WO2009148962A1

WO2009148962A1 - Design of dynamic experiments for the modeling and optimization of batch process

Info

Publication number: WO2009148962A1
Application number: PCT/US2009/045684
Authority: WO
Inventors: Christos Georgakis
Original assignee: Tufts University
Priority date: 2008-05-29
Filing date: 2009-05-29
Publication date: 2009-12-10

Abstract

The present invention relates to systems and methods for the dynamic design of experiments for modeling and optimization of batch processes in a wide variety of fields. In particular, the present invention provides systems and methods for identifying, selecting, implementing, and improving upon batch, semi-batch, and/or fed-batch processes.

Description

DESIGN OF DYNAMIC EXPERIMENTS FOR THE MODELING AND OPTIMIZATION OF BATCH PROCESS

The present application claims priority to United States Provisional Patent Application Serial Numbers 61/057,034, filed May 29, 2008 and 61/115,336, filed November 17, 2008, the disclosures of which are herein incorporated by reference in their entireties.

FIELD OF THE INVENTION The present invention relates to systems and methods for the design of dynamic experiments for modeling and optimization of batch, semi-batch, or fed- batch processes in a wide variety of fields. In particular, the present invention provides systems and methods for identifying, selecting, implementing, and improving upon batch, semi-batch, and/or fed-batch processes.

BACKGROUND OF THE INVENTION

Processes in the chemical, pharmaceutical, food processing, and other industries often employ a complex set of operating conditions that can be altered to affect the outcome, as well as the efficiency of achieving the desired outcome. For centuries, certain processes have been honed by trial-and-error. In more modern times, experimental testing or modeling of processes has led to improvements in performance and efficiency. However, with many processes, it is simply not practical, or possible, to test every combination of operating conditions to select the best outcome or most efficient route of achieving a suitable outcome. What are needed are systems and methods for identifying, selecting, implementing, and improving upon processes without requiring extensive or prohibitive experimentation or testing.

SUMMARY OF THE INVENTION In certain embodiments, the present invention provides novel systems and methods for designing time varying experiments to enable rapid input-output modeling for chemical, pharmaceutical, biopharmaceutical and other processes so that they can be optimized. Many chemical, pharmaceutical, biopharmaceutical, and a variety of other processes are not understood in the greatest detail for an accurate fundamental model, describing their inner workings, to be developed and used to optimize their operation. Such optimization defines the best operation, which leads to the highest product quality or maximum process profit or the lowest processing cost.

Embodiments of the present invention provide systems and methods that assess processing conditions over time to identify, select, implement, and improve upon processes. In most batch, semi-batch, and/or fed-batch operations, properly varying the processing conditions over time will result in better performance. For example, in the operation of a reactor that produces a certain chiral compound through a hydrogenation step, one can imagine that a change over time of the reactor temperature or the feeding rate of the catalyst or the feeding rate of a reactant, might lead to a larger yield of the desired molecule. The same can be said about the feeding profile of the single substrate or multiple ones whether simple or complex in nature or composition in the operation of a bioprocess. In crystallization or precipitation processes, a change in temperature is necessary to crystallize out the desired molecule or protein. In most cases, a constant cooling rate is used. A cooling rate that changes with time, to keep a constant supersaturation will be more desirable. At present, calculating the optimum time dependence of such operations can only be done by the use of a fundamental model describing the internal workings of the reactor or the crystallizer. The development of such a fundamental model requires a substantial investment in effort and time and it is not frequently undertaken, especially in the most complex of processes.

In some embodiments of the present invention, the factors or operating conditions vary with time instead of being kept constant, as in the traditional approachs (see e.g., Box and Lucas, Design of Experiments in Non-Linear Situations. Biometrika, 1959. 46(1-2): p. 77-90; Kuehl, Design of Experiments: Statistical

Principles of Research Design and Analysis 1999: Brooks/Cole; Box, Fisher, R. A. and the Design of Experiments, 1922-1926. American Statistician, 1980. 34(1): p. 1-7, herein incorporated by reference in their entireties). Besides assigning to a factor or operating condition a maximum, minimum or intermediate constant value, one can also examine the effect of several time -varying functionalities or patterns of the same factor or operating condition. For example, one can consider a linearly decreasing and a linearly increasing dependence on time, a dependence that is decreasing and then increasing or increasing and then decreasing, along with more involved time dependent factors or operating conditions. The data collected from such experiments enables the calculation of a response surface model (RSM). This model can be used to optimize the process by selecting the appropriate values of the traditional fixed factors as well as the appropriate combination of the dynamic patterns defining each novel dynamic factor. For example, one might calculate that in order to optimize the yield of the desired compound, the initial concentration of the reactant (traditional factor) should be between the maximum and minimum values examined, and that the batch reactor temperature should start at the maximum value, be reduced to a value close to the minimum at the end of the first hour of the batch remaining constant at that value for half an hour and then rise very slowly to the mid-point temperature value by the end of the three hour batch run. Discovering such a time varying profile of the temperature can only be achieved at present with a detailed model of the process. Such models are indeed very useful as they provide valuable insight in the process. However, they require some understanding of the physical, chemical and biological phenomena taking place. Furthermore, they need the investment of significant amount of time and effort for their development.

