Modelling glioblastoma resistance to temozolomide. A mathematical model to simulate cellular adaptation in vitro.

Drug resistance is one of the biggest challenges in the fight against cancer. In particular, in the case of glioblastoma, the most lethal brain tumour, resistance to temozolomide (the standard of care drug for chemotherapy in this tumour) is one of the main reasons behind treatment failure and hence responsible for the poor prognosis of patients diagnosed with this disease. In this work, we combine the power of three-dimensional in vitro experiments of treated glioblastoma spheroids with mathematical models of tumour evolution and adaptation. We use a novel approach based on internal variables for modelling the acquisition of resistance to temozolomide that was observed in experiments for a group of treated spheroids. These internal variables describe the cell's phenotypic state, which depends on the history of drug exposure and affects cell behaviour. We use model selection to determine the most parsimonious model and calibrate it to reproduce the experimental data, obtaining a high level of agreement between the in vitro and in silico outcomes. A sensitivity analysis is carried out to investigate the impact of each model parameter in the predictions. More importantly, we show how the model is useful for answering biological questions, such as what is the intrinsic adaptation mechanism, or for separating the sensitive and resistant populations. We conclude that the proposed in silico framework, in combination with experiments, can be useful to improve our understanding of the mechanisms behind drug resistance in glioblastoma and to eventually set some guidelines for the design of new treatment schemes.

Modeling the Growth and Size Distribution of Human Pluripotent Stem Cell Clusters in Culture.

Human pluripotent stem cells (hPSCs) hold promise for regenerative medicine to replace essential cells that die or become dysfunctional. In some cases, these cells can be used to form clusters whose size distribution affects the growth dynamics. We develop models to predict cluster size distributions of hPSCs based on several plausible hypotheses, including (0) exponential growth, (1) surface growth, (2) Logistic growth, and (3) Gompertz growth. We use experimental data to investigate these models. A partial differential equation for the dynamics of the cluster size distribution is used to fit parameters (rates of growth, mortality, etc.). A comparison of the models using their mean squared error and the Akaike Information criterion suggests that Models 1 (surface growth) or 2 (Logistic growth) best describe the data.

Modelling the spatial spread of COVID-19 in a German district using a diffusion model.

In this study, we focus on modeling the local spread of COVID-19 infections. As the pandemic continues and new variants or future pandemics can emerge, modelling the early stages of infection spread becomes crucial, especially as limited medical data might be available initially. Therefore, our aim is to gain a better understanding of the diffusion dynamics on smaller scales using partial differential equation (PDE) models. Previous works have already presented various methods to model the spatial spread of diseases, but, due to a lack of data on regional or even local scale, few actually applied their models on real disease courses in order to describe the behaviour of the disease or estimate parameters. We use medical data from both the Robert-Koch-Institute (RKI) and the Birkenfeld district government for parameter estimation within a single German district, Birkenfeld in Rhineland-Palatinate, during the second wave of the pandemic in autumn 2020 and winter 2020-21. This district can be seen as a typical middle-European region, characterized by its (mainly) rural nature and daily commuter movements towards metropolitan areas. A basic reaction-diffusion model used for spatial COVID spread, which includes compartments for susceptibles, exposed, infected, recovered, and the total population, is used to describe the spatio-temporal spread of infections. The transmission rate, recovery rate, initial infected values, detection rate, and diffusivity rate are considered as parameters to be estimated using the reported daily data and least square fit. This work also features an emphasis on numerical methods which will be used to describe the diffusion on arbitrary two-dimensional domains. Two numerical optimization techniques for parameter fitting are used: the Metropolis algorithm and the adjoint method. Two different methods, the Crank-Nicholson method and a finite element method, which are used according to the requirements of the respective optimization method are used to solve the PDE system. This way, the two methods are compared and validated and provide similar results with good approximation of the infected in both the district and the respective sub-districts.

Gait parameter fitting and adaptive enhancement based on cerebral blood oxygen information.

