Hyperbolastic growth models: theory and application

Mathematical Modeling of Alpha-Tocopherol Early Degradation Kinetics to Predict the Shelf-Life of Bulk Oils

The kinetics of lipid oxidation includes a lag phase followed by an exponential increase in oxidation products, which cause rancidity. Current models focus on the slope of this exponential curve for shelf-life estimation, which still requires the measurement of full oxidation kinetics. In this paper, we analyzed the formation of lipid oxidation products in stripped soybean oil containing different levels of α-tocopherol. The lag phases of lipid hydroperoxides and headspace hexanal formation were found to have a strong positive correlation with the α-tocopherol depletion time. We propose that the kinetics of antioxidant (α-tocopherol) depletion occur during the lag phase and could serve as an early shelf-life indicator. Our results showed that α-tocopherol degradation can be described by Weibull kinetics over a wide range of initial concentrations. Furthermore, we conducted in silico investigations using Monte Carlo simulations to critically evaluate the feasibility and sensitivity of the shelf-life prediction using early antioxidant degradation kinetics. Our results revealed that the shelf life of soybean oil may be accurately predicted as early as 20% of the overall shelf life. This innovative approach provides a more efficient and faster assessment of shelf life, ultimately reducing waste and enhancing product quality.

A Growth Performance and Nonlinear Growth Curve Functions of Large- and Normal-Sized Japanese Quail (Coturnix japonica)

This study aimed to evaluate the differences between the growth patterns of large- and normal-sized Japanese quail strains and their F₁ progeny, by fitting their growth parameter values to five nonlinear regression growth models (Weibull, Logistic, Gompertz, Richards, and Brody). The Richards model presented the best fit for both sexes of the large-sized quail strain, whereas the Gompertz model presented the best fit for both sexes of the normal-sized quail strain, based on goodness-of-fit criteria (higher adjusted R² and lower Akaike and Bayesian information criteria). Both sexes of F₁ birds derived from the cross between normal-sized females and large-sized males were best fitted by the Richards model. In contrast, growth parameters of the F₁ birds derived from the cross between large-sized females and normal-sized males were best fitted to the Gompertz model. The data could be fitted nearly as well to the Weibull and Logistic models as to the Richards and Gompertz models. The Brody model presented the poorest fit for the growth parameter values. The results indicated that the Richards and Gompertz models could best describe the growth characteristics of both large- and normal-sized quails. Moreover, the observed growth pattern of the F₁ birds was likely inherited from the male parental strain. To the best of our knowledge, this is the first study comparing the growth curves of the reciprocal F₁ generations with their parental strains in quails.

Optimal minimum variance-entropy control of tumour growth processes based on the Fokker-Planck equation

The authors demonstrated an optimal stochastic control algorithm to obtain desirable cancer treatment based on the Gompertz model. Two external forces as two time-dependent functions are presented to manipulate the growth and death rates in the drift term of the Gompertz model. These input signals represent the effect of external treatment agents to decrease tumour growth rate and increase tumour death rate, respectively. Entropy and variance of cancerous cells are simultaneously controlled based on the Gompertz model. They have introduced a constrained optimisation problem whose cost function is the variance of a cancerous cells population. The defined entropy is based on the probability density function of affected cells was used as a constraint for the cost function. Analysing growth and death rates of cancerous cells, it is found that the logarithmic control signal reduces the growth rate, while the hyperbolic tangent-like control function increases the death rate of tumour growth. The two optimal control signals were calculated by converting the constrained optimisation problem into an unconstrained optimisation problem and by using the real-coded genetic algorithm. Mathematical justifications are implemented to elucidate the existence and uniqueness of the solution for the optimal control problem.

A computational diffusion model to study antibody transport within reconstructed tumor microenvironments

Background

Antibodies revolutionized cancer treatment over the past decades. Despite their successfully application, there are still challenges to overcome to improve efficacy, such as the heterogeneous distribution of antibodies within tumors. Tumor microenvironment features, such as the distribution of tumor and other cell types and the composition of the extracellular matrix may work together to hinder antibodies from reaching the target tumor cells. To understand these interactions, we propose a framework combining in vitro and in silico models. We took advantage of in vitro cancer models previously developed by our group, consisting of tumor cells and fibroblasts co-cultured in 3D within alginate capsules, for reconstruction of tumor microenvironment features.

