beta_{i+2m-2}\end{array}} \right] , \quad i\in\mathbb N. \end{equation}

    Then,

    \begin{equation} \nonumber Z_1 = \left[ {\begin{array}{*{20}{c}}\beta_1&\beta_2&\cdots&\beta_m\\ \beta_2&\beta_3&\cdots&\beta_{m+1}\\ \vdots\\ \beta_m&\beta_{m+1}&\cdots&\beta_{2m-1}\end{array}} \right] . \end{equation}

    where

    \begin{equation} \nonumber \beta_j = \sum\limits_{k = 1}^m p_kS_k^{j-1} = \sum\limits_{k = 1}^m (p_{k1}+ip_{k2})(x_k+iy_k)^{j-1}, \quad j = 1, 2, ..., 2m-1. \end{equation}
    \begin{eqnarray*} &&\det(Z_1) = \left| {\begin{array}{*{20}{c}} \beta_1&\beta_2&\cdots&\beta_m\\ \beta_2&\beta_3&\cdots&\beta_{m+1}\\ \vdots\\ \beta_m&\beta_{m+1}&\cdots&\beta_{2m-1}\end{array}} \right| \\ & = &\left| {\begin{array}{*{20}{c}} \sum p_k&\sum p_kS_k&\cdots&\sum p_kS_k^{m-1}\\ \sum p_kS_k&\sum p_kS_k^2&\cdots&\sum p_kS_k^m\\ \vdots\\ \sum p_kS_k^{m-1}&\sum p_kS_k^m&\cdots&\sum p_kS_k^{2m-2}\end{array}} \right| \\ & = &\sum\limits_{m_1\neq m_2\neq\cdots\neq m_m}\tau(m_1, m_2, ..., m_m)\cdot p_{m_1}p_{m_2}\cdots p_{m_m}\left| {\begin{array}{*{20}{c}} 1&S_{m_2}&\cdots&S_{m_m}^{m-1}\\ S_{m_1}&S_{m_2}^2&\cdots&S_{m_m}^m\\ \vdots\\ S_{m_1}^{m-1}&S_{m_2}^m&\cdots&S_{m_m}^{2m-2}\end{array}} \right| \\ \end{eqnarray*}
    \begin{eqnarray*} & = &\sum\limits_{m_1\neq m_2\neq\cdots\neq m_m}\tau(m_1, m_2, ..., m_m)\cdot p_{m_1}p_{m_2}\cdots p_{m_m}\left| {\begin{array}{*{20}{c}} 1&1&\cdots&1\\ S_{m_1}&S_{m_2}&\cdots&S_{m_m}\\ \vdots\\ S_{m_1}^{m-1}&S_{m_2}^{m-1}&\cdots&S_{m_m}^{m-1}\end{array}} \right| \cdot\\ &&\cdot S_{m_1}^0S_{m_2}^1\cdots S_{m_m}^{m-1}\\ & = &p_1p_2\cdots p_m\left| {\begin{array}{*{20}{c}} 1&1&\cdots&1\\ S_{m_1}&S_{m_2}&\cdots&S_{m_m}\\ \vdots\\ S_{m_1}^{m-1}&S_{m_2}^{m-1}&\cdots&S_{m_m}^{m-1}\end{array}} \right| \cdot\\ &&\cdot\left(\sum\limits_{m_1\neq m_2\neq\cdots\neq m_m}\tau(m_1, m_2, ..., m_m)\cdot S_{m_1}^0S_{m_2}^1\cdots S_{m_m}^{m-1}\right)\\ & = &p_1p_2\cdots p_m\left| {\begin{array}{*{20}{c}} 1&1&\cdots&1\\ S_{m_1}&S_{m_2}&\cdots&S_{m_m}\\ \vdots\\ S_{m_1}^{m-1}&S_{m_2}^{m-1}&\cdots&S_{m_m}^{m-1}\end{array}} \right| \cdot\left| {\begin{array}{*{20}{c}} 1&1&\cdots&1\\ S_{m_1}&S_{m_2}&\cdots&S_{m_m}\\ \vdots\\ S_{m_1}^{m-1}&S_{m_2}^{m-1}&\cdots&S_{m_m}^{m-1}\end{array}} \right| \\ & = &p_1p_2\cdots p_m\prod\limits_{1\le i < j\le m}(S_i-S_j)^2. \end{eqnarray*}

    Here, (m_1, m_2, ..., m_m) is any permutation of (1, 2, ..., m) and \tau(m_1, m_2, ..., m_m) is the sign determined by the permutation.

