Title - METHOD AND SYSTEM FOR ESTIMATING A MECHANICAL UNCERTAINTY OF A CHROMATOGRAPHY MODEL

Abstract

A non-transient, computer-readable method, system, and means for estimating the mechanical uncertainty of a chromatography model. A mechanical chromatography model comprising a plurality of parameters is given. For each of the plurality of parameters, a corresponding region of values ​​is identified based on a relationship between values ​​for the plurality of parameters. A sample of each parameter from the plurality of parameters is taken within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets. An uncertainty for the mechanical model is quantified using the plurality of simulation sets.

Description

MECHANICAL UNCERTAINTY ESTIMATION OF CHROMATOGRAPHY MODEL Inventors: Jessica Yang Lyall {Redwood City, CA, USA, US nationality} Connor James Thompson {San Francisco, CA, USA, US nationality} Sean Mackenzie Burgess {San Francisco, CA, USA, US nationality} FIELD

[0001] In general terms, this description is directed to the mechanical modeling of chromatography. More specifically, this description provides methods and systems for estimating the uncertainty associated with a mechanical model of chromatography. INTRODUCTION

[0002] In general, chromatography is the primary process used to purify biopharmaceutical products. Mechanical modeling can be used to improve chromatographic processes, investigate problems related to these processes, and perform chromatography simulations. A mechanical model is based on the premise that a complex system or process can be understood by analyzing how the individual parts of the system or process function and how those parts interact. The mechanical model then represents this complex system or process mathematically in a simplified manner that captures the underlying principles. A barrier to the widespread or systematic application of mechanical models to processes such as chromatography can be the difficulty associated with qualifying and establishing confidence in mechanical models.For example, some currently available methodologies for estimating mechanical model uncertainty are more time-consuming and computationally expensive than desired. SYNTHESIS

[0003] In one or more embodiments, a method is provided for estimating the mechanical uncertainty of a chromatography model. A mechanical chromatography model comprising a plurality of parameters is received. For each of the pluralities of parameters, a region of values ​​1 238488 is identified. 1723673 of 26 corresponding to a relationship between values ​​for the plurality of parameters. A sample of each parameter is taken from the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets. An uncertainty for the mechanical model is quantified using the plurality of simulation sets.

[0004] In one or more embodiments, a system is provided for estimating the mechanical uncertainty of a chromatography model. The system comprises a data source configured to obtain a mechanical chromatography model; and a processor configured to receive the mechanical chromatography model from the data source, wherein the mechanical model includes a plurality of parameters. The processor is further configured to: identify, for each of the plurality of parameters, a corresponding region of values ​​based on a relationship between values ​​for the plurality of parameters; generate a sample of each parameter of the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets; and quantify an uncertainty for the mechanical model using the plurality of simulation sets.

[0005] In one or more embodiments, a non-transient, computer-readable medium is provided on which a program is stored, wherein the program is configured to enable a computer to perform a method for estimating the mechanical uncertainty of a chromatography model. The method comprises receiving a mechanical chromatography model comprising a plurality of parameters; identifying, for each of the plurality of parameters, a corresponding region of values ​​based on a relationship between values ​​for the plurality of parameters; generating a sample of each parameter of the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets; and quantifying an uncertainty for the mechanical model using the plurality of simulation sets. 238488 1723673 of 26 BRIEF DESCRIPTION OF THE DRAWINGS

[0006] For a further understanding of the principles disclosed herein, and their advantages, reference is made to the following descriptions taken in conjunction with the accompanying drawings, where:

[0007] Figure 1 is a schematic diagram of a chromatography system according to various embodiments.

[0008] Figure 2 is a block diagram of a model analysis system according to various forms of implementation.

[0009] Figure 3 is a flow diagram of a process for estimating the mechanical uncertainty of a model according to various forms of embodiment.

[0010] Figure 4 is a flow diagram of a process for predicting the mechanical uncertainty of a chromatography model according to various embodiments.

[0011] Figure 5 is a flowchart of a process for calculating a covariance matrix for a mechanical model according to various forms of embodiment.

[0012] Figure 6 is a side-by-side comparison of two different types of multiparameter charts generated for a mechanical model according to various embodiments.

[0013] Figure 7 is an illustration of a table comparing uncertainty values ​​generated using a Hessian approach and uncertainty values ​​generated using a Markov Chain Monte Carlo approach.

[0014] Figure 8 is a block diagram of a computer system according to various forms of implementation.

[0015] It should be noted that the figures are not necessarily drawn to scale, nor are the objects in the figures necessarily drawn to scale in relation to one another. The figures are representations intended to clarify and explain various embodiments of the apparatus, systems, and methods disclosed herein. Wherever possible, the same reference numbers will be used in all drawings to refer to identical or similar parts. Furthermore, it should be noted that the drawings are not intended to limit the scope of these instructions in any way. 238488 1723673 of 26 DETAILED DESCRIPTION I. Overview

[0016] Mechanical modeling is an important tool for understanding how various systems and processes work. For example, mechanical modeling of chromatography can be an important tool for a variety of biological, pharmaceutical, and biopharmaceutical applications. Mechanical modeling of chromatography allows for the non-destructive, real-time measurement of molecular attributes during a simulated chromatographic process (e.g., simulated chromatographic purification), which can provide significant information about the simulated chromatographic process.

[0017] When verified or otherwise justified, mechanical chromatography modeling can describe chromatography in a way that allows for confidence in the interpolation and extrapolation of results generated through mechanical chromatography modeling. Furthermore, mechanical chromatography modeling can support process optimization, real-time process monitoring, control of chromatographic processes, and other operations that provide information about the chromatographic process. Moreover, this type of modeling can support the identification of specific operating parameters that are critical to the chromatographic process.

