Differences in predictions of ODE models of tumor growth: a cautionary example

Curating models from BioModels: Developing a workflow for creating OMEX files

The reproducibility of computational biology models can be greatly facilitated by widely adopted standards and public repositories. We examined 50 models from the BioModels Database and attempted to validate the original curation and correct some of them if necessary. For each model, we reproduced these published results using Tellurium. Once reproduced we manually created a new set of files, with the model information stored by the Systems Biology Markup Language (SBML), and simulation instructions stored by the Simulation Experiment Description Markup Language (SED-ML), and everything included in an Open Modeling EXchange (OMEX) file, which could be used with a variety of simulators to reproduce the same results. On the one hand, the validation procedure of 50 models developed a manual workflow that we would use to build an automatic platform to help users more easily curate and verify models in the future. On the other hand, these exercises allowed us to find the limitations and possible enhancement of the current curation and tooling to verify and curate models.

Assessing tumor volumetric reduction with consideration for setup errors based on mathematical tumor model and microdosimetric kinetic model in single-isocenter VMAT for brain metastases

The volumetric reduction rate (VRR) was evaluated with consideration for six degrees-of-freedom (6DoF) patient setup errors based on a mathematical tumor model in single-isocenter volumetric modulated arc therapy (SI-VMAT) for brain metastases. Simulated gross tumor volumes (GTV) of 1.0 cm and dose distribution were created (27 Gy/3 fractions). The distance between the GTV center and isocenter (d) was set at 0-10 cm. The GTV was translated within 0-1.0 mm (Trans) and rotated within 0-1.0° (Rot) in the three axis directions using affine transformation. The tumor growth volume was calculated using a multicomponent mathematical model (MCTM), and lethal effects of irradiation and repair from damage during irradiation were calculated by a microdosimetric kinetic model (MKM) for non-small cell lung cancer (NSCLC) A549 and NCI-H460 (H460) cells. The VRRs were calculated 5 days after the end of irradiation using the physical dose to the GTV for varying d and 6DoF setup errors. The tolerance value of VRR, the GTV volume reduction rate, was set at 5%, based on the pre-irradiation GTV volume. With the exception of the only one A549 condition where (Trans, Rot) = (1.0 mm, 1.0°) was repeated for 3 fractions, all conditions met all the tolerance VRR values for A549 and H460 cells with varying d from 0 to 10 cm. Evaluation based on the mathematical tumor model suggested that if the 6DoF setup errors at each irradiation could be kept within 1.0 mm and 1.0°, there would be little effect on tumor volume regardless of the distance from the isocenter in SI-VMAT.

Modeling Physical Forces Experienced by Cancer and Stromal Cells Within Different Organ-Specific Tumor Tissue

Mechanical force exerted on cancer cells by their microenvironment have been reported to drive cells toward invasive phenotypes by altering cells' motility, proliferation, and apoptosis. These mechanical forces include compressive, tensile, hydrostatic, and shear forces. The importance of forces is then hypothesized to be an alteration of cancer cells' and their microenvironment's biophysical properties as the indicator of a tumor's malignancy state. Our objective is to investigate and quantify the correlation between a tumor's malignancy state and forces experienced by the cancer cells and components of the microenvironment. In this study, we have developed a multicomponent, three-dimensional model of tumor tissue consisting of a cancer cell surrounded by fibroblasts and extracellular matrix (ECM). Our results on three different organs including breast, kidney, and pancreas show that: A) the stresses within tumor tissue are impacted by the organ specific ECM's biophysical properties, B) more invasive cancer cells experience higher stresses, C) in pancreas which has a softer ECM (Young modulus of 1.0 kPa) and stiffer cancer cells (Young modulus of 2.4 kPa and 1.7 kPa) than breast and kidney, cancer cells experienced significantly higher stresses, D) cancer cells in contact with ECM experienced higher stresses compared to cells surrounded by fibroblasts but the area of tumor stroma experiencing high stresses has a maximum length of 40 μm when the cancer cell is surrounded by fibroblasts and 12 μm for when the cancer cell is in vicinity of ECM. This study serves as an important first step in understanding of how the stresses experienced by cancer cells, fibroblasts, and ECM are associated with malignancy states of cancer cells in different organs. The quantification of forces exerted on cancer cells by different organ-specific ECM and at different stages of malignancy will help, first to develop theranostic strategies, second to predict accurately which tumors will become highly malignant, and third to establish accurate criteria controlling the progression of cancer cells malignancy. Furthermore, our in silico model of tumor tissue can yield critical, useful information for guiding ex vivo or in vitro experiments, narrowing down variables to be investigated, understanding what factors could be impacting cancer treatments or even biomarkers to be looking for.

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.

Dual therapy of cancer using optimal control supported by swarm intelligence

BACKGROUND

The scientific revolution in the treatment of many illnesses has been significantly aided by stem cells. This paper presents an optimal control on a mathematical model of chemotherapy and stem cell therapy for cancer treatment.

OBJECTIVE

To develop effective hybrid techniques that combine the optimal control theory (OCT) with the evolutionary algorithm and multi-objective swarm algorithm. The developed technique is aimed to reduce the number of cancerous cells while utilizing the minimum necessary chemotherapy medications and minimizing toxicity to protect patients' health.

METHODS

Two hybrid techniques are proposed in this paper. Both techniques combined OCT with the evolutionary algorithm and multi-objective swarm algorithm which included MOEA/D, MOPSO, SPEA II and PESA II. This study evaluates the performance of two hybrid techniques in terms of reducing cancer cells and drug concentrations, as well as computational time consumption.

RESULTS

In both techniques, MOEA/D emerges as the most effective algorithm due to its superior capability in minimizing tumour size and cancer drug concentration.

CONCLUSION

This study highlights the importance of integrating OCT and evolutionary algorithms as a robust approach for optimizing cancer chemotherapy treatment.

Copper isotope ratios in serum do not track cancerous tumor evolution, but organ failure

Relative to healthy controls, lighter copper isotopic compositions have been observed in the serum of breast cancer and end-stage liver disease patients, raising the possibility that Cu isotope ratios could be used as a tracer for disease progression. Here, we assess the potential of natural Cu isotopic variations (expressed as δ65Cu) as diagnostic tools for cancer progression and/or liver failure by performing a first-order analysis of Cu isotopic cycling in the human body. Using a box model, we simulate the kinetics of Cu mass transfer throughout significant reservoirs in the body, allowing isotopic fractionation to occur during Cu uptake/release from these reservoirs. With this model, we determine under which conditions the serum δ65Cu values would reflect perturbation related to cancer growth and/or liver failure at a level resolvable with modern mass spectrometry. We find that tumor growth alone is unable to explain the light isotopic signature observed in serum. Instead, we find that metabolic changes to the liver function resulting in a ∼1‰ isotope fractionation during Cu uptake from the blood into the liver can readily explain the long-term serum isotopic shift of ∼0.2‰ observed in cancer patients. A similar fractionation (∼1.3‰) during Cu uptake into the liver also readily explains the -1.2‰ shift observed in the serum of cirrhosis patients with ascites, suggesting a potentially common driver of isotopic fractionation in both cases. Using this model, we then test hypotheses put forward by previous studies and begin to probe the mechanisms behind the measured isotopic compositions.

