Designing cost-effective biopharmaceutical facilities using mixed-integer optimization

Authors


  • Songsong Liu and Ana S. Simaria are joint first authors.

Abstract

Chromatography operations are identified as critical steps in a monoclonal antibody (mAb) purification process and can represent a significant proportion of the purification material costs. This becomes even more critical with increasing product titers that result in higher mass loads onto chromatography columns, potentially causing capacity bottlenecks. In this work, a mixed-integer nonlinear programming (MINLP) model was created and applied to an industrially relevant case study to optimize the design of a facility by determining the most cost-effective chromatography equipment sizing strategies for the production of mAbs. Furthermore, the model was extended to evaluate the ability of a fixed facility to cope with higher product titers up to 15 g/L. Examination of the characteristics of the optimal chromatography sizing strategies across different titer values enabled the identification of the maximum titer that the facility could handle using a sequence of single column chromatography steps as well as multi-column steps. The critical titer levels for different ratios of upstream to dowstream trains where multiple parallel columns per step resulted in the removal of facility bottlenecks were identified. Different facility configurations in terms of number of upstream trains were considered and the trade-off between their cost and ability to handle higher titers was analyzed. The case study insights demonstrate that the proposed modeling approach, combining MINLP models with visualization tools, is a valuable decision-support tool for the design of cost-effective facility configurations and to aid facility fit decisions. © 2013 The Authors. Published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers Biotechnol. Prog., 29:1472–1483, 2013

Introduction

As the monoclonal antibody (mAb) sector has matured, it has become critical to rapidly identify the most cost-effective purification processes that can handle increasing upstream productivities in a timely manner and overcome existing purification bottlenecks.[1-3] Chromatography operations are identified as critical steps in a mAb purification process and can represent a significant proportion of the purification material costs, particularly due to the use of expensive affinity matrices as well as the high amounts of buffer reagents required. Higher product titers allow meeting larger demands and decreasing the relative cost of upstream activities. However they increase the protein load on chromatography steps resulting in an increase in the number of cycles or further investment in larger columns and hence the relative cost of downstream increases.[3, 4] Although alternatives to traditional column chromatography platforms are emerging (e.g., non-chromatography operations, membrane adsorbers), industry practitioners are still reluctant to perform major process changes.[1-3] At the same time, it is important to determine how best to use existing installed production capacity for mAbs.[5, 6] In this context, continuous improvement of existing processes, particularly the optimization of chromatography operations, is a valuable approach to address the current challenges. The development of computer-based decisional tools for the bioprocess sector is an emerging area[7-11] and frameworks have been developed to assess different solutions for the design and operation of chromatography steps. Joseph et al.[12] present a simulation model to identify windows of operation for a chromatography step, using productivity and cost of goods (COG) as performance criteria. A model to find combinations of protein load and loading flow rate that meet yield and throughput constraints has been developed by Chhatre et al.[13] The discrete-event simulation framework proposed by Stonier et al.[14] allows the selection of optimal chromatography column sizes over a range of titers by brute force simulation. However, such an approach may not be feasible for very large decision spaces, and, particularly, when the variables have integer domain, as is the case in the problem addressed in the present article.

There are a large number of possible permutations and trade-offs related to running packed-bed chromatography operations such as opting for a smaller column run for several cycles so as to reduce resin costs vs. a large column run for fewer cycles so as to save time and labor costs. Decision makers usually have empirical approaches to come to a solution, mainly based on previous experience, and so may be missing good opportunities for improvement. The combinatorial optimization (CO) nature of the decision problem consists of selecting the most appropriate sizing strategy for the chromatography operation. In this article, the decisions are addressed using mixed-integer programming (MIP) techniques due to their widely recognized ability to handle CO problems.

Mixed-integer linear (MILP) and non-linear programming (MINLP) models have been developed to address capacity planning problems in the pharmaceutical[15, 16] and biopharmaceutical[17, 18] industries. At the process level, MIP models have focused on determining optimal purification sequences, using physicochemical data of protein mixtures and mathematical correlations of the separation techniques.[19-21] In some cases, the process synthesis optimization has also considered product loss by incorporating the decisions on the time of product collection and the start and finishing cut-points.[22, 23] More recently, efficient MILP models were developed using the discretization[24] and piecewise linearization approximation[25] to overcome the computational expense of MINLP models. These models use the number of chromatography steps, purity, and yield as performance metrics, but do not account for overall process costs.

