# Modeling of Filtration Processes—Microfiltration and Depth Filtration for Harvest of a Therapeutic Protein Expressed in Pichia pastoris at Constant Pressure

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Theory

#### 1.2. Standard Blocking Model

_{0}is the initial flux and Ks is the standard blocking constant (m

^{−1}).

#### 1.3. Complete Blocking Model

_{b}has the unit of (s

^{−1}).

#### 1.4. Intermediate Blocking Model

_{i}has the unit of (m

^{−1}).

#### 1.5. Cake Filtration

_{c}has a unit of (sm

^{−2}).

#### 1.6. Combined Models for Membrane Fouling

## 2. Experimental Section

#### 2.1. Materials

#### 2.2. Experimental Setup

^{2}area, were attached to the feed vessel as needed and the supernatant was introduced in the feed vessel. Different micro filters and depth filters used in the investigation are shown in Table 1. Experiments were performed in batch mode using a laboratory-scale setup. Pressure was maintained at 1.5 bar throughout the experiment. After processing, the permeate flowed to a permeate vessel and its weight was measured as a function of time. The weights were converted to volumes using density correlations. Once the initial flux value had declined to about 90%, the microfiltration or depth filtration membrane that was used was discarded and the filtration feed vessel was emptied. The initial flux was calculated based on the initial permeate volume per time during the initial stage of the filtration. Solute concentrations of the filtrate and feed were measured using reversed phase high performance liquid chromatography (RP HPLC). All experiments were conducted at ambient temperature.

#### 2.3. Particle Size Distribution Measurement

S.No. | Filter Type | Filter Name | Nominal Retention Rating (µm) | Description |
---|---|---|---|---|

1. | Micro filters SuporGrade (Micro filter) | EKV | 0.2 | Filter Media is Supor EKV membrane (hydrophilic polyethersulfone) |

2. | Supra Cap 60 HP (Depth filter) | PDK5 | 1.5–20.0 | The HP-series depth filter sheets are comprised of two full thickness, graded, high-efficiency P-Series depth filter sheets in combination |

3. | Supra Cap P Series (Depth filter) | KS 50P | 0.4–0.8 | P series depth filter, combination of cellulose fibers, DE and perlite, pyrogen removal capability |

#### 2.4. Turbidity Measurement

#### 2.5. RP-HPLC

## 3. Results and Discussion

**Figure 2.**(

**A**) Average mean particle size distribution of the fermentation broth. (

**B**) Average mean particle size distribution of the centrifuged process stream.

**Figure 4.**Volume vs. time data for individual models fit against the filter data for EKV. (

**A**) Individual models. (

**B**) Combined models.

#### 3.1. Direct Microfiltration (Option 1)

#### 3.2. Centrifugation Followed by Microfiltration (Option 2)

Model | Component mechanism | Equation | Parameters | Model |
---|---|---|---|---|

Cake-complete (6) | Cake filtration, complete blocking | $V=\frac{{J}_{0}}{{K}_{b}}\left(1-\text{exp}\left(\frac{-{K}_{b}}{{K}_{c}{{J}_{0}}^{2}}\left(\sqrt{1+2{K}_{c}{{J}_{0}}^{2}t}-1\right)\right)\right)$ | K_{c} (s/m^{2}), | Cake-complete (6) |

Cake-intermediate (7) | Cake filtration, intermediate blocking | $V=\frac{1}{{K}_{i}}\text{ln}\left(1+\frac{{K}_{i}}{{K}_{c}{J}_{0}}\left(\left(\sqrt{1+2{K}_{c}{{J}_{0}}^{2}t}\right)-1\right)\right)$ | K_{c} (s/m^{2}), | Cake-intermediate (7) |

Complete-standard (8) | Complete blocking, standard blocking | $V=\frac{{J}_{0}}{{K}_{b}}\left(1-\text{exp}\left(\frac{-2{K}_{b}t}{2+{K}_{s}{J}_{0}t}\right)\right)$ | K_{b} (s^{−1}), | Complete-standard (8) |

Intermediate-standard (9) | Intermediate blocking, standard blocking | $V=\frac{1}{{K}_{i}}\text{ln}\left(1+\frac{2{k}_{i}{j}_{0}t}{2+{k}_{s}{j}_{0}t}\right)$ | K_{i} (m^{−1}), | Intermediate-standard (9) |

_{c}and K

_{b}, and it was observed that K

_{c}is higher. Thus, cake filtration is the major contributor to fouling in this case.

**Table 3.**Comparison of the different models and values of the related parameters and the errors. The best fit for a given filter is shown in bold.

Mechanism | EKV | PDK5 | PDH4 | PDE2 | EKMP | KS50P | EKSP | PDD1 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | Parameters | Error fit (SSR) | |

Standard blocking | K_{S} = 87.98 | 169.17 | K_{S} = 29.91 | 2611.7 | K_{S} = 32.61 | 11,429 | K_{S} = 41.91 | 3642.73 | K_{S} = 45.84 | 5310.9 | K_{S} = 32.61 | 11,429 | K_{S} = 107.38 | 849.32 | K_{S} = 50.63 | 1794.8 |

Intermediate blocking | K_{i} = 188 | 174.97 | K_{i} = 84 | 2756.1 | K_{i} = 58 | 18,375 | K_{i} = 100 | 322.59 | K_{i} = 148 | 1460.5 | K_{i} = 58 | 18,735 | K_{i} = 297 | 36.07 | K_{i} = 100 | 534.43 |

