Estimating Uncertainty in Mechanistic Chromatography Models

Abstract

A method, system, and non-transitory computer-readable medium for estimating uncertainty in a mechanistic chromatography model. A mechanistic model of chromatography including a plurality of parameters is received (302). For each of the plurality of parameters, a corresponding range of values ​​is identified (304) based on a relationship between values ​​of the plurality of parameters. Each parameter of the plurality of parameters is sampled within the corresponding range of values ​​of each parameter to form a plurality of simulation sets (306). The uncertainty of the mechanistic model is quantified using the plurality of simulation sets (308).

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims priority to and the benefit of U.S. Provisional Application No. 63 / 166,939, filed March 26, 2021, the contents of which are incorporated herein in their entirety.

[0002] Field of Disclosure The present specification relates generally to mechanistic chromatography modeling. More specifically, the present specification provides methods and systems for estimating uncertainties associated with mechanistic chromatography models. [Background technology]

[0003] Introduction In general, chromatography is the main process used to purify biopharmaceuticals. Mechanistic modeling can be used to improve chromatographic processes, investigate problems related to these processes, and perform chromatographic simulations. A mechanistic model assumes that a complex system or process can be understood by examining how the individual parts of the system or process work and how those parts are combined. The mechanistic model then mathematically represents this complex system or process in a simplified manner that still captures the fundamental principles of the complex system or process. One barrier to widespread or systematic application of mechanistic models for processes such as chromatographic processes can be the difficulties associated with qualifying and establishing confidence in the mechanistic model. For example, some currently available methods for estimating the uncertainty of a mechanistic model are more time-consuming and computationally expensive than desired. Summary of the Invention

[0004] overview In one or more embodiments, a method is provided for estimating uncertainty of a mechanistic chromatography model. A mechanistic model of chromatography is received that includes a plurality of parameters. For each of the plurality of parameters, a corresponding range of values ​​is identified based on a relationship between values ​​for the plurality of parameters. Each parameter of the plurality of parameters is sampled in the corresponding range of values ​​for each parameter to form a plurality of simulation sets. Uncertainty for the mechanistic model is quantified using the plurality of simulation sets.

[0005] In one or more embodiments, a system for estimating uncertainty of a mechanistic chromatography model is provided. The system includes a data source configured to acquire a mechanistic model of chromatography, and a processor configured to receive the mechanistic model of chromatography from the data source, the mechanistic model including a plurality of parameters. The processor is further configured to: identify, for each of the plurality of parameters, a corresponding range of values ​​based on a relationship between values ​​for the plurality of parameters, sample each of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets, and quantify the uncertainty for the mechanistic model using the plurality of simulation sets.

[0006] In one or more embodiments, a non-transitory computer-readable medium is provided having a program stored thereon, the program configured to cause a computer to execute a method for estimating uncertainty of a mechanistic chromatography model, the method including receiving a mechanistic model of chromatography including a plurality of parameters, identifying for each of the plurality of parameters a corresponding range of values ​​based on a relationship between values ​​for the plurality of parameters, sampling each of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets, and quantifying uncertainty for the mechanistic model using the plurality of simulation sets. [Brief description of the drawings]

[0007] For a more complete understanding of the principles disclosed herein and their advantages, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:

[0008] [Figure 1] 1 is a schematic diagram of a chromatography system according to various embodiments.

[0009] [Diagram 2] FIG. 1 is a block diagram of a model analysis system in accordance with various embodiments.

[0010] [Diagram 3] 1 is a flowchart of a process for estimating uncertainty in a mechanistic model in accordance with various embodiments.

[0011] [Figure 4] 1 is a flowchart of a process for predicting uncertainty in a mechanistic chromatography model in accordance with various embodiments.

[0012] [Diagram 5]1 is a flowchart of a process for calculating a covariance matrix for a mechanistic model in accordance with various embodiments.

[0013] [Figure 6] 1 is a side-by-side comparison of two different types of multi-parametric plots generated for a mechanistic model according to various embodiments.

[0014] [Figure 7] 1 shows a table comparing uncertainty values ​​generated by the Hessian method with uncertainty values ​​generated using the Markov Chain Monte Carlo method.

[0015] [Figure 8] FIG. 1 is a block diagram of a computer system according to various embodiments.

[0016] It should be understood that the figures are not necessarily drawn to scale, and that objects in the figures are not necessarily drawn to scale relative to each other. The figures are intended to bring clarity and understanding to the various embodiments of the devices, systems, and methods disclosed herein. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. Furthermore, it should be understood that the drawings are not intended to limit the scope of the present teachings in any way. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0017] Detailed Description I. Overview Mechanistic modeling is an important tool for understanding how various systems and processes work. For example, mechanistic chromatography modeling can be an important tool for various biological, pharmaceutical, and biopharmaceutical applications. Mechanistic chromatography modeling allows for non-destructive, real-time measurement of molecular attributes during simulated chromatographic processes (e.g., simulated chromatographic purification), and can provide meaningful insight into the simulated chromatographic processes.

[0018] When verified or otherwise justified, mechanistic chromatography modeling may describe chromatography in a manner that allows reliable interpolation and extrapolation from results generated by mechanistic chromatography modeling. Furthermore, mechanistic chromatography modeling may enable process optimization, real-time process monitoring, control chromatographic processes, and other operations that provide insight into the chromatographic process. Still further, this type of modeling may allow identification of specific operating parameters that are important to the chromatographic process.

[0019] In order to be able to use a reliable mechanistic chromatography model for a given application, it may be necessary to understand the predictive power of the mechanistic chromatography model. For example, it may be important to understand the accuracy of the predictions made by the mechanistic chromatography model. Some currently available methods for evaluating such mechanistic chromatography models evaluate or estimate the uncertainty associated with individual parameters of the model. However, the parameter uncertainties may not provide an indication of the predictive power of the overall mechanistic chromatography model. Furthermore, some currently available methods for evaluating the uncertainty of a mechanistic chromatography model are computationally expensive and time consuming. For example, one currently available method for evaluating the uncertainty of a mechanistic chromatography model requires several days of processing resources (e.g., 10 5 or 10 6 function calls) required for such methods. Thus, the time, cost, and processing resources required for such methods may make their use practically infeasible for certain applications.

