Estimating the Growth Kinetics of Experimental Tumors From as Few as Two Determinations of Tumor Size: Implications for Clinical Oncology

Deep reinforcement learning identifies personalized intermittent androgen deprivation therapy for prostate cancer

The evolution of drug resistance leads to treatment failure and tumor progression. Intermittent androgen deprivation therapy (IADT) helps responsive cancer cells compete with resistant cancer cells in intratumoral competition. However, conventional IADT is population-based, ignoring the heterogeneity of patients and cancer. Additionally, existing IADT relies on pre-determined thresholds of prostate-specific antigen to pause and resume treatment, which is not optimized for individual patients. To address these challenges, we framed a data-driven method in two steps. First, we developed a time-varied, mixed-effect and generative Lotka-Volterra (tM-GLV) model to account for the heterogeneity of the evolution mechanism and the pharmacokinetics of two ADT drugs Cyproterone acetate and Leuprolide acetate for individual patients. Then, we proposed a reinforcement-learning-enabled individualized IADT framework, namely, I$^{2}$ADT, to learn the patient-specific tumor dynamics and derive the optimal drug administration policy. Experiments with clinical trial data demonstrated that the proposed I$^{2}$ADT can significantly prolong the time to progression of prostate cancer patients with reduced cumulative drug dosage. We further validated the efficacy of the proposed methods with a recent pilot clinical trial data. Moreover, the adaptability of I$^{2}$ADT makes it a promising tool for other cancers with the availability of clinical data, where treatment regimens might need to be individualized based on patient characteristics and disease dynamics. Our research elucidates the application of deep reinforcement learning to identify personalized adaptive cancer therapy.

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.

Physics-based tissue simulator to model multicellular systems: A study of liver regeneration and hepatocellular carcinoma recurrence

We present a multiagent-based model that captures the interactions between different types of cells with their microenvironment, and enables the analysis of the emergent global behavior during tissue regeneration and tumor development. Using this model, we are able to reproduce the temporal dynamics of regular healthy cells and cancer cells, as well as the evolution of their three-dimensional spatial distributions. By tuning the system with the characteristics of the individual patients, our model reproduces a variety of spatial patterns of tissue regeneration and tumor growth, resembling those found in clinical imaging or biopsies. In order to calibrate and validate our model we study the process of liver regeneration after surgical hepatectomy in different degrees. In the clinical context, our model is able to predict the recurrence of a hepatocellular carcinoma after a 70% partial hepatectomy. The outcomes of our simulations are in agreement with experimental and clinical observations. By fitting the model parameters to specific patient factors, it might well become a useful platform for hypotheses testing in treatments protocols.

Liquid biopsies for early diagnosis of brain tumours: in silico mathematical biomarker modelling

Brain tumours are the biggest cancer killer in those under 40 and reduce life expectancy more than any other cancer. Blood-based liquid biopsies may aid early diagnosis, prediction and prognosis for brain tumours. It remains unclear whether known blood-based biomarkers, such as glial fibrillary acidic protein (GFAP), have the required sensitivity and selectivity. We have developed a novel in silico model which can be used to assess and compare blood-based liquid biopsies. We focused on GFAP, a putative biomarker for astrocytic tumours and glioblastoma multi-formes (GBMs). In silico modelling was paired with experimental measurement of cell GFAP concentrations and used to predict the tumour volumes and identify key parameters which limit detection. The average GBM volumes of 449 patients at Leeds Teaching Hospitals NHS Trust were also measured and used as a benchmark. Our model predicts that the currently proposed GFAP threshold of 0.12 ng ml-1 may not be suitable for early detection of GBMs, but that lower thresholds may be used. We found that the levels of GFAP in the blood are related to tumour characteristics, such as vasculature damage and rate of necrosis, which are biological markers of tumour aggressiveness. We also demonstrate how these models could be used to provide clinical insight.

Radiological evaluation of ex novo high grade glioma: velocity of diametric expansion and acceleration time study

Background

One of the greatest neuro-oncological concern remains the lack of knowledge about the etiopathogenesis and physiopathology of gliomas. Several studies reported a strict correlation between radiological features and biological behaviour of gliomas; in this way the velocity of diametric expansion (VDE) correlate with lower grade glioma aggressiveness. However, there are no the same strong evidences for high grade gliomas (HGG) because of the lack of several preoperative MRI.

