Residence time distribution in continuous virus filtration

Regulatory authorities recommend using residence time distribution (RTD) to address material traceability in continuous manufacturing. Continuous virus filtration is an essential but poorly understood step in biologics manufacturing in respect to fluid dynamics and scale‐up. Here we describe a model that considers nonideal mixing and film resistance for RTD prediction in continuous virus filtration, and its experimental validation using the inert tracer NaNO3. The model was successfully calibrated through pulse injection experiments, yielding good agreement between model prediction and experiment ( R 2 > ${R}^{2}\gt $  0.90). The model enabled the prediction of RTD with variations—for example, in injection volumes, flow rates, tracer concentrations, and filter surface areas—and was validated using stepwise experiments and combined stepwise and pulse injection experiments. All validation experiments achieved R 2 > ${R}^{2}\gt $  0.97. Notably, if the process includes a porous material—such as a porous chromatography material, ultrafilter, or virus filter—it must be considered whether the molecule size affects the RTD, as tracers with different sizes may penetrate the pore space differently. Calibration of the model with NaNO3 enabled extrapolation to RTD of recombinant antibodies, which will promote significant savings in antibody consumption. This RTD model is ready for further application in end‐to‐end integrated continuous downstream processes, such as addressing material traceability during continuous virus filtration processes.


| INTRODUCTION
Virus filtration is mandatory to efficiently and robustly eliminate viral particles, without compromising product safety.Typically, virus filters are single-use devices operated under either constant pressure or constant flux conditions (Fan et al., 2021;Goodrich et al., 2020;Wickramasinghe et al., 2010).Their separation capability relies on a size-based nanofiltration membrane structure, featuring a retentive region with a pore size of 15-20 nm, that effectively captures similarly sized viral particles.On the other hand, biopharmaceuticals-such as monoclonal antibodies with dimensions of 9-12 nm-can readily pass through these membranes (Shirataki & Wickramasinghe, 2023;Suh et al., 2023).
A fully end-to-end continuous biomanufacturing process must include continuous virus filtration (David et al., 2019).Research on continuous virus filtration processes is still in early stages (Bohonak et al., 2021;Fan et al., 2021;Lute et al., 2020;Shirataki et al., 2023).It is crucial to elucidate the differences between batch and continuous operations, and to master the methods for designing operation parameters of continuous processes based on batch operations.Notably, there is a significant distinction between batch and continuous operations, with regard to material traceability and understanding how disturbances propagate and affect material quality (Chen, Mao, et al., 2024;Lin et al., 2021).Regulatory authorities strongly advocate traceability in continuous manufacturing, with ICH-Q13 (2023) calling for the application of residence time distribution (RTD) to tackle this challenge.The concept of RTD originates from chemical processes, and describes the probability distribution of the time a material is present inside a reactor (Pereira & Leib, 2019).The RTD is used to determine the durations of start-up and shut-down phases, and when steady-state is achieved, and facilitates real-time control of process parameters in continuous operation.
Experimental measurement of RTD is typically achieved via tracer experiments, in which an inert material is injected into a reactor and monitored over time.However, it is not always easy to find an inert tracer that is easily detectable, inexpensive, and nontoxic, and that closely mimics the properties of the target product.Malakian et al. (2022) recently reported the use of a NaCl solution and a fluorescent dye as inert tracers to measure RTD in virus filtration processes.In another study, Lali et al. (2022) utilized a fluorescent-labeled antibody as an inert tracer in protein A affinity chromatography.If a process includes a porous material-such as a porous chromatography material, ultrafilter, or virus filter-it must be considered whether the molecule size affects the RTD, as tracers with different sizes may penetrate the pore space differently.
Mathematical models describing RTD are available for variety of reactors (Pereira & Leib, 2019).For simple standard processes, RTD can be estimated using models for continuously stirred tanks (CST), tanks in series (TIS), and dispersed plug flow (DPF), with the choice and combination depending on the extent of back-mixing in the system (Sencar, Hammerschmidt, & Jungbauer et al., 2020).Processes involving dead zones require modeling of nonideal mixing (Pancholi et al., 2022).More complex processes need more sophisticated mathematical models, and analytical solutions are not always available.These models offer greater extrapolation capabilities compared to data-driven or statistical models (Dürauer et al., 2023;Tang et al., 2023).Establishing robust and efficient models require proper model calibration and suitable calibration experiments (Chen et al., 2022;Chen, Chen, et al., 2023;Chen, Yao, et al., 2023;Yang et al., 2024).In virus filtration, pressure operations have been primarily applied and modeled, rather than considering constant flux processes (Shirataki et al., 2023(Shirataki et al., , 2021)) Our research group has conducted extensive investigations of RTD in downstream biopharmaceutical processes (Lali et al., 2021(Lali et al., , 2022;;Sencar, Hammerschmidt, & Jungbauer et al., 2020;Sencar, Hammerschmidt, Martins, et al., 2020).However, these studies have primarily focused on either tracer experiments or model development.
In this study, our primary focus is the development and application of an RTD model for virus filtration processes.The model is ought to be robust under various perturbations, including changes in loading volumes, flow rates, tracer concentrations, and filter surface areas.First, we calibrated the RTD model by pulse injection experiments using NaNO 3 as a tracer.The tracer can be monitored by conductivity and UV absorbance at 280 nm, using the sensors built into the workstation.Then, we validated the model through stepwise experiments, and combined stepwise and pulse injection experiments.We also conducted tracer experiments with antibodies to confirm the suitability of NaNO 3 as an inexpensive replacement for antibodies, because the different sizes might affect the RTD in a virus filter.Finally, we performed a shortcut continuous virus filtration process to investigate the applicability of the RTD model calibrated under batch conditions.

