CALIBRACION DE MODELO DE CROMATOGRAFIA DE INTERCHANGE IONICO MECANISTICO

Abstract

A method is provided, comprising: obtaining, for a chromatography machine including a first dispersed plug flow reactor (DPFR) and a continuous stirred tank reactor (CSTR) upstream of a column, and a second DPFR downstream of the column, geometric measurements associated with the second DPFR; generating, by means of a processor, transport model parameters for a transport model associated with the second DPFR based on the geometric measurements; feeding a tracer molecule into the chromatography machine; capturing one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine;and estimate, using the processor, based on the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more transport model parameters for a transport model associated with the first DPFR and the CSTR.

Description

CALIBRATION OF A MECHANICAL ION EXCHANGE CHROMATOGRAPHY MODEL CROSS REFERENCE TO RELATED APPLICATIONS This application claims priority over the pending U.S. provisional application No. 63 / 123,170 entitled “Mechanistic Ion Exchange Chromatography Model Calibration”, filed on December 9, 2020, the full disclosure of which is incorporated herein by reference. FIELD OF DISCLOSURE This disclosure generally relates to the calibration of a mechanical ion exchange chromatography model for a chromatography machine. BACKGROUND The biopharmaceutical industry has made significant strides in digital biofabrication over the past few decades with the introduction of powerful processors and large hard drives capable of storing and interpreting vast amounts of data. Digital tools that aid in process development and analysis have evolved rapidly to improve process performance and efficiency. The application of predictive models enables in silico exploration of design spaces during process development and improved real-time control strategies in a manufacturing environment. Chromatography is a method used for the separation and purification of molecules during the downstream processing of biomolecules and is an important step in the purification of biopharmaceutical products. Using chromatography, solutions of biomolecules can be purified and concentrated using porous packed beds that separate different molecules based on their differences in mass, size, or charge. Ion-exchange chromatography (IEX) is a purification step based on ionic interactions between the chromatographic material and the molecules. It involves using charged sites on the surface of the packing material to adsorb biomolecules based on the charged sites on its surface. Cation-exchange chromatography (CEX) is a mode of ion-exchange chromatography that uses negatively charged sites on the surface of the adsorbent to adsorb biomolecules based on positively charged sites on its surface.Adsorbed molecules can be eluted by decreasing the surface affinity of the molecules through the use of a buffer solution of higher ionic strength. The Quality by Design (QbD) approach, described by the U.S. Food and Drug Administration (FDA) in 2004, introduced the concept of building product quality requirements into the production process. In subsequent years, the International Conference of IVIA / a / ¿U¿ó / UUODO I Harmonisation (ICH) approved the QbD approach and introduced guidelines that define systematic approaches and scientific principles for drug development processes. Along with the regulatory scrutiny requirements for biopharmaceutical process development in the CEX stages, there are also technical challenges, which make technical transfers to larger scales, process improvements, and process characterizations resource-intensive tasks. Tight, accelerated timelines leave little room for a thorough exploration of the process design space to find process parameters that allow for maximum yield and purity, while simultaneously meeting stringent constraints for process robustness and maximum molecule recovery. Processes developed under such conditions in early stages for first-in-human (FIH) studies meet all the required constraints at that time, but their scalability may be insufficient when quality, quantity, or cost requirements change later on. Iterative experimentation, which requires significant resources, can be the key to identifying critical process parameters that reduce material consumption and increase process performance, such as yield and molecular purity. Methods that can be integrated into the early stages of process development, enabling targeted experimentation around the most promising process parameter ranges, can help guide the development toward more optimal processes in terms of performance and scalability under accelerated timelines. Mechanistic models are a tool used for knowledge-based process development, process optimization, predictive process control, and process compression in chromatographic separation processes. Mechanistic models provide predictions of important process outcomes and process performance indicators across a wide range of processes. In particular, mechanistic modeling of CEX chromatography is a useful first-principles-based tool for describing the transport and elution behavior of molecules and predicting the behavior of a chromatography step. Applying a CEX chromatography model in process development to explore the design space of process parameters, such as loading factors or elution gradients, can reduce the effort required for iterative experimentation.Furthermore, mechanistic modeling provides an opportunity to understand the mechanisms that affect the separation of a therapeutic protein from its impurities. This knowledge can be used to guide decision-making during development and to optimize product quality and process performance. Essentially, mechanistic modeling provides a useful and foundational tool. IVIA / a / ZUZÓ / UUODO I scientific for process design and optimization. For example, mechanistic modeling of CEX chromatography can be used to make decisions about CEX stage parameters such as gradient slope, collection criteria, and loading factor. However, mechanistic models require prior knowledge and a meticulous calibration process to provide sufficient predictability in both small- and large-scale separation processes. In particular, a mechanistic model requires a well-defined mathematical description of each process stage. Several studies use and suggest workflows for mechanistic models for process optimization, process research, and performance evaluation or model-based control. Several models are available that can describe chromatographic steps in biopharmaceutical products. One of the first mathematical descriptions of chromatographic processes described the adsorption of CO2 to carbon and a silica gel using empirical equations. Since then, chromatography models have been developed to also include many possible contributions to mass transfer kinetics based on first principles. An important step is the development of the general velocity model (GRM). The GRM describes combinations of kinetic transport phenomena, such as forced convection (by pumping), diffusion, and adsorption of molecules in liquid chromatography systems. This allows for more detailed descriptions of transport phenomena in chromatographic columns than previous simplified versions (e.g., the ideal model and the Thomas model).For molecules that are small compared to the pore size of the chromatographic material, the clustered pore model (CPM) may be a suitable simplification of the GRM. The adsorption of biomolecules to the adsorbent can be described using adsorption models. In this case, too, a large number of models are available that have proven applicable to chromatography. An early adsorption model is the Langmuir model, which allows for concentration-dependent adsorption to surfaces with a finite capacity. A further milestone in modeling adsorption behavior is the stentic mass action (SMA) model. Its equations allow the determination of the influence of ionic strength, spherical shielding of binding sites, and the affinity of biomolecules for the chromatographic material caused by ionic interactions on the adsorption behavior of mixtures of molecules. Several studies have demonstrated the successful application of the SMA model to ion-exchange chromatography.Even under less-than-ideal process conditions, such as high column loading, detailed models are available that can describe non-intuitive adsorption behavior. Model development activities for industrial chromatography stages can utilize these models. IVIA / a / ¿U¿ó / UUODO I of transport and adsorption already exist. However, before their application, it is necessary to calibrate the mechanistic models. That is, it is necessary to identify appropriate model parameters so that the model explains the desired processes. However, the calibration process, that is, the development of the model and the estimation of the model parameters, is still a time-consuming stage. In other words, the model parameters can generally be obtained through measurement, such as using optical microscopy to determine pore transport coefficients. However, direct measurements of model parameters are often not possible or are very laborious; therefore, recursive parameter estimation is a method frequently used in chromatography modeling. In addition to the influence of competitive adsorption and ionic strength, mass transfer in the extracolumn volume affects the absolute elution time and the shape of the peaks in simulated chromatograms. Furthermore, the configuration of the tubes, valves, or mixing chambers impacts the retention time of each component. During adsorption, parameter estimation involves using a recursive method over the time elapsed from the start of elution until a respective eluted component is reflected in the adsorption parameter. Therefore, parameters estimated without considering the extracolumn volume are scale-dependent and may not be suitable for larger or smaller system scales or varying system configurations. This effect can have a significant impact on the chromatogram, especially when the column size is small compared to the extracolumn volume.However, in many cases, this effect has not been taken into account when estimating the adsorption model parameters. Scale-up research is an application of ion-exchange chromatography models that has been previously investigated. One of the main findings in this field is that mass transfer coefficients are flow-rate dependent and therefore need to be matched to the flow rate. Studies have compared model calibration methods for identifying parameters in an SMA adsorption model and have found that the inverse method is more than adequate for this purpose. In one example, adsorption model parameters can be identified using neural networks, which, after a training period, accelerates parameter identification compared to inverse chromatogram fitting. A robust model calibration workflow involves a set of 14 experiments and is suggested for calibrating a multi-component SMA model. IVIA / a / ZUZÓ / UUODO I for a bispecific antibody. These 14 experiments are used for the calibration of the adsorption model and are added to the experiments required for the characterization of the column and extracolumn characteristics. Based on this workflow, the value of the mechanistic model for investigating the robustness of the process can be demonstrated. However, a description of how extracolumn transport behavior, inherent to chromatographic equipment, is considered in the calibration workflow is not available. Many physical processes influence the chromatographic behavior of a molecule. Along with the influence of competitive adsorption and ionic strength, extracolumn mass transfer affects the absolute elution time and the shape of the peaks in simulated chromatograms. The configuration of tubes, valves, or mixing chambers impacts the absolute retention time of each component. Consequently, when estimating adsorption parameters using a recursive method, the time elapsed from the start of elution until a respective component elutes will affect the estimated value of the adsorption model parameters.Therefore, parameters estimated without considering the extracolumn volume are scale-dependent and may not be suitable for larger or smaller system scales or varying system configurations. This effect can significantly impact the chromatogram, especially when the column size is small compared to the extracolumn volume. Previous publications have demonstrated the importance of accurately representing the extracolumn volume. However, many contributions have not addressed how this effect can be considered in the model calibration workflow. In the interest of time and effort in the development of industrial processes, it is favorable to reduce the number of additional experiments required for model calibration to the fewest possible. SUMMARY This disclosure provides a workflow for calibrating the CEX model using a minimal set of experiments. The model provided herein accounts for extracolumn mass transfer and involves a modeling sequence that combines dead-volume models adjusted upstream of a chromatography column with models configured solely from tube geometry values ​​downstream of a chromatography column. Furthermore, in the model provided herein, the model input parameters reflecting mass transfer and those reflecting molecule-specific adsorption behavior are determined in a decoupled manner.This disclosure demonstrates that the adsorption parameter iviA / a / ¿u¿d / uuoooi can be scaled across the process size and shows sufficient predictive quality for scale-up process runs. Advantageously, this disclosure provides a chromatography model calibration approach with increased model accuracy and agility. The calibration is structured in three parts for sequential parameter estimation that decouples parts of the transport model and the adsorption model from each other. First, a unit operation representation of a chromatography skid and its transport parameters are identified. Next, a transport model for the packed bed is identified. Finally, the adsorption model parameters are estimated. Figure 1A illustrates an example flow path representation commonly used in chromatographic machines and skids, such as the ÁKTA™ Avant. As shown in Figure 1A, there are variable dead volumes upstream of the column. Since this dead volume influences the peak position relative to the elution starting point, it is necessary to account for the dead volume in the model as accurately as possible to decouple this effect from the adsorption behavior, which also impacts the peak position. The possible representations range from simple time displacements to mechanistic representations of the volume of the incorporated tubes and valves. For a BiTE® antibody construct (bispecific T-cell coupler), a combination of a DPFR model and a CSTR model is used to represent the dead volume upstream of the column. The dead volume between the column outlet and the UV and conductivity sensor is also represented using DPFR models. A combination of tube and valve models is used to represent the flow path from each sample pump to the UV sensor and from the