Accurate recognition of patients' movement intentions and real-time adjustments are crucial in rehabilitation exoskeleton robots. However, some patients are unable to utilize electromyography (EMG) signals for this purpose due to poor or missing signals in their lower limbs. In order to address this issue, we propose a novel method that fits gait parameters using cerebral blood oxygen signals. Two types of walking experiments were conducted to collect brain blood oxygen signals and gait parameters from volunteers. Time domain, frequency domain, and spatial domain features were extracted from brain hemoglobin. The AutoEncoder-Decoder method is used for feature dimension reduction. A regression model based on the long short-term memory (LSTM) model was established to fit the gait parameters and perform incremental learning for new individual data. Cross-validation was performed on the model to enhance individual adaptivity and reduce the need for individual pre-training. The coefficient of determination (R2) for the gait parameter fit was 71.544%, with a mean square error (RMSE) of less than 3.321%. Following adaptive enhancement, the coefficient of R2 increased by 6.985%, while the RMSE decreased by 0.303%. These preliminary results indicate the feasibility of fitting gait parameters using cerebral blood oxygen information. Our research offers a new perspective on assisted locomotion control for patients who lack effective myoelectricity, thereby expanding the clinical application of rehabilitation exoskeleton robots. This work establishes a foundation for promoting the application of Brain-Computer Interface (BCI) technology in the field of sports rehabilitation.

End-to-End Bubble Size Distribution Detection Technique in Dense Bubbly Flows Based on You Only Look Once Architecture.

Accurate measurements of the bubble size distribution (BSD) are crucial for investigating gas-liquid mass transfer mechanisms and describing the characteristics of chemical production. However, measuring the BSD in high-density bubbly flows remains challenging due to limited image algorithms and high data densities. Therefore, an end-to-end BSD detection method in dense bubbly flows based on deep learning is proposed in this paper. The bubble detector locates the positions of dense bubbles utilizing objection detection networks and simultaneously performs ellipse parameter fitting to measure the size of the bubbles. Different You Only Look Once (YOLO) architectures are compared, and YOLOv7 is selected as the backbone network. The complete intersection over union calculation method is modified by the circumferential horizontal rectangle of bubbles, and the loss function is optimized by adding L2 constraints of ellipse size parameters. The experimental results show that the proposed technique surpasses existing methods in terms of precision, recall, and mean square error, achieving values of 0.9871, 0.8725, and 3.8299, respectively. The proposed technique demonstrates high efficiency and accuracy when measuring BSDs in high-density bubbly flows and has the potential for practical applications.

SIRSi-vaccine dynamical model for the Covid-19 pandemic.

Covid-19, caused by severe acute respiratory syndrome coronavirus 2, broke out as a pandemic during the beginning of 2020. The rapid spread of the disease prompted an unprecedented global response involving academic institutions, regulatory agencies, and industries. Vaccination and nonpharmaceutical interventions including social distancing have proven to be the most effective strategies to combat the pandemic. In this context, it is crucial to understand the dynamic behavior of the Covid-19 spread together with possible vaccination strategies. In this study, a susceptible-infected-removed-sick model with vaccination (SIRSi-vaccine) was proposed, accounting for the unreported yet infectious. The model considered the possibility of temporary immunity following infection or vaccination. Both situations contribute toward the spread of diseases. The transcritical bifurcation diagram of alternating and mutually exclusive stabilities for both disease-free and endemic equilibria were determined in the parameter space of vaccination rate and isolation index. The existing equilibrium conditions for both points were determined in terms of the epidemiological parameters of the model. The bifurcation diagram allowed us to estimate the maximum number of confirmed cases expected for each set of parameters. The model was fitted with data from São Paulo, the state capital of SP, Brazil, which describes the number of confirmed infected cases and the isolation index for the considered data window. Furthermore, simulation results demonstrate the possibility of periodic undamped oscillatory behavior of the susceptible population and the number of confirmed cases forced by the periodic small-amplitude oscillations in the isolation index. The main contributions of the proposed model are as follows: A minimum effort was required when vaccination was combined with social isolation, while additionally ensuring the existence of equilibrium points. The model could provide valuable information for policymakers, helping define disease prevention mitigation strategies that combine vaccination and non-pharmaceutical interventions, such as social distancing and the use of masks. In addition, the SIRSi-vaccine model facilitated the qualitative assessment of information regarding the unreported infected yet infectious cases, while considering temporary immunity, vaccination, and social isolation index.

RNA Folding Based on 5 Beads Model and Multiscale Simulation.