Modeling of oncolytic viruses in a heterogeneous cell population to predict spread into non-cancerous cells

New cancer treatment modalities that limit patient discomfort need to be developed. One possible new therapy is the use of oncolytic (cancer-killing) viruses. It is only recently that our ability to manipulate viral genomes has allowed us to consider deliberately infecting cancer patients with viruses. One key consideration is to ensure that the virus exclusively targets cancer cells and does not harm nearby non-cancerous cells. Here, we use a mathematical model of viral infection to determine the characteristics a virus would need to have in order to eradicate a tumor, but leave non-cancerous cells untouched. We conclude that the virus must differ in its ability to infect the two different cell types, with the infection rate of non-cancerous cells needing to be less than one hundredth of the infection rate of cancer cells. Differences in viral production rate or infectious cell death rate alone are not sufficient to protect non-cancerous cells.

Mathematical Modeling of Clonal Interference by Density-Dependent Selection in Heterogeneous Cancer Cell Lines

Many cancer cell lines are aneuploid and heterogeneous, with multiple karyotypes co-existing within the same cell line. Karyotype heterogeneity has been shown to manifest phenotypically, thus affecting how cells respond to drugs or to minor differences in culture media. Knowing how to interpret karyotype heterogeneity phenotypically would give insights into cellular phenotypes before they unfold temporally. Here, we re-analyzed single cell RNA (scRNA) and scDNA sequencing data from eight stomach cancer cell lines by placing gene expression programs into a phenotypic context. Using live cell imaging, we quantified differences in the growth rate and contact inhibition between the eight cell lines and used these differences to prioritize the transcriptomic biomarkers of the growth rate and carrying capacity. Using these biomarkers, we found significant differences in the predicted growth rate or carrying capacity between multiple karyotypes detected within the same cell line. We used these predictions to simulate how the clonal composition of a cell line would change depending on density conditions during in-vitro experiments. Once validated, these models can aid in the design of experiments that steer evolution with density-dependent selection.

Mathematical model combined with microdosimetric kinetic model for tumor volume calculation in stereotactic body radiation therapy

We proposed a new mathematical model that combines an ordinary differential equation (ODE) and microdosimetric kinetic model (MKM) to predict the tumor-cell lethal effect of Stereotactic body radiation therapy (SBRT) applied to non-small cell lung cancer (NSCLC). The tumor growth volume was calculated by the ODE in the multi-component mathematical model (MCM) for the cell lines NSCLC A549 and NCI-H460 (H460). The prescription doses 48 Gy/4 fr and 54 Gy/3 fr were used in the SBRT, and the effect of the SBRT on tumor cells was evaluated by the MKM. We also evaluated the effects of (1) linear quadratic model (LQM) and the MKM, (2) varying the ratio of active and quiescent tumors for the total tumor volume, and (3) the length of the dose-delivery time per fractionated dose (tinter) on the initial tumor volume. We used the ratio of the tumor volume at 1 day after the end of irradiation to the tumor volume before irradiation to define the radiation effectiveness value (REV). The combination of MKM and MCM significantly reduced REV at 48 Gy/4 fr compared to the combination of LQM and MCM. The ratio of active tumors and the prolonging of tinter affected the decrease in the REV for A549 and H460 cells. We evaluated the tumor volume considering a large fractionated dose and the dose-delivery time by combining the MKM with a mathematical model of tumor growth using an ODE in lung SBRT for NSCLC A549 and H460 cells.

Mathematical prediction with pretreatment growth rate of metastatic cancer on outcomes: implications for the characterization of oligometastatic disease

Background

Oligometastatic disease (OMD) represents an indolent cancer status characterized by slow tumor growth and limited metastatic potential. The use of local therapy in the management of the condition continues to rise. This study aimed to investigate the advantage of pretreatment tumor growth rate in addition to baseline disease burden in characterizing OMDs, generally defined by the presence of ≤ 5 metastatic lesions.

Methods

The study included patients with metastatic melanoma treated with pembrolizumab. Gross tumor volume of all metastases was contoured on imaging before (TP_-1) and at the initiation of pembrolizumab (TP₀). Pretreatment tumor growth rate was calculated by an exponential ordinary differential equation model using the sum of tumor volumes at TP_-1 and TP₀ and the time interval between TP_-1. and TP₀. Patients were divided into interquartile groups based on pretreatment growth rate. Overall survival, progression-free survival, and subsequent progression-free survival were the study outcomes.

Results

At baseline, median cumulative volume and number of metastases were 28.4 cc (range, 0.4-1194.8 cc) and 7 (range, 1-73), respectively. The median interval between TP_-1 and TP₀ was -90 days and pretreatment tumor growth rate (×10^-2 days^-1) was median 4.71 (range -0.62 to 44.1). The slow-paced group (pretreatment tumor growth rate ≤ 7.6 ×10^-2 days^-1, the upper quartile) had a significantly higher overall survival rate, progression-free survival, and subsequent progression-free survival compared to those of the fast-paced group (pretreatment tumor growth rate > 7.6 ×10^-2 days^-1). Notably, these differences were prominent in the subgroup with >5 metastases.

Conclusion

Pretreatment tumor growth rate is a novel prognostic metric associated with overall survival, progression-free survival, and subsequent progression-free survival among metastatic melanoma patients, especially patients with >5 metastases. Future prospective studies should validate the advantage of disease growth rate plus disease burden in better defining OMDs.

Calibrating agent-based models to tumor images using representation learning

Agent-based models (ABMs) have enabled great advances in the study of tumor development and therapeutic response, allowing researchers to explore the spatiotemporal evolution of the tumor and its microenvironment. However, these models face serious drawbacks in the realm of parameterization-ABM parameters are typically set individually based on various data and literature sources, rather than through a rigorous parameter estimation approach. While ABMs can be fit to simple time-course data (such as tumor volume), that type of data loses the spatial information that is a defining feature of ABMs. Tumor images provide spatial information; however, such images only represent individual timepoints, limiting their utility in calibrating the tumor dynamics predicted by ABMs. Furthermore, it is exceedingly difficult to compare tumor images to ABM simulations beyond a qualitative visual comparison. Without a quantitative method of comparing the similarity of tumor images to ABM simulations, a rigorous parameter fitting is not possible. Here, we present a novel approach that applies neural networks to represent both tumor images and ABM simulations as low dimensional points, with the distance between points acting as a quantitative measure of difference between the two. This enables a quantitative comparison of tumor images and ABM simulations, where the distance between simulated and experimental images can be minimized using standard parameter-fitting algorithms. Here, we describe this method and present two examples to demonstrate the application of the approach to estimate parameters for two distinct ABMs. Overall, we provide a novel method to robustly estimate ABM parameters.

Multicomponent mathematical model for tumor volume calculation with setup error using single-isocenter stereotactic radiotherapy for multiple brain metastases

We evaluated the tumor residual volumes considering six degrees-of-freedom (6DoF) patient setup errors in stereotactic radiotherapy (SRT) with multicomponent mathematical model using single-isocenter irradiation for brain metastases. Simulated spherical gross tumor volumes (GTVs) with 1.0 (GTV 1), 2.0 (GTV 2), and 3.0 (GTV 3)-cm diameters were used. The distance between the GTV center and isocenter (d) was set at 0-10 cm. The GTV was simultaneously translated within 0-1.0 mm (T) and rotated within 0°-1.0° (R) in the three axis directions using affine transformation. We optimized the tumor growth model parameters using measurements of non-small cell lung cancer cell lines' (A549 and NCI-H460) growth. We calculated the GTV residual volume at the irradiation's end using the physical dose to the GTV when the GTV size, d, and 6DoF setup error varied. The d-values that satisfy tolerance values (10%, 35%, and 50%) of the GTV residual volume rate based on the pre-irradiation GTV volume were determined. The larger the tolerance value set for both cell lines, the longer the distance to satisfy the tolerance value. In GTV residual volume evaluations based on the multicomponent mathematical model on SRT with single-isocenter irradiation, the smaller the GTV size and the larger the distance and 6DoF setup error, the shorter the distance that satisfies the tolerance value might need to be.