inlet pump to the conductivity sensor. Both flow paths are a combination of an adjusted DPFR model and a CSTR model upstream of the column, and a DPFR model configured from geometric specifications provided by the supplier. After identifying the separate model components for the column model, the two dead volume representations of the inlet flow path and the sample flow path are combined into a more complex model. The two downstream DPFR models are specified by the diameter and length of the tubes specified by the supplier. Only the dispersion coefficient of these two DPFR models is set to the estimated value for the upstream DPFR. IVIA / a / ¿U¿ó / UUODO IThe specific sequence of steps involves first obtaining the geometric values ​​of the downstream tubes (diameter, length), for example, by measurement or based on supplier specifications, and then running experiments that feed a tracer molecule into the system from the desired location. Next, a DPFR-CSTR-DPFR model sequence is configured, with the downstream DPFR model specified according to the specified tube geometric values. That is, while the geometric specifications of this downstream DPFR are fixed, the parameter specifying the dispersion properties of the DPFR is not fixed and is estimated along with the properties of the upstream DPFR and CSTR. In total, the estimation includes estimating the geometric parameters of the upstream DPFR and CSTR, and transport coefficients, such as dispersion, of the upstream and downstream DPFR.The parameters not included in the estimate are the DPFR geometric parameters downstream of the column. This step is performed for both the sample flow path (blue in Figure 1A) and the inlet flow path (green in Figure 1A). Once all transport parameters are estimated, the binding parameters can be estimated. The model can include as many components as it needs to simulate peaks. Parameters can then be estimated for each of the modeled components. The range of estimated parameters can be used to include additional components that form one or more of the chromatogram peaks, and the binding parameters can be re-estimated using narrow bounds for the parameters. The first innovation in this sequence lies in specifying the DPFR model downstream of the column according to the geometric values ​​of the specific tubes. This entails using a combination of fitted models and models configured based on geometric specifications to identify the level of transport effects impacting the column at the outset. Consequently, the point at which the sample and the elution buffer impact the column can be identified more accurately than by pooling both upstream and downstream flow paths without performing more complex bypass experiments to separate the paths. Furthermore, using the methods provided herein, it is possible to separate the transport behavior of the buffer from the transport behavior of the dissolved molecules. In the previous technique, the focus of model-based process development lies in how the model supports process design and optimization. The model calibration procedure is often not described. If the focus is placed on model calibration, the greatest IVIA / a / ¿U¿ó / UUOOO Part of the previous technique involves new ways of estimating the adsorption parameter, but not new ways of estimating the dead volume before the column. A second innovation in this sequence is that the adsorption model parameters are initially estimated using a model with a reduced number of components. Once this stage is complete, more components are added to the model. This is necessary for BiTE® and often for antibodies, as several molecule species elute in such close proximity that their summed UV signal appears as a peak. This more complex model then needs to be recalibrated with a larger number of components, resulting in a much larger number of estimated adsorption parameters. However, because the adsorption parameters were previously estimated approximately from the less complex model, a smaller parameter space can be applied to the parameter estimation this time, which speeds up the estimation procedure. Decoupling transport and adsorption effects in the chromatogram allows for more accurate chromatography models. Furthermore, the models are more flexible with respect to their application domain. This means that, as long as the resin remains the same, the adsorption model parameters can remain constant even if other process specifications change. This is an insight that is often overlooked. However, it allows for a shorter model calibration procedure compared to performing new model calibrations every time a process changes. Model-based process development can benefit from this speed improvement. Additionally, the models are more accurate compared to those where transport effects are difficult to distinguish from adsorption parameters. The method provided herein is more flexible and accurate than previous methods for modeling dead volumes in chromatography machines and skids, such as shifting the elution start point according to the dead volume, or determining the dead volume based on derivation experiments. Both of the previous methods treat the dead volume after the column outlet as the dead volume before the column. Finally, the method provided herein allows for the estimation of adsorption parameters with less influence from transport effects. This approach decouples transport and adsorption effects in the chromatogram more effectively than previous methods. This is beneficial when process specifications that influence molecule transport are changed, such as in scale-up applications. In such cases, the impact of increased tubing, flow rates, and other factors is significantly reduced. IVIA / a / ZUZÓ / UUODO I can be considered as a separate part of the model and the adsorption model remains the same and does not need to be recalibrated under the changed process conditions. In one aspect, a method is provided, comprising: obtaining, for a chromatography machine including a first dispersed plug flow reactor (DPFR) and a continuous stirred tank reactor (CSTR) upstream of a column, and a second DPFR downstream of the column, geometric measurements associated with the second DPFR; generating, by means of a processor, transport model parameters for a transport model associated with the second DPFR based on the geometric measurements; feeding a tracer molecule into the chromatography machine; capturing one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine;and estimate, using the processor, based on the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more transport model parameters for a transport model associated with the first DPFR and the CSTR. In some examples, the method may include feeding an experimental sample into the chromatography machine; capturing one or more experimental measurements based on the experimental sample moving through the chromatography machine; and estimating, by means of the processor, based on the one or more experimental measurements based on the experimental sample moving through the chromatography machine, one or more transport model parameters estimated for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, one or more adsorption model parameters for an adsorption model associated with the experimental sample. Additionally, in some examples, geometric measurements may include pipe diameter measurements and pipe length measurements associated with the second DPFR. Furthermore, in some examples, one or more tracer molecule measurements are captured based on a chromatogram associated with the tracer molecule moving through the chromatography machine. Similarly, in some examples, one or more experimental measurements are captured based on a chromatogram associated with the experimental sample moving through the chromatography machine. Additionally, in some examples, the method may include identifying, using the processor, the experimental sample based on the adsorption model associated with the experimental sample. IVIA / a / ZUZÓ / UUODO I Furthermore, in some examples, the estimation of one or more adsorption model parameters for the adsorption model associated with the experimental sample may be a first estimate of one or more adsorption parameters for a first adsorption model associated with the experimental sample. The method may also include a second estimation, using the processor, of one or more adsorption parameters for a second adsorption model associated with the experimental sample, based on a range associated with the first binding parameters for the first adsorption model associated with the experimental sample. Additionally, in some examples, the method may include identifying the experimental sample using the processor, based on the second adsorption model associated with the experimental sample. Furthermore, in some examples, the first DPFR and a CSTR before the column, and the second DPFR after the column are part of an inlet flow path of the chromatography machine, and wherein the chromatography machine further includes a sample flow path having a first DPFR and a CSTR before a sample flow path column, and a second DPFR after the sample flow path column, and the method steps are further performed for the first DPFR and the CSTR before the sample flow path column, and the second DPFR after the sample flow path column. Additionally, in some examples, the experimental sample is a first experimental sample, and the method may further include feeding a second experimental sample into the chromatography machine; capturing one or more second experimental measurements based on the second experimental sample moving through the chromatography machine; and estimating, by means of the processor, based on the one or more second experimental measurements based on the second experimental sample moving through the chromatography machine, one or more transport model parameters estimated for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, one or more adsorption model parameters for an adsorption model associated with the second experimental sample.For example, in some cases, the second experimental sample is different from the first experimental sample. In addition, in some examples, the transport model parameters include one or more of the following: DPFR dispersion coefficient, DPFR volume, DPFR cross-sectional area, and CSTR volume. Furthermore, in some examples, the adsorption model parameters include one or more of the following: adsorption coefficient, desorption coefficient, characteristic charge, and shielding factor. IVIA / a / ZUZÓ / UUODO I Additionally, in some examples, the method also includes estimating, using the processor, based on the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR, and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more parameters of the column-specific transport model for a column-specific transport model associated with the chromatography machine column. Furthermore, in some examples, the column-specific transport model parameters include one or more of the following: column porosity and column dispersion. Furthermore, in some examples, the method also includes estimating, using the processor, based on the column-specific transport model, the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR, and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more resin transport parameters for a resin transport model associated with resin particles in the chromatography machine. Additionally, in some examples, the resin transport parameters include one or more of the following: film transport coefficient for each component and pore porosity. Furthermore, in some examples, estimating one or more adsorption model parameters for an adsorption model associated with the experimental sample is further based on one or more of a column-specific transport model or a resin transport model. Additionally, in some examples, the tracer molecule is dextran. In some examples, the tracer molecule is NaCl. Furthermore, in some examples, the tracer molecule is a DNA molecule. Additionally, in some examples, the tracer molecule is a nanoparticle. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1A illustrates a representation of the flow path of a chromatography machine (e.g., such as the ÁKTA™ Avant). As shown in Figure 1A, the flow paths from different locations, depending on the line priming procedures, indicate varying time points for when the sample or elution buffer enters the column. Figure 1B illustrates an example of how the mobile phase moves through a packed bed column. Figure 1C illustrates a visualization of the steric mass action model, including the meaning of the parameters. IVIA / a / ZUZÓ / UUODO I Figure 2 illustrates a graph of a sum-of-differences-of-squares (SSD) alignment problem using synthetic dice, according to some examples. Figure 3 illustrates a graph comparing conventional position penalty metrics and initially reduced position penalty metrics, according to some examples. Figure 4 illustrates a step-by-step approach as an example for model development. Figure 5 illustrates a diagram of the flow paths used for gradient elution, according to some examples. Figure 6 illustrates a representation of a chromatography machine system for a CEX model. The sequence of unit operations for a general model construction includes the dispersed plug flow reactor (DPFR) and the continuous stirred tank reactor (CSTR). These represent peak delay and peak broadening induced in the system, excluding the column (valves, sensors, mixing chamber). Salt and molecules could experience different valve or tubing systems, so the parameters associated with these unit operations could vary between proteins and salt species. Figures 7A and 7B illustrate the calibration of pre-column models for transport within the chromatography skid. Figure 7A illustrates the advance of the dextran tracer from the sample pump, while Figure 7B illustrates the NaCl tracer from the inlet pump. Figures 8A, 8B, and 8C illustrate the calibration of the column transport model parameters. Figure 8A illustrates the advancement of the dextran tracer (which does not penetrate the pores), Figure 8B illustrates the dextran pulse tracer (which does not penetrate the pores), and Figure 8C illustrates FVIP for the advancement of a BiTE® antibody construct (bispecific T-cell coupler) (which does not penetrate the pores). Figures 9A and 9B illustrate a comparison of experimental and simulated chromatograms and fractionation datasets under elution conditions. Figure 9A illustrates the best fit for a 5 mM / CV gradient run, while Figure 9B illustrates the best fit for an 11 mM / CV gradient run. Figures 10A, 10B, 10C, and 10D illustrate the validation of the model in process-scale experiments with varying loading factor, gradient, and collection stop criteria. Figure 10A illustrates the validation of the model for a gradient of 8 mM / CV and a loading factor of 25 g / L in a benchtop-scale column. Figure 10B illustrates the validation of the model for a gradient of 8 mM / CV and a loading factor of 25 g / L at process scale with a collection stop criterion of 30%. Figure 10C illustrates the validation of the model for