Optimization of chromatography equipment sizing strategies for a sequence of chromatography steps on the basis of a global criterion, such as cost of goods per gram (COG/g), requires the use of either MINLP approaches or heuristic search methods such as evolutionary algorithms to handle the complex model dependencies. Meta-heuristic methods have been developed that integrate evolutionary algorithms with detailed process economics models to determine the most cost-effective purification sequences and chromatography sizing strategies that meet purity constraints.[26, 27] MINLP approaches have the advantage of providing exact solutions in the cases where commercial solvers or linearization techniques allow a feasible solution to be identified. However, an MINLP model for this problem domain does not exist in the literature. Hence, this article presents a novel mathematical programming model based on an MINLP formulation to determine the best chromatography equipment sizing strategies for the production of mAbs. The CO model addresses the challenge of optimizing the chromatography sizing strategy for a sequence of chromatography steps in a downstream purification train whilst considering several key decision variables for each step, including column bed height, column diameter, number of columns, and number of cycles. Furthermore, the model is used also to determine the optimal facility fit configuration for products with higher titers. A related problem has been previously addressed by Stonier et al.[28] using a stochastic simulation framework and multivariate analysis to identify root causes of facility mismatches.

The problem under study in this work—optimization of chromatography sizing strategies for facility design and facility fit—is formulated as an MINLP model, which can be solved to global optimality using commercially available global optimization solvers.

Problem Description

The problem addressed in this article is to determine the optimal equipment sizing strategies for a sequence of packed-bed chromatography columns used in the purification of mAbs. A typical mAb platform process is used in this study (as shown in Figure 1). In upstream processing (USP), mammalian cells expressing the mAb of interest are cultured in bioreactors. Then the broth moves to downstream processing (DSP), where the mAb is recovered, purified, and cleared from viruses using a variety of operations, such as different types of filtration, and a number of chromatography steps. The chromatography sequence includes three packed-bed chromatography steps, namely affinity (AFF) chromatography for product capture followed by cation-exchange (CEX) chromatography for intermediate purification, and anion-exchange (AEX) chromatography for polishing.

Figure 1.

A typical mAb platform process.

The problem addresses the challenges of dealing with multiple decisions, criteria, and constraints. This is further complicated by the sequential nature of decisions and their interdependencies, e.g. in a multi-step purification process the amount of resin required for a particular chromatography step depends on the equipment sizing strategy that was selected for the previous chromatography step. A schematic of the decision choices in this chromatography sizing problem is shown in Figure 2. The decisions at each chromatography step include the bed height, diameter, number of cycles, and number of columns to run in parallel at each step. The strategy selected has a direct impact on key metrics related to cost, time, and annual product output. This captures the trade-offs of using large columns with a single cycle vs. smaller columns with multiple cycles as illustrated in the schematic. Small changes in bed height were also accommodated to account for typical ranges seen in industrial applications and the use of multiple parallel columns per step was incorporated so as to determine whether this offered significant advantages that might outweigh current preferences to avoid parallel columns due to validation burdens.

Figure 2.

Comparison of alternative chromatography column sizing strategies in terms of the decision variables of the optimization problem (bed height, diameter, number of cycles, and number of columns) and the corresponding performance metrics of each configuration (step time, resin cost, and equipment depreciation cost). All the configurations allow processing the same amount of product (volume and mass). Column configurations with dotted outline indicate multiple cycles.

In this problem, the USP trains work constantly so it is necessary to monitor the real DSP time such that the scheduling of batches between the USP and DSP trains occurs as originally planned. This model feature is very important as scenarios of multiple USP trains feeding a single DSP train are considered. It is assumed that multiple bioreactors operate in staggered mode and feed the DSP trains intermittently. Ideally, as soon as the cell culture is complete the product should enter the DSP train, hence an increase in the USP:DSP trains ratio corresponds not only to a decrease in the bioreactor(s) size, and hence on the batch size, but also to a decrease in the DSP window, i.e. the time available to perform the DSP operations. This might be a challenging scenario which requires an appropriate column sizing strategy in order to ensure that the DSP operations are performed within the DSP window.

The COG comprises both direct costs based on resource utilization (e.g., resin costs, buffer costs, and variable labor costs associated with DSP time) and indirect costs (e.g. facility-dependent overheads and capital costs). The total cost is then divided by the product output to compute the cost of goods per gram (COG/g). This is a standard approach[7] which allows the incorporation of multiple process features into a single metric. Particularly relevant to the current work is the relationship between annual product output and COG/g, as process configurations which result in lower product outputs are automatically penalized in terms of COG/g. The COG/g was used by the proposed mathematical programming model as the objective function to be minimized.

In this problem, the annual demand is an input of the model and it is used to calculate the bioreactor size and required number of batches, which is an upper bound of the number of completed batches. However, if a particular equipment sizing strategy leads to long processing times, it may not be possible to meet the required number of batches and hence the annual product output would be below the production target. This issue is indirectly addressed by the use of COG/g as objective function, which favors solutions with higher product output values. Thus, the annual product demand can be met, unless the DSP time exceeds the DSP window.