Mathematical Modelling of Cervical Precancerous Lesion Grade Risk Scores: Linear Regression Analysis of Cellular Protein Biomarkers and Human Papillomavirus E6/E7 RNA Staining Patterns

The current practice of determining histologic grade with a single molecular biomarker can facilitate differential diagnosis but cannot predict the risk of lesion progression. Cancer is caused by complex mechanisms, and no single biomarker can both make accurate diagnoses and predict progression risk. Modelling using multiple biomarkers can be used to derive scores for risk prediction. Mathematical models (MMs) may be capable of making predictions from biomarker data. Therefore, this study aimed to develop MM–based scores for predicting the risk of precancerous cervical lesion progression and identifying precancerous lesions in patients in northern Thailand by evaluating the expression of multiple biomarkers. The MMs (Models 1–5) were developed in the test sample set based on patient age range (five categories) and biomarker levels (cortactin, p16INK4A, and Ki–67 by immunohistochemistry [IHC], and HPV E6/E7 ribonucleic acid (RNA) by in situ hybridization [ISH]). The risk scores for the prediction of cervical lesion progression (“risk biomolecules”) ranged from 2.56–2.60 in the normal and low–grade squamous intraepithelial lesion (LSIL) cases and from 3.54–3.62 in cases where precancerous lesions were predicted to progress. In Model 4, 23/86 (26.7%) normal and LSIL cases had biomolecule levels that suggested a risk of progression, while 5/86 (5.8%) cases were identified as precancerous lesions. Additionally, histologic grading with a single molecular biomarker did not identify 23 cases with risk, preventing close patient monitoring. These results suggest that biomarker level–based risk scores are useful for predicting the risk of cervical lesion progression and identifying precancerous lesion development. This multiple biomarker–based strategy may ultimately have utility for predicting cancer progression in other contexts.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Modeling cancer's ecological and evolutionary dynamics

In this didactic paper, we present a theoretical modeling framework, called the G-function, that integrates both the ecology and evolution of cancer to understand oncogenesis. The G-function has been used in evolutionary ecology, but has not been widely applied to problems in cancer. Here, we build the G-function framework from fundamental Darwinian principles and discuss how cancer can be seen through the lens of ecology, evolution, and game theory. We begin with a simple model of cancer growth and add on components of cancer cell competition and drug resistance. To aid in exploration of eco-evolutionary modeling with this approach, we also present a user-friendly software tool. By the end of this paper, we hope that readers will be able to construct basic G function models and grasp the usefulness of the framework to understand the games cancer plays in a biologically mechanistic fashion.

Hybrid computational models of multicellular tumour growth considering glucose metabolism

Cancer cells metabolize glucose through metabolic pathways that differ from those used by healthy and differentiated cells. In particular, tumours have been shown to consume more glucose than their healthy counterparts and to use anaerobic metabolic pathways, even under aerobic conditions. Nevertheless, scientists have still not been able to explain why cancer cells evolved to present an altered metabolism and what evolutionary advantage this might provide them. Experimental and computational models have been increasingly used in recent years to understand some of these biological questions. Multicellular tumour spheroids are effective experimental models as they replicate the initial stages of avascular solid tumour growth. Furthermore, these experiments generate data which can be used to calibrate and validate computational studies that aim to simulate tumour growth. Hybrid models are of particular relevance in this field of research because they model cells as individual agents while also incorporating continuum representations of the substances present in the surrounding microenvironment that may participate in intracellular metabolic networks as concentration or density distributions. Henceforth, in this review, we explore the potential of computational modelling to reveal the role of metabolic reprogramming in tumour growth.

Analysis of tumor-immune functional responses in a mathematical model of neoantigen cancer vaccines

Cancer neoantigen vaccines have emerged as a promising approach to stimulating the immune system to fight cancer. We propose a simple model including key elements of cancer-immune interactions and conduct a phase plane analysis to understand the immunological mechanisms of cancer neoantigen vaccines. Analytical results are obtained for two widely used functional forms that represent the killing rate of tumor cells by immune cells: the law of mass action (LMA) and the dePillis-Radunskaya Law (LPR). Using the LMA, our results reveal that a slowly growing tumor can escape the immune surveillance and that there is a unique periodic solution. The LPR offers richer dynamics, in which tumor elimination and uncontrolled tumor growth are both present. We show that tumor elimination requires sufficient number of initial activated T cells in relationship to the malignant cells, which lends support to using the neoantigen cancer vaccine as an adjuvant therapy after the primary tumor is surgically removed or treated using radiotherapy. We also derive a sufficient condition for uncontrolled tumor growth under the assumption of the LPR. The juxtaposition of analyses with these two different choices for the killing rate function highlights their importance on model behavior and biological implications, by which we hope to spur further theoretical and experimental work to understand mechanisms underlying different functional forms for the killing rate.

Reinforcement learning strategies in cancer chemotherapy treatments: A review

BACKGROUND AND OBJECTIVE

Cancer is one of the major causes of death worldwide and chemotherapies are the most significant anti-cancer therapy, in spite of the emerging precision cancer medicines in the last 2 decades. The growing interest in developing the effective chemotherapy regimen with optimal drug dosing schedule to benefit the clinical cancer patients has spawned innovative solutions involving mathematical modeling since the chemotherapy regimens are administered cyclically until the futility or the occurrence of intolerable adverse events. Thus, in this present work, we reviewed the emerging trends involved in forming a computational solution from the aspect of reinforcement learning.

METHODS

Initially, this survey in-depth focused on the details of the dynamic treatment regimens from a broad perspective and then narrowed down to inspirations from reinforcement learning that were advantageous to chemotherapy dosing, including both offline reinforcement learning and supervised reinforcement learning.

RESULTS

The insights established in the chemotherapy-planning problem associated with the Reinforcement Learning (RL) has been discussed in this study. It showed that the researchers were able to widen their perspectives in comprehending the theoretical basis, dynamic treatment regimens (DTR), use of the adaptive control on DTR, and the associated RL techniques.

CONCLUSIONS

This study reviewed the recent researches relevant to the topic, and highlighted the challenges, open questions, possible solutions, and future steps in inventing a realistic solution for the aforementioned problem.

Practical Understanding of Cancer Model Identifiability in Clinical Applications

Mathematical models are a core component in the foundation of cancer theory and have been developed as clinical tools in precision medicine. Modeling studies for clinical applications often assume an individual's characteristics can be represented as parameters in a model and are used to explain, predict, and optimize treatment outcomes. However, this approach relies on the identifiability of the underlying mathematical models. In this study, we build on the framework of an observing-system simulation experiment to study the identifiability of several models of cancer growth, focusing on the prognostic parameters of each model. Our results demonstrate that the frequency of data collection, the types of data, such as cancer proxy, and the accuracy of measurements all play crucial roles in determining the identifiability of the model. We also found that highly accurate data can allow for reasonably accurate estimates of some parameters, which may be the key to achieving model identifiability in practice. As more complex models required more data for identification, our results support the idea of using models with a clear mechanism that tracks disease progression in clinical settings. For such a model, the subset of model parameters associated with disease progression naturally minimizes the required data for model identifiability.