Results

In this work, an experimental-computational framework of antibody transport within alginate capsules was established, assuming a purely diffusive transport, combined with an exponential saturation effect that mimics the saturation of binding sites on the cell surface. Our tumor microenvironment in vitro models were challenged with a fluorescent antibody and its transport recorded using light sheet fluorescence microscopy. Diffusion and saturation parameters of the computational model were adjusted to reproduce the experimental antibody distribution, with root mean square error under 5%. This computational framework is flexible and can simulate different random distributions of tumor microenvironment elements (fibroblasts, cancer cells and collagen fibers) within the capsule. The random distribution algorithm can be tuned to follow the general patterns observed in the experimental models.

Conclusions

We present a computational and microscopy framework to track and simulate antibody transport within the tumor microenvironment that complements the previously established in vitro models platform. This framework paves the way to the development of a valuable tool to study the influence of different components of the tumor microenvironment on antibody transport.

The recent advances in the mathematical modelling of human pluripotent stem cells

Human pluripotent stem cells hold great promise for developments in regenerative medicine and drug design. The mathematical modelling of stem cells and their properties is necessary to understand and quantify key behaviours and develop non-invasive prognostic modelling tools to assist in the optimisation of laboratory experiments. Here, the recent advances in the mathematical modelling of hPSCs are discussed, including cell kinematics, cell proliferation and colony formation, and pluripotency and differentiation.

Comparative Study between Generalized Maximum Entropy and Bayes Methods to Estimate the Four Parameter Weibull Growth Model

The Weibull growth model is an important model especially for describing the growth instability; therefore, in this paper, three methods, namely, generalized maximum entropy, Bayes, and maximum a posteriori, for estimating the four parameter Weibull growth model have been presented and compared. To achieve this aim, it is necessary to use a simulation technique to generate the samples and perform the required comparisons, using varying sample sizes (10, 12, 15, 20, 25, and 30) and models depending on the standard deviation (0.5). It has been shown from the computational results that the Bayes method gives the best estimates.

A hyperbolastic type-I diffusion process: Parameter estimation by means of the firefly algorithm

A stochastic diffusion process, whose mean function is a hyperbolastic curve of type I, is presented. The main characteristics of the process are studied and the problem of maximum likelihood estimation for the parameters of the process is considered. To this end, the firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stagewise procedure. Some examples based on simulated sample paths and real data illustrate this development.

A predictive estimation method for carbon dioxide transport by data-driven modeling with a physically-based data model

In this study, a data-driven method for predicting CO₂ leaks and associated concentrations from geological CO₂ sequestration is developed. Several candidate models are compared based on their reproducibility and predictive capability for CO₂ concentration measurements from the Environment Impact Evaluation Test (EIT) site in Korea. Based on the data mining results, a one-dimensional solution of the advective-dispersive equation for steady flow (i.e., Ogata-Banks solution) is found to be most representative for the test data, and this model is adopted as the data model for the developed method. In the validation step, the method is applied to estimate future CO₂ concentrations with the reference estimation by the Ogata-Banks solution, where a part of earlier data is used as the training dataset. From the analysis, it is found that the ensemble mean of multiple estimations based on the developed method shows high prediction accuracy relative to the reference estimation. In addition, the majority of the data to be predicted are included in the proposed quantile interval, which suggests adequate representation of the uncertainty by the developed method. Therefore, the incorporation of a reasonable physically-based data model enhances the prediction capability of the data-driven model. The proposed method is not confined to estimations of CO₂ concentration and may be applied to various real-time monitoring data from subsurface sites to develop automated control, management or decision-making systems.

Racial and Gender Disparities in Incidence of Lung and Bronchus Cancer in the United States: A Longitudinal Analysis

Background

Certain population groups in the United States carry a disproportionate burden of cancer. This work models and analyzes the dynamics of lung and bronchus cancer age-adjusted incidence rates by race (White and Black), gender (male and female), and prevalence of daily smoking in 38 U.S. states, the District of Columbia, and across eight U.S. geographic regions from 1999 to 2012.

Methods

Data, obtained from the U.S. Cancer Statistics Section of the Centers for Disease Control and Prevention, reflect approximately 77% of the U.S. population and constitute a representative sample for making inferences about incidence rates in lung and bronchus cancer (henceforth lung cancer). A longitudinal linear mixed-effects model was used to study lung cancer incidence rates and to estimate incidence rate as a function of time, race, gender, and prevalence of daily smoking.