Embodiments of the systems and methods of the present invention simplify the dynamic optimization of batch processes without the need for a fundamental processes model. For example, one can denote with Jo the best operation of the process of interest achieved with the classical Design of Experiment (DoE or DoX) approach and the use no prior fundamental model. One can also denote with Joe the best performance of the process. This can only be achieved with a detailed fundamental model at hand. Under the approaches provided by embodiments of the systems and methods of the present invention, and without the use of a fundamental model, the result of the best performance of the process achieved is denoted by J_D. If one wishes to maximize (or minimize) the value of J, then one has Joe > Jo (or Joe < Jo). One could expect that under all circumstances the methodologies described herein will result in J_D > Jo (or J_D < Jo), as the systems and methods are utilizing additional degrees of operational freedom. More importantly, one can expect that the methodologies achieve, for example, 60-70% of the total possible process improvement {|JD - Do|> (0.6 - 0.7)* | Joe - Jo| } with the information the additional time dependent experiments have provided. For complex industrial processes, the time and expenses that are required for the additional experiments of the methodologies described herein are much smaller than what it is needed to develop a fundamental mathematical model to use in calculating the optimal strategy.

In some embodiments, an aspect of the methodologies is the proper systematic design of the dynamic experiments. Let us assume that the time -varying input factor (control variable) is u(t) and the batch time is fixed to a value of tb. The goal is to eventually calculate a u(t) vs. t function that maximizes a certain performance characteristic J(u) of the process that depends on the choice of the function u(t). To design the dynamic experiment that reveals the approximate dependence of the J functional on u(t), one expands the unknown u(t) function in terms of shifted Legendre (or other) orthogonal polynomials in the interval (0, tb): _u^ ₌ y^N _a p _(t) .

Each Legendre (or other) polynomial P_n(t) or any other appropriate, linearly independent (orthogonal or non-orthogonal) set of non-polynomial functions is treated as an independent dynamic factor. For all practical purposes N is smaller than 10 and likely between 3 and 5, although the present invention is not limited by the value of N. In some embodiments, N is equal to 2, 3, 4, 5, or 6. If one limits oneself, for now, to N=3, the three first Legendre polynomial will be involved: P₀(t)=l, P₁(X)= 1- (t/tb); P2(t)=+l+6(t/tb)²-6(t/tb). A novel component of this problem treats the coefficients a_n as the dynamic factors of the input function u(t). If one assumes that the maximum values the a_n coefficients can take are -1 and +1, then there are 8 dynamic experiments to be +performed (=2 ) according to the classical DoX approach are: (+, +, +), (+, +, -), (+, -, +),(+, -, -), ... (-, -, -). For example the eighth experiment, (-,-,-), will have as an input function: uβ(t)=-P_o-Pi(t)-P2(t)=-3+8(t/tb)- 6(t/t_b)².

This is a profile that starts low (-3), rises linearly initially (by 8(t/tb)), levels off and drops slightly (by 6(t/tb)²) to the final value ofi-l. This is a very versatile profile obtained with the use of just three orthogonal polynomials. In the case that the overall dynamic coded variable u(t) need be constrained between -1 and +1 the initial values (-1, -1, -1) of the a_n coefficients are reduced appropriately and simultaneously, in the above example to (-1/3, -1/3, -1/3). Once the experiments are performed, a response surface is calculated treating each of the polynomials used to describe the time-dependence as independent factors fi and f₂ for N=2. Let us assume that the response surface model obtained from a three (3) level full factorial design is given by: J(M) = 2.2+1.49a_o+2.54a₁+1.41a_oa₁-3.22ao,-3.34af

