Mathematical modelling of the formation of gold nanoparticles via the citrate synthesis method

This work presents a new model for predicting the evolution of the size of gold nanoparticles (GNPs) in the citrate synthesis method. In this method, the precursor is an acid solution of tetrachloroauric acid, while the reducing agent is a base solution of sodium citrate. The acid-base properties of the solutions influence how the size of the particles evolves during the synthesis. In the literature, various mechanistic theories have been proposed to explain this evolution. Turkevich et al. (1951), who pioneered this synthesis method, suggested the “organizer theory” also known as “nucleation-growth” mechanism. Recently, however, Wuithschick et al. (2015) proposed a “seed-mediated” mechanism, a nucleation-aggregation-growth mechanism. In investigating the synthesis, while Turkevich et al. (1951) used the conventional techniques such as the transmission electron microscopy (TEM), the scanning electron microscopy (SEM) and the UV-vis spectroscopy, Wuithschick et al. (2010) used a combination of X-ray absorption near edge spectroscopy (XANES) and small angle X-ray scattering (SAXS) along with the conventional techniques. This setup provides time-resolved in situ information on the formation of GNPs, thereby yielding reliable accounts of the synthesis mechanism. Nevertheless, only one mathematical model has been developed, that advanced by Kumar et al. (2007), which is based on the nucleation-growth theory proposed by Turkevich et al. (1951). This model had not been thoroughly tested. In a part of this work, we investigate the model of Kumar et al. (2007) for different conditions of pH, temperature and initial reactant concentrations. To solve the model, we use the numerical code Parsival, which is developed for solving population balance equations. We test the model for different synthesis conditions studied experimentally by various researchers, for which experimental data are available in the literature. The model poorly predicts these data, because the Turkevich organizer theory does not account for the acid-base properties of chloroauric acid and sodium citrate. Thereafter, we present a novel kinetic model based on the synthesis seed-mediated mechanistic description proposed by Wuithschick et al. (2015). In this description, the precursor concurrently reduces into gold atoms and hydroxylates into a passive form. The gold atoms then aggregate into seed particles, which finally react with the passive form of the precursor in a growth step. We validate the model using experimental data from the literature obtained for conditions in which the seed-mediated mechanism is valid. The predicted GNP final sizes closely agree with those obtained experimentally. Finally, we present a modelling approach for the aggregation process in metal nanoparticles syntheses based on the theory proposed by Polte (2015). In this theory, metal atoms formed by reducing the precursor solution aggregate to larger sizes due to the Van der Waals’ forces of attraction. Then, due to the electrostatic forces of repulsion induced by the “potential determining” ions, the nanoparticles eventually stop aggregating and become stabilized. Based on this theory, we develop a model for the aggregation process resulting from the interplay of the attractive and repulsive forces in the evolution of nanoparticles. Using this model, we describe how gold atoms aggregate into seed particle in the citrate synthesis method. Then, we couple this aggregation model with the model developed for the seed-mediated mechanism. To validate the model predictions, we employ the experimental data used to validate the seed-mediated mechanism. In addition to the GNP final sizes, this integrated model correctly predicts the size polydispersity and completely describes the final GNP particle size distribution

Integrated Modelling of Sugar Manufacturing Plant for Nigeria Cane Plantation

Nigeria is a major importer of brown sugar, an important food source that can be manufactured from sugarcane locally available. To serve the purpose of designing a sugar manufacturing plant, this paper develops a plant-wide model for obtaining brown sugar from sugarcane. This model comprising material balance equations accounts for various processes involved in the manufacturing such as milling, filtration, evaporation, crystallization and drying. GAMS, an algebraic modelling tool, was employed to solve the model. From a basis of 100 tonnes of cane per day, a simulation result of 2000 tonnes of brown sugar per year compared excellently with literature. Therefore employing the model with a basis of 13 million tonnes of sugarcane, Nigeria’s sugarcane plantation potential capacity, showed that the country can produce 500,000 tonnes per year, compared to the current capacity of less than 10,000

