Influence of Substrate Concentrations on the Performance of Fed-Batch and Perfusion Bioreactors: Insights from Mathematical Modelling
Abstract
1. Introduction
- Product Yield: This is the mass of product produced from the substrate added. It is desirable to have a higher value.
- Productivity: This is the mass of product produced per bioreactor volume per unit time. This has a major influence on the size of the bioreactor required to produce a specified mass of product in a given time. A higher productivity is more desirable because this results in a smaller bioreactor size.
- Titer: This is the product concentration in the final working volume. It is desirable to have a higher value.
- Wasted Substrate: This is related to the mass of substrate added that is not used in the bioreaction and ends up being wasted, or requiring wastewater treatment. It is desirable to have a lower value of wasted substrate.
- Product Residence Time: This is the mean time that product spends in the bioreactor. It is desirable to have lower product residence times, especially for sensitive products that may be degraded over time.
2. Mathematical Modelling
2.1. Bioreaction Kinetics
2.2. Bioreactor Mass Balances
2.2.1. Fed-Batch
2.2.2. Perfusion
2.3. Product Recovery
2.4. Bioreactor Performance Parameters
- Yield
- Wasted Substrate
- Productivity
- Titer (product concentration)
- Mean Product Residence Time
2.4.1. Yield
2.4.2. Wasted Substrate
2.4.3. Titer
2.4.4. Productivity
2.4.5. Product Mean Residence Time
2.5. Initial Conditions and Model Implementation
3. Results and Discussion
3.1. Effect of Substrate Concentrations on Fed-Batch Bioreactor Performance
3.1.1. Effect of Substrate Concentrations on the Evolution of Component Concentrations over Time
3.1.2. Effect of Substrate Concentrations on Performance Parameters
3.1.3. Effect of Substrate Concentrations on the Performance of a Fed-Batch Bioreactor Operated in Continuous Mode of Media Addition
3.1.4. Effect of Rate-Limiting Bioreactor Substrate Concentration
3.2. Effect of Substrate Concentrations on Perfusion Bioreactor Performance
3.2.1. Effect of Substrate Concentrations on the Evolution of Component Concentrations over Time
3.2.2. Effect of Substrate Concentrations on Performance Parameters
3.2.3. Effect of Rate-Limiting Bioreactor Substrate Concentration
3.3. Effect of Bioreaction Time on Bioreactor Performance
3.3.1. Fed-Batch
- Titer, yield. and %WS: Final cell (), final product, and final metabolite () concentrations all increase over time and eventually reach constant values. This is reflected in titer and yield behaving similarly and in %WS decreasing over time to a constant value.
- Bioreactor working volume: Bioreaction time has a major effect on the final bioreactor working volume (), which is an exponential effect as illustrated in Figure 9. This is due to the mode of operation of fed-batch, whereby a greater volume of media addition is required to maintain the substrate concentration as the working volume increases over time. This increase in bioreactor working volume shows that the bioreaction time will be constrained in practice by the size of the bioreactor.
- Mean product residence time: Increasing the bioreaction time increases the residence time, but it does not affect it as one might intuitively expect, as illustrated in Figure 9. For example, one might expect that increasing the bioreaction time from 240 h to 960 h would significantly increase the product residence time. However, the product residence time for the 960 h bioreaction time is 33.9 h, which is only a little greater than the 32.3 h at the 240 h bioreaction time. The reason for this is due to the exponential increase in the bioreactor working volume; thus, most of the product is being produced in the latter part of the bioreaction time.
- Productivity: This initially increases as titer increases, achieves a maximum value, and then progressively decreases with increasing bioreaction time, in particular after 240 h, with productivity at 480 h being roughly half that at 240 h. The decrease in productivity is a consequence of the definition of productivity in Equation (27). As the bioreaction time doubles from 240 h to 480 h and then again to 960 h and the titer changes minimally, the net effect is that the productivity is reduced by half for each doubling of time. Figure 9 highlights that as the bioreaction time increases, more and more of the available working volume in a constant volume bioreactor is not being used, and this is contributing to the reduction in productivity over longer bioreaction times.