[0020] Recognizing and considering the importance of having confidence in the mechanistic chromatography model used for a given application, various embodiments described herein provide methods and systems for evaluating mechanistic chromatography models. For example, various embodiments described herein provide methods and systems for determining how accurate a mechanistic chromatography model is based on an estimation of the uncertainty associated with the mechanistic chromatography model. The methods and systems described herein allow for estimating the uncertainty of a mechanistic chromatography model in a manner that is faster and less computationally expensive than at least some of the currently available methods and systems. For example, without limitation, various embodiments described herein may allow for calculating the uncertainty of a mechanistic chromatography model in a few hours (e.g., in some cases, less than 6 hours) compared to the several days required by some currently available methods. In one or more embodiments, these time savings may be due to the use of fewer function calls (e.g., 10 or 10 function calls) compared to the 10 or 10 function calls required by some currently available methods. 3 ~10 4 Accordingly, savings in time, cost, and processing resources may be achieved.

[0021] II. Definition The present disclosure is not limited to these exemplary embodiments and applications, nor to the manner of operation of the exemplary embodiments and applications, or to the manner of operation of the exemplary embodiments and applications described herein. Further, the figures may show simplified or partial views, and the dimensions of elements in the figures may be exaggerated or not to scale.

[0022] Additionally, when the terms "on," "attached to," "connected to," "coupled to," or similar terms are used herein, an element (e.g., a component, material, layer, substrate, etc.) may be "on," "attached to," "connected to," or "coupled to" another element, whether the element is directly on, directly attached to, directly connected to, or directly coupled to another element, or whether there are one or more intervening elements between the element and the other element. Additionally, when referring to a list of elements (e.g., elements a, b, c), such reference is intended to include any one of the listed elements alone, any combination of less than all of the listed elements, and / or a combination of all of the listed elements. The division of sections herein is merely for ease of review and does not limit any combination of elements discussed.

[0023] Unless otherwise specified, scientific and technical terms used in conjunction with the teachings described herein shall have the meanings commonly understood by those skilled in the art.Furthermore, unless otherwise required by context, singular terms shall include plurals and plural terms shall include singulars.In general, the nomenclatures and techniques utilized in connection with chemistry, biochemistry, molecular biology, pharmacology, and toxicology described herein are those nomenclatures and techniques well known and commonly used in the art.

[0024] As used herein, "substantially" means sufficient to function for the intended purpose. Thus, the term "substantially" allows for small, inconsequential variations from absolute or perfect conditions, dimensions, measurements, results, etc., that would be expected by one of ordinary skill in the art, but that do not appreciably affect overall performance. When used in reference to a numerical value, or a parameter or characteristic that can be expressed as a numerical value, "substantially" means within 10 percent.

[0025] The term "plurality" means two or more.

[0026] As used herein, the term "plurality" can mean 2, 3, 4, 5, 6, 7, 8, 9, 10, or more.

[0027] As used herein, the term "set of" means one or more. For example, a set of items includes one or more items.

[0028] As used herein, the phrase "at least one of" when used in conjunction with a list of items means that various combinations of one or more of the listed items may be used, and only one of the items in the list may be required. An item may be a specific object, thing, step, action, process, or category. In other words, "at least one of" means that any combination of items, or some items from the list may be used, but not all items in the list are required. For example, without limitation, "at least one of item A, item B, or item C" means 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 item C. In some cases, "at least one of item A, item B, or item C" means, without limitation, 2 item A, 1 item B, and 10 item C; 4 item B, and 7 item C, or some other suitable combination.

[0029] 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.

[0030] As used herein, "analyte" refers to a mixture containing one or more individual components. In the context of chromatography, an analyte may be a mixture from which individual components or molecules are to be separated and analyzed.

[0031] As used herein, "chromatography" refers to a technique or process for separating an analyte into various components (e.g., molecules) of interest. The analyte is typically dissolved in a fluid (e.g., gas, solvent, water, etc.), commonly referred to as the "mobile phase" or carrier. The mobile phase carries the solute through a system (e.g., column, capillary, plate, sheet, etc.) onto which a material, commonly referred to as the stationary phase or adsorbent, is immobilized. The stationary phase may include, but is not limited to, for example, silica gel beads, or any other type of particle capable of immobilization and packing. Various molecules of the analyte may have different affinities for the stationary phase and may have different interactions with the stationary phase that can be analyzed. Additionally, chromatography involves various fluid dynamic principles, such as, for example, convection, dispersion, diffusion, and adsorption. Diffusion includes, for example, membrane diffusion and pore diffusion.

[0032] As used herein, "convection" refers to the mechanism of mass transfer by bulk motion of a fluid. This fluid movement is induced by an external force. In the context of chromatography, convection means that the analytes in the mobile phase move towards the stationary phase as they are 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 analytes towards the stationary phase is driven by a concentration gradient.

[0033] As used herein, "dispersion" refers to a mechanism of mass transfer due to non-ideal fluid flow patterns that result in the diffusion of mass from regions of high concentration to regions of low concentration. In chromatography, the packing in a column consists of particles (e.g., beads) that form flow paths between the particles. Differences in packing and particle shape can cause differences in the velocity of the mobile phase in different flow paths. Furthermore, analyte molecules flowing in the mobile phase can move at different velocities along different flow paths. Velocity differences and other flow disturbances result in axial mass spreading.

[0034] As used herein, "diffusion" refers to the mechanism of mass transfer from regions of high concentration to regions of low concentration due to random movement of particles in a fluid (Brownian motion). This movement is a microscopic effect independent of fluid flow and is driven by concentration gradients within the fluid. Chromatography columns can be packed with porous particles (or beads). In chromatography, the mobile phase enters the pores of these beads and a stagnant layer of mobile phase (fluid) forms a "membrane" around the beads. Diffusion in the mobile phase is described by convection. Membrane diffusion occurs when molecules pass through the membrane of the beads, for example, entering the pores. Pore diffusion is the movement of molecules in the pores.

