The inverse problem in mathematical biology

Invariances in relations between aging, exposure to external hazards, and mortality reflected in life table aging rate (LAR) patterns examined through the lens of generalized Gompertz-Makeham law

According to the Gompertz law, the age-dependent change in the logarithm of mortality (life-table aging rate, LAR) is equal to the population-averaged age-independent biological aging rate (γ), and LAR would be constant if aging were the only cause of mortality increase. However, LAR is influenced by population exposures to the external hazards. If they were constant, according to the Gompertz-Makeham law (GML), LAR would be below γ at lower ages and asymptotically and monotonically approach γ with increasing age. Actually, LAR trajectories derived from data on mortality in different countries and historical periods feature systematic undulations. In the present investigation, mortality-vs.-age trajectories were modeled based on a generalized GML (gGML). Unlike the canonical GML terms, which are population-specific constants, the respective terms of the gGML are represented with some population-specific functions of age. Invariant in gGML are the modes of translation of these functions into the dependency of mortality on age: linear for population exposure to the irresistible external hazards or exponential for population-averaged ability to withstand the resistible external and internal hazards. Modeling suggests that, at earlier ages, LAR undulations are attributable to changes in population exposures to the former hazards. However, only their unrealistically high levels can produce the transient increase in LAR at about 65 to 90 years. This pervasive undulation of LAR-vs.-age trajectory is rather caused by an increment in γ. Reasons to regard gGML as a genuine natural law, which defines relations between mortality, aging and environment, are discussed.

Description of an activity-based enzyme biosensor for lung cancer detection

BACKGROUND

Lung cancer is associated with the greatest cancer mortality as it typically presents with incurable distributed disease. Biomarkers relevant to risk assessment for the detection of lung cancer continue to be a challenge because they are often not detectable during the asymptomatic curable stage of the disease. A solution to population-scale testing for lung cancer will require a combination of performance, scalability, cost-effectiveness, and simplicity.

METHODS

One solution is to measure the activity of serum available enzymes that contribute to the transformation process rather than counting biomarkers. Protease enzymes modify the environment during tumor growth and present an attractive target for detection. An activity based sensor platform sensitive to active protease enzymes is presented. A panel of 18 sensors was used to measure 750 sera samples from participants at increased risk for lung cancer with or without the disease.

RESULTS

A machine learning approach is applied to generate algorithms that detect 90% of cancer patients overall with a specificity of 82% including 90% sensitivity in Stage I when disease intervention is most effective and detection more challenging.

CONCLUSION

This approach is promising as a scalable, clinically useful platform to help detect patients who have lung cancer using a simple blood sample. The performance and cost profile is being pursued in studies as a platform for population wide screening.

PhysiCOOL: A generalized framework for model Calibration and Optimization Of modeLing projects

In silico models of biological systems are usually very complex and rely on a large number of parameters describing physical and biological properties that require validation. As such, parameter space exploration is an essential component of computational model development to fully characterize and validate simulation results. Experimental data may also be used to constrain parameter space (or enable model calibration) to enhance the biological relevance of model parameters. One widely used computational platform in the mathematical biology community is PhysiCell, which provides a standardized approach to agent-based models of biological phenomena at different time and spatial scales. Nonetheless, one limitation of PhysiCell is the lack of a generalized approach for parameter space exploration and calibration that can be run without high-performance computing access. Here, we present PhysiCOOL, an open-source Python library tailored to create standardized calibration and optimization routines for PhysiCell models.

A dynamically structured matrix population model for insect life histories observed under variable environmental conditions

Various environmental drivers influence life processes of insect vectors that transmit human disease. Life histories observed under experimental conditions can reveal such complex links; however, designing informative experiments for insects is challenging. Furthermore, inferences obtained under controlled conditions often extrapolate poorly to field conditions. Here, we introduce a pseudo-stage-structured population dynamics model to describe insect development as a renewal process with variable rates. The model permits representing realistic life stage durations under constant and variable environmental conditions. Using the model, we demonstrate how random environmental variations result in fluctuating development rates and affect stage duration. We apply the model to infer environmental dependencies from the life history observations of two common disease vectors, the southern (Culex quinquefasciatus) and northern (Culex pipiens) house mosquito. We identify photoperiod, in addition to temperature, as pivotal in regulating larva stage duration, and find that carefully timed life history observations under semi-field conditions accurately predict insect development throughout the year. The approach we describe augments existing methods of life table design and analysis, and contributes to the development of large-scale climate- and environment-driven population dynamics models for important disease vectors.