Results

Between 1999 and 2012, age-adjusted incidence rates in lung cancer have decreased in all states and regions. However, racial and gender disparities remain. Whites continue to have lower age-adjusted incidence rates for this cancer than Blacks in all states and in five of the eight U.S. geographic regions. Disparities in incidence rates between Black and White men are significantly larger than those between Black and White women, with Black men having the highest incidence rate of all subgroups. Assuming that lung cancer incidence rates remain within reasonable range, the model predicts that the gender gap in the incidence rate for Whites would disappear by mid-2018, and for Blacks by 2026. However, the racial gap in lung cancer incidence rates among Black and White males will remain. Among all geographic regions, the Mid-South has the highest overall lung cancer incidence rate and the highest incidence rate for Whites, while the Midwest has the highest incidence rate for Blacks. Between 1999 and 2012, there was a downward trend in the prevalence of daily smokers in both genders. However, males have significantly higher rates of cigarette smoking than females at all time points. The highest and lowest prevalence of daily smoking are found in the Mid-South and New England, respectively. There was a significant correlation between lung cancer incidence rates and smoking prevalence in all geographic regions, indicating a strong influence of cigarette smoking on regional lung cancer incidence rates.

Conclusion

Although age-adjusted incidence rates in lung cancer have decreased throughout the U.S., racial and gender disparities remain. This longitudinal model can help health professionals and policy makers make predictions of age-adjusted incidence rates for lung cancer in the U.S. in the next five to ten years.

Modeling the Dynamics of the U.S. Healthcare Expenditure Using a Hyperbolastic Function

In this paper, we investigate the dynamics of the U.S. national healthcare expenditures from 1960 to 2011. The data were obtained from the U.S. Department of Health and Human Services, Center for Medicare and Medicaid Services. The analytical method allows extracting the long-run deterministic trend from the cyclical and the random components. The long-run trend is estimated using six classical growth models and three more recent growth curves called Hyperbolastic growth models of types I, II, and III, denoted by H1, H2, and H3, respectively. The statistical results indicate that the H1 model provides the best fit of the data. The study is complemented by a mathematical analysis of the deterministic long-run component of the national healthcare expenditure (NHE) as modeled by H1. This analysis is performed by examining the behavior of the absolute growth rate (pace of increase curve), the relative growth rate, and the acceleration of the U.S. NHE over the 52-year time frame. To the best of our knowledge, this paper provides the first application of Hyperbolastic models to economics data. This study can be used by researchers and policy makers as a descriptive as well as a predictive tool.

Disparities in cervical cancer mortality rates as determined by the longitudinal hyperbolastic mixed-effects type II model

Background

The main purpose of this study was to model and analyze the dynamics of cervical cancer mortality rates for African American (Black) and White women residing in 13 states located in the eastern half of the United States of America from 1975 through 2010.

Methods

The cervical cancer mortality rates of the Surveillance, Epidemiology, and End Results (SEER) were used to model and analyze the dynamics of cervical cancer mortality. A longitudinal hyperbolastic mixed-effects type II model was used to model the cervical cancer mortality data and SAS PROC NLMIXED and Mathematica were utilized to perform the computations.

Results

Despite decreasing trends in cervical cancer mortality rates for both races, racial disparities in mortality rates still exist. In all 13 states, Black women had higher mortality rates at all times. The degree of disparities and pace of decline in mortality rates over time differed among these states. Determining the paces of decline over 36 years showed that Tennessee had the most rapid decline in cervical cancer mortality for Black women, and Mississippi had the most rapid decline for White Women. In contrast, slow declines in cervical cancer mortality were noted for Black women in Florida and for White women in Maryland.

Conclusions

In all 13 states, cervical cancer mortality rates for both racial groups have fallen. Disparities in the pace of decline in mortality rates in these states may be due to differences in the rates of screening for cervical cancers. Of note, the gap in cervical cancer mortality rates between Black women and White women is narrowing.