Then the optimum value of J is sought by constraining the a; value in the interval (-1, +1) as well as their four (4) combinations (+ao+a_l5 +ao-a_ls -ao+a_l5 -ao-ai) to be less than 1. This will ensure that the time varying function u(t)=aoP₀(t)+aiPi(t) will be constrained between the values of - 1 and + 1. The maximum value of J(u) is calculated, using a constrained optimization algorithm, to be equal to 3.0171 and is attained by the operation that is characterized by ao=O.33 and ai= 0.45. The corresponding time-dependent optimal profile u(t) is equal to 0.33P_o(t)+ 0.45Pi(t)=0.74-0.9t varying linearly from 0.74 to -0.16. If one were to explore a finer time dependence of u(t), a larger value of N should be considered. However a full factorial design would require 2^N experiments, a number which quickly grows with N. One could anticipate that a full factorial design with N=3 or 4 or a fractional factorial design with N=4 or 5 is sufficient for a substantial dynamic optimization of the process, although the present invention is not limited to these ranges. The methodologies of the present invention find use in a wide variety of different phases of process selection and management. In some embodiments, the methodologies are used to identify and select a process for initial implementation. In some embodiments, the methodologies are used to optimize an existing process. In some embodiments, multiple iterations of the methodologies are employed to continuously improve a process.

The present invention is not limited in manner in which the methodologies are carried out. In some embodiments, a system comprising a computer processor is employed. In some embodiments, the system comprises software that encodes one or more of the functionalities described herein. In some embodiments, a system comprises a user interface that permits the user to select and test experimental operating conditions. In some embodiments, the user interface displays results of experiments. In some embodiments, the user interface displays the best solution to a particular problem, based on user inputted criteria. In some embodiments, the system is included with a control component for implementing a process. Thus, in some such embodiments, optimized processes identified by the methodologies are implemented by the system. In some embodiments, the system collects and/or stores data of implemented processes. In some embodiments, the system further includes additional functionality. For example, in some embodiments, the system further include DoE or DoX software (e.g., DESIGN EXPERT, JMP, MINITAB, SAS, SPSS, etc.) or other desired functionality. In some embodiments, the system is configured to conduct a large number of parallel experiments for use in optimizing, for example, examination of different reaction or crystallization conditions or any other process that may involve a large number of alternative reaction conditions.

For example, in some embodiments, the present invention provides methods for industrial process design comprising one or more of the steps of: a) defining a process by one or more input variables or conditions and one or more output variables or conditions, wherein one or more of the input variables or conditions are dynamic variables or conditions over the process duration; b) simulating the process through a set of experiments, wherein one or more said dynamic variables or conditions is varied over the process duration (e.g., according to a systematically selected linear combination of a limited number of dynamic patterns of interest); c) collecting data from the experiments; wherein the data comprises values for the input variables or conditions (e.g., including the characteristics of the dynamic ones) and the output variables or conditions over the duration of the process; and d) determining optimized values for the input variables or conditions (e.g., and the related optimal combination of the dynamic patters) over the process duration which yield optimized values for the output variables or conditions. In some embodiments, the method further comprises the step of e) designing the industrial process based on the optimized values for the one or more input variables or conditions over the process duration. In some embodiments, the method further comprises the step of f) conducting the industrial process with the optimized input variables or conditions over the process duration. In some embodiments, the industrial process comprises a batch process or a semi-batch process or a fed-batch process. The present invention is not limited by the nature of the industrial process. In some embodiments, the industrial process is employed in an industry selected from the list comprising: chemicals industry, specialty chemicals industry, pharmaceutical industry, biopharmaceutical industry, alternative energy industry, bio-fuels industry, food industry, beverage industry, and manufacturing industry. The present invention is not limited by the nature of the input or output variables or conditions. In some embodiments, the input variables or conditions are selected from the list comprising: temperature, pressure, rate, flow, volume, reagent concentration, catalyst concentration, pH, level, and force. In some embodiments, the output variables or conditions are selected from the list comprising cost, energy, product amount, efficiency, and rate. The methods may employ a wide variety of approaches in the design of the selection of the family of dynamic patterns and the maximum finite number of such dynamic patterns within the selected family that are used in the characterization of the dynamic input valuables or conditions to be used in the design of the experiments. In some embodiments, the number of dynamic patterns used for each dynamic input variable or condition is limited to a finite set so that the number planned experiments remains finite. The data thus collected is used to calculate a finite set of unknown coefficients or parameters of an input-output model, such as the Response Surface Model (RSM), for identifying the optimal input variables or conditions that will yield the desired optimal output variables or conditions. In some embodiments, the number of dynamic patterns considered for each dynamic input variable or condition is 2, 3, 4, 5 or 6, although the present invention is not limited to this range and a skilled artisan may select an appropriate number based on the problem being solved and the level of optimization desired.