Nonlinear model-based cardiac arrhythmia diagnosis using the optimization-based inverse problem solution

This study investigates a nonlinear model-based feature extraction approach for the accurate classification of four types of heartbeats. The features are the morphological parameters of ECG signal derived from the nonlinear ECG model using an optimization-based inverse problem solution. In the model-based methods, high feature extraction time is a crucial issue. In order to reduce the feature extraction time, a new structure was employed in the optimization algorithms. Using the proposed structure has considerably increased the speed of feature extraction. In the following, the effectiveness of two types of optimization methods (genetic algorithm and particle swarm optimization) and the McSharry ECG model has been studied and compared in terms of speed and accuracy of diagnosis. In the classification section, the adaptive neuro-fuzzy inference system and fuzzy c-mean clustering methods, along with the principal component analysis data reduction method, have been utilized. The obtained results reveal that using an adaptive neuro-fuzzy inference system with data obtained from particle swarm optimization will have the shortest process time and the best diagnosis, with a mean accuracy of 99% and a mean sensitivity of 99.11%.

On the effects of diabetes mellitus on the mechanical properties of DRG sensory neurons and their possible relation with diabetic neuropathy

Diabetic neuropathy (DN) is one of the principal complications of diabetes mellitus (DM). Dorsal root ganglion (DRG) neurons are the primary sensory neurons that transduce mechanical, chemical, thermal, and pain stimuli. Diabetes-caused sensitivity alterations and presence of pain are due to cellular damage originated by persistent hyperglycemia, microvascular insufficiency, and oxidative and nitrosative stress. However, the underlying mechanisms have not been fully clarified. The present work addresses this problem by hypothesizing that sensitivity changes in DN result from mechanotransduction-system alterations in sensory neurons; especially, plasma membrane affectations. This hypothesis is tackled by means of elastic-deformation experiments performed on DGR neurons from a murine model for type-1 DM, as well a mathematical model of the cell mechanical structure. The obtained results suggest that the plasma-membrane fluidity of DRG sensory neurons is modified by the induction of DM, and that this alteration may correlate with changes in the cell calcium transient that results from mechanical stimuli.

Bespoke Turing Systems

Reaction-diffusion systems are an intensively studied form of partial differential equation, frequently used to produce spatially heterogeneous patterned states from homogeneous symmetry breaking via the Turing instability. Although there are many prototypical "Turing systems" available, determining their parameters, functional forms, and general appropriateness for a given application is often difficult. Here, we consider the reverse problem. Namely, suppose we know the parameter region associated with the reaction kinetics in which patterning is required-we present a constructive framework for identifying systems that will exhibit the Turing instability within this region, whilst in addition often allowing selection of desired patterning features, such as spots, or stripes. In particular, we show how to build a system of two populations governed by polynomial morphogen kinetics such that the: patterning parameter domain (in any spatial dimension), morphogen phases (in any spatial dimension), and even type of resulting pattern (in up to two spatial dimensions) can all be determined. Finally, by employing spatial and temporal heterogeneity, we demonstrate that mixed mode patterns (spots, stripes, and complex prepatterns) are also possible, allowing one to build arbitrarily complicated patterning landscapes. Such a framework can be employed pedagogically, or in a variety of contemporary applications in designing synthetic chemical and biological patterning systems. We also discuss the implications that this freedom of design has on using reaction-diffusion systems in biological modelling and suggest that stronger constraints are needed when linking theory and experiment, as many simple patterns can be easily generated given freedom to choose reaction kinetics.

Characterizing the type 2 diabetes mellitus epidemic in Jordan up to 2050

We aimed to characterize the type 2 diabetes mellitus (T2DM) epidemic and the role of key risk factors in Jordan between 1990-2050, and to forecast the T2DM-related costs. A recently-developed population-level T2DM mathematical model was adapted and applied to Jordan. The model was fitted to six population-based survey data collected between 1990 and 2017. T2DM prevalence was 14.0% in 1990, and projected to be 16.0% in 2020, and 20.6% in 2050. The total predicted number of T2DM cases were 218,326 (12,313 were new cases) in 1990, 702,326 (36,941 were new cases) in 2020, and 1.9 million (79,419 were new cases) in 2050. Out of Jordan's total health expenditure, 19.0% in 1990, 21.1% in 2020, and 25.2% in 2050 was forecasted to be spent on T2DM. The proportion of T2DM incident cases attributed to obesity was 55.6% in 1990, 59.5% in 2020, and 62.6% in 2050. Meanwhile, the combined contribution of smoking and physical inactivity hovered around 5% between 1990 and 2050. Jordan's T2DM epidemic is predicted to grow sizably in the next three decades, driven by population ageing and high and increasing obesity levels. The national strategy to prevent T2DM needs to be strengthened by focusing it on preventive interventions targeting T2DM and key risk factors.