    The maximum absolute row sum norm is defined by

    \begin{equation} \nonumber \|A\|_\infty = \max\limits_i\sum\limits_{j}|a_{ij}|, \end{equation}

    where A is a matrix. When A is a vector, \|A\|_\infty = \max_i|a_i| .

    In the following proof we will use an important inequality:

    \begin{equation} \nonumber \|a(x)b(x)-a(y)b(y)\|_\infty\le\|a(x)-a(y)\|_\infty\cdot\|b(x)\|_\infty+\|b(x)-b(y)\|_\infty\cdot\|a(x)\|_\infty \end{equation}

    where a(x) and b(x) can be scalar, vector, or matrix.

    By Cauchy-Schwarz inequality, we have

    \begin{eqnarray*} &&R(v_j)\\ & = &\langle{\varphi}, {{v_j}}\rangle-\langle{f}, {{\frac{\partial {v_j}}{\partial {\nu}}}}\rangle\\ & = &\int_\Gamma\varphi\cdot v_jds-\int_\Gamma f\cdot\frac{\partial {v_j}}{\partial {\nu}}ds\\ & = &\int_\Gamma\varphi\cdot(x+iy)^jds-\int_\Gamma f\cdot\frac{ \partial (x+iy)^j}{\partial {\nu}}ds\\ &\le&\left(\int_\Gamma\varphi^2ds\right)^{1/2}\left(\int_\Gamma(x+iy)^{2j}ds\right)^{1/2}+\left(\int_\Gamma f^2ds\right)^{1/2}\left(\int_\Gamma\left(\frac{\partial (x+iy)^j}{\partial {\nu}}\right)^2ds\right)^{1/2}\\ &\le&\left(\int_\Gamma\varphi^2ds\right)^{1/2}R^j\sqrt{2\pi R}+\left(\int_\Gamma f^2ds\right)^{1/2}jR^{j-1}\sqrt{2\pi R}\\ &\le&j\|\varphi\|_2R^j\sqrt{2\pi R}+j\|f\|_2R^{j-1}\sqrt{2\pi R}. \end{eqnarray*}
    \begin{eqnarray*} |\beta_j|& = &\left|\frac{R(v_j)}{-j}\right|\\ &\le&\|\varphi\|_2R^j\sqrt{2\pi R}+\|f\|_2R^{j-1}\sqrt{2\pi R}\\ &\le&\|\varphi\|_2R^{2m}\sqrt{2\pi R}+\|f\|_2R^{2m}\sqrt{2\pi R} \end{eqnarray*}

    where R > 1 .

    Let

    \begin{equation} \nonumber T = Z_2Z_1^{-1} = Z_2\frac{\text{adj}(Z_1)}{\det(Z_1)} \end{equation}

    where Z_1 = \left[{\begin{array}{*{20}{c}}\beta_1 & \beta_2 & \cdots & \beta_m\\ \beta_2 & \beta_3 & \cdots & \beta_{m+1}\\ \vdots\\ \beta_m & \beta_{m+1} & \cdots & \beta_{2m-1}\end{array}} \right] and Z_2 = \left[{\begin{array}{*{20}{c}}\beta_2 & \beta_3 & \cdots & \beta_{m+1}\\ \beta_3 & \beta_4 & \cdots & \beta_{m+2}\\ \vdots\\ \beta_{m+1} & \beta_{m+2} & \cdots & \beta_{2m}\end{array}} \right] .

    We can view R(v_j) as the measurement obtained by the "detector" v_j , while \beta_j is just a constant multiple of R(v_j) . So, \beta_j is still a measurement of another form, which contains the information about the moment and the position of the dipole source. Since Z_1 and Z_2 are constructed by different measurements \beta_j , T is also a matrix of measurements.

    Assume T is the measurements without noise, and \tilde T is the measurements with noise. Then,

    \begin{eqnarray*} \|T-\tilde T\|_\infty& = &\|Z_2Z_1^{-1}-\tilde Z_2\tilde Z_1^{-1}\|_\infty\\ &\le&\|Z_2-\tilde Z_2\|_\infty\|Z_1^{-1}\|_\infty+\|Z_1^{-1}-\tilde Z_1^{-1}\|_\infty\|Z_2\|_\infty. \end{eqnarray*}

    We will analyse the four norms in the above inequality one by one.

    \begin{eqnarray*} \|Z_2-\tilde Z_2\|_\infty &\le&m\|\varphi-\tilde\varphi\|_2R^{2m}\sqrt{2\pi R}+m\|f-\tilde f\|_2R^{2m}\sqrt{2\pi R}. \end{eqnarray*}

    To find \|Z_1^{-1}\|_\infty we need to estimate \|\text{adj}(Z_1)\|_\infty . We first observe the results for m = 3 , then prove the results to the arbitrary m using mathematical induction.

