1 IJCPE Vol.11 No.1 (March 2010) Iraqi Journal of Chemical and Petroleum Engineering Vol.11 No.1 (March 2010) 1-9 ISSN: 1997-4884 Separation of Bovine Serum Albumin Using Chromatographical Column: Parameters and Simulation Cecilia Khoshaba Haweel * , Ahmed J. Ali and Samar N. Alias * Chemical Engineering Department - College of Engineering - University of Baghdad – Iraq Abstract A liquid-solid chromatography of Bovine Serum Albumin (BSA) on (diethylaminoethyl-cellulose) DEAE-cellulose adsorbent is worked experimentally, to study the effect of changing the influent concentration of (0.125, 0.25, 0.5, and 1 mg/ml) at constant volumetric flow rate Q=1ml/min. And the effect of changing the volumetric flow rate (1, 3, 5, and 10 ml/min) at constant influent concentration of Co=0.125mg/ml. By using a glass column of (1.5cm) I.D and (50cm) length, packed with adsorbent of DEAE-cellulose of height (7cm). The influent is introduced in to the column using peristaltic pump and the effluent concentration is investigated using UV-spectrophotometer at 30oC and 280nm wavelength. A spread (steeper) break-through curve is gained at lean feed concentration of 0.125mg/ml, while the flow rate greater than (3ml/min) is almost the same. So it is butter to work at low volumetric flow rate between (1-3) ml/min. The equilibrium-dispersion model of liquid-solid chromatography for a binary mixture, related with Langmuir isotherm correlation is used in the modeling of this work. The resulting model is solved numerically by using MATLAB V.6.5 program. Introduction Chromatography is a very special separation process for a multitude of reasons! First of all, it can separate complex mixtures with great precision. Even very similar components, such as proteins that may only vary by a single amino acid. Second, chromatography can be used to separate delicate products since the conditions under which it is performed are not typically serving (Hubble, 2004). A mixture of various components enters a chromatography process and is allowed to come into contact with two phases, the stationary phase is contained in a column where the mobile phase moves in a controlled manner relative to the stationary phase, carrying with it any material that may prefer to mix with it. To determine the differences in the concentration of mobile phase (solute), a detector is used which can give a curve of the separated substance. So, the name chromatography stems from this, where each separated substance has different color on the stationary phase and gives different curve from the detector (Stock, 1974, Encyclopedia, 1992 and Hubble, 2004). Frontal chromatography analysis has widely used for large scale separations of industry. It is made in which conditions are chosen that the desired component very quickly saturates the column and emerges from the column shortly after the feed is saturated. Impurities, on the other hand, are strongly retained by the adsorbent and feeding is continued until the front of the adsorbed impurities approaches the column outlet (Simpson, 2001). Ion-exchange technique of chromatography is used to represent the frontal analysis. That was because of the simplicity of this technique where separation take place according to charge of both solid adsorbent and the separated molecules. There can be two types of functional groups of exchangers, covalently attached to the support beads. These are called anion exchangers (resin with positive functional groups) or cation University of Baghdad College of Engineering Iraqi Journal of Chemical and Petroleum Engineering Separation of Bovine Serum Albumin Using Chromatographical Column: Parameters and Simulation 2 IJCPE Vol.11 No.1 (March 2010) exchangers (resin with negative functional groups) (Raply and Walker, 1998, and Freifelder, 1998). Representing the frontal analysis with Ion-exchange chromatography by a mathematical model is an important feature not only for experimental