Predictive Modelling of Covid-19 Pandemic: World Perspective

COVID-19, a disease caused by coronavirus, is a global pandemic currently ravaging the world. From tens of cases reported in January 2020, the disease as of August 2020 has infected over 16 million people worldwide, thus becoming a concern to everyone. Modelling can be used to describe the pandemic and assess the effectiveness of various control measures. This paper presents results from a model developed to predict the time evolution of total confirmed cases of COVID-19. The model, which is an exponential equation, is derived from the balance equation modelling, where the time evolution of cases is the accumulation term. While COVID-19 carriers’ cross-border migration can affect the cases within a society, the developed model considers a closed border scenario and mimics a batch system. Therefore, confirmed cases can only be affected by transmission within a society as well as death or recovery rate of infected cases. While death or recovery occurs days after, a carrier can begin transmission as soon as coming in contact with the disease. China, the country where the disease was first reported, did not record any death or recovery from COVIDd-19 until mid-January 2020 (Ravelo and Jerving, 2020). Neglecting death and recovery terms, the model resulted in an exponential equation. Some of the COVID-19 data reported in China from 01 – 11 January 2020 were used to obtain the exponential rate constant as 0.1706 〖day〗^(-1) at an r-squared value of 0.9963. In fourteen days, the reported period for the disease to manifest, the value of this pseudo-rate constant is 2.39. This value is comparable to the transmission rate of 2.2 reported by Sun et al. (2020). The remaining data from 01 – 11 January 2020 were used to validate the model. Model predictions yielded good agreement with reported data

Predictive Modelling of COVID-19 Pandemic Evolution in Nigeria

COVID-19 is a pandemic that has defined human life, shutting down economic activities. Various measures have been implemented to curb its spread. However, in many places, confirmed cases have continued to increase. Many believe that unless its vaccine is discovered, the pandemic has come to stay. This article aims to develop a model for the evolution of the total confirmed cases, which in the early stage increase exponentially and in the later stage flatten out. Following the balance equation modelling and employing assumptions similar to the typical pandemic modelling, an exponential model was developed. In order to describe the whole trajectory of the pandemic, following the general trend of COVID-19 data, the exponential model was modified with an inverse exponential expression, comparable to the Arrhenius Equation in chemical reaction engineering. After parameter estimation, the resulting model was validated using data from Italy and Nigeria. The model predictions compare reasonably well with the data. The model was then employed to predict the future of COVID-19 in Nigeria. The final equilibrium total confirmed cases would be 81,292 and the time for the country to record very low new daily cases would be in March 2021

Modelling Colloidal Stability in Gold Nanoparticles Synthesis - A Review

Nanoparticles are commonly synthesized in colloidal systems, a liquid with suspended macromolecules. By the Brownian motion, these particles can collide and aggregate, leading to larger particles. To stabilize the aggregation process, charges induced by ions cause repulsion among the particles. Through the works of researchers in the past, various models for describing particles aggregation and stabilization have been developed and modified. These models are based on the popular DLVO theory, named after Derjaguin, Landau, Verwey and Overbreek. In this work, evidence illustrating aggregation and stabilization in gold nanoparticles synthesis is reported. Thereafter, models for describing aggregation by the Brownian motion and stabilization by the electrostatic effect are reviewed. The electrostatic effect among the particles is mathematically expressed as the cumulative sum of the Van der Waal's energy of interaction and the electrostatic energy of repulsion. As the resulting stabilization model is too complex to solve, past researchers reported a simplified stabilization submodel and employed it in describing gold nanoparticles synthesis. Unfortunately, as shown in this review, the submodel failed to describe the synthesis as the aggregation process never stopped, thus making a case for a new modelling approach

Cloud-based MBDoE applications for optimal design of experiments to accelerate kinetic model identification

