Mechanistic modelling of dynamic MRI data predicts that tumour heterogeneity decreases therapeutic response

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.

Whole solid tumor volume histogram parameters for predicting the recurrence in patients with epithelial ovarian carcinoma: a feasibility study on quantitative DCE-MRI

BACKGROUND

Preoperative prediction of the recurrence of epithelial ovarian carcinoma (EOC) can guide the clinical treatment and improve the prognosis. However, there are still no reliable predictive biomarkers.

PURPOSE

To evaluate whether whole solid tumor volume histogram parameters measured from quantitative dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) can predict the recurrence in patients with EOC.

MATERIAL AND METHODS

We followed up 56 patients with surgical and histopathologically diagnosed EOC who underwent quantitative DCE-MRI scans. The differences of the histogram parameters between patients with and without recurrence were compared. Mann-Whitney U test, Pearson's Chi-squared test, or Fisher's exact test, and receiver operating characteristic (ROC) curves were used for statistical analysis.

RESULTS

All histogram parameters of K^trans, k_ep, and v_e were not significantly different between EOC patients with and without recurrence (P>0.05). For 30 patients with high-grade serous ovarian carcinoma (HGSOC), the histogram parameters of K^trans (mean and 5th, 10th, 25th, 50th, 75th percentiles) and k_ep (mean and 50th percentile) in 12 patients with recurrence were significantly lower than those in 18 patients without recurrence (all P<0.05). ROC curves showed that the 5th percentile of K^trans had the largest area under the curve (AUC) of 0.792 for predicting the recurrence in patients with HGSOC. When the threshold value was ≤0.0263/min, the sensitivity, specificity, and accuracy were 100%, 66.7%, and 80%, respectively.

CONCLUSION

Instead of predicting the recurrence of EOC, whole solid tumor volume quantitative DCE-MRI histogram parameters could predict the recurrence of HGSOC and may be potential biomarkers for the prediction of HGSOC recurrence.

Efficient DCE-MRI Parameter and Uncertainty Estimation Using a Neural Network

Quantitative DCE-MRI provides voxel-wise estimates of tracer-kinetic parameters that are valuable in the assessment of health and disease. These maps suffer from many known sources of variability. This variability is expensive to compute using current methods, and is typically not reported. Here, we demonstrate a novel approach for simultaneous estimation of tracer-kinetic parameters and their uncertainty due to intrinsic characteristics of the tracer-kinetic model, with very low computation time. We train and use a neural network to estimate the approximate joint posterior distribution of tracer-kinetic parameters. Uncertainties are estimated for each voxel and are specific to the patient, exam, and lesion. We demonstrate the methods' ability to produce accurate tracer-kinetic maps. We compare predicted parameter ranges with uncertainties introduced by noise and by differences in post-processing in a digital reference object. The predicted parameter ranges correlate well with tracer-kinetic parameter ranges observed across different noise realizations and regression algorithms. We also demonstrate the value of this approach to differentiate significant from insignificant changes in brain tumor pharmacokinetics over time. This is achieved by enforcing consistency in resolving model singularities in the applied tracer-kinetic model.

Quantitative dynamic contrast-enhanced MR imaging for differentiating benign, borderline, and malignant ovarian tumors

PURPOSE

This study aimed to investigate the diagnostic performance of quantitative DCE-MRI for characterizing ovarian tumors.

METHODS

We prospectively assessed the differences of quantitative DCE-MRI parameters (K^trans, k_ep, and v_e) among 15 benign, 28 borderline, and 66 malignant ovarian tumors; and between type I (n = 28) and type II (n = 29) of epithelial ovarian carcinomas (EOCs). DCE-MRI data were analyzed using whole solid tumor volume region of interest (ROI) method, and quantitative parameters were calculated based on a modified Tofts model. The non-parametric Kruskal-Wallis test, Mann-Whitney U test, Pearson's chi-square test, intraclass correlation coefficient (ICC), variance test, and receiver operating characteristic curves (ROC) were used for statistical analysis.

RESULTS

The largest K^trans and k_ep values were observed in ovarian malignant tumors, followed by borderline and benign tumors (all P < 0.001). K_ep was the better parameter for differentiating benign tumors from borderline and malignant tumors, with a sensitivity of 89.3% and 95.5%, a specificity of 86.7% and 100%, an accuracy of 88.4% and 96.3%, and an area under the curve (AUC) of 0.94 and 0.992, respectively, whereas K^trans was better for differentiating borderline from malignant tumors with a sensitivity of 60.7%, a specificity of 78.8%, an accuracy of 73.4%, and an AUC of 0.743. In addition, a combination with k_ep could further improve the sensitivity to 78.9%. The median K^trans and k_ep values were significantly higher in type II than in type I EOCs.