data but, also for the process design and optimization. Experiments need to be carried out to get the mass-transfer parameters for modeling and simulation and comparisons with experimental results indicated that the prediction was acceptable. Experimental Work A system of Bovine Serum Albumin (BSA) protein adsorbed onto DEAE-cellulose was chosen to study the frontal analysis and compare it with the theoretical work (Alias, 2007). Bovine Serum Albumin protein of 66430 molecular weight from Sigma Chemical Company dissolved in (5mg/ml) Tris-HCl buffer of 98% purity mead by BBH Company to produce the mobile phase of the chromatography system at (pH 7). DEAE-cellulose anion-exchanger from Sigma Chemical Company was used as stationary phase of the chromatography system. Five samples of different weights of DEAE-cellulose was prepared to evaluate the equilibrium isotherm, each one of them is put in a beaker with (30ml) of BSA solution of concentration (0.12mg/ml) and pH of 7. The samples were shaked at (30oC) for about (26 hours) then the concentrations were calculated using UV- spectrophotometer. The amount of solute adsorbed into DEAE-cellulose q in (mg/g DEAE-cellulose) was found according to the equilibrium relation: A Lo W VCC q )(   (1) The adsorption isotherm was obtained by plotting q vs. C A glass beaker of (250ml) capacity was prepared for batch experiment. A (50ml) of BSA-solution was added to the beaker containing (3.37mg) of DEAE-cellulose. At zero time, DEAE-cellulose of certain weight was added. Samples were taken every 10 minutes during the experiment and concentration is measured using UV- spectrophotometer. Frontal chromatography experiments were carried out to find the breakthrough curve of BSA adsorbed to DEAE- cellulose bed. To study the effect of changing the initial concentration (0.125, 0.25, 0.5, and 1 mg/ml) at constant volumetric flow rate of (1ml/min) and changing the volumetric flow rate (1, 3, 5, and 10 ml/min) at constant initial concentration of (0.125mg/ml). The procedures of all run experiments are as follows: DEAE-cellulose was poured into the column at (7cm) length. BSA-solution was prepared at the desired concentration for each run using Tris-HCl buffer. The solution was introduced to the top of the column using peristaltic pump at a certain flow rate. Samples were collected directly from the bottom of the column using tubes for every (1ml) and then the concentration was measured using UV-spectrophotometer at 280 nm wavelength as in (fig. 1). The resulting breakthrough curves were determined by plotting effluent concentration (C/Co) against dimensionless time τ. (Alias, 2007). Fig.1 the experimental apparatus of BSA adsorbed in DEAE-cellulose bed. Modeling and Simulation Batch Adsorber Batch adsorber analyses are important to calculate the external mass transfer coefficient kf (m/s) of batch experiment and pore diffusivity DP (m 2 /s). The formulation of batch adsorber based on (Ahmed, 2006). For batch adsorber, the mathematical model with pore diffusion model is:  Mass balance in fluid-bulk phase: 0)( 3 Pr   PRbif PP Ai L CCk R W dt dC V  (2) Where: VL = volume of fluid in the batch adsorber WA = mass of adsorbent Cecilia Khoshaba Haweel and Ahmed J. Ali and Samar N. Alias IJCPE Vol.11 No.1 (March 2010) 3  Mass balance inside particle phase: )( 1 2 2 r C r rr D dt dq t C p ppP p P         (3)  Initial conditions and boundary conditions ][ 00:0 00:0 , 00 p p Rrp bf Rr p ppp rr p pob CCk r C DRr dr dq r C r qCCCt                                     In case of using Langmuir isotherm correlation: bC bC qq p p m   1 (4) Where Cb and Cp are the solute concentration in the fluid- bulk and particle phase, respectively. The external mass transfer coefficient for the solute adsorbed at certain particle size and optimum agitation speed, can be obtained by the analytical solution for equation (2.29) where at t = 0, Cp,r=Rp = 0 and Cb = Co, hence:          o t A Lpp f C C tW VR k ln 3  (5) Where Co, Ct are the solute concentration at time zero and time (t) and obtained from the typical concentration decay curve. Modeling of Liquid-Solid Chromatographical Column The formulation of Chromatographical column is Based on (Volker, 1999, and Eggers, 2003), the governing equations can be obtained from differential mass balances of the bulk-fluid phase and the particle phase, respectively, for component i.  