[0035] As used herein, "adsorption" refers to the process by which an analyte present in the pores of the stationary phase attaches to the inner surface of the beads. Adsorption can be driven by a variety of mechanisms, including but not limited to charge, hydrophobicity, and polarity, depending on the properties of the molecule and the stationary phase.

[0036] As used herein, a "mechanistic model" refers to a model based on fundamental laws of natural science. Physical and biochemical principles make up the model equations that make up the mechanistic model. For example, a mechanistic model may be composed of mathematical equations that represent a complex system or process, its individual parts, and how those parts are connected or used with each other. A mechanistic model may require a small number of experimental data or data points to calibrate the model and determine unknown model parameters. For example, in some cases, parameters for a mechanistic model can be determined in about 3-10 experiments. Generally, the parameters of a mechanistic model have real physical meaning, which can help to facilitate the interpretation of predictions made by the mechanistic model. Furthermore, because the parameters have real physical meaning, a mechanistic model allows the parameters to be easily changed to model different processes. Thus, a single mechanistic model may be used to capture a wide variety of model applications and, in some cases, ensure quality obligations through design.

[0037] As used herein, a "mechanistic chromatography model" is a mechanistic model used to represent a chromatographic process. A mechanistic chromatography model may include various parameters, such as, but not limited to, adsorption coefficients, diffusion properties, material properties, other types of properties, or combinations thereof. Relying on the laws of natural science, a mechanistic chromatography model represents various effects involved in chromatography, including, for example, fluid dynamics, mass transfer phenomena, and thermodynamics of phase equilibria. For example, a mechanistic chromatography model may take into account convection, dispersion, diffusion (membrane diffusion and pore diffusion), adsorption, or combinations thereof. In general, a mechanistic chromatography model includes many process parameters directly in the model equations. Furthermore, a number of different process quality attributes can be calculated from the simulation results generated from the mechanistic chromatography model. In this manner, a mechanistic chromatography model may be used to test or analyze effects on a chromatographic process in a computer.

[0038] III. Mechanistic Chromatography Modeling 1 is a schematic diagram of a chromatography system 100 according to various embodiments. Chromatography system 100 includes a column 102, which is an example of a type of column or system that may be used in chromatography system 100. Column 102 is filled with a fluid 104 that is packed with particles 106 (e.g., beads, silica gel beads) to form a packed bed.

[0039] A molecule of interest 108 may be injected into the column 102. In various embodiments, the molecule 108 takes the form of a protein. In various embodiments, the chromatography system 100 may be used to perform protein purification. In one or more embodiments, the molecule 108 includes an antibody, an antibody fragment, an antibody conjugate, a nucleic acid, and / or other types of molecules.

[0040] After being injected into the column 102, the molecules 108 are transported by convection within the fluid 104 within the column 102 in the direction of arrow 110. This flow may be induced using, for example, but not limited to, pressure, force, or both. In one or more embodiments, a pump is connected to the column 102 to promote convection. Higher pumping speeds may result in greater convection within the column 102.

[0041] Within the chromatography system 100, molecules 108 move within the interstitial spaces formed between the particles 106 due to the principles of dispersion. These interstitial spaces are the flow paths or pathways created between the particles 106. Various factors can affect the interstitial velocity of the molecules 108 within the column 102.

[0042] Various molecules 108 may pass through the membranes of various particles 106 and enter the pores of these particles by membrane diffusion. For example, molecule 112 passes through membrane 114 of particle 116. The movement of these different molecules within the pores is governed by pore diffusion. For example, the movement of molecule 118 within pore 120 of particle 116 is governed by pore diffusion. Additionally, molecules such as molecule 118 may attach to the interior surface 122 of particle 116 by adsorption.

[0043] IV. Predicting Uncertainty in Mechanistic Modeling 2 is a block diagram of a model analysis system 200 in accordance with various embodiments. The model analysis system 200 is used to analyze and provide information about a mechanistic model 202. The mechanistic model 202 is a mechanistic chromatography model, which may also be referred to as a mechanistic model of chromatography. In one or more embodiments, the mechanistic model 202 is used to study, analyze, simulate, control, modify, or otherwise evaluate a chromatography system or process, such as, but not limited to, the chromatography system 100 of FIG. 1.

[0044] In various embodiments, model analysis system 200 may be implemented using hardware, software, firmware, or a combination thereof. In various embodiments, model analysis system 200 may be implemented using a computing platform 204. Computing platform 204 may take a variety of forms. In one or more embodiments, computing platform 204 includes a single computer (or computer system) or multiple computers in communication with each other. In other examples, computing platform 204 takes the form of a cloud computing platform.

[0045] In one or more embodiments, the computing platform 204 may be communicatively coupled to a data storage device 206, a display system 208, a set of input devices 210, or a combination thereof. In various embodiments, the data storage device 206, the display system 208, the set of input devices 210, or a combination thereof may be considered part of the computing platform 204 or may be otherwise integrated with the computing platform 204. Thus, in some examples, the computing platform 204, the data storage device 206, and the display system 208 may be separate components that communicate with each other, while in other examples some combination of these components may be integrated.

[0046] The model analysis system 200 is used to analyze the mechanistic model 202. In one or more embodiments, the model analysis system 200 receives the mechanistic model 202 for processing. For example, but not limited to, the model analysis system 200 may receive the mechanistic model 202 from a remote source (e.g., another computing platform). In one or more embodiments, the model analysis system 200 receives the mechanistic 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 mechanistic model 202 is retrieved from a data storage device 206.

[0047] In various embodiments, model analysis system 200 is used to generate mechanistic model 202 based on experimental data 212. Experimental data 212 may include, for example, data obtained or generated by conducting one, two, three, or some other number of experiments. In one or more embodiments, experimental data 212 is generated using data from about three to about ten experiments. In one or more embodiments, experimental data 212 is stored in data storage device 206.