Cloud technologies offer the enabling platform for remote computing and interconnection of machines globally, thus increasing and accelerating industrial production. This work introduces a novel cloud-based platform driven by online model-based design of experiments (MBDoE) techniques to remotely optimise experimentation in a smart flow reactor situated at the University of Leeds. Using the dynamics of a physical system, the MBDoE algorithm can optimally design experiments for model discrimination and improve parameter precision to identify a reliable mathematical model for kinetics describing reaction processes occurring in the physical system. In MBDoE for parameter precision, the purpose is to design experiments to reduce uncertainties in model parameters for reliable model predictions. Uncertain parameters are used in the algorithm as prior estimates to search for the posterior experiment to conduct for the highest reduction in model parameter uncertainties. In this work, we have demonstrated the applicability of the cloud-based platform in a pharmaceutically relevant case study, homogeneous amide formation, where the synthesis can be described using reversible chemical steps. Prior uncertain estimates for kinetic parameters in the reversible kinetics were calculated from preliminary experiments designed using Latin hypercube sampling. Using these prior estimates, MBDoE for parameter precision designed a single additional experiment, which updated preliminary experiments to yield a statistically precise parameter estimation of parameters after just one additional run. The kinetic model with the precisely estimated parameters produced reliable predictions when tested against an unseen validation data set from experiments designed using a full factorial approach

Application of a novel cloud-based platform for kinetics model identification in pharmaceutical processes

The pharmaceutical industry has recently implemented a new framework to incorporate industry 4.0 technologies into drug manufacturing to guarantee quality and accelerate commercialization. These technologies include cloud computing for speeding up information sharing among research collaborators, across companies and with health authorities worldwide, artificial intelligence, digitalization and mathematical modelling as accelerators for drug discovery, and the use of emerging data-driven and physics-based modelling technologies in pharmaceutical development and manufacturing for advanced process design, monitoring and control in continuous systems. In accordance, we present in this work a novel cloud-based platform driven by optimal experimental design software deployed from University College London to remotely control experimentation in a smart flow reactor system situated at University of Leeds. Employing hybrid modelling, the software initially designs experiments using model-free DoE techniques such as Latin hypercube sampling (LHS) to obtain process information and thereafter designs experiments using model-based design of experiments (MBDoE) techniques to identify reaction kinetics. The model is then validated online using statistical tests to achieve a probabilistic description of model reliability across the experimental design space. We have demonstrated this platform on pharmaceutical-related processes including homogenous amide formation, and heterogeneous hydrogen borrowing. In the homogeneous amide formation, following initial process understanding and experimentation driven by LHS, the platform tested 2 alternative kinetic models representing a single-forward chemical reaction and a reversible chemical reaction, respectively. While rejecting the former as inadequate in describing the process system, the cloud-based platform proceeded with the latter to design a new experiment, the most informative experiment in the design space, which by updating the experimental conditions ensured a precise estimation of kinetic model parameters. In heterogeneous hydrogen borrowing, a synthesis protocol being explored for new drug discovery, the platform via sequential parameter estimation and MBDoE for model discrimination, reduced 12 initially tested candidate kinetic models to 2 models with identifiable parameters and tested the latter models in silico for distinguishability

Application of MBDoE techniques to a cloud-based platform for automated chemical manufacturing in flow reactor systems

Industry 4.0 has birthed a new era for the chemical manufacturing sector, transforming reactor design and automating process control. Towards autonomous chemistry development, on-demand manufacturing, and real-time optimization, we developed a cloud-based platform driven by model-based design of experiment (MBDoE) to coordinate remotely smart flow reactors, also known as “LabBots”, sited in different locations. A cloud-based iterative MBDoE framework was proposed characterised by five elemental stages: i) formulation of candidate models, ii) preliminary design of experiments (DoE), iii) parameter estimation using the experimental data acquired from designed experiments, iv) MBDoE for parameter precision, and v) model validation. The framework has been applied to two pharmaceutically relevant case studies: 1) homogeneous amide formation from amine and 2) nucleophilic aromatic substitution of 2,4-difluoronitrobenzene with morpholine. The first case study involves a single-forward reaction step and yields products that can also react in the reverse direction, presenting two alternative mechanisms. After employing rate expressions to model the two mechanisms and then estimating their Arrhenius parameters through a preliminary DoE-driven experimental investigation, the MBDoE framework was used to analyse the relative model performance. While rejecting the single-forward model, a χ^2 lack-of-fit test accepted the single-reversible model as the best model for the amide formation. With one single additional experiment designed using MBDoE for parameter precision, the numerical values of the reversible model parameters were estimated with statistically satisfactory confidence intervals. In the model validation stage, the reversible model performed well in describing new data from experiments designed by a full factorial design of experiments. The second case study (nucleophilic aromatic substitution) is characterised by a unique structure but involves a more complex reaction mechanism. Analogous to the amide formation, the MBDoE framework employed rate equations to describe the parallel and consecutive steps and identified a mechanistic kinetic model with the minimum number of runs. Following MBDoE for parameter precision, the resulting model predictions improved in terms of uncertainty, particularly in unexplored regions of the experimental design space