In some embodiments, the methods employ a set of experiments equal, at most, to 2^{n+m*(N 1)} for level-two full factorial designs, wherein n in the total number of input variables or conditions examined, m (m<n) is a subset of these variables that is varied over time and N is the maximum number of dynamic patterns considered; and ^^n+m*(N-ⁱ⁾ fø_{r a} [gygi.tøj-gg experiment. In the case classical DoE designs with n variables, none of which changes with time, the number of experiments is 2ⁿ and 3ⁿ, respectively. For the case of n=4 this is 2⁴=16 and 3⁴=81. If one of these input variables is considered to be a time varying one and two dynamic patterns are considered, then m=l and N=2 and the number of experiments needed in a full factorial design are 2⁴⁺¹= 15 or 3⁴⁺¹=243. The above upper limits on the number of experiments can be drastically reduced by considering a wide variety of fractional factorial designs, central composite t designs, response surface designs as well as a variety of other design such as D-optimal designs.

The methods may be applied to one or more processes in an industrial setting. In some embodiments, the methods are applied to a partial or incompletely modeled process to optimize the process. In some embodiments, the methods are applied to any process that does not have a developed fundamental model. In some embodiments, the methods are used to optimize a process that interacts with a second, well-modeled process, wherein the method is not applied to the second process, but only to the first. The present invention further provides systems for industrial process design comprising a processor configured for carrying out any of the above described methods.

DESCRIPTION OF THE FIGURES Figure 1 shows experiments conducted using embodiments of the DoDE (also referred to as DDoX) approach for level-two (A, N=2; B, N=3; C, N=4; D, N=5) and level-three full factorial experiments (E, N=2; F, N=3; and G, N=4). Figure IH shows a comparison of the 24 experiments of central composite design for N=3, compared to the 27 of the three-level full factorial design above of Fig. IF. For N=4, the central composite design requires 36 experiments (Figure II) compared to the 81 of the full factorial design (Figure IG).

Figure 2 shows a Model-Based Optimal Concentration and Temperature Profile (in terms of ⁰C and the coded variable w(t)) for the optimization of the operation of a batch reactor in which a reversible reaction between reactant A and product B takes place with the following characteristics:

Ic₁ = k ₁₀ exp(-E_[ I RT) [1/hr]; k₂ = k₂₀ exp(-E₂ I RT) [1/hr] k_lo=1.32xlθ⁷; k₂₀=5.24xl0¹³; .E₁ = IO₁OOO; E₁ = 20,000

Figure 3 shows the time dependencies of the feeding flowrate of a substrate considered in a level-two DoDE (or DDoX) design where the only two dynamic patterns examined are those described by the first two shifted Legendre polynomials. The number of experiments is 16, (16=2²x2²) with 4=2² variations of the feed rate in the growth phase (0 <t<t_f) and 4=2² additional federate variations for the production phase (t_f <t<t_b).

Figure 4 shows one of the above (Fig. 3) 16 simulated penicillin fermentation process, denoted in Table 3 of Example 4 below as run DBl 1. This run produced 86.08 g of product and was the best of the 16 experiments designed and run. Figure 5 shows a simulated penicillin fermentation process that represents the optimal operation within the constraints examined.

DETAILED DESCRIPTION OF THE INVENTION The selection and design of different process steps in, for example, a manufacturing process, is a great challenge. One strives to balance achieving a desired output, while managing resources in obtaining the output. In an ideal world, one could test every combination of variables or operating conditions experimentally to determine the best combination. In practice, this level of experimentation can be impossible or imprudent. At the other end of the spectrum, selecting process steps based on simple, incomplete, or inaccurate models can lead to non-optimized outcomes. In some cases, this may result in a loss of time or money or product quality. In some cases, it many may spell the end of a business or the inability to make a desired product. To date, there is insufficient modeling for selecting the optimal variables or operating conditions of batch, semi-batch, and/or fed-batch processes, without over-committing resources to the identification of those optimized processes. The present invention provides systems and methods for selecting highly optimized processes, without the need for committing undue resources to the analysis. One prior approach to selecting and optimizing processes is called the Design of Experiments (DoE) methodology (also known as DoX). The DoE methodology has aimed and has succeeded in optimizing the condition of a very wide host of processes by systematically changing the conditions of operation and recording the results of the process. Despite its wide success, the DoE approach is limited in a very important way. It considers only conditions, or factors, that are constant with time. In contrast to this approach, the systems and methods of embodiments of the present invention systematically designs a set of time evolving experiments. The result of these experiments is used to calculate the optimum time evolving conditions of operation to permit selection of appropriate conditions for running a process. A major advantage of embodiments of the present invention is that the methodology achieves optimum time evolving conditions without the use of an a priori model that describes in some accuracy the process characteristics.