Continuity equation in the bulk-fluid phase: Using Cpi, the concentration in the stagnant fluid-phase (in the macropores), and writing the expression of interfacial flux leads to:   pRRpibi p fii CC R k t q    , 3 (7)    013 ,2 2              pRRpibi pb bfibibibi bi CC R k t C Z C Z C D    (8)  Continuity equation inside the macropores: The particle phase continuity equation in spherical coordinates is:  Initial and boundary conditions The initial and boundary conditions may be represented by the following equations: Initial condition (t = 0): 0),0(  ZCC bibi (10) 0),,0(  ZRCC pipi (11) Separation of Bovine Serum Albumin Using Chromatographical Column: Parameters and Simulation 4 IJCPE Vol.11 No.1 (March 2010) Boundary conditions: Z = 0:  oibi bi bi CC DZ C     (12) Z =L: 0   Z Cbi (13) R = 0: 0   R C pi (14) R = Rp:   pRRpibi pip fipi CC D k R C    ,  (15) Model Parameters of Fixed-Bed Chromatography Column a) Correlations to estimate (molecular diffusivity) Dm: Polson proposed the equation for high molecular weight solutes: ) . (104.9 3/1 15 w m M T D    (16) Where: T is temperature (K), µ is the viscosity of the solution (Pa·s) and Mw is the molecular weight of the solute (Polson, 1950). b) Correlation to estimate mass transfer coefficient kf in fixed-bed adsorption: )100(ReRe45.10.2 5.03/1  ScSh (17) Where: Sh=2Rpkf/Dm (Sherwood number), Sc = µ/(ρDm) (Schmidt number), and Re=2Rpρν/µ (Reynolds number). µ is the fluid (water) viscosity, and ρ is the fluid (water) density (Jian, 2004, and Montesinos, 2005). c) Correlation to estimate Db the axial dispersion coefficient can be estimated from the following correlation: )10(Re Re011.00.2 2 48.0    p b R D (18) Where: ν is the interstitial velocity along the column = 4Q/επd 2 (Q is the volumetric flow rate) and Re=2Rpρν/µ (Jian, 2004, and Montesinos, 2005). Simulation Program The model equations must be solved numerically. The numerical method of lines (MOL) is used in order to solve the coupled system of PDE obtained ether from batch adsorber or chromatography column. The bulk- fluid phase and the particle equations are first discretized using the FE (finite element) and the OC (orthogonal collocation) methods, respectively. The resulting ODE system is solved using an existing ODE solver provided by MATLAB v-6.5.The code of MATLAB m-files for both batch adsorber and chromatography column are presented in (Ahmed, 2006). This program is used for both single and multicomponent systems after changing some parameters. The model parameters used in the simulation program are listed in tables 1 and 2. Results and Discussion Isotherm Determinations and Constants The adsorption isotherm of BSA adsorbed onto DEAE- cellulose of 0.35mm-particle size at (30 o C and 5mg/ml Tris-HCl buffer pH 7) is shown in (fig. 2), were the initial concentration of BSA-solution was (0.12 mg/ml). By fitting the equilibrium data with Langmuir isotherm correlation, the results were good (qm = 2.022 mg/g) and (b = 13.083 ml/mg). Fig. 2: Adsorption isotherm of BSA onto DEAE- cellulose at (30 o C) Intraparticle Diffusivity coefficient Dp Estimations In order to find Intraparticle Diffusivity coefficient Dp, it was needed to find the mass transfer coefficient kf in batch adsorption process and then the numerical solution of batch adsorber model is applied. The external mass transfer kf, coefficient in batch adsorber was computed from initial rate data, (i.e. from the concentration decay curve fig. 3) using equation (5):          o t A Lpp f C C