Distinguishing alternative kinetic models for hydrogen borrowing within the model-based design of experiment framework for model discrimination

Hydrogen borrowing is a catalytic synthetic protocol gaining popularity for producing a variety of new drug entities as the synthetic protocol can elongate the entity molecular structure when repeated in several cycles. While various kinetic models can be developed to describe the hydrogen borrowing cycles, available experimental data constrain the model space to a few kinetic models with estimable parameters. The best model can then be selected using model-based design of experiments for model discrimination (MBDoE-MD), hindered only where models are indistinguishable. Focussing on a synthesis case study that can be described using two hydrogen borrowing cycles, we employ in this work a software system called “SimBot”, which within a cloud-based platform controls a remotely located smart flow reactor system for physical experimentation and comprises modules for kinetic model development, parameter estimation and MBDoE-MD. Among six candidate kinetic models developed for hydrogen borrowing, Simbot identified only two candidate kinetic models as statistically acceptable after employing available experimental data by assessing the models adequacy using the χ^2 test and parameter identifiability using Fisher information matrix-based identifiability criteria. Using in-silico experiments manipulating inlet flow conditions and temperature only, these two models (Model 1, Model 2) could not be distinguished within the MBDoE framework for model discrimination as their corresponding discrimination probabilities were close to 0.50. However, exploiting the model structural differences: Model 1 being zeroth order with respect to the catalyst amount while Model 2 being first order, the SimBot software allowed to identify the catalyst amount as key model discrimination driver to be used in further kinetic experiments. In silico testing showed that assuming 1, 2 and 3% decrease in catalyst amount, Model 1 differed from Model 2, and these experiments would allow to achieve discrimination probabilities of 0.52, 0.78 and 0.99, respectively. Future validation experiments in the automated platforms will be needed to confirm the adequacy of Model 2 on representing reaction kinetics in the hydrogen borrowing system

Stroke genetics informs drug discovery and risk prediction across ancestries

Previous genome-wide association studies (GWASs) of stroke — the second leading cause of death worldwide — were conducted predominantly in populations of European ancestry1,2. Here, in cross-ancestry GWAS meta-analyses of 110,182 patients who have had a stroke (five ancestries, 33% non-European) and 1,503,898 control individuals, we identify association signals for stroke and its subtypes at 89 (61 new) independent loci: 60 in primary inverse-variance-weighted analyses and 29 in secondary meta-regression and multitrait analyses. On the basis of internal cross-ancestry validation and an independent follow-up in 89,084 additional cases of stroke (30% non-European) and 1,013,843 control individuals, 87% of the primary stroke risk loci and 60% of the secondary stroke risk loci were replicated (P < 0.05). Effect sizes were highly correlated across ancestries. Cross-ancestry fine-mapping, in silico mutagenesis analysis3, and transcriptome-wide and proteome-wide association analyses revealed putative causal genes (such as SH3PXD2A and FURIN) and variants (such as at GRK5 and NOS3). Using a three-pronged approach4, we provide genetic evidence for putative drug effects, highlighting F11, KLKB1, PROC, GP1BA, LAMC2 and VCAM1 as possible targets, with drugs already under investigation for stroke for F11 and PROC. A polygenic score integrating cross-ancestry and ancestry-specific stroke GWASs with vascular-risk factor GWASs (integrative polygenic scores) strongly predicted ischaemic stroke in populations of European, East Asian and African ancestry5. Stroke genetic risk scores were predictive of ischaemic stroke independent of clinical risk factors in 52,600 clinical-trial participants with cardiometabolic disease. Our results provide insights to inform biology, reveal potential drug targets and derive genetic risk prediction tools across ancestries