RNA folding prediction is very meaningful and challenging. The molecular dynamics simulation (MDS) of all atoms (AA) is limited to the folding of small RNA molecules. At present, most of the practical models are coarse grained (CG) model, and the coarse-grained force field (CGFF) parameters usually depend on known RNA structures. However, the limitation of the CGFF is obvious that it is difficult to study the modified RNA. Based on the 3 beads model (AIMS_RNA_B3), we proposed the AIMS_RNA_B5 model with three beads representing a base and two beads representing the main chain (sugar group and phosphate group). We first run the all atom molecular dynamic simulation (AAMDS), and fit the CGFF parameter with the AA trajectory. Then perform the coarse-grained molecular dynamic simulation (CGMDS). AAMDS is the foundation of CGMDS. CGMDS is mainly to carry out the conformation sampling based on the current AAMDS state and improve the folding speed. We simulated the folding of three RNAs, which belong to hairpin, pseudoknot and tRNA respectively. Compared to the AIMS_RNA_B3 model, the AIMS_RNA_B5 model is more reasonable and performs better.

A mathematical model of the within-host kinetics of SARS-CoV-2 neutralizing antibodies following COVID-19 vaccination.

Compelling evidence continues to build to support the idea that SARS-CoV-2 Neutralizing Antibody (NAb) levels in an individual can serve as an important indicator of the strength of protective immunity against infection. It is not well understood why NAb levels in some individuals remain high over time, while in others levels decline rapidly. In this work, we present a two-state mathematical model of within-host NAb dynamics in response to vaccination. By fitting only four host-specific parameters, the model is able to capture individual-specific NAb levels over time as measured by the AditxtScore™ for NAbs. The model can serve as a foundation for predicting NAb levels in the long-term, understanding connections between NAb levels, protective immunity, and breakthrough infections, and potentially guiding decisions about whether and when a booster vaccination may be warranted.

Improvements to the ADM1 based Process Simulation Model: Reaction segregation, parameter estimation and process optimization.

Anaerobic digestion is a sustainable organic waste treatment technique with energy recovery via biogas generation. This work presents a novel Aspen Plus ADM1-based flowsheet for this process. Three reactor segments were chosen: stoichiometric for the hydrolysis step, kinetic for acido-aceto-methanogenesis, and equilibrium for hydrogenotrophic methane production. Selected parameters- conversion ratios, kinetic pre-exponent and inhibitor factors- were controlled to best fit model and experimental results. The parity plot fitting had an R2 = 0.999, a slope of 1.0058 and an intercept of -0.8651. Obtained parameter values stressed the importance of inhibitions, and simulation results showcased the bell-shaped curve for acetic and volatile fatty acid reduction. The model was used for a subsequent sensitivity analysis as well as an optimization runs, leading to a 50% higher methane production ratio. The proposed model presents itself as a significant contribution for optimal anaerobic digestion process design.

Time-Scale Analysis and Parameter Fitting for Vector-Borne Diseases with Spatial Dynamics.

Vector-borne diseases are progressively spreading in a growing number of countries, and it has the potential to invade new areas and habitats. From the dynamical perspective, the spatial-temporal interaction of models that try to adjust to such events is rich and challenging. The first challenge is to address the dynamics of vectors (very fast and local) and the dynamics of humans (very heterogeneous and non-local). The objective of this work is to use the well-known Ross-Macdonald models, identifying different time scales, incorporating human spatial movements and estimate in a suitable way the parameters. We will concentrate on a practical example, a simplified space model, and apply it to dengue spread in the state of Rio de Janeiro, Brazil.

An improved parameter fitting approach of a planar biaxial test including the experimental prestretch.

Planar biaxial testing is a popular experimental technique for characterizing and comparing biological soft tissues. A correct identification of the different stress states of the tissue sample is therefore essential. However, the difference between the zero-stress reference state and the sample state prior to the loading cycle caused by the mounting, preconditioning and preloading is often not considered. The importance of this difference, caused by prestretch, is investigated by simulating virtual planar biaxial experiments, either assuming an ideal test with a single deformation gradient or using finite element modeling to simulate a rake-based experiment. Multiple parameter fitting methods are used to estimate the material properties based on the available experimental data. These methods vary based on how they approximate the zero-stress state: either the prestretch is ignored, or the loads are zeroed after the preload has been reached, or the unknown prestretch values are included into the optimization function. The results reveal the high necessity of assessing the stress-free state when analyzing a planar biaxial test. The material fitting including the prestretch outperforms the other methods in terms of correctly describing the mechanical behavior of the tested material. It can be extended to correct for the boundary effects induced by the gripping mechanisms, providing a more accurate, yet more computationally expensive estimate of the material properties.