Time-continuous and time-discrete SIR models revisited: theory and applications

Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people-infectious people-recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran.

Model-Based Quantification of Left Ventricular Diastolic Function in Critically Ill Patients with Atrial Fibrillation from Routine Data: A Feasibility Study

Introduction

Left ventricular diastolic dysfunction (LVDD) and atrial fibrillation (AF) are connected by pathophysiology and prevalence. LVDD remains underdiagnosed in critically ill patients despite potentially significant therapeutic implications since direct measurement cannot be performed in routine care at the bedside, and echocardiographic assessment of LVDD in AF is impaired. We propose a novel approach that allows us to infer the diastolic stiffness, β, a key quantitative parameter of diastolic function, from standard monitoring data by solving the nonlinear, ill-posed inverse problem of parameter estimation for a previously described mechanistic, physiological model of diastolic filling. The beat-to-beat variability in AF offers an advantageous setting for this.

Methods

By employing a global optimization algorithm, β is inferred from a simple six parameter and an expanded seven parameter model of left ventricular filling. Optimization of all parameters was limited to the interval ]0, 400[ and initialized randomly on large intervals encompassing the support of the likelihood function. Routine ECG and arterial pressure recordings of 17 AF and 3 sinus rhythm (SR) patients from the PhysioNet MGH/MF Database were used as inputs.

Results

Estimation was successful in 15 of 17 AF patients, while in the 3 SR patients, no reliable estimation was possible. For both models, the inferred β (0.065 ± 0.044 ml⁻¹ vs. 0.038 ± 0.033 ml⁻¹ (p=0.02) simple vs. expanded) was compatible with the previously described (patho) physiological range. Aortic compliance, α, inferred from the expanded model (1.46 ± 1.50 ml/mmHg) also compared well with literature values.

Conclusion

The proposed approach successfully inferred β within the physiological range. This is the first report of an approach quantifying LVDF from routine monitoring data in critically ill AF patients. Provided future successful external validation, this approach may offer a tool for minimally invasive online monitoring of this crucial parameter.

Parameter estimation in models of biological oscillators: an automated regularised estimation approach

BACKGROUND

Dynamic modelling is a core element in the systems biology approach to understanding complex biosystems. Here, we consider the problem of parameter estimation in models of biological oscillators described by deterministic nonlinear differential equations. These problems can be extremely challenging due to several common pitfalls: (i) a lack of prior knowledge about parameters (i.e. massive search spaces), (ii) convergence to local optima (due to multimodality of the cost function), (iii) overfitting (fitting the noise instead of the signal) and (iv) a lack of identifiability. As a consequence, the use of standard estimation methods (such as gradient-based local ones) will often result in wrong solutions. Overfitting can be particularly problematic, since it produces very good calibrations, giving the impression of an excellent result. However, overfitted models exhibit poor predictive power. Here, we present a novel automated approach to overcome these pitfalls. Its workflow makes use of two sequential optimisation steps incorporating three key algorithms: (1) sampling strategies to systematically tighten the parameter bounds reducing the search space, (2) efficient global optimisation to avoid convergence to local solutions, (3) an advanced regularisation technique to fight overfitting. In addition, this workflow incorporates tests for structural and practical identifiability.

RESULTS

We successfully evaluate this novel approach considering four difficult case studies regarding the calibration of well-known biological oscillators (Goodwin, FitzHugh-Nagumo, Repressilator and a metabolic oscillator). In contrast, we show how local gradient-based approaches, even if used in multi-start fashion, are unable to avoid the above-mentioned pitfalls.

CONCLUSIONS

Our approach results in more efficient estimations (thanks to the bounding strategy) which are able to escape convergence to local optima (thanks to the global optimisation approach). Further, the use of regularisation allows us to avoid overfitting, resulting in more generalisable calibrated models (i.e. models with greater predictive power).