IVIA / a / ZUZÓ / UUODO I a gradient of 8 mM / CV and a loading factor of 15 g / L at process scale with collection stop criteria of 45%. Figure 10D illustrates the validation of the model for a gradient of 8 mM / CV and a loading factor of 5 g / L at process scale with collection stop criteria of 30%. Figure 11 illustrates a flowchart of a method as an example, according to some examples provided in this document. DETAILED DESCRIPTION DEFINITIONS As used herein, “chromatography” refers to a method for the separation and purification of molecules in a downstream process. As used herein, “ion exchange chromatography” (“IEX”) refers to a chromatography step based on ionic interactions between the molecule and the chromatographic material. As used herein, “cation exchange” (“CEX”) refers to a chromatography step based on cationic interactions between the molecule and the chromatographic material. As used herein, “anion exchange” (“AEX”) refers to a chromatography step based on anionic interactions between the molecule and the chromatographic material. As used herein, “ÁKTA™” refers to the product name of the Cytiva™ chromatography machines. As used herein, “dispersed plug flow reactor” (“DPFR”) refers to a cylindrical tube with convective laminar flow with a plug flow profile instead of a smooth slope flow profile. As used herein, “continuous stirred tank reactor” (“CSTR”) refers to a tank filled with fluid that is thoroughly mixed. As used herein, “mixing chamber” refers to a small tank (-1-10 mL) for mixing buffer solutions in a chromatography machine, such as the ÁKTA™. As used herein, “valve” refers to a valve that couples all the parts in a chromatography machine such as the ÁKTA™. These valves can switch between numerous inlets and outlets and have a small volume (-1-1000 pL). As used herein, “model calibration” refers to a procedure for estimating unknown parameters of the model. IVIA / a / ZUZÓ / UUODO I As used herein, “adsorption” refers to the process by which a solid holds a molecule of a gas or liquid or solute as a thin film. As used herein, “dead volume” refers to gaps within the chromatography machine that are filled with liquid. As used herein, “line priming” refers to filling tubes / valves with the necessary fluid (e.g., buffers) before the process begins. As used herein, “tightening” refers to the procedure of recursively estimating the unknown parameters of the model by comparing the simulation with the measured values. As used herein, “elution” refers to the process of removing adsorbed molecules from the chromatographic material. THEORY In general, the column model provided herein describes important transport and adsorption effects of counterions and molecules present in the column in both their solute and adsorbed states. A variety of adsorption models can be selected, but the most important one describes how salt ions, as mobile phase modulators, influence the adsorption kinetics of molecules. Such processes are described on a timescale of seconds and on length scales of micrometers and meters. A simplified general velocity model (GRM) and the spherical mass action model are used to simulate the transport and elution behavior of acid, major, and basic charge variants of the target molecule and high molecular weight (HMW) impurity species. TRANSPORT MODEL As shown in Figure 1B, the mobile phase moves through a packed-bed column of length L. The movement can be described by the volumetric flow rate, the cross-sectional area of ​​the column, and the column porosity (void ratio between packed layers). This movement is forced convection and is described by a convection term along the column length z. There are interstitial volume effects (void between beads in the liquid mobile phase), such as Fickian diffusion, wall effects, or molecules moving along different paths through the bed, leading to peak broadening. These peaks cluster together and are described in the model by a dispersion term.The flow of molecules from the interstitial volume to the porous particle volume of the bed can be hindered, which is considered by the model using a film mass transfer model. IVIA / a / ZUZÓ / UUODO I Within the pores of the beads, diffusion transport is assumed to be much faster than mass transfer to the pores and is therefore assumed to be instantaneous. Consequently, no concentration gradients are present along the bead radius, leading to simplified general velocity model (GRM) equations with clustered pores, the clustered-pore diffusion model (COP). The model describing transport is independent of the model describing the adsorption of molecules to the adsorbed volume, or solid phase s. The column packing and bead sizes are assumed to be highly homogeneous, leading to the assumptions that, firstly, a concentration gradient in the radial direction of the column is neglected, and secondly, the porosity of the packed bed and that of the beads does not vary along the column axis and bead radius. The GRM is used to describe the transport of biomolecules through a porous packed bed due to convection, scattering, and diffusion. Convection is the movement of a species through a system due to the movement of the surrounding bulk fluid. Scattering describes the broadening of peaks due to sample viscosity, collisions with tube walls or particles, and displacement through a packed bed. Additional mass transfer: Transport from the bulk liquid to the pore and diffusion within the pore are described by the GRM. Diffusion into the pore from the bulk liquid can be a rate-limiting step due to spatial confinement, depending on the molecule-to-pore size ratio. The rate at which this transport occurs is described as proportional to the film mass transfer coefficient.The second component is diffusion through the pore, which is described by the diffusion coefficient of the respective species. This typically occurs more rapidly than film mass transport. Due to this assumption, the GRM equations are simplified by neglecting a radial concentration gradient in the pores, leading to the clustered-pore model. The simplified GRM for any species concentration is c¿e [0, Nc], where Nc denotes the number of components, t denotes the time in s, z denotes the axial coordinate along the length of the column L, and r denotes the radial coordinate of the bead along IVIA / a / ZUZÓ / UUODO I of the radius of the pearl rp. (1)rp dedt (2) The Danckwerts boundary conditions apply: iíc'ín(t,z = 0) = ucli(t,z = 0)-Dax^ (t,z = 0) (3) ^(t,z = L) = 0 (4) The initial conditions are set at: Λ / a / ZUZÓ / UUODO I c'(t = 0,z) = c'o (5) cf(t = 0,z) = cfo (6) c?(t = 0,z) = cf0 (7) l, p, and s denote the reference volumes of interstitial volume (void between the beads), pore volume (void within the beads), and solid volume (solid portion including adsorbed molecules). The interstitial flow rate is denoted by u in m / s, the axial dispersion coefficient is denoted by Dax in m² / s, the column porosity (ratio of interstitial void volume to total bed volume) is denoted by sc, the bead porosity (ratio of liquid pore volume to total bead volume) is denoted by ερ, and the film mass transfer coefficient is denoted by kfi in m / s. A dispersed plug flow reactor (DPFR) model was used to describe the transport of molecules in the tubes. The equations for a DPFR describe convective laminar flow in a cylindrical tube, neglecting the flow profile. The model equations are described by: In this case, c-uh denotes the concentration inside the tubes, utuby D^b denote the interstitial velocity and the dispersion coefficient inside the tubes. A continuous stirred-tank reactor (CSTR) model was used to describe the mixing effects within the valve and mixing chamber chromatography system. That is, a tank model with inlets and outlets and constant volume is used to describe how the concentrations of multiple components change when they are mixed together. The model equations are described by: dci” _1y ^inside _ yWoutraym^w = l '< in,wLin,i,w ^w=l V out'WLi,w The initial conditions are given by cfub(t = 0,z) = c$b(10)C¡m(ü = 0) = c™ (11) (9) In this case, cf denotes the concentration inside the tank, Vmen m3 denotes the volume of the tank, Qintro,w and Qfuera denote the volumetric flow rate in m3 / s inside and outside the tank. MODEL OF STEREO MASS ACTION The spherical mass action (SMA) model describes the adsorption, desorption, and spherical interactions of a protein with a resin. These interactions with the resin are characterized by four parameters: an adsorption coefficient, a desorption coefficient, a spherical factor, and a characteristic charge. These parameters are specific to each modeled species. Unlike transport parameters, binding parameters are translatable across different scales and systems as long as the resin and the molecule are the same. The adsorption and desorption coefficients describe the rate at which molecules adsorb (bind) and desorb (elute) from the resin. They describe the binding kinetics of the modeled species and are affected by the salt concentration. The spherical factor and characteristic charge parameters describe the influence of a molecule's size and surface charge on binding kinetics, respectively. The spherical factor describes the number of spaces a molecule "shields" from binding on the resin. The characteristic charge describes the number of charge-based interactions a molecule has with a resin. The sum of the spherical factor and the characteristic charge quantifies the total number of binding sites occupied by the molecule on the resin. A visualization of these four factors is shown in Figure 1C. In addition to the spherical factor and characteristic charge, the ionic capacity of the resin is required to determine the protein-to-resin binding capacity. The SMA model is based on a neutral charge equilibrium on the resin, such that all charged sites are occupied by either counterions or an oppositely charged protein. The ionic capacity can be determined from a titration experiment, which is detailed in the Methods section. The system of equations of the SMA model is described as follows: IVIA / a / ZUZÓ / UUODO I - L· rPrsVi - k cpVics gtc0 -d.Ao Cí (12) A = c0 + Σ^Οί + a,)c? (13) cos = o + (14) c?(t = 0, z, p) = c?0 (15) In this case, A in mol / m3 denotes the total capacity of the column, v¡ denotes the characteristic load, a¡ denotes the spherical factor, kai in m3 / mf s, kdi in 1 / s, c0 denotes the concentration of counterions available for the adsorption of (unprotected) molecules, cj is the total amount of counterions adsorbed to the resin. A key limitation of the SMA isotherm is that the bonding parameter values ​​are only applicable at the pH at which they were determined. This limits the robustness of a predictive mechanistic model since pH is an important process parameter in the CEX stage. To expand the current SMA model and improve its robustness, a function that can address the pH dependence is required. Experiments were performed to calibrate the model according to the conventional process, the only difference being the pH of the buffers and the loading material. NUMERICAL SOLVERISTS Numerical solutions are calculated for all the above systems of equations since analytical solutions are not available. The general approach is to discretize the partial differential equations (PDEs) along the special variable, resulting in several ordinary differential equations (ODEs). The ODE systems can then be solved using time-step algorithms. Special discretization of chromatography models can be achieved using a finite volume method. In this case, mass equilibrium equations are set up for several uniform finite volume elements along the column length and bead radius. The average concentrations in each cell volume are calculated, resulting in a new set of state variables associated with each cell volume. Numerical integration involves solving the discretized equations for the averaged volume concentrations. The boundary conditions at the column inlet and outlet (equations (3) and (4)) are integrated into the discretized equation at the first and last finite volumes. The weighted essentially non-oscillatory (WENO) scheme is used to approximate the concentrations at the finite volume boundaries. The time-step solver, or ODE solver, used in this case is part of the CADET framework and uses a back differential formula (BFD) method for time integration. The CADET implementation uses the implicit differential-algebraic (IDA) solver contained in the SUNDIALS package. The solver tolerances were set with Abstol at 10⁶, Algtol at 10⁻¹, Reltol at 10⁻¹, and the initial stage size as 10⁻⁴. PARAMETER ESTIMATION ALGORITHM GOAL SYSTEM This disclosure introduces a new goal system for estimating chromatography model parameters. In this context, "goal" refers to a set of metrics. IVIA / a / ZUZÓ / UUODO I shape-sensitive. Each metric is a single scalar value, such as the time difference between the simulation and the peak measurement, the height difference at the peak, etc. Metrics are defined based on specific knowledge of the modeled process and typical errors in the measurement data. Metrics in a goal can be passed to a multi-goal search algorithm or combined into a single goal and passed to a single-goal search algorithm. Multiple metrics can guide multi-objective search strategies to the desired optimum much better than single-objective sum of squared differences (SSD). Metrics are grouped into scores to organize the goal specification for different parameter estimation procedures in CADET-Match. A suitable goal should have the property that, as the quality of fit improves, the value of at least one metric decreases, and as the quality of fit worsens, the value of at least one metric increases. A goal lacking this property can lead search algorithms in the wrong direction. This might seem trivial, but it is critically important and is the reason why new goals had to be created. Due to competitive binding and other complex mechanisms, many model parameters influence simulated chromatograms in linearly uncoupled modes.Therefore, some commonly applied metrics, such as SSD, may increase as the model parameters approach their correct values. Furthermore, chromatograms measured from large-scale industrial applications are often affected by systematic errors such as pump delays, which can cause a time lag between the measured and simulated signals, unless the model captures the cause of the delay, which is often not possible in practice. A good target should take this into account, since otherwise, the simulated peaks end up in the correct location but with the wrong shape. Wrong shapes generally indicate errors in the underlying physics of the model. Therefore, good metrics should favor peaks with a nearly perfect shape but small lags over peaks with no lag but with wrong shapes. SUM OF DIFFERENCE OF SQUARES For the SSD, the squared differences between the simulated and measured chromatograms are summed across the time points (see Equation 16 below). For the normalized root mean square difference (NRMSD), the SSD is divided by the number of time points before taking the square root and dividing the result by the maximum of the measurement data (see Equation 