How quickly does a wound heal? Bayesian calibration of a mathematical model of venous leg ulcer healing.

Chronic wounds, such as venous leg ulcers, are difficult to treat and can reduce the quality of life for patients. Clinical trials have been conducted to identify the most effective venous leg ulcer treatments and the clinical factors that may indicate whether a wound will successfully heal. More recently, mathematical modelling has been used to gain insight into biological factors that may affect treatment success but are difficult to measure clinically, such as the rate of oxygen flow into wounded tissue. In this work, we calibrate an existing mathematical model using a Bayesian approach with clinical data for individual patients to explore which clinical factors may impact the rate of wound healing for individuals. Although the model describes group-level behaviour well, it is not able to capture individual-level responses in all cases. From the individual-level analysis, we propose distributions for coefficients of clinical factors in a linear regression model, but ultimately find that it is difficult to draw conclusions about which factors lead to faster wound healing based on the existing model and data. This work highlights the challenges of using Bayesian methods to calibrate partial differential equation models to individual patient clinical data. However, the methods used in this work may be modified and extended to calibrate spatiotemporal mathematical models to multiple data sets, such as clinical trials with several patients, to extract additional information from the model and answer outstanding biological questions.

A Parameter Representing Missing Charge Should Be Considered when Calibrating Action Potential Models.

Computational models of the electrical potential across a cell membrane are longstanding and vital tools in electrophysiology research and applications. These models describe how ionic currents, internal fluxes, and buffering interact to determine membrane voltage and form action potentials (APs). Although this relationship is usually expressed as a differential equation, previous studies have shown it can be rewritten in an algebraic form, allowing direct calculation of membrane voltage. Rewriting in this form requires the introduction of a new parameter, called Γ0 in this manuscript, which represents the net concentration of all charges that influence membrane voltage but are not considered in the model. Although several studies have examined the impact of Γ0 on long-term stability and drift in model predictions, there has been little examination of its effects on model predictions, particularly when a model is refit to new data. In this study, we illustrate how Γ0 affects important physiological properties such as action potential duration restitution, and examine the effects of (in)correctly specifying Γ0 during model calibration. We show that, although physiologically plausible, the range of concentrations used in popular models leads to orders of magnitude differences in Γ0, which can lead to very different model predictions. In model calibration, we find that using an incorrect value of Γ0 can lead to biased estimates of the inferred parameters, but that the predictive power of these models can be restored by fitting Γ0 as a separate parameter. These results show the value of making Γ0 explicit in model formulations, as it forces modellers and experimenters to consider the effects of uncertainty and potential discrepancy in initial concentrations upon model predictions.

Mathematical Modelling Highlights the Potential for Genetic Manipulation as an Adjuvant to Counter Efflux-Mediated MDR in Salmonella.

Bacteria have developed resistance to antibiotics by various mechanisms, notable amongst these is the use of permeation barriers and the expulsion of antibiotics via efflux pumps. The resistance-nodulation-division (RND) family of efflux pumps is found in Gram-negative bacteria and a major contributor to multidrug resistance (MDR). In particular, Salmonella encodes five RND efflux pump systems: AcrAB, AcrAD, AcrEF, MdsAB and MdtAB which have different substrate ranges including many antibiotics. We produce a spatial partial differential equation (PDE) model governing the diffusion and efflux of antibiotic in Salmonella, via these RND efflux pumps. Using parameter fitting techniques on experimental data, we are able to establish the behaviour of multiple wild-type and efflux mutant Salmonella strains, which enables us to produce efflux profiles for each individual efflux pump system. By combining the model with a gene regulatory network (GRN) model of efflux regulation, we simulate how the bacteria respond to their environment. Finally, performing a parameter sensitivity analysis, we look into various different targets to inhibit the efflux pumps. The model provides an in silico framework with which to test these potential adjuvants to counter MDR.