    If Z_1 = \left[{\begin{array}{*{20}{c}}\beta_1 & \beta_2 & \beta_3\\ \beta_2 & \beta_3 & \beta_4\\ \beta_3 & \beta_4 & \beta_5\end{array}} \right] , then the absolute value of the first element of \text{adj}(Z_1) would be

    \begin{eqnarray*} &&\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4\\ \beta_4&\beta_5\end{array}} \right| \right)\\ & = &|\beta_3\beta_5-\beta_4^2|\le|\beta_3|\cdot|\beta_5|+|\beta_4^2|\\ & = &(p_1S_1^2+p_2S_2^2+p_3S_3^2)(p_1S_1^4+p_2S_2^4+p_3S_3^4)+(p_1S_1^3+p_2S_2^3+p_3S_3^3)^2\\ &\le&(3p_{max}R^2)(3p_{max}R^4)+(3p_{max}R^3)^2 = 2(3p_{max}R^3)^2\\ & = &(3-1)!3^{3-1}p_{max}^{3-1}R^{3(3-1)}\\ & = :&\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4\\ \beta_4&\beta_5\end{array}} \right| \right)\right). \end{eqnarray*}

    Then,

    \begin{eqnarray*} &&\|\text{adj}(Z_1)\|_{\infty}\\ &\le&\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4\\ \beta_4&\beta_5\end{array}} \right| \right)\right)+\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_2&\beta_4\\ \beta_3&\beta_5\end{array}} \right| \right)\right)+\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_2&\beta_3\\ \beta_3&\beta_4\end{array}} \right| \right)\right)\\ &\le&3\cdot\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4\\ \beta_4&\beta_5\end{array}} \right| \right)\right)\\ & = &3\cdot(3-1)!3^{3-1}p_{max}^{3-1}R^{3(3-1)}\\ & = &3!3^{3-1}p_{max}^{3-1}R^{3(3-1)}. \end{eqnarray*}

    Assume when m = n-1 , we have

    \begin{equation} \nonumber \text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4&\cdots&\beta_{n+1}\\ \beta_4&\beta_5&\cdots&\beta_{n+2}\\ \vdots\\ \beta_{n+1}&\beta_{n+2}&\cdots&\beta_{2n-1}\end{array}} \right| \right)\le(n-1)!n^{n-1}p_{max}^{n-1}R^{n(n-1)}. \end{equation}

    In fact, this inequality is also true for other minors with matrix size (n-1)\times(n-1) .

    Then, when m = n we have

    \begin{eqnarray*} &&\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4&\cdots&\beta_{n+1}&\beta_{n+2}\\ \beta_4&\beta_5&\cdots&\beta_{n+2}&\beta_{n+3}\\ \vdots\\ \beta_{n+1}&\beta_{n+2}&\cdots&\beta_{2n-1}&\beta_{2n}\\ \beta_{n+2}&\beta_{n+3}&\cdots&\beta_{2n}&\beta_{2n+1}\end{array}} \right| \right)\\ &\le&\max|\beta_{2n+1}|\cdot \max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4&\cdots&\beta_{n+1}\\ \beta_4&\beta_5&\cdots&\beta_{n+2}\\ \vdots\\ \beta_{n+1}&\beta_{n+2}&\cdots&\beta_{2n-1}\end{array}} \right| \right)\right)+\cdots\\ &&+\max|\beta_{n+2}|\cdot \max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_4&\beta_5&\cdots&\beta_{n+2}\\ \beta_5&\beta_6&\cdots&\beta_{n+3}\\ \vdots\\ \beta_{n+2}&\beta_{n+3}&\cdots&\beta_{2n}\end{array}} \right| \right)\right)\\ &\le&n\cdot\max|\beta_{2n+1}|\cdot \max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4&\cdots&\beta_{n+1}\\ \beta_4&\beta_5&\cdots&\beta_{n+2}\\ \vdots\\ \beta_{n+1}&\beta_{n+2}&\cdots&\beta_{2n-1}\end{array}} \right| \right)\right)\\ &\le&n\cdot\max|p_1S_1^{2n}+p_2S_2^{2n}+\cdots+p_nS_n^{2n}|\cdot (n-1)!n^{n-1}p_{max}^{n-1}R^{n(n-1)}\\ &\le&n\cdot np_{max}R^{2n}\cdot (n-1)!n^{n-1}p_{max}^{n-1}R^{n^2-n)}\\ & = &n!n^np_{max}^nR^{(n+1)n}\\ &\le&n!(n+1)^np_{max}^nR^{(n+1)n}. \end{eqnarray*}

