CHROMATOGRAPHY METHOD, METHOD OF DETERMINING THE CONCENTRATION OF AT LEAST ONE COMPOUND IN A CHROMATOGRAPHY METHOD, METHOD OF OBTAINING AN ADSORPTION ISOTHERM, METHOD OF OBTAINING AT LEAST ONE STATIONARY PHASE AND METHOD OF EVALUATING THE ACCURACY OF A PREDETERMINED ADSORPTION ISOTHERM

Abstract

The present invention relates to a chromatography method, a method of determining the concentration of at least one compound in a chromatography method, a method of obtaining an adsorption isotherm, a method of obtaining at least one stationary phase and a method of evaluating the accuracy of a predetermined adsorption isotherm.

Claims

1. A method of obtaining an adsorption isotherm of at least one compound comprising the steps of: (ia) selecting at least one compound; (ib) selecting a stationary phase; (ic) selecting a mobile phase; (id) selecting a temperature of the at least one compound, the stationary phase and the mobile phase; wherein at least one parameter, selected from parameters capable of characterizing the composition of the mobile phase and the temperature, varies; (ii) obtaining a plurality of parameter value sets including the at least one varying parameter and the concentration of the at least one compound in the mobile phase; (iii) obtaining a plurality of binding capacity values of the at least one compound by the stationary phase for each of the plurality of parameter value sets obtained in step (ii); and (iv) obtaining the adsorption isotherm based on the plurality of binding capacity values obtained in step (iii) by means of multivariate data analysis.

2. The method according to claim 1, wherein the parameters capable of characterizing the composition of the mobile phase are selected from the group consisting of concentrations of solvents included in the mobile phase, pH, salt concentration, and/or concentration of the at least one compound.

3. The method according to claim 1, wherein the number of the at least one varying parameter is two or more.

4. The method according to claim 3, wherein the at least two varying parameters include the pH and the salt concentration of the mobile phase.

5. The method according to claim 1, wherein the plurality of parameter value sets is obtained in step (ii) by means of experimental design.

6. The method according to claim 5, wherein the experimental design is a full factorial design or a fractional factorial design.

7. The method according to claim 1, wherein step (iii) involves one or more laboratory experiments.

8. The method of claim 1 further comprising: determining a concentration of at least one compound in a chromatography method by: (v) selecting a chromatography device having a chromatography bed comprising the stationary phase and the mobile phase; and (vi) calculating a concentration c(z, t) of the at least one compound in the mobile phase at a predetermined location of the chromatography device and at a predetermined time t based on the adsorption isotherm obtained in step (iv).

9. The method according to claim 8, wherein the method further includes a step of: (v′) determining at least the flow velocity of the mobile phase in the chromatography bed v and the bulk porosity of the chromatography bed ε.sub.b; wherein in step (vi), the calculation of the concentration c(z, t) is further based on the flow velocity and the bulk porosity ε.sub.b.

10. A method of obtaining at least one stationary phase comprising the steps of: (I) executing the method according to claim 1 for m times, wherein m is an integer of 2 or more and the m executions differ from one another with respect to step (ib); and (II) selecting the at least one stationary phase based on the result of step (I).

11. A method of evaluating the accuracy of a predetermined adsorption isotherm comprising the method according to claim 1 and a step of comparing the predetermined adsorption isotherm with the adsorption isotherm obtained in step (iv).

12. The method of claim 8, further comprising the step of: (vii) performing a chromatography method after performing the steps of claim 8.

13. The chromatography method according to claim 12, wherein step (vii) includes: monitoring the values of the at least one varying parameter in the chromatography method.

14. The chromatography method according to claim 13, wherein step (vii) further includes determining for one or more of the monitored values whether a respective predetermined criterion is satisfied and, in case the respective predetermined criterion is not satisfied, modulating one or more of the at least one varying parameters in the chromatography method based on the adsorption isotherm obtained in step (iv).

15. The method of claim 10, further comprising: (III) performing the chromatography method after performing the steps of claim 8, using the at least one stationary phase selected in step (II).

Description

[0177] FIG. 1 shows schematically the reversible swelling behavior of a charged hydrogel layer depending on the salt concentration of the mobile phase.

[0178] FIG. 2 displays porosity values ε(c.sub.S) determined for a dextrane 2000 kDa molecule on a Sartobind® Q Nano module depending on the salt concentration c.sub.S.