A phenomenological model to infer the microbial growth: A case study for psychrotrophic pathogenic bacteria.

AIMS: The two-parameter (α and ß) Schiraldi's model reliably fits growth curves of psychrotrophic pathogens and suggests a different description of the latency phase. METHODS AND RESULTS: Data obtained at various temperatures and different starting cell densities for Aeromonas hydrophila, Listeria monocytogenes and Yersinia enterocolitica have been fitted with the Baranyi and Roberts' model and the new one. On average, the former showed higher standard error and R2 values (0.140 and 0.991) than the Schiraldi's one (0.079 and 0.983). Around 15℃, the increase of temperature showed a milder effect on the growth rate than that expected. Y. enterocolitica showed a practically null duration of the lag phase, no matter the value of the starting density, whereas A. hydrophila and L. monocytogenes revealed slower onset trends. CONCLUSIONS: Parameter ß defines the number of cell duplications and appears independent on temperature, while (ß/α)1/2 is proportional to the maximum specific growth rate. The α-1/2 versus temperature trend directly reflects the corresponding behaviour of the growth rate and does not require the use of Arrhenius plots. SIGNIFICANCE AND IMPACT OF THE STUDY: Values of the parameters α and ß, as well as the duration of the latency phase, allowed some considerations about the effect of storage temperature in terms of food safety, especially for psychrotrophic bacteria of concern.

Stochastic models for single-cell data: Current challenges and the way forward.

Although the quantity and quality of single-cell data have progressed rapidly, making quantitative predictions with single-cell stochastic models remains challenging. The stochastic nature of cellular processes leads to at least three challenges in building models with single-cell data: (a) because variability in single-cell data can be attributed to multiple different sources, it is difficult to rule out conflicting mechanistic models that explain the same data equally well; (b) the distinction between interesting biological variability and experimental variability is sometimes ambiguous; (c) the nonstandard distributions of single-cell data can lead to violations of the assumption of symmetric errors in least-squares fitting. In this review, we first discuss recent studies that overcome some of the challenges or set up a promising direction and then introduce some powerful statistical approaches utilized in these studies. We conclude that applying and developing statistical approaches could lead to further progress in building stochastic models for single-cell data.

A program for the fitting of up to three Havriliak-Negami dispersions to dielectric data.

A new version of the DielParamFit program [J. Colloid Interface Sci. 419 (2014) 102-106] for the fitting of a superposition of dispersion terms to measured dielectric data is presented. It extends its applicability to a wide range of new systems by allowing the use of up to three Havriliak-Negami dispersions, each of which can be readily transformed into Debye, Cole-Cole, or Cole-Davidson terms. Moreover, it greatly enhances its usability by means of a graphic interface that displays the measured data together with the model spectra allowing to iteratively adjust the chosen terms and guess parameter values guided by a live view of the obtained results. The DielParamFit_2 program executable file for Microsoft Windows is available upon request from the author.

Passive mechanical properties in healthy and infarcted rat left ventricle characterised via a mixture model.

During the cardiac cycle, electrical excitation is coupled with mechanical response of the myocardium. Besides the active contraction, passive mechanics plays an important role, and its behaviour differs in healthy and diseased hearts as well as among different animal species. The aim of this study is the characterisation of passive mechanical properties in healthy and infarcted rat myocardium by means of mechanical testing and subsequent parameter fitting. Elasticity assessments via uniaxial extension tests are performed on healthy and infarcted tissue samples from left ventricular rat myocardium. In order to fully characterise the orthotropic cardiac tissue, our experimental data are combined with other previously published tests in rats - shear tests on healthy myocardium and equibiaxial tests on infarcted tissue. In a first step, we calibrate the Holzapfel-Ogden strain energy function in the healthy case. Sa far, this orthotropic constitutive law for the passive myocardium has been fitted to experimental data in several species, however there is a lack of an appropriate parameter set for the rat. With our determined parameters, a finite element simulation of the end-diastolic filling is performed. In a second step, we propose a model for the infarcted tissue. It is represented as a mixture of intact myocardium and a transversely isotropic scar structure. In our mechanical experiments, the tissue after myocardial infarction shows significantly stiffer behaviour than in the healthy case, and the stiffness correlates with the amount of fibrosis. A similar relationship is observed in the computational simulation of the end-diastolic filling. We conclude that our new proposed material model can capture the behaviour of two kinds of tissues - healthy and infarcted rat myocardium, and its calibration with the fitted parameters represents the experimental data well.