[0018] Being able to use a mechanical chromatography model with confidence for a given application may require understanding the model's predictive power. For example, it may be important to understand the accuracy of the predictions made by the mechanical chromatography model. Some currently available methods for evaluating such mechanical chromatography models assess or estimate the uncertainties associated with individual model parameters. However, parameter uncertainties may not provide any indication of the overall predictive power of the mechanical chromatography model. Furthermore, some currently available methodologies for evaluating the uncertainty of a mechanical chromatography model are computationally expensive and time-consuming.For example, a currently available method for assessing the uncertainty of a mechanical chromatography model may require days of processing resources (e.g., 10⁵ or 10⁶ function calls). Therefore, the time, costs, and processing resources required for such methodologies may render their use practically unfeasible for certain applications. 238488 1723673 of 26

[0019] Recognizing and considering the importance of having confidence in the mechanical chromatography models used for a given application, the various embodiments described herein provide methods and systems for evaluating a mechanical chromatography model. For example, the various embodiments described herein provide methods and systems for determining the accuracy of a mechanical chromatography model based on estimates of the uncertainty associated with the model. The methods and systems described herein allow for estimating the uncertainty of a mechanical chromatography model more quickly and computationally less expensively compared to at least some of the currently available methods and systems.For example, among others, the various embodiments described herein can allow the calculation of the uncertainty of a mechanical chromatography model in a matter of hours (e.g., less than six hours in some cases) compared to the several days required with some currently available methods. In one or more embodiments, these time savings may be at least partly due to a smaller number of function calls (e.g., 10³ to 10⁴ function calls) compared to the 10⁵ or 10⁶ function calls required with some currently available methods. Therefore, savings in time, costs, and processing resources can be achieved. II. Definitions

[0020] Disclosure is not limited to these example embodiments and applications or to the manner in which the example embodiments and applications operate or are described herein. In addition, the figures may show simplified or partial views, and the dimensions of elements in the figures may be exaggerated or otherwise out of proportion.

[0021] Furthermore, in the manner in which the expressions “on,” “attached to,” “connected to,” “coupled to,” or similar words are used herein, an element (for example, a component, a material, a layer, a substrate, etc.) may be “on,” “attached to,” “connected to,” or “coupled to” another element regardless of whether that element is directly on, attached to, connected to, or coupled to the other element or whether there are one or more intervening elements between that element and the other element. Furthermore, when a list of elements is referenced (for example, elements a, b, c), such reference is intended to include any of the listed elements themselves, any combination of less than the total of the 5 238488 1723673 of 26 detailed elements and / or a combination of all detailed elements. The divisions into sections in the descriptive report are only for the purpose of facilitating review and do not limit any combination of the elements analyzed.

[0022] Unless otherwise defined herein, the scientific and technical terms used in connection with the teachings described herein have the meanings commonly given to them by practitioners of a mid-level skill. Furthermore, unless the context indicates otherwise, singular terms include the plural, and plural terms include the singular. In general, the nomenclatures used in connection with and in techniques of chemistry, biochemistry, molecular biology, pharmacology, and toxicology described herein are those commonly known and used in the prior art.

[0023] As used herein, “substantially” means sufficient to work for the intended purpose. The term “substantially” thus permits minor, insignificant variations from a state, dimension, measurement, absolute or perfect result, or the like, as might be expected by a person of average skill, but which do not noticeably affect overall performance. When used with respect to numerical values ​​or parameters or characteristics that can be expressed as numerical values, “substantially” means within ten percent.

[0024] The term “ones” means more than one.

[0025] As used herein, the term “plurality” may be 2, 3, 4, 5, 6, 7, 8, 9, 10 or more.

[0026] As used herein, the expression “set of” means one or more. For example, a set of items includes one or more items.

[0027] As used herein, the phrase “at least one of,” when used with a list of items, means that different combinations of one or more of the detailed items may be used, and only one of the items on the list may be required. The item may be a particular object, thing, stage, operation, process, or category. In other words, “at least one of” means that any combination of items or number of items may be used from the list, but not every item on the list may be required. For example, among others, “at least one of item A, item B, or item C” refers to item A; item A and item B; item B; item A, item B, and item C; item B and item C; or item A and C. In some cases, “at least one of item A, item B, or item C” refers, among others, to two of item A, one of item B, and ten of item C; four of item B and seven of item C; or some other suitable combination. 238488 1723673 of 26

[0028] As used herein, a “model” may include one or more algorithms, one or more mathematical techniques, one or more machine learning algorithms, or a combination thereof.

[0029] As used herein, an “analyte” refers to a mixture comprising one or more individual components. In the context of chromatography, an analyte may be a mixture whose individual components or molecules are to be separated and analyzed.

[0030] As used herein, “chromatography” refers to a technique or process for separating an analyte into various components (e.g., molecules) of interest. In general, an analyte is dissolved in a fluid (e.g., gas, solvent, water, etc.), which is usually called the “mobile phase” or carrier. The mobile phase carries the solute through a system (e.g., a column, capillary tube, plate, sheet, etc.) on or to which a material, usually called the stationary phase or adsorbent, is attached. The stationary phase may include, for example, silica gel microspheres or some other type of particle that can be attached and packed. Different analyte molecules may have different affinities for the stationary phase and may have different interactions with the stationary phase that can be analyzed.Furthermore, chromatography incorporates various principles of fluid dynamics, including convection, dispersion, diffusion, and adsorption. Diffusion includes, for example, film diffusion and pore diffusion.