If, for example, one considers a case with a complex reaction system in a batch reactor, the classical design of experiments methodology defines experiments in which the initial compositions of the reacting mixture as well as the reactor temperature are assigned low, middle or high, albeit constant- with-time, values in each of the experiments. Systems and methods of embodiments of the present invention, in contrast, employ experiments in which some or all of the considered factors are varying with time, as necessary or desired. Because more than one pattern of time variation is examined, the number of experiments increases. Likewise, better information is obtained. However, the amount of experimentation required is quite manageable. Indeed, as shown in the examples below, a surprising amount of optimized information can be achieved with a limited number of experiments.

The present invention is not limited by the nature of the process utilized with the systems and methods described herein. One of skill in the art will recognize a wide variety of industries and process type to which the systems and methods may be employed. Several non-limiting examples include:

Chemicals and Specialty Chemicals Industry: there are many processes that are operated in batch or semi-batch mode. Such processes are used to perform chemical transformations, separations, solid and powder handling steps and many others.

Pharmaceutical Industry: similar chemical transformations and separations, via crystallization and many other steps, are used to produce the active pharmaceutical ingredient. They are followed by many solid handling steps that prepare the final product in which the medicine is delivered to the patient, in the form of a solid pill, a semi-liquid gel, or liquid syrup.

Biopharmaceutical Industry: where fermentation and other processing steps are used to grow populations of cells that then secrete a compounds such as proteins of certain therapeutic utility. Alternative Energy Industry I Bio-fuels Industry: where the there is substantial need to quickly design and optimally operate new processes about which the understanding of their inner workings is rather limited at present.

Food Industry: where production of the processed food products invariably following a batch or semi-batch operation. Beverage Industry: where the preparation of a product takes place in a batch processing environment where a process step could last from minutes, to hours, to even years, as it is the case with wine, whiskey, brandy and others products.

Plastics, Rubber, Textiles, Tobacco as well as other Industries: which employ a variety of batch or semi-batch processing steps. The overwhelming majority of these processes are operated in batch or semi- batch or fed-batch utilizing procedures and recipes that have been arrived at empirically. The systems and methods of the present invention enable the systematic examination of new time varying operation and procedures that can substantially improve the quality of the products manufactured and/or reduce the operational and other expenses.

In some embodiments, computer systems are provided that are configured to carry out any one or more of the methods described herein. For example, in some embodiments, computing devices such as desktop computers, hand-held computers, or similar devices comprise a processor configured to run a program that conducts any of the methods and processes described herein. In some embodiments, computer memory and software are provided with the system. The system may comprise data and command entry components for receiving information and commands from a user and a display for displaying results and/or a user interface to a user. In some embodiments, the user is queried by the user interface to select variables or operating conditions that are to be assessed. The processor then conducts the calculations and displays results. In some embodiments, a computer memory is provided for storing input variables or operating conditions, data, experimental results, or other desired information.

DEFINITIONS

To facilitate an understanding of the present invention, a number of terms and phrases are defined below: As used herein, the terms "computer memory" and "computer memory device" refer to any storage media readable by a computer processor. Examples of computer memory include, but are not limited to, RAM, ROM, computer chips, digital video disc (DVDs), compact discs (CDs), hard disk drives (HDD), and magnetic tape.

As used herein, the term "computer readable medium" refers to any device or system for storing and providing information (e.g., data and instructions) to a computer processor. Examples of computer readable media include, but are not limited to, DVDs, CDs, hard disk drives, magnetic tape and servers for streaming media over networks. As used herein, the terms "processor" and "central processing unit" or "CPU" are used interchangeably and refer to a device that is able to read a program from a computer memory (e.g., ROM or other computer memory) and perform a set of steps according to the program. As used herein the term "encode" refers to the process of converting one type of information or signal into a different type of information or signal to, for example, facilitate the transmission and/or interpretability of the information or signal. For example, audio sound waves can be converted into (i.e., encoded into) electrical or digital information. Likewise, light patterns can be converted into electrical or digital information that provides and encoded video capture of the light patterns.