    Then, for any m we have

    \begin{eqnarray*} &&\|\text{adj}(Z_1)\|_\infty = \left\|\text{adj}\left(\left[ {\begin{array}{*{20}{c}}\beta_1&\beta_2&\cdots&\beta_{m-1}&\beta_m\\ \beta_2&\beta_3&\cdots&\beta_{m}&\beta_{m+1}\\ \vdots\\ \beta_{m-1}&\beta_{m}&\cdots&\beta_{2m-3}&\beta_{2m-2}\\ \beta_{m}&\beta_{m+1}&\cdots&\beta_{2m-2}&\beta_{2m-1}\end{array}} \right] \right)\right\|_\infty\\ &\le&\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4&\cdots&\beta_{m+1}\\ \beta_4&\beta_5&\cdots&\beta_{m+2}\\ \vdots\\ \beta_{m+1}&\beta_{m+2}&\cdots&\beta_{2m-1}\end{array}} \right| \right)\right)+\cdots\\ &&+\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_2&\beta_3&\cdots&\beta_{m}\\ \beta_3&\beta_4&\cdots&\beta_{m+1}\\ \vdots\\ \beta_{m}&\beta_{m+1}&\cdots&\beta_{2m-2}\end{array}} \right| \right)\right)\\ &\le&m\cdot\max\left(\text{abs}\left(\left| {\begin{array}{*{20}{c}} \beta_3&\beta_4&\cdots&\beta_{m+1}\\ \beta_4&\beta_5&\cdots&\beta_{m+2}\\ \vdots\\ \beta_{m+1}&\beta_{m+2}&\cdots&\beta_{2m-1}\end{array}} \right| \right)\right)\\ & = &m\cdot(m-1)!m^{m-1}p_{max}^{m-1}R^{m(m-1)}\\ & = &m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}. \end{eqnarray*}

    Thus,

    \begin{eqnarray*} \|Z_1^{-1}\|_\infty& = &\left\|\frac{\text{adj}(Z_1)}{\det(Z_1)}\right\|_\infty\\ &\le&\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_1p_2\cdots p_m\prod _{1\le i < j\le m}(S_i-S_j)^2}\\ &\le&\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}} \end{eqnarray*}

    where d is the smallest distance between any two dipoles.

    Notice that

    \begin{equation} \nonumber Z_1(Z_1^{-1}-\tilde Z_1^{-1})+(Z_1-\tilde Z_1)\tilde Z_1^{-1} = 0. \end{equation}
    \begin{equation} \nonumber Z_1^{-1}-\tilde Z_1^{-1} = -Z_1^{-1}(Z_1-\tilde Z_1)\tilde Z_1^{-1}. \end{equation}
    \begin{eqnarray*} &&\|Z_1^{-1}-\tilde Z_1^{-1}\|_\infty\\ &\le&\|Z_1^{-1}\|_\infty\cdot\|Z_1-\tilde Z_1\|_\infty\cdot\|\tilde Z_1^{-1}\|_\infty\\ &\le&\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)^2\cdot\left(m\|\varphi-\tilde\varphi\|_2R^{2m}\sqrt{2\pi R}+m\|f-\tilde f\|_2R^{2m}\sqrt{2\pi R}\right). \end{eqnarray*}