[0048] The mechanistic model 202 includes a number of parameters 214. In various embodiments, at least some of the parameters 214 may have an actual physical meaning. In one or more embodiments, each of the number of parameters 214 has an actual physical meaning that corresponds to a chromatographic process, fluid dynamics, mass transfer phenomena, or other properties or factors. In this manner, each of the parameters 214 may provide a way to relate the output of the mechanistic model 202 to an actual chromatographic process.

[0049] Model analysis system 200 first identifies an initial parameter set 216. In one or more embodiments, initial parameter set 216 is a randomly or near-randomly selected set of values ​​for parameters 214. In various embodiments, initial parameter set 216 is the result of a previous parameter set process. Initial parameter set 216 may also be referred to as initial parameter values. In one or more embodiments, model analysis system 200 randomly or near-randomly selects initial parameter set 216 using Latin hypercube sampling (LHS) (also called Latin hypercube screen). In various embodiments, a loss function or minimization algorithm is used to identify initial parameter set 216.

[0050] The model analysis system 200 finds the extreme value 218 for the mechanistic model 202 using a selected loss function. In one or more embodiments, the selected loss function takes the form of a maximum log-likelihood function, a negative log-likelihood function, or a maximum likelihood function. An optimization algorithm may be selected to identify the extreme value 218 for the selected loss function. The selected loss function is generally selected to ensure that the extreme value can be reached given the initial parameter set 216. The optimization algorithm may include any number of algorithms or combinations of algorithms. In one or more embodiments, the optimization algorithm may include a Levenberg-Marquardt (LM) minimization algorithm. In various embodiments, the optimization algorithm may include a steepest descent method, a Gauss-Newton method, a Broden-Fletcher-Goldfarb-Shanno method (BFGS), or other gradient-based non-heuristic optimization algorithm.

[0051] The model analysis system 200 makes assumptions about the relationships between the parameters 214. In various embodiments, the model analysis system 200 assumes that the underlying distribution is a multi-parametric Gaussian distribution. This assumption is made to provide savings in terms of time, cost, processing resources, or a combination thereof. This assumption may narrow the range of values ​​to be sampled from each parameter 214, improving the likelihood that sampling is performed for each of the parameters 214 where the parameters are most correlated when viewed together. That is, sampling is performed from more likely values ​​of the parameters 214 based on the multi-parametric Gaussian distribution.

[0052] Based on this assumption, a covariance matrix 220 may be calculated at the extreme values ​​218. The covariance matrix 220 describes the multi-parametric (multivariate) Gaussian distribution over the parameters 214 and helps to identify a "narrowed space" over which the parameters 214 may be sampled to reliably determine the uncertainty in the mechanistic model 202.

[0053] 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 mechanistic model 202 using the simulation sets 222 and generates various predictions using the mechanistic model 202. These predictions are used to quantify the uncertainty for the mechanistic model 202. For example, in one or more embodiments, the predictions are used to generate an uncertainty output 224 for the mechanistic model 202. The uncertainty output 224 may include, for example, but not limited to, an indication of the accuracy of the mechanistic model 202. In one or more embodiments, the uncertainty output 224 identifies a confidence interval or one or more confidence values ​​for the mechanistic model 202. For example, the uncertainty output 224 may identify a value for 95% confidence, 99.7% confidence, some other level of confidence, or a combination thereof. By providing information about the accuracy associated with the mechanistic model 202, the model analysis system 200 can provide confidence in the mechanistic model 202.

[0054] Uncertainty output 224 may take a variety of forms. For example, uncertainty output 224 may be one or more values, a report including a confidence interval, an alert identifying the confidence interval, a plot, some other type of visual representation of the uncertainty, or a combination thereof. In one or more embodiments, model analysis system 200 displays uncertainty output 224 on display system 208. In various embodiments, model analysis system 200 displays an output generated based on uncertainty output 224 on a display system. For example, uncertainty output 224 may include one or more confidence values ​​(e.g., a 95% confidence value, 5% and 95% confidence values, etc.). Model analysis system 200 may display an alert or report on display system 208 indicating whether the confidence values ​​are acceptable (e.g., above or below a selected threshold).

[0055] Figure 3 is a flow chart of a process 300 for estimating uncertainty of a mechanistic model according to various embodiments. In various embodiments, the process 300 is implemented using the model analysis system 200 described in Figure 2. In particular, the process 300 may be used to generate an uncertainty output that provides an indication of the uncertainty associated with predictions from a mechanistic chromatography model, such as, for example, but not limited to, the mechanistic model 202 of Figure 2.

[0056] Step 302 includes receiving a mechanistic model of chromatography that includes a number of parameters. The mechanistic model may include a number of parameters. In one or more embodiments, each of these parameters may have an actual physical meaning. In various embodiments, a single mechanistic model may be used for a wide variety of applications.

[0057] Step 304 includes, for each of the plurality of parameters, identifying a corresponding region of values ​​based on the relationship between the values ​​for the plurality of parameters. In one or more embodiments, step 304 includes assuming that the distribution for the parameters is Gaussian or approximately Gaussian. For example, step 304 may include assuming that the distribution of each parameter of the mechanistic model is Gaussian or similar to a Gaussian distribution. Thus, across the parameters, in one or more embodiments, the mechanistic model may have a multi-parametric (multivariate) Gaussian distribution. In one or more embodiments, step 304 includes identifying a peak of the distribution for each parameter and a range of values ​​for that parameter centered on or otherwise located around the peak.

[0058] By identifying corresponding value regions based on the relationships between the parameters, the parameters can be looked at simultaneously to obtain a more complete and thorough picture of the parameters and where the maximum likelihood values ​​of the parameters are expected. Furthermore, time that may be wasted analyzing values ​​for less likely parameters can be avoided. The identified corresponding value regions provide a framework that allows the uncertainty of the model to be determined using a new sampling space.