tW VR k ln 3  (19) Cecilia Khoshaba Haweel and Ahmed J. Ali and Samar N. Alias IJCPE Vol.11 No.1 (March 2010) 5 Where Co, Ct are the solute concentration at time zero and time t. For accurate estimation of kf, samples were taken after 10, 20, and 30 minutes and analyzed immediately. The average calculated value of kf was 1.0683×10 -6 m/s. The pore diffusivity coefficient was derived from the typical concentration decay curve by iterative search technique predicted on the minimization of the difference between experimental and predicted data from pore diffusion model (Ahmed, 2006). The results are shown in (fig. 3). The pore diffusivity coefficient for BSA solute according to (fig 3) is 0.078×10 -10 m 2 /s. Fig. 3 Comparison of the measured concentration-time data at (30 o C) with that predicted by pore diffusion model in batch adsorber. Chromatography Experiments  Figures (4 to 7) were obtained from the adsorption of BSA solution on DEAE-cellulose adsorbent at different concentrations of influent Co that at Q = 1ml/min. As follows: Fig. 4 Break-through curve of BSA on DEAE-cellulose at Co = 0.125mg/ml Fig. 5 Break-through curve of BSA on DEAE-cellulose at Co = 0.25mg/ml Fig. 6 Break-through curve of BSA on DEAE-cellulose at Co = 0.5mg/ml Figure 8 represents the effect of changing the initial concentration at (0.125, 0.25, 0.5, and 1 mg/ml). From this figure, the change in inlet solute concentration markedly affects the shape and position of the break- through curve. The higher the solute concentration the faster the break-through. However, the quantity of solution to be supplied is large when the solute concentration is low to arrive the optimum value. Fig. 7 Break-through curve of BSA on DEAE-cellulose at Co = 1mg/ml Separation of Bovine Serum Albumin Using Chromatographical Column: Parameters and Simulation 6 IJCPE Vol.11 No.1 (March 2010) Fig. 8 Break-through curve of BSA on DEAE-cellulose at different initial concentrations  Figures (9 to 12) were obtained from the adsorption of BSA solution on DEAE-cellulose adsorbent at different volumetric flow rates Q that at Co = 0.125mg/ml initial concentration. As follows: Fig. 9 Break-through curve of BSA on DEAE-cellulose at Q = 1ml/min Fig. 10 Break-through curve of BSA on DEAE-cellulose at Q = 3ml/min Fig. 11 Break-through curve of BSA on DEAE-cellulose at Q = 5ml/min Fig. 12 Break-through curve of BSA on DEAE-cellulose at Q = 10ml/min Figure 13 represents the effect of changing the volumetric flow rate of the mobile phase for (1, 3, 5, and 10 ml/min). Flow rate affects the film mass transfer coefficient. If Rp is unchanged, for correlation 17, kf is proportional to ν 0.5 . The flow rate also affects the axial dispersion coefficient according to correlation 18. For higher flow rates, Break-through is faster and poor adsorption efficiencies will result. When the flow rate is low, an increase in the spreading of the break-through curve will occur and an increase in the time that the solute is in contact with the stationary phase, allowing more time for adsorption and permitting near-local equilibrium conditions. That the break point is unchanged which is about 0.6 of initial concentration. Cecilia Khoshaba Haweel and Ahmed J. Ali and Samar N. Alias IJCPE Vol.11 No.1 (March 2010) 7 Fig. 13 Break-through curve of BSA on DEAE-cellulose at different volumetric flow rates  All the resulting break-through curves obtained to the adsorption of BSA on DEAE-cellulose bed, are almost sharp and the break point occur about 0.557 from the initial concentration for high volumetric flow rates, and the adsorption process is faster at about (30- 40) minutes. From the observations of fig. 13, the break-through curves obtained for volumetric flow rates of (3-10 ml/min) are almost the same (i.e. the break-through curves are almost compatible).  