[0179] FIG. 3 shows the dependence of the accessible volume fraction depending on the tracer molecule size, here pullulane in 10 mM KPi buffer at pH 7, 10 mM NaCl (see Example 2), wherein the hydrodynamic radius r.sub.H of the respective compound was determined as described in S. Viel, D. Capitani, L. Mannina, A. Segre, Diffusion-ordered NMR spectroscopy: A versatile tool for the molecular weight determination of uncharged polysaccharides, Biomacromolecules. 4 (2003) 1843-1847. doi:10.1021/bm0342638.

[0180] FIG. 4 displays stationary phase porosity values ε.sub.sp determined for Fractogel EMD SO.sub.3.sup.− (M) depending on the salt concentration and a corresponding fit to a Boltzmann function (see Example 2).

[0181] FIG. 5 displays bulk porosity values ε.sub.b determined for Fractogel EMD SO.sub.3.sup.− (M) depending on the salt concentration and a corresponding fit to a Boltzmann function (see Example 2).

[0182] FIG. 6 shows the isotherm determination workflow of Example 3.

[0183] FIG. 7 shows equilibrium adsorption data for bovine serum albumin on Sartobind® Q of Example 3.

[0184] FIG. 8 shows a size exclusion chromatogram for a cell culture supernatant fraction eluted from Sartobind® S after equilibration of 8 hours of Example 3a.

[0185] FIG. 9 shows equilibrium adsorption data for multiple components at a salt concentration of 20 mM for a hydrogel grafted chromatographic medium (hydrogel grafted membrane adsorber Sartobind® S) of Example 3a. The graphs represent Langmuir isotherm fits.

[0186] FIG. 10 shows an equilibrium adsorption data map for the monomer of a monoclonal antibody depending on conductivity and pH of Example 3b.

[0187] FIG. 11 shows an equilibrium adsorption data map for a dimer of a monoclonal antibody depending on conductivity and pH of Example 3b.

[0188] FIG. 12 shows the axial dispersion coefficient for the chromatographic medium depending on the linear flow velocity of Example 5.

[0189] FIG. 13 shows the axial dispersion coefficient D.sub.ax.sup.DPF of the DPF of the external system “ST+DPF” (see FIG. 17) depending on the linear flow velocity of Example 5.

[0190] FIG. 14 shows simulation results at low salt concentrations of Example 7.

[0191] FIG. 15 shows simulation results at high salt concentrations of Example 7.

[0192] FIG. 16 (a) shows equilibrium adsorption data including a Langmuir isotherm fit of Example 8.

[0193] FIG. 16 (b) shows the dependence of the equilibrium binding constant on the salt concentration of Example 8.

[0194] FIG. 16 (c) shows the dependence of the maximum adsorbent capacity on the salt concentration of Example 8.

[0195] FIG. 17 shows a schematization of a chromatography device in accordance with a preferred embodiment of the present invention.

[0196] FIG. 18 shows an exemplary chromatographic set-up.

[0197] FIG. 19 schematically shows a preferred embodiment of the present invention.

[0198] FIG. 20 shows results of Example 10.

[0199] The present invention is further illustrated by means of the following non-limiting Examples.

EXAMPLES

Example 1: Determination of Porosity Data Depending on Process Conditions (Step (v′))

[0200] The determination of a reversible swelling behavior can be easily performed using inverse size exclusion chromatography (iSEC) while varying the desired process conditions. Here, a specific example is given.

[0201] The stationary phase was a Sartobind® S membrane adsorber. Being a membrane adsorber, Sartobind® S has no internal porosity (ε.sub.b=ε.sub.T). It shows a pronounced reversible swelling behavior originating from its charged hydrogel surface modification. The chromatography device was a 3 mL (V.sub.b=3 mL) Sartobind® Nano with 8 mm bed height.

[0202] The dead volume V.sub.Dead of the chromatography apparatus was determined by using 5 μL injections 0.25 g/L dextran (M.sub.w=2000 kDa, determined by size exclusion chromatography), determined by the RI detector without the chromatographic device with 0.319±0.03 mL. The peak maxima and the first momentum analysis, respectively, was used to determine the dead volume. The chromatography device had a dead volume of 1 mL.

[0203] For the iSEC experiments the used buffer was 10 mM potassium phosphate buffer (KPi) buffer at pH 7. The salt concentration (NaCl) was varied from 0.01 to 0.8 M. The membrane adsorber (MA) was equilibrated for 15 membrane volumes with the desired salt concentration before being loaded with 50 μL injections 0.5 g/L dextran 2000 kDa. The resulting peak response was recorded using an IR detector.