[0031] As used herein, “convection” refers to the mass transfer mechanism due to the general motion of a fluid. The motion of this fluid is induced by an external force. With respect to chromatography, convection means the movement of an analyte within the mobile phase toward the stationary phase as the analyte is transported through the column by the movement of the mobile phase. The movement of the mobile phase is driven by an external force, such as a pressure gradient, while the movement of the analyte toward the stationary phase is driven by a concentration gradient.

[0032] As used herein, “dispersion” refers to the mass transfer mechanism due to non-ideal fluid flow patterns that cause mass propagation from areas of high concentration to areas of low concentration. In chromatography, the packing in a column consists of particles (e.g., microspheres) with flow channels formed between these particles. Differences in packing and particle shape can cause differences in phase velocity. 7 238488 1723673 of 26 mobile in the different flow channels. Furthermore, the analyte molecules flowing within the mobile phase can travel at different speeds along the different flow channels. The difference in speeds, as well as other flow disturbances, generates mass propagation in the axial direction.

[0033] As used herein, “diffusion” refers to the mechanism of mass transfer from areas of high concentration to areas of low concentration due to the random motion of particles (Brownian motion) in a fluid. This motion is a microscopic effect independent of fluid flow and is driven by a concentration gradient within the fluid. A chromatography column can be packed with porous particles (or microspheres). In chromatography, the mobile phase enters the pores in these microspheres, and the static layer of the mobile phase (fluid) creates a “film” around the microsphere. Diffusion in the mobile phase is described by convection. Film diffusion occurs when a molecule passes through the microsphere film into, for example, a pore. Pore diffusion is the movement of the molecule within the pore.

[0034] As used herein, “adsorption” refers to the process by which the analyte present in the stationary phase pore adheres to the inner surface of the microsphere. Adsorption can be driven by various mechanisms depending on the properties of the molecule and the stationary phase, such as, but not limited to, charge, hydrophobicity, and polarity.

[0035] As used herein, a “mechanical model” refers to a model based on the fundamental laws of natural science. Physical and biochemical principles constitute the model equations that comprise the mechanical model. For example, a mechanical model may consist of mathematical equations that represent a complex system or process, its individual parts, and how those parts are coupled or used together. A mechanical model may require only a few experimental data points to calibrate the model and determine its unknown parameters. For example, in some cases, the parameters for a mechanical model can be determined from about three to ten experiments. Generally, the parameters of a mechanical model have real physical meaning, which can help facilitate the interpretation of the predictions made by the mechanical model.Furthermore, because the parameters have real physical meanings, mechanical models allow for easy parameter changes to model different processes. Therefore, a single mechanical model can be used to capture a wide variety of model applications. 1723673 of 26 and, in some cases, to guarantee quality obligations through design.

[0036] As used herein, a “mechanical model of chromatography” is a mechanical model used to represent a chromatographic process. A mechanical model of chromatography may include various parameters, such as, but not limited to, adsorption coefficients, diffusivity properties, material properties, other types of properties, or a combination thereof. By relying on the laws of natural science, a mechanical model of chromatography represents the different effects involved in chromatography, including, for example, fluid dynamics, mass transfer phenomena, and phase equilibrium thermodynamics. For instance, a mechanical model of chromatography may take into account convection, dispersion, diffusion (film diffusion and pore diffusion), adsorption, or a combination thereof. Generally, a mechanical model of chromatography incorporates many process parameters directly into the model equations.Furthermore, many of the various process quality attributes can be calculated from the simulation results generated by the mechanical chromatography model. In this way, a mechanical chromatography model can be used to examine or analyze the effects on the in silico chromatographic process. III. Mechanical modeling of chromatography

[0037] Figure 1 is a schematic diagram of a chromatography system 100 according to various embodiments. The chromatography system 100 includes column 102, which is an example of a type of column or system that can be used in the chromatography system 100. Column 102 is filled with fluid 104 in which particles 106 (e.g., microspheres, silica gel microspheres) are packed to form a packed bed.

[0038] The molecules 108 of interest can be injected into column 102. In various embodiments, the molecules 108 take the form of proteins. In various embodiments, the chromatography system 100 can be used to carry out protein purification. In one or more embodiments, the molecules 108 include antibodies, antibody fragments, antibody complexes, nucleic acids, and / or other types of molecules.

[0039] After being injected into column 102, the molecules 108 are transported by convection within the fluid 104 in column 102 in the direction of arrow 110. This flow can be induced by, for example, but not limited to, the use of pressure, force, or both. In one or more embodiments, a pump is connected to column 102 to facilitate convection. Pumping with a 9 238488 1723673 of 26 higher speed can generate greater convection within column 102.

[0040] Within the chromatography system 100, the molecules 108 move within the interstitial spaces formed between particles 106 by means of the principles of scattering. These interstitial spaces are channels or flow pathways created between the particles 106. Various factors can influence the interstitial velocity of the molecules 108 within the column 102.

[0041] Several of the molecules 108 can pass through the film of several of the particles 106 by film diffusion to enter the pores of these particles. For example, molecule 112 passes through the film 114 of particle 116. The movement of these different molecules within the pores is dominated by pore diffusion. For example, the movement of molecule 118 within the pore 120 of particle 116 is dominated by pore diffusion. In addition, a molecule, such as molecule 118, can adhere to the inner surface 122 of particle 116 by adsorption. IV. Prediction of uncertainty in mechanical modeling

[0042] Figure 2 is a block diagram of a model analysis system 200 according to various embodiments. The model analysis system 200 is used to analyze and provide information about mechanical model 202. Mechanical model 202 is a mechanical chromatography model, which may also be referred to as a chromatographic mechanical model. In one or more embodiments, mechanical model 202 is used to study, analyze, simulate, control, modify, or otherwise evaluate a chromatographic system or process, such as, among others, the chromatography system 100 in Figure 1.