    Based on the above results, we have

    \begin{eqnarray} &&\|T-\tilde T\|_\infty \\ &\le&\|Z_2-\tilde Z_2\|_\infty\|Z_1^{-1}\|_\infty+\|Z_1^{-1}-\tilde Z_1^{-1}\|_\infty\|Z_2\|_\infty\\ &\le&\left(m\|\varphi-\tilde\varphi\|_2R^{2m}\sqrt{2\pi R}+m\|f-\tilde f\|_2R^{2m}\sqrt{2\pi R}\right)\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)\\ &&+\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)^2\cdot\left(m\|\varphi\|_2R^{2m}\sqrt{2\pi R}+m\|f\|_2R^{2m}\sqrt{2\pi R}\right)\\ &&\cdot\left(m\|\varphi-\tilde\varphi\|_2R^{2m}\sqrt{2\pi R}+m\|f-\tilde f\|_2R^{2m}\sqrt{2\pi R}\right)\\ &\le&2m\left(\|\varphi\|_2R^{2m}\sqrt{2\pi R}+\|f\|_2R^{2m}\sqrt{2\pi R}\right)\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)\\ &&+2m^2\left(\|\varphi\|_2R^{2m}\sqrt{2\pi R}+\|f\|_2R^{2m}\sqrt{2\pi R}\right)^2\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right)^2. \end{eqnarray} (3.3)

    We can further simplify it as

    \begin{equation} \|T-\tilde T\|_\infty \le E+E^2 \end{equation} (3.4)

    where

    \begin{equation} E = 2mR^{2m}\sqrt{2\pi R}\left(\|f\|_2+\|\varphi\|_2\right)\left(\frac{m!m^{m-1}p_{max}^{m-1}R^{m(m-1)}}{p_{min}^md^{m(m-1)}}\right). \end{equation} (3.5)

    When 0 < E < 1 , the error in the position estimate is mainly controlled by E ; when E > 1 , the error in the position estimate is mainly controlled by E^2 .

    Let \Omega be a circular disk centered at the origin and of radius r = 1 . Then, the numerical implementation can be simplified as follow.

    \begin{equation} \notag \frac{\partial {{v_j}}}{\partial {\nu}} = \frac{\partial {{(x+iy)^j}}}{\partial {r}} = \frac{\partial {{\left(re^{i\theta}\right)^j}}}{\partial {r}} = jr^{j-1}e^{i\theta j} = \frac{jr^je^{i\theta j}}{r} = \frac{jv_j}r. \end{equation}
    \begin{eqnarray*} R(v_j)& = &-\langle{{f}}, {{\frac{\partial {v_j}}{\partial {\nu}}}}\rangle\\ & = &-\int_\Gamma f\cdot\frac{\partial {v_j}}{\partial {\nu}}d\Gamma\\ & = &-\int_0^{2\pi} f\cdot\frac{jv_j}{r}\cdot rd\theta\\ & = &-j\int_0^{2\pi} f\cdot v_jd\theta\\ & = &-j\int_0^{2\pi} f\cdot\left(re^{i\theta}\right)^jd\theta, \end{eqnarray*}

    where f is a function of \theta on the boundary. We do not need to know the explicit form of f , but we can measure as many points as possible on the boundary to get enough discretized function values of f . Then, the above integral can be approximated by a Riemann sum.

    The measurable values we want to use in the following are

    \begin{equation} \notag \beta_j = -\frac{R(v_j)}j = \int_0^{2\pi} f\cdot\left(re^{i\theta}\right)^jd\theta. \end{equation}

    The Romberg algorithm is used to calculate the integral numerically.

    We compare the efficacy of the harmonic function method in dipolar source reconstruction when the perturbation level is 0, 0.001, 0.01, 0.1 and the number of dipoles is 1, 2, 3, 4, 5 . It is shown that as the perturbation level increases, the reconstruction error increases (see Figures 15).

    Figure 1.  The effect of the perturbation level on the reconstruction error of 1 dipole. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is \sigma , then the perturbed measurement is the exact measurement times (1\pm\sigma) , where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 2.  The effect of the perturbation level on the reconstruction error of 2 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is \sigma, then the perturbed measurement is the exact measurement times (1\pm\sigma), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 3.  The effect of the perturbation level on the reconstruction error of 3 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is \sigma, then the perturbed measurement is the exact measurement times (1\pm\sigma), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 4.  The effect of the perturbation level on the reconstruction error of 4 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is \sigma, then the perturbed measurement is the exact measurement times (1\pm\sigma), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 5.  The effect of the perturbation level on the reconstruction error of 5 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is \sigma , then the perturbed measurement is the exact measurement times (1\pm\sigma) , where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.

    In the following we show the results of source estimation, assuming there are 3 dipolar sources ( m = 3 ).

    ● Dipole 1: position (0.3, -0.3) and moment (0, 1) .

    ● Dipole 2: position (0.6, 0.2) and moment (1, 1) .

    ● Dipole 3: position (-0.5, 0.4) and moment (2, 2) .

    In the graphs (see Figure 6) we use a small circle and a red line segment to indicate the true value, and use a cross sign and a green line segment to indicate the reconstructed values.