Table 1 Batch adsorber of BSA on DEAE- cellulose system at 30 o C. Simulation Data Parameter Value Rp 0.3510 -3 m εp 0.6 b 13.083 m 3 /kg qm 0.002022 kg/kg Dp 0.07810 -10 m 2 /s kf 1.068310 -6 m/s ρp 309.5 kg/m 3 VL 0.0510 -3 m 3 WA 3.3710 -3 kg Co 0.12 kg/m 3 Table 2 Data used in the simulation program of chromatography column Parameter Value Column length L 0.07 m Column diameter d 0.015 m Particle diameter Rp 0.3510 -3 m Column porosity ε 0.775 Particle porosity εp 0.6 Axial dispersion Db Correlation (18) Pore diffusivity Dp 0.07810 -10 m 2 /s Molecular diffusivity Dm Correlation (16) External mass transfer coefficient kf Correlation (17) Maximum adsorption capacity qm 0.002022 kg/kg Langmuir isotherm constant b 13.083 m 3 /kg Bed density ρp 309.5 kg/m 3 Conclusions 1. Comparisons were made between the results obtained from the model program and that gained from experimental work data, the agreement was very good, because all the parameters used in the model program were evaluated ether experimentally or by empirical relations of grate accuracy. This program is applied in all cases of liquid-solid chromatography systems with a very good accuracy. 2. For the experimental work data or literature data, Langmuir isotherm correlation gives good results. This correlation is a good assumption for the system of very low concentrations of influent. 3. From the experimental worked data it was concluded that it is preferable to work with lean feed and low volumetric flow rate. In those conditions, a steeper or spread break-through curve will be gained which lead to increase the solute recovery efficiency. Also the effect of increasing the volumetric flow rat more than (3ml/min) is not efficient. 4. Whenever the break-through curve is spread (steeper), the break-through time point will occur with less C/Co ratio 5. Axial dispersion coefficient Db, which is affected by the flow rate of the system is more significant at lower flow rate and is less noticeable as the rate is increased, but the pore diffusivity Dp and external mass transfer coefficient kf is not that influential parameter because it represent the diffusivity inside the pore of particle, which its size is very small. Separation of Bovine Serum Albumin Using Chromatographical Column: Parameters and Simulation 8 IJCPE Vol.11 No.1 (March 2010) Nomenclature Notation ai Constant from Langmuir correlation (qm . b) bi Langmuir isotherm constant (m 3 /kg) Bii Biot number of mass transfer for component i, (kfRp/εpDpi). ( - ) Coi Concentration used for nondimensionalization, max{Cfi(t)}. Ct Concentration of solute at any time. (kg/m 3 ) Cbi Bulk-Fluid phase concentration of component i, (kg/m 3 ) Cfi Feed concentration profile of component i a time dependant variable (kg/m 3 ) Cpi Concentration of component i in the stagnant fluid phase inside theparticle macropores (kg/m 3 ) C*pi Concentration of component i in the solid phase of particle (kg/m 3 ) cbi = Cbi /Coi ( - ) cpi = Cpi/Coi ( - ) c*pi = C*pi / Coi ( - ) Dbi Axial dispersion coefficient of component i (m 2 /s) Dpi Pore diffusivity coefficient of component i (m 2 /s) Dm Molecular diffusivity (m 2 /s) d Inner diameter of a column (m) kfi, External mass transfer coefficient of component i (m/s) L Column length (m) MW Molecular weight. PeLi Peclet number of axial dispersion for component i, νL/Dbi. Q Volumetric flow rate of mobile phase (m 3 /s) q Concentration of solute in stationary phase bed (kg solute/kg sorbent) qm Maximum adsorption capacity (kg solute/kg sorbent) R Radial coordinates for particle Re Reynolds number for sphere, (2Rpρν/μ). Rp Particle radios (m) r = R/Rp Sc Schmidt number, (μ/ρDm). Sh Sherwood number for sphere, (2Rpkf/Dm). t Time (s) T Temperature (C) VL the final volume of the sample put in beaker (ml) z Dimensionless axial coordinate Z/L. Greek εb Column or bed porosity. εP Particle porosity. ν Interstitial velocity, (4Q/εbπd 2 ). τ Dimensionless time. μ Fluid (water) viscosity (Pa.s). ρ Fluid (water) density (Kg/m 3 ). ρp Particle density (Kg/m 3 ). Subscripts and superscript P Particle phase. m Mono layer. i i-th component. j j-th component. * Particle phase concentration Abbreviations BSA Bovine Serum Albumin. FDM Finite Difference Method. FEM Finite Element Method. OCM Orthogonal Collocation Method. ODE Ordinary Differential Equation. PDE Partial Differential Equation. References 1. Alias S. N., 2007, "Separation of Biological Materials Using Chromatographical Column", M Sc thesis submitted to Baghdad University. 