[0059] Step 306 includes sampling each of the multiple parameters within a corresponding range of values ​​for each parameter to form a plurality of simulation sets. Step 306 includes, for example, but not limited to, generating N simulation sets, each of which includes a sample value for each parameter of the mechanistic model. By sampling each parameter from within a corresponding range of values ​​rather than the entire available space for each parameter, time, cost, and processing resources can be saved.

[0060] Step 308 includes quantifying the uncertainty for the mechanistic model using a plurality of simulation sets. In various embodiments, step 308 includes generating an uncertainty output for the mechanistic model, identifying a confidence interval for the mechanistic model, or a combination thereof. The sampling performed in step 306 is performed in a precise manner that helps ensure that the uncertainty quantification performed in step 308 is a reliable measure of the accuracy of the predictions of the mechanistic model.

[0061] 4 is a flow chart of a process 400 for predicting the uncertainty of a mechanistic chromatography model according to various embodiments. In various embodiments, the process 400 is implemented using the model analysis system 200 described in FIG 2. In particular, the process 400 may be used to predict the uncertainty of a mechanistic chromatography model, such as, for example, but not limited to, the mechanistic model 202 of FIG 2.

[0062] Step 402 includes receiving a mechanistic model of chromatography that includes a plurality of parameters. In one or more embodiments, the mechanistic model is a model generated from fewer than 20 experiments.

[0063] Step 404 includes calculating a covariance matrix for a mechanistic model that describes a multi-parametric probability distribution for the multiple parameters. In one or more embodiments, the multi-parametric probability distribution takes the form of a multi-parametric (multivariate) multi-parametric Gaussian distribution.

[0064] Step 406 includes, for each of the plurality of parameters, identifying a corresponding range of values ​​from the multi-parametric probability distribution of the plurality of parameters based on the selected accuracy criteria to form a plurality of simulation sets. The selected accuracy criteria may include various criteria for narrowing the range of values ​​sampled for a given parameter. In one or more embodiments, the selected accuracy criteria includes, for each parameter of the plurality of parameters, a range of values ​​that meets a threshold likelihood of occurrence.

[0065] Step 408 includes sampling each parameter of the plurality of parameters within a corresponding range of values ​​for each parameter to form a plurality of simulation sets.

[0066] Step 410 includes generating a model prediction distribution for the mechanistic model using the multiple simulation sets, the model prediction distribution capturing various predictions generated by the mechanistic model based on the simulation sets.

[0067] Step 412 includes generating an uncertainty output for the mechanistic model using the model prediction distribution. The uncertainty output generated in step 410 provides an indication of how accurate the predictions of the mechanistic model are. The uncertainty output may include, for example, without limitation, a confidence interval for the mechanistic model (e.g., a 95% confidence interval, a 99.7% confidence interval, etc.), some other representation of the uncertainty in the mechanistic model, an alert including the confidence interval, a report including the confidence interval, some other type of output, or a combination thereof.

[0068] 5 is a flowchart of a process 500 for calculating a covariance matrix for a mechanistic model according to various embodiments. The process 500 of FIG. 5 may be an example of a process used to calculate a covariance matrix for a mechanistic model and may include one or more steps that may be used to implement step 404 of FIG.

[0069] Step 502 samples a number of parameters of the mechanistic model to form a number of parameter sets. In one or more embodiments, step 502 includes performing a near-random sampling process (or a random parameter screen) to find an initial parameter set for processing. In one or more embodiments, Latin hypercube sampling may be used to perform the near-random sampling process.

[0070] Step 504 includes selecting an initial parameter set from a plurality of parameter sets for the mechanistic model. For example, in step 504, a best-fit parameter set may be used as the initial parameter set. In one or more embodiments, step 504 may be referred to as a global optimization step, and may be performed, for example, using any of several 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-likelihood (or log-likelihood) algorithm, a negative log-likelihood algorithm, or a maximum likelihood algorithm. The initial parameter set formed in step 504 is a parameter set that allows further optimization to be performed. In one or more embodiments, step 504 includes identifying a search region in which extrema should be identified.

[0071] Step 506 includes calculating an extreme value for a selected loss function using an initial parameter set. In one or more embodiments, the selected loss function is a maximum log-likelihood (or log-likelihood), a negative log-likelihood, or a maximum likelihood algorithm. Thus, depending on the case, the selected loss function used in step 506 may be the same as or different from the loss function used in step 504. If the selected loss function is a negative log-likelihood (or log-likelihood), the calculated extreme value is a local minimum. If the selected loss function is a maximum log-likelihood (or log-likelihood), the calculated extreme value is a local maximum. If the selected loss function is a maximum likelihood, the calculated extreme value is a local maximum. In one or more embodiments, one or more different types of optimization algorithms may be used to identify the extreme value. For example, a minimization algorithm such as a Levenberg-Marquardt (LM) minimization algorithm may be used to identify the desired extreme value for the selected loss function.

[0072] Step 508 includes calculating a covariance matrix for the mechanistic model based on the selected loss function. The covariance matrix describes the underlying multi-parametric Gaussian distribution of the multiple parameters.

[0073] If the selected loss function is negative log-likelihood, the Hessian of the selected loss function is calculated at the local minimum and inverted to obtain the covariance matrix. If the selected loss function is maximum log-likelihood (or log-likelihood), the Hessian of the selected loss function is calculated at the local maximum and the inverted negative of this Hessian is used to obtain the covariance matrix. When the selected loss function is maximum likelihood, the Hessian is calculated for the log of the selected loss function at the local maximum and the inverted negative of this Hessian is used to obtain the covariance matrix.

[0074] V. Examples / Results FIG. 6 is a side-by-side comparison of two different types of multi-parametric plots generated for a mechanistic model according to various embodiments. In FIG. 6, a first multi-parametric plot 602 is generated using a Hessian method. A second multi-parametric plot 604 is generated using a Markov Chain Monte Carlo (MCMC) method. The first multi-parametric plot 602 and the second multi-parametric plot 604 identify distributions 606 and 608, respectively, for the parameters of the mechanistic model. Additionally, the first multi-parametric plot 602 and the second multi-parametric plot 604 identify covariances 610 and 612, respectively, between the parameters of the mechanistic model.

