A new method to design energy-conserving surrogate models for the coupled, nonlinear responses of intervertebral discs.

The aim of this study was to design physics-preserving and precise surrogate models of the nonlinear elastic behaviour of an intervertebral disc (IVD). Based on artificial force-displacement data sets from detailed finite element (FE) disc models, we used greedy kernel and polynomial approximations of second, third and fourth order to train surrogate models for the scalar force-torque -potential. Doing so, the resulting models of the elastic IVD responses ensured the conservation of mechanical energy through their structure. At the same time, they were capable of predicting disc forces in a physiological range of motion and for the coupling of all six degrees of freedom of an intervertebral joint. The performance of all surrogate models for a subject-specific L4 | 5 disc geometry was evaluated both on training and test data obtained from uncoupled (one-dimensional), weakly coupled (two-dimensional), and random movement trajectories in the entire six-dimensional (6d) physiological displacement range, as well as on synthetic kinematic data. We observed highest precisions for the kernel surrogate followed by the fourth-order polynomial model. Both clearly outperformed the second-order polynomial model which is equivalent to the commonly used stiffness matrix in neuro-musculoskeletal simulations. Hence, the proposed model architectures have the potential to improve the accuracy and, therewith, validity of load predictions in neuro-musculoskeletal spine models.

Improving Determination of Pigment Contents in Microalgae Suspension with Absorption Spectroscopy: Light Scattering Effect and Bouguer-Lambert-Beer Law.

The Bouguer-Lambert-Beer (BLB) law serves as the fundamental basis for the spectrophotometric determination of pigment content in microalgae. Although it has been observed that the applicability of the BLB law is compromised by the light scattering effect in microalgae suspensions, in-depth research concerning the relationship between the light scattering effect and the accuracy of spectrophotometric pigment determination remains scarce. We hypothesized that (1) the precision of spectrophotometric pigment content determination using the BLB law would diminish with increasing nonlinearity of absorbance, and (2) employing the modified version of the BLB (mBLB) law would yield superior performance. To assess our hypotheses, we cultivated Phaeodactylum tricornutum under varying illumination conditions and nitrogen supplies in controlled indoor experiments, resulting in suspensions with diverse pigment contents. Subsequently, P. tricornutum samples were diluted into subsamples, and spectral measurements were conducted using different combinations of biomass concentrations and path lengths. This was carried out to assess the applicability of the BLB law and the nonlinearity of absorbance. The chlorophyll a and fucoxanthin contents in the samples were analyzed via high-performance liquid chromatography (HPLC) and subsequently used in our modeling. Our findings confirm our hypotheses, showing that the modified BLB law outperforms the original BLB law in terms of the normalized root mean square error (NRMSE): 6.3% for chlorophyll a and 5.8% for fucoxanthin, compared to 8.5% and 7.9%, respectively.

Improving microalgae growth modeling of outdoor cultivation with light history data using machine learning models: A comparative study.

Accurate prediction of microalgae growth is crucial for understanding the impacts of light dynamics and optimizing production. Although various mathematical models have been proposed, only a few of them have been validated in outdoor cultivation. This study aims to investigate the use of machine learning algorithms in microalgae growth modeling. Outdoor cultivation data of Phaeodactylum tricornutum in flat-panel airlift photobioreactors for 50 days were used to compare the performance of Long Short-Term Memory (LSTM) and Support Vector Regression (SVR) with traditional models, namely Monod and Haldane. The results indicate that the machine learning models outperform the traditional models due to their ability to utilize light history as input. Moreover, the LSTM model shows an excellent ability to describe the light acclimation effect. Last, two potential applications of these models are demonstrated: 1) use as a biomass soft sensor and 2) development of an optimal harvest strategy for outdoor cultivation.

A novel model extended from the Bouguer-Lambert-Beer law can describe the non-linear absorbance of potassium dichromate solutions and microalgae suspensions.

Introduction: The Bouguer-Lambert-Beer law is widely used as the fundamental equation for quantification in absorption spectroscopy. However, deviations from the Bouguer-Lambert-Beer law have also been observed, such as chemical deviation and light scattering effect. While it has been proven and shown that the Bouguer-Lambert-Beer law is valid only under very restricted limitations, there are only a few alternatives of analytical models to this law. Based on the observation in the experiments, we propose a novel model to solve the problem of chemical deviation and light scattering effect. Methods: To test the proposed model, a systematic verification was conducted using potassium dichromate solutions and two types of microalgae suspensions with varying concentrations and path lengths. Results: Our proposed model demonstrated excellent performance, with a correlation coefficient ( R 2 ) exceeding 0.995 for all tested materials, significantly surpassing the Bouguer-Lambert-Beer law, which had an R 2 as low as 0.94. Our results confirm that the absorbance of pure pigment solutions follows the Bouguer-Lambert-Beer law, while the microalgae suspensions do not due to the light scattering effect. We also show that this scattering effect leads to huge deviations for the commonly used linear scaling of the spectra, and we provide a better solution based on the proposed model. Discussion: This work provides a powerful tool for chemical analysis and especially for the quantification of microorganisms, such as the concentration of biomass or intracellular biomolecules. Not only the high accuracy but also the simplicity of the model makes it a practical alternative to the existing Bouguer-Lambert-Beer law.

Gaussian Process Regression for Minimum Energy Path Optimization and Transition State Search.

We implemented a gradient-based algorithm for finding minimum energy paths (MEPs) using Gaussian process regression (GPR). A subsequent search for transition states can be performed very fast. We describe the algorithm in detail and compare its performance to the nudged elastic band (NEB) method in 27 test systems. Additionally, three different possibilities for an initial guess of the path are evaluated. We found the new optimizer to considerably decrease the number of required energy and gradient evaluations.

Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods.

In this work, we consider 2 kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery, i.e., in an artery that is located far away from the heart. For our simulations, we place the stenosis in one of the tibial arteries belonging to the right lower leg (right posterior tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore, it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.

Efficient parametric analysis of the chemical master equation through model order reduction.

BACKGROUND: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. RESULTS: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. CONCLUSIONS: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.

Kernel discriminant analysis for positive definite and indefinite kernels.

Kernel methods are a class of well established and successful algorithms for pattern analysis thanks to their mathematical elegance and good performance. Numerous nonlinear extensions of pattern recognition techniques have been proposed so far based on the so-called kernel trick. The objective of this paper is twofold. First, we derive an additional kernel tool that is still missing, namely kernel quadratic discriminant (KQD). We discuss different formulations of KQD based on the regularized kernel Mahalanobis distance in both complete and class-related subspaces. Secondly, we propose suitable extensions of kernel linear and quadratic discriminants to indefinite kernels. We provide classifiers that are applicable to kernels defined by any symmetric similarity measure. This is important in practice because problem-suited proximity measures often violate the requirement of positive definiteness. As in the traditional case, KQD can be advantageous for data with unequal class spreads in the kernel-induced spaces, which cannot be well separated by a linear discriminant. We illustrate this on artificial and real data for both positive definite and indefinite kernels.