Tumor growth modeling: Parameter estimation with Maximum Likelihood methods

Impact of stroma remodeling on forces experienced by cancer cells and stromal cells within a pancreatic tumor tissue

Remodeling (re-engineering) of a tumor's stroma has been shown to improve the efficacy of anti-tumor therapies, without destroying the stroma. Even though it still remains unclear which stromal component/-s and what characteristics hinder the reach of nanoparticles deep into cancer cells, we hypothesis that mechanisms behind stroma's resistance to the penetration of nanoparticles rely heavily on extrinsic mechanical forces on stromal cells and cancer cells. Our hypothesis has been formulated on the basis of our previous study which has shown that changes in extracellular matrix (ECM) stiffness with tumor growth influence stresses exerted on fibroblasts and cancer cells, and that malignant cancer cells generate higher stresses on their stroma. This study attempts to establish a distinct identification of the components' remodeling on the distribution and magnitude of stress within a tumor tissue which ultimately will impact the resistance of stroma to treatment. In this study, our objective is to construct a three-dimensional in silico model of a pancreas tumor tissue consisting of cancer cells, stromal cells, and ECM to determine how stromal remodeling alters the stresses distribution and magnitude within the pancreas tumor tissue. Our results show that changes in mechanical properties of ECM significantly alter the magnitude and distribution of stresses within the pancreas tumor tissue. Our results revealed that these stresses are more sensitive to ECM properties as we see the stresses reaching to a maximum of 22,000 Pa for softer ECM with Young's modulus of 250 Pa. The stress distribution and magnitude within the pancreas tumor tissue does not show high sensitivity to the changes in mechanical properties of stromal cells surrounding stiffer cancer cells (PANC-1 with Young's modulus of 2400 Pa). However, softer cancer cells (MIA-PaCa-2 with (Young's modulus of 500 Pa) increase the stresses experienced by stiffer stromal cells and for stiffer ECM. By providing a unique platform to dissect and quantify the impact of individual stromal components on the stress distribution within a tumor tissue, this study serves as an important first step in understanding of which stromal components are vital for an efficient remodeling. This knowledge will be leveraged to overcome a tumor's resistance against the penetration of nanoparticles on a per-patient basis.

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.

A calibration and uncertainty quantification analysis of classical, fractional and multiscale logistic models of tumour growth

BACKGROUND AND OBJECTIVE

The validation of mathematical models of tumour growth is frequently hampered by the lack of sufficient experimental data, resulting in qualitative rather than quantitative studies. Recent approaches to this problem have attempted to extract information about tumour growth by integrating multiscale experimental measurements, such as longitudinal cell counts and gene expression data. In the present study, we investigated the performance of several mathematical models of tumour growth, including classical logistic, fractional and novel multiscale models, in terms of quantifying in-vitro tumour growth in the presence and absence of therapy. We further examined the effect of genes associated with changes in chemosensitivity in cell death rates.

METHODS

The multiscale expansion of logistic growth models was performed by coupling gene expression profiles to the cell death rates. State-of-the-art Bayesian inference, likelihood maximisation and uncertainty quantification techniques allowed a thorough evaluation of model performance.

RESULTS

The results suggest that the classical single-cell population model (SCPM) was the best fit for the untreated and low-dose treatment conditions, while the multiscale model with a cell death rate symmetric with the expression profile of OCT4 (Sym-SCPM) yielded the best fit for the high-dose treatment data. Further identifiability analysis showed that the multiscale model was both structurally and practically identifiable under the condition of known OCT4 expression profiles.

CONCLUSIONS

Overall, the present study demonstrates that model performance can be improved by incorporating multiscale measurements of tumour growth when high-dose treatment is involved.

Development and Validation of an α-Fetoprotein Tumor Burden Score Model to Predict Postrecurrence Survival among Patients with Hepatocellular Carcinoma

BACKGROUND

The purpose of this study is to establish a prognostic model to predict postrecurrence survival (PRS) probability after initial resection of hepatocellular carcinoma (HCC).

STUDY DESIGN

Patients with recurrent HCC after curative resection were identified through a multicenter consortium (training cohort, TC); data were from a separate institution were used as validation cohort (VC). The α-fetoprotein (AFP) tumor burden score (ATS) was defined as the distance from the origin on a 3-dimensional Cartesian coordinate system that incorporated 3 variables: largest tumor diameter ( x axis), number of tumors ( y axis), and ln AFP ( z axis). ATS was calculated using the Pythagorean theorem: ATS 2 = (largest tumor diameter) 2 + (number of tumors) 2 + (ln AFP) 2 , where ATS d and ATS r represent ATS at the time of initial diagnosis and at the time of recurrence, respectively. The final model was ATS m = ATS d + 4 × ATS r . Predictive performance and discrimination of the ATS model were evaluated and compared with traditional staging systems.