Patients and methods

We describe a series of 4 patients affected by HGG followed from 2014 to January 2019. Two patients are male and two female; two had a pathological diagnosis of glioblastoma (GBM), one of anaplastic astrocytoma (AA) and one had a neuroradiological diagnosis of GBM. The VDE and the acceleration time (AT) was calculated for fluid attenuated inversion recovery (FLAIR) volume and for the enhancing nodule (EN). Every patients underwent sequential MRI study along a mean period of 413 days.

Results

Mean VDE evaluated on FLAIR volume was 39.91 mm/year. Mean percentage ratio between peak values and mean value of acceleration was 282.7%. Median appearance time of EN after first MRI scan was 432 days. Mean VDE was 45.02 mm/year. Mean percentage ratio between peak values and mean value of acceleration was 257.52%.

Conclusions

To our knowledge, this is the first report on VDE and acceleration growth in HGG confirming their strong aggressiveness. In a case in which we need to repeat an MRI, time between consecutive scans should be reduced to a maximum of 15–20 days and surgery should be executed as soon as possible.

The initial engraftment of tumor cells is critical for the future growth pattern: a mathematical study based on simulations and animal experiments

Background

Xenograft mouse tumor models are used to study mechanisms of tumor growth and metastasis formation and to investigate the efficacy of different therapeutic interventions. After injection the engrafted cells form a local tumor nodule. Following an initial lag period of several days, the size of the tumor is measured periodically throughout the experiment using calipers. This method of determining tumor size is error prone because the measurement is two-dimensional (calipers do not measure tumor depth). Primary tumor growth can be described mathematically by suitable growth functions, the choice of which is not always obvious. Growth parameters provide information on tumor growth and are determined by applying nonlinear curve fitting.

Methods

We used self-generated synthetic data including random measurement errors to research the accuracy of parameter estimation based on caliper measured tumor data. Fit metrics were investigated to identify the most appropriate growth function for a given synthetic dataset. We studied the effects of measuring tumor size at different frequencies on the accuracy and precision of the estimated parameters. For curve fitting with fixed initial tumor volume, we varied this fixed initial volume during the fitting process to investigate the effect on the resulting estimated parameters. We determined the number of surviving engrafted tumor cells after injection using ex vivo bioluminescence imaging, to demonstrate the effect on experiments of incorrect assumptions about the initial tumor volume.

Results

To select a suitable growth function, measurement data from at least 15 animals should be considered. Tumor volume should be measured at least every three days to estimate accurate growth parameters. Daily measurement of the tumor volume is the most accurate way to improve long-term predictability of tumor growth. The initial tumor volume needs to have a fixed value in order to achieve meaningful results. An incorrect value for the initial tumor volume leads to large deviations in the resulting growth parameters.

Conclusions

The actual number of cancer cells engrafting directly after subcutaneous injection is critical for future tumor growth and distinctly influences the parameters for tumor growth determined by curve fitting.

Mathematical Approach to Differentiate Spontaneous and Induced Evolution to Drug Resistance During Cancer Treatment

Purpose

Drug resistance is a major impediment to the success of cancer treatment. Resistance is typically thought to arise from random genetic mutations, after which mutated cells expand via Darwinian selection. However, recent experimental evidence suggests that progression to drug resistance need not occur randomly, but instead may be induced by the treatment itself via either genetic changes or epigenetic alterations. This relatively novel notion of resistance complicates the already challenging task of designing effective treatment protocols.

Materials and Methods

To better understand resistance, we have developed a mathematical modeling framework that incorporates both spontaneous and drug-induced resistance.

Results

Our model demonstrates that the ability of a drug to induce resistance can result in qualitatively different responses to the same drug dose and delivery schedule. We have also proven that the induction parameter in our model is theoretically identifiable and propose an in vitro protocol that could be used to determine a treatment’s propensity to induce resistance.

Individualized growth prediction of mice skin tumors with maximum likelihood estimators

BACKGROUND & OBJECTIVE

In this work, we focus on estimating the parameters of the Gompertz model in order to predict tumor growth. The estimation is based on measurements from mice skin tumors of de novo carcinogenesis. The main objective is to compare the Maximum Likelihood estimator with the best performance from our previous work with the Non-linear Least Squares estimator which is commonly used in the literature to estimate the growth parameters of the Gompertz model.

METHODS

To describe tumor growth, we propose a stochastic model which is based on the Gompertz growth function. The principle of Maximum Likelihood is used to estimate both the growth rate and the carrying capacity of the Gompertz function, along with the characteristics of the additive Gaussian process and measurement noise. Moreover, we examine whether a Maximum A Posteriori estimator is able to utilize any available prior knowledge in order to improve the predictions.