Overall, the problem addressed in this work is described as follows. The following parameters are inputs: the process sequence of a mAb product, the annual demand, the product titer, the ratio of USP to DSP trains, the key operating parameters of the chromatography operations (e.g., yield, linear velocity, buffer usage, resin dynamic binding capacity), the processing times of non-chromatography unit operations, cost data (e.g., reference equipment costs, labor rate, resin, buffer, and media prices), the column diameter and height candidates, and the maximum number of cycles and columns. Given these inputs, the goal is to determine the column sizing strategies (i.e., column diameter and height, the number of cycles, number of columns at each step), the number of completed batches, the total product output and the total annual cost so as to minimize COG/g.

Mathematical Formulation

An MINLP model was developed for the chromatography column sizing problem described in the previous section. Only the equations most relevant to the case study discussion are presented in the main text; the complete set of constraints is shown in Table A.1 (Appendix).

Calculation of input parameters

To initialize the model, the required number of batches to meet the annual demand was estimated, given the number of production bioreactors existing in the facility. This corresponds to the maximum number of batches that can be completed within the planning horizon:

display math(1)

where math formula is the annual operating time, math formula is the bioreaction time, and math formula is the number of bioreactors. Then, the volume of a single bioreactor was estimated by the following expression:

display math(2)

where math formula is the annual demand, math formula is the yield of product at unit operation s, math formula is the bioreactor working volume ratio, “ math formula” is the titer of the product and math formula is the batch success rate. With the above two parameters defined and calculated, the proposed mathematical programming model is presented in the next section.

Product mass constraints

In each batch, the initial product mass entering the DSP train depends on the titer of the product and the working volume of production bioreactor (Eq. (3)). The product mass after each unit operation s depends on its yield (Eq. (4)).

display math(3)
display math(4)

The annual product output was determined using the product mass after the bulk fill operation (last operation in the process) per batch multiplied by the number of batches and the batch success rate.

display math(5)

math formulawas limited by the required number of batches to meet the demand, i.e., its upper bound given by Eq. (1).

display math(6)

Chromatography operation constraints

Resin Volume

The total column volume for chromatography step s was defined as the number of columns multiplied by the corresponding column volume.

display math(7)

where math formula is the volume of the candidate column size i for chromatography step s, determined by specific diameter math formula and height math formula. Thus, if a column size i was selected, the corresponding diameter and bed height were both known. Here, it was assumed that only one column size could be selected for each step, for ease of validation, as defined by:

display math(8)
display math(9)

where math formula is a binary variable to indicate whether column size i is selected for step s and math formula represents its maximum number of columns.

The total amount of resin available must be sufficient to process all product mass entering this operation, so the number of cycles multiplied by the total column volume should be greater than the minimum required resin volume:

display math(10)

The amount of resin required per batch, for a particular chromatography step, depends on the mass of product to be processed, the dynamic binding capacity of the resin used in that step, and the resin utilization factor:

display math(11)

Also, the number of cycles for each chromatography operation cannot exceed its upper bound:

display math(12)

Product and Buffer Volume

In both AFF and CEX operations, which operated in bind-and-elute mode (Eq. (13)), the volume of the output product was equal to the eluate volume. In the flow-through AEX operation, it was assumed that the product volume did not change from the previous step (Eq. (14)). The total buffer volume necessary to run a chromatography cycle was given by the buffer usage ratio multiplied by the total column volume (Eq. (15)).

display math(13)
display math(14)
display math(15)

Processing Time

In each chromatography step, the total processing time per batch was the summation of time for adding buffer and loading product.

display math(16)

When there were parallel columns, the product volume loaded to each column was the total product volume from the previous operation divided by the number of columns. The processing time for loading product was the product volume loaded to each column divided by the volumetric flow rate.

display math(17)

The volumetric flow rate (L/h) at a chromatographic operation was determined by the velocity ( math formula) and column diameter as follows:

display math(18)

The processing time (h) for adding buffer is given by:

display math(19)

Batch time

As other non-chromatographic DSP operations are not the main concern of this problem, it was assumed that their operating times were constant. The total DSP time per batch was defined as the sum of the processing times of all DSP operations converted into days such that it reflects the shift pattern of DSP operators.

display math(20)

where math formula indicates the number of hours per shift and math formula indicates the number of shifts per day. If math formula is greater than the DSP window, math formula, the required number of batches cannot be completed and the annual demand cannot be met.