    Figure 6.  The effect of the perturbation level on the reconstruction error of 3 dipoles. As the perturbation level increases, the reconstruction error increases.

    From error estimates we know that as the distance between two dipoles gets closer, the reconstruction error for the positions of dipoles gets larger (see Table 1 and Figure 7). This is verified by the numerical simulations.

    Table 1.  The effect of dipole distance on the reconstruction error. As two dipoles get closer, the mean reconstruction error in the positions of the dipoles gets larger, which is consistent with the result in the error estimate.
    Exact Dipole Distance Reconstructed Dipole Distance
    0.03 0.7429
    0.05 0.3084
    0.10 0.1200

     | Show Table
    DownLoad: CSV
    Figure 7.  The effect of dipole distance on the reconstruction error. As two dipoles get closer, the reconstruction error in the positions of the dipoles gets larger, which is consistent with the theoretical analysis in the error estimate. When d_{exact} = 0.10 , \overline{d_{est}} = 0.1200 ; when d_{exact} = 0.05 , \overline{d_{est}} = 0.3084 ; when d_{exact} = 0.03 , \overline{d_{est}} = 0.7429 .

    We randomly assign two dipoles with fixed distance, say 0.1, in the unit disk, then reconstruct their positions. We fix the noise level for all experiments at \sigma = 0.001 .

    Let d_i (i = 1, 2) be the distance between the i th exact dipole and the i th estimated dipole, and d_{max} be the largest d .

    We repeat the experiment 10 times and show their performance on average over different dipole distances.

    The above experiment also provides a numerical example to show that the estimate given by us in Theorem 3.1 provides a better error bound when the two poles are very close.

    When the number of dipoles is m = 2 , Chafik's estimate is bounded by \frac{C_1}{d} (see Inequality (3.1)), while our estimate is bounded by \frac{C_2}{d^2} (see Inequality (3.4) and Eq (3.5)) where d is the smallest distance between two dipoles and C_i ( i = 1, 2 ) are constants independent of d . That is, when the distance is halved, the error bound will be amplified by 2 in Chafik's estimate and by 4 in our estimate.

    From the data simulation, we see that

    \begin{equation} \nonumber \frac{0.05}{0.03} = 1.67 < \frac{0.7429}{0.3084} = 2.41 < 1.67^2 = 2.79. \end{equation}
    \begin{equation} \nonumber \frac{0.10}{0.05} = 2 < \frac{0.3084}{0.1200} = 2.57 < 2^2 = 4. \end{equation}
    \begin{equation} \nonumber \frac{0.10}{0.03} = 3.33 < \frac{0.7429}{0.1200} = 6.19 < 3.33^2 = 11.09. \end{equation}

    For example, when the distance between the two dipoles is reduced from 0.10 to 0.05, by Chafik's estimate the error should be amplified by 2, but in fact, the error is amplified by 2.57, which is bounded by 4 in our estimate.

    In this paper we studied a harmonic function method for the dipolar source reconstruction, derived error estimate for the harmonic function method and compared our result with Chafik's estimate. By numerical simulations it is shown that the harmonic function method can quickly and accurately locate active regions in EEG source reconstruction. In the future, we plan to extend the harmonic function method to 3D case and applied this method to some real EEG data. The brain's conductance variation in different brain regions also leads to additional challenges in source localization [35]. Although these tissue properties can be quantified through MRI methods, numerical methods such as finite element method will be needed to solve the inverse problems. Since the estimation of the number of dipoles relies on the calculation of the rank of the measurement matrix, which is significantly affected by the noise, we hope to find some way to solve or circumvent this problem. In addition, the situation that the number of exact dipoles is not equal to the estimated value could also be considered. Furthermore, when two dipoles get close enough, it may be better to regard them as an equivalent dipole to avoid increased error.

    We thank anonymous reviewers for providing us with valuable suggestions.

    All authors declare no conflicts of interest in this paper.



