Modelling of Excipient Concentration during the Final
Diafiltration step of Therapeutic MAbs
Suhas Shalome RajeevaSouhardya Roy
Objective• Parameter Determination and Calculation.• Developing a numerical model to calculate the electric potential around an
Electric Double Layer (EDL) developed on a protein surface. • Prediction of Charged Excipient concentration inside the EDL. • Prediction of Charged and Uncharged Excipients in the overall Retentate
fraction. • Generate Conductivity and pH vs Protein Concentration Data for Diafiltration
Runs. • Prediction of pH in the Overall Retentate Fraction. • Flow Pattern Modelling of a self-designed TFF stirrer.
Parameter Determination: Known • Constants:
• Ɛ : Electric Charge = 1.602x10-23 C• D : Dielectric Constant of Protein in Buffer ≈ 80.4 (taken as that of water)• K : Boltzmann’s Constant = 1.38064852 × 10-23 m2 kg s-2 K-1
• T : Temperature of the system = 298 K
• Parameters (known or solved for): • Co : Overall Histidine Concentration in Diafiltration Buffer• Ψ : Electric Potential inside the EDL• r : Distance from the Protein Surface• Cp : Protein Concentration
Parameter Determination: Protein Mass • Mp: Protein Mass.
• Mass and composition of Protein Molecule can be determined from the sequence of the protein.
• Formula used:• Molecular weight = sum of individual residues weights - water molecular weight x ( number of
residues - 1 )• Mass Determined:
• (49326.47 + 23317.65 + 49198.3 + 23317.65) D = 145.16 KD
Composition
Parameter Determination: Protein Charge• From sequence analysis we have:
• Total number of Amino acids = 1325• Theoretical pI = 8.1737• Extinction Coefficient = 1.35
• Protein Charge: • Total number of Positively charged residues = 137• Total number of Positively charged residues = 124• Net Charge = + 23 (pH : 6.04 as of Histidine Buffer)
• Formula Used: • q = 1/(1 + 10(pH-pK)) : Protonisation of uncharged group• q = -1/(1 + 10(pH-pK)) : Dissociation of uncharged group• Q(pH) = q1 + ... + qn : Net Charge Calculated
Parameter Determination : Unknown • Vp : Protein Partial Volume = 0.97 ml/mg
• Calculated from the Density Meter Experiment.• Key parameter, from which remaining parameters can be estimated.
• Ɵ : Protein Volume Fraction : Function of the Protein Concentration.• Ɵ = (Cp/Mp)(Na)(4/3πa3)
• Na = 6.023x1023 : Avogadro's Number• a = Effective radius of the protein
• Vδ : Partial Volume of EDL: Depends on the Initial Histidine concentration.
Parameter Determination : Calculated • a : Effective Radius of Protein = 3.84 nm
• R: Distance from centre point to midpoint of two closest protein molecules : Function of the protein concentration.
• δ : EDL thickness : Function of the initial histidine concentration in buffer.
Calculation of Electric Potential in EDL• Model based on Poisson Boltzmann Equation, which predicts for the excipient-colloid charge
interaction. • Assumptions:
• Proteins as impermeable sphere• Uniform distribution of charge
• Distribution of charged excipients:• Uneven within the EDL• Homogeneous outside the EDL
• Graph: • Electric Potential vs Distance from Protein Surface for concentrations:
• 50mg/ml• 100 mg/ml • 150mg/ml
• Potential varied as a function of Protein Concentration and not Histidine concentration.
• Equation Solved: ODE
• Boundary Conditions:
• Reduced Form (Non dimensional):
• Boundary Condition:
Substitution:
Methodology and AnalysisCode: • Electric Field can be obtained by solving the given ordinary differential equation and the
accompanying boundary conditions. • MatLab ODE solver function: ‘ode45’ has been used to solve the Non-dimensional form of above
equation.
