Cell population balance, ensemble and continuum modeling frameworks: Conditional equivalence and hybrid approaches

The cell population balance, ensemble and continuum modeling frameworks are widely used for the mathematical description of microbial populations. Each of these approaches focuses on different aspects of the processes of growth, division and intracellular reaction occurrence. Therefore, each framework can output different information at different computational expense. Continuum models assume lumped biomasses, capture the dynamics of bulk intracellular concentrations and are easy to simulate. Ensemble models account for heterogeneity due to different initial conditions or kinetic constants, and are more computationally expensive. Finally, cell population balances capture the partitioning of the intracellular contents in detail, but can quickly become intractable, as the number of biochemical species taken into consideration increases. It is thus natural to ask whether one can adequately simulate cell populations with a simpler approach. In this paper, starting from an (n+1) cell population balance, for n biochemical species and the cell volume, we investigate biologically plausible conditions, under which two exact hybrid models are derived. Provided that the conditions hold true, both hybrid models are exact but much simpler alternatives of the cell population balance; no approximation is involved. In the first, the evolution of the species concentrations is captured by an ensemble model, and in the second by a continuum model. Both hybrid models also contain a 1-dimensional population balance for the cell volume. This work should provide a guideline for choosing the modeling approach which is the most appropriate for the particular application. © 2009 Elsevier Ltd. All rights reserved

Integrating the physics with data analytics for the hybrid modeling of the granulation process

A hybrid model based on physical and data interpretations to investigate the high shear granulation (HSG) process is proposed. This model integrates three separate component models, namely, a computational fluid dynamics model, a population balance model, and a radial basis function model, through an iterative procedure. The proposed hybrid model is shown to provide the required understanding of the HSG process, and to also accurately predict the properties of the granules. Furthermore, a new fusion model based on integrating fuzzy logic theory and the Dempster-Shafer theory is also developed. The motivation for such a new modeling framework stems from the fact that integrating predictions from models which are elicited using different paradigms can lead to a more robust and accurate topology. As a result, significant improvements in prediction performance have been achieved by applying the proposed framework when compared to single models

Measurement, modelling, and closed-loop control of crystal shape distribution: Literature review and future perspectives

Crystal morphology is known to be of great importance to the end-use properties of crystal products, and to affect down-stream processing such as filtration and drying. However, it has been previously regarded as too challenging to achieve automatic closed-loop control. Previous work has focused on controlling the crystal size distribution, where the size of a crystal is often defined as the diameter of a sphere that has the same volume as the crystal. This paper reviews the new advances in morphological population balance models for modelling and simulating the crystal shape distribution (CShD), measuring and estimating crystal facet growth kinetics, and two- and three-dimensional imaging for on-line characterisation of the crystal morphology and CShD. A framework is presented that integrates the various components to achieve the ultimate objective of model-based closed-loop control of the CShD. The knowledge gaps and challenges that require further research are also identified

Modelling of biological systems using multidimensional population balances

Biological systems are intrinsically heterogeneous and, consequently, their mathematical descriptions should account for this heterogeneity as it often influences the dynamic behaviour of the individual cells. For example, in the cell cycle dependent production ofproteins, it is necessary to account for the distribution of the individual cells with respect to their position in the cell cycle as this has a strong influence on protein production. A second notable example is the formation of cancerous cells. In this case, the failure of regulatory mechanisms results in the transition of somatic cells to their cancerous state. Therefore, in developing the corresponding mathematical model, it is necessary to consider both the different states of the cells as well as their regulation. In this regard, the population balance equation is the ideal mathematical framework to capture cell population heterogeneity as it elegantly takes into account the distribution of cell populations with respect to their intracellular state together with the phenomena of cell birth, division, differentiation and recombination. Recent developments in solution algorithms together with the exponential increase in computational abilities now permit the efficient solution of one-dimensional population balance models which attribute the heterogeneity of cell populations to differences in the age or mass of individual cells. The inherent complexity of biological systems implies that the differentiation of cells based on a single characteristic alone may not be sufficient to capture the underlying biological phenomena. Therefore, current research is focussing on the development of multi-dimensional population balances that consider the differentiation of cells based on multiple characteristics, most notably, the state of cells with respect to key intracellular metabolites. However, conventional numerical techniques are inefficient for the solution of the formulated population balance models and this warrants the development of novel, tailor-made algorithms. This thesis presents one such solution algorithm and demonstrates its application to the study of several biological systems. The algorithm developed herein employs a finite-volume technique to convert the partial-differential equation comprising the population balance model into a set of ordinary differential equations. A two-tier technique based on the solution technique for inhomogeneous differential equations is then developed to solve the system of ordinary differential equations. This approach has two main advantages: (a) the decomposition technique considerably reduces the stiffness of the system of equations enabling more efficient solution, and (b) semianalytical solutions for the integrals employed in the modelling of cell division and differentiation can be obtained further reducing computation times. Further improvements in solution efficiency are obtained by the formulation of a two-level discretisation algorithm. In this approach, processes such as cell growth which are more sensitive to the discretisation are solved using a fine grid whereas less sensitive processes such as cell' division - which are usually more computationally expensive - are solved using a coarse grid at a higher level. Thus, further improvements are obtained in the efficiency of the technique. The solution algorithm is applied to various multi-dimensional population balance models of biological systems. The technique is first demonstrated on models of oscillatory dynamics in yeast glycolysis, cell-cycle related oscillations in eukaryotes, and circadian oscillations in crayfish. A model of cell division and proliferation control in eukaryotes is an example of a second class of problems where extracellular phenomena influence the behaviour of cells. As a third case for demonstration, a hybrid model of biopolymer accumulation in bacteria is formulated. In this case, cybernetic modelling principles are used to account for intracellular competitions while the population balance framework takes into consideration the heterogeneity of the cell population. Another important aspect in the formulation ofmulti-dimensional population balances is the development of the intracellular models themselves. While research in the biological sciences is permitting the formulation of detailed dynamic models of various bioprocesses, the accurate estimation of the kinetic parameters in these models can be difficult due to the unavailability of sufficient experimental data. This can result in considerable parametric uncertainty as is demonstrated on a simple cybernetic' model of biopolymer accumulation in bacteria. However, it is shown that, via the use of systems engineering tools, experiments can be designed that permit the accurate estimation of all model parameters even when measurements pertaining to all modelled quantities are unavailable.Imperial Users onl