17 below). Due to the monotonicity of this transformation, SSD and NRMSD have the same minima. These metrics can IVIA / a / ¿U¿ó / UUODO I can be applied to the entire chromatogram j = {1; ...;Nd}, or to a subset of the data, / c {1; ...;Nd}. The SSD is most commonly applied with gradient descent search algorithms. Therefore, it is included herein for comparison. The theory is well established within the framework of estimating the maximum probability for independent and identically distributed random measurement errors. However, these preconditions are not generally valid for modeling large-scale preparative chromatography where systematic errors such as feed variations, pump delays, and flow rate variations typically dominate detector noise. The NRMSD is more suitable than the SSD for interpreting results because the numerator has the same unit as the data and is related to the maximum concentration by the denominator. SSD^Y^ -n / )2(16) hehe Jrn ^·>ει(χM ~ NRMSD (XitY¿j = ¡----- (17) The SSD requires sufficient overlap between the simulated and measured chromatograms to be sensitive to parameter changes and guide the search algorithm toward the optimum. This can complicate the selection of suitable starting points, particularly for sharp and / or small peaks. An additional disadvantage of the SSD is illustrated in Figure 2 using a synthetic example with parameters shown in Table 1. The parameters of scenario 2 are much closer to reality, with only a relatively small deviation in the characteristic load, v, even though scenario 1 has a smaller SSD and would therefore normally be considered a better fit. Furthermore, the peak shape of scenario 2 is more similar to reality but out of alignment. In real-world experiments, such time lags are often caused by pump delays that the mechanistic model cannot explain. In this case, the SSD favors peaks that are in the correct position even when it is obvious to the naked eye that the peak shape is completely wrong. The model can also reproduce the correct peak shape but not in the correct position with a much larger SSD. Since the peak shape is determined primarily by the junction model parameters, the SSD would lead to non-physical parameter values. Therefore, alternative metrics will now be introduced that favor peak shape over position and are less demanding in the choice of suitable starting points. IVIA / a / ¿U¿ó / UUODO I Table 1 Reality Scenario 1 Scenario 2 ka 2.00 2.9e02 2.00 10.0 3.7e03 10.0 V 7.00 9.60 6.00 σ 50.0 99.0 50.0 SSD 4.3e+00 1.5e+01 NRMSD 4.2e-03 7.7e-03 ALTERNATIVE METRICSThe shape and position of a chromatogram are determined by mass transport throughout the entire system, including column and external volumes, and by binding to the functionalized resin. The disadvantages of solid-state diffusion (SSD) are avoided by separately measuring the shape, position, and height of individual peaks without requiring baseline separation. Peak position metrics are sensitive to changes in the respective model parameters, regardless of peak overlaps between simulation and measured data. This provides flexibility and robustness regarding the selection of starting points for search algorithms, which is critically important for automation in industrial applications. Focusing on individual peaks reduces the impact of process variations and additional components not fully included in the model.For example, pump flushes or pressure alarms can cause spurious peaks, and industrial feeds typically contain large numbers of more or less uncharacterized impurities. In such cases, separate metrics can be assigned to distinct but partially separate peaks of target components and high- and low-molecular-weight impurities. Separate metrics also help provide (multi-target) search algorithms with more accurate information about which component impacts which peak. All metrics yield zero for a perfect match between simulation and experiment. BEAK SHAPE The shape metric is the most innovative of the new metrics and a core component of almost all scores. It is the difference between one and the maximum Pearson correlation between measured and simulated chromatograms over a continuous range of time lags (see Equation 18 below). To evaluate this metric, the simulated chromatogram is time-shifted, Yj(t) = Y¡(t - τ'). The maximum in Equation (18) is determined by a search on the initial grid followed by a method of Powell. Although it is advisable for the SSD to simulate the chromatogram on the same grid as the measurement data, continuous time lags require interpolation of the simulated data. This is implemented in CADET-Match using the 5th-order spline. Interpolation can be improved by simulating the chromatogram on a denser grid than that required for the SSD. Allowing continuous time lags that are independent of the discrete measurement grid is crucial for creating a smooth metric. However, it complicates the analytical monitoring of parameter sensitivities. The shape metric is typically applied to individual peaks that are clipped from the chromatogram. As required, it is zero for perfect match between simulation and experiment and increases for poorer agreement.By design, this metric only takes into account shape similarity and requires two other metrics to measure time lag and height difference between simulation and measurement data. (cov(X,, Form(X¿, K,), = 1 - maxI-----------) (18)τ\σΧί,;σητ,7 / PEAK POSITION The position metric can be more complex than it first appears. It is based on the time lag, ts, obtained from maximizing equation (19), below. / cov(X¿, Kτ) Λ LA,T)}= arg max --------- (19) τ \ AL'j / The conventional position metric provides an immediate penalty for a time lag, with a linear increase to one when out of alignment by tr (see Equation 20 below). In this case, tr is the length of the measurement time interval. It can be replaced by the retention time of a non-binding tracer if sufficient starting points are provided to the search algorithm. As will be shown in the Results section, this metric is a good choice for estimating column and particle porosity. However, it requires great care when running experiments to ensure that there are as few delays as possible and that alarms are canceled immediately. Such delays affect the chromatogram almost exactly as changes in column and particle porosities do.As discussed earlier, in the presence of such delays, it may be advantageous for the parameter estimation procedure to compromise between aligning the simulated and measured peaks while also matching their shape and height. Therefore, an alternative position metric is introduced that initially reduces the penalty by 1 / 2 in intervals of less than 1 / 10 try, then increases linearly to one when out of alignment by tr (see section 1). IVIA / a / ZUZÓ / UUODO I equation (21), below. Figure 3 illustrates the difference between the conventional and initially reduced position penalty metrics. The initial reduction, 1 / 2, and interval, 1 / 10, are chosen by experience and can be changed by the user. ίΛΧ,,YA, PositiontXi.Yi)j =-------- (20) PositionfX,·, YOj tsíXi.Yj),<£ tr' tr“10 19^,19; _ £ ts(Xi,Ydj £ 18 tr18' tr>10 PEAK HEIGHT (21) The peak height metric refers to the maximum concentration of the simulated chromatogram relative to that of the measured data (see equation (18)). This metric increases to one when the difference in either direction is greater than 100%. Height(Xi, Y¿)j — maxY;, 1-^—í maxX,· ,· je; (22) COMBINED SCORES The metrics previously introduced serve as building blocks for creating scores that quantify the difference between simulated chromatograms and measurement data. Scores are defined for individual components and can target the entire chromatogram, individual peaks, or parts thereof, such as just the leading edge of a peak. Each score is a set of metrics that depend on the component index and the set of time points considered. Goals will consist of one or more scores that can then be combined into a single target or passed to a multi-target search algorithm. SUM OF DIFFERENCE OF SQUARES The SSD score is the set of differences between simulated and measured chromatogram data, see equation (23) below. For technical reasons, each difference is interpreted as a separate metric. In section 8, a target will be defined as the sum of the squares of these metrics. SssD={Xí.j-YiJ\j^j} (23) FULL BEAK The simplest of the new scores is the combination of shape, position, and peak height, SCauss. This score is typically applied to a time interval, specified by the index set / , that contains a single peak of almost Gaussian shape. IVIA / a / ZUZÓ / UUODO I This time interval is not automatically detected; it must be specified by the user. A more elaborate score, SPeak, also takes into account the shape, minimum, and maximum of the time derivative and is better suited for fitting non-Gaussian peaks. By combining the peak shape with the shape of its derivative, this score is highly sensitive to chromatogram curvature. The time lags in the peak and the slope are not technically required to be equal. In practice, they hardly differ, except in the first iterations of search algorithms with few starting points. The SCaus and SPeak scores are defined analogously with Position(Xi,Yi)j instead of Position(Xi,Yi)j.As will be shown in the results section, the SPeakcon score with conventional position penalty is particularly useful for estimating transport parameters, while the SPeakcon score with initially reduced position penalty is more suitable for estimating bonding parameters. SGauss={ShapeíXi.YOpPositiontX^YOpHeightíXí.YOj}(24) Slope = {Shape^Xi, Y¡)Height(-X¡,-Yí)j·, Height(Xi, ή);](24) Spico FGaussU SPending(25) PEAK FRONT In some cases, only the leading edge of a peak can be used for parameter estimation, while other parts of the peak are degraded by nonspecific interactions of a tracer molecule with the column or tubing. Dextran is a relevant example of such non-ideal behavior, leading to strong tailing and reduced peak height. On the other hand, dextran is commonly used as a tracer that does not penetrate particle pores. Errors in experimental execution can also render the trailing edge of a peak unusable for parameter estimation. These situations can be addressed using a score, sFront, which considers the shape and position but not the peak height. This score is typically used with relatively short time intervals and few data points. SFront={Shape(Xi,Yi')j; Position(Xi,Yi')j] (26) IVIA / a / ¿U¿ó / UUODO I Peak front scoring is designed to extract as much useful information as possible from the chromatogram. Unattended application of this scoring requires automatically determining the useful time interval while robustly removing non-ideal portions with high precision at the cutoff points. For dextran data, the lower end of this interval is chosen at the first inflection point of the measured chromatogram; that is, the upper cutoff point is at the first maximum of the time derivative. Experience shows this to be a good choice since non-ideal interactions primarily impact peak height and tailing. The lower cutoff point is chosen when the measured chromatogram begins to differ from the baseline by more than 0.1% of the concentration at the upper cutoff point. Experience shows that 0.1% is a robust choice for this threshold.The exact positions of these cut-off points are determined using Powell's method on the continuous spline approximation of section 5. The nearest time points from the discrete measurement data are then used as limits of the time interval specified by J. SUBDIVISION DATA Optical detectors typically used to measure chromatograms cannot distinguish between different chemical components. Instead, they provide a single summation signal where the contributions of individual components are weighted by their extinction coefficients. Such signals cannot be used alone for parameter estimation unless the peaks of the relevant components are sufficiently separated. For example, the acidic, major, and basic components of a monoclonal antibody often overlap completely in a single peak. This situation is typically addressed by fractionation, that is, pooling the column effluent into a series of vials. Each of these vials is then analyzed offline to quantify the components of interest, providing additional information for establishing a dedicated parameter estimation score. The metrics introduced above can be generally applied to the concentrations in each vial using the centers of the corresponding collection intervals as time points. For accurate comparison, the corresponding component simulations are averaged over the same collection intervals when a metric is applied to the fractionation data. The resulting information is often sparse, with 5 to 10 fractions per peak, and can be affected by additional errors in fractionation times and volumes. Small shifts in the collection intervals can cause significant changes in the component distribution among the analyzed fractions, particularly for sharp peaks. The smoothing procedure in Section 5 is not suitable for such sparse measurements.However, the spline approximation must be applied to the original simulated data before it is shifted and virtually split to determine the time lag, ts, in order to maintain the accuracy of the. IVIA / a / ZUZÓ / UUODO I subgrid. Based on this offset, the Saussy Speak scores can be calculated, as well as their immediately penalized versions. Similarly, the SSD score can be applied to the fractionation data by averaging the simulations over the collection intervals. SEARCH STRATEGIES The scores described above are defined per component, but they can also be applied to the sum signal. Scores from multiple components can be combined to create a goal for the applied search strategy. It is advisable to use all available information, including the scores in the sum signal. CADETMatch uses two alternative search strategies: gradient descent and a multi-target genetic algorithm. For gradient descent, all metrics must be combined into a single scalar value, while the genetic algorithm can operate on multiple metrics. GRADIENT DESCENT Gradient descent algorithms seek a local optimum for the goal function using derived information about the parameters being sought. Gradient descent has long been used for parameter estimation in chromatography. It is very efficient near the target optimum, but can fail if the goal function is not smooth or if the Jacobian becomes singular. Furthermore, this algorithm is prone to getting stuck at local optima, which may be far from the starting point. This can be avoided using basin hopping or multiple-start strategies. The latter is often applied when refining the results of population-based search strategies. GENETIC ALGORITHM (GA) Genetic algorithms (GAs) are an example of biomimicry. In essence, they function like a bacterial colony that adapts to an external environment and shares many of the same characteristics. Unfortunately, GAs are parallel. An initial population is created, often using quasi-random methods such as Latin hypercube sampling or Sobol sequencing. Each member of the population is then evaluated based on one or more objectives. At