Analysis: • The electric field decreases with distance from the protein surface, the nature of decrease being
convex. • For higher concentration of protein in buffer, relatively higher electric fields are obtained. • EDL strength as well as charge separation increases with increasing protein concentration.
Prediction of Charged Excipients inside the EDL• Positively charged Histidine concentration inside the EDL: (Boltzmann’s Law)• Average charged Histidine concentration inside the EDL:
• Overall charged Histidine Concentration:
• Overall uncharged Histidine Concentration:
• Total Histidine Concentration in Retentate:
Methodology and AnalysisCode: • Histidine concentration obtained as a function of Protein Concentration in Buffer. • Corresponding plots obtained, for three different values of the protein effective radius, ‘a’: 3,3.5
and 4 nm.
Analysis: • Histidine concentration decreases linearly with increasing protein concentration.• Since both Histidine and Protein (Mab) are positively charged, owing to electrostatic repulsion,
histidine concentration is supposed to decrease, there by justifying the model. • The extent of decrease however goes on decreasing as the protein effective radius increases,
thereby suggesting a weakening in the corresponding electric field of the EDL.
Histidine Concentration Prediction• Using Curve Fitting in Matlab, the following equation predicts the Histidine concentration in
Product as a function of the Histidine concentration in Buffer. • Equation (Linear): C_His_Pdt = 0.9272 x C_His_Buf – 3.625• Using the above equation, following Histidine concentrations are predicted for higher Histidine
concentrations:
0 20 40 60 80 100 120 140 1600
20
40
60
80
100
120
140
160
Predicted Histidine Concentration
Buffer Histidine Concentration
Rete
ntat
e Hi
stidi
ne C
once
ntra
tion
Analysis• The equation predicting the Histidine concentration in Product as a function of the corresponding Buffer
concentration suggests: • Slope close to 1 (0.9272)• Intercept: negative (-3.625)
• Result: • At lower Histidine Buffer concentrations, intercept will be more dominant than the slope.
Consequently, there will be a significant difference in the two histidine concentrations. • At higher Histidine Buffer concentrations, slope will be much more dominant than the negative
intercept. Also, since it is closed to 1, it suggests that the Product Histidine Concentration will very closely approach the Buffer Histidine Concentration.
• Conclusion: Thus, we can conclude that at higher Excipient concentrations, the effect of Protein
concentration or charge on the subsequent excipient concentration in the retentate will not be that significant.
Experiment DesignDesign:• 25 mL of Retentate samples collected during the diafiltration run of a positively charged MAb in Histidine Buffer at specific
expected concentration intervals:
Analysis: • Subsequent Analysis for:
• Density ~ Densitometer.• Viscosity ~ Viscometer.• Concentration ~ UV Spectrometer. • pH ~ pH Probe. • Conductivity ~ Conductivity Probe.
Formulation Buffer
• Expected Concentration:• 9 • 10• 20• 30• 40
Buffer Exchange
• At concentration: • 40 mg/mL• 8
Diavolumes of Buffer required
Conductivity Trend
5 10 15 20 25 30 35 402.1
2.12
2.14
2.16
2.18
2.2
2.22
2.24
2.26
2.28
Formulation Buffer
Protein Concentration (mg/mL)
Cond
uctiv
ity (m
S)
30 40 50 60 70 80 900
100
200
300
400
500
600
700
Histidine Buffer
Protein Concentration (mg/mL)
Cond
uctiv
ity (u
S)
0 10 20 30 40 50 60 70 80 900
500
1000
1500
2000
2500
Conductivity Trend
Protein Concentration (mg/mL)
Cond
uctiv
ity (u
S)
pH Trend
5 10 15 20 25 30 35 407.5
7.5057.51
7.5157.52
7.5257.53
7.5357.54
7.5457.55
7.555
Formulation Buffer
Protein Concentration (mg/ml)
pH
30 40 50 60 70 80 906.25
6.3
6.35
6.4
6.45
6.5
6.55
6.6
Histidine Buffer
Protein Concentration (mg/mL)
pH
0 10 20 30 40 50 60 70 80 905.5
6
6.5
7
7.5
8
pH Trend
Protein Concentration (mg/mL)
pH
Analysis: Conductivity and pH
• Conductivity• Conductivity decreases with Protein concentration for Formulation Buffer, but in an insignificant fashion. • For Histidine Buffer, conductivity increases in a linear fashion, with a significant variation. • There is a significant difference in the conductivity of the two buffers. It is much lower in the histidine buffer,
compared to Formulation Buffer. • The linearly increasing value of Conductivity can be accounted for due to increasing protein concentration (positively
charged) and a slight decrease in the histidine concentration (positively charged). • The dip in conductivity is due to the lower dissolution of protein ions in Histidine buffer, compared to the
formulation buffer.