A Three-Dimensional Population Balance model of Granulation Processes Employing Mechanistic Kernels

Granulation is a process for agglomeration where powder material is combined with liquid binder solution to facilitate the formation of larger, free-flowing granules. Granulation has become a mainstream process amongst the industries with applicability in numerous areas, which include the pharmaceuticals, mineral processing, fertilisers and in the production of a range of commodity products. A major driVing force for the production of granules from their ungranulated counterparts arises from the economic savings Le., increased bulk density permits savings to be made in transportation. and storage. Furthermore, granules may be tailored to possess certain desirable attributes that will suit the final application of the granules. Granulation is an example of a process that exhibits complex interactions between the underlying granulation phenomena such as nucleation, consolidation, aggregation and breakage. In addition, the granUle properties are distributed heterogeneously across the entire particle population posing as a particular challenge in generating a mathematical model that is able to accurately describe the granulation behaviour. The modelling approach used in this study is different from common practices, which tend to rely on heuristics and empiricism for the operation of the granulation process. This empirical approach signifies a disconnect from our understanding of the underlying physics of the process, which poses as a impediment towards the efficient operation of granulation processes. The work presented in this thesis attempts to address this disconnect by applying a threedimensional population balance with mechanistic representations for the underlying granulation rate processes. The population balance framework is ideally suited for this particular process, as it enables the evolution of the granules to be tracked with respect to differentiating particle traits, e.g. the granule size distribution. The selection of the desired properties is influenced by the importance of these particle properties on the end granule product, and also by their influence on key process mechanisms. A novel mechanistic nucleation kernel is developed incorporating fundamental material properties pertaining to the powder substrate and the liquid binder solution. The model form of the nucleation kernel is formulated by drawing a parallel with the collision/transition state theory. There are few literature reports on the inclusion of nucleation phenomena in the population balance models of granulation processes, let alone a mechanistic nucleation model. This study is one of the first in this regard. The recent recognition of the importance of the wetting kinetics and the nucleation thermodynamics on the nucleation phenomenon has been factored into the nucleation kernel by explicitly accounting for the effects of the liquid flow rate and the physicochemical properties of the material properties (surface tension, contact angle, and spreading coefficient). Batch granulation experiments were conducted obtaining granule measurements with respect to the size distribution, porosity and fractional binder content. Preliminary results for the validation of the population balance model with the experiment-measurements showed a good agreement, providing partial albeit valuable validation of the population balance model. This is also one of the first studies to model and validate a three-dimensional population balance model for granulation. Model based analyses were also carried out under a variety of processing conditions, which included the effects of changing formulations, droplet size effects, feed size distribution and the effects of powder and binder properties. The proposed model demonstrated the interactions for a range of feed formulations in tandem with granulating operating conditions, establishing qualitative agreement with similar findings derived from past experimental studies.Imperial Users onl

Optimal control of particle size distribution in semi-batch emulsion polymerisation

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Multi-scale population balance modelling and controllability of granulation processes

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Experimental and model-based analysis of twin-screw wet granulation in pharmaceutical processes