[0043] In various embodiments, the Model 200 analysis system can be implemented using hardware, software, firmware, or a combination thereof. In various embodiments, the Model 200 analysis system can be implemented using the Computing Platform 204. The Computing Platform 204 can take several forms. In one or more embodiments, the Computing Platform 204 includes a single computer (or computing system) or multiple computers communicating with each other. In other examples, the Computing Platform 204 takes the form of a cloud computing platform.

[0044] In one or more embodiments, the computing platform 204 can be communicatively coupled with the data storage 206, display system 208, input device set 210, or a combination thereof. In various embodiments, the data storage 206, system 10 238488 1723673 of 26 display 208, input device set 210 or a combination thereof may be considered part of, or otherwise be integrated with, the computing platform 204. Thus, in some examples, the computing platform 204, data storage 206 and display system 208 may be separate components in communication with each other, but in other examples, some combination of these components may be integrated with each other.

[0045] The model analysis system 200 is used to analyze the mechanical model 202. In one or more embodiments, the model analysis system 200 receives the mechanical model 202 for processing. For example, among others, the model analysis system 200 may receive the mechanical model 202 from a remote source (e.g., another computing platform). In one or more embodiments, the model analysis system 200 receives the mechanical model 202 via one or more wired communication links, one or more wireless communication links, one or more optical communication links, or a combination thereof. In various embodiments, the mechanical model 202 was retrieved from data storage 206.

[0046] In various embodiments, the model analysis system 200 is used to generate the mechanical model 202 based on experimental data 212. The experimental data 212 may include, for example, data obtained or generated by performing one, two, three, or some other number of experiments. In one or more embodiments, the experimental data 212 is generated using data from approximately three to approximately ten experiments. In one or more embodiments, the experimental data 212 is stored in data storage 206.

[0047] The mechanical model 202 includes a plurality of parameters 214. In various embodiments, at least a portion of the parameters 214 may have actual physical meanings. In one or more embodiments, each of the plurality of parameters 214 has an actual physical meaning corresponding to the chromatography process, fluid dynamics, mass transfer phenomena, or other properties or factors. In this way, each of the parameters 214 can provide a means of relating the result of the mechanical model 202 to the actual chromatography process.

[0048] The model 200 analysis system first identifies the initial parameter set 216. In one or more embodiments, the initial parameter set 216 is a set of randomly selected values ​​or 11 238488 1723673 of 26 almost random for parameters 214. In various embodiments, the initial parameter set 216 is the result of a previous round of parameter set processing. The initial parameter set 216 may also be referred to as initial parameter values. In one or more embodiments, the model analysis system 200 uses Latin hypercube sampling (LHS) (also called Latin hypercube screening) to randomly or almost randomly select the initial parameter set 216. In various embodiments, a loss-of-function or minimization algorithm is used to identify the initial parameter set 216.

[0049] The model analysis system 200 finds a local extremum 218 for the mechanical model 202 by using a selected loss of function. In one or more embodiments, the selected loss of function takes the form of a maximum log probability function, a negative log probability function, or a maximum probability function. An optimization algorithm can be selected to identify the local extremum 218 for the selected loss of function. The selected loss of function is chosen to generally guarantee that a local extremum can be reached given the initial parameter set 216. The optimization algorithm can include any number or combination of algorithms. In one or more embodiments, the optimization algorithm can include a Levenberg-Marquardt (LM) minimization algorithm.In various forms of implementation, the optimization algorithm may include gradient descent, Gauss-Newton, Broden-Fletcher-Goldfarb-Shanno (BFGS), or another non-heuristic gradient-based optimization algorithm.

[0050] The analysis system of model 200 is based on a premise regarding the relationship between the parameters 214. In various implementations, the analysis system of model 200 assumes that the underlying distribution is a multiparametric (multivariable) Gaussian distribution. This premise is intended to provide savings in terms of time, costs, processing resources, or a combination thereof. Based on this premise, the range of values ​​to be sampled for each of the parameters 214 can be reduced to improve the probability of generating a sample for each of the parameters 214 where the parameters, when observed simultaneously, are most correlated. In other words, sampling is performed from the most probable values ​​of the parameters 214 according to the multiparametric (multivariable) Gaussian distribution.

[0051] Based on this premise, the covariance matrix 220 can be 238488 1723673 of 26 calculate at the local extreme 218. The covariance matrix 220 describes a multiparametric (multivariable) Gaussian distribution on the parameters 214 and helps to identify the “reduced space” from which samples of the parameters 214 can be taken to reliably determine the uncertainty of the mechanical model 202.

[0052] The model analysis system 200 samples each of the parameters 214 to form a plurality of simulation sets 222. The model analysis system 200 runs simulations of the mechanical model 202 using simulation sets 222 to generate various predictions using the mechanical model 202. These predictions are used to quantify an uncertainty for the mechanical model 202. For example, in one or more embodiments, the predictions are used to generate an uncertainty result 224 for the mechanical model 202. The uncertainty result 224 may include, for example, but not limited to, an indication of the accuracy of the mechanical model 202. In one or more embodiments, the uncertainty result 224 identifies a confidence interval or one or more confidence values ​​for the mechanical model 202.For example, the uncertainty result 224 can identify values ​​with 95% confidence, 99.7% confidence, some other confidence level, or a combination thereof. By providing information about the accuracy associated with mechanical model 202, the model analysis system 200 can provide confidence in mechanical model 202.