Selection and tuning of a fast and simple phase-contrast microscopy image segmentation algorithm for measuring myoblast growth kinetics in an automated manner

Acquiring and processing phase-contrast microscopy images in wide-field long-term live-cell imaging and high-throughput screening applications is still a challenge as the methodology and algorithms used must be fast, simple to use and tune, and as minimally intrusive as possible. In this paper, we developed a simple and fast algorithm to compute the cell-covered surface (degree of confluence) in phase-contrast microscopy images. This segmentation algorithm is based on a range filter of a specified size, a minimum range threshold, and a minimum object size threshold. These parameters were adjusted in order to maximize the F-measure function on a calibration set of 200 hand-segmented images, and its performance was compared with other algorithms proposed in the literature. A set of one million images from 37 myoblast cell cultures under different conditions were processed to obtain their cell-covered surface against time. The data were used to fit exponential and logistic models, and the analysis showed a linear relationship between the kinetic parameters and passage number and highlighted the effect of culture medium quality on cell growth kinetics. This algorithm could be used for real-time monitoring of cell cultures and for high-throughput screening experiments upon adequate tuning.

T model of growth and its application in systems of tumor-immune dynamics

In this paper we introduce a new growth model called T growth model. This model is capable of representing sigmoidal growth as well as biphasic growth. This dual capability is achieved without introducing additional parameters. The T model is useful in modeling cellular proliferation or regression of cancer cells, stem cells, bacterial growth and drug dose-response relationships. We recommend usage of the T growth model for the growth of tumors as part of any system of differential equations. Use of this model within a system will allow more flexibility in representing the natural rate of tumor growth. For illustration, we examine some systems of tumor-immune interaction in which the T growth rate is applied. We also apply the model to a set of tumor growth data.

Simulating radiotherapy effect in high-grade glioma by using diffusive modeling and brain atlases

Applying diffusive models for simulating the spatiotemporal change of concentration of tumour cells is a modern application of predictive oncology. Diffusive models are used for modelling glioblastoma, the most aggressive type of glioma. This paper presents the results of applying a linear quadratic model for simulating the effects of radiotherapy on an advanced diffusive glioma model. This diffusive model takes into consideration the heterogeneous velocity of glioma in gray and white matter and the anisotropic migration of tumor cells, which is facilitated along white fibers. This work uses normal brain atlases for extracting the proportions of white and gray matter and the diffusion tensors used for anisotropy. The paper also presents the results of applying this glioma model on real clinical datasets.

Oscillabolastic model, a new model for oscillatory dynamics, applied to the analysis of Hes1 gene expression and Ehrlich ascites tumor growth

This paper introduces a new dynamical model, called the oscillabolastic model, to analyze the dynamical behavior of biomedical data when one observes oscillatory behavior. The proposed oscillabolastic model is sufficiently flexible to represent various types of oscillatory behavior. The oscillabolastic model is applied to two sets of data. The first data set deals with the oscillabolastic modeling of Ehrlich ascites tumor cells and the second one is the oscillabolastic modeling of the mean signal intensity of Hes1 gene expression in response to serum stimulation. A generalized oscillabolastic model is also suggested to accommodate cases in which predictor variables other than time are also involved.

High-grade glioma diffusive modeling using statistical tissue information and diffusion tensors extracted from atlases

Glioma, especially glioblastoma, is a leading cause of brain cancer fatality involving highly invasive and neoplastic growth. Diffusive models of glioma growth use variations of the diffusion-reaction equation in order to simulate the invasive patterns of glioma cells by approximating the spatiotemporal change of glioma cell concentration. The most advanced diffusive models take into consideration the heterogeneous velocity of glioma in gray and white matter, by using two different discrete diffusion coefficients in these areas. Moreover, by using diffusion tensor imaging (DTI), they simulate the anisotropic migration of glioma cells, which is facilitated along white fibers, assuming diffusion tensors with different diffusion coefficients along each candidate direction of growth. Our study extends this concept by fully exploiting the proportions of white and gray matter extracted by normal brain atlases, rather than discretizing diffusion coefficients. Moreover, the proportions of white and gray matter, as well as the diffusion tensors, are extracted by the respective atlases; thus, no DTI processing is needed. Finally, we applied this novel glioma growth model on real data and the results indicate that prognostication rates can be improved.

Hyperbolastic modeling of tumor growth with a combined treatment of iodoacetate and dimethylsulphoxide

BACKGROUND

An understanding of growth dynamics of tumors is important in understanding progression of cancer and designing appropriate treatment strategies. We perform a comparative study of the hyperbolastic growth models with the Weibull and Gompertz models, which are prevalently used in the field of tumor growth.