The total annual DSP time was calculated by:

display math(21)

where math formula cannot exceed its upper bound:

display math(22)

Cost calculation

The total annual operating cost, math formula, comprises both direct costs based on resource utilization (e.g., resin costs, buffer costs, and variable labor costs associated with DSP time) and indirect costs (e.g., facility-dependent overheads and capital costs). Due to space constraints, the description of the cost calculation constraints will be limited to the resin and equipment costs. The full calculation is presented in Table A.1 (Appendix).

Resin Cost

In this work, only the resin was considered to calculate the consumables cost, as the resin volume is a key decision in this problem. The cost for other consumables was ignored. Assuming that the resin would be re-used until it reaches its lifetime, the annual resin cost was calculated by:

display math(23)

where math formula is the resin price, A is the over-packing factor for resin, and L is the resin lifetime (in terms of number of cycles).

Equipment Cost

Given the nature of the optimization problem addressed as well as the case study scenarios analyzed in this article, the two types of equipment considered for the calculation of indirect costs were the production bioreactors and chromatography columns. For different sizes of chromatography columns and bioreactors, the costs were calculated by using the values of reference equipment sizes and costs to scale-up the equipment cost:

display math(24)
display math(25)

where math formula is the cost of a single chromatography column with a diameter of math formula, and math formula is the cost of a single bioreactor with a volume of math formula. Both reference costs are used to scale up the costs of chromatography columns and bioreactors with different sizes, using math formula and math formula as the scale-up factors for columns and bioreactors, respectively. The equipment cost was then used to calculate the capital investment value.

Objective function

In the work, the objective was to minimize COG/g, which equals to the annual total cost divided by the annual product output. Thus, COG/g can be expressed as:

display math(26)

Overall, the problem was formulated as an MINLP model with Eqs. (3)-(25) as key constraints and with Eq. (26) as the objective function. A number of constraints in the proposed model are nonlinear, which involve bilinear (e.g., Eq. (5)) or trilinear terms (e.g., Eq. (23)). Also, the objective function is a fraction of two variables. The nature of the nonlinearity in the model leads to high computational complexity to find the global optimum. The complete set of the model constraints is presented in Table A.1 (Appendix).

Case Study Setup

The MINLP model was applied to an industrially relevant case study, based on a biopharmaceutical company using a platform process for mAb purification to manufacture a single product with a demand of 500 kg/year and a titer of 3 g/L. The key parameters of the considered three packed-bed chromatography steps are shown in Table 1 and the candidate values of the chromatography equipment sizing decision variables are shown in Table 2. As there are 11 possible bed heights and 10 possible diameters, a single column has 110 possible volumes. The number of cycles can be up to 10, while at most 4 columns are allowed to be used in parallel. The complete set of data used in the MINLP model for the case study is shown in Table A.1 (Appendix), alongside the corresponding model constraints. The parameter values used in this case study were similar to the ones presented in Simaria et al.[26]

Table 1. Characteristics of the Three Packed-Bed Chromatography Steps in the Case Study
Chromatography StepYield (Yd, %)Resin Dynamic Binding Capacity (DBC, g/L)Resin Price (Pcresin, £/L)Linear Velocity (Vel, cm/h)Eluate Volume (EluCV, CV)Buffer Volume (BuffCV, CV)
AFF91306,4003002.337
CEX92404003001.426
AEX9510070030010
Table 2. Candidate Values of the Column Sizing Decision Variables in the Case Study
Decision VariableCandidate Value
Bed height (Hcol, cm)15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25
Diameter (DMcol, cm)50, 60, 70, 80, 90, 100, 120, 160, 180, 200
Number of cycles (Ncyc)1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Number of columns (Ncol)1, 2, 3, 4

The goal was to design a new facility that was able to (a) manufacture the product in a cost-effective manner and (b) cope with predicted future higher titers. To address the first goal, the MINLP model described in Section “Mathematical Formulation” was run using the current product titer (3 g/L) and this model was named MINLPDesign. In order to address the second goal, the original model was modified to account for facility fit constraints, resulting in a second model MINLPFacility-fit. In this model, the size of the production bioreactor(s) was fixed to represent an existing facility (instead of calculated by Eq. (2)) and the maximum column diameter was dictated by the existing facility and therefore the degrees of freedom of the model to achieve the minimum COG/g were the column bed height and number of cycles. Future products in the pipeline were expected to have titers up to 15 g/L and so the values 6, 9, 12, and 15g/L were considered in the study. Given the fixed bioreactor volume, higher titers meant higher product masses entering into each chromatographic purification operation, increasing the required resin volume, as given by Eq. (11). However, as the number of columns and column diameters were fixed, and the maximum bed height and number of cycles were limited, it was important to account for product being discarded when the required resin volumes could not be met.