Bayesian forecasting of tumor size metrics and overall survival

The tumor size ratio (TSR), time-to-tumor growth (TTG), and tumor growth rate (kG) are frequently suggested as model-based predictors of overall survival (OS) for different types of tumors. When the tumor metrics are applied in forecasting of the outcome for individual patients at an early stage, the tumor data might be sparse resulting in imprecise prediction. This simulation study aimed to investigate how the tumor follow-up data and estimation approaches influence the accuracy in the tumor size metrics and the predicted hazard of death for individual patients. Longitudinal tumor size and OS data were simulated using tumor growth inhibition and Weibull distribution models, respectively. Based on the model and increasing measurement durations, the accuracy (defined as 80-125% of the simulated "true" value) in individual metrics and hazard was computed. TSR week 6 (TSRw6) accuracy was adequate for 91% of the patients when tumor size was measured up to 12 weeks. For TTG and kG metrics, the highest accuracy observed was lower (43 and 77%, respectively) and occurred later (42 and 60 weeks, respectively). The simultaneous (joint) and sequential estimation approaches resulted in similar accuracies, however, in general, the sequential approach where individual tumor size parameters are fixed, demonstrated inferior estimation properties. The TSRw6 and the model-predicted tumor time course (absolute or relative change) had better forecasting properties than TTG or kG. The population pharmacokinetic (PK) parameters and data approach performed similarly or better than the simultaneous approach and had a better accuracy in estimating individuals' hazard of death than the individual PK parameters method.

Computational model of CAR T-cell immunotherapy dissects and predicts leukemia patient responses at remission, resistance, and relapse

BACKGROUND

Adaptive CD19-targeted chimeric antigen receptor (CAR) T-cell transfer has become a promising treatment for leukemia. Although patient responses vary across different clinical trials, reliable methods to dissect and predict patient responses to novel therapies are currently lacking. Recently, the depiction of patient responses has been achieved using in silico computational models, with prediction application being limited.

METHODS

We established a computational model of CAR T-cell therapy to recapitulate key cellular mechanisms and dynamics during treatment with responses of continuous remission (CR), non-response (NR), and CD19-positive (CD19+) and CD19-negative (CD19-) relapse. Real-time CAR T-cell and tumor burden data of 209 patients were collected from clinical studies and standardized with unified units in bone marrow. Parameter estimation was conducted using the stochastic approximation expectation maximization algorithm for nonlinear mixed-effect modeling.

RESULTS

We revealed critical determinants related to patient responses at remission, resistance, and relapse. For CR, NR, and CD19+ relapse, the overall functionality of CAR T-cell led to various outcomes, whereas loss of the CD19+ antigen and the bystander killing effect of CAR T-cells may partly explain the progression of CD19- relapse. Furthermore, we predicted patient responses by combining the peak and accumulated values of CAR T-cells or by inputting early-stage CAR T-cell dynamics. A clinical trial simulation using virtual patient cohorts generated based on real clinical patient datasets was conducted to further validate the prediction.

CONCLUSIONS

Our model dissected the mechanism behind distinct responses of leukemia to CAR T-cell therapy. This patient-based computational immuno-oncology model can predict late responses and may be informative in clinical treatment and management.

“Song of Life”: A Comprehensive Evaluation of Biographical Music Therapy in Palliative Care by the EMW-TOPSIS Method

The “Song of Life (SOL)” is a kind of music therapy in palliative care for addressing emotional and existential needs in terminally ill patients nearing the end of life. Few previous studies focus on objective data analysis methods to validate the effectiveness of psychotherapy therapy for patients’ overall state. This article combines the entropy weighting method (EWM) and the technique for order preference by similarity to the ideal solution (TOPSIS) method to evaluate the effectiveness of SOL music therapy and the treatment satisfaction of the patients and family members. Firstly, the collaborative filtering algorithm (CFA) machine learning algorithm is used to predict the missing ratings a patient might have given to a variable. Secondly, the EWM determines the weights of quality of life, spiritual well-being, ego-integrity, overall quality of life, and momentary distress. Thirdly, the EWM method is applied for the TOPSIS evaluation model to evaluate the patient’s state pre- and post-intervention. Finally, we obtain the state change in patients and recognition based on the feedback questionnaire. The multiple criteria decision making (MCDM) comprehensive evaluation method objectively validated the overall effectiveness of SOL music therapy. Based on MCDM method, we provide a new approach for judging the overall effect of psychological intervention and accurately recommend psychotherapy that fits the symptoms of psychological disorders.

Model-informed experimental design recommendations for distinguishing intrinsic and acquired targeted therapeutic resistance in head and neck cancer

The promise of precision medicine has been limited by the pervasive resistance to many targeted therapies for cancer. Inferring the timing (i.e., pre-existing or acquired) and mechanism (i.e., drug-induced) of such resistance is crucial for designing effective new therapeutics. This paper studies cetuximab resistance in head and neck squamous cell carcinoma (HNSCC) using tumor volume data obtained from patient-derived tumor xenografts. We ask if resistance mechanisms can be determined from this data alone, and if not, what data would be needed to deduce the underlying mode(s) of resistance. To answer these questions, we propose a family of mathematical models, with each member of the family assuming a different timing and mechanism of resistance. We present a method for fitting these models to individual volumetric data, and utilize model selection and parameter sensitivity analyses to ask: which member(s) of the family of models best describes HNSCC response to cetuximab, and what does that tell us about the timing and mechanisms driving resistance? We find that along with time-course volumetric data to a single dose of cetuximab, the initial resistance fraction and, in some instances, dose escalation volumetric data are required to distinguish among the family of models and thereby infer the mechanisms of resistance. These findings can inform future experimental design so that we can best leverage the synergy of wet laboratory experimentation and mathematical modeling in the study of novel targeted cancer therapeutics.

A Spatially Resolved Mechanistic Growth Law for Cancer Drug Development Predicting Tumor Growing Fractions

Mathematical models used in preclinical drug discovery tend to be empirical growth laws. Such models are well suited to fitting the data available, mostly longitudinal studies of tumor volume; however, they typically have little connection with the underlying physiologic processes. This lack of a mechanistic underpinning restricts their flexibility and potentially inhibits their translation across studies including from animal to human. Here we present a mathematical model describing tumor growth for the evaluation of single-agent cytotoxic compounds that is based on mechanistic principles. The model can predict spatial distributions of cell subpopulations and account for spatial drug distribution effects within tumors. Importantly, we demonstrate that the model can be reduced to a growth law similar in form to the ones currently implemented in pharmaceutical drug development for preclinical trials so that it can integrated into the current workflow. We validate this approach for both cell-derived xenograft and patient-derived xenograft (PDX) data. This shows that our theoretical model fits as well as the best performing and most widely used models. However, in addition, the model is also able to accurately predict the observed growing fraction of tumours. Our work opens up current preclinical modeling studies to also incorporating spatially resolved and multimodal data without significant added complexity and creates the opportunity to improve translation and tumor response predictions.

Significance

This theoretical model has the same mathematical structure as that currently used for drug development. However, its mechanistic basis enables prediction of growing fraction and spatial variations in drug distribution.