As used herein the term "in electronic communication" refers to electrical devices (e.g., computers, processors, communications equipment, etc.) that are configured to communicate with one another through direct or indirect signaling. For example, a computer configured to transmit (e.g., through cables, wires, infrared signals, telephone lines, satellite, etc) information to another computer or device, is in electronic communication with the other computer or device.

As used herein the term "transmitting" refers to the movement of information (e.g., data) from one location to another (e.g., from one device to another) using any suitable means.

Certain exemplary implementations of the invention are described in more detail below. These examples are not intended to limit the scope of the invention.

EXAMPLE 1 Optimization design methodology

The following provides an example application of the systems and methods of the present invention for modeling and optimization of a generic batch process. In the following example, a set of experiments is designed in which, for example, one of the dynamic factors is changed with time. A finite number of specific time dependences are selected from an infinite number of possibilities in order to quickly and effectively select the time dependent operation that optimizes the process results. The duration of the batch is assumed to be tb (e.g., tb hours or any other pertinent units of time). The variable τ is assigned to be a dimensionless time, where τ=t/t_b, representing the fraction of the process duration. The dimensionless variable u(t), referred to as the coded variable, that varies between -1 and +1, characterizes the time dependent process variable, or dynamic factor. If, for example, the dynamic factor is the reactor temperature and it is allowed to very between T_max and Tm_1n, then the coded variable u(t) is defined by:

T(t) - 0 5 * (T + T )

( VT max - T mm λ ^

An appropriate functional basis {(pi(t)| with i=l , 2, 3, ... } defined in the interval (0, 1) is selected. The set of functions is a linearly independent set so that it serves as a functional basis. The functional basis could be either an orthogonal or a non- orthogonal basis. The selection of this basis is not necessarily influenced by the expected character of the problem's solution, although such influence may aid in reducing the number of needed experiments. The unknown value of the dynamic factors u(t) that maximizes a certain performance index of the process J(u(t)) is denoted by, u*(t):

J(«*) = maxJ(«) The unknown value is expanded in terms of an linear combination of the basis functions (pi(t), which linear combination is often truncated to a finite sum of N terms:

l

In the case that u is not the optimal profile the * superscript is dropped. Depending in the selected value of N, the optimal value C₁ of the unknown coefficients C₁; z-1 ,2,3, ... ,N is determined. These initially unknown constants, are considered the N sub-factors that characterize the unknown dynamic factor u(t).

The methodology can precede utilizing tools already developed for classical DoE methodology. An appropriate response surface for J from the performed experiments is determined, and the optimal values of the C₁ coefficients that determine the optimal operation of the process are calculated.

EXAMPLE 2 Full factorial experiments

In the following examples, Legendre polynomials are used as the basis functions to design two-level full factorial experiments. The following examples represent the set of experiments that are performed in order to collect the necessary information to optimize the given processes. For N=2, 2²= 4 experiments are required, and the 4 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure IA. For N=3, 2³= 8 experiments are required, and the 8 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure IB. For N=4, 2⁴= 16 experiments are required, and the 16 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure 1C. For N=5, 2⁵= 32 experiments are required, and the 32 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure ID. In the following examples, Legendre polynomials are also used as the basis functions to design three-level full factorial experiments. The following examples represent the set of experiments that are performed in order to collect the necessary information to optimize the given processes. For N=2, 3²= 9 experiments are required, and the 9 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure IE. For N=3, 3 = 27 experiments are required, and the 27 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure IF. For N=4, 3⁴= 81 experiments are required, and the 81 different time variations of the u(t) dynamic factors in these experiments are illustrated in Figure IG. In order to examine the rich dynamic behaviors (high N) and also reduce the number of experiments needed, fractional factorial designs or response surface designs are used. The 24 experiments of central composite design for N=3, compared to the 27 of the three-level full factorial design above, is demonstrated in the Figure IH. For N=4, the central composite design requires 36 experiments (Figure II) compared to the 81 of the full factorial design (Figure IG). Similar reductions in the number of dynamic experiments are achieved by using such fractional factorial designs, as is the case of classical DoE with static factors.

EXAMPLE 3 Optimization of a reversible reaction with regard to the temperature profile

The following exemplary embodiment provides optimization calculations using the systems and methods of the present invention related to a simulated batch chemical reactor in which a reversible reaction is considered. The optimum reactor temperature profile over time, that will maximize the conversion at the end of a batch, is calculated. The reaction and its overall reaction rate are as follows:

A < ' > B; r = k,C_A-k₂C_B with k_l = k _l0 CXPi-E₁ 1 RT) [1/hr]; k₂ = k₂₀ exp(-E₂ / RT) [1/hr] k_lo=5.24xl0¹³; k₂₀=1.32xl0⁷; E₁ = IO₅OOO; E₂ = 20,000

For a batch time tb = 2.5 hours, the dimensionless time θ=t/tb.is defined. The reactor temperature is bound by 20<T<50 ⁰C. The design utilizes the shifted Legendre polynomials P_n(t)as the Cp₁(I;) basis functions.