CONCLUSION

DCE-MRI with volume quantification is a technically feasible method, and can be used for the differentiation of ovarian tumors and for discriminating between type I and type II EOCs.

Precision Medicine with Imprecise Therapy: Computational Modeling for Chemotherapy in Breast Cancer

Medical oncology is in need of a mathematical modeling toolkit that can leverage clinically-available measurements to optimize treatment selection and schedules for patients. Just as the therapeutic choice has been optimized to match tumor genetics, the delivery of those therapeutics should be optimized based on patient-specific pharmacokinetic/pharmacodynamic properties. Under the current approach to treatment response planning and assessment, there does not exist an efficient method to consolidate biomarker changes into a holistic understanding of treatment response. While the majority of research on chemotherapies focus on cellular and genetic mechanisms of resistance, there are numerous patient-specific and tumor-specific measures that contribute to treatment response. New approaches that consolidate multimodal information into actionable data are needed. Mathematical modeling offers a solution to this problem. In this perspective, we first focus on the particular case of breast cancer to highlight how mathematical models have shaped the current approaches to treatment. Then we compare chemotherapy to radiation therapy. Finally, we identify opportunities to improve chemotherapy treatments using the model of radiation therapy. We posit that mathematical models can improve the application of anticancer therapeutics in the era of precision medicine. By highlighting a number of historical examples of the contributions of mathematical models to cancer therapy, we hope that this contribution serves to engage investigators who may not have previously considered how mathematical modeling can provide real insights into breast cancer therapy.

Incorporating drug delivery into an imaging-driven, mechanics-coupled reaction diffusion model for predicting the response of breast cancer to neoadjuvant chemotherapy: theory and preliminary clinical results

Clinical methods for assessing tumor response to therapy are largely rudimentary, monitoring only temporal changes in tumor size. Our goal is to predict the response of breast tumors to therapy using a mathematical model that utilizes magnetic resonance imaging (MRI) data obtained non-invasively from individual patients. We extended a previously established, mechanically coupled, reaction-diffusion model for predicting tumor response initialized with patient-specific diffusion weighted MRI (DW-MRI) data by including the effects of chemotherapy drug delivery, which is estimated using dynamic contrast-enhanced (DCE-) MRI data. The extended, drug incorporated, model is initialized using patient-specific DW-MRI and DCE-MRI data. Data sets from five breast cancer patients were used-obtained before, after one cycle, and at mid-point of neoadjuvant chemotherapy. The DCE-MRI data was used to estimate spatiotemporal variations in tumor perfusion with the extended Kety-Tofts model. The physiological parameters derived from DCE-MRI were used to model changes in delivery of therapy drugs within the tumor for incorporation in the extended model. We simulated the original model and the extended model in both 2D and 3D and compare the results for this five-patient cohort. Preliminary results show reductions in the error of model predicted tumor cellularity and size compared to the experimentally-measured results for the third MRI scan when therapy was incorporated. Comparing the two models for agreement between the predicted total cellularity and the calculated total cellularity (from the DW-MRI data) reveals an increased concordance correlation coefficient from 0.81 to 0.98 for the 2D analysis and 0.85 to 0.99 for the 3D analysis (p  <  0.01 for each) when the extended model was used in place of the original model. This study demonstrates the plausibility of using DCE-MRI data as a means to estimate drug delivery on a patient-specific basis in predictive models and represents a step toward the goal of achieving individualized prediction of tumor response to therapy.

A DCE-MRI Driven 3-D Reaction-Diffusion Model of Solid Tumor Growth

Predicting tumor growth and its response to therapy remains a major challenge in cancer research and strongly relies on tumor growth models. In this paper, we introduce, calibrate, and verify a novel image-driven reaction-diffusion model of avascular tumor growth. The model allows for proliferation, death and spread of tumor cells, and accounts for nutrient distribution and hypoxia. It is constrained by longitudinal time series of dynamic contrast-enhancement-MRI images. Tumor specific parameters are estimated from two early time points and used to predict the spatio-temporal evolution of the tumor volume and cell densities at later time points. We first test our parameter estimation approach on synthetic data from 15 generated tumors. Our in silico study resulted in small volume errors (<5%) and high Dice overlaps (>97%), showing that model parameters can be successfully recovered and used to accurately predict the tumor growth. Encouraged by these results, we apply our model to seven pre-clinical cases of breast carcinoma. We are able to show promising preliminary results, especially for the estimation for early time points. Processes like angiogenesis and apoptosis should be included to further improve predictions for later time points.