Combining Mechanisms of Growth Arrest in Solid Tumours: A Mathematical Investigation

The processes underpinning solid tumour growth involve the interactions between various healthy and tumour tissue components and the vasculature, and can be affected in different ways by cancer treatment. In particular, the growth-limiting mechanisms at play may influence tumour responses to treatment. In this paper, we propose a simple ordinary differential equation model of solid tumour growth to investigate how tumour-specific mechanisms of growth arrest may affect tumour response to different combination cancer therapies. We consider the interactions of tumour cells with the physical space in which they proliferate and a nutrient supplied by the tumour vasculature, with the aim of representing two distinct growth arrest mechanisms. More specifically, we wish to consider growth arrest due to (1) nutrient deficiency, which corresponds to balancing cell proliferation and death rates, and (2) competition for space, which corresponds to cessation of proliferation without cell death. We perform numerical simulations of the model and a steady-state analysis to determine the possible tumour growth scenarios described by the model. We find that there are three distinct growth regimes: the nutrient- and spatially limited regimes and a bi-stable regime, in which both growth arrest mechanisms are simultaneously active. Thus, the proposed model has the features required to investigate and distinguish tumour responses to different cancer treatments.

Slow-Fast Model and Therapy Optimization for Oncolytic Treatment of Tumors

The present work studies models of oncolytic virotherapy without space variable in which virus replication occurs at a faster time scale than tumor growth. We address the questions of the modeling of virus injection in this slow-fast system and of the optimal timing for different treatment strategies. To this aim, we first derive the asymptotic of a three-species slow-fast model and obtain a two-species dynamical system, where the variables are tumor cells and infected tumor cells. We fully characterize the behavior of this system depending on the various biological parameters. In the second part, we address the modeling of virus injection and its expression in the two-species system, where the amount of virus does not appear explicitly. We prove that the injection can be described by an instantaneous jump in the phase plane, where a certain amount of tumors cells are transformed instantly into infected tumor cells. This description allows discussing qualitatively the timing of different injections in the frame of successive treatment strategies. This work is illustrated by numerical simulations. The timing and amount of injected virus may have counterintuitive optimal values; nevertheless, the understanding is clear from the phase space analysis.

Model Calibration of Pharmacokinetic-Pharmacodynamic Lung Tumour Dynamics for Anticancer Therapies

Individual curves for tumor growth can be expressed as mathematical models. Herein we exploited a pharmacokinetic-pharmacodynamic (PKPD) model to accurately predict the lung growth curves when using data from a clinical study. Our analysis included 19 patients with non-small cell lung cancer treated with specific hypofractionated regimens, defined as stereotactic body radiation therapy (SBRT). The results exhibited the utility of the PKPD model for testing growth hypotheses of the lung tumor against clinical data. The model fitted the observed progression behavior of the lung tumors expressed by measuring the tumor volume of the patients before and after treatment from CT screening. The changes in dynamics were best captured by the parameter identified as the patients’ response to treatment. Median follow-up times for the tumor volume after SBRT were 126 days. These results have proven the use of mathematical modeling in preclinical anticancer investigations as a potential prognostic tool.

Mathematical modeling of therapeutic neural stem cell migration in mouse brain with and without brain tumors

Neural stem cells (NSCs) offer a potential solution to treating brain tumors. This is because NSCs can circumvent the blood-brain barrier and migrate to areas of damage in the central nervous system, including tumors, stroke, and wound injuries. However, for successful clinical application of NSC treatment, a sufficient number of viable cells must reach the diseased or damaged area(s) in the brain, and evidence suggests that it may be affected by the paths the NSCs take through the brain, as well as the locations of tumors. To study the NSC migration in brain, we develop a mathematical model of therapeutic NSC migration towards brain tumor, that provides a low cost platform to investigate NSC treatment efficacy. Our model is an extension of the model developed in Rockne et al. (PLoS ONE 13, e0199967, 2018) that considers NSC migration in non-tumor bearing naive mouse brain. Here we modify the model in Rockne et al. in three ways: (i) we consider three-dimensional mouse brain geometry, (ii) we add chemotaxis to model the tumor-tropic nature of NSCs into tumor sites, and (iii) we model stochasticity of migration speed and chemosensitivity. The proposed model is used to study migration patterns of NSCs to sites of tumors for different injection strategies, in particular, intranasal and intracerebral delivery. We observe that intracerebral injection results in more NSCs arriving at the tumor site(s), but the relative fraction of NSCs depends on the location of injection relative to the target site(s). On the other hand, intranasal injection results in fewer NSCs at the tumor site, but yields a more even distribution of NSCs within and around the target tumor site(s).

Using mathematical modeling to estimate time-independent cancer chemotherapy efficacy parameters

One of the primary cancer treatment modalities is chemotherapy. Unfortunately, traditional anti-cancer drugs are often not selective and cause damage to healthy cells, leading to serious side effects for patients. For this reason more targeted therapeutics and drug delivery methods are being developed. The effectiveness of new treatments is initially determined via in vitro cell viability assays, which determine the IC 50  of the drug. However, these assays are known to result in estimates of IC 50  that depend on the measurement time, possibly resulting in over- or under-estimation of the IC 50 . Here, we test the possibility of using cell growth curves and fitting of mathematical models to determine the IC 50  as well as the maximum efficacy of a drug ( ε max ). We measured cell growth of MCF-7 and HeLa cells in the presence of different concentrations of doxorubicin and fit the data with a logistic growth model that incorporates the effect of the drug. This method leads to measurement time-independent estimates of IC 50  and ε max , but we find that ε max  is not identifiable. Further refinement of this methodology is needed to produce uniquely identifiable parameter estimates.

Resource competition and technological diversity

The work develops and investigates a mathematical model for evolution of the technological structure of an economic system where different technologies compete for the common essential resources. The model is represented by a system of consumer-resource rate equations. Consumers are technologies formalized as populations of weakly differentiated firms producing a similar commodity with like average output. Firms are characterized by the Leontief-Liebig production function in stock-flow representation. Firms self-replicate with a rate proportional to production output of the respective technology and dissolve with a constant rate of decay. The resources are supplied to the system from outside and consumed by concerned technologies; the unutilized resource amounts are removed elsewhere. The inverse of a per firm break-even resource availability is proposed to serve as a measure for competitiveness towards a given resource. The necessary conditions for coexistence of different technologies are derived, according to which each contender must be a superior competitor for one specific resource and an inferior competitor for the others. The model yields a version of the principle of competitive exclusion: in a steady state, the number of competing technologies cannot exceed the number of limiting resources. Competitive outcomes (either dominance or coexistence) in the general system of multiple technologies feeding on multiple essential resources are shown to be predictable from knowledge of the resource-dependent consumption and growth rates of each technological population taken alone. The proposed model of exploitative competition with explicit resource dynamics enables more profound insight into the patterns of technological change as opposed to conventional mainstream models of innovation diffusion.

How translational modeling in oncology needs to get the mechanism just right

Translational model‐based approaches have played a role in increasing success in the development of novel anticancer treatments. However, despite this, significant translational uncertainty remains from animal models to patients. Optimization of dose and scheduling (regimen) of drugs to maximize the therapeutic utility (maximize efficacy while avoiding limiting toxicities) is still predominately driven by clinical investigations. Here, we argue that utilizing pragmatic mechanism‐based translational modeling of nonclinical data can further inform this optimization. Consequently, a prototype model is demonstrated that addresses the required fundamental mechanisms.