Mathematical Modeling of a Supramolecular Assembly for Pyrophosphate Sensing.

The power of sensing molecules is often characterized in part by determining their thermodynamic/dynamic properties, in particular the binding constant of a guest to a host. In many studies, traditional nonlinear regression analysis has been used to determine the binding constants, which cannot be applied to complex systems and limits the reliability of such calculations. Supramolecular sensor systems include many interactions that make such chemical systems complicated. The challenges in creating sensing molecules can be significantly decreased through the availability of detailed mathematical models of such systems. Here, we propose uncovering accurate thermodynamic parameters of chemical reactions using better-defined mathematical modeling-fitting analysis is the key to understanding molecular assemblies and developing new bio/sensing agents. The supramolecular example we chose for this investigation is a self-assembled sensor consists of a synthesized receptor, DPA (DPA = dipicolylamine)-appended phenylboronic acid (1) in combination with Zn2+(1.Zn) that forms various assemblies with a fluorophore like alizarin red S (ARS). The self-assemblies can detect multi-phosphates like pyrophosphate (PPi) in aqueous solutions. We developed a mathematical model for the simultaneous quantitative analysis of twenty-seven intertwined interactions and reactions between the sensor (1.Zn-ARS) and the target (PPi) for the first time, relying on the Newton-Raphson algorithm. Through analyzing simulated potentiometric titration data, we describe the concurrent determination of thermodynamic parameters of the different guest-host bindings. Various values of temperatures, initial concentrations, and starting pHs were considered to predict the required measurement conditions for thermodynamic studies. Accordingly, we determined the species concentrations of different host-guest bindings in a generalized way. This way, the binding capabilities of a set of species can be quantitatively examined to systematically measure the power of the sensing system. This study shows analyzing supramolecular self-assemblies with solid mathematical models has a high potential for a better understanding of molecular interactions within complex chemical networks and developing new sensors with better sensing effects for bio-purposes.

A 3-D constitutive model for finite element analyses of agarose with a range of gel concentrations.

Hydrogels have seen widespread application across biomedical sciences and there is considerable interest in using hydrogels, including agarose, for creating in vitro three-dimensional environments to grow cells and study mechanobiology and mechanotransduction. Recent advances in the preparation of agarose gels enable successful encapsulation of viable cells at gel concentrations as high as 5%. Agarose with a range of gel concentrations can thus serve as an experimental model mimicking changes in the 3-D microenvironment of cells during disease progression and can facilitate experiments aimed at probing the corresponding mechanobiology, e.g. the evolving mechanobiology of chondrocytes during the progression of osteoarthritis. Importantly, whether stresses (forces) or strains (displacements) drive mechanobiology and mechanotransduction is currently unknown. We can use experiments to quantify mechanical properties of hydrogels, and imaging to estimate microstructure and even strains; however, only computational models can estimate intra-gel stresses in cell-seeded agarose constructs because the required in vitro experiments are currently impossible. Finite element modeling is well-established for (computational) mechanical analyses, but accurate constitutive models for modeling the 3-D mechanical environments of cells within high-stiffness agarose with varying gel concentrations are currently unavailable. In this study we aimed to establish a 3-D constitutive model of high-stiffness agarose with a range of gel concentrations. We applied a multi-step, physics-based optimization approach to separately fit subsets of model parameters and help achieve robust convergence. Our constitutive model, fitted to experimental data on progressive stress-relaxations, was able to predict reaction forces determined from independent experiments on cyclical loading. Our model has broad applications in finite element modeling aimed at interpreting mechanical experiments on agarose specimens seeded with cells, particularly in predicting distributions of intra-gel stresses. Our model and fitted parameters enable more accurate finite element simulations of high-stiffness agarose constructs, and thus better understanding of experiments aimed at mechanobiology, mechanotransduction, or other applications in tissue engineering.