[0053] The uncertainty result 224 can take several forms. For example, the uncertainty result 224 can consist of one or more values, a report containing a confidence interval, an alert identifying a confidence interval, a graph, some other type of visual representation of uncertainty, or a combination thereof. In one or more embodiments, the model analysis system 200 displays the uncertainty result 224 in the visualization system 208. In various embodiments, the model analysis system 200 displays a result generated from the uncertainty result 224 in the visualization system. For example, the uncertainty result 224 can include one or more confidence values ​​(for example, a 95% confidence value, a 5% and a 95% confidence value, etc.).The model 200 analysis system can display an alert or report in the 208 visualization system indicating whether these confidence values ​​are acceptable (e.g., above or below a selected threshold).

[0054] Figure 3 is a flow diagram of a 300 process for estimating the 238488 1723673 of 26 mechanical uncertainty of model according to various embodiments. In various embodiments, process 300 is implemented using the model analysis system 200 described in Figure 2. In particular, process 300 can be used to generate an uncertainty result that provides an indication of uncertainty associated with predictions of a mechanical chromatography model such as, for example, mechanical model 202 in Figure 2.

[0055] Step 302 includes receiving a chromatography mechanical model comprising a plurality of parameters. The mechanical model may include a plurality of parameters. In one or more embodiments, each of these parameters may have an actual physical meaning. In various embodiments, a single mechanical model may be used for a wide variety of applications.

[0056] Step 304 includes identifying, for each of the pluralities of parameters, a corresponding region of values ​​as a function of a relationship between values ​​for the plurality of parameters. In one or more embodiments, step 304 includes assuming that the distributions for the parameters are Gaussian or nearly Gaussian. For example, step 304 may include assuming that the distribution of each parameter of the mechanical model is, or is similar to, a Gaussian distribution. Thus, in one or more embodiments, the mechanical model may have a multiparameter (multivariable) Gaussian distribution. In one or more embodiments, step 304 includes identifying a peak in the distribution of each parameter and a range of values ​​for that parameter centered around, or otherwise surrounding, the peak.

[0057] Identifying the corresponding region of values ​​based on the relationship between parameters results in observing the parameters simultaneously to obtain a more complete and general picture of the parameters and where the most probable parameter values ​​are expected to be found. Furthermore, it saves time that would otherwise be spent analyzing less probable parameter values. The identified corresponding region of values ​​provides a structure from which a new sample space can be generated to determine the model's uncertainty.

[0058] Step 306 includes generating a sample of each parameter from the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets. Step 306 includes generating, for example, among others, N simulation sets. Each simulation set includes a sample value for each of the parameters in model 14 238488 1723673 of 26 mechanical. Generating a sample of each parameter within the corresponding value region, as opposed to the entire available space for each parameter, can save time, costs, and processing resources.

[0059] Step 308 involves quantifying an uncertainty for the mechanical model using a plurality of simulation sets. In various embodiments, step 308 includes generating an uncertainty result for the mechanical model, identifying a confidence interval for the mechanical model, or a combination of these. The sample generation performed in step 306 occurs in a manner that ensures that the uncertainty quantification performed in step 308 is a reliable measure of the accuracy of the mechanical model's predictions.

[0060] Figure 4 is a flow diagram of a process 400 for predicting the mechanical uncertainty of a chromatography model according to various embodiments. In various embodiments, process 400 is implemented using the model analysis system 200 described in Figure 2. In particular, process 400 can be used to predict the uncertainty of a mechanical chromatography model such as, for example, mechanical model 202 in Figure 2.

[0061] Step 402 includes receiving a mechanical chromatography model that includes a plurality of parameters. In one or more embodiments, the mechanical model is a model generated from fewer than 20 experiments.

[0062] Step 404 includes calculating a covariance matrix for the mechanical model that describes a multiparametric probability distribution for the plurality of parameters. In one or more embodiments, the multiparametric probability distribution takes the form of a multiparametric (multivariable) Gaussian distribution.

[0063] Step 406 includes identifying, for each of the pluralities of parameters, a corresponding region of values ​​from the multiparameter probability distribution of the plurality of parameters based on selected accuracy criteria to form a plurality of simulation sets. The selected accuracy criteria may include various criteria for reducing the range of values ​​to be sampled for a given parameter. In one or more embodiments, the selected accuracy criteria include, for each parameter of the plurality of parameters, a range of values ​​that satisfy a threshold probability of occurrence.

[0064] Step 408 includes generating a sample of each parameter of the 238488 1723673 of 26 plurality of parameters within the corresponding value region so that each parameter forms a plurality of simulation sets.

[0065] Step 410 involves generating a model prediction distribution for the mechanical model using the plurality of simulation sets. This model prediction distribution captures the various predictions generated by the mechanical model based on the simulation sets.

[0066] Step 412 involves generating an uncertainty result for the mechanical model using the model's prediction distribution. The uncertainty result generated in step 410 provides an indication of how accurate the mechanical model's predictions are. The uncertainty result may include, for example, but is not limited to, a confidence interval for the mechanical model (e.g., a 95% confidence interval, a 99.7% confidence interval, etc.), some other representation of uncertainty in the mechanical model, an alert that includes the confidence interval, a report that includes the confidence interval, some other type of result, or a combination thereof.

[0067] Figure 5 is a flowchart of a process 500 for calculating a covariance matrix for a mechanical model according to various embodiments. The process 500 in Figure 5 can be an example of a process used to calculate a covariance matrix for a mechanical model and may include one or more steps that can be used to implement step 404 in Figure 4.