Deep Reinforcement Learning for Fractionated Radiotherapy in Non-Small Cell Lung Carcinoma

Lung cancer is by far the leading cause of cancer death among both men and women. Radiation therapy is one of the main approaches to lung cancer treatment, and its planning is crucial for the therapy outcome. However, the current practice that uniformly delivers the dose does not take into account the patient-specific tumour features that may affect treatment success. Since radiation therapy is by its very nature a sequential procedure, Deep Reinforcement Learning (DRL) is a well-suited methodology to overcome this limitation. In this respect, in this work we present a DRL controller optimizing the daily dose fraction delivered to the patient on the basis of CT scans collected over time during the therapy, offering a personalized treatment not only for volume adaptation, as currently intended, but also for daily fractionation. Furthermore, this contribution introduces a virtual radiotherapy environment based on a set of ordinary differential equations modelling the tissue radiosensitivity by combining both the effect of the radiotherapy treatment and cell growth. Their parameters are estimated from CT scans routinely collected using the Particle Swarm Optimization algorithm. This permits the DRL to learn the optimal behaviour through an iterative trial and error process with the environment. We performed several experiments considering three rewards functions modelling treatment strategies with different tissue aggressiveness and two exploration strategies for the exploration-exploitation dilemma. The results show that our DRL approach can adapt to radiation therapy treatment, optimizing its behaviour according to the different reward functions and outperforming the current clinical practice.

A Quantitative Systems Pharmacology Framework for Optimal Doxorubicin Granulocyte Colony-Stimulating Factor Regimens in Triple-Negative Breast Cancer

INTRODUCTION

To mitigate the risk of neutropenia during chemotherapy treatment of triple-negative breast cancer, prophylactic and supportive therapy with granulocyte colony-stimulating factor (G-CSF) is administered concomitant to chemotherapy. The proper timing of combined chemotherapy and G-CSF is crucial for treatment outcomes.

METHODS

Leveraging our established mathematical model of neutrophil production by G-CSF, we developed quantitative systems pharmacology (QSP) framework to investigate how modulating chemotherapy dose frequency and intensity can maximize antitumour effects. To establish schedules that best control tumour size while minimizing neutropenia, we combined Gompertzian tumour growth with pharmacokinetic/pharmacodynamic models of doxorubicin and G-CSF, and our QSP model of neutrophil production.

RESULTS

We optimized a range of chemotherapeutic cycle lengths and dose sizes to establish regimens that simultaneously reduced tumour burden while minimizing neutropenia. Our results suggest that cytotoxic chemotherapy with doxorubicin 45 mg/m2 every 14 days provides effective control of tumour growth while mitigating neutropenic risks.

CONCLUSION

This work suggests future avenues for optimal regimens of chemotherapy with prophylactic G-CSF support. Importantly, the algorithmic approach that we developed can aid in balancing the anticancer and the neutropenic effects of both drugs, and therefore contributes to rational considerations in clinical decision-making in triple-negative breast cancer.

Modeling the effect of tumor compression on airflow dynamics in trachea using contact simulation and CFD analysis

Malignant central airway obstruction can cause severe breathing difficulty in a patient that requires surgical intervention or stent implantation to alleviate it. A predictive model to identify the onset of this event as the central airway is progressively compressed by tumor growth will be helpful for clinicians to plan for medical intervention. We present such a model to simulate tumor compression of the trachea and the resulting change in airflow dynamics to estimate the level of stenosis that will cause severe breathing difficulties. A patient-specific model of trachea was generated from acquired Computed Tomography (CT) scans for the simulations. The compression of this trachea due to tumor growth is modeled using nonlinear contact simulations of ellipsoidal tumors with the trachea. Computational fluid dynamics (CFD) is employed to simulate the turbulent airflow during inhalation in the stenosed trachea. From the CFD simulated flow fields, the power loss due to airflow through the domain is calculated. The results show that when the obstruction in the trachea reaches 50%, compared to the undeformed model, the power loss can rise to more than 66%. A measure of breathing difficulty can be derived by correlating it with the power loss. Thus, medical intervention can be predicted based on the degree of stenosis if the induced power loss exceeds a threshold that causes severe breathing discomfort.

CARTmath-A Mathematical Model of CAR-T Immunotherapy in Preclinical Studies of Hematological Cancers

Immunotherapy has gained great momentum with chimeric antigen receptor T cell (CAR-T) therapy, in which patient's T lymphocytes are genetically manipulated to recognize tumor-specific antigens, increasing tumor elimination efficiency. In recent years, CAR-T cell immunotherapy for hematological malignancies achieved a great response rate in patients and is a very promising therapy for several other malignancies. Each new CAR design requires a preclinical proof-of-concept experiment using immunodeficient mouse models. The absence of a functional immune system in these mice makes them simple and suitable for use as mathematical models. In this work, we develop a three-population mathematical model to describe tumor response to CAR-T cell immunotherapy in immunodeficient mouse models, encompassing interactions between a non-solid tumor and CAR-T cells (effector and long-term memory). We account for several phenomena, such as tumor-induced immunosuppression, memory pool formation, and conversion of memory into effector CAR-T cells in the presence of new tumor cells. Individual donor and tumor specificities are considered uncertainties in the model parameters. Our model is able to reproduce several CAR-T cell immunotherapy scenarios, with different CAR receptors and tumor targets reported in the literature. We found that therapy effectiveness mostly depends on specific parameters such as the differentiation of effector to memory CAR-T cells, CAR-T cytotoxic capacity, tumor growth rate, and tumor-induced immunosuppression. In summary, our model can contribute to reducing and optimizing the number of in vivo experiments with in silico tests to select specific scenarios that could be tested in experimental research. Such an in silico laboratory is an easy-to-run open-source simulator, built on a Shiny R-based platform called CARTmath. It contains the results of this manuscript as examples and documentation. The developed model together with the CARTmath platform have potential use in assessing different CAR-T cell immunotherapy protocols and its associated efficacy, becoming an accessory for in silico trials.

Revisiting a Null Hypothesis: Exploring the Parameters of Oligometastasis Treatment

PURPOSE

In the treatment of patients with metastatic cancer, the current paradigm states that metastasis-directed therapy does not prolong life. This paradigm forms the basis of clinical trial null hypotheses, where trials are built to test the null hypothesis that patients garner no overall survival benefit from targeting metastatic lesions. However, with advancing imaging technology and increasingly precise techniques for targeting lesions, a much larger proportion of metastatic disease can be treated. As a result, the life-extending benefit of targeting metastatic disease is becoming increasingly clear.

METHODS AND MATERIALS

In this work, we suggest shifting this qualitative null hypothesis and describe a mathematical model that can be used to frame a new, quantitative null. We begin with a very simple formulation of tumor growth, an exponential function, and illustrate how the same intervention (removing a given number of cells from the tumor) at different times affects survival. Additionally, we postulate where recent clinical trials fit into this parameter space and discuss the implications of clinical trial design in changing these quantitative parameters.

RESULTS

Our model shows that although any amount of cell kill will extend survival, in many cases the extent is so small as to be unnoticeable in a clinical context or is outweighed by factors related to toxicity and treatment time.

CONCLUSIONS

Recasting the null in these quantitative terms will allow trialists to design trials specifically to increase understanding of the circumstances (patient selection, disease burden, tumor growth kinetics) that can lead to improved overall survival when targeting metastatic lesions, rather than whether targeting metastases extends survival for patients with (oligo-) metastatic disease.