| Virus filter and control system
We used Planova BioEX filters (Asahi Kasei Medical) with three distinct surface areas: 3, 10, or 100 cm 2 .These dead-end filters were connected to the column valve of an ÄKTA pure 25 (Cytiva) (Figure 1).The 3-, 10-, and 100-cm 2 filters were operated at flow rates of 0. integrated sensors of the ÄKTA system.Table 1 lists the manufacturer's specifications for all equipment considered for modeling.

| Mathematical models
To predict RTD characteristics of the virus filtration system, we applied several well-established RTD models.These models include not only the filtration units themselves, but also the peripheral equipment (Chen, Lu, et al., 2024), such as sample loops, tubing, valves, monitors, mixers, pumps, and the injection system.Symbols and units are provided in the Nomenclature section.

| Models of connections and workstation
For the workstation parts and tubing with given diameter and length -such as sample loops and tubes-we used the DPF model to calculate RTD, as follows: For equipment, such as valves, mixer, and monitors, we used the CST model (Table 1) with c c c

| Models of virus filters
The hold-up volume of the dead-end filters can be divided into three distinct compartments: hollow spaces and headers (V I ); within the hollow fiber walls (V wall ), also referred to as retentate space and T A B L E 1 Description of filters and peripheral equipment in the ÄKTA system.permeate space (V O ), as illustrated in Figure 2.For each compartment, the choice of models varies based on their fluid dynamics characteristics.We made the following assumptions: 1.The fiber bundle was considered as a single hollow fiber, which was assumed to be identical among different filters.
2. The 3-cm 2 filter comprised a single fiber, while the 10-and 100-cm² membranes comprised unknown numbers of fibers, which were calculated using A A / 10 3 and A A / 100 3 , respectively.3. Compartment V I was modeled as an equivalent cylinder with a uniform length (L =10.8 cm) but varying diameters , j {3, 10, 100} ∈ ).Compartments V I and V wall were characterized using the DPF and CST models, respectively.
To calculate the concentration change over time in the permeate space V O , we developed a model that accounted for nonideal mixing in both the radial and axial directions, as well as film resistance.For radial nonideal mixing, the TIS model was employed.Permeate space V O was divided into a cascade of CSTs arranged in an annular fashion.
The concentration change of each tank was calculated using Equation (2).
For axial nonideal mixing, we used a model in which two interconnected well-mixed CSTs were divided along the axial direction (Pancholi et al., 2022).In the first axial CST, it was assumed that all solutions entered and exited.The second axial CST was assumed to have no inflow or outflow, but there was an exchange of mass between the two CSTs at a defined flow rate denoted as ηV ˙.
Consequently, the mass balance equation for the two CSTs can be derived as follows: Here, the volumes of the two CSTs, εV l O , are time-dependent, and their sum equals V O .This volume variation was explained by the flux due to film resistance.The Graetz-Leveque correlation (van den Berg et al., 1989) confirmed that the film mass transfer coefficient was proportional to u 1/3 .Velocity was calculated through the side area of the equivalent cylinder, . Thus, the flux could be defined as follows: The total flux contributed by all components was calculated as follows: m,eq max ∆ , ε could be derived as a time-dependent linear relationship as follows: For a binary system comprising equilibration and tracer solutions, t m,eq was defined.Combining this with Equations ( 5) and ( 6), Equation ( 7) can be rewritten as follows: F I G U R E 2 Schematic representation of virus filtration residence time distribution (RTD) models accounting for nonideal mixing and film resistance.The hold-up volume of the filters was divided into three distinct compartments V I , V wall , and V O .The tanks in series (the number of cascaded was l) and two interconnected well-mixed CST was employed to account for the radial and axial nonideal mixing, respectively.The volumes of the two CSTs were εV l O , respectively, with an exchange of mass between the two CSTs defined as ηV ˙.
The domain of ε clearly fell within the range of 0-1.Therefore, we had the following equation:

| Initial conditions and models of injection system
Before tracer injection, all system equipment was connected in series and flushed with low-salt buffer.Therefore, the output of the preceding equipment constituted the input for the subsequent one.
For experiments involving injection via a sample loop, the loop was equilibrated using the sample itself.The salt concentration in the system before tracer injection served as the initial conditions to complete the DPF (Equation 1) and CST (Equation 2) models.
Injection via pumps was represented as a rectangular pulse.

| Numerical solution and software
There were no available analytical solutions for these models under complex initial conditions.Therefore, numerical methods were used for their solution.For equations involving spatial variables (e.g., the DPF model), spatial discretization was achieved using the discontinuous Galerkin finite element method (Breuer et al., 2023;Meyer et al., 2020).Its semidiscrete form, along with the CST model, can be efficiently solved by SciPy (Virtanen et al., 2020) using Python 3.10.In this study, we also used several other Python packages, including NumPy (Walt et al., 2011), scikit-learn (Pedregosa et al., 2011), Pandas, and Matplotlib (Hunter, 2007).

| Model calibration
The developed model included four unknown parameters (l, η, α, and ).We used the inverse method to estimate these unknown parameters based on experimental data.If UV and conductivity signals held equal weight in the objective function, the inverse method could be formulated as an optimization problem as follows: Computing Equation ( 10) required a numerical conversion of concentration units.Considering the 2-mm path length of the UV monitor, and applying Beer's law, the contribution of UV absorbance to Equation ( 10) is given by Equation ( 11): Similarly, the contribution of conductivity could be converted using Kohlrausch's law, as follows: which was suitable for calculating strong electrolytes under ideal diluted conditions (Carta & Jungbauer, 2020).We initially used the heuristic algorithm to explore solutions for the above optimization problem over a wide range, and then applied a deterministic algorithm to obtain the final solution.

| Model qualification
The coefficient of determination, often referred to as R 2 , indicated how well the model replicated observed outcomes.It was calculated using the r2_score function in the scikit-learn package.

| Chemicals, reagents, and protein
All chemicals and reagents were purchased from Merck or Sigma Aldrich, unless otherwise specified.Before use, all buffers were filtered using a 0.22-μm filter (Merck), and degassed in an ultrasonic bath.In this study, we used the recombinant antibody IgG1 (commercial name, trastuzumab).
To prepare purified and aggregate-free antibody for virus filtration experiments, this antibody was captured via preparative protein A affinity chromatography and purified using preparative size-exclusion chromatography (details provided in the Supporting Information).

| Pulse injection experiments for model calibration
Pulse injection experiments were conducted as follows.First, the system was equilibrated with an equilibration buffer via pump A, until achieving stable UV and conductivity signals.Next, 0.5 M NaNO 3 was loaded into a 260-μL sample loop via the ÄKTA sample pump.Finally, the sample within the sample loop was introduced into the system via pump A. The above steps were performed once for each of the three different connections: ÄKTA bypass (Figure 1a), connector (Figure 1b), and three virus filters with distinct surface areas (Figure 1c).Detailed experimental conditions are provided in  3 | RESULTS