RESULTS

Experimental data from a total of 24 tumors in 8 mice (3 tumors each) were used to study the performance of the proposed methods with respect to prediction accuracy. Our results show that the Maximum Likelihood estimator is able to provide, in most cases, more accurate predictions. Moreover, the Maximum A Posteriori estimator has the potential to correct potentially non-realistic estimates for the carrying capacity at early growth stages.

CONCLUSION

In most cases, the Maximum Likelihood estimator is able to provide more reliable predictions for the tumor's growth on individual test subjects. The Maximum A Posteriori estimator, it has the potential to improve the prediction when the available experimental data do not provide adequate information by utilizing prior knowledge about the unknown parameters.

Cancer recurrence times from a branching process model

As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. This is the main cause of cancer related mortality. Here we investigate a conceptually simple model of metastasis formation where metastatic lesions are initiated at a rate which depends on the size of the primary tumor. The evolution of each metastasis is described as an independent branching process. We assume that the primary tumor is resected at a given size and study the earliest time at which any metastasis reaches a minimal detectable size. The parameters of our model are estimated independently for breast, colorectal, headneck, lung and prostate cancers. We use these estimates to compare predictions from our model with values reported in clinical literature. For some cancer types, we find a remarkably wide range of resection sizes such that metastases are very likely to be present, but none of them are detectable. Our model predicts that only very early resections can prevent recurrence, and that small delays in the time of surgery can significantly increase the recurrence probability.

The majority of cancer related deaths are due to the development of secondary tumors called metastases. However, the dynamics of metastases establishment and growth and their relation with the primary tumor evolution are still not clear. A standard treatment starts with the resection of the primary tumor. At this time metastases may have already formed and still be too small to be detected. The presence of only undetectable metastases poses a challenge for deciding on the follow-up therapy. These small metastases could grow to a detectable size—thus leading to a recurrence of the disease—some time after surgery. We are interested in this time until cancer relapse. We present a mathematical model of metastases formation using tools from probability theory and estimate the model parameters for five different cancer types. Our predictions for the probability of visible metastases present at surgery and the mean time to relapse when no visible metastases are found at surgery are both in agreement with clinical data.

Recursive model for dose-time responses in pharmacological studies

Background

Clinical studies often track dose-response curves of subjects over time. One can easily model the dose-response curve at each time point with Hill equation, but such a model fails to capture the temporal evolution of the curves. On the other hand, one can use Gompertz equation to model the temporal behaviors at each dose without capturing the evolution of time curves across dosage.

Results

In this article, we propose a parametric model for dose-time responses that follows Gompertz law in time and Hill equation across dose approximately. We derive a recursion relation for dose-response curves over time capturing the temporal evolution and then specify a regression model connecting the parameters controlling the dose-time responses with individual level proteomic data. The resultant joint model allows us to predict the dose-response curves over time for new individuals.

Conclusion

We have compared the efficacy of our proposed Recursive Hybrid model with individual dose-response predictive models at desired time points. We note that our proposed model exhibits a superior performance compared to the individual ones for both synthetic data and actual pharmacological data. For the desired dose-time varying genetic characterization and drug response values, we have used the HMS-LINCS database and demonstrated the effectiveness of our model for all available anticancer compounds.

Electronic supplementary material

The online version of this article (10.1186/s12859-019-2831-4) contains supplementary material, which is available to authorized users.

A mathematical model of tumour growth with Beddington-DeAngelis functional response: a case of cancer without disease

A previously published mathematical model, governing tumour growth with mixed immunotherapy and chemotherapy treatments, is modified and studied. The search time, which is assumed to be neglectable in the previously published model, is incorporated into the functional response for tumour-cell lysis by effector cells. The model exhibits bistability where a tumour-cell population threshold exists. A tumour with an initial cell population below the threshold can be controlled by the immune system and remains microscopic and asymptomatic called cancer without disease while that above the threshold grows to lethal size. Bifurcation analysis shows that (a) the chemotherapy-induced damage may cause a microscopic tumour, which would never grow to become lethal if untreated, to grow to lethal size, (b) applying chemotherapy alone requires a large dosage to be successful,

Tumor growth modeling: Parameter estimation with Maximum Likelihood methods

BACKGROUND & OBJECTIVE

In this work, we focus on estimating the parameters of the widely used Gompertz tumor growth model, based on measurements of the tumor's volume. Being able to accurately describe the dynamics of tumor growth on an individual basis is very important both for growth prediction and designing personalized, optimal therapy schemes (e.g. when using model predictive control).