3.3.2. Perfusion
- Titer, yield, and %WS: Final cell (), final product, and final metabolite () concentrations all increase over time and eventually reach constant steady-state values. This is reflected in titer and yield behaving similarly and in %WS decreasing over time to a constant value.
- Mean product residence time: This decreases over time, with residence times being much higher early on and then approaching a constant value. Bioreaction rates are very slow initially and increase over time. This leads to higher feed/harvest flowrates over time, which reduces residence times. These flowrates eventually attain steady-state values, leading to mean residence times approaching constant values.
- Productivity: The bioreaction time has significant impact on the productivity, with it increasing significantly over time. This is due to longer bioreaction times having higher titers and also higher harvest flowrates (which is evidence by the shorter residence times).
3.4. Comparison of Fed-Batch and Perfusion Bioreactor Performance
3.5. Sensitivity Analysis
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
List of Symbols | |
media feed volumetric flowrate [perfusion] (m3 h−1) | |
mean media feed volumetric flowrate [perfusion] (m3 h−1) | |
metabolite concentration (g L−1) | |
metabolite concentration at end of bioreaction (g L−1) | |
harvest volumetric flowrate [perfusion] (m3 h−1) | |
Monod kinetic constant for substrate (g L−1) | |
mass of product produced during bioreaction time (kg) | |
specific maintenance coefficient (h−1) | |
product concentration (g L−1) | |
product concentration at end of bioreaction (g L−1) | |
product concentration before media addition [fed-batch] (g L−1) | |
metabolite production rate (g L−1 h−1) | |
product production rate (g L−1 h−1) | |
substrate utilization rate (g L−1 h−1) | |
cell growth rate (g L−1 h−1) | |
cell death rate (g L−1 h−1) | |
intrinsic cell growth rate (g L−1 h−1) | |
substrate concentration (g L−1) | |
lower limit substrate concentration in bioreactor [fed-batch] (g L−1) | |
substrate concentration in the media added (g L−1) | |
initial substrate concentration in the bioreactor (g L−1) | |
steady-state substrate concentration (g L−1) | |
upper limit substrate concentration in bioreactor [fed-batch] (g L−1) | |
time (h) | |
bioreaction time at the ith time increment (h) | |
hydraulic residence time [perfusion] (h) | |
bioreaction time from the start to end of bioreaction process (h) | |
mean product residence time (h) | |
bioreactor working volume (m3) | |
volume of cell concentrate after cell separation [fed-batch] (m3) | |
final bioreactor working volume [fed-batch] (m3) | |
working volume before media addition [fed-batch] (m3) | |
volume of media added [fed-batch] (m3) | |
initial bioreactor working volume [fed-batch] (m3) | |