[0068] Step 502 includes generating a sample of a plurality of parameters from a mechanical model to form a plurality of parameter sets. In one or more embodiments, step 502 includes performing a quasi-random sample generation process (or random parameter screening) to search for an initial parameter set for processing. In one or more embodiments, Latin hypercube sampling may be used to implement the quasi-random sample generation process.

[0069] Step 504 involves selecting an initial parameter set from the plurality of parameter sets for the mechanical model. For example, in step 504, the most adaptable parameter set may be used as the initial parameter set. In one or more embodiments, step 504 may be called the global optimization step and may be carried out using, for example, any number of different types of loss functions. Examples of loss functions that may be used include, but are not limited to, a root mean square error (RMSE) algorithm, a maximum log probability algorithm (or 16 238488 1723673 of 26 registration probability), a negative registration probability algorithm or a maximum probability algorithm. The initial parameter set formed in step 504 is the parameter set that can be further optimized. In one or more embodiments, step 504 includes identifying the search area from which a local endpoint can be identified.

[0070] Step 506 involves calculating a local extremum for a selected loss function using the initial parameter set. In one or more embodiments, the selected loss function is a maximum log probability (or log probability), a negative log probability, or a maximum probability algorithm. Thus, in some cases, the selected loss function used in Step 506 may be the same as, or different from, the loss function used in Step 504. When the selected loss function is a negative log probability, the calculated local extremum is the local minimum. When the selected loss function is a maximum log probability (or log probability), the calculated local extremum is the local maximum. When the selected loss function is a maximum probability, the calculated local extremum is the local maximum.In one or more embodiments, one or more different types of optimization algorithms can be used to identify the local extremum. For example, a minimization algorithm such as the Levenberg-Marquardt (LM) minimization algorithm can be used to identify the desired local extremum for the selected loss function.

[0071] Step 508 involves calculating a covariance matrix for the mechanical model as a function of a selected loss function. The covariance matrix describes the underlying multiparametric (multivariable) Gaussian distribution of the plurality of parameters.

[0072] When the selected loss function is negative log probability, a Hessian matrix of the selected loss function is computed at the local minimum and inverted to obtain the covariance matrix. When the selected loss function is maximum log probability (or log probability), the Hessian matrix of the selected loss function is computed at the local maximum, and the inverted negative of this Hessian matrix is ​​used to obtain the covariance matrix. When the selected loss function is maximum log probability, the Hessian matrix is ​​computed for the log of the selected loss function at the local maximum, and the inverted negative of this Hessian matrix is ​​used to obtain the covariance matrix. 238488 1723673 of 26 V. Examples / Results:

[0073] Figure 6 is a side-by-side comparison of two different types of multiparameter graphs generated for a mechanical model according to various embodiments. In Figure 6, the first multiparameter graph 602 is generated using a Hessian approach. The second multiparameter graph 604 is generated using a Markov Chain Monte Carlo (MCMC) approach. The first multiparameter graph 602 and the second multiparameter graph 604 identify distributions 606 and distributions 608, respectively, for the parameters of the mechanical model. Furthermore, the first multiparameter graph 602 and the second multiparameter graph 604 identify covariances 610 and covariances 612, respectively, between the parameters of the mechanical model.

[0074] As shown in Figure 6, the plots of both approaches appear to be similar. More specifically, the distributions 606 of the first multiparameter plot 602 and the distributions 608 of the second multiparameter plot 604 appear to be similar. Furthermore, the correlations 610 of the first multiparameter plot 602 and the correlations 612 of the second multiparameter plot 604 appear to be similar. These similarities reinforce the idea that the Hessian approach can be used to generate a sample for estimating the uncertainty of predictions made using the mechanical model. In addition, the Hessian approach can provide savings in time, cost, and processing resources. For example, the second multiparameter plot 604 appears to be denser than the first multiparameter plot 602 because it requires more samples and data points, which in turn requires more processing time and resources.

[0075] Figure 7 illustrates Table 700, which compares uncertainty values ​​generated using the Hessian approach and uncertainty values ​​generated using an MCMC approach. The best values ​​and 95% confidence levels shown in Table 700 indicate that the Hessian and MCMC approaches produce similar data. Accordingly, the Hessian approach can be used in various applications to reduce the time, cost, and processing resources associated with determining mechanical model uncertainty, and in particular, chromatography model mechanical uncertainty. VI. Computer-implemented system

[0076] Figure 8 is a block diagram of a computer system according to various embodiments. The 800 computer system can be an example of an implementation for the described 204 computer platform. 238488 1723673 of 26 previously in Figure 2. In one or more examples, the 800 computer system may include an 802 bus or other communication mechanism for transmitting information, and an 804 processor coupled to an 802 bus for processing information. In various embodiments, the 800 computer system may also include a memory, which may be an 806 random-access memory (RAM) or other dynamic storage device, coupled to the 802 bus to determine the instructions to be executed by the 804 processor. The memory may also be used to store temporary variables or other intermediate information during the execution of instructions to be executed by the 804 processor. In various embodiments, the 800 computer system may further include an 808 read-only memory (ROM) or other static storage device coupled to the 802 bus to store static information and instructions for the 804 processor.An 810 storage device, such as a magnetic disk or optical disk, can be provided and coupled to the 802 bus for the storage of information and structures.