    [1] Pinheiro LRS, Gradissimo DG, Xavier LP, et al. (2022) Degradation of azo dyes: Bacterial potential for bioremediation. Sustainability 14: 1510. https://doi.org/10.3390/su14031510. doi: 10.3390/su14031510
    [2] Rauf MA, Ashraf SS (2009) Review: Fundamental principles and application of heterogeneous photocatalytic degradation of dyes in solution. Chem Eng J 151: 10–18. https://doi.org/10.1016/j.cej.2009.02.026 doi: 10.1016/j.cej.2009.02.026
    [3] Zhuang Y, Zhu Q, Li G, et al. (2022) Photocatalytic degradation of organic dyes using covalent triazine-based framework. Mater Res Bulletin 146: 111619. https://doi/org/10/1016/j/materresbull.2021.111619. doi: 10.1016/j/materresbull.2021.111619
    [4] Sibhatu AS, Weldegebrieal KG, Sgaradevan S (2022) Photocatalytic activity of CuO nanoparticles for organic and inorganic pollutants removal in wastewater remediation. Chemosphere 300: 134623. https://doi.org/10.1016/j.chemosphere.2022.134623 doi: 10.1016/j.chemosphere.2022.134623
    [5] Fouda A, Salam S, Wassel AR, et al. (2020) Optimization of green biosynthesized visible light active CuO/ZnO nano-photocatalysts for the degradation of organic methylene blue dye. Hélion 6: e04896. https://doi.org/10.1016/jhelion.2020.e04896.
    [6] Lacombe S, Tran-thi T, Guillard C, et al. (2007) La photocalyse pour l'elimination des polluants. Actualités chimique 308: 79–93.
    [7] Liu X, C Chen, Zhao Z, et al. (2013) A review on the synthesis of manganese oxide nanomaterials and their applications on lithium-ion batteries. J Nanomater 2013: 736375. http://dx.doi.org/10.1155/2013/736375 doi: 10.1155/2013/736375
    [8] Naika HR, Lingaraju K, Manjunath K, et al. (2015) Green synthesis of CuO nanoparticles using Gloriosa superba L. extract and their antibacterial activity. J Taibah Univ Sci 9: 7–12. https://doi.org/10.1016/j.jtusci.2014.04.006. doi: 10.1016/j.jtusci.2014.04.006
    [9] Ahmad MM, Kotb HM, Mushta S, et al. (2022) Green synthesis of Mn + Cu bimetallic nanoparticles using vinca rosea extract and their antioxidant, antibacterial, and catalytic activities. Crystals 12: 72. https://doi.org/10.3390/cryst12010072. doi: 10.3390/cryst12010072
    [10] Basavegowda N, Baek K (2021) Multimetallic nanoparticles as alternative antimicrobial agents: Challenges and perspectives. Molecules 26: 912. https://doi.org/10.3390/molecules26040912. doi: 10.3390/molecules26040912
    [11] Iqbal M, Thebo AA, Shah AH, et al. (2016) Influence of Mn-doping on the photocatalytic and solar cell efficiency of CuO nanowires. Inorg Chem Commun 76: 71–76. https://doi.org/10.1016/j.inoche.2016.11.023. doi: 10.1016/j.inoche.2016.11.023
    [12] Pramothkumar A, Senthilkumar N, Mercy Gnana Malar KC, et al. (2019) A comparative analysis on the dye degradation efciency of pure, Co, Ni and Mn‑doped CuO nanoparticles. J Mater Sci-Mater El 30: 19043–19059. https://doi.org/10.1007/s10854-019-02262-4. doi: 10.1007/s10854-019-02262-4
    [13] Vindhya PS, Kavitha VT (2022) Leaf extract-mediated synthesis of Mn-doped CuO nanoparticles for antimicrobial, antioxidant and photocatalytic applications. Chem Pap. https://doi.org/10.1007/s11696-022-02631-0.
    [14] Kabengele CN, Kasiama GN, Ngoyi EM, et al. (2022) Secondary metabolites and mineral elements of Manotes expansa and Aframomum alboviolaceum leaves collected in the democratic republic of Congo. ARRB 37: 57–63.
    [15] Rizwana H, Alwhibi MS, Al-Judaie RA, et al. (2022) Sunlight-mediated green synthesis of silver nanoparticles using the berries of Ribes rubrum (Red Currants): characterization and evaluation of their antifungal and antibacterial activities. Molecules 27: 2186. https://doi.org/10.3390/molecules27072186. doi: 10.3390/molecules27072186
    [16] Chen LQ, Li Fang, Ling J, et al. (2015) Nanotoxicity of silver nanoparticles to red blood cells: size dependent adsorption, uptake, and hemolytic activity. Chem Res Toxicol 28: 501–509. https://doi.org/10.1021/tx500479 doi: 10.1021/tx500479