Tissue transport affects how treatment scheduling increases the efficacy of chemotherapeutic drugs

A method to predict the effect of tissue transport on the scheduling of chemotherapeutic treatment could increase efficacy. Many drugs with desirable pharmacokinetic properties fail in vivo due to poor transport through tissue. To predict the effect of treatment schedule on drug efficacy we developed an in silico method that integrates diffusion through tissue and cell binding into a pharmacokinetic model. The model was evaluated with an array of theoretical drugs that had different rates of diffusivity, binding, and clearance. The efficacy of each drug, quantified as the fraction of cells killed, was calculated for twenty dosage schedules. Simulations showed that efficacy strongly depended on tissue transport, with a range of 0.00 to 99.99%, despite each drug having equal plasma areas under the curve (AUC). For most drugs, schedules that increased exposure also increased efficacy. Drugs with fast clearance benefited the most from increasing the number of doses and this was most effective for those with intermediary binding. All drugs with slow diffusivity were ineffective. For a subset of drugs, increasing the number of doses decreased efficacy. This phenomenon was unexpected because, when considering uptake into tissue, sustained plasma levels from multiple doses are generally assumed to be more effective. This counterintuitive decrease in efficacy was caused by drug retention within tumor tissue. These results established a set of rules that suggests how transport parameters affect the efficacy of drugs at different schedules. The two most predominant rules are (1) multiple doses improve efficacy for drugs with fast clearance, fast diffusivity and low to intermediate cell binding; and (2) one dose is most effective for drugs with slow clearance, slow diffusivity or strong cell binding. Understanding the role of tissue transport when determining drug treatment schedules would improve the outcome of preclinical animal experiments and early clinical trials.

Exosome is a mechanism of intercellular drug transfer: Application of quantitative pharmacology

PURPOSE

Exosomes are small membrane vesicles (30-100nm in diameter) secreted by cells into extracellular space. The present study evaluated the effect of chemotherapeutic agents on exosome production and/or release, and quantified the contribution of exosomes to intercellular drug transfer and pharmacodynamics.

METHODS

Human cancer cells (breast MCF7, breast-to-lung metastatic LM2, ovarian A2780 and OVCAR4) were treated with paclitaxel (PTX, 2-1000nM) or doxorubicin (DOX, 20-1000nM) for 24-48h. Exosomes were isolated from the culture medium of drug-treated donor cells (Donor cells) using ultra-centrifugation, and analyzed for acetylcholinesterase activity, total proteins, drug concentrations, and biological effects (cytotoxicity and anti-migration) on drug-naïve recipient cells (Recipient cells). These results were used to develop computational predictive quantitative pharmacology models.

RESULTS

Cells in exponential growth phase released ~220 exosomes/cell in culture medium. PTX and DOX significantly promoted exosome production and/or release in a dose- and time-dependent manner, with greater effects in ovarian cancer cells than in breast cancer cells. Exosomes isolated from Donor cells contained appreciable drug levels (2-7pmole/10⁶ cells after 24h treatment with 100-1000nM PTX), and caused cytotoxicity and inhibited migration of Recipient cells. Quantitative pharmacology models that integrated cellular PTX pharmacokinetics with PTX pharmacodynamics successfully predicted effects of exosomes on intercellular drug transfer, cytotoxicity of PTX on Donor cells and cytotoxicity of PTX-containing exosomes on Recipient cells. Additional model simulations indicate that within clinically achievable PTX concentrations, the contribution of exosomes to active drug efflux increased with drug concentration and exceeded the p-glycoprotein efflux when the latter was saturated.

CONCLUSIONS

Our results indicate (a) chemotherapeutic agents stimulate exosome production or release, and (b) exosome is a mechanism of intercellular drug transfer that contributes to pharmacodynamics of neighboring cells.