the end of each generation, the fittest members survive and reproduce to form the next generation. The next generation is created through a combination of reproduction and mutation of the surviving members. There are variations on this procedure that maintain population diversity, select the members to be included in the next generation, and change how reproduction and mutation are implemented. These variations result in different algorithms such as NSGA2. IVIA / a / ZUZÓ / UUODO I NSGA3 and SPEA2. In view of the no-free-lunch theorem, different variants of GA were tested and optimized on a variety of problems before settling on NSGA2 for single-objective problems and NSGA3 for multi-objective problems. ONLINE MONITORING Building complex models correctly, properly processing experimental data, and determining suitable starting points can be difficult and tedious tasks. Based on experience, it is unlikely that new models or concepts will be successfully implemented from scratch. However, such issues can only be tested by attempting to fit the model to the data. Since errors can often be identified in the early stages of the parameter estimation process, CADET-Match provides functionality to monitor the progress of specific indicators such as peak height, shape, mass, etc. This allows observation of whether the starting points are producing reasonable results and whether the search algorithm is continuously improving the target. Online monitoring allows for timely aborts if progress is slow or if the results are already good enough.This is essential for quickly testing models, goals, starting points, and stopping criteria. Since determining suitable starting points can be difficult, a GA with a fairly large population size is generally a good choice for initial testing. Multi-start gradient search is not a good alternative, as parallel iterative processes are more difficult to monitor. PARAMETER TRANSFORMATION Most search strategies struggle when the search parameters span orders of magnitude or are correlated with each other. Parameter transformation can help mitigate these challenges. CADET-Match provides several transformation rules—that is, biuniic maps between model parameters, p, passed to the chromatography simulator and estimated parameters, p', passed to the search algorithm. These transformations are based on upper bounds, p, and lower bounds, p', of the model parameters. The linear transformation (see Equation 27 below) maps the original interval [p,p] to [0,1]. This is usually sufficient when the upper and lower bounds of the parameters are separated by less than three orders of magnitude. For wider intervals, a nonlinear transformation is advisable (see Equation 28 below). This transformation automatically adjusts the search algorithm's step width to the magnitude of the respective model parameter. Otherwise, the same step could be enormous for one parameter but minuscule for another. IVIA / a / ZUZÓ / UUODO I Ρ = (.Ρ ~Ρ) ·ρ' +ρ (27) Ρ = exp((Zo,g(p) - Ζο^(ρ)) ρ' + logíp)} (28) Nonlinear parameter correlations are difficult to detect and require specific attention. For example, the adsorption and equilibrium constants, ka and keq, are typically much less correlated than the adsorption and desorption constants, ka and kd. The relationship keq = ka / kd allows ka and kd to be passed through the simulator while the search algorithm operates on ka and keq. The corresponding transformation (see Equations 29 and 30 below) also accounts for large parameter ranges. This decouples the binding rate from the concentration equilibrium. ka = exp [(.logíka) ~ logfka)} k'a + log(ka^ (29) exp [(log(£a) - log( / cQ)) k'a+ log^j) kd — f~, (3θ) '^eq SENSITIVITY ANALYSIS The sensitivity of a system to an input at a working point is defined as the derivative of the system's solution. Such an input can be model parameters or feed concentrations. Sensitivity measures how much a change in a certain model input affects the model's output. The sensitivity of any model input can be calculated using the CADET library and written to the .h5 output. CADET uses algorithmic differentiation. Numerical approximations of the analytical derivative can be obtained using a finite difference method, which is also available in the MoChA tool. PARAMETER IDENTIFICABILITY Model parameters can generally be obtained through measurement, such as using optical microscopy to determine pore transport coefficients. However, direct measurements of model parameters are often not feasible or are very laborious; therefore, recursive parameter estimation is a frequently used method in chromatography modeling. Parameter identifiability describes the ability to find model parameters that are unique for a given input and a known model equation. In chromatography, model parameters have poor identifiability when different sets of parameters result in the same chromatogram, or simulated chromatograms show only slight deviations. This makes parameter estimation challenging. Parameter identifiability can A / a / ¿U¿ó / UUODO I visualize outlining the calculated target value for the quality of fit assessment for a specific parameter value. MATERIALS AND METHODS A novel modality, a BiTE® (bispecific T-cell coupler) antibody construct, is used as an example to illustrate the methodical approach presented herein. The molecule is an extended-half-life BiTE®, a member of a class of bispecific antibodies that holds great promise for the treatment of cancer and other serious diseases. The BiTE® molecule was produced using Chinese hamster ovary cell lines developed by Amgen and then captured via a protein A affinity step. The BiTE® was captured via a protein A affinity step followed by viral inactivation and deep filtration, resulting in a filtered inactivated virus pool (FVIP). Sequentially, virus inactivation was applied by adding 1 M formic acid. The pool was neutralized to pH 5.0 using 2 M Tris base, resulting in an unfiltered virus inactivated pool (nVIP). The nVIP was filtered using a Millistak+® HC Pod depth filter, resulting in a filtered virus inactivated pool (FVIP). The experiments were performed using an ÁKTA™ Avant 150 chromatography machine with Unicom 7.3 software. Online measurements of pH, UV radiation, and conductivity were taken using the conventional UV, pH, and conductivity probes installed in the chromatography machine. To confirm the readings, offline measurements of UV radiation, pH, and conductivity were taken using a SoloVPE instrument (C Technologies, Inc.), a pH probe (ThermoFisher Scientific), and a conductivity probe (ThermoFisher Scientific), respectively. WORKBENCH EXPERIMENTS For benchtop runs, Capto SP ImpRes (Cytiva™) resin was packed into a Millipore Vantage column (1.15 cm diameter and 20.7 cm height) with a compression ratio of 1.11, resulting in a column volume of 21.5 mL. The column skewness factor was 0.91 and the HETP was 0.0203 cm. Dextran Blue 2000 (Cytiva™) solution was used as a tracer molecule for dead volume and dispersion experiments. A 2 L solution was prepared for the dextran advance and pulse experiments at a concentration of 0.1 g / L dextran in MilliQ water. The ionic capacity of Capto SP ImpReS resin was obtained using a titration method introduced by Thiemo Huuk in 2016. A small column with a CV of 1.7 mL packed with the same batch of CaptoSP ImpRes was exposed to 0.5 mM hydrochloric acid. IVIA / a / ZUZÓ / UUODO I (HCl) for -500 CV. The column was then washed with water to remove the remaining acid. The column was exposed to 0.01 M sodium hydroxide (NaOH). Finally, the volume of NaOH required to completely exchange the counterions was used to determine the ionic capacity of the column according to Thiemo Huuk 2016. The filtered viral inactivated pool (FVIP) material was used for the benchtop chromatography runs. The feed material was from the same production batch, and the feed characteristics are shown below in Table 2. IVIA / a / ZUZÓ / UUODO I Table 2 Execution: Concentration (mg / ml) Conductivity (mS / cm) pH 8 mM / CV 5.06 7.18 5.05 5 mM / CV 5.08 6.15 5.02 11 mM / CV 4.79 6.32 5.03 Stepwise elution 4.8 6.93 5.05 Non-binding pulse 4.25 44.00 4.92 The conductivity of the feed material for the junctionless pulse was adjusted using 4 M sodium chloride at 180 mS / cm. All benchtop experiments were run at a flow rate of 2.60 mL / min. A benchtop bypass experiment was performed by connecting the inlet and outlet tubes of the chromatography machine column with a zero-volume connector. Dextran solution was run from the sample pump at a flow rate of 2.60 mL / min until the UV signal stabilized. The same procedure was used with 1 M NaCl solution to determine the retention from the sample pump to the conductivity meter. A dextran pulse-and-advance tracer experiment involving the column was performed. The dextran solution was loaded from the sample line. Initially, 3 CVs of 18% ethanol were flowed through to establish a baseline UV and conductivity reading. 10 mL of dextran were loaded for the pulse experiment, while loading continued until a stable UV signal was achieved for the advance experiment. A protein advancement experiment was performed using an equilibration buffer at a conductivity of 44 mS / cm and the FVIP conductivity-adjusted assembly, such that the column was equilibrated and loaded under non-binding conditions. The column was first equilibrated with the high-conductivity equilibration buffer for 3 CVs and then run with mL of the FVIP conductivity-adjusted assembly. Regeneration was performed after loading to confirm that no protein bound to the column. Gradient elution runs were performed, first equilibrating for 5 CV, then loading FVIP material at 25 g / L of resin. A wash step was performed using equilibration buffer for 3 CV. Elution was performed at gradients of 5, 8, and 11 mM / CV, and a stepwise elution was carried out. The chromatography buffers are listed below in Table 3. Table 3 IVIA / a / ZUZÓ / UUODO I pH / Conductivity Buffer (mS / cm) Component Quantity Anhydrous Sodium Acetate Sodium Chloride Glacial Acetic Acid, 17.4 M Water Equilibrium Buffer 4.98 / 17.1 10.12 g 15.778 g 2.988 mL 2000 mL 4.997 / 17.45 5.06 g 7.89 g 1.494 mL 1000 mL 4.99 / 20 5.060 g 7.889 g 1.494 mL 5.0 / 19.6 10.120 g 15.778 g 2.988 mL 2000 mL Elution Buffer 5.01 / 29.4 12.322 g 29.22 g 2.862 mL 2000 mL 5.009 / 29.16 6.16 g 14.61 g 1.431 mL 1000 mL 5.05 / 31.8 12.322 g 29.22 g 2.862 mL 2000 mL PROCESS ROBUSTNESS EXPERIMENTS For process robustness tests, unfiltered viral inactivated pool (NVIP) material was used. The same batch material was injected at a concentration of 4.80 g / L for varying loads and gradients. All process robustness experiments were performed at a flow rate of 4.33 mL / min. These experiments were conducted using a 44 mL column (1.6 cm diameter, 22 cm length). Unless otherwise stated, collection stop criteria were applied at 30% of the main peak value, halting the collection mode and immediately switching to 100% elution buffer upon detection of this event. This approach is part of the automated control system of the chromatography machine at Unicom. Experiments were performed with proteins using different elution gradients, loading factors, and collection criteria. Table 4 shows the parameter combinations used for the process robustness experiments. Table 4 Experiment Gradient slope (mM / CV) Column loading (g / L) Collection stop (% of peak) Experiment A 8 5 30 Experiment B 10.4 15 30 Experiment C 8 25 30 Experiment D 8 15 30 Experiment E 8 15 45 Experiment F 8 15 15 Experiment G 5.6 15 30 ALGORITHMS The model equations were solved using the Chromatography Analysis and Design Toolkit (CADET), an open-source simulator. This software allows for the combination of variable transport and adsorption models for chromatographic steps, as well as CSTR and DPFR transport models, to describe unit operation sequences. Furthermore, the software provides tools for model-based analysis, such as providing sensitivities with respect to the model's input parameters. The software package is available at https: / / github.com / modsim / CADET. CADET was accessed via a Python interface. Parameter search was performed using the open-source software package CADET-Match, which is based on the CADET engine and is available at https: / / github.com / modsim / CADET-Match. The estimation algorithm used in this tool employs multi-objective maximization, measuring the similarity between simulated and measured chromatograms.The best fit was identified using the result with the highest product of these target values. RESULTS AND DISCUSSION A stepwise approach was followed for the development of the model. This approach is used to separate transport effects, such as peak delay or peak broadening caused by the transport of molecules in tubes and valves, from adsorption effects in the model. This is especially important since the aforementioned transport effects could be grouped into the adsorption coefficient through recursive calibration of the model, which reduces the model's applicability to varying column scales. The stepwise approach is shown in Figure 4. STAGE 1: CAPTURE PROCESS SYSTEM Figure 1A illustrates a representation of the flow path of commonly used chromatographic skids and machines, such as the ÁKTA™. Figure 1A illustrates variable dead volumes upstream of the column. Since this dead volume influences peak shift during elution relative to the start of elution, it is necessary to account for the dead volume in the model. Possible representations range from simple time shifts of when elution begins to mechanistic representations of the volume of incorporated tubing and valves. A simplified representation of the paths experienced by molecules in the chromatography machine is outlined in Figure 5. This combination of unit operations was used to develop a representation of the process required for gradient elution. The system resulting from the unit operations models is configured by combining DPFR and CSTR models with a column model. The model sequence involving multiple unit operations is shown in Figure 4. A piping model and a tank are used to simulate the effects of the pipes, mixer, and valves. The column model equations provided in equations (1) to (7) are used. DETERMINATION OF EXTRACOLUMN TRANSPORT PARAMETERS A combination of tube model equations and valve model equations provided in equations (11) and (12) is used to represent the flow path from the sample pump to the UV sensor and from the inlet pump to the conductivity sensor. All paths are configured as a combination of a DPFR and CSTR model adjusted upstream of the column (inlet pump path (buffer) and sample pump path (molecule)) and DPFR models configured solely from the geometric specifications (length and diameter) provided by the supplier. As described herein, the tube model and mixer / valve models upstream of the column are adjusted to the dextran advance bypass and salt bypass configurations individually. In addition, tube models 6 and 7 are configured according to the ÁKTA™ Avant 150 manual with a 1 mm diameter and lengths of 17 cm and 10 cm, respectively.The dispersion coefficients for those two models fit the estimated dispersion coefficient