| RTD model calibration using pulse injection experiments
First, the virus filtration RTD model was calibrated by performing pulse injection experiments, with NaNO 3 as a tracer.The ÄKTA bypass Experiment C1 (Figure 1a) was conducted to verify the ÄKTA dead volume.For modeling, we considered the following system equipment: the sample loop; UV monitor; conductivity monitor; and tubes 5, 6, and 7. Figure 3a shows the comparison between simulated results, generated after feeding the dead volume data into the model, and experimental results.We found that R = 2 0.905, which indicated a good match but did not fully elucidate the RTD behavior.The experimental profiles revealed more pronounced tailing that did not fully match the simulated UV and conductivity signals.The flow rate curve indicated that the system pump required approximately a 6s delay to reach the preset during the flow rate increase phase.These effects are dominating at very small scale and give rise to this peculiar shape.
To determine the dead volume of the tubes, we conducted pulse injection experiments, for which the filters had to be connected to the column valve.Compared to the bypass setup (Figure 1a), this configuration included the addition of C-VF and C-VF tubes, along with a connector (Figure 1b).The experimental data exhibited good agreement with the simulation, in terms of both conductivity and UV signals (Figure 3b-d).One commonly overlooked phenomenon is that at a low flow rate, the pump reached the specified flow rate with only a negligible delay, while at a high flow rate, the pulse was eluted before the specified flow rate was reached (Figure 3d).Although our model accounted for the delayed flow, Figure 3d exhibited the lowest R 2 (0.944).
The virus filtration RTD model was calibrated using pulse injections (Tables 2, C3-1 to C3-3) within the system where the filters were connected to the ÄKTA system (Figure 1c).Reasonable model parameters were obtained by solving the optimization problem in Equation ( 10), with l, η, α, and c max ∆ values of 3, 0.13, 1.14, and 2.17E−7 M/(s/m), respectively.These parameters indicated that the RTD of permeate space could be described by three cascaded CSTs radially, with roughly 13% mass exchange between two CSTs axially, and a mass transfer ratio of 1.14 between tris-acetate and NaNO 3 .
The simulation results based on these parameters are presented in Figure 3e-g.All experimental results showed good agreement with the simulation (R > 2 0.9).The filter resulted in more pronounced peak tailing, due to an increased space for exponential wash-out.
For the 0.05 M NaNO 3 experiment V2 (Figure 4d-f), we observed good agreement between simulation and experiment, with R > 2 0.99.For the 0.1 M NaNO 3 experiments V4 (Figure 4g-i), the results remained highly satisfactory, although we observed slight distinctions during the start-up and shut-down phases in the experiment using the 10-cm 2 filter.
Compared to the 0.05 M NaNO 3 experiments, the 0.1 M NaNO 3 experiments required a longer time to reach a steady state, with a smooth start-up phase.In particular, the system did not fully reach a steady state in the experiments with the 10-cm 2 filter.

| RTD model validation using combined stepwise and pulse injection experiments
To emulate a process perturbation, we combined a stepwise injection with a pulse injection.We conducted combined experiments for ÄKTA bypass (Table 2, V5) and filters (Table 2, V6).
To determine whether the tracer size influenced the RTD, and whether NaNO 3 can be used as a surrogate for antibodies, we conducted a combined experiment (Table 2, 11) and ( 12) as follows: Solving this optimization problem yielded β = 4.49 g/M.This implied that the RTD for 0.5 and 2.0 g/L mAb solutions equaled that of 0.11 and 0.45 M NaNO 3 , respectively.Figure 4m displays the predicted RTD in the mAb-tris-acetate system based on this equivalence relationship.We found that R = 2 0.99, indicating excellent agreement between model predictions and experiments, in terms of stepwise and pulse injection.

| RTD model prediction in continuous virus filtration
Finally, the RTD model was calibrated using batch conditions and NaNO 3 was applied to predict continuous virus filtration of antibodies.We conducted experiment V8 using two 100-cm 2 filters connected in parallel (Figure 1d).After reaching steady state, we switched to the second filter (Figure 4n).Our results showed excellent agreement between the experiment and simulation during transitions between cycles, start-up, and shut-down, with R = 2 0.99.
The R 2 values for all experiments in this study are presented in Table 3 for ease of comparison.