Modelling drug transport during intraperitoneal chemotherapy

Despite a strong rationale for intraperitoneal (IP) chemotherapy, the actual use of the procedure is limited by the poor penetration depth of the drug into the tissue. Drug penetration into solid tumours is a complex mass transport process that involves multiple parameters not only related to the used cytotoxic agent but also to the tumour tissue properties and even the therapeutic setup. Mathematical modelling can provide unique insights into the different transport barriers that occur during IP chemotherapy as well as offer the possibility to test different protocols or drugs without the need for in vivo experiments. In this work, a distinction is made between three different types of model: the lumped parameter model, the distributed model and the cell-based model. For each model, we discuss which steps of the transport process are included and where assumptions are made. Finally, we focus on the advantages and main limitations of each category and discuss some future perspectives for the modelling of IP chemotherapy.

Multiscale Modeling of Antibody-Drug Conjugates: Connecting Tissue and Cellular Distribution to Whole Animal Pharmacokinetics and Potential Implications for Efficacy

Antibody-drug conjugates exhibit complex pharmacokinetics due to their combination of macromolecular and small molecule properties. These issues range from systemic concerns, such as deconjugation of the small molecule drug during the long antibody circulation time or rapid clearance from nonspecific interactions, to local tumor tissue heterogeneity, cell bystander effects, and endosomal escape. Mathematical models can be used to study the impact of these processes on overall distribution in an efficient manner, and several types of models have been used to analyze varying aspects of antibody distribution including physiologically based pharmacokinetic (PBPK) models and tissue-level simulations. However, these processes are quantitative in nature and cannot be handled qualitatively in isolation. For example, free antibody from deconjugation of the small molecule will impact the distribution of conjugated antibodies within the tumor. To incorporate these effects into a unified framework, we have coupled the systemic and organ-level distribution of a PBPK model with the tissue-level detail of a distributed parameter tumor model. We used this mathematical model to analyze new experimental results on the distribution of the clinical antibody-drug conjugate Kadcyla in HER2-positive mouse xenografts. This model is able to capture the impact of the drug-antibody ratio (DAR) on tumor penetration, the net result of drug deconjugation, and the effect of using unconjugated antibody to drive ADC penetration deeper into the tumor tissue. This modeling approach will provide quantitative and mechanistic support to experimental studies trying to parse the impact of multiple mechanisms of action for these complex drugs.

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using in vivo experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug vs. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.

The Added Prognostic Value of Preoperative Dynamic Contrast-Enhanced MRI Histogram Analysis in Patients with Glioblastoma: Analysis of Overall and Progression-Free Survival

BACKGROUND AND PURPOSE

The prognostic value of dynamic contrast-enhanced MR imaging in patients with glioblastoma is controversial. We investigated the added prognostic value of dynamic contrast-enhanced MR imaging to clinical parameters and molecular biomarkers in patients with glioblastoma by using histogram analysis.

MATERIALS AND METHODS

This retrospective study consisted of 61 patients who underwent preoperative dynamic contrast-enhanced MR imaging for glioblastoma. The histogram parameters of dynamic contrast-enhanced MR imaging, including volume transfer constant, extravascular extracellular volume fraction, and plasma volume fraction, were calculated from entire enhancing tumors. Univariate analyses for overall survival and progression-free survival were performed with preoperative clinical and dynamic contrast-enhanced MR imaging parameters and postoperative molecular biomarkers. Multivariate Cox regression was performed to build pre- and postoperative models for overall survival and progression-free survival. The performance of models was assessed by calculating the Harrell concordance index.

RESULTS

In univariate analysis, patients with higher volume transfer constant and extravascular extracellular volume fraction values showed worse overall survival and progression-free survival, whereas plasma volume fraction showed no significant correlation. In multivariate analyses for overall survival, the fifth percentile value of volume transfer constant and kurtosis of extravascular extracellular volume fraction were independently prognostic in the preoperative model, and kurtosis of volume transfer constant and extravascular extracellular volume fraction were independently prognostic in the postoperative model. For progression-free survival, independent prognostic factors were minimum and fifth percentile values of volume transfer constant and kurtosis of extravascular extracellular volume fraction in the preoperative model and kurtosis of extravascular extracellular volume fraction in the postoperative model. The performance of preoperative models for progression-free survival was significantly improved when minimum or fifth percentile values of volume transfer constant and kurtosis of extravascular extracellular volume fraction were added.

CONCLUSIONS

Higher volume transfer constant and extravascular extracellular volume fraction values are associated with worse prognosis, and dynamic contrast-enhanced MR imaging may have added prognostic value in combination with preoperative clinical parameters, especially in predicting progression-free survival.