RESULTS

The ATS model demonstrated strong predictive performance of PRS in both the TC (area under the curve [AUC] 0.70) and VC (AUC 0.71). An ATS-based nomogram was able to stratify patients accurately into low- and high-risk categories relative to PRS (TC: ATS m ≤ 27, 74.9 months vs. ATS m ≥ 28, 23.3 months; VC: ATS m ≤ 27, 59.4 months vs. ATS m ≥ 28, 15.1 months; both p < 0.001). The ATS model predicted PRS among patients undergoing curative or noncurative treatment of HCC recurrence (both p < 0.05). Of note, the ATS model outperformed the Barcelona Clinic Liver Cancer (BCLC), China Liver Cancer (CNLC), and American Joint Committee on Cancer (AJCC) staging systems relative to 1-, 2-, 3-, 4- and 5-year PRS (AUC 0.70, vs. BCLC, AUC 0.50, vs. CNLC, AUC 0.54, vs. AJCC, AUC 0.51).

CONCLUSIONS

The ATS model had excellent prognostic discriminatory power to stratify patients relative to PRS.

Modeling of glioma growth using modified reaction-diffusion equation on brain MR images

BACKGROUND AND OBJECTIVE

Modeling of glioma growth and evolution is of key importance for cancer diagnosis, predicting clinical progression and improving treatment outcomes of neurosurgery. However, existing models are unable to characterize spatial variations of the proliferation and infiltration of tumor cells, making it difficult to achieve accurate prediction of tumor growth.

METHODS

In this paper, a new growth model of brain tumor using a reaction-diffusion equation on brain magnetic resonance images is proposed. Both the heterogeneity of brain tissue and the density of tumor cells are used to estimate the proliferation and diffusion coefficients of brain tumor cells. The diffusion coefficient that characterizes tumor diffusion and infiltration is calculated based on the ratio of tissues (white and gray matter), while the proliferation coefficient is evaluated using the spatial gradient of tumor cells. In addition, a parameter space is constructed using inverse distance weighted interpolation to describe the spatial distribution of proliferation coefficient.

RESULTS

The glioma growth predicted by the proposed model were tested by comparing with the real magnetic resonance images of the patients. Experiments and simulation results show that the proposed method achieves accurate modeling of glioma growth. The interpolation-based growth model has higher average dice score of 0.0647 and 0.0545, and higher average Jaccard index of 0.0673 and 0.0573, respectively, compared to the uniform- and gradient-based growth models.

CONCLUSIONS

The experimental results demonstrate the feasibility of calculating the proliferation and diffusion coefficients of the growth model based on patient-specific anatomy. The parameter space that characterizes spatial distribution of proliferation and diffusion coefficients is established and incorporated into the simulation of glioma growth. It enables to obtain patient-specific models about glioma growth by estimating and calibrating only a few model parameters.

Development of a Model to Estimate the Association Between Delay in Cancer Treatment and Local Tumor Control and Risk of Metastases

IMPORTANCE

The coronavirus disease 2019 (COVID-19) pandemic has led to treatment delays for many patients with cancer. While published guidelines provide suggestions on which cases are appropriate for treatment delay, there are no good quantitative estimates on the association of delays with tumor control or risk of new metastases.

OBJECTIVES

To develop a simplified mathematical model of tumor growth, control, and new metastases for cancers with varying doubling times and metastatic potential and to estimate tumor control probability (TCP) and metastases risk as a function of treatment delay interval.

DESIGN, SETTING, AND PARTICIPANTS

This decision analytical model describes a quantitative model for 3 tumors (ie, head and neck, colorectal, and non-small cell lung cancers). Using accepted ranges of tumor doubling times and metastatic development from the clinical literature from 2001 to 2020, estimates of tumor growth, TCP, and new metastases were analyzed for various treatment delay intervals.

MAIN OUTCOMES AND MEASURES

Risk estimates for potential decreases in local TCP and increases in new metastases with each interval of treatment delay.

RESULTS

For fast-growing head and neck tumors with a 2-month treatment delay, there was an estimated 4.8% (95% CI, 3.4%-6.4%) increase in local tumor control risk and a 0.49% (0.47%-0.51%) increase in new distal metastases risk. A 6-month delay was associated with an estimated 21.3% (13.4-30.4) increase in local tumor control risk and a 6.0% (5.2-6.8) increase in distal metastases risk. For intermediate-growing colorectal tumors, there was a 2.1% (0.7%-3.5%) increase in local tumor control risk and a 2.7% (2.6%-2.8%) increase in distal metastases risk at 2 months and a 7.6% (2.2%-14.2%) increase in local tumor control risk and a 24.7% (21.9%-27.8%) increase in distal metastases risk at 6 months. For slower-growing lung tumors, there was a 1.2% (0.0%-2.8%) increase in local tumor control risk and a 0.19% (0.18%-0.20%) increase in distal metastases risk at 2 months, and a 4.3% (0.0%-10.6%) increase in local tumor control risk and a 1.9% (1.6%-2.2%) increase in distal metastases risk at 6 months.