A shift from batch to continuous processing is challenging but equally rewarding for the pharmaceutical sector. This opportunity for moving beyond traditional batch processing is possible due to a change of attitude in the regulatory environment by the publication of the process analytical technology (PAT) guidance. However, in order to utilise this opportunity, detailed process understanding about the key processes in pharmaceutical manufacturing is required to turn this transformation to the continuous mode into a success. Continuous wet granulation is a crucial part of future continuous manufacturing of solid dosage forms. Continuous high shear wet granulation is performed using a twin-screw granulator (TSG) which is characterised by a modular screw profile including a sequence of different screw elements with various shapes, orientations and functions. A TSG achieves mixing and granulation by a complex interplay between the screw configuration and process settings (e.g. feed rate, screw speed, etc.) to produce granules with certain specifications in a short time. Therefore, a fundamental understanding of these complex phenomena is required to optimise and control this new technology. Analysing the twin-screw wet granulation to a satisfactory degree is only possible when sufficient information on the rheo-kinetic characteristics of the granulation mixture is available. Thus an investigation of residence time distribution (RTD), the solid-liquid mixing, and the resulting granule size distribution (GSD) evolution governed by the field conditions in the TSG contain interesting information about mixing and different granulation rate processes such as aggregation and breakage. For this purpose, a combination of experimental and mathematical techniques/approaches was applied in this work. Additionally, a single placebo formulation based on α-lactose monohydrate was granulated in the experimental studies performed to verify the hypothesis proposed in this work. The characterisation of wetted material transport and mixing inside the confined spaces of the rotating screws was performed by the experimental determination of the residence time distribution at different process conditions and screw configurations using near infrared chemical imaging. The experimental data was later compared with a conceptual model based on classical chemical engineering methods to estimate the parameters of the model and to analyse the effects of changes in number of kneading discs and their stagger angle, screw speed, material throughput and liquid-to-solid ratio (L/S) on RTD. According to this study, increased screw speed resulted in a low mean residence time mean residence time and wider RTD, i.e. more axial mixing. Increasing powder feed rate increased mean residence time by higher throughput force while increasing L/S increased mean residence time by raising the sluggishness or inertia of the material in the barrel. The material transport in the mixing zone(s) of the TSG became more plug-flow like. Thus, an increase in the number of kneading discs reduced the axial mixing in the barrel. In addition, to understand the GSD dynamics as a function of individual screw modules along the TSG barrel, the change in GSD was investigated both experimentally and mathematically. Using a TSG which allows the opening of the barrel, samples from several locations inside the TSG barrel were collected after granulation at different process conditions and screw configurations. A detailed experimental investigation was hence performed to understand the granule size and shape dynamics in the granulator. The experimental data from this study together with the residence time measurements was then used for calibrating a population balance model for each kneading disc module in the twin-screw granulator in order to obtain an improved insight into the role of the kneading discs at certain locations inside the TSG. The study established that the kneading block in the screw configuration acts as a plug-flow zone inside the granulator. It was found that a balance between the throughput force and conveying rate is required to obtain a good axial mixing inside the twin-screw granulator. Also, a high throughput can be achieved by increasing the liquid-solid ratio and screw speed. Furthermore, the study indicated that the first kneading block after wetting caused an increased aggregation rate, which was reduced after the material processing by the second kneading block. In contrast, the breakage rate in the increased successively along the length of the granulator. Such a reversion in physical phenomena indicated potential separation between the granulation regimes, which can be promising for future design and advanced control of the continuous twin-screw granulation process. In another experimental study the transport and mixing (both axial and bulk mixing of solid-liquid) was linked to the GSD of the produced granules. This study demonstrated that insufficient solid-liquid mixing due to inability of the currently used kneading discs is the reason behind the inferior performance of the TSG in terms of yield. It was shown that other factors which support mixing such as higher axial mixing at a high screw speed and a low fill ratio support an increase in the yield. However, more effort is required to explore non-conventional screw elements with modified geometries to find screws which can effectively mix the solid-liquid material. Furthermore, in order to generalise the TSG knowledge, a regime map based approach was applied. Herewith, the scale independent parameters, L/S and specific mechanical energy (SME) were correlated. It was shown that an increasing L/S strongly drives the GSD towards a larger mean granule size. However, an increasing energy input to the system can effectively be used to lower the mean granule size and also narrow the width of the size distribution. Along with this, particle-scale simulations for the characterisation of liquid distribution in the mixing zone of the granulator were performed. It was found that the agglomeration is rather a delayed process which takes place by redistribution of liquid once the excess liquid on the particle surface is transferred to the liquid bridges. Moreover, the transfer of liquid from particle surface to liquid bridges, i.e. initialisation of agglomeration, is most dominant in the intermeshing region of the kneading discs. Besides the major outcomes of this work, i.e. building fundamental knowledge on pharmaceutical twin-screw wet granulation by combining experimental and theoretical approaches to diagnose the transport, mixing and constitutive mechanisms, several gaps and potential research needs were identified as well. As the regulators have opened up to increasingly rely on the science- and risk-based holistic development of pharmaceutical processes and products for commercialisation, the opportunity as well as responsibility lies with academic and industrial partners to develop a systematic framework and scientific approach to utilise this opportunity efficiently