for the derivation experiments. IVIA / a / ZUZÓ / UUODO I STAGE 2: CAPTURE TRANSPORT AND COLUMN TRANSPORT PROPERTIES DETERMINATION OF SUBTRACT PARAMETERS The parameters describing transport behavior outside the column can be isolated from adsorption effects by running dextran and salt tracer experiments that bypass the column. The column's inlet and outlet tubes are connected by a zero-volume connector so that the column is removed from the flow path for the bypass experiments. The flow paths important for gradient elution runs are shown in Figure 6. A list of the parameters estimated at this stage is provided. Table 5 provides a list of names, descriptions, and parameter ranges estimated for each trajectory experienced by the molecules. The ranges correspond to the limits used for the estimation algorithm. Table 5 IVIA / a / ZUZÓ / UUODO I Parameter Description Interval ntub ax Dispersion coefficient in DPFR 10-20 — 10-3 m2 / s yDPFR Volume of DPFR 10“8 — 1 m 3 A Cross-sectional area of ​​DPFR 10“10- lm2 ym Volume of CSTR 10-10 - 10“3m 3 It is important to note that dextran has exhibited non-ideal behavior. Possible causes include interactions with the tube walls or pressure deviations caused by strong density gradients. This non-ideal behavior can be observed at various inflection points in the rise of a chromatogram (see Figure 7A). Due to this behavior, the parameter estimation algorithm adjusts the simulations to the beginning of a chromatogram's front or peak and disregards subsequent points. In CADET-Match, this approach is fully automated. The results of this model calibration stage are shown in Figures 7A and 7B. Final parameters are provided in Tables 6A–6C. In Figure 7A and Figure 7B, the effect of a tracer molecule such as dextran is evident and shows that fitting the first ascending portion of the dextran advance, as shown in Figure 7A, leads to different parameters compared to fitting the entire dextran advance. The small pre-peak in the dextran advance rise is assumed to be an artifact and has been largely ignored by the estimation algorithm. The fit in Figure 7A provides the model parameter for the unit operations of the sample path (reference Figure 6). The calibration result in Figure 7B indicates the model representation of the inlet path (reference Figure 6). Due to the near-ideal behavior of the salt tracer, only negligible deviations are observed between the fitted model and the conductivity signal. Table 6A IVIA / a / ¿U¿ó / UUODO I Trajectory [m2 / s] VDPFR [mi] Vm [mi] Sample trajectory 5.32E-14 0.68 0.09 Inlet trajectory 3.66E-14 1.58 5.35 Table 6B Dax [m2 / s] [-] £p [-] [m / s] 7.5E-14 0.38 0.86 1.87E-6 Table 6C Component ka [1 / 5] kd [1 / s] v[-] σ[-] Acid charge variant 0.36 0.042 12.322 8.98 Main target molecule 1.74 0.099 12.54 11.56 Basic charge variant 9.33 0.566 14.84 8.56 HMW 0.23 0.002 23.02 25.25 DETERMINATION OF COLUMN TRANSPORT PARAMETERS Column-specific transport parameters were determined using dextran advance and pulse experiments performed with the column in-line. Knowledge of the retention and dispersion volumes caused by the tubes before and after the column allows for more precise parameter determination for column-specific transport effects. Dextran Blue 2000 acts as a tracer molecule that does not penetrate the pores for Capto SP ImpReS; therefore, in this first stage, the column's interstitial transport is calibrated. The transport parameters estimated from the dextran experiments are provided in Table 7. Table 7 lists the parameters subjected to interstitial transport calibration. IVIA / a / ZUZÓ / UUODO I Table 7 Parameter Value / range Note «c 0.2 - 0.4 [-] Column porosity D <ix 10"20- 10"12m2 s dispersión de la columnaAgain, the CADET-Match tool was used for this estimation stage. The final adjustments are shown in Figures 8A and 8B. The simulated tracer advance and pulse have been fitted to the dextran experiments on the chromatogram fronts and indicate very good fits. The resulting model parameters are provided in Tables 6A-6C. As a next step, the transport parameters related to the resin particles are estimated. These parameters refer to the mass transport from the interstitial volume to the particles and the particle porosity. Table 7 provides the parameters and their ranges used for the estimation procedure. The result of the particle transport calibration step is shown in Figure 8C. The resulting model parameter values ​​are provided in Tables 6A-6C. Table 7 Parameter Value / range Note kf 10"12 - 10"2 m / s Film transport coefficient for each component ερ 0.3 - 0.85 [-] Pore porosity Table 7 provides a list of names and parameter ranges subject to particle transport. The non-ideality at the top of the chromatogram in Figure 8C is caused by a pressure warning due to the column being loaded to full capacity and is ignored for parameter estimation. Again, the estimation engine only uses the first part of the leading edge. For this stage of model calibration, FVIP material is used, which contains both the target molecule and impurities. Therefore, the parameters obtained from this stage describe pore transport for the actual molecules used, not for the tracer molecules. STAGE 3: QUANTITATIVE CAPTURE ADSORPTION BEHAVIOR The description of the unit operations system model of the chromatography machine can now be used in step 3 to determine the adsorption behavior of the target molecules. For the experimental chromatography, runs were performed with different gradient slopes (5 mM / CV, 8 mM / CV, 11 mM / CV, and a "step-through" elution). The elution volume was fractionated at 0.5 CV intervals, and the fractions were analyzed by analytical size exclusion chromatography (SEC-HPLC) and cation exchange chromatography (CEX-HPLC) to determine the content of the molecules of interest. MODELED SPECIES Fractionation data were used to determine the modeled species. The area of ​​interest in the target chromatogram is the main peak and the following peak containing impurities. Based on the analyzed fractions, the main peak is represented in the model by grouping all charge variants of the target molecule into three charge variant species: the acidic, the main, and the basic variants. The molecular weight is considered equal for all three variants. Based on the SEC analysis of the fractions, all species contained in the impurity peak are described in the model as a single component. Finally, a five-component model is obtained: one salt species describing sodium as counterions, and four protein components describing the acidic, main, and basic charge variants, as well as the high molecular weight (HMW) component. The model uses the concentration of each modeled species in the FVIP material as an input parameter. This means that, for an accurate representation of the molecule species entering the column, analytical experiments are required to determine the protein concentrations in the feed for each modeled species. Due to time constraints, the protein concentrations in the feed were identified using experimental fractionation sets along with UV chromatograms instead of analytical profiles of the feed solution. The ratios between peak areas in the fractionation set and the ratios between peaks in the UV chromatogram are calculated. Then, considering the total protein concentration present in the FVIP material, the concentration of each species in the feed material is calculated. CALIBRATION RESULTS Gradient runs of 5 mM / CV and 11 mM / CV were used to estimate the parallel junction parameters. The UV signal as well as the fractionation data were used for the estimation. The parameters and their estimated ranges are listed in the table. IVIA / a / ZUZÓ / UUODO I Table 9 provides a list of names and ranges of adsorption parameters subject to estimation. Each parameter is estimated for each modeled species. Table 9 IVIA / a / ZUZÓ / UUODO I Parameter Description Range ka Adsorption coefficient 103 - 102 1 / s kd Desorption coefficient 10-8 _ 108 V Characteristic load 1 - 45 [-] σ Shielding factor 1 - 60 [-] The ionic capacity A was set at 1339 mM according to the procedure described earlier in the "Benchtop Experiments" section. The best fits obtained are shown in Figures 9A and 9B. The parameter values ​​of the final adsorption model are listed in Tables 6A-6C. MODEL EVALUATION To measure model predictability for model validation, process performance indicators (PPIs) such as pool volume, yield, and pool concentration were used. Additional gradient experiments were run at both process scale (44 mL column) and bench scale (21.5 mL column) with varying loading and gradients. The model was scaled up by adjusting the column diameter and length to the values ​​specified for the process-scale column. Additionally, the volumetric flow rate was changed from 2.60 mL / min to 4.33 mL / min. All model parameters were kept the same as those estimated by the model calibration. The PPI thresholds for model acceptability are listed in Figure 9. Visualizations of the model predictions and process-scale experimental data are presented in Figures 10A, 10B, 10C, and 10D. Table 10 PPI Range (%) of the experimentally obtained value Yield 91 < experimental value < 109 Volume of the assembly 84 < experimental value < 115 Concentration of the assembly 64 < experimental value < 135 Benchtop and process-scale gradient runs are used for model validation. First, the PPIs for runs at a load factor of 25 g / L are compared, and the results are shown in Table 10. IVIA / a / ¿U¿ó / UUODO I Table 11 Column Volume (mL) Yield (%) Pool Volume (mL) Pool Concentration (g / L) Experiment 21 79.3 82.5 5.1 Model 21 77.8 84 4.9 Deviation (%) N / A 1.9 1.8 3.6 Experiment 44 76.6 179.51 4.72 Model 44 83.1 173.6 5.3 Deviation (%) N / A 9.6 3.2 13.1 As shown in Table 11, the model predicts the PPI for the benchtop-scale process within a 5% deviation range. The model's prediction for the full-scale process run is within a 13% deviation range. Both 5% and 13% fall within the predefined acceptable ranges provided in Table 10. Additional model validation activities include comparing the model's prediction at process scale with variable load factors, gradient slopes, and harvest stop criteria. The results of the model validation are compared in 12 and 13. Table 12 Gradient Load Stop Performance Volume Concentration (mM / CV) (g / L) Collection (%) of the set (%) of the set (g / L) (CV) Prediction 8 15 45 77.4 2.9 4.0 Experiment 8 15 45 72.0 2.8 3.8 Deviation 107.5% 103.6% 105.3% Prediction 5.6 15 30 87 4.6 2.8 Experiment 5.6 15 30 45 3.8 2.9 Deviation 115.5% 121.0% 92.5% Prediction 10.4 15 30 82.7 2.6 4.8 Experiment 10.4 15 30 73.5 2.5 4.4 Deviation 112.5% ​​104.0% 109.1% Prediction 8 5 30 Experiment IVIA / a / ¿U¿ó / UUODO I IVIA / a / ¿U¿ó / UUODO I Table 13 Main Basic (%) (%) 16.9 16.3 18.7 O 00 T- 18.3 17.4 79.7 79.7 77.9 78.9 CJ 00 78.9 Acid (%) 3.2 3.6 2.6 3.1 3.1 3.7 Main SEC (%) 8.66 99.5 99.34 99.5 99.7 99.25 HMW SEC (%) 0.21 0.52 0.65 0.54 0.3 0.75 Pickup Stop (%) 45 30 30 30 Load (g / L) lo LO LO T- LO Gradient (mM / CV) 00 5.6 10.4 00 Prediction Experiment Deviation Prediction Experiment Deviation Prediction Experiment Deviation Prediction Experiment Deviation From Tables 12 and 13, it can be observed that, when varying the loading factor, the model predictions fall within the defined PPI ranges. However, the model appears to be less predictive of yield when the gradient slope varies, even when the gradient speed is close to the values ​​used for model calibration. A trend can be observed with these runs indicating a general overestimation of the yield and stack volume. The reason for the yield overestimation may lie in the general overestimation of the peak area, which is related to a limitation of the model that is a general challenge in mechanistic chromatography modeling. This indicates that the mass loaded onto the simulated column is greater than that measured by the UV detector in the chromatogram, resulting in a peak area deviation of approximately 10%.The model appears to be highly sensitive to protein loading. More precise feed concentrations, which determine feed mass, are not available for this dataset. Table 12 compares portions of the load variants in the collected dataset for the main, basic, and acid load variants between predicted and measured values. All simulated portions deviate between 0 and 16.2% from the experimentally obtained values. These values ​​indicate sufficient accuracy of the model to predict load variant profiles for varying load factors and gradient slopes. This suggests that the model calibration workflow is suitable for model calibration activities for the development of industrial CEX processes. It should be noted that, in order to convert the UV signal from AU units to SI units, a molecular weight must be assigned to each modeled species. SI units are convenient for simulation activities. The molecular weight of the HMW species is not precisely known. Therefore, the amount of HMW species in the feed is also not precisely determined. As a step to improve the accuracy of the model, the molecular weight of the HMW could be considered more precisely. Further improvement in model accuracy can be achieved by combining the UV280 and UV300 measurement signals in a more integrated manner. It is observed that the maximum value of the main peak in some of the presented chromatograms is very close to the upper end of the linear range of the UV280 signal. This could contribute to the observed deviations in the peak area of ​​the chromatograms and simulated peaks. In this case, a flexible data integration process for combining UV280 and UV300 data signals would be valuable for improving model accuracy. IVIA / a / ZUZÓ / UUODO I SUMMARY AND CONCLUSION A model development approach for a pharmaceutical therapeutic molecule was introduced that suggests a step-by-step method of calibrating parts of the model in three stages to build the final model. This approach can be used to decouple the transport model descriptions from the adsorption model descriptions. A model representation was developed for a chromatographic system containing multiple unit operations. The system describes the paths that either sample molecules or buffer molecules undergo during a salt elution step, both of which involve valves and tubing separate from the column. In the next step, model parameters describing interstitial column transport are estimated. The following step estimates parameters describing pore mass transfer. Finally, adsorption model parameters are estimated in the last stage of model calibration. A separate set of experiments is used for each parameter estimation step. The model's accuracy is evaluated using experiments performed at different column scales. The model predicts elution behavior for process robustness experiments with acceptable quality and can be used to explore process parameter ranges, such as gradient slope, loading factor, and collection stop criteria. Therefore, it can be concluded that the suggested workflow includes the necessary steps for calibrating parts of the model for extra-column volumes, in-column transport, and adsorption behavior. The beneficial features of the