CONCLUSIONS AND RELEVANCE

This study proposed a model to quantify the association of treatment delays with local tumor control and risk of new metastases. The detrimental associations were greatest for tumors with faster rates of proliferation and metastasis. The associations were smaller, but still substantial, for slower-growing tumors.

Bezerra P, de Oliveira MS, da Costa WA, Ferreira GC, de Carvalho RN. Bioactive Compounds and Biological Activity of Croton Species (Euphorbiaceae): An Overview

Background:

Croton species are widely spread around the world, and present a varied chemical composition distributed in many classes of secondary metabolites, such as terpenoides, alkaloids, phenolic compounds and phenylpropanoids. These compounds can be obtained by different extraction methods, and more recently, with supercritical fluids. The crude and isolated extracts may have applications due to their biological activities in animals and humans.

Methods:

The text was written based on literature data from 1996 onwards.

Results:

The research showed in a concise way the botanical and taxonomic aspects of Croton and the success of its application is in studies related to the biological activities of the plant parts. It was also related to the chemical composition of its extracts and isolated compounds, obtained by many methods.

Conclusion:

In summary, the review feature studies reported the use of extracts and isolated Croton compounds due to their biological effects with antioxidant, antimicrobial, anti-inflammatory, neuroprotective, antitumor, anticancer, cytotoxic, insecticidal and allelopathic activities, with potential application in food, cosmetics, pharmaceuticals, and agrochemicals products.

Integrated Biophysical Modeling and Image Analysis: Application to Neuro-Oncology

Central nervous system (CNS) tumors come with vastly heterogeneous histologic, molecular, and radiographic landscapes, rendering their precise characterization challenging. The rapidly growing fields of biophysical modeling and radiomics have shown promise in better characterizing the molecular, spatial, and temporal heterogeneity of tumors. Integrative analysis of CNS tumors, including clinically acquired multi-parametric magnetic resonance imaging (mpMRI) and the inverse problem of calibrating biophysical models to mpMRI data, assists in identifying macroscopic quantifiable tumor patterns of invasion and proliferation, potentially leading to improved (a) detection/segmentation of tumor subregions and (b) computer-aided diagnostic/prognostic/predictive modeling. This article presents a summary of (a) biophysical growth modeling and simulation,(b) inverse problems for model calibration, (c) these models' integration with imaging workflows, and (d) their application to clinically relevant studies. We anticipate that such quantitative integrative analysis may even be beneficial in a future revision of the World Health Organization (WHO) classification for CNS tumors, ultimately improving patient survival prospects.

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.

Effect of fractal-shaped outer boundary of glioblastoma multiforme on drug delivery

BACKGROUND AND OBJECTIVE

The present study investigates the transport of drugs in the proximity of the glioblastoma multiforme, a brain neoplasm which is regarded to be the most aggressive type of cancer. In such a small distance from the tumor, diffusion dominates and is driven by the concentration gradient of drug that acquires its maximum at regions where the drugs are released and its minimum at the cancer cell boundary. Undoubtedly, the morphology of the aforementioned boundary is going to play a crucial role in the drug delivery and should be taken into account for the optimal design of the treatment. As first step in order to simulate the topography of glioblastoma multiforme, a fractal boundary is examined which mimics an acceptable model-problem for prognosis and diagnosis of a number of cancer tumors in breast, lungs and brain.

METHODS

The drug diffusion is investigated for two concentrations, namely a strong and a mild diffusion, while the outer boundary of the glioblastoma multiforme is approximated via triangular Von Koch shapes. Besides, a Finite Element Method is utilized via FEniCS, which is a Python-based open-source computing platform. Finally, after ascertaining the accuracy of the present numerical model, the concentration of the drug, the entropy production and the mass fluxes in the horizontal and vertical directions are estimated up to the fifth order of Von Koch fractal iterations.

RESULTS

It is ascertained that as the boundaries become more and more irregular, the entropy production in specific areas increases and as a consequence the delivery of the drug is facilitated. Hence, the mass fluxes in these sites appear to be larger comparing to the rest of the boundary and increase, as expected, for the case of strong diffusion.

CONCLUSIONS

These active regions, which are referred as "hot spots", are of great importance since they seem to be the sites where the drug ultimately penetrates the glioblastoma. This first-principles investigation is anticipated to shed light on a very significant part of drug delivery, which deals with the vicinity of the glioblastoma multiforme, stress the importance of the topography and give rise to future studies to be conducted based on subject-specific geometries.