Table 1

If the model of the batch reactor was known and sophisticated optimization techniques used, the calculated true optimum conversion is 77.68% with a decreasing temperature profile (SEE FIG. 2). The coded variable, u(t), the reactor temperature in ⁰C (=35+15u(t)) and the reactant and product concentrations are plotted against the reaction time t (hr). One performs some experiments to define the model and estimate the model parameters. Even in the case that one correctly assumes the kinetics are first-order, then a minimum of four experiments are used to estimate the parameters k₁₀, k₂o, Ei and E₂. As demonstrated in Table 1, the design DA6 with only 3 classical DoE experiments , none of which is characterized by an input variable or condition that is changing with time, provided a suboptimum solution that is just 4.82% different from the model-based optimum one. The difference from the model based optimum of 77.68% decreases to 1.43% with the suboptimum calculation achieved with 4 experiments in design DA2 or to 0.14% with 9 DoDE experiments in design DA7.

The impact of measurement error can be calculated by performing a systematic Analysis of Variance (ANOVA) and by calculating the P-values related to the coefficients of the response surface model both well known in the classical DoE methodology.

Table 2 displays the optimal conversion predicted by the RSM and the resulting "experimental" conversion values obtained through the nonlinear simulation of the optimal profile. These results are for a level-three full factorial design, and three dynamic patterns (N=3). The impact of the percent measurement error in the final conversion measurement at the end of the reaction has been examined. One hundred runs were simulated.

Table 2

As the measurement error increases, it is seen that there is an increasing difference between the average of the RSM-predicted optimal conversion and the resulting average "experimental" optimal conversion. Furthermore, the standard deviation of the RSM-predicted as well as the standard deviation of the "experimental" optimum conversion is much smaller than the standard deviation of the measurement error. In fact, with the exception of the 10% case, the above difference is much larger: 0.83% and 1.42% compared to corresponding measurement error of 5%. This demonstrates that the measurement error does not impact at all the power of the method.

EXAMPLE 4 Optimization of penicillin fermentation

The following exemplary embodiment simulates optimization of the penicillin fermentation model of Bajpai and Reuss (Biotechnology & Bioengineering, 28, (1984) p.1396, herein incorporated by reference in its entirety) using the systems and methods of the present invention. This model has been the center of attention in several model-based optimizations (see for example: Riascos & Pinto. Chem. Engng, Sci. 99, (2004), pp 23-44.; herein incorporated by reference in its entirety). The model used to simulate the experiment consists of the equations:

dV x, = V : dt /^s f dx dt s_fV k_Λ x - ?

X₄ = /? : - dfp = px - k ,d i — P ^- s * t > pmax ^ - - -_: - dt s V A \ << - \ k

The batch time is fixed to t_b=135 hrs and the growth phase of the biomass to tf=30 hrs when the substrate is feed with a high flow rate. Experiments with different values for these two non-dynamic factors (tf tt,,) can be defined. Such design would follow the classical DoE methodology and could easily be combined with the dynamic design discussed here . The systems and methods of the present invention (DoDE) can be used to optimize the process using 16=2²x2² experiments, with 4=2² variations for the growth phase (0 <t<tf) and 4=2² additional variations for the production phase (tf <t<tb) (SEE FIG. 3). However some of these designs result in increasing the bioreactor volume by more than 4 liters from the initial value of 7 liters. To constrain the final volume value to a maximum of 11 It, some of the a; coefficients of the dynamic sub-factors need be scaled back further, as shown in experiments DB 12, DB 13 , DB 15 , and DB 16 of Table 3. The definition of the above feeding profiles and results of the 16 experiments with respect to the performance index which is the total amount of the product at the end of the fermentation are given in Table 3.