methods provided herein include: □ A low number of experiments is required for calibration The extracolumn volumes are characterized, and the respective parts of the model can be reused for other molecules. By relative decoupling of the transport and adsorption model calibration, the model reflects changes in equipment and can also be used at different scales. The model predicts the process at a larger scale with acceptable quality, but it reveals general limitations. One such limitation is that the feed concentration of each modeled species is often unknown, requiring additional analytical experiments. For the sake of time and effort, an alternative method for identifying feed concentrations from fractionation data and UV chromatograms is suggested. No previous study has revealed this drawback of mechanistic models or described how to address this issue. This disclosure IVIA / a / ZUZÓ / UUODO I provides an approach for identifying feed concentrations from fractionation data and UV chromatograms that suggests no additional experiments are required. For the sake of time and effort, a minimal set of experiments was used for model calibration and validation. A set of 12 experiments was used for the complete workflow, including the determination of extra-column transport model parameters for model validation. A set of 9 experiments was used for model calibration under binding conditions at a specific pH value. An additional set of 4 experiments was required to estimate the transport model parameters for intra- and extra-column transport. Additional model validation experiments were also used. It is generally known that model acceptability and predictive power increase with the amount of knowledge and experiments used for model calibration.However, with increased experimental effort, the acceptability of integrating model-based methods into platform process development approaches also increases. Therefore, identifying a minimum experimental set for model calibration supports the integration of chromatography modeling into industrial process development activities. Furthermore, the model has been shown to be more sensitive to the amount of loaded protein than experiments suggest. Consequently, the PPIs are often overestimated. In this case, two approaches can help improve the model's accuracy. First, increase analytical effort to improve the characterization of the feed material. Second, include an estimate of the parameter uncertainty in the model evaluation. In this case, the estimated parameters could be investigated with respect to variations in feed material concentration. Such knowledge is valuable for guiding process optimization and process robustness testing. The methods provided herein can be used and reproduced by other modelers, as the solvers and algorithms are publicly available. Furthermore, the model includes a set of unit operations available for use in HDF5 format, and the simulations are ready to be reproduced. The results described herein can be used to further investigate and enhance the capabilities of the model and workflow, making a significant contribution to the advancement of the chromatography community. METHOD AS AN EXAMPLE Figure 11 illustrates a flow diagram of an example method as described herein. For a chromatography machine including a first dispersed plug flow reactor (DPFR) and a continuous stirred tank reactor (CSTR) before From a column, and a second DPFR downstream of the column, geometric measurements associated with the second DPFR (block 102) can be obtained. For example, the geometric measurements associated with the second DPFR can include tube diameter measurements and tube length measurements. These measurements can be obtained, for example, based on tube diameter and tube length measurements, and / or based on specifications from a known manufacturer for the chromatography machine. Transport model parameters for a transport model associated with the second DPFR (block 104) can be generated, for example, by a processor, based on geometric measurements. For example, the transport model parameters might include the dispersion coefficient of the DPFR, the volume of the DPFR, the cross-sectional area of ​​the DPFR, etc. A tracer molecule (e.g., dextran, NaCl, or another suitable tracer molecule) (block 106) can be fed into the chromatography machine. One or more tracer molecule measurements (block 108) can be captured based on the tracer molecule moving through the chromatography machine. For example, one or more tracer molecule measurements can be captured based on a chromatogram associated with the tracer molecule moving through the chromatography machine. Based on the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more transport model parameters can be estimated for a transport model associated with the first (and second) DPFR and the CSTR (block 110). For example, transport model parameters may include dispersion coefficient in DPFR, DPFR volume, DPFR cross-sectional area, CSTR volume, etc. Furthermore, in some examples, one or more parameters of the column-specific transport model can be estimated for a column-specific transport model associated with the chromatography machine column based on the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR, and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine. For example, the parameters of the column-specific transport model may include column porosity, column dispersion, etc. Additionally, in some examples, one or more resin transport parameters can be estimated for a resin transport model associated with resin particles from the chromatography machine based on the column-specific transport model. IVIA / a / ¿U¿ó / UUODO I the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR, and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine. Optionally, an experimental sample (block 112) can be fed into the chromatography machine. One or more experimental measurements can be captured based on the experimental sample moving through the chromatography machine (block 114). For example, one or more experimental measurements can be captured based on a chromatogram associated with the experimental sample moving through the chromatography machine. Based on one or more experimental measurements of the experimental sample moving through the chromatography machine, one or more transport model parameters estimated for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, one or more adsorption model parameters can be estimated for an adsorption model associated with the experimental sample (block 116), for example, using a processor. In some examples, the column-specific transport model and / or the resin transport model may also be factors used in estimating the adsorption model parameters for the adsorption model. For example, the adsorption model parameters may include one or more of the following: adsorption coefficient, desorption coefficient, characteristic charge, shielding factor, etc.In some examples, a second set of adsorption parameters can be estimated for a second adsorption model associated with the experimental sample based on intervals associated with the initially estimated adsorption parameters. In some examples, the experimental sample can be identified based on the adsorption model associated with the experimental sample (and / or based on the second adsorption model associated with the experimental sample). Furthermore, in some examples, the chromatography machine may include an inlet flow path that includes a first DPFR and a CSTR before an inlet flow path column, and a second DPFR after the inlet flow path column, as well as a sample flow path that includes a first DPFR and a CSTR before a sample flow path column, and a second DPFR after the sample flow path column, and Method 100 may be performed for both the inlet flow path and the sample flow path. Additionally, in some examples, blocks 112, 114, and 116 can be performed again, using a second experimental sample (e.g., different from the first sample). IVIA / a / ZUZÓ / UUODO I experimental), but using the same estimated transport model parameters. That is, once a transport model has been developed for the chromatography machine, the transport model can be used to determine adsorption model parameters for multiple experimental samples. ASPECTS 1. A method comprising: obtaining, for a chromatography machine including a first dispersed plug flow reactor (DPFR) and a continuous stirred tank reactor (CSTR) upstream of a column, and a second DPFR downstream of the column, geometric measurements associated with the second DPFR; generating, by means of a processor, transport model parameters for a transport model associated with the second DPFR based on the geometric measurements; feeding a tracer molecule into the chromatography machine; capturing one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine;and estimate, using the processor, based on the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more transport model parameters for a transport model associated with the first DPFR and the CSTR. 2. The method of aspect 1, further comprising: feeding an experimental sample into the chromatography machine; capturing one or more experimental measurements based on the experimental sample moving through the chromatography machine; and estimating, by means of the processor, based on the one or more experimental measurements based on the experimental sample moving through the chromatography machine, one or more transport model parameters estimated for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, one or more adsorption model parameters for an adsorption model associated with the experimental sample. 3. The method of any of aspects 1-2, wherein the geometric measurements include measurements of the diameter of the tubes and measurements of the length of the tubes associated with the second DPFR. 4. The method of any of aspects 1-3, wherein one or more tracer molecule measurements are captured based on a chromatogram associated with the tracer molecule moving through the chromatography machine. 5. The method of any of aspects 2-4, wherein one or more experimental measurements are captured based on a chromatogram associated with the experimental sample moving through the chromatography machine. IVIA / a / ZUZÓ / UUODO I 6. The method of any of aspects 2-5, further comprising: identifying, by means of the processor, the experimental sample based on the adsorption model associated with the experimental sample. 7. The method of any of aspects 2-6, wherein the estimation of one or more adsorption model parameters for the adsorption model associated with the experimental sample is a first estimate of a first one or more adsorption parameters for a first adsorption model associated with the experimental sample, and further comprising: a second estimate, by means of the processor, of a second one or more adsorption parameters for a second adsorption model associated with the experimental sample based on a range associated with the first one or more binding parameters for the first adsorption model associated with the experimental sample. 8. The method of aspect 7, which further comprises: identifying, by means of the processor, the experimental sample based on the second adsorption model associated with the experimental sample. 9. The method of any of aspects 1-8, wherein the first DPFR and a CSTR upstream of the column, and the second DPFR downstream of the column are part of an inlet flow path of the chromatography machine, and wherein the chromatography machine further includes a sample flow path having a first DPFR and a CSTR upstream of a sample flow path column, and a second DPFR downstream of the sample flow path column, and wherein the steps of claim 1 are further performed for the first DPFR and the CSTR upstream of the sample flow path column, and the second DPFR downstream of the sample flow path column. 10. The method of any of aspects 2-9, wherein the experimental sample is a first experimental sample, and further comprising: feeding a second experimental sample into the chromatography machine; capturing one or more second experimental measurements based on the second experimental sample moving through the chromatography machine; and estimating, by means of the processor, based on the one or more second experimental measurements based on the second experimental sample moving through the chromatography machine, one or more estimated transport model parameters for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, and one or more adsorption model parameters for an adsorption model associated with the second experimental sample. 11. The aspect 10 method, where the second experimental sample is different from the first experimental sample. IVIA / a / ¿U¿ó / UUODO I 12. The method of any of aspects 1-11, wherein the transport model parameters include one or more of: dispersion coefficient in DPFR, DPFR volume, DPFR cross-sectional area, and CSTR volume. 13. The method of any of aspects 2-12, wherein the adsorption model parameters include one or more of: adsorption coefficient, desorption coefficient, characteristic charge, and shielding factor. 14. The method of any of aspects 1-13, further comprising estimating, by means of the processor, based on the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR, and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more column-specific transport model parameters for a column-specific transport model associated with the column of the chromatography machine. 15. The method of aspect 14, wherein the parameters of the column-specific transport model include one or more of: column porosity and column dispersion. 16. The method of any of aspects 14-15, further comprising estimating, by means of the processor, based on the column-specific transport model, the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more resin transport parameters for a resin transport model associated with resin particles of the chromatography machine. 17. The method of aspect 16, wherein the resin transport parameters include one or more of: film transport coefficient for each component and pore porosity. 18. The method of any of aspects 2-17, wherein the estimation of one or more adsorption model parameters for an adsorption model associated with the experimental sample is further based on one or more of a column-specific transport model or a resin transport model. 19. The method of any of aspects 1-18, wherein the tracer molecule is dextran. 20. The method of any of aspects 1-19, wherein the tracer molecule is NaCl. IVIA / a / ZUZÓ / UUODO I 21. The method of any of aspects 1-19, wherein the tracer molecule is a DNA molecule. 22. The method of any of aspects 1-19, wherein the tracer molecule is a nanoparticle. 23. A computer system that includes a processor and one or more memories that store instructions that, when executed by the processor, cause the computer system to perform the steps of the method in any of aspects 1-22. 24. A non-transient, computer-readable storage medium that stores instructions which, when executed by a processor, cause the processor to perform the steps of the method in any of aspects 1-22.< / ix>