Complete blocking | K_{b} = 0.0358 | 1102.9 | K_{b} = 0.0194 | 8465.9 | K_{b} = 0.003 | 3295.9 | K_{b} = 0.0042 | 7255.56 | K_{b} = 0.0074 | 5947 | K_{b} = 0.003 | 9295.9 | K_{b} = 0.007 | 2075.2 | K_{b} = 0.0039 | 2115.39 |

Cake filtration | K_{c} = 1.1 × 10^{6} | 1337.3 | K_{c} = 7.02 × 10^{5} | 32,325 | K_{c} = 1.0 × 10^{6} | 43,477 | K_{c} = 3.49 × 10^{6} | 1147.116 | K_{c} = 9.8 × 10^{6} | 8372.8 | K_{c} = 1.45 × 10^{6} | 43,477 | K_{c} = 2.6 × 10^{7} | 587.15 | K_{c} = 3.0 × 10^{6} | 1351.2 |

Cake- Complete | K_{b} = 0.031 | 5.17 | K_{b} = 0.018 | 1393.3 | K_{b} = 0.0028 | 10,962 | K_{b} = 0.031 | 1059.88 | K_{b} = 0.007 | 1.45 × 10^{3} | K_{b} = 0.0028 | 1.106 × 10^{2} | K_{b} = 0.006 | 35.93 | K_{b} = 0.003 | 189.63 |

Cake-intermediate | K_{c} = 2.08 × 10^{5} | 294.96 | K_{i} = 50 | 1.23 × 10^{3} | K_{i} = 45 | 1.8 × 10^{2} | K_{i} = 35 | 3.66 | K_{c} = 9.7 × 10^{4} | 393.59 | K_{c} = 1.15 × 10^{4} | 31,831 | K_{c} = 2.6 × 10^{6} | 37.87 | K_{c} = 1.28 × 10^{4} | 297.81 |

Complete-standard | K_{b} = 0.0034 | 147.41 | K_{b} = 0.017 | 2169.8 | K_{b} = 0.0026 | 10,098 | K_{b} = 0.0028 | 2673.53 | K_{b} = 0.007 | 4055.1 | K_{b} = 0.0026 | 10,091 | K_{b} = 0.006 | 672.45 | K_{b} = 0.0028 | 1930 |

Intermediate-standard | K_{i} = 62 | 154.73 | K_{i} = 21 | 1241.8 | K_{i} = 20 | 12,040 | K_{i} = 50 | 2508.66 | K_{i} = 45 | 2544 | K_{i} = 20 | 12,101 | K_{i} = 150 | 419.28 | K_{i} = 50 | 896.01 |

#### 3.3. Depth Filtration (Option 3)

**Figure 5.**Volume vs. time data for individual models fit against the individual and combined models. (

**A**) Individual models for PDK5. (

**B**) Combined models for PDK5. (

**C**) Individual models for PDH4. (

**D**) Combined models for PDH4. (

**E**) Individual models for PDE2. (

**F**) Combined models for PDE2.

**Figure 6.**Volume vs. time data for individual models fit against the individual and combined models. (

**A**) Individual models for EKMP. (

**B**) Combined models for EKMP. (

**C**) Individual models for KS50P. (

**D**) Combined models for KS50P. (

**E**) Individual models for EKSP. (

**F**) Combined models for EKSP. (

**G**) Individual models for PDD1. (

**H**) Combined models for PDD1.

#### 3.4. Discussion

_{b}is about 10

^{7}times smaller than K

_{c}). Hence, in these situations, cake filtration dominates. However, for cases when the pore size of the depth filter is large (0.2–20 µm), a combination of intermediate blocking and cake filtration provides the best fit. The difference in the coefficients for these cases is significantly smaller (10

^{3}times), thus indicating that in these cases both mechanisms contribute significantly. The work presented here can be used to predict the blocking mechanism based on the filter pore size and this model can then be used for filter sizing.

## 4. Conclusions

## Nomenclature

A | Available membrane frontal area (m ^{2}) |

J_{0} | Initial flux (m/s) |

K_{b} | Complete blocking constant (s ^{−1}) |

K_{c} | Cake filtration constant (s/m ^{2}) |

K_{i} | Intermediate blocking constant (m ^{−1}) |

K_{s} | Standard blocking constant (m ^{−1}) |

T | Time (s) |

V | Volume filtered through available membrane area (m ^{3}/m^{2}) |

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Sampath, M.; Shukla, A.; Rathore, A.S.
Modeling of Filtration Processes—Microfiltration and Depth Filtration for Harvest of a Therapeutic Protein Expressed in *Pichia pastoris *at Constant Pressure. *Bioengineering* **2014**, *1*, 260-277.
https://doi.org/10.3390/bioengineering1040260

**AMA Style**

Sampath M, Shukla A, Rathore AS.
Modeling of Filtration Processes—Microfiltration and Depth Filtration for Harvest of a Therapeutic Protein Expressed in *Pichia pastoris *at Constant Pressure. *Bioengineering*. 2014; 1(4):260-277.
https://doi.org/10.3390/bioengineering1040260

**Chicago/Turabian Style**

Sampath, Muthukumar, Anupam Shukla, and Anurag S. Rathore.
2014. "Modeling of Filtration Processes—Microfiltration and Depth Filtration for Harvest of a Therapeutic Protein Expressed in *Pichia pastoris *at Constant Pressure" *Bioengineering* 1, no. 4: 260-277.
https://doi.org/10.3390/bioengineering1040260