[0075] As shown in FIG. 6, the plots of both approaches appear to be similar. More specifically, the distribution 606 of the first multi-parametric plot 602 and the distribution 608 of the second multi-parametric plot 604 appear to be similar. Furthermore, the correlation 610 of the first multi-parametric plot 602 and the correlation 612 of the second multi-parametric plot 604 appear to be similar. These similarities reinforce the idea that the Hessian method may be used for sampling to estimate the uncertainty of predictions made using mechanistic models. Furthermore, the Hessian method may provide savings in terms of time, cost, and processing resources. For example, the second multi-parametric plot 604 may require more samples and data points and therefore be denser and require more time and processing resources than the first multi-parametric plot 602.

[0076] 7 shows a table 700 comparing the uncertainty values ​​generated by the Hessian method with those generated using the MCMC method. The best and 95% confidence values ​​shown in table 700 indicate that the Hessian and MCMC methods generate similar data. Thus, the Hessian method may be used in a variety of applications to reduce the time, cost, and processing resources associated with determining the uncertainty of mechanistic models, particularly mechanistic chromatography models.

[0077] VI. Computer-implemented systems FIG. 8 is a block diagram of a computer system according to various embodiments. Computer system 800 may be an example of one implementation of computing platform 204 described above in FIG. 2. In one or more examples, computer system 800 may include a bus 802 or other communication mechanism for communicating information and a processor 804 coupled to bus 802 for processing information. In various embodiments, computer system 800 may further include memory, which may be a random access memory (RAM) 806 or other dynamic storage device, coupled to bus 802 for determining instructions to be executed by processor 804. The memory may also be used to store temporary variables or other intermediate information during execution of instructions to be executed by processor 804. In various embodiments, computer system 800 may further include a read only memory (ROM) 808 or other static storage device coupled to bus 802 for storing static information and instructions for processor 804. A storage device 810, such as a magnetic disk or optical disk, may be provided and coupled to bus 802 for storing information and instructions.

[0078] In various embodiments, the computer system 800 may be coupled via the bus 802 to a display device 812, such as a cathode ray tube (CRT) or liquid crystal display (LCD), for displaying information to a computer user. An input device 814, including alphanumeric and other keys, may be coupled to the bus 802 for communicating information and command 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, eye gaze-based input device, or cursor direction keys, for communicating directional information and command selections to the processor 804 and controlling cursor movement on the display 812. This input device 814 typically has two degrees of freedom in two axes, a first axis (e.g., x) and a second axis (e.g., y), that allow the device to specify a position in a plane. However, it should be understood that input devices 814 that allow cursor movement in three dimensions (e.g., x, y, and z) are also contemplated herein.

[0079] According to certain implementations of the present teachings, results may be provided by computer system 800 in response to execution by processor 804 of one or more sequences of one or more instructions contained in RAM 806. Such instructions may be read into RAM 806 from another computer-readable medium or computer-readable storage medium, such as storage device 810. Execution of the sequences of instructions contained in RAM 806 may cause processor 804 to perform the processes described herein. Alternatively, hard-wired circuitry may be used in place of or in combination with software instructions to implement the present teachings. Thus, implementations of the present teachings are not limited to any specific combination of hardware circuitry and software.

[0080] As used herein, the term "computer-readable medium" (e.g., data store, data storage, storage device, data storage device, etc.) or "computer-readable storage medium" refers to any medium that participates in providing instructions to the processor 804 for execution. Such media may take many 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, and magnetic disks, such as storage device 810. Examples of volatile media may include, but are not limited to, dynamic memory, such as RAM 806. Examples of transmission media may include, but are not limited to, coaxial cables, copper wire, and fiber optics, such as the wires that comprise bus 802.

[0081] Common forms of computer-readable media include, for example, floppy disks, flexible disks, hard disks, magnetic tape or any other magnetic medium, CD-ROMs, any other optical medium, punch cards, paper tape, any other physical medium with a pattern of holes, RAM, PROMs, and EPROMs, flash EPROMs, any other memory chip or cartridge, or any other tangible medium from which a computer can read.

[0082] In addition to computer-readable media, instructions or data may be provided as signals on a transmission medium included in a communication device or system to provide a sequence of one or more instructions to the processor 804 of the computer system 800 for execution. For example, a communication device may include a transceiver having signals representing instructions and data. The instructions and data are configured to cause one or more processors to perform the functions outlined in the disclosure herein. Representative examples of data communication transmission connections may include, but are not limited to, a telephone modem connection, a wide area network (WAN), a local area network (LAN), an infrared data connection, an NFC connection, an optical communication connection, and the like.

[0083] It should be understood that the methods described in this specification, flowcharts, figures, and accompanying disclosure can be implemented using computer system 800 as a stand-alone device or on a distributed network of shared computer processing resources, such as a cloud computing network.

[0084] The methods described herein may be implemented by various means depending on the application. For example, the methods may be implemented in hardware, firmware, software, or any combination thereof. In the case of a hardware implementation, the processing unit may be implemented in 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 combinations thereof.

[0085] In various embodiments, the methods of the present teachings may be implemented as firmware and / or software programs and applications written in conventional programming languages ​​such as C, C++, Python, etc. When implemented as firmware and / or software, the embodiments described herein may be implemented on a non-transitory computer-readable medium having stored thereon a program for causing a computer to perform the above-mentioned methods. It should be understood that by providing the various engines described herein on a computer system such as computer system 800, processor 804 will perform the analysis and judgments provided by these engines according to instructions provided by any one or combination of memory components RAM 806, ROM 808, or storage device 810, as well as user input provided via input device 814.