METHODS

Our analysis aims to compute both the growth rate and the carrying capacity of the Gompertz function, along with the characteristics of the additive Gaussian process and measurement noise of the system. Three methods based on Maximum Likelihood estimation are proposed. The first utilizes an assumption regarding the measurement noise that simplifies the problem, the second combines the Extended Kalman Filter and Maximum Likelihood estimation, and the third is a nonstandard exact form of Maximum Likelihood estimation, where numerical integration is used to approximate the likelihood of the measurements, along with a novel way to reduce the required computations.

RESULTS

Synthetic data were used in order to perform extensive simulations aiming to compare the methods' effectiveness, with respect to the accuracy of the estimation. The proposed methods are able to estimate the growth dynamics, even when the noise characteristics are not estimated accurately. Another very important finding is that the methods perform best in the case that corresponds to the problem needed to be solved when dealing with experimental data.

CONCLUSION

Using nonstandard, problem specific techniques can improve the estimation accuracy and best exploit the available data.

The thyroid cancer policy model: A mathematical simulation model of papillary thyroid carcinoma in The U.S. population

Background

Thyroid cancer affects over ½ million people in the U.S. and the incidence of thyroid cancer has increased worldwide at a rate higher than any other cancer, while survival has remained largely unchanged. The aim of this research was to develop, calibrate and verify a mathematical disease model to simulate the natural history of papillary thyroid cancer, which will serve as a platform to assess the effectiveness of clinical and cancer control interventions.

Methods

Herein, we modeled the natural pre-clinical course of both benign and malignant thyroid nodules with biologically relevant health states from normal to detected nodule. Using established calibration techniques, optimal parameter sets for tumor growth characteristics, development rate, and detection rate were used to fit Surveillance Epidemiology and End Results (SEER) incidence data and other calibration targets.

Results

Model outputs compared to calibration targets demonstrating sufficient calibration fit and model validation are presented including primary targets of SEER incidence data and size distribution at detection of malignancy. Additionally, we show the predicted underlying benign and malignant prevalence of nodules in the population, the probability of detection based on size of nodule, and estimates of growth over time in both benign and malignant nodules.

Conclusions

This comprehensive model provides a dynamic platform employable for future comparative effectiveness research. Future model analyses will test and assess various clinical management strategies to improve patient outcomes related to thyroid cancer and optimize resource utilization for patients with thyroid nodules.

Modeling the Dynamics of High-Grade Serous Ovarian Cancer Progression for Transvaginal Ultrasound-Based Screening and Early Detection

High-grade serous ovarian cancer (HGSOC) represents the majority of ovarian cancers and accounts for the largest proportion of deaths from the disease. A timely detection of low volume HGSOC should be the goal of any screening studies. However, numerous transvaginal ultrasound (TVU) detection-based population studies aimed at detecting low-volume disease have not yielded reduced mortality rates. A quantitative invalidation of TVU as an effective HGSOC screening strategy is a necessary next step. Herein, we propose a mathematical model for a quantitative explanation on the reported failure of TVU-based screening to improve HGSOC low-volume detectability and overall survival.We develop a novel in silico mathematical assessment of the efficacy of a unimodal TVU monitoring regimen as a strategy aimed at detecting low-volume HGSOC in cancer-positive cases, defined as cases for which the inception of the first malignant cell has already occurred. Our findings show that the median window of opportunity interval length for TVU monitoring and HGSOC detection is approximately 1.76 years. This does not translate into reduced mortality levels or improved detection accuracy in an in silico cohort across multiple TVU monitoring frequencies or detection sensitivities. We demonstrate that even a semiannual, unimodal TVU monitoring protocol is expected to miss detectable HGSOC. Lastly, we find that circa 50% of the simulated HGSOC growth curves never reach the baseline detectability threshold, and that on average, 5-7 infrequent, rate-limiting stochastic changes in the growth parameters are associated with reaching HGSOC detectability and mortality thresholds respectively. Focusing on a malignancy poorly studied in the mathematical oncology community, our model captures the dynamic, temporal evolution of HGSOC progression. Our mathematical model is consistent with recent case reports and prospective TVU screening population studies, and provides support to the empirical recommendation against frequent HGSOC screening.