Claims

1. A method comprising: obtaining, for a chromatography machine including a first dispersed plug flow reactor (DPFR) and a continuous stirred tank reactor (CSTR) upstream of a column, and a second DPFR downstream of the column, geometric measurements associated with the second DPFR; generating, by means of a processor, transport model parameters for a transport model associated with the second DPFR based on the geometric measurements; feeding a tracer molecule into the chromatography machine; capturing one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine;and estimate, using the processor, based on the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more transport model parameters for a transport model associated with the first DPFR and the CSTR.

2. The method of claim 1, further comprising: feeding an experimental sample into the chromatography machine; capturing one or more experimental measurements based on the experimental sample moving through the chromatography machine; and estimating, by means of the processor, based on the one or more experimental measurements based on the experimental sample moving through the chromatography machine, one or more estimated transport model parameters for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, and one or more adsorption model parameters for an adsorption model associated with the experimental sample.

3. The method of claim 1, wherein the geometric measurements include measurements of the diameter of the tubes and measurements of the length of the tubes associated with the second DPFR.

4. The method of claim 1, wherein one or more tracer molecule measurements are captured based on a chromatogram associated with the tracer molecule moving through the chromatography machine.

5. The method of claim 2, wherein one or more experimental measurements are captured based on a chromatogram associated with the experimental sample moving through the chromatography machine.

6. The method of claim 2, further comprising: IVIA / a / ZUZÓ / UUODO I identifying, by means of the processor, the experimental sample based on the adsorption model associated with the experimental sample.

7. The method of claim 2, wherein the estimation of one or more adsorption model parameters for the adsorption model associated with the experimental sample is a first estimation of a first one or more adsorption parameters for a first adsorption model associated with the experimental sample, and further comprising: a second estimation, by means of the processor, of a second one or more adsorption parameters for a second adsorption model associated with the experimental sample based on a range associated with the first one or more binding parameters for the first adsorption model associated with the experimental sample.

8. The method of claim 7, further comprising: identifying, by means of the processor, the experimental sample based on the second adsorption model associated with the experimental sample.

9. The method of claim 1, wherein the first DPFR and a CSTR upstream of the column, and the second DPFR downstream of the column are part of an inlet flow path of the chromatography machine, and wherein the chromatography machine further includes a sample flow path having a first DPFR and a CSTR upstream of a sample flow path column, and a second DPFR downstream of the sample flow path column, and wherein the steps of claim 1 are further performed for the first DPFR and the CSTR upstream of the sample flow path column, and the second DPFR downstream of the sample flow path column.

10. The method of claim 2, wherein the experimental sample is a first experimental sample, and further comprising: feeding a second experimental sample into the chromatography machine; capturing one or more second experimental measurements based on the second experimental sample moving through the chromatography machine; and estimating, by means of the processor, based on the one or more second experimental measurements based on the second experimental sample moving through the chromatography machine, one or more estimated transport model parameters for the transport model associated with the first DPFR and the CSTR, and the transport parameters for the transport model associated with the second DPFR, and one or more adsorption model parameters for an adsorption model associated with the second experimental sample.

11. The method of claim 10, wherein the second experimental sample is different from the first experimental sample. IVIA / a / ZUZÓ / UUODO I 12. The method of claim 1, wherein the transport model parameters include one or more of: dispersion coefficient in DPFR, DPFR volume, DPFR cross-sectional area, and CSTR volume.

13. The method of claim 2, wherein the adsorption model parameters include one or more of: adsorption coefficient, desorption coefficient, characteristic charge, and shielding factor.

14. The method of claim 1, further comprising estimating, by means of the processor, based on the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR, and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more column-specific transport model parameters for a column-specific transport model associated with the column of the chromatography machine.

15. The method of claim 14, wherein the column-specific transport model parameters include one or more of: column porosity and column dispersion.

16. The method of claim 14, further comprising estimating, by means of the processor, based on the column-specific transport model, the transport model associated with the first DPFR and the CSTR, the transport model associated with the second DPFR and one or more tracer molecule measurements based on the tracer molecule moving through the chromatography machine, one or more resin transport parameters for a resin transport model associated with resin particles of the chromatography machine.

17. The method of claim 16, wherein the resin transport parameters include one or more of: film transport coefficient for each component and pore porosity.

18. The method of claim 2, wherein the estimation of one or more adsorption model parameters for an adsorption model associated with the experimental sample is further based on one or more of a column-specific transport model or a resin transport model.

19. The method of claim 1, wherein the tracer molecule is dextran.

20. The method of claim 1, wherein the tracer molecule is NaCl.

21. The method of claim 1, wherein the tracer molecule is a DNA molecule. IVIA / a / ZUZÓ / UUODO I 22. The method of claim 1, wherein the tracer molecule is a nanoparticle.

23. A computer system comprising a processor and one or more memories that store instructions that, when executed by the processor, cause the computer system to perform the steps of the method of any of claims 1-22.

24. A non-transient, computer-readable storage medium that stores instructions that, when executed by a processor, cause the processor to perform the steps of the method of any of claims 1-22.

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