[0077] In various embodiments, the computer system 800 can be coupled via the bus 802 to a display 812, such as a cathode ray tube (CRT) or liquid crystal display (LCD), to display information to a computer user. An input device 814, including alphanumeric keyboards and others, can be coupled to the bus 802 to communicate information and order selections to the processor 804. Another type of user input device is a cursor control 816, such as a mouse, joystick, trackball, gesture input device, gaze-based input device, or cursor direction keypad, to communicate direction information and order selections to the processor 804 and to control the movement of the cursor on the display 812.This 814 input device typically has two degrees of release on two axes, a first axis (e.g., x) and a second axis (e.g., y), allowing the device to specify positions in a plane. However, it should be noted that 814 input devices that allow three-dimensional cursor movement (e.g., x, y, yz) are also covered herein.

[0078] According to some implementations of the present teachings, the results can be provided by computer system 800 in response to processor 804 executing one or more sequences of one or more instructions contained in RAM 806. Such instructions can be read into RAM 806 from another computer-readable medium or storage medium readable by 19 238488 1723673 of 26 computer, such as a storage device 810. Execution of the instruction sequences contained in RAM 806 may cause the processor 804 to carry out the processes described herein. Alternatively, the programmed integrated circuit may be used instead of or in combination with software instructions to implement these instructions. Thus, the implementations of these instructions are not limited to any specific combination of hardware and software circuits.

[0079] The term “computer-readable medium” (for example, data storage, storage device, data storage device, etc.) or “computer-readable storage medium” as used herein refers to any medium that participates in providing instructions to the 804 processor for execution. Such medium may take various forms, including, but not limited to, non-volatile media, volatile media, and transmission media. Examples of non-volatile media may include, but are not limited to, optical disks, solid-state disks, magnetic disks, such as the 810 storage device. Examples of volatile media may include, but are not limited to, dynamic memory, such as the 806 RAM. Examples of transmission media may include, but are not limited to, coaxial cables, copper cables, and optical fiber, including the cables comprising the 802 bus.

[0080] Common forms of computer-readable media include, for example, a floppy disk, a hard disk, a magnetic tape or any other magnetic medium, a CD-ROM, any other optical media, punched cards, paper tape and any other physical media with hole patterns, RAM, PROM and EPROM, a FLASH-EPROM, any other memory chip or cartridge, or any other tangible media from which a computer can read.

[0081] In addition to computer-readable media, instructions or data may be provided as signals on the transmission media included in a communications apparatus or system to provide sequences of one or more instructions to the computer system's processor for execution. For example, a communications apparatus may include a transceiver with signals indicating instructions and data. The instructions and data are configured so that one or more processors implement the functions detailed in this disclosure. Representative examples of data communications transmission connections may include, but are not limited to, telephone modem connections, wide area networks (WANs), local area networks (LANs), infrared data connections, and 20 238488 1723673 of 26 NFC, optical communication connections, etc.

[0082] It is worth noting that the methodologies described herein, flowcharts, diagrams and the accompanying disclosure, can be implemented using an 800 computer system as a standalone device or in a distributed network of shared computing processing resources, such as a cloud computing network.

[0083] The methodologies described herein can be implemented in various ways depending on the application. For example, these methodologies can be implemented in hardware, firmware, software, or any combination thereof. For a hardware implementation, the processing unit can be implemented within one or more application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, electronic devices, other electronic units designed to perform the functions described herein, or a combination thereof.

[0084] In various embodiments, the methods of these teachings can be implemented as firmware and / or software programs and applications written in conventional programming languages ​​such as C, C++, Python, etc. If implemented as firmware and / or software, the embodiments described herein can be implemented on a non-transient, computer-readable medium in which a program is stored for the computer to execute the methods described above.It is worth noting that the various devices described herein may be provided in a computer system, such as a computer system 800, wherein the processor 804 executes the analyses and determinations provided by these devices, subject to instructions provided by any or a combination of the RAM memory component 806, ROM, 808, or storage device 810 and user inputs provided through the input device 814.

[0085] Although these teachings are described in conjunction with various forms of implementation, they are not intended to be limited to these various forms of implementation. On the contrary, these teachings encompass various alternatives, modifications, and equivalents, as those in the mid-level trade will appreciate.

[0086] In describing the various forms of realization, the descriptive report 238488 1723673 of 26 may have presented a method and / or process as a particular sequence of steps. However, insofar as the method or process does not depend on the particular order of the steps indicated herein, the method or process should not be limited to the particular sequence of steps described, and a person of average skill can readily appreciate that the sequences can be varied and still remain within the spirit and scope of the various forms of realization. STATEMENT OF THE FORMS OF REALIZATION

[0087] Embodiment 1. A method for estimating the mechanical uncertainty of a chromatography model, wherein the method comprises receiving a mechanical chromatography model comprising a plurality of parameters; identifying for each of the plurality of parameters, a corresponding region of values ​​as a function of a relationship between values ​​for the plurality of parameters; generating a sample of each parameter of the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets; and quantifying an uncertainty for the mechanical model using the plurality of simulation sets.

[0088] Implementation Form 2. The method of implementation form 1, wherein identifying for each of the pluralities of parameters, the corresponding region of values ​​comprises calculating a covariance matrix for the mechanical model as a function of a selected loss function.

[0089] Implementation Form 3. The method of Implementation Form 2, wherein the selected loss function comprises at least one of a negative log probability algorithm, a maximum log probability algorithm, or a maximum probability algorithm.

[0090] Implementation Form 4. The method of Implementation Form 2, wherein calculating the covariance matrix for the mechanical model as a function of the selected loss function comprises identifying a search area using at least one of the selected loss function or another loss function and calculating a local extremum for the selected loss function with respect to the search area.