Optimizing clinical benefit with targeted treatment in mRCC: "Tumor growth rate" as an alternative clinical endpoint

Tumor growth rate (TGR), usually defined as the ratio between the slope of tumor growth before the initiation of treatment and the slope of tumor growth during treatment, between the nadir and disease progression, is a measure of the rate at which tumor volume increases over time. In patients with metastatic renal cell carcinoma (mRCC), TGR has emerged as a reliable alternative parameter to allow a quantitative and dynamic evaluation of tumor response. This review presents evidence on the correlation between TGR and treatment outcomes and discusses the potential role of this tool within the treatment scenario of mRCC. Current evidence, albeit of retrospective nature, suggests that TGR might represent a useful tool to assess whether treatment is altering the course of the disease, and has shown to be significantly associated with progression-free survival and overall survival. Therefore, TGR may represent a valuable endpoint for clinical trials evaluating new molecularly targeted therapies. Most importantly, incorporation of TGR in the assessment of individual patients undergoing targeted therapies may help clinicians decide if a given agent is no longer able to control disease growth and whether continuing therapy beyond RECIST progression may still produce clinical benefit.

Estimating Tumor Growth Rates In Vivo

In this paper, we develop methods for inferring tumor growth rates from the observation of tumor volumes at two time points. We fit power law, exponential, Gompertz, and Spratt’s generalized logistic model to five data sets. Though the data sets are small and there are biases due to the way the samples were ascertained, there is a clear sign of exponential growth for the breast and liver cancers, and a 2/3’s power law (surface growth) for the two neurological cancers.

Growth dynamics of untreated glioblastomas in vivo

BACKGROUND

Glioblastomas are primary malignant brain tumors with a dismal prognosis. Knowledge of growth rates and underlying growth dynamics is useful for understanding basic tumor biology, developing realistic tumor models, and planning treatment logistics.

METHODS

By using repeated pretreatment contrast-enhanced T1-weighted MRI scans from 106 patients (aged 26-83 years), we studied the growth dynamics of untreated glioblastomas in vivo. Growth rates were calculated as specific growth rates and equivalent volume doubling times. The fit of different possible growth models was assessed using maximum likelihood estimations.

RESULTS

There were large variations in growth rates between patients. The median specific growth rate of the tumors was 1.4% per day, and the equivalent volume doubling time was 49.6 days. Exploring 3 different tumor growth models showed similar statistical fit for a Gompertzian growth model and a linear radial growth model and worse fit for an exponential growth model. However, large tumors had significantly lower growth rates than smaller tumors, supporting the assumption that glioblastomas reach a plateau phase and thus exhibit Gompertzian growth.

CONCLUSION

Based on the fast growth rate of glioblastoma shown in this study, it is evident that poor treatment logistics will influence tumor size before surgery and can cause significant regrowth before adjuvant treatment. Since there is a known association between tumor volume, extent of surgical resection, and response to adjuvant therapy, it is likely that waiting times play a role in patient outcomes.

Mammary gland tumor promotion by chronic administration of IGF1 and the insulin analogue AspB10 in the p53R270H/⁺WAPCre mouse model

INTRODUCTION

Insulin analogues are structurally modified molecules with altered pharmaco-kinetic and -dynamic properties compared to regular human insulin used by diabetic patients. While these compounds are tested for undesired mitogenic effects, an epidemiological discussion is ongoing regarding an association between insulin analogue therapy and increased cancer incidence, including breast cancer. Standard in vivo rodent carcinogenesis assays do not pick up this possible increased carcinogenic potential.

METHODS

Here we studied the role of insulin analogues in breast cancer development. For this we used the human relevant mammary gland specific p53R270H/⁺WAPCre mouse model. Animals received life long repeated treatment with four different insulin (-like) molecules: normal insulin, insulin glargine, insulin X10 (AspB10) or insulin-like growth factor 1 (IGF1).

RESULTS

Insulin-like molecules with strong mitogenic signaling, insulin X10 and IGF1, significantly decreased the time for tumor development. Yet, insulin glargine and normal insulin, did not significantly decrease the latency time for (mammary gland) tumor development. The majority of tumors had an epithelial to mesenchymal transition phenotype (EMT), irrespective of treatment condition. Enhanced extracellular signaling related kinase (Erk) or serine/threonine kinase (Akt) mitogenic signaling was in particular present in tumors from the insulin X10 and IGF1 treatment groups.

CONCLUSIONS

These data indicate that insulin-like molecules with enhanced mitogenic signaling increase the risk of breast cancer development. Moreover, the use of a tissue specific cancer model, like the p53R270H/⁺WAPCre mouse model, is relevant to assess the intrinsic pro-carcinogenic potential of mitogenic and non-mitogenic biologicals such as insulin analogues.

Interactions between the immune system and cancer: a brief review of non-spatial mathematical models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.