[0204] The porosity ε was determined based on the following equation.

[00037]$\epsilon =\frac{V}{F/{\mu }_p}$

[0205] The obtained porosity values depending on the used salt concentration are shown in FIG. 2.

[0206] The data set was fitted using a Boltzmann function as shown in the following.

[00038]$y=\frac{A_1-A_2}{1+e^{\left(x-x_0\right)/\mathrm{dx}}}+A_2$

[0207] The fit parameters were determined as follows:

TABLE-US-00001 Parameter Value A.sub.1 −54.022 A.sub.2 0.759 x.sub.0 −0.865 dx 0.125

Example 2: Determination of Porosity Data Depending on Process Conditions (Step (v′))

[0208] In the case of chromatographic media having an external and internal porosity, both values can be determined using reverse size exclusion chromatography as described above. Chromatographic media having an internal porosity show a different accessible volume fraction of the chromatographic bed depending on the tracer molecule size. An example for Fractogel EMD SO.sub.3.sup.− (M) is shown in FIG. 3.

[0209] The stationary phase porosity ε.sub.sp can be calculated using the total porosity ε.sub.T and the external porosity ε.sub.b (voidage).

[00039]${\epsilon }_{\mathrm{sp}}=\frac{{\epsilon }_T-{\epsilon }_b}{1-{\epsilon }_b}$

[0210] The total porosity ε.sub.T is accessible by a tracer molecule with complete accessibility of the internal porosity. In this example, acetone was used. A tracer molecule completely excluded can determine the bulk porosity ε.sub.b. In this example, dextran having a molecular weight M.sub.w of 2000 kDa was used. Using the above equation, the values obtained for ε.sub.T and ε.sub.b can be used to calculate the stationary phase porosity ε.sub.sp depending on the salt concentration. The obtained values for ε.sub.sp (stationary phase porosity) and ε.sub.b (external porosity) including Boltzmann fit and the corresponding parameters are shown in FIGS. 4 and 5.

[0211] Stationary phase porosity ε.sub.sp:

TABLE-US-00002 Parameter Value A.sub.1 0.365 A.sub.2 0.674 x.sub.0 0.122 dx 0.140

[0212] Bulk porosity ε.sub.b:

TABLE-US-00003 Parameter Value A.sub.1 −72.89 A.sub.2 0.539 x.sub.0 −2.548 dx 0.31

[0213] An excellent fit was obtained for both porosity values ε.sub.b and ε.sub.sp when using a Boltzmann function.

Example 3: Acquiring Equilibrium Adsorption Data (Step (iii))

[0214] In Example 3, equilibrium adsorption data for bovine serum albumin (BSA) on Sartobind® Q was obtained.

[0215] Determination of equilibrium adsorption data based on batch experiments as shown in Antibodies 2018, 7(1), 13; https://doi.org/10.3390/antib7010013, “Evaluation of Continuous Membrane Chromatography Concepts with an Enhanced Process Simulation Approach”, Zobel, Stein, Strube, for Sartobind® Q, mean pore diameter of 3 μm and a ligand density of

[00040]$2-5\frac{\mu \mathrm{eq}}{{\mathrm{cm}}^2},$

was carried out with 0.1-5 g/L bovine serum albumin (BSA). The used buffer was 20 mM TRIS HCl buffer, at pH=7, with NaCl concentrations of 0 to 0.3 M NaCl. The pH-value was adjusted using HCl or NaOH. The round Sartobind® Q membrane adsorber (MA) sample with a diameter of 20 cm and a height of 0.024-0.028 cm was equilibrated 30 min in 20 mM TRIS HCl buffer with the respective pH and salt concentration. The volume of the buffer was 200 times the volume of the MA. After an equilibration time of 30 minutes, the MA was dabbed with paper and transferred in a 12 well plate cavity. The BSA was dissolved in TRIS HCl buffer corresponding to the experiment pH and salt concentration. The concentration of the BSA feed solution was measured by UV/Vis spectroscopy at 280 nm and added with 4 mL to the MA in the 12 well plate. After a residence time of at least 8 h the supernatant concentration was measured, the MA was again paper dabbed and transferred in a new well plate. Subsequently, the MA was eluted with 4 mL 20 mM TRIS HCl and 1 M NaCl for at least 4 h. The supernatant concentration was measured after the 4 h elution time. The foregoing process is schematically depicted in FIG. 6.