Preoperative prognostic value of dynamic contrast-enhanced MRI-derived contrast transfer coefficient and plasma volume in patients with cerebral gliomas

BACKGROUND AND PURPOSE

The prognostic value of dynamic contrast-enhanced MR imaging-derived plasma volume obtained in tumor and the contrast transfer coefficient has not been well-established in patients with gliomas. We determined whether plasma volume and contrast transfer coefficient in tumor correlated with survival in patients with gliomas in addition to other factors such as age, type of surgery, preoperative Karnofsky score, contrast enhancement, and histopathologic grade.

MATERIALS AND METHODS

This prospective study included 46 patients with a new pathologically confirmed diagnosis of glioma. The contrast transfer coefficient and plasma volume obtained in tumor maps were calculated directly from the signal-intensity curve without T1 measurements, and values were obtained from multiple small ROIs placed within tumors. Survival curve analysis was performed by dichotomizing patients into groups of high and low contrast transfer coefficient and plasma volume. Univariate analysis was performed by using dynamic contrast-enhanced parameters and clinical factors. Factors that were significant on univariate analysis were entered into multivariate analysis.

RESULTS

For all patients with gliomas, survival was worse for groups of patients with high contrast transfer coefficient and plasma volume obtained in tumor (P < .05). In subgroups of high- and low-grade gliomas, survival was worse for groups of patients with high contrast transfer coefficient and plasma volume obtained in tumor (P < .05). Univariate analysis showed that factors associated with lower survival were age older than 50 years, low Karnofsky score, biopsy-only versus resection, marked contrast enhancement versus no/mild enhancement, high contrast transfer coefficient, and high plasma volume obtained in tumor (P < .05). In multivariate analysis, a low Karnofsky score, biopsy versus resection in combination with marked contrast enhancement, and a high contrast transfer coefficient were associated with lower survival rates (P < .05).

CONCLUSIONS

In patients with glioma, those with a high contrast transfer coefficient have lower survival than those with low parameters.

Cancer systems biology: embracing complexity to develop better anticancer therapeutic strategies

The transformation of normal cells into cancer cells and maintenance of the malignant state and phenotypes are associated with genetic and epigenetic deregulations, altered cellular signaling responses and aberrant interactions with the microenvironment. These alterations are constantly evolving as tumor cells face changing selective pressures induced by the cells themselves, the microenvironment and drug treatments. Tumors are also complex ecosystems where different, sometime heterogeneous, subclonal tumor populations and a variety of nontumor cells coexist in a constantly evolving manner. The interactions between molecules and between cells that arise as a result of these alterations and ecosystems are even more complex. The cancer research community is increasingly embracing this complexity and adopting a combination of systems biology methods and integrated analyses to understand and predictively model the activity of cancer cells. Systems biology approaches are helping to understand the mechanisms of tumor progression and design more effective cancer therapies. These approaches work in tandem with rapid technological advancements that enable data acquisition on a broader scale, with finer accuracy, higher dimensionality and higher throughput than ever. Using such data, computational and mathematical models help identify key deregulated functions and processes, establish predictive biomarkers and optimize therapeutic strategies. Moving forward, implementing patient-specific computational and mathematical models of cancer will significantly improve the specificity and efficacy of targeted therapy, and will accelerate the adoption of personalized and precision cancer medicine.

Multichannel imaging to quantify four classes of pharmacokinetic distribution in tumors

Low and heterogeneous delivery of drugs and imaging agents to tumors results in decreased efficacy and poor imaging results. Systemic delivery involves a complex interplay of drug properties and physiological factors, and heterogeneity in the tumor microenvironment makes predicting and overcoming these limitations exceptionally difficult. Theoretical models have indicated that there are four different classes of pharmacokinetic behavior in tissue, depending on the fundamental steps in distribution. In order to study these limiting behaviors, we used multichannel fluorescence microscopy and stitching of high-resolution images to examine the distribution of four agents in the same tumor microenvironment. A validated generic partial differential equation model with a graphical user interface was used to select fluorescent agents exhibiting these four classes of behavior, and the imaging results agreed with predictions. BODIPY-FL exhibited higher concentrations in tissue with high blood flow, cetuximab gave perivascular distribution limited by permeability, high plasma protein and target binding resulted in diffusion-limited distribution for Hoechst 33342, and Integrisense 680 was limited by the number of binding sites in the tissue. Together, the probes and simulations can be used to investigate distribution in other tumor models, predict tumor drug distribution profiles, and design and interpret in vivo experiments.