To model the above situation, a new continuous variable, math formula for the protein mass loss at chromatography step s was introduced in the MINLPFacility-fit model. The product mass entering chromatography operation s should be the product mass after the previous operation, math formula, minus the product mass loss at this chromatography operation, math formula, due to the lack of resin capacity at step s. The product mass after operation s, math formula, and the amount of resin required per batch, math formula, was formulated considering the mass loss, using two new constraints to replace Eqs. (4) and (11), as shown in Table 3. Also, an alternative objective to penalize the product loss was introduced (Table 3), in which U was developed as a penalty weight for mass loss. Computational tests on the case study showed that when U< 100, the optimal solution could be obtained. In this case study U was set to 10 in all scenarios.

Table 3. Characteristics of the MINLP Models
ModelMINLPDesignMINLPFacility-fit
Version AB
GoalDesign a new facility for current titer.Fit process with higher titers to existing facility.Increase DSP capacity
DescriptionAs described in section 3.Bioreactor and column sizes given by MINLPDesign; bed height and number of cycles can change.Bioreactor and column sizes given by MINLPDesign; bed height and number of cycles can change and additional columns can be installed.
Model variables   
Bioreactor sizeCalculatedGiven by MINLPDesign
Column diameterDecision variableGiven by MINLPDesign
Number of columnsDecision variableGiven by MINLPDesignDecision variable
Handling of mass lossNo mass loss occursNew continuous variable: math formula

Modified constraints to replace Eqs. (4) and (11):

math formula (4a)

math formula (11a)

Modified objective function to replace Equation (26):

math formula (26a)

where U is a penalty weight for mass loss

In order to assess different strategies of increasing the facility's capacity, two versions of the MINLPFacility-fit model were developed as presented in Table 3. In version A, the number of columns running in parallel at each chromatography step, math formula, was fixed to the values obtained from the MINLPDesign model, while version B allowed the installation of additional columns by letting the decision variable math formula change. In the latter situation it was assumed that the parallel columns were equally sized given industry preferences for ease of validation and operation.

Three different ratios of USP to DSP trains were considered (1:1, 2:1, and 4:1) so as to evaluate which configuration would be most suitable in terms of cost-effectiveness and robustness to cope with higher titers. The different USP:DSP configurations will have an impact of the size of the bioreactor(s) as well as on the DSP window, i.e. the time available to perform the DSP operations, as it is assumed that multiple bioreactors are operated in a staggered mode, feeding a single DSP train intermittently.

Results and Discussion

The proposed optimization models were implemented in GAMS 23.9[29] using the global MINLP solver BARON on a 64-bit Windows 7 based machine with 3.20 GHz six-core Intel Xeon processor W3670 and 12.0 GB RAM. The MINLPDesign model, with 403 constraints, 73 continuous variables, and 664 discrete variables, took less than 1,200 sec to find the optimal solution for each scenario. The number of variables in the MINLPFacility-fit model depended on the solution of the MINLPDesign model, and its CPU time was tens of seconds for all scenarios investigated in the case study.

New facility design for current titers (MINLPDesign)

Figure 3 summarizes the characteristics of the optimal solutions provided by MINLPDesign model for the different USP:DSP ratios analyzed in the case study, in terms of the volume of the columns and number of cycles of each chromatography step (Figure 3a), cost metrics (Figure 3b), and kg product output metrics (Figure 3c). In all the scenarios examined, single columns were selected for each purification step. With the increasing number of USP trains, the optimal solutions were characterized by using similar column volumes but running for fewer cycles to shorten the DSP time such that it fitted within tighter DSP windows. The DSP windows were 15, 7.5, and 3.8 days for 1USP:1DSP, 2USP:1DSP, and 4USP:1DSP configurations, respectively.

Figure 3.

Comparison of the characteristics of the optimal solutions provided by the MINLP model for the different USP:DSP scenarios in terms of (a) column volume and number of cycles, (b) COG/g with corresponding breakdown, (c) product output and number of batches manufactured per year. Results are shown for the scenario where a new facility was designed for manufacturing a product with titer of 3 g/L and demand of 500 kg/year.

The value of the objective function COG/g was lowest for the scenario of 1USP:1DSP, where a single bioreactor was used, and it increased with the number of USP trains as illustrated in Figure 3b. This can be attributed to the higher investment cost and hence indirect costs (e.g., depreciation) per gram associated with installing multiple smaller bioreactors vs. a single large bioreactor due to economies of scale as well as the increased labor costs associated with running multiple bioreactors. Hence a trade-off exists between the lower COG/g with a single bioreactor set-up vs. the lower risk (and hence mass loss) consequences, greater equipment utilization, and potentially greater agility with smaller staggered bioreactor set-ups. Besides the value of COG/g, other criteria were used to assess the different USP:DSP ratios, and provide a more complete decision-support framework for biopharmaceutical facility design. The next section presents the results and discussion regarding the ability of a facility to cope with higher titer values.