METHODS

The hyperbolastic growth models H1, H2, and H3 are applied to growth of solid Ehrlich carcinoma under several different treatments. These are compared with results from Gompertz and Weibull models for the combined treatment.

RESULTS

The growth dynamics of the solid Ehrlich carcinoma with the combined treatment are studied using models H1, H2, and H3, and the models are highly accurate in representing the growth. The growth dynamics are also compared with the untreated tumor, the tumor treated with only iodoacetate, and the tumor treated with only dimethylsulfoxide, and the combined treatment.

CONCLUSIONS

The hyperbolastic models prove to be effective in representing and analyzing the growth dynamics of the solid Ehrlich carcinoma. These models are more precise than Gompertz and Weibull and show less error for this data set. The precision of H3 allows for its use in a comparative analysis of tumor growth rates between the various treatments.

Quantifying and Testing Indirect Effects in Simple Mediation Models When the Constituent Paths Are Nonlinear

Most treatments of indirect effects and mediation in the statistical methods literature and the corresponding methods used by behavioral scientists have assumed linear relationships between variables in the causal system. Here we describe and extend a method first introduced by Stolzenberg (1980) for estimating indirect effects in models of mediators and outcomes that are nonlinear functions but linear in their parameters. We introduce the concept of the instantaneous indirect effect of X on Y through M and illustrate its computation and describe a bootstrapping procedure for inference. Mplus code as well as SPSS and SAS macros are provided to facilitate the adoption of this approach and ease the computational burden on the researcher.

Hypoxia enhances proliferation of mouse embryonic stem cell-derived neural stem cells

Neural stem (NS) cells can provide a source of material with potential applications for neural drug testing, developmental studies, or novel treatments for neurodegenerative diseases. Herein, the ex vivo expansion of a model system of mouse embryonic stem (mES) cell-derived NS cells was characterized and optimized, cells being cultivated under adherent conditions. Culture was first optimized in terms of initial cell plating density and oxygen concentration, known to strongly influence brain-derived NS cells. To this end, the growth of cells cultured under hypoxic (2%, 5%, and 10% O(2)) and normoxic (20% O(2)) conditions was compared. The results showed that 2-5% oxygen, without affecting multipotency, led to fold increase values in total cell number about twice higher than observed under 20% oxygen (20-fold vs. 10-fold, respectively) this effect being more pronounced when cells were plated at low density. With an optimal cell density of 10(4) cells/cm(2), the maximum growth rates were 1.9 day(-1) under hypoxia versus 1.7 day(-1) under normoxia. Cell division kinetics analysis by flow cytometry based on PKH67 tracking showed that when cultured in hypoxia, cells are at least one divisional generation ahead compared to normoxia. In terms of cell cycle, a larger population in a quiescent G(0) phase was observed in normoxic conditions. The optimization of NS cell culture performed here represents an important step toward the generation of a large number of neural cells from a reduced initial population, envisaging the potential application of these cells in multiple settings.

Multicellular tumor spheroids: an underestimated tool is catching up again

The present article highlights the rationale, potential and flexibility of tumor spheroid mono- and cocultures for implementation into state of the art anti-cancer therapy test platforms. Unlike classical monolayer-based models, spheroids strikingly mirror the 3D cellular context and therapeutically relevant pathophysiological gradients of in vivo tumors. Some concepts for standardization and automation of spheroid culturing, monitoring and analysis are discussed, and the challenges to define the most convenient analytical endpoints for therapy testing are outlined. The potential of spheroids to contribute to either the elimination of poor drug candidates at the pre-animal and pre-clinical state or the identification of promising drugs that would fail in classical 2D cell assays is emphasised. Microtechnologies, in the form of micropatterning and microfluidics, are also discussed and offer the exciting prospect of standardized spheroid mass production to tackle high-throughput screening applications within the context of traditional laboratory settings. The extension towards more sophisticated spheroid coculture models which more closely reflect heterologous tumor tissues composed of tumor and various stromal cell types is also covered. Examples are given with particular emphasis on tumor-immune cell cocultures and their usefulness for testing novel immunotherapeutic treatment strategies. Finally, tumor cell heterogeneity and the extraordinary possibilities of putative cancer stem/tumor-initiating cell populations that can be maintained and expanded in sphere-forming assays are introduced. The relevance of the cancer stem cell hypothesis for cancer cure is highlighted, with the respective sphere cultures being envisioned as an integral tool for next generation drug development offensives.