Table 3

Figure 4 presented here describes the time evolution of the main characteristics of the simulated fermentation process for run DBl 1 described in Table 3. This is the best of the set of 16 DoDE experiments designed and run. This run (DBl 1) produces 86.08 g of product. Construction of the response surface function and a constrained (V(tb)<l 1) optimization yields an optimum of the penicillin process with the production of J=V(tf)p(tf)= V(tf)x₄(tf)=102.30 grams of product with a feeding profile that is characterized by the following values of the a^ parameters:

If the optimum experiment were run, simulating the process through its model, 104.17 g of product is produced (See FIG. 5). The value is 1.83% larger than the prediction of response surface. The difference (a welcome positive one) is caused by the nonlinear character of the fermentation process and the use of a level-two experimental design which necessitates that the response surface has only linear and interaction terms . No quadratic terms are allowed. A more accurate response surface could have been constructed if one used a level-three design or a response surface design, using systems and method of the present invention. Nevertheless, the optimal run (Fig. 5) produced 16.96% more product that the best of the initial 16 runs, and this by using only two dynamic patterns described by the first two shifted Legendre polynomials.

All publications and patents mentioned in the present application are herein incorporated by reference. Various modification and variation of the described methods and compositions of the invention will be apparent to those skilled in the art without departing from the scope and spirit of the invention. Although the invention has been described in connection with specific preferred embodiments, it should be understood that the invention as claimed should not be unduly limited to such specific embodiments.

Claims

CLAIMSI claim:

1. A method of industrial process design comprising: a) defining a process by one or more input variables or operating conditions and one or more output variables or operating conditions, wherein one or more of said input variables or operating conditions are dynamic variables or operating conditions over the process duration; b) simulating the process through a set of experiments, wherein one or more said dynamic variables or operating conditions is varied over the process duration; c) collecting data from said experiments; wherein said data comprises values for said input variables or operating conditions and said output variables or operating conditions over the duration of said process; and d) determining optimized values for said input variables or operating conditions over the process duration which yield optimized values for said output variables or operating conditions.

2. The method of claim 1, further comprising: e) designing said industrial process based on said optimized values for said one or more input variables or operating conditions over said process duration.

3. The method of claim 2, further comprising: f) conducting said industrial process with said optimized input variables or operating conditions over said process duration.

4. The method of claim 1, wherein said industrial process comprises a batch process.

5. The method of claim 1, wherein said industrial process comprises a semi-batch or fed-batch process.

6. The method of claim 1, wherein said industrial process is employed in an industry selected from the list comprising chemicals industry, specialty chemicals industry, pharmaceutical industry, biopharmaceutical industry, alternative energy industry, bio-fuels industry, food industry, beverage industry, and manufacturing industry.

7. The method of claim 1 , wherein said input variables or operating conditions are selected from the list comprising: temperature, pressure, rate, flow, volume, reagent concentration, catalyst concentration, pH, level, and force.

8. The method of claim 1 , wherein said output variables or operating conditions are selected from the list comprising cost, energy, product amount, efficiency, and rate.

9. The method of claim 1 , wherein said set of experiments comprises a number of experiments equal, at most, to 2^n+m*(-^N"1^ for level-two full factorial designs, wherein n is the total number of input variables or conditions examined, m (m<n) is a subset of these variables that is varied over time and N is the maximum number of dynamic patterns considered; and 3^n+m*(N"1) f_{or a} level-three experiment.

10. The method of claim 9, wherein N is 2, 3, 4, 5, or 6.

11. The method of claim 9, wherein n is 1 , 2, 3, 4, 5, or 6.

12. The method of claim 9, wherein m is 1, 2, or 3.

13. The method of claim 1 , wherein said industrial process design is performed without a fundamental model of said industrial process.

14. The method of claim 1, wherein said process comprises a partial or incomplete model.

15. The method of claim 14, wherein said process is a first process that interacts with a second, well-modeled, process, wherein said method is not applied to said second process.

16. A system for industrial process design comprising a processor configured for: a) defining a process by one or more input variables or operating conditions and one or more output variables or operating conditions, wherein one or more of said input variables or operating conditions are dynamic variables or operating conditions over the process duration; b) simulating the process through a set of experiments, wherein one or more said dynamic variables or operating conditions are allowed to vary over the process duration; c) collecting data from said experiments; wherein said data comprises values for said input variables or operating conditions and said output variables or operating conditions over the duration of said process; and d) determining optimized values for said input variables or operating conditions over the process duration which yield optimized values for said output variables or operating conditions.

17. The system of claim 16, wherein said processor resides in a computer having a display.

18. The system of claim 16, wherein said determined optimized values are displayed on said display.

19. The system of claim 16, wherein said display displays a user interface that permits a user to select said input and output variables or operating conditions.

20. The system of claim 16, wherein said computer comprises a computer memory.

21. The system of claim 20, wherein said collected data is stored in said computer memory.

22. The system of claim 20, wherein said determined optimized values are stored in said computer memory.