[0091] Implementation Form 5. The method of implementation form 4, wherein calculating the covariance matrix for the mechanical model as a function of the selected loss function further comprises calculating the covariance matrix for the local extremum. 238488 1723673 of 26

[0092] Implementation Form 6. The method of implementation forms 1 to 5, wherein identifying, for each of the pluralities of parameters, the corresponding region of values ​​comprises generating a sample of the plurality of parameters to form a plurality of parameter sets; selecting an initial parameter set from the plurality of parameter sets for the mechanical model; and calculating a covariance matrix for the mechanical model as a function of a selected loss function using the initial parameter set.

[0093] Implementation Form 7. The method of any of implementation forms 1 to 6, wherein quantifying uncertainty comprises generating a model prediction distribution for the mechanical model using the plurality of simulation sets.

[0094] Implementation Form 8. The method of implementation form 7, wherein quantifying uncertainty further comprises identifying a confidence interval for the mechanical model using the model's prediction distribution.

[0095] Embodiment Form 9. The method of any of embodiment forms 1 to 8, further comprising receiving experimental data; and generating the mechanical model using experimental data.

[0096] Embodiment 10. A system for estimating the mechanical uncertainty of a chromatography model, wherein the system comprises a data source configured to obtain a mechanical chromatography model; and a processor configured to receive the mechanical chromatography model from the data source wherein the mechanical model includes a plurality of parameters and wherein the processor is further configured to identify, for each of the pluralities of parameters, a corresponding region of values ​​based on a relationship between values ​​for the plurality of parameters; generate a sample of each parameter of the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets; and quantify an uncertainty for the mechanical model using the plurality of simulation sets.

[0097] Implementation Form 11. The system of implementation form 10, wherein the processor is further configured to calculate a covariance matrix for the mechanical model as a function of a selected loss function.

[0098] Implementation Form 12. The system of implementation form 11, wherein the selected loss function comprises at least one of an algorithm 23 238488 1723673 of 26 negative registration probability, a maximum registration probability algorithm or a maximum probability algorithm.

[0099] Implementation Form 13. The system of any of Implementation Forms 10 to 12, wherein the processor is further configured to identify a search area using at least one of the selected loss function or another loss function and compute a local extremum for the selected loss function with respect to the search area.

[0100] Implementation Form 14. The system of implementation form 13, wherein the processor is further configured to calculate the covariance matrix for the mechanical model as a function of the selected loss function and to calculate the covariance matrix for the local extremum.

[0101] Implementation Form 15. The system of any of implementation forms 10 to 14, wherein the processor is further configured to generate a sample of the plurality of parameters to form a plurality of parameter sets, select an initial parameter set from the plurality of parameter sets for the mechanical model, and compute a covariance matrix for the mechanical model as a function of a selected loss function using the initial parameter set.

[0102] Implementation Form 16. The system of Implementation Form 15, wherein the processor is further configured to generate a model prediction distribution for the mechanical model using the plurality of simulation sets and to identify a confidence interval for the mechanical model using the model prediction distribution.

[0103] Implementation Form 17. The system of any of the implementation forms 10 to 16, wherein the processor is further configured to receive experimental data; and generate the mechanical model using experimental data.

[0104] Embodiment 18. A non-transient, computer-readable medium in which a program is stored, wherein the program is configured to enable a computer to carry out a method for estimating the mechanical uncertainty of a chromatography model, wherein the method comprises receiving a mechanical chromatography model comprising a plurality of parameters; identifying, for each of the plurality of parameters, a corresponding region of values ​​based on a relationship between values ​​for the plurality of parameters; generating a sample of each parameter of the plurality of parameters within the corresponding region of values ​​so that each parameter forms a 24 238488 1723673 of 26 plurality of simulation sets; and quantify an uncertainty for the mechanical model using the plurality of simulation sets.

[0105] Embodiment Form 19. The non-transient, computer-readable means of embodiment form 18, wherein the method further comprises calculating a covariance matrix for the mechanical model as a function of a selected loss function.

[0106] Implementation Form 20. The non-transient computer-readable medium of Implementation Form 19, wherein the selected loss function comprises at least one of a negative record probability algorithm, a maximum record probability algorithm, or a maximum probability algorithm.

[0107] Embodiment 21. The non-transient, computer-readable means of embodiment 19, wherein the method further comprises identifying a search area using at least one of the selected loss function or another loss function and calculating a local extremum for the selected loss function with respect to the search area.

[0108] Implementation Form 22. The non-transient, computer-readable means of Implementation Form 21, wherein the method further comprises calculating the covariance matrix for the local extremum.

[0109] Embodiment Form 23. The non-transient, computer-readable means of any of embodiment forms 18 to 22, wherein the method further comprises generating a sample of the plurality of parameters to form a plurality of parameter sets, selecting an initial parameter set from the plurality of parameter sets for the mechanical model, and calculating a covariance matrix for the mechanical model as a function of a selected loss function using the initial parameter set.

[0110] Implementation Form 24. The non-transient, computer-readable means of any of Implementation Forms 18 to 23, wherein the method further comprises generating a model prediction distribution for the mechanical model using the plurality of simulation sets.

[0111] Implementation Form 25. The non-transient, computer-readable means of Implementation Form 24, wherein the method further comprises identifying a confidence interval for the mechanical model using the model's prediction distribution.

Claims

1. A method for estimating a mechanical uncertainty of a chromatography model, the method characterized in that it comprises: receiving a mechanical chromatography model comprising a plurality of parameters; identifying, for each of the plurality of parameters, a corresponding region of values ​​as a function of a relationship between values ​​for the plurality of parameters; generating a sample of each parameter of the plurality of parameters within the corresponding region of values ​​so that each parameter forms a plurality of simulation sets; and quantifying an uncertainty for the mechanical model using the plurality of simulation sets. Sixteen claims follow.