[0086] While the present teachings have been described in conjunction with various embodiments, the present teachings are not limited to such embodiments, but rather encompass various alternatives, modifications, and equivalents, as will be appreciated by those skilled in the art.

[0087] In describing various embodiments, the specification may present the method and / or process as a particular sequence of steps. However, to the extent that the method or process does not depend on the particular order of steps described herein, the method or process should not be limited to the particular sequence of steps described, and as one of ordinary skill in the art will readily appreciate, the sequence may be varied and still remain within the spirit and scope of the various embodiments.

[0088] Enumeration of embodiments Embodiment 1. A method for estimating uncertainty in a mechanistic chromatography model, comprising: receiving a mechanistic model of chromatography including a plurality of parameters; for each of the plurality of parameters, identifying a corresponding range of values ​​based on a relationship between values ​​for the plurality of parameters; sampling each of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets; and quantifying uncertainty for the mechanistic model using the plurality of simulation sets.

[0089] Embodiment 2. The method of embodiment 1, wherein identifying the corresponding range of values ​​for each of the plurality of parameters includes calculating a covariance matrix for the mechanistic model based on a selected loss function.

[0090] Embodiment 3. The method of embodiment 2, wherein the selected loss function comprises at least one of a negative log-likelihood algorithm, a maximum log-likelihood algorithm, or a maximum likelihood algorithm.

[0091] Embodiment 4. The method of embodiment 2, wherein calculating the covariance matrix for the mechanistic model based on the selected loss function includes identifying a search space using at least one of the selected loss function or another loss function, and calculating an extreme value for the selected loss function over the search space.

[0092] Embodiment 5. The method of embodiment 4, wherein calculating the covariance matrix for the mechanistic model based on the selected loss function further comprises calculating the covariance matrix for the extreme values.

[0093] Embodiment 6. A method as described in any one of embodiments 1 to 5, wherein identifying the corresponding value range for each of the plurality of parameters includes sampling the plurality of parameters to form a plurality of parameter sets, selecting an initial parameter set from the plurality of parameter sets for the mechanistic model, and calculating a covariance matrix for the mechanistic model based on a selected loss function using the initial parameter set.

[0094] Embodiment 7. The method of any one of embodiments 1 to 6, wherein quantifying the uncertainty includes generating a model prediction distribution for the mechanistic model using the multiple simulation sets.

[0095] Embodiment 8. The method of embodiment 7, wherein quantifying the uncertainty further comprises identifying a confidence interval for the mechanistic model using the model prediction distribution.

[0096] Embodiment 9. The method of any one of embodiments 1 to 8, further comprising receiving experimental data and using said experimental data to generate said mechanistic model.

[0097] Embodiment 10. A system for estimating uncertainty in a mechanistic chromatography model, comprising: a data source configured to acquire a mechanistic model of chromatography; and a processor configured to receive the mechanistic model of chromatography from the data source, the mechanistic model including a plurality of parameters, the processor further configured to: identify, for each of the plurality of parameters, a corresponding range of values ​​based on a relationship between values ​​for the plurality of parameters; sample each of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets; and quantify uncertainty for the mechanistic model using the plurality of simulation sets.

[0098] Embodiment 11. The system of embodiment 10, wherein the processor is further configured to calculate a covariance matrix for the mechanistic model based on a selected loss function.

[0099] Embodiment 12. The system of embodiment 11, wherein the selected loss function includes at least one of a negative log-likelihood algorithm, a maximum log-likelihood algorithm, or a maximum likelihood algorithm.

[0100] Embodiment 13. The system of any one of embodiments 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 to calculate an extreme value for the selected loss function with respect to the search area.

[0101] Embodiment 14. The system of embodiment 13, wherein the processor is further configured to calculate the covariance matrix for the mechanistic model based on the selected loss function by calculating the covariance matrix for the extreme values.

[0102] Embodiment 15. A system described in any one of embodiments 10 to 14, wherein the processor is further configured to sample the plurality of parameters to form a plurality of parameter sets, select an initial parameter set from the plurality of parameter sets for the mechanistic model, and calculate a covariance matrix for the mechanistic model based on a selected loss function using the initial parameter set.

[0103] Embodiment 16. The system of embodiment 15, wherein the processor is further configured to generate a model prediction distribution for the mechanistic model using the multiple simulation sets, and to identify a confidence interval for the mechanistic model using the model prediction distribution.

[0104] Embodiment 17. A system described in any one of embodiments 10 to 16, wherein the processor is further configured to receive experimental data and generate the mechanistic model using the experimental data.

[0105] Embodiment 18. A non-transitory computer-readable medium having a program stored thereon, the program being configured to cause a computer to execute a method for estimating uncertainty in a mechanistic chromatography model, the method including: receiving a mechanistic model of chromatography including a plurality of parameters; for each of the plurality of parameters, identifying a corresponding range of values ​​based on a relationship between values ​​for the plurality of parameters; sampling each of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets; and quantifying uncertainty for the mechanistic model using the plurality of simulation sets.

[0106] Embodiment 19. The non-transitory computer-readable medium of embodiment 18, wherein the method further comprises calculating a covariance matrix for the mechanistic model based on a selected loss function.

[0107] Embodiment 20. The non-transitory computer-readable medium of embodiment 19, wherein the selected loss function includes at least one of a negative log-likelihood algorithm, a maximum log-likelihood algorithm, or a maximum likelihood algorithm.

[0108] Embodiment 21. The non-transitory computer-readable medium of embodiment 19, wherein the method further includes identifying a search space using at least one of the selected loss function or another loss function, and calculating an extremum for the selected loss function with respect to the search space.

[0109] Embodiment 22. The non-transitory computer-readable medium of embodiment 21, wherein the method further comprises calculating the covariance matrix for the extreme values.

[0110] Embodiment 23. A non-transitory computer-readable medium described in any one of embodiments 18 to 22, wherein the method further includes sampling the plurality of parameters to form a plurality of parameter sets, selecting an initial parameter set from the plurality of parameter sets for the mechanistic model, and calculating a covariance matrix for the mechanistic model based on a selected loss function using the initial parameter set.