volume of recovered product solution [fed-batch] (m3) | |
viable cell concentration (g L−1) | |
total cell concentration in cell concentrate after cell separation (g L−1) | |
dead cell concentration (g L−1) | |
cell concentration at end of bioreaction (g L−1) | |
physiological maximum cell concentration (g L−1) | |
total cell concentration (g L−1) | |
yield of product produced from substrate added (%) | |
yield of product recovered from substrate added (%) | |
yield coefficient for biomass | |
yield coefficient for product | |
percentage wasted substrate | |
, ,, | bioreaction model kinetic constants |
mass of product produced during the ith time increment (kg) | |
specific growth rate (h−1) | |
maximum specific growth rate (h−1) |
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(g L−1) | 0.1 | (h−1) | 0.01 |
(h−1) | 0.06 | (h−1 | 0.03 |
(g L−1) | 50 | (h−1) | 0.04 |
(g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (%) | (g L−1) | |
---|---|---|---|---|---|---|---|
6 | 0.38 | 1.55 | 26.6 | 6.3 | 6.2 | 62.4 | 1.42 |
10 | 0.95 | 3.84 | 28.0 | 9.6 | 9.2 | 42.6 | 3.39 |
20 | 2.50 | 9.58 | 30.7 | 12.7 | 11.7 | 24.0 | 7.98 |
(g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (%) | (m3) | |
---|---|---|---|---|---|---|---|
= 1 g L−1 and = 0.99 g L−1 | |||||||
2 | 0.17 | 0.69 | 27.8 | 8.4 | 8.3 | 49.7 | 46.9 |
6 | 0.83 | 3.36 | 29.0 | 13.9 | 13.5 | 16.7 | 7.93 |
10 | 1.48 | 5.87 | 30.1 | 15.0 | 14.2 | 10.1 | 3.91 |
20 | 3.06 | 11.59 | 32.3 | 15.8 | 14.4 | 5.1 | 1.52 |
= 5 g L−1 and = 4.99 g L−1 | |||||||
2 | - | - | - | - | - | - | - |
6 | 0.17 | 0.69 | 25.8 | 2.8 | 2.8 | 83.3 | 85.4 |
10 | 0.83 | 3.35 | 27.1 | 8.3 | 8.1 | 50.1 | 13.8 |
20 | 2.46 | 9.43 | 29.8 | 12.5 | 11.4 | 25.3 | 3.30 |
(g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (%) | (L) | (m3) | |
---|---|---|---|---|---|---|---|---|
= 1 g L−1 and = 0.99 g L−1 | ||||||||
2 | 0.17 | 0.69 | 27.8 | 8.4 | 8.3 | 49.7 | 0.60 | 46.9 |
20 | 3.06 | 11.6 | 32.3 | 15.8 | 14.4 | 5.1 | 9.22 | 1.52 |
= 0.05 g L−1 and = 0.04 g L−1 [rate-limiting] | ||||||||
2 | 0.27 | 1.11 | 67.9 | 16.2 | 16.2 | 2.6 | 0.35 | 0.28 |
20 | 1.02 | 4.21 | 68.2 | 16.5 | 16.3 | 0.8 | 1.32 | 0.07 |
(g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (%) | |
---|---|---|---|---|---|---|
= 1 g L−1 | ||||||
2 | 0.17 | 148 | 0.45 | 8.3 | 8.3 | 50.1 |
6 | 0.81 | 147 | 1.87 | 13.8 | 13.7 | 17.0 |
10 | 1.44 | 145 | 3.09 | 14.9 | 14.6 | 10.4 |
20 | 2.90 | 142 | 5.82 | 15.8 | 15.1 | 5.4 |
= 5 g L−1 | ||||||
2 | - | - | - | - | - | - |
6 | 0.17 | 172 | 0.40 | 2.8 | 2.8 | 83.4 |
10 | 0.82 | 170 | 1.65 | 8.3 | 8.2 | 50.5 |
20 | 2.35 | 167 | 4.27 | 12.3 | 12.0 | 26.1 |
(g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (%) | (g L−1) | |
---|---|---|---|---|---|---|---|
= 1 g L−1 | |||||||
2 | 0.17 | 148 | 0.45 | 8.3 | 8.3 | 50.1 | 46.3 |
20 | 2.90 | 142 | 5.82 | 15.8 | 15.1 | 5.4 | 46.3 |
