A mathematical model of tumor oxygen and glucose mass transport and metabolism with complex reaction kinetics

Computational modelling of the therapeutic outputs of photodynamic therapy on spheroid-on-chip models

Photodynamic therapy (PDT) is a medical radio chemotherapeutic method that uses light, photosensitizing agents, and oxygen to produce cytotoxic compounds, which eliminate malignant cells. Recently, Microfluidic systems have been used to analyse photosensitizers (PSs) due to their potential to replicate in vivo environments. While prior studies have established a strong correlation between reacted singlet oxygen concentration and PDT-induced cellular death, the effects that the ambient fluid flow might have on the concentration of oxygen and PS have been disregarded in many, which limits the reliability of the results. Herein, we coupled the transport of oxygen and PS throughout the ambient medium and within the spheroidal multicellular aggregate to initially study the profiles of oxygen and PS concentration alongside PDT-induced cellular death throughout the spheroid before and after radiation. The attained results indicate that the PDT-induced cellular death initiates on the surface of the spheroids and subsequently spreads to the neighbouring regions, which is in great accordance with experimental results. Afterward, the effects that drug-light interval (DLI), fluence rate, PS composition, microchannel height, and inlet flow rate have on the therapeutic outcomes are studied. The findings show that adequate DLI is critical to ensure uniform distribution of PS throughout the medium, and a value of 5 h was found to be sufficient. The composition of PS is critical, as ALA-PpIX induces earlier cell death but accelerates oxygen consumption, especially in the outer layers, depriving the inner layers of oxygen necessary for PDT, which in turn disrupts and prolongs the exposure time compared to mTHPC and Photofrin. Despite the fluence rate directly influencing the singlet oxygen generation rate, increasing the fluence rate by 189 mW/cm2 would not significantly benefit us. Microwell height and inlet flow rate involve competing phenomena-increasing height or decreasing flow reduces oxygen supply and increases PS "washout" and its concentration.

Radial Flow Perfusion Enables Real-Time Profiling of Cellular Metabolism at Low Oxygen Levels with Hyperpolarized 13C NMR Spectroscopy

In this study, we describe new methods for studying cancer cell metabolism with hyperpolarized ¹³C magnetic resonance spectroscopy (HP ¹³C MRS) that will enable quantitative studies at low oxygen concentrations. Cultured hepatocellular carcinoma cells were grown on the surfaces of non-porous microcarriers inside an NMR spectrometer. They were perfused radially from a central distributer in a modified NMR tube (bioreactor). The oxygen level of the perfusate was continuously monitored and controlled externally. Hyperpolarized substrates were injected continuously into the perfusate stream with a newly designed system that prevented oxygen and temperature perturbations in the bioreactor. Computational and experimental results demonstrated that cell mass oxygen profiles with radial flow were much more uniform than with conventional axial flow. Further, the metabolism of HP [1-¹³C]pyruvate was markedly different between the two flow configurations, demonstrating the importance of avoiding large oxygen gradients in cell perfusion experiments.

Mimicking and surpassing the xenograft model with cancer-on-chip technology

Organs-on-chips are in vitro models in which human tissues are cultured in microfluidic compartments with a controlled, dynamic micro-environment. Specific organs-on-chips are being developed to mimic human tumors, but the validation of such 'cancer-on-chip' models for use in drug development is hampered by the complexity and variability of human tumors. An important step towards validation of cancer-on-chip technology could be to first mimic cancer xenograft models, which share multiple characteristics with human cancers but are significantly less complex. Here we review the relevant biological characteristics of a xenograft tumor and show that organ-on-chip technology is capable of mimicking many of these aspects. Actual comparisons between on-chip tumor growth and xenografts are promising but also demonstrate that further development and empirical validation is still needed. Validation of cancer-on-chip models to xenografts would not only represent an important milestone towards acceptance of cancer-on-chip technology, but could also improve drug discovery, personalized cancer medicine, and reduce animal testing.

Determining the parameter space for effective oxygen depletion for FLASH radiation therapy

There has been a recent revival of interest in the FLASH effect, after experiments have shown normal tissue sparing capabilities of ultra-high-dose-rate radiation with no compromise on tumour growth restraint. A model has been developed to investigate the relative importance of a number of fundamental parameters considered to be involved in the oxygen depletion paradigm of induced radioresistance. An example eight-dimensional parameter space demonstrates the conditions under which radiation may induce sufficient depletion of oxygen for a diffusion-limited hypoxic cellular response. Initial results support experimental evidence that FLASH sparing is only achieved for dose rates on the order of tens of Gy/s or higher, for a sufficiently high dose, and only for tissue that is slightly hypoxic at the time of radiation. We show that the FLASH effect is the result of a number of biological, radiochemical and delivery parameters. Also, the threshold dose for a FLASH effect occurring would be more prominent when the parameterisation was optimised to produce the maximum effect. The model provides a framework for further FLASH-related investigation and experimental design. An understanding of the mechanistic interactions producing an optimised FLASH effect is essential for its translation into clinical practice.

Oxygen in the Tumor Microenvironment: Mathematical and Numerical Modeling

There are many reasons to try to achieve a good grasp of the distribution of oxygen in the tumor microenvironment. The lack of oxygen - hypoxia - is a main actor in the evolution of tumors and in their growth and appears to be just as important in tumor invasion and metastasis. Mathematical models of the distribution of oxygen in tumors which are based on reaction-diffusion equations provide partial but qualitatively significant descriptions of the measured oxygen concentrations in the tumor microenvironment, especially when they incorporate important elements of the blood vessel network such as the blood vessel size and spatial distribution and the pulsation of local pressure due to blood circulation. Here, we review our mathematical and numerical approaches to the distribution of oxygen that yield insights both on the role of the distribution of blood vessel density and size and on the fluctuations of blood pressure.