Impact of higher titers on facility design (MINLPFacility-fit)

In order to evaluate the impact of higher titers on the facility design, the results obtained by the model MINLPDesign (namely, bioreactor volumes and column diameters) were used as inputs to the MINLPFacility-fit model (versions A and B). The set of optimal solutions found by the MINLP models for the different case study scenarios is presented in Table 4 and visually displayed in Figure 5. The values in bold in Table 4 represent scenarios where mass loss could only be avoided by the installation of parallel chromatography columns. For these situations, at least one of the chromatography steps reached the maximum limit of all column sizing variables allowed to change in the MINLPFacility-fit_A (bed height and number of cycles) and any excess mass entering the column was lost. The model MINLPFacility-fit_B was solved in order to obtain a solution without mass loss. This was observed for the 1USP:1DSP scenario at titers equal to or greater than 6g/L and for the 2USP:1DSP configuration for titers 12 and 15 g/L.

Table 4. Characteristics of the Optimal Solutions Found by the MINLP Models for the Different Case-Study Scenarios
MINLP ModelDesignFacility-Fit AFacility-Fit BFacility-Fit AFacility-Fit BFacility-Fit A
  1. Note: The values in bold represent scenarios where mass loss could only be avoided by the installation of parallel chromatography columns. The underlined values represent input data of the corresponding model

USP:DSP1:12:14:11:11:12:12:14:1
Titer (g/L)33361561561515615
DSP window (days)157.53.815157.57.53.8
Maximum number of batches/year2040802020404080
Bioreactor volume (L)21,66810,8345,41721,66821,66810,83410,8345,417
AFF bed height (cm)161616252516161625201620
AFF number of cycles4215581045424
AFF diameter (cm)180180180180180180180180180180180180
AFF number of columns111111211211
CEX bed height (cm)151516171718251817161616
CEX number of cycles63210101095107410
CEX diameter (cm)120120100120120120120120120120100100
CEX number of columns111111211211
AEX bed height (cm)222224252522222225212424
AEX number of cycles6311010610610825
AEX diameter (cm)606070606060606060607070
AEX number of columns111112311211
Mass loss (kg/year)000702849000907000
DSP time (days)5.94.13.48.78.78.29.85.78.66.44.36.9
Number of batches/year204080202020204034406942
COG (£/g)74.584.8100.944.144.142.923.947.931.925.959.637.1
Product output (kg/year)500500500962962100025001000163625008631313

Figure 4 shows a comparison between the optimal solutions of MINLPFacility-fit_A (Figure 4a) and MINLPFacility-fit_B (Figure 4b) models in terms of COG/g and product output, for different titer values in the 1USP:1DSP scenario. As titer increased more product mass was processed per batch hence increasing the total annual product output. However, due to the mass loss that occurred in the optimal solutions of the model MINLPFacility-fit_A, a flattening of the output and COG/g curves was observed in Figure 4a. In this situation, the increase in titer did not translate into a reduction in COG/g. The penalization in the objective function given by Eq. 26a in Table 3 minimizes the amount of mass loss but it does not avoid its occurrence. The absence of mass loss was achieved by the optimal solutions of the MINLPFacility-fit_B model and this was obtained by increasing the number of columns running in parallel in the bottleneck steps, as shown in Table 4 and Figure 5. This resulted in an increase of product output and consequent reduction in COG/g, depicted in Figure 4b. For the scenario 2USP:1DSP, mass loss occurred for titers of 12 g/L and above. For the scenario of 4USP:1DSP, there was no mass loss even for the highest titer values, and so both MINLPFacility-fit_A and MINLPFacility-fit_B models produced the same optimal solutions This was due to the low number of cycles initially determined by the MINLPDesign model which allowed the increase to a higher value without reaching the maximum limit. Note that although there was no mass loss at higher titers in the 4USP:1DSP scenario, the annual product output was not fully achieved. The increase in the number of cycles required to meet resin constraints led to DSP times which exceeded the DSP window, reducing the total number of batches that could be produced in a year. This is shown in the last two columns of Table 4.

Figure 4.

COG/g and annual product output of the optimal solutions of the models (a) MINLPFacility-fit_A and (b) MINLPFacility-fit_B for the 1USP:1DSP scenario across different titer values. (c) Total number of columns of the optimal solutions obtained by the MINLPFacility-fit_B model for different USP:DSP ratios. The model MINLPFacility-fit_A allows the increase of bed height and number of cycles of a chromatography step to cope with higher titers, while the model MINLPFacility-fit_B also allows increasing the number of columns to run in parallel. The full details of these models are presented in Table 3.