| DISCUSSION
The developed RTD model was calibrated by pulse injection experiments with tracer, and was validated using three types of experiments: stepwise, a combination of stepwise and pulse injection, and continuous injection.
Our results demonstrated that the model could characterize the RTD behavior of filters under six different conditions, including different injection volumes, flow rates, tracer concentrations, filter surface areas, and operating modes (e.g., continuous or batch), and with the use of NaNO 3 as a tracer for antibodies.
Pulse injection and stepwise injection were performed with slightly different experimental set-up, leading to varying injection volumes.Specifically, pulse injection was performed using a sample loop, while a pump was used for stepwise injection.The proposed model could predict the RTD behavior under different loading volumes (Figures 3 and 4).Our results showed a good agreement between the areas under the experimental curves and the simulated ones, indicating that the calculated results were consistent with Beer's and Kohlrausch's law.
Flow rates significantly affectes the proposed RTD model.Our present study includes two types of flow rate variations.In the first type, the flow rate of the pump increases after the pump starts, which does not occur immediately, but rather with a significant delay.
This phenomenon is particularly pronounced in Figure 3a,b, in which the delayed flow rate increases in a gradient pattern.In the second type, the flow rate increases in a sawtooth pattern as shown in Figures 3g,4a-c,f,i,l.All of these experiments were conducted at a higher flow rate of 10 mL/min.An explanation for this abnormal observation is that when the column valve was switched at a higher flow rate, the introduction of an additional tracer resulted in flow interruption.This overpressure phenomenon could be avoided or made negligible by reducing the flow rates.Including the flow rate change in the model enables highly accurate prediction of RTD under these complex conditions.This is because this modification can account for (a) the effects of flow rates and their impact on the axial diffusion coefficient in the DPF model, (b) the influence of flow rates in the CST model, and (c) the effects of flow rates and their impact on the mass transfer coefficient in the virus filter model.
To predict the effects of tracer concentration on RTD behavior in virus filtration processes, the DPF and CST models had to be modified.The most significant modification involves nonideal mixing, as introduced in Equations (5-7).We observed changes of the UV profile according to the sequence of buffers or tracer introduced to the system (Figure 5).This change is dependent on tracer concentration.Specifically, in validation experiments, we observed a sharper transition in the start-up phase when using 0.05 M NaNO 3 , compared to 0.1 M NaNO 3 .In contrast, in the shut-down phase, the transition was sharper when the higher concentration was used.We and where l → ∞ (no back-mixing), respectively (Toson et al., 2019).
The calibrated result, with l = 3, implies the presence of partial back- mixing in the filter.Thakur and Rathore (2021)  dynamics simulations would be a better choice (Francis et al., 2006).
The modification introduced in Equation ( 7), with a relative filter surface area, enables the characterization of RTD behavior for different filter surface areas in experiments C3, V2, V4, and V6.The modification simplifies all fibers as cylinders, with different diameters but equal lengths (Figure 1).It also assumes that the well-mixed volume is the same for all fibers, and equal to the volume of the simplified fiber.This assumption is simple and efficient, but ignores the interactions between fibers and assumes that the distance between fibers is much smaller than the filter diameter.Our present results confirmed that these assumptions are valid for filters with surface area in the range of 3-100 cm 2 ; however, their applicability to larger surfaces remains to be confirmed.

The results in
. Recently, Malakian et al. (2022) developed a PF-CST-CST model to describe the RTD of virus filtration in constant flux processes.However, discrepancies are observed between the model simulations and experimental results.
35, 1, and 10 mL/min, respectively, yielding corresponding fluxes of 70, 60, and 60 liters per square meter and hour.Before operation, all air was displaced from the filters with equilibration buffer.UV 280 nm and conductivity were monitored using the F I G U R E 1 Schematic depiction of the ÄKTA system and the flow path for different experiments.(a) Bypass.(b) Experiments to measure the residence time distribution (RTD) of the chromatography workstation.(c) Experiments for batch operations with one filter.(d) Set-up for continuous operation.
u and D ax were obtained by V A ˙/ where A πd = the absence of a second CST, where ε was equal to 1. On the other hand, when the difference was sufficient to maximize the flux, there was no first CST, such that ε was equal to 0. If we denoted the maximum flux as k c the flux contribution due to film resistance remained consistently positive, with the maximum value at c max ∆ .Up to this point, we had obtained an RTD model for virus filtration processes, based on considerations of nonideal mixing and film resistance.
above-described virus filtration RTD model was calibrated using pulse injection experiments with 0.5 M NaNO 3 .To assess the extrapolation ability, we performed stepwise experiments with 0.05 and 0.1 M NaNO 3 .The model developed for the sample loop in pulse injection experiments was not applicable to the stepwise experiments, due to the use of the ÄKTA sample pump for tracer injection.We first conducted the bypass experiments V1 (0.05 M NaNO 3 ) and V3 (0.1 M NaNO 3 ) to demonstrate that the model could predict the stepwise injection of different NaNO 3 concentrations with R ≈ 2 0.98.
V7) in which we loaded antibodies instead of NaNO 3 .In the developed model, α was defined as describing the relative magnitude of mass transfer resistance between the equilibration and tracer solutions.It was assumed that α remained consistent despite the system variations.Thus, it was important to answer the following question: what NaNO 3 concentration is equivalent to 0.5 and 2.0 g/L antibody solutions if c βc = mAb NaNO 3 ?To address this issue, we modified the objective functions Equations ( attribute this sharp transition to radial nonideal mixing, which can be characterized using the TIS model.In this model, an increased number of cascaded CSTs (l) results in altered RTD behavior, similar to in the DPF model.A single CST model and the DPF model corresponded to specific cases of the TIS model: where l = 1 (complete back-mixing) reported a similar start-up phase phenomenon in single-pass tangential flow ultrafiltration processes, which was attributed to changes in flux induced by time-dependent membrane resistance.In our model developed for dead-end filtration, this phenomenon is explained by flux contributions from film resistance in the axial direction.Based on the contrasting RTD behavior observed in experiments using 0.05 and 0.1 M NaNO 3 , the ratio of the film mass transfer coefficient should fall between 1 and 2. Thus, the calibrated result, α = 1.14, is considered reasonable.In addition to the validation experiments, the 0.5 M NaNO 3 pulse injection experiments (Figure 3e-g) performed for model calibration also exhibited the above-described concentration-dependent phenomena, with tris-acetate displacing 0.5 M NaNO 3 , and 0.5 M NaNO 3 displacing tris-acetate.This is why the simple model calibrated with pulse injection experiments can accurately predict outcomes for both stepwise experiments, and combined stepwise and pulse injection experiments.In summary, the RTD model developed based on the observations of concentrationdependent nonideal mixing provides a simple and efficient means of describing the effects of tracer concentration on the RTD behavior of filtration processes.In situations requiring a more detailed investigation of fluid flow behavior within the filter, computational fluid