[0216] The obtained data sets for three different salt concentrations are shown in FIG. 7.

Example 3a: Acquiring Equilibrium Adsorption Data for Several Compounds

[0217] The method of Example 3 is also viable for multicomponent analysis. A prominent example is the simultaneous determination of equilibrium adsorption data for monoclonal antibodies (mAb) as well as their aggregates and contaminants. The batch experiments can be carried out in the same way but the supernatant and the elution has to be analyzed in a way that allows distinguishing between all components (mAb monomer, aggregates and further contaminants). For example, this can be achieved by size exclusion chromatography.

[0218] The resulting peaks of the obtained size exclusion chromatogram (see FIG. 8 for this Example) can be evaluated using a proper calibration to determine the concentration of the target components during the batch experiments as shown in FIG. 6. This leads to equilibrium adsorption data. An example is shown in FIG. 9 for a given salt concentration.

Example 3b: Acquiring Equilibrium Adsorption Data for Varying Salt Concentrations and Varying pH

[0219] In the above Example 3, only the salt concentration was varied. Following the same approach, other influencing factors like the pH can also be varied, resulting in multidimensional adsorption data maps. FIGS. 10 and 11 show equilibrium adsorption data for the monoclonal antibody IgG and its dimer on Sartobind® Q depending on pH and salt concentration.

Example 4: Fitting of Data Sets to an SMA Isotherm (Step (iv))

[0220] The data sets obtained in Example 3 were used to calculate protein characteristic charge v.sub.i and equilibrium constant K.sub.i, furthermore the steric factor σ.sub.i was fitted using a computer-assisted least square regression at the three different salt concentrations (c.sub.S=0.05,0.15,0.25 M) to an SMA adsorption isotherm to obtain the necessary adsorption model parameters in the investigated salt concentration area. The ionic capacity Λ was 0.97 mol/L (Sartobind® Q).

[00041]$c_i=\frac{q_i}{K_i}.Math.{\left(\frac{c_1}{\Lambda -{.Math.}_{i=2}^{n+1}\left(v_i+{\sigma }_i\right)q_i}\right)}^{v_i}$

[0221] Adsorption constant: K.sub.i=7.55

[0222] Steric factor: σ.sub.i=46.04

[0223] Characteristic charge: v.sub.i=2.72

[0224] Salt concentration: c.sub.1=0.05 M

Example 5: Determining the Fluid Dynamic Behavior and Axial Dispersion Coefficient

[0225] An Åkta™ Explorer from GE Healthcare was selected as the chromatography device (step (v)).

[0226] The parameters V.sub.SYS, V.sub.ST, V.sub.DPF and D.sub.ax.sup.DPF were determined as follows. Pulse injections of acetone (2 vol % in reverse-osmosis-water) or mAb (4 mg/mL in potassium phosphate (KPi) buffer, 10 mM, 20 mM NaCl, pH=6) were carried out in the absence of the chromatographic medium. The experiments were performed using different volumetric flow rates and buffer conditions. The resulting peak signals were evaluated following the method of moments and regressed using a least squares fitting procedure to obtain the desired values. The results of this procedure are given in FIGS. 12 and 13.

Example 6: Determination of Kinetic and Diffusion Coefficients

[0227] In accordance with the lumped pore model, the effective film-diffusion coefficient D.sub.eff was determined using mathematical correlations for a hydrogel modified membrane adsorber Sartobind® S:

[0228] The bulk diffusion coefficient was calculated using the Einstein-Stokes equation. In particular, intravenous immunoglobulin (IVIG, human γ-Globulin, SeraCare; r=5.2 nm) was dissolved in an aqueous sodium phosphate buffer (20 mM, pH=7) having a viscosity η of 1.05 mPa s at a temperature of 298 K.

[0229] The porosity values were determined using inverse size exclusion chromatography (iSEC). Briefly, the chromatographic bed was equilibrated for 50 column volumes (CVs) of the desired buffer (sodium phosphate buffer (20 mM, pH=7)) before being loaded with injections (100 μL) of a solution containing pullulan molecules (2 mg/mL) with a narrow molecular weight distribution. The mean molecular weight, directly linked to the mean hydrodynamic radius of the applied pullulan samples, was varied for every injection covering a wide range (Mn=320-740,000 g/moL). The elution profile was recorded and analyzed by an RI detector.

[0230] The effective diffusion coefficients were calculated using the following correlations.