Spheroid-based drug screen: considerations and practical approach

Although used in academic research for several decades, 3D culture models have long been regarded expensive, cumbersome and unnecessary in drug development processes. Technical advances, coupled with recent observations showing that gene expression in 3D is much closer to clinical expression profiles than those seen in 2D, have renewed attention and generated hope in the feasibility of maturing organotypic 3D systems to therapy test platforms with greater power to predict clinical efficacies. Here we describe a standardized setup for reproducible, easy-handling culture, treatment and routine analysis of multicellular spheroids, the classical 3D culture system resembling many aspects of the pathophysiological situation in human tumor tissue. We discuss essential conceptual and practical considerations for an adequate establishment and use of spheroid-based drug screening platforms and also provide a list of human carcinoma cell lines, partly on the basis of the NCI-DTP 60-cell line screen, that produce treatable spheroids under identical culture conditions. In contrast to many other settings with which to achieve similar results, the protocol is particularly useful to be integrated into standardized large-scale drug test routines as it requires a minimum number of defined spheroids and a limited amount of drug. The estimated time to run the complete screening protocol described herein--including spheroid initiation, drug treatment and determination of the analytical end points (spheroid integrity, and cell survival through the acid phosphatase assay)--is about 170 h. Monitoring of spheroid growth kinetics to determine growth delay and regrowth, respectively, after drug treatment requires long-term culturing (> or =14 d).

Experimental anti-tumor therapy in 3-D: spheroids--old hat or new challenge?

PURPOSE

To give a state-of-the-art overview on the promise of three-dimensional (3-D) culture systems for anticancer drug development, with particular emphasis on multicellular tumor spheroids (MCTS).

RESULTS AND CONCLUSIONS

Cell-based assays have become an integral component in many stages of routine anti-tumor drug testing. However, they are almost always based on homogenous monolayer or suspension cultures and thus represent a rather artificial cellular environment. 3-D cultures--such as the well established spheroid culture system--better reflect the in vivo behavior of cells in tumor tissues and are increasingly recognized as valuable advanced tools for evaluating the efficacy of therapeutic intervention. The present article summarizes past and current applications and particularly discusses technological challenges, required improvements and recent progress with the use of the spheroid model in experimental therapeutics, as a basis for sophisticated drug/therapy screening. A brief overview is given focusing on the nomenclature of spherical 3-D cultures, their potential to mimic many aspects of the pathophysiological situation in tumors, and currently available protocols for culturing and analysis. A list of spheroid-forming epithelial cancer cell lines of different origin is provided and the recent trend to use spheroids for testing combination treatment strategies is highlighted. Finally, various spheroid co-culture approaches are presented that have been established to study heterologous cell interactions in solid tumors and thereby are able to reflect the cellular tumor environment with increasing accuracy. The intriguing observation that in order to retain certain tumor initiating cell properties, some primary tumor cell populations must be maintained exclusively in 3-D culture is mentioned, adding a new but fascinating challenge for future therapeutic campaigns.

A general bilinear model to describe growth or decline time profiles

Linear models are widely used because of their unrivaled simplicity, but they cannot be applied for data that have a turning-or rate-change-point, even if the data show good linearity sufficiently far from this point. To describe such bilinear-type data, a completely generalized version of a linearized biexponential model (LinBiExp) is proposed here to make possible smooth and fully parametrizable transitions between two linear segments while still maintaining a clear connection with the linear models. Applications and brief conclusions are presented for various time profiles of biological and medical interest including growth profiles, such as those of human stature, agricultural crops and fruits, multicellular tumor spheroids, single fission yeast cells, or even labor productivity, and decline profiles, such as age-effects on cognition in patients who develop dementia and lactation yields in dairy cattle. In all these cases, quantitative model selection criteria such as the Akaike and the Schwartz Bayesian information criteria indicated the superiority of the bilinear model compared to adequate less parametrized alternatives such as linear, parabolic, exponential, or classical growth (e.g., logistic, Gompertz, Weibull, and Richards) models. LinBiExp provides a versatile and useful five-parameter bilinear functional form that is convenient to implement, is suitable for full optimization, and uses intuitive and easily interpretable parameters.