    [17] Pandey S, Singh S (2020) Eco-friendly nanocomposite and properties of manganese nanoparticles using UV-vis and IR fourier spectrum. IJISRT 5: 770–773.
    [18] Shah M, Fawcett D, Sharma S, et al. (2015) Review green synthesis of metallic nanoparticles via biological entities. Materials 8: 7278–7308. https://doi.org/10.3390/ma8115377. doi: 10.3390/ma8115377
    [19] El-seedi, El-Shabasy RM, Khalifa SAM, et al. (2019) Metal nanoparticles fabricated by green chemistry using natural extracts: biosynthesis, mechanisms, and applications. RSC Adv 24539–24559. https://doi.org/10.1039/C9RA02225B
    [20] Makarov VV, Love AJ, Sinitsyna OV, et al. (2014) Green nanotechnologies: Synthesis of metal nanoparticles using plants. Acta Naturae 6: 35–44. https://doi.org/10.32607/20758251-2014-6-1-35-44 doi: 10.32607/20758251-2014-6-1-35-44
    [21] Desai R, Mankad V, Gupta SG, et al. (2012) Size distribution of silver nanoparticles: UV-visible spectroscopic assessment. Nanosci Nanotechnol Let 4: 30–34. https://doi.org/10.1166/nnl.2012.1278 doi: 10.1166/nnl.2012.1278
    [22] Yeshchenko OA, Bondarchuk IS, Gurin VS, et al. (2013) Temperature dependence of the surface plasmon resonance in gold nanoparticles. Surf Sci 608: 275–281, http://dx.doi.org/10.1016/j.susc.2012.10.019. doi: 10.1016/j.susc.2012.10.019
    [23] Seifipour R, Nozari M, Pishkar L (2020) Green synthesis of silver nanoparticles using Tragopogon Collinus leaf extract and study of their antibacterial effects. JIOPM 30: 2926–2936. https://doi.org/10.1007/s10904-020-01441-9 doi: 10.1007/s10904-020-01441-9
    [24] Vidhu VK, Aromal SA, Philip D (2011) Green synthesis of silver nanoparticles using Macrotyloma uniform. Spectrochim Acta A 83: 392–397. https://doi.org/10.1016/j.saa.2011.08.051 doi: 10.1016/j.saa.2011.08.051
    [25] Berta L, Coman NA, Rusu A, et al. (2021) A review on plant-Mediated synthesis of Bimetallic nanoparticles, characterization and their biological applications. Materials 14: 7677. https://doi.org/10.3390/ma14247677 doi: 10.3390/ma14247677
    [26] Pinto VV, Ferreira MJ, Silva R, et al. (2010) Long time effect on the stability of silver nanoparticles in aqueous medium: effect of synthesis and storage conditions. Colloid Surface A 364: 19–25. https://doi.org/10.1016/j.colsurfa.2010.04.015 doi: 10.1016/j.colsurfa.2010.04.015
    [27] Azeez F, Al-Hetlani E, Arafa M (2018) The effect of surface charge on photocatalytic degradation of Methylene Blue dye using chargeable titania nanoparticles. Sci Rep 2018: 7104. https://doi.org/10.1038/s4158-018-15673-5. doi: 10.1038/s4158-018-15673-5
    [28] Taylor MG, Augustin N, Gounaris CE, et al. (2015) Catalyst design based on morphology and environment dependent adsorption on metal nanoparticles. ACS Catal 20155: 6296–6301. https://doi.org/10.1021/acscatal.5b01696 doi: 10.1021/acscatal.5b01696
    [29] Chanu LA, Singh WJ, Singh KJ, et al. (2019) Effect of operational parameters on the photocatalytic degradation of Methylene blue dye solution using manganese doped ZnO nanoparticles. Results Phys 12: 1230–1237. https://doi.org/10.1016/j.rinp.2018.12.089 doi: 10.1016/j.rinp.2018.12.089
    [30] Dobrovolskaia MA, Clogston JD, Neun BW (2008) Method for analysis of nanoparticle hemolytic properties in vitro. Nano Lett 8: 2180–2187. https://doi.org/10.1021/nl0805615 doi: 10.1021/nl0805615
    [31] Gabor F (2011) Characterization of nanoparticles intended for drug delivery. Sci Pharm 79: 701–702.
    [32] Gul A, Shaheen A, Ahmad I, et al. (2021) Green synthesis, characterization, enzyme inhibition, antimicrobial potential, and cytotoxic activity of plant mediated silver nanoparticle using Ricinus communis leaf and root extracts. Biomolecules 11: 206. https://doi.org/10.3390/biom11020206. doi: 10.3390/biom11020206
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