F
Figure 4m-along with a comparison to experiments V5, V6, and V7-demonstrate the suitability of NaNO 3 as a tracer for monitoring antibody RTD behavior during virus filtration processes.Additionally, we obtained an equivalent relationship between their concentrations, expressed as c βc = mAb NaNO 3 , where β =4.49g/M.As a small-molecule tracer, the validity of this equivalence is assured under conditions characterized by minimal fouling and low membrane resistance.Considering the comprehensive factors incorporated within the developed model, this equivalence can be applied to investigate antibody RTD behavior across varying conditions, including injection volumes, flow rates, tracer T A B L E 3 R 2 for calibration (C) and validation (V) experiments.I G U R E 4 Model validation results.(a) 0.05 M NaNO3 stepwise loading; (b) 0.1 M NaNO3 stepwise loading; (c) 0.05 M NaNO3 stepwise loading with a 0.5 M NaNO3 pulse into ÄKTA bypass at 10 mL/min.0.05 M NaNO3 stepwise loading into (d) 3-cm2 filter at 0.35 mL; (e) 10-cm2 filter at 1 mL/min; (f) 100-cm2 filter at 10 mL/min.0.1 M NaNO3 stepwise loading into (g) 3-cm2 filter at 0.35 mL; (h) 10-cm2 filter at 1 mL/min; (i) 100-cm2 filter at 10 mL/min.0.05 M NaNO3 stepwise loading with a 0.5 M NaNO3 pulse into (j) 3-cm2 filter at 0.35 mL; (k) 10-cm2 filter at 1 mL/min; (l) 100-cm2 filter at 10 mL/min; (m) 0.5 g/L antibody stepwise loading with a 2.0 g/L antibody pulse into 3-cm2 filter at 0.35 mL/min; (n) emulation of continuous operation by loading of 0.05 M NaNO3 on two 100-cm2 filter at 10 mL/min.The curves of the system and sample pumps were integrated.concentrations, filter surface areas, and operating modes.As a costeffective tracer, NaNO 3 reduces antibody consumption in RTD investigation experiments, thereby reducing experimental costs.This inert tracer has significant potential for use beyond virus filtration processes into integrated continuous downstream processes of antibody production.A good example of the proposed equivalence relationship is the use of 0.05 M NaNO 3 as a tracer to investigate the RTD behavior of 0.22 g/L antibodies in continuous filtration, thereby reducing the antibody consumption required for multiple cycles in continuous operation.The transition from batch to continuous conditions brings various benefits, including enhanced productivity and cost reduction.

Figure
Figure4nindicates that the continuous process with two cycles achieves approximately a 16.7% increase in productivity, compared to the batch process.A remarkable aspect of the developed model lies in its ability to describe both batch and continuous filtration processes.Based on this, there is potential for further applications of continuous virus filtration RTD models, such as in process development, optimization, characterization, and real-time control-for instance, the model-assisted steady state prediction for experimental design demonstrated in this work.Additionally, the model predicts that the productivity of the continuous process after 10 cycles will be 150% higher than that of the batch process.The ability to characterize how perturbations will affect material quality is