2. Ahmed Kawther W., 2006,”Experimental and modeling for the removal of multi pollutants by adsorption”. A PhD thesis submitted to Baghdad university. 3. Eggers R., 2003, “Simulation of Frontal Adsorption”. REPORT submitted to the Technische Universitat Hamburg-Harburg (TUHH). 4. “ENCYCLOPEDIA OF CHEMICAL TECHNOLOGY”, 1992. Forth edition, volume 6, A Wily-Interscience Publication. 5. Freifelder David, 1998, “Physical Biochemistry”, second edition, W. H. Freeman and Company, New York. 6. Jian-gang L. U., Feb. 28, 2004, “Mathematical modeling of salt-gradient ion-exchange simulated moving bed chromatography for protein separations”. Journal of Zhejiang University SCIENCE (JZUS). http://www.zju.edu.cn/jzus E-mail: jzus@zju.edu.cn 7. Hubble John, 2004, “Adsorption and Chromatographic separations”. Literature (lecture) http://people.bath.ac.uk/cesjh/adsorb.htm E- mail: mailto:cesjh@bath.ac.uk http://www.zju.edu.cn/jzus mailto:jzus@zju.edu.cn http://people.bath.ac.uk/cesjh/adsorb.htm mailto:cesjh@bath.ac.uk Cecilia Khoshaba Haweel and Ahmed J. Ali and Samar N. Alias IJCPE Vol.11 No.1 (March 2010) 9 8. Montesinos-Cisneros R. M., Guzmلn-Zamudio R. Ortega-Lpَez J., Tejeda-Mansir Y. A., Marzo 2005, “FRONTAL CHROMATOGRAPHY OF PLASMIDS: PARAMETERS ESTIMATION AND SIMULATION”. Revista Mexicana De Ingenieria QUIMICA Vol. 4 (2005) 47-58 9. Rapley Ralph and walker John M., 1998, “Molecular Biomethods Handbook”. Humana Press, Totowa, NJ.s 10. Simpson John M., 2001, “PROTEIN PURIFICATION PROCESS ENGINEERING, Marcel Dekker, Inc. 11. Stock R. and Rice C. B. F., 1974, “Chromatographic Methods”, Halsted Press New York. 12. Volker Kasche, 1999, “Simulation of Liquid Chromatography and Simulated Moving Bed (SMB) Systems”. A PhD Thesis submitted to Hamburg University. 13. Polson, A. (1950). “Some aspects of diffusion in solution and a definition of a colloidal particle”. J. Phys. Colloid Chem., 54, 649-652. المعامالت والمحاكات: فصل مصل الذم الحيواني باستخذام عمود الكروماتوكراف طيظيهيا خٕشاتا ْأيم 1 حًذ خٕاد ػهيو, 2 طًزَداذ انياصٔ خايؼح , كهيح انُٓذطح. 3لظى ُْذطح انكيًياء انحياذيح , خايؼح تغذاد, كهيح انُٓذطح. 2لظى انُٓذطح انكيًيأيح , خايؼح تغذاد, كهيح انُٓذطح. 1 لظى ُْذطح انكيًياء انحياذيح, تغذاد الخالصة -Diethyleaminoethiel) ػهٗ يادج (Bovine Serum Albumin)انظائم نًصم انذو انحيٕاَي -كزٔياذٕكزاف انصهة cellulose)ٔا (DEAE-cellulose)0.5 ,0.25 ,0.125)نذراطح ذأثيز انرغيز انثرزكيش انذاخم انٗ انؼًٕد . لذ ذى اشرغانّ ػًهيا, and 1mg/ml) تثثٕخ يؼذل اندزياٌ انحدًي (1ml/min) 5 ,3 ,1) ثى دراطح ذأثيز انرغيز تًؼذل اندزياٌ انحدًي, and 10ml/min) ترثٕخ انرزكيش انذاخم (0.125mg/ml) . انؼًم انردزيثي لذ ذى تاطرخذاو ػًٕد سخاخي َصف عٕل لغزِ انذاخهي انظائم . ( طى7) تغٕل DEAE-cellulose (diethylaminoethyl-cellulose), يحشٕ تًادج ( طى50)ٔعٕنّ (طى1.5) , ٔذزكيش انظائم انخارج ذى لياطّ تاطرخذاو خٓاس األشؼح peristaltic pump انذاخم يذخم إنٗ انؼًٕد تاطرخذاو يضخح يٍ َٕع 30) تذرخح حزارج UV-spectrophotometer فٕق انثُفظديح o C) تغٕل يٕخي (280 nm) . كاٌ ذزكيش انظائم انذاخم (0.125mg/ml) األفضم الَّ يؼغيbreak-through point لهيهح يغ break-through curveْذا انرزكيش . يُثظظ (0.125mg/ml) ُْا, يؼذالخ اندزياٌ انحدًي انري ْي الم يٍ . اطرخذو في ذغثيك ذدارب أخزٖ نًؼذالخ خزياٌ يخرهفح (3ml/min) نٓذا يٍ األفضم نٓذا انُظاو أٌ يؼًم ػهٗ يؼذل خزياٌ حدًي لهيم . ذكٌٕ ذمزيثا يرغاتمح نُفض انظزٔف انرشغيهيح انصهة نخهيظ ثُائي انًزتٕط _يٕديالخ انرٕاسٌ انًرشرد نهكزٔياذٕغزاف انظائم. تُفض انظزٔف (3ml/min-1)يرزأذ تيٍ انًٕديم انُاذح لذ حم ػذديًا تاطرخذاو تزَايح . تؼاللح الَكًٕر ايشٔثيزو لذ اطرخذو في اشرماق انًٕديم انزياضي نٓذا انؼًم MATLAB v.6.5 .