= 0.05 g L−1 [rate limiting] | |||||||
2 | 0.27 | 7.2 | 15.5 | 16.2 | 16.1 | 3.0 | 2.34 |
20 | 1.13 | 7.1 | 128 | 16.5 | 16.2 | 0.73 | 2.34 |
(h) | (g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (g L−1) | (g L−1) | (m3) |
---|---|---|---|---|---|---|---|---|
48 | 0.1 | 2.57 | 17.4 | 7.1 | 57.2 | 0.54 | 0.37 | 0.05 |
120 | 1.2 | 9.78 | 26.7 | 14.1 | 11.9 | 4.29 | 3.68 | 0.08 |
240 | 3.1 | 11.6 | 32.3 | 14.4 | 5.1 | 9.22 | 9.19 | 1.52 |
480 | 3.2 | 5.98 | 33.9 | 14.4 | 5.0 | 9.36 | 9.50 | 1799 |
960 | 3.2 | 2.99 | 33.9 | 14.4 | 5.0 | 9.36 | 9.50 | 2.6(109) |
(h) | (g L−1) | Productivity (mg L−1 h−1) | (h) | (%) | (%) | (g L−1) | (g L−1) |
---|---|---|---|---|---|---|---|
48 | 0.1 | 2.7 | 48 | 7.1 | 57.3 | 0.57 | 0.38 |
120 | 1.2 | 16 | 43 | 13.9 | 12.0 | 6.8 | 4.42 |
240 | 2.9 | 142 | 5.8 | 15.1 | 5.4 | 46.3 | 9.50 |
480 | 3.1 | 319 | 3.8 | 15.7 | 5.1 | 50.0 | 9.50 |
960 | 3.1 | 409 | 3.4 | 15.8 | 5.0 | 50.0 | 9.50 |
All −20% | Base- Case | All +20% | +20% | +20% | +20% | +20% | +20% | |
---|---|---|---|---|---|---|---|---|
Fed-batch | ||||||||
−8 | - | 3 | 17 | −2 | 3 | 0 | −14 | |
−1 | - | 0 | 0 | 0 | 0 | 20 | −17 | |
Productivity | −9 | - | 2 | 1 | 0 | 0 | 20 | −15 |
−9 | - | 3 | 2 | 0 | 0 | 20 | −16 | |
11 | - | −2 | −2 | 0 | 0 | 0 | 0 | |
20 | - | −15 | −12 | 1 | −2 | 0 | −2 | |
Perfusion | ||||||||
−35 | - | 27 | 6 | −1 | 18 | 0 | 0 | |
−1 | - | 0 | 0 | 0 | 0 | 20 | −17 | |
Productivity | −58 | - | 87 | 39 | −4 | 12 | 20 | 1 |
−11 | - | 4 | 2 | 0 | 1 | 20 | −16 | |
10 | - | −4 | −2 | 0 | −1 | 0 | −1 | |
106 | - | −40 | −20 | 3 | −11 | 0 | −14 |
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Fitzpatrick, J.J.; O'Leary, F.; Hill, A.; Daly, J.; Lalor, F.; Byrne, E.P. Influence of Substrate Concentrations on the Performance of Fed-Batch and Perfusion Bioreactors: Insights from Mathematical Modelling. ChemEngineering 2025, 9, 48. https://doi.org/10.3390/chemengineering9030048
Fitzpatrick JJ, O'Leary F, Hill A, Daly J, Lalor F, Byrne EP. Influence of Substrate Concentrations on the Performance of Fed-Batch and Perfusion Bioreactors: Insights from Mathematical Modelling. ChemEngineering. 2025; 9(3):48. https://doi.org/10.3390/chemengineering9030048
Chicago/Turabian StyleFitzpatrick, John J., Fionn O'Leary, Ali Hill, James Daly, Fergal Lalor, and Edmond P. Byrne. 2025. "Influence of Substrate Concentrations on the Performance of Fed-Batch and Perfusion Bioreactors: Insights from Mathematical Modelling" ChemEngineering 9, no. 3: 48. https://doi.org/10.3390/chemengineering9030048
APA StyleFitzpatrick, J. J., O'Leary, F., Hill, A., Daly, J., Lalor, F., & Byrne, E. P. (2025). Influence of Substrate Concentrations on the Performance of Fed-Batch and Perfusion Bioreactors: Insights from Mathematical Modelling. ChemEngineering, 9(3), 48. https://doi.org/10.3390/chemengineering9030048