• pH• pH change is insignificant in the formulation buffer. • For histidine buffer, there is a slight increase in pH. It is because with increasing protein concentration, its basic
nature neutralizes Histidine acidic nature, and hence pH starts approaching 7. • However, there is a dip in pH below 7 from formulation buffer to histidine buffer, clearly due to the acidic nature of
Histidine buffer, whose pH should be 6.0.
Density Trend
5 10 15 20 25 30 35 400.998
1
1.002
1.004
1.006
1.008
1.01
1.012
Formulation Buffer
Protein Concentration (mg/mL)
Dens
ity (g
/L)
30 40 50 60 70 80 901
1.005
1.01
1.015
1.02
1.025
Histidine Buffer
Protein Concentration (mg/mL)
Dens
ity (g
/L)
0 10 20 30 40 50 60 70 80 900.99
0.995
1
1.005
1.01
1.015
1.02
1.025
Density Trend
Protein Concentration (mg/mL)
Dens
ity (g
/L)
Viscosity Trend
5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
Formulation Buffer
Protein Concentration (mg/mL)
Visc
osity
30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Histidine Buffer
Protein Concentration (mg/mL)
Visc
osity
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Viscosity Trend
Protein Concentration (mg/mL)
Visc
osity
Analysis: Density and Viscosity
• Density• Density increases with a linear trend for both the Formulation Buffer and the Histidine Buffer.• The linear density trend rather proves the goodness of the Concentration TFF run. • However, there is a slight dip in the density owing to the buffer exchange.
• Viscosity• Viscosity increases overall as we go on increasing the concentration. • However, there seems to be no established trend, which might be due to experimental errors. • However, on an approximation, viscosity should be follow a trend parallel to he density, and hence
should be monotonously increasing with concentration.
pH Prediction in overall Retentate fraction
0 10 20 30 40 50 600
1
2
3
4
5
6
pH
Histidine Concentration (mg/mL)
pH
pH of the retentate goes on increasing, thereby approaching 7 as the Histidine Concentration is increased in the Buffer.
The nature of increase follows the Logarithmic trend, thereby justifying the following Logarithmic relation used to calculate the pH of the Retentate:
i.e. the Henderson Hasselbalch equation.
Velocity Profile: TFF Stirrer Flow Pattern
TFF Stirrer Flow Pattern• Simulation and Modelling:
• Velocity profile has been obtained for a custom TFF stirrer, using COMSOL Multiphysics 4.4.• Simulation results are presented as static velocity profiles, which give some rough idea about the mixing pattern inside
the stirrer.
• An analysis of the simulation results shows:• Mixing is more near close to the inlet, outlet and near the stirrer blades. • The distance of the stirrer bottom from the tank bottom serves as an important parameter in homogenising the
mixture. • Adding Baffles designs to the tank wall will also increase the mixing capacity. • However, every attempt to increase mixing increases randomness, and hence there is a higher chance to reach
turbulence even at slightly lower velocities than the turbulent limit.
• Future Goals:• For a clearer view of the same, CFD analysis is required, with higher complexity in simulation. This can be considered
as a target to achieve in near future.
Thank You.
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