[00042]$D_{\mathrm{eff}}=\frac{D_{\mathrm{bulk}}}{\tau }\tau =\frac{{\left(2-{\epsilon }_{\mathrm{sp}}\right)}^2}{{\epsilon }_{\mathrm{sp}}^2}$

[0231] The values of the internal porosity ε.sub.p were determined by iSEC as described above.

[0232] For a membrane adsorber the following values were calculated depending on the respective target compound NaCl, acetone, and the monoclonal antibody IgG:

TABLE-US-00004 Molecular diffusion k.sub.ef f, A/ (coefficient/(m.sup.2/s) (1/s) NaCl 11.99 .Math. 10.sup.−9 56.63 Acetone  1.14 .Math. 10.sup.−9 23.67 Monoclonal   4.00 .Math. 10.sup.−11  5.47 antibody

Example 7: Prediction of a Protein Purification Process (Step (vi))

[0233] The model parameter sets obtained in Examples 1 and 5 were used to predict the fluid-dynamic behavior of an acetone tracer pulse signal. The adsorption isotherm of acetone on the used membrane adsorber was found to be 0 (acetone does not bind to the stationary phase). The liquid chromatography (LC) system and membrane adsorber (MA) were simulated by the equilibrium dispersive model (EDM). Moreover, the chromatography apparatus was considered as a combination of an ST, DPF and the chromatography column, as described above and as shown in the following equations (see FIG. 17).

[00043]$$\begin{gathered} \frac{\partial c_{\mathrm{out}}^{\mathrm{ST}}}{\partial t}=\frac{F}{V_{\mathrm{ST}}}\left(c_{\mathrm{in}}^{\mathrm{ST}}-c_{\mathrm{out}}^{\mathrm{ST}}\right) \\ \frac{\partial c^{\mathrm{DPF}}\left(z^{\mathrm{DPF}},t\right)}{\partial t}=-v\frac{\partial c^{\mathrm{DPF}}\left(z^{\mathrm{DPF}},t\right)}{\partial z^{\mathrm{DPF}}}+D_{\mathrm{ax}}^{\mathrm{DPF}}\frac{{\partial }^2{c}^{\mathrm{DPF}}\left(z^{\mathrm{DPF}},t\right)}{\partial {\left(z^{\mathrm{DPF}}\right)}^2} \\ \frac{\partial c\left(z,t\right)}{\partial t}=-\frac{\left(1-{\epsilon }_b\right)}{{\epsilon }_b}.Math.\frac{\partial q\left(z,t\right)}{\partial t}-v\frac{\partial c\left(z,t\right)}{\partial z}+D_{\mathrm{ax}}\frac{{\partial }^{2}c\left(z,t\right)}{\partial z^2} \end{gathered}$$

[0234] wherein

V.sub.SYS=V.sub.ST+V.sub.DPF

c(t=0,z)=0

c.sub.out.sup.ST=c.sup.DPF(z.sup.DPF=0,t)=c.sub.in.sup.DPF(t)

c.sup.DPF(z.sup.DPF=z.sub.max.sup.DPF,t)=c.sub.out.sup.DPF=c(z=0,t)

[0235] Overall LC System had a pipe length of 2943 mm, a volume of 1.3 mL and a porosity of 1. The overall MA device is represented by 5.73 mm chromatography bed height, 3.5 mL chromatography bed volume and a porosity of 76-71% (0.76-0.71). The used MA was Sartobind® S in 10 mM KPi buffer at pH 7. Tracer experiments were carried out with a 2 mL injection volume of KPi buffer containing 5% acetone with 0 or 0.8 M additional sodium chloride at 4 mL/min. In FIG. 14, the low salt concentration simulation results and experimental data are compared. At low salt concentrations, conventional and improved model approach gave similar results. At high salt concentrations, taking the reversible swelling of the stationary phase into account yielded much better results than a method were the reversible swelling of the stationary phase was not taken into account (see FIG. 15). FIGS. 14 and 15 show the dramatic influence of a varying porosity value with respect to the fluid dynamic behavior.

[0236] The following table shows a comparison of a conventional simulation approach with the method according to the present invention corresponding to the half peak width FWHM (Full Width at Half Maximum) and the peak center. The inventive method gave a smaller deviation from the experimental value as the conventional approach.

TABLE-US-00005 FWHM Center Deviation conventional 2.2 10.6  simulation to experiment/% Deviation inventive 2.0 2.6 simulation to experiment/%

Example 8: Fitting of Data Sets to a Langmuir Isotherm (Step (iv))

[0237] A data set of equilibrium adsorption data obtained for IVIG on a hydrogel grafted chromatographic membrane was fitted to a Langmuir isotherm taking account of different NaCl concentrations. The results are shown in FIGS. 16 (a) to 16 (c).