A tipping point in cancer-immune dynamics leads to divergent immunotherapy responses and hampers biomarker discovery

BACKGROUND

Predicting treatment response or survival of cancer patients remains challenging in immuno-oncology. Efforts to overcome these challenges focus, among others, on the discovery of new biomarkers. Despite advances in cellular and molecular approaches, only a limited number of candidate biomarkers eventually enter clinical practice.

METHODS

A computational modeling approach based on ordinary differential equations was used to simulate the fundamental mechanisms that dictate tumor-immune dynamics and to investigate its implications on responses to immune checkpoint inhibition (ICI) and patient survival. Using in silico biomarker discovery trials, we revealed fundamental principles that explain the diverging success rates of biomarker discovery programs.

RESULTS

Our model shows that a tipping point-a sharp state transition between immune control and immune evasion-induces a strongly non-linear relationship between patient survival and both immunological and tumor-related parameters. In patients close to the tipping point, ICI therapy may lead to long-lasting survival benefits, whereas patients far from the tipping point may fail to benefit from these potent treatments.

CONCLUSION

These findings have two important implications for clinical oncology. First, the apparent conundrum that ICI induces substantial benefits in some patients yet completely fails in others could be, to a large extent, explained by the presence of a tipping point. Second, predictive biomarkers for immunotherapy should ideally combine both immunological and tumor-related markers, as a patient's distance from the tipping point can typically not be reliably determined from solely one of these. The notion of a tipping point in cancer-immune dynamics helps to devise more accurate strategies to select appropriate treatments for patients with cancer.

Statistical analysis of comparative tumor growth repeated measures experiments in the ovarian cancer patient derived xenograft (PDX) setting

Repeated measures studies are frequently performed in patient-derived xenograft (PDX) models to evaluate drug activity or compare effectiveness of cancer treatment regimens. Linear mixed effects regression models were used to perform statistical modeling of tumor growth data. Biologically plausible structures for the covariation between repeated tumor burden measurements are explained. Graphical, tabular, and information criteria tools useful for choosing the mean model functional form and covariation structure are demonstrated in a Case Study of five PDX models comparing cancer treatments. Power calculations were performed via simulation. Linear mixed effects regression models applied to the natural log scale were shown to describe the observed data well. A straight growth function fit well for two PDX models. Three PDX models required quadratic or cubic polynomial (time squared or cubed) terms to describe delayed tumor regression or initial tumor growth followed by regression. Spatial(power), spatial(power) + RE, and RE covariance structures were found to be reasonable. Statistical power is shown as a function of sample size for different levels of variation. Linear mixed effects regression models provide a unified and flexible framework for analysis of PDX repeated measures data, use all available data, and allow estimation of tumor doubling time.

Study of Combinatorial Drug Synergy of Novel Acridone Derivatives With Temozolomide Using in-silico and in-vitro Methods in the Treatment of Drug-Resistant Glioma

Drug resistance is one of the critical challenges faced in the treatment of Glioma. There are only limited drugs available in the treatment of Glioma and among them Temozolomide (TMZ) has shown some effectiveness in treating Glioma patients, however, the rate of recovery remains poor due to the inability of this drug to act on the drug resistant tumor sub-populations. Hence, in this study three novel Acridone derivative drugs AC2, AC7, and AC26 have been proposed. These molecules when combined with TMZ show major tumor cytotoxicity that is effective in suppressing growth of cancer cells in both drug sensitive and resistant sub-populations of a tumor. In this study a novel mathematical model has been developed to explore the various drug combinations that may be useful for the treatment of resistant Glioma and show that the combinations of TMZ and Acridone derivatives have a synergistic effect. Also, acute toxicity studies of all three acridone derivatives were carried out for 14 days and were found safe for oral administration of 400 mg/kg body weight on albino Wistar rats. Molecular Docking studies of acridone derivatives with P-glycoprotein (P-gp), multiple resistant protein (MRP), and O6-methylguanine-DNA methyltransferase (MGMT) revealed different binding affinities to the transporters contributing to drug resistance. It is observed that while the Acridone derivatives bind with these drug resistance causing proteins, the TMZ can produce its cytotoxicity at a much lower concentration leading to the synergistic effect. The in silico analysis corroborate well with our experimental findings using TMZ resistant (T-98) and drug sensitive (U-87) Glioma cell lines and we propose three novel drug combinations (TMZ with AC2, AC7, and AC26) and dosages that show high synergy, high selectivity and low collateral toxicity for the use in the treatment of drug resistant Glioma, which could be future drugs in the treatment of Glioblastoma.

Fractional Mathematical Oncology: On the potential of non-integer order calculus applied to interdisciplinary models

Mathematical Oncology investigates cancer-related phenomena through mathematical models as comprehensive as possible. Accordingly, an interdisciplinary approach involving concepts from biology to materials science can provide a deeper understanding of biological systems pertaining the disease. In this context, fractional calculus (also referred to as non-integer order) is a branch in mathematical analysis whose tools can describe complex phenomena comprising different time and space scales. Fractional-order models may allow a better description and understanding of oncological particularities, potentially contributing to decision-making in areas of interest such as tumor evolution, early diagnosis techniques and personalized treatment therapies. By following a phenomenological (i.e. mechanistic) approach, the present study surveys and explores different aspects of Fractional Mathematical Oncology, reviewing and discussing recent developments in view of their prospective applications.

Beyond the pair approximation: Modeling colonization population dynamics

The process of range expansion (colonization) is one of the basic types of biological dynamics, whereby a species grows and spreads outwards, occupying new territories. Spatial modeling of this process is naturally implemented as a stochastic cellular automaton, with individuals occupying nodes on a rectangular grid, births and deaths occurring probabilistically, and individuals only reproducing onto unoccupied neighboring spots. In this paper we derive several approximations that allow prediction of the expected range expansion dynamics, based on the reproduction and death rates. We derive several approximations, where the cellular automaton is described by a system of ordinary differential equations that preserves correlations among neighboring spots (up to a distance). This methodology allows us to develop accurate approximations of the population size and the expected spatial shape, at a fraction of the computational time required to simulate the original stochastic system. In addition, we provide simple formulas for the steady-state population densities for von Neumann and Moore neighborhoods. Finally, we derive concise approximations for the speed of range expansion in terms of the reproduction and death rates, for both types of neighborhoods. The methodology is generalizable to more complex scenarios, such as different interaction ranges and multiple-species systems.

Effects of Doxorubicin Delivery by Nitrogen-Doped Graphene Quantum Dots on Cancer Cell Growth: Experimental Study and Mathematical Modeling

With 18 million new cases diagnosed each year worldwide, cancer strongly impacts both science and society. Current models of cancer cell growth and therapeutic efficacy in vitro are time-dependent and often do not consider the E_max value (the maximum reduction in the growth rate), leading to inconsistencies in the obtained IC₅₀ (concentration of the drug at half maximum effect). In this work, we introduce a new dual experimental/modeling approach to model HeLa and MCF-7 cancer cell growth and assess the efficacy of doxorubicin chemotherapeutics, whether alone or delivered by novel nitrogen-doped graphene quantum dots (N-GQDs). These biocompatible/biodegradable nanoparticles were used for the first time in this work for the delivery and fluorescence tracking of doxorubicin, ultimately decreasing its IC₅₀ by over 1.5 and allowing for the use of up to 10 times lower doses of the drug to achieve the same therapeutic effect. Based on the experimental in vitro studies with nanomaterial-delivered chemotherapy, we also developed a method of cancer cell growth modeling that (1) includes an E_max value, which is often not characterized, and (2), most importantly, is measurement time-independent. This will allow for the more consistent assessment of the efficiency of anti-cancer drugs and nanomaterial-delivered formulations, as well as efficacy improvements of nanomaterial delivery.