[0111] Embodiment 24. A non-transitory computer-readable medium according to any one of embodiments 18 to 23, wherein the method further comprises generating a model prediction distribution for the mechanistic model using the plurality of simulation sets.

[0112] Embodiment 25. The non-transitory computer-readable medium of embodiment 24, wherein the method further comprises identifying a confidence interval for the mechanistic model using the model predictive distribution.

Claims

1. 1. A method for estimating uncertainty in a mechanistic chromatography model, comprising: receiving a mechanistic model of chromatography comprising a plurality of parameters; identifying, for each of the plurality of parameters, a corresponding range of values ​​based on relationships between values ​​for the plurality of parameters; Sampling each parameter of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets; Quantifying uncertainty about the mechanistic model using the plurality of simulation sets; and A method comprising:

2. The method of claim 1 , wherein identifying the corresponding range of values ​​for each of the plurality of parameters comprises calculating a covariance matrix for the mechanistic model based on a selected loss function.

3. The method of claim 2 , wherein the selected loss function comprises at least one of a negative log-likelihood algorithm, a maximum log-likelihood algorithm, or a maximum likelihood algorithm.

4. Calculating the covariance matrix for the mechanistic model based on the selected loss function includes: identifying a search space using at least one of the selected loss function or another loss function; calculating an extremum of the selected loss function over the search region; The method of claim 2 , comprising:

5. The method of claim 4 , wherein calculating the covariance matrix for the mechanistic model based on the selected loss function further comprises calculating the covariance matrix for the extreme values.

6. Identifying the corresponding range of values ​​for each of the plurality of parameters comprises: Sampling the plurality of parameters to form a plurality of parameter sets; selecting an initial parameter set from the plurality of parameter sets for the mechanistic model; calculating a covariance matrix for the mechanistic model based on a selected loss function using the initial parameter set; The method of claim 1 , comprising:

7. The method of claim 1 , wherein quantifying the uncertainty comprises generating a model prediction distribution for the mechanistic model using the multiple simulation sets.

8. The method of claim 7 , wherein quantifying the uncertainty further comprises identifying a confidence interval for the mechanistic model using the model prediction distribution.

9. Receiving experimental data; and generating the mechanistic model using the experimental data; and The method of claim 1 further comprising:

10. 1. A system for estimating uncertainty in a mechanistic chromatography model, comprising: a data source configured to obtain a mechanistic model of chromatography; a processor configured to receive the chromatographic mechanistic model from the data source; Equipped with The mechanistic model includes a plurality of parameters; The processor, identifying, for each of the plurality of parameters, a corresponding range of values ​​based on relationships between values ​​for the plurality of parameters; Sampling each parameter of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets; Quantifying uncertainty about the mechanistic model using the plurality of simulation sets. The system further comprises:

11. The system of claim 10 , wherein the processor is further configured to calculate a covariance matrix for the mechanistic model based on a selected loss function.

12. The system of claim 11 , wherein the selected loss function comprises at least one of a negative log-likelihood algorithm, a maximum log-likelihood algorithm, or a maximum likelihood algorithm.

13. 11. The system of claim 10, wherein the processor is further configured to identify a search space using at least one of the selected loss function or another loss function and to calculate an extremum of the selected loss function over the search space.

14. 14. The system of claim 13, wherein the processor is further configured to calculate the covariance matrix for the mechanistic model based on the selected loss function by calculating the covariance matrix for the extreme values.

15. 11. The system of claim 10, wherein the processor is further configured to sample the plurality of parameters to form a plurality of parameter sets, select an initial parameter set from the plurality of parameter sets for the mechanistic model, and calculate a covariance matrix for the mechanistic model based on a selected loss function using the initial parameter set.

16. 16. The system of claim 15, wherein the processor is further configured to generate a model prediction distribution for the mechanistic model using the multiple simulation sets, and to identify a confidence interval for the mechanistic model using the model prediction distribution.

17. The system of claim 10 , wherein the processor is further configured to receive experimental data and generate the mechanistic model using the experimental data.

18. A non-transitory computer readable medium having stored thereon a program configured to cause a computer to execute a method for estimating uncertainty in a mechanistic chromatography model, the method comprising: The method further comprising: receiving a mechanistic model of chromatography comprising a plurality of parameters; identifying, for each of the plurality of parameters, a corresponding range of values ​​based on relationships between values ​​for the plurality of parameters; Sampling each parameter of the plurality of parameters within the corresponding range of values ​​for each parameter to form a plurality of simulation sets; Quantifying uncertainty about the mechanistic model using the plurality of simulation sets; and A non-transitory computer readable medium comprising:

19. 20. The non-transitory computer-readable medium of claim 18, wherein the method further comprises calculating a covariance matrix for the mechanistic model based on a selected loss function.

20. 20. The non-transitory computer-readable medium of claim 19, wherein the selected loss function comprises at least one of a negative log-likelihood algorithm, a maximum log-likelihood algorithm, or a maximum likelihood algorithm.

21. The method comprises: identifying a search space using at least one of the selected loss function or another loss function; calculating an extremum of the selected loss function over the search region; 20. The non-transitory computer readable medium of claim 19, further comprising:

22. 22. The non-transitory computer-readable medium of claim 21, the method further comprising calculating the covariance matrix for the extreme values.

23. The method comprises: Sampling the plurality of parameters to form a plurality of parameter sets; selecting an initial parameter set from the plurality of parameter sets for the mechanistic model; calculating a covariance matrix for the mechanistic model based on a selected loss function using the initial parameter set; 20. The non-transitory computer-readable medium of claim 18, further comprising:

24. 20. The non-transitory computer-readable medium of claim 18, wherein the method further comprises generating a model prediction distribution for the mechanistic model using the multiple simulation sets.

25. 25. The non-transitory computer-readable medium of claim 24, wherein the method further comprises identifying a confidence interval for the mechanistic model using the model predictive distribution.

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