A dynamic model of lignin biosynthesis in Brachypodium distachyon

BACKGROUND

Lignin is a crucial molecule for terrestrial plants, as it offers structural support and permits the transport of water over long distances. The hardness of lignin reduces plant digestibility by cattle and sheep; it also makes inedible plant materials recalcitrant toward the enzymatic fermentation of cellulose, which is a potentially valuable substrate for sustainable biofuels. Targeted attempts to change the amount or composition of lignin in relevant plant species have been hampered by the fact that the lignin biosynthetic pathway is difficult to understand, because it uses several enzymes for the same substrates, is regulated in an ill-characterized manner, may operate in different locations within cells, and contains metabolic channels, which the plant may use to funnel initial substrates into specific monolignols.

RESULTS

We propose a dynamic mathematical model that integrates various datasets and other information regarding the lignin pathway in Brachypodium distachyon and permits explanations for some counterintuitive observations. The model predicts the lignin composition and label distribution in a BdPTAL knockdown strain, with results that are quite similar to experimental data.

CONCLUSION

Given the present scarcity of available data, the model resulting from our analysis is presumably not final. However, it offers proof of concept for how one may design integrative pathway models of this type, which are necessary tools for predicting the consequences of genomic or other alterations toward plants with lignin features that are more desirable than in their wild-type counterparts.

The best models of metabolism

Biochemical systems are among of the oldest application areas of mathematical modeling. Spanning a time period of over one hundred years, the repertoire of options for structuring a model and for formulating reactions has been constantly growing, and yet, it is still unclear whether or to what degree some models are better than others and how the modeler is to choose among them. In fact, the variety of options has become overwhelming and difficult to maneuver for novices and experts alike. This review outlines the metabolic model design process and discusses the numerous choices for modeling frameworks and mathematical representations. It tries to be inclusive, even though it cannot be complete, and introduces the various modeling options in a manner that is as unbiased as that is feasible. However, the review does end with personal recommendations for the choices of default models. WIREs Syst Biol Med 2017, 9:e1391. doi: 10.1002/wsbm.1391 For further resources related to this article, please visit the WIREs website.

Kalman filter parameter estimation for a nonlinear diffusion model of epithelial cell migration using stochastic collocation and the Karhunen-Loeve expansion

Several Kalman filter algorithms are presented for data assimilation and parameter estimation for a nonlinear diffusion model of epithelial cell migration. These include the ensemble Kalman filter with Monte Carlo sampling and a stochastic collocation (SC) Kalman filter with structured sampling. Further, two types of noise are considered -uncorrelated noise resulting in one stochastic dimension for each element of the spatial grid and correlated noise parameterized by the Karhunen-Loeve (KL) expansion resulting in one stochastic dimension for each KL term. The efficiency and accuracy of the four methods are investigated for two cases with synthetic data with and without noise, as well as data from a laboratory experiment. While it is observed that all algorithms perform reasonably well in matching the target solution and estimating the diffusion coefficient and the growth rate, it is illustrated that the algorithms that employ SC and KL expansion are computationally more efficient, as they require fewer ensemble members for comparable accuracy. In the case of SC methods, this is due to improved approximation in stochastic space compared to Monte Carlo sampling. In the case of KL methods, the parameterization of the noise results in a stochastic space of smaller dimension. The most efficient method is the one combining SC and KL expansion.

Robust and efficient parameter estimation in dynamic models of biological systems

Background

Dynamic modelling provides a systematic framework to understand function in biological systems. Parameter estimation in nonlinear dynamic models remains a very challenging inverse problem due to its nonconvexity and ill-conditioning. Associated issues like overfitting and local solutions are usually not properly addressed in the systems biology literature despite their importance.

Here we present a method for robust and efficient parameter estimation which uses two main strategies to surmount the aforementioned difficulties: (i) efficient global optimization to deal with nonconvexity, and (ii) proper regularization methods to handle ill-conditioning. In the case of regularization, we present a detailed critical comparison of methods and guidelines for properly tuning them. Further, we show how regularized estimations ensure the best trade-offs between bias and variance, reducing overfitting, and allowing the incorporation of prior knowledge in a systematic way.

Results

We illustrate the performance of the presented method with seven case studies of different nature and increasing complexity, considering several scenarios of data availability, measurement noise and prior knowledge. We show how our method ensures improved estimations with faster and more stable convergence. We also show how the calibrated models are more generalizable. Finally, we give a set of simple guidelines to apply this strategy to a wide variety of calibration problems.

Conclusions

Here we provide a parameter estimation strategy which combines efficient global optimization with a regularization scheme. This method is able to calibrate dynamic models in an efficient and robust way, effectively fighting overfitting and allowing the incorporation of prior information.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0219-2) contains supplementary material, which is available to authorized users.