Figure 5.

Characteristics of the optimal column sizing strategies (bed height (x-axis), number of cycles (y-axis) and number of columns (hollow bubbles represent a single column and crossed bubbles represent a solution where two columns are used in parallel) obtained by the MINLP models for the AFF step, for different titers and USP:DSP train ratios of (a) 1:1, (b) 2:1, and (c) 4:1. *For 6 g/L a single column is used and for 12 g/L two columns are used. The column diameter of the optimal solutions is 180 cm for all scenarios. A schematic of the optimal configurations obtained for titer 15 g/L is shown in (d).

The results of the case study were used to predict the critical titer levels where multiple parallel columns were needed to remove bottlenecks. In a 2USP:1DSP configuration (2 × 10,834 L bioreactors) parallel columns were required for titers of 12 g/L and above (harvest mass = 130 kg) while for 1USP:1DSP configurations this occurred at titer values of 6 g/L (1 × 21,668 L bioreactor, harvest mass = 130 kg). For the 4USP:1DSP facility configuration, the titer would need to be over 20 g/L for multiple columns to be required per step, given the column sizes installed. This value exceeds expected titer values for routine performance in the near future. Thus, it can be seen that although the configurations with multiple smaller bioreactors are more expensive to run they will be more robust to titer increases that could be expected in the future.

Facility design selection

The case study considered different independent USP:DSP scenarios that were used as model input parameters. In this section, that approach was taken further and the results of the case study were used to generate a decision-making framework for selecting the best USP:DSP configuration.

Assuming that no mass loss is desired, there are three different facility designs generated by the MINLP models described in this work, one for each USP:DSP scenario considered in the study, as shown in Figure 5. Each design is adapted to the product titer (e.g., increasing the number of cycles or using additional columns) but the facility is designed with that flexibility inbuilt. For example, multiple columns in parallel are installed and only used when necessary, as opposed to retrofitting the facility in the future. In light of these assumptions, an analysis of the trade-offs between the different alternatives is displayed in Figure 6. The line represents the Pareto front with the three solutions that establish a compromise between the average COG/g (calculated over the titer range) and the number of columns in the facility necessary to cope with the highest titer value considered in the case study. Larger columns with fewer batches per year offer economies of scale (e.g., 1USP:1DSP) but high titers will increase the load on the DSP stage and additional columns will be required to avoid mass loss. If companies are not keen to operate multiple parallel columns for a particular step due to validation concerns, then a facility design with smaller batches (e.g., 4USP:1DSP) could be selected leading to higher COG/g values.

Figure 6.

Comparison between the facility designs that result in no mass loss obtained by the MINLP models in the case study. Average COG/g = average COG/g of the MINLP optimal solutions of a particular USP:DSP ratio across the titer range. The line represents the Pareto front, i.e. solutions that present a trade-off between the average COG/g and the number of columns to be installed in the facility.

Conclusion

In this work, an MINLP modeling framework was proposed and applied to an industrially relevant case study to optimize the design of a facility by determining the most cost-effective chromatography equipment sizing strategies for the production of mAbs. Furthermore, the framework was used to evaluate the ability of the facility to cope with higher product titers, and to explore the trade-offs between alternative facility designs. The case study insights demonstrate that the proposed modeling approach can act as a valuable decision-support tool for the design of cost-effective facility configurations and to aid facility fit decisions. Future work will focus on extending the models to address chromatography sequencing decisions, incorporate uncertainty, and consider multi-objective optimization.

Acknowledgments

Funding from the UK Engineering & Physical Sciences Research Council (EPSRC) for the EPSRC Centre for Innovative Manufacturing in Emergent Macromolecular Therapies hosted by University College London is gratefully acknowledged. Financial support from the consortium of industrial and governmental users is also acknowledged.

Notation

The mathematical formulation of the MINLP model is presented with the following notation:

Indices
aex

anion-exchange chromatography step

aff

affinity chromatography step

bf

bulk fill step

cex

cation-exchange chromatography step

i

column size

s

downstream step

Sets
CS

set of chromatography steps, math formula

Parameters
A

overpacking factor for resin

math formula

annual product demand, g

math formula

total buffer usage of resin at chromatography step s, column volume (CV)

