New formulation of the Gompertz equation to describe the kinetics of untreated tumors

Proper likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth

The analysis of unperturbed tumor growth kinetics, particularly the estimation of parameters for S-shaped equations used to describe growth, requires an appropriate likelihood function that accounts for the increasing error in solid tumor measurements as tumor size grows over time. This study aims to propose suitable likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth. Five different likelihood functions are evaluated and compared using three Bayesian criteria (the Bayesian Information Criterion, Deviance Information Criterion, and Bayes Factor) along with hypothesis tests on residuals. These functions are applied to fit data from unperturbed Ehrlich, fibrosarcoma Sa-37, and F3II tumors using the Gompertz equation, though they are generalizable to other S-shaped growth models for solid tumors or analogous systems (e.g., microorganisms, viruses). Results indicate that error models with tumor volume-dependent dispersion outperform standard constant-variance models in capturing the variability of tumor measurements, particularly the Thres model, which provides interpretable parameters for tumor growth. Additionally, constant-variance models, such as those assuming a normal error distribution, remain valuable as complementary benchmarks in analysis. It is concluded that models incorporating volume-dependent dispersion are preferred for accurate and clinically meaningful tumor growth modeling, whereas constant-dispersion models serve as useful complements for consistency and historical comparability.

Advancing cancer drug development with mechanistic mathematical modeling: bridging the gap between theory and practice

Quantitative predictive modeling of cancer growth, progression, and individual response to therapy is a rapidly growing field. Researchers from mathematical modeling, systems biology, pharmaceutical industry, and regulatory bodies, are collaboratively working on predictive models that could be applied for drug development and, ultimately, the clinical management of cancer patients. A plethora of modeling paradigms and approaches have emerged, making it challenging to compile a comprehensive review across all subdisciplines. It is therefore critical to gauge fundamental design aspects against requirements, and weigh opportunities and limitations of the different model types. In this review, we discuss three fundamental types of cancer models: space-structured models, ecological models, and immune system focused models. For each type, it is our goal to illustrate which mechanisms contribute to variability and heterogeneity in cancer growth and response, so that the appropriate architecture and complexity of a new model becomes clearer. We present the main features addressed by each of the three exemplary modeling types through a subjective collection of literature and illustrative exercises to facilitate inspiration and exchange, with a focus on providing a didactic rather than exhaustive overview. We close by imagining a future multi-scale model design to impact critical decisions in oncology drug development.

Modeling tumor growth using fractal calculus: Insights into tumor dynamics

Important concepts like fractal calculus and fractal analysis, the sum of squared residuals, and Aikaike's information criterion must be thoroughly understood in order to correctly fit cancer-related data using the proposed models. The fractal growth models employed in this work are classified in three main categories: Sigmoidal growth models (Logistic, Gompertz, and Richards models), Power Law growth model, and Exponential growth models (Exponential and Exponential-Lineal models)". We fitted the data, computed the sum of squared residuals, and determined Aikaike's information criteria using Matlab and the web tool WebPlotDigitizer. In addition, the research investigates "double-size cancer" in the fractal temporal dimension with respect to various mathematical models.

Simulations of surface charge density changes during the untreated solid tumour growth

Understanding untreated tumour growth kinetics and its intrinsic behaviour is interesting and intriguing. The aim of this study is to propose an approximate analytical expression that allows us to simulate changes in surface charge density at the cancer-surrounding healthy tissue interface during the untreated solid tumour growth. For this, the Gompertz and Poisson equations are used. Simulations reveal that the unperturbed solid tumour growth is closely related to changes in the surface charge density over time between the tumour and the surrounding healthy tissue. Furthermore, the unperturbed solid tumour growth is governed by temporal changes in this surface charge density. It is concluded that results corroborate the correspondence between the electrical and physiological parameters in the untreated cancer, which may have an essential role in its growth, progression, metastasis and protection against immune system attack and anti-cancer therapies. In addition, the knowledge of surface charge density changes at the cancer-surrounding healthy tissue interface may be relevant when redesigning the molecules in chemotherapy and immunotherapy taking into account their polarities. This can also be true in the design of completely novel therapies.

A quantitative systems pharmacology model of hyporesponsiveness to erythropoietin in rats

Recombinant human erythropoietin (rHuEPO) is effective in managing chronic kidney disease and chemotherapy-induced anemia. However, hyporesponsiveness to rHuEPO treatment was reported in about 10% of the patients. A decreased response in rats receiving a single or multiple doses of rHuEPO was also observed. In this study, we aimed to develop a quantitative systems pharmacology (QSP) model to examine hyporesponsiveness to rHuEPO in rats. Pharmacokinetic (PK) and pharmacodynamic (PD) data after a single intravenous dose of rHuEPO (100 IU/kg) was obtained from a previous study (Yan et al. in Pharm Res, 30:1026-1036, 2013) including rHuEPO plasma concentrations, erythroid precursors counts in femur bone marrow and spleen, reticulocytes (RETs), red blood cells (RBCs), and hemoglobin (HGB) in circulation. Parameter values were obtained from literature or calibrated with experimental data. Global sensitivity analysis and model-based simulations were performed to assess parameter sensitivity and hyporesponsiveness. The final QSP model adequately characterizes time courses of rHuEPO PK and nine PD endpoints in both control and treatment groups simultaneously. The model indicates that negative feedback regulation, neocytolysis, and depletion of erythroid precursors are major factors leading to hyporesponsiveness to rHuEPO treatment in rats.