[0238] As can be taken from FIG. 16 (a), a Langmuir fit approximates the isotherm adsorption data with high accuracy.

Example 9: Investigation of an Ion-Exchange Chromatography (IEX) Method by Means of Experimental Design (Step (ii))

[0239] A conventional batch isotherm determination experimental set up for a one component ionexchange chromatography (IEX) step could be: measuring the bound mass at equilibrium conditions with 6 different feed concentrations c.sub.Feed, 4 different salt concentrations c.sub.Salt and 5 pH values. For instance, the 6 different feed concentrations c.sub.Feed could be values ranging from 0.5 to 2.5 in 0.5 g/L steps, the 5 pH values could be values ranging from 5 to 7 in 0.5 steps and the ε4 salt concentrations c.sub.Salt could be selected as 0.001 M, 0.1 M, 0.2 M, 0.3 M and 0.4 M. This results in a total number of 120 experiments.

[0240] Contrary thereto, by using experimental design (design of experiments, DoE), a total number of 27 experiments would be required for a full factorial (fullfac) model or a total number of 17 experiments would be required for a CCF model with star distance 1. The CCF model results are as follows:

TABLE-US-00006 pH .sub.CSalt .sub.CFeed 5 0.001 0.5 5 0.4 0.5 5 0.001 2.5 5 0.4 2.5 5 0.2005 1.5 6 0.001 1.5 6 0.4 1.5 6 0.2005 0.5 6 0.2005 2.5 6 0.2005 1.5 6 0.2005 1.5 6 0.2005 1.5 7 0.001 0.5 7 0.4 0.5 7 0.001 2.5 7 0.4 2.5 7 0.2005 1.5

[0241] Thus, the above Example demonstrates that the number of binding capacity values to be obtained in step (iii) for obtaining the adsorption isotherm in step (iv) can be substantially reduced by carrying out step (ii) based on experimental design.

Example 10: Obtaining, a Global Adsorption Isotherm for Varying pH and Varying Salt Concentration (Step (iv))

[0242] Based on the adsorption data acquired in step (iii), a global adsorption isotherm in the applied design space (i.e. the respective range of the concentration, pH and salt concentration) can be obtained. This can be achieved by an evaluation of the underlying statistical experimental plan. Various commercial software is available for the statistical plan design and for the following evaluation, such as MODDE® from Sartorius AG.

[0243] Based on experimental binding capacity values obtained in step (iii), a plurality of preliminary adsorption isotherms was obtained (iv-2a), where for each preliminary adsorption isotherm, one of the at least one varying parameter was kept constant. Here, the pH was kept constant while a varying salt concentration is taken account by the preliminary adsorption isotherms. Each of the preliminary fittings was performed to an SMA isotherm. As a result, the following SMA isotherm parameters were obtained.

TABLE-US-00007 equilibrium steric characteristic pH constant K.sub.i factor σ.sub.i charge v.sub.i 5 90.49 1549.96 0.404 5.5 39.67 1401.18 0.292 6 33.29 1506.95 0.266 6.5 26.03 1581.41 0.2 7 21.89 1706.3 0.1831

[0244] Based on the results obtained in the preliminary fitting as summarized in the above table, a main fitting step (iv-2b) was carried out. That is, in the main fitting step (iv-2b), a global adsorption isotherm was obtained by fitting one parametric equation to the numerical values of each of the parameters of the SMA isotherms obtained in the preliminary fitting step. The results are displayed in FIG. 20.

[0245] The following fitted parametric equations, considering linear, quadratic and cubic terms, were obtained.

log(K.sub.i)=43.90−20.00.Math.pH+3.16.Math.pH.sup.2−0.17.Math.pH.sup.3

σ.sub.i=34891−16251.60.Math.pH+2596.88.Math.pH.sup.2−136.08.Math.pH.sup.3

v.sub.i=7.57−3.23.Math.pH+0.48.Math.pH.sup.2−0.02.Math.pH.sup.3

[0246] As can be taken from the results displayed in FIG. 20, the effect of a variation of the pH on the SMA isotherm parameters K.sub.i (“K”) and σ.sub.i (“steric”) is highly non-linear. A conventional linear interpolation would not take this non-linearities into account and thus yield inaccurate results.