math formula

dynamic binding capacity of resin at chromatography step s, g/L

math formula

diameter of column size i at step s, cm

math formula

eluate volume of resin at chromatography step s, CV

math formula

bed height of column size i at step s, cm

L

resin life time, number of cycles

math formula

maximum number of batches per year

math formula

maximum number of columns at chromatography step s

math formula

maximum number of cycles at chromatography step s

math formula

number of production bioreactors

math formula

number of hours per shift

math formula

number of shifts per day

math formula

resin price at chromatography step s, £/L

math formula

reference cost of bioreactor, £

math formula

reference cost of column, £

math formula

reference diameter of column, cm

math formula

reference volume of bioreactor, L

math formula

scale-up factor of bioreactor

math formula

scale-up factor of column

math formula

annual operating time, days

math formula

bioreaction time, days

math formula

product titer, g/L

math formula

volume of column size i at step s, L

math formula

linear velocity of resin at chromatography step s, cm/h

math formula

DSP window, days

math formula

product yield of operation s

math formula

bioreactor working volume ratio

math formula

chromatography resin utilization factor

math formula

batch success rate

Continuous Variables
math formula

annual cost, £

math formula

annual product output, g

math formula

annual downstream processing time, days

math formula

processing time of each batch, days

math formula

bioreactor cost, £

math formula

cost of column of size i at chromatography step s, £

math formula

resin cost, £

math formula

minimum resin volume required at chromatography step s, L

math formula

initial product mass entering DSP, g

math formula

product mass after operation s, g

math formula

processing time of operation s, h

math formula

processing time for loading product at chromatography step s, h

math formula

processing time for adding buffer at chromatography step s, h

math formula

total column volume at chromatography step s, L

math formula

volumetric flow rate at chromatographic step s, L/h

math formula

buffer volume used in chromatography step s, L

math formula

product volume after operation s, L

math formula

optimization objective, cost of goods per gram, £/g

Binary Variables
math formula

1 if column size i is selected for chromatography operation s; 0 otherwise

Integer Variables
math formula

number of completed batches

math formula

number of columns of size i at chromatography step s

math formula

number of cycles at chromatography step s

Appendix:  

Notation (additional to Section “Mathematical Formulation”)

Indices
h

harvest step

ufdf

ultrafiltration/diafiltration step

vf

virus filtration step

vi

virus inactivation step

Parameters
a,b,c

utilities cost coefficients

math formula

diafiltration volume

math formula

final concentration of product

math formula

flush volume

math formula

general equipment factor

math formula

general utility unit cost

math formula

Lang factor

math formula

number of operators for DSP

math formula

number of operators per bioreactor in USP

math formula

neutralization volume

ny

project length, year

math formula

buffer price

math formula

cell culture media price

math formula

interest rate

math formula

labor rate

math formula

media overfill allowance

math formula

other equipment cost factor

math formula

other indirect costs factor

math formula

other labor cost factor

math formula

miscellaneous material cost factor

Continuous variables
math formula

annual buffer volume

math formula

buffer cost

math formula

capital cost

math formula

direct labor cost

math formula

labor cost

math formula

media cost

math formula

other indirect costs

math formula

utilities cost

math formula

fixed capital investment

math formula

initial product volume entering DSP

Table A.1. Complete Set of Constraints of the MINLPDesign Model and Parameter Values Used in Case Study
Model Constraint Parameter Values Used in Case Study
Product mass  
math formula(A.1) math formula
math formula(A.2) math formula, math formula, math formula, math formula, math formula, See Table 1 for math formula values
Annual product output  
math formula(A.3) math formula
math formula(A.4) 
Product volume  
math formula(A.5) math formula
math formula(A.6) math formula
math formula(A.7)See Table 1 for math formula values
math formula(A.8) 
math formula(A.9) math formula
math formula(A.10) math formula
math formula(A.11) math formula
Chromatography resin volume  
math formula(A.12) 
math formula(A.13) 
math formula(A.14) 
math formula(A.15) math formula, see Table 1 for math formula values
math formula(A.16) math formula
Buffer usage  
math formula(A.17) math formula
math formula(A.18)See Table 1 for math formula values
math formula(A.19) math formula
math formula(A.20) math formula
math formula(A.21) math formula
math formula(A.22) 
Processing time  
math formula(A.23) 
math formula(A.24) 
math formula(A.25)See Table 1 for math formula values
math formula(A.26) 
math formula(A.27) math formula math formula=1.5, math formula
math formula(A.28) 
math formula(A.29) math formula
Materials cost  
math formula(A.30) math formula see Table 1 for math formula values
math formula(A.31) math formula
math formula(A.32) math formula
math formula(A.33) math formula
Labor cost  
math formula(A.34) math formula
math formula  
math formula(A.35) math formula
Utilities cost  
math formula(A.36) math formula
Capital cost  
math formula(A.37) math formula
math formula(A.38) math formula
math formula(A.39) math formula
math formula(A.40) math formula
Other indirect costs  
math formula(A.41) math formula
Total annual cost  
math formula(A.42) 

Ancillary