Effect of small-molecule modification on single-cell pharmacokinetics of PARP inhibitors

The heterogeneous delivery of drugs in tumors is an established process contributing to variability in treatment outcome. Despite the general acceptance of variable delivery, the study of the underlying causes is challenging, given the complex tumor microenvironment including intra- and intertumor heterogeneity. The difficulty in studying this distribution is even more significant for small-molecule drugs where radiolabeled compounds or mass spectrometry detection lack the spatial and temporal resolution required to quantify the kinetics of drug distribution in vivo. In this work, we take advantage of the synthesis of fluorescent drug conjugates that retain their target binding but are designed with different physiochemical and thus pharmacokinetic properties. Using these probes, we followed the drug distribution in cell culture and tumor xenografts with temporal resolution of seconds and subcellular spatial resolution. These measurements, including in vivo permeability of small-molecule drugs, can be used directly in predictive pharmacokinetic models for the design of therapeutics and companion imaging agents as demonstrated by a finite element model.

Rapid uptake of glucose and lactate, and not hypoxia, induces apoptosis in three-dimensional tumor tissue culture

The spatial arrangement of cellular metabolism in tumor tissue critically affects the treatment of cancer. However, little is known about how diffusion and cellular uptake relate to intracellular metabolism and cell death in three dimensions. To quantify these mechanisms, fluorescent microscopy and multicellular tumor cylindroids were used to measure pH and oxygen profiles, and quantify the distribution of viable, apoptotic and necrotic cells. Spheroid dissociation, enzymatic analysis, and mass spectrometry were used to measure concentration profiles of glucose, lactate and glutamine. A mathematical model was used to integrate these measurements and calculate metabolic rate parameters. It was found that large cylindroids, >500 μm in diameter, contained apoptotic and necrotic cells, whereas small cylindroids contained apoptotic but not necrotic cells. The center of cylindroids was found to be acidic but not hypoxic. From the edge to the center, concentrations of glucose, lactate and glutamine decreased rapidly. Throughout the cell masses lactate was consumed and not produced. These measurements indicate that apoptosis was the primary mechanism of cell death; acidity was not caused by lactic acid; and cell death was caused by depletion of carbon sources and not hypoxia. The mathematical model showed that the transporter enzymes for glucose and lactate were not saturated; oxygen uptake was limited by intracellular metabolism; and oxygen uptake was not limited by membrane-transport or diffusion. Unsaturated transmembrane uptake may be the cause of both proliferative and apoptotic regimes in cancer. These results suggest that transporter enzymes are excellent targets for treating well oxygenated tumors.

Current advances in mathematical modeling of anti-cancer drug penetration into tumor tissues

Delivery of anti-cancer drugs to tumor tissues, including their interstitial transport and cellular uptake, is a complex process involving various biochemical, mechanical, and biophysical factors. Mathematical modeling provides a means through which to understand this complexity better, as well as to examine interactions between contributing components in a systematic way via computational simulations and quantitative analyses. In this review, we present the current state of mathematical modeling approaches that address phenomena related to drug delivery. We describe how various types of models were used to predict spatio-temporal distributions of drugs within the tumor tissue, to simulate different ways to overcome barriers to drug transport, or to optimize treatment schedules. Finally, we discuss how integration of mathematical modeling with experimental or clinical data can provide better tools to understand the drug delivery process, in particular to examine the specific tissue- or compound-related factors that limit drug penetration through tumors. Such tools will be important in designing new chemotherapy targets and optimal treatment strategies, as well as in developing non-invasive diagnosis to monitor treatment response and detect tumor recurrence.

Evolutionary dynamics of carcinogenesis and why targeted therapy does not work

All malignant cancers, whether inherited or sporadic, are fundamentally governed by Darwinian dynamics. The process of carcinogenesis requires genetic instability and highly selective local microenvironments, the combination of which promotes somatic evolution. These microenvironmental forces, specifically hypoxia, acidosis and reactive oxygen species, are not only highly selective, but are also able to induce genetic instability. As a result, malignant cancers are dynamically evolving clades of cells living in distinct microhabitats that almost certainly ensure the emergence of therapy-resistant populations. Cytotoxic cancer therapies also impose intense evolutionary selection pressures on the surviving cells and thus increase the evolutionary rate. Importantly, the principles of Darwinian dynamics also embody fundamental principles that can illuminate strategies for the successful management of cancer.