On multistep tumor growth models of fractional variable-order

Fractional mathematical oncology is a research topic that applies non-integer order calculus to tackle cancer problems such as tumor growth analysis or its optimal treatment. This work proposes a multistep exponential model with a fractional variable-order representing the evolution history of a tumor. Model parameters are tuned according to variable fractional order profiles while assessing their capability of fitting a clinical time series. The results point to the superiority of the proposed model in describing the experimental data, thus providing new perspectives for modeling tumor growth.

Bayesian Information-Theoretic Calibration of Radiotherapy Sensitivity Parameters for Informing Effective Scanning Protocols in Cancer

With new advancements in technology, it is now possible to collect data for a variety of different metrics describing tumor growth, including tumor volume, composition, and vascularity, among others. For any proposed model of tumor growth and treatment, we observe large variability among individual patients’ parameter values, particularly those relating to treatment response; thus, exploiting the use of these various metrics for model calibration can be helpful to infer such patient-specific parameters both accurately and early, so that treatment protocols can be adjusted mid-course for maximum efficacy. However, taking measurements can be costly and invasive, limiting clinicians to a sparse collection schedule. As such, the determination of optimal times and metrics for which to collect data in order to best inform proper treatment protocols could be of great assistance to clinicians. In this investigation, we employ a Bayesian information-theoretic calibration protocol for experimental design in order to identify the optimal times at which to collect data for informing treatment parameters. Within this procedure, data collection times are chosen sequentially to maximize the reduction in parameter uncertainty with each added measurement, ensuring that a budget of n high-fidelity experimental measurements results in maximum information gain about the low-fidelity model parameter values. In addition to investigating the optimal temporal pattern for data collection, we also develop a framework for deciding which metrics should be utilized at each data collection point. We illustrate this framework with a variety of toy examples, each utilizing a radiotherapy treatment regimen. For each scenario, we analyze the dependence of the predictive power of the low-fidelity model upon the measurement budget.

Generalized Additive Mixed Modeling of Longitudinal Tumor Growth Reduces Bias and Improves Decision Making in Translational Oncology

Scientists working in translational oncology regularly conduct multigroup studies of mice with serially measured tumors. Longitudinal data collected can feature mid-study dropouts and complex nonlinear temporal response patterns. Parametric statistical models such as ones assuming exponential growth are useful for summarizing tumor volume over ranges for which the growth model holds, with the advantage that the model's parameter estimates can be used to summarize between-group differences in tumor volume growth with statistical measures of uncertainty. However, these same assumed growth models are too rigid to recapitulate patterns observed in many experiments, which in turn diminishes the effectiveness of their parameter estimates as summary statistics. To address this problem, we generalized such models by adopting a nonparametric approach in which group-level response trends for logarithmically scaled tumor volume are estimated as regression splines in a generalized additive mixed model. We also describe a novel summary statistic for group level splines over user-defined, experimentally relevant time ranges. This statistic reduces to the log-linear growth rate for data well described by exponential growth and also has a sampling distribution across groups that is well approximated by a multivariate Gaussian, thus facilitating downstream analysis. Real-data examples show that this nonparametric approach not only enhances fidelity in describing nonlinear growth scenarios but also improves statistical power to detect interregimen differences when compared with the simple exponential model so that it generalizes the linear mixed effects paradigm for analysis of log-linear growth to nonlinear scenarios in a useful way. SIGNIFICANCE: This work generalizes the statistical linear mixed modeling paradigm for summarizing longitudinally measured preclinical tumor volume studies to encompass studies with nonlinear and nonmonotonic group response patterns in a statistically rigorous manner.

Mutant Evolution in Spatially Structured and Fragmented Expanding Populations

Mutant evolution in spatially structured systems is important for a range of biological systems, but aspects of it still require further elucidation. Adding to previous work, we provide a simple derivation of growth laws that characterize the number of mutants of different relative fitness in expanding populations in spatial models of different dimensionalities. These laws are universal and independent of "microscopic" modeling details. We further study the accumulation of mutants and find that, with advantageous and neutral mutants, more of them are present in spatially structured, compared to well-mixed colonies of the same size. The behavior of disadvantageous mutants is subtle: if they are disadvantageous through a reduction in division rates, the result is the same, and it is the opposite if the disadvantage is due to a death rate increase. Finally, we show that in all cases, the same results are observed in fragmented, nonspatial patch models. This suggests that the patterns observed are the consequence of population fragmentation, and not spatial restrictions per se We provide an intuitive explanation for the complex dependence of disadvantageous mutant evolution on spatial restriction, which relies on desynchronized dynamics in different locations/patches, and plays out differently depending on whether the disadvantage is due to a lower division rate or a higher death rate. Implications for specific biological systems, such as the evolution of drug-resistant cell mutants in cancer or bacterial biofilms, are discussed.

Tumor vascular status controls oxygen delivery facilitated by infused polymerized hemoglobins with varying oxygen affinity

Oxygen (O₂) delivery facilitated by hemoglobin (Hb)-based O₂ carriers (HBOCs) is a promising strategy to increase the effectiveness of chemotherapeutics for treatment of solid tumors. However, the heterogeneous vascular structures present within tumors complicates evaluating the oxygenation potential of HBOCs within the tumor microenvironment. To account for spatial variations in the vasculature and tumor tissue that occur during tumor growth, we used a computational model to develop artificial tumor constructs. With these simulated tumors, we performed a polymerized human hemoglobin (hHb) (PolyhHb) enhanced oxygenation simulation accounting for differences in the physiologic characteristics of human and mouse blood. The results from this model were used to determine the potential effectiveness of different treatment options including a top load (low volume) and exchange (large volume) infusion of a tense quaternary state (T-State) PolyhHb, relaxed quaternary state (R-State) PolyhHb, and a non O₂ carrying control. Principal component analysis (PCA) revealed correlations between the different regimes of effectiveness within the different simulated dosage options. In general, we found that infusion of T-State PolyhHb is more likely to decrease tissue hypoxia and modulate the metabolic rate of O₂ consumption. Though the developed models are not a definitive descriptor of O₂ carrier interaction in tumor capillary networks, we accounted for factors such as non-uniform vascular density and permeability that limit the applicability of O₂ carriers during infusion. Finally, we have used these validated computational models to establish potential benchmarks to guide tumor treatment during translation of PolyhHb mediated therapies into clinical applications.

High rates of oxygen consumption and abnormal vascularization lead to low oxygen levels within solid tumors. The lack of oxygen results in resistance to chemotherapies and increased rates of cancer progression. Hemoglobin-based oxygen carriers have the potential to increase the amount of oxygen delivered to tumors, which may make chemotherapies more effective. Unfortunately, translating experimental results from mice to humans is complicated by allometric scaling between mice and humans. To predict how these therapies may perform differently between human and murine systems, we computationally predicted how hemoglobin-based oxygen delivery varies between the two organisms. Our model accounts for how variations in the tumor vascular network impact the performance of hemoglobin-based oxygen carriers. This model also allows us to assess how the oxygen affinity of hemoglobin-based oxygen carriers affects the oxygenation of hypoxic tissue. The results of these models help us predict how results from murine models may translate to humans. Also, our models help to highlight what clinically-measurable tumor properties should be measured to predict the effectiveness of hemoglobin-based oxygen carriers in biological systems.