Pulsation-limited oxygen diffusion in the tumour microenvironment

Hypoxia is central to tumour evolution, growth, invasion and metastasis. Mathematical models of hypoxia based on reaction-diffusion equations provide seemingly incomplete descriptions as they fail to predict the measured oxygen concentrations in the tumour microenvironment. In an attempt to explain the discrepancies, we consider both the inhomogeneous distribution of oxygen-consuming cells in solid tumours and the dynamics of blood flow in the tumour microcirculation. We find that the low-frequency oscillations play an important role in the establishment of tumour hypoxia. The oscillations interact with consumption to inhibit oxygen diffusion in the microenvironment. This suggests that alpha-blockers-a class of drugs used to treat hypertension and stress disorders, and known to lower or even abolish low-frequency oscillations of arterial blood flow -may act as adjuvant drugs in the radiotherapy of solid tumours by enhancing the oxygen effect.

Computational Model for Tumor Oxygenation Applied to Clinical Data on Breast Tumor Hemoglobin Concentrations Suggests Vascular Dilatation and Compression

We present a computational model for trans-vascular oxygen transport in synthetic tumor and host tissue blood vessel networks, aiming at qualitatively explaining published data of optical mammography, which were obtained from 87 breast cancer patients. The data generally show average hemoglobin concentration to be higher in tumors versus host tissue whereas average oxy-to total hemoglobin concentration (vascular segment RBC-volume-weighted blood oxygenation) can be above or below normal. Starting from a synthetic arterio-venous initial network the tumor vasculature was generated by processes involving cooption, angiogenesis, and vessel regression. Calculations of spatially resolved blood flow, hematocrit, oxy- and total hemoglobin concentrations, blood and tissue oxygenation were carried out for ninety tumor and associated normal vessel networks starting from various assumed geometries of feeding arteries and draining veins. Spatial heterogeneity in the extra-vascular partial oxygen pressure distribution can be related to various tumor compartments characterized by varying capillary densities and blood flow characteristics. The reported higher average hemoglobin concentration of tumors is explained by growth and dilatation of tumor blood vessels. Even assuming sixfold metabolic rate of oxygen consumption in tumorous versus host tissue, the predicted oxygen hemoglobin concentrations are above normal. Such tumors are likely associated with high tumor blood flow caused by high-caliber blood vessels crossing the tumor volume and hence oxygen supply exceeding oxygen demand. Tumor oxy- to total hemoglobin concentration below normal could only be achieved by reducing tumor vessel radii during growth by a randomly selected factor, simulating compression caused by intra-tumoral solid stress due to proliferation of cells and extracellular matrix. Since compression of blood vessels will impede chemotherapy we conclude that tumors with oxy- to total hemoglobin concentration below normal are less likely to respond to chemotherapy. Such behavior was recently reported for neo-adjuvant chemotherapy of locally advanced breast tumors.

Histological Image Processing Features Induce a Quantitative Characterization of Chronic Tumor Hypoxia

Hypoxia in tumors signifies resistance to therapy. Despite a wealth of tumor histology data, including anti-pimonidazole staining, no current methods use these data to induce a quantitative characterization of chronic tumor hypoxia in time and space. We use image-processing algorithms to develop a set of candidate image features that can formulate just such a quantitative description of xenographed colorectal chronic tumor hypoxia. Two features in particular give low-variance measures of chronic hypoxia near a vessel: intensity sampling that extends radially away from approximated blood vessel centroids, and multithresholding to segment tumor tissue into normal, hypoxic, and necrotic regions. From these features we derive a spatiotemporal logical expression whose truth value depends on its predicate clauses that are grounded in this histological evidence. As an alternative to the spatiotemporal logical formulation, we also propose a way to formulate a linear regression function that uses all of the image features to learn what chronic hypoxia looks like, and then gives a quantitative similarity score once it is trained on a set of histology images.

Multilayer spheroids to quantify drug uptake and diffusion in 3D

There is a need for new quantitative in vitro models of drug uptake and diffusion to help assess drug toxicity/efficacy as well as new more predictive models for drug discovery. We report a three-dimensional (3D) multilayer spheroid model and a new algorithm to quantitatively study uptake and inward diffusion of fluorescent calcein via gap junction intercellular communication (GJIC). When incubated with calcein-AM, a substrate of the efflux transporter P-glycoprotein (Pgp), spheroids from a variety of cell types accumulated calcein over time. Accumulation decreased in spheroids overexpressing Pgp (HEK-MDR) and was increased in the presence of Pgp inhibitors (verapamil, loperamide, cyclosporin A). Inward diffusion of calcein was negligible in spheroids that lacked GJIC (OVCAR-3, SK-OV-3) and was reduced in the presence of an inhibitor of GJIC (carbenoxolone). In addition to inhibiting Pgp, verapamil and loperamide, but not cyclosporin A, inhibited inward diffusion of calcein, suggesting that they also inhibit GJIC. The dose response curves of verapamil’s inhibition of Pgp and GJIC were similar (IC₅₀: 8 μM). The method is amenable to many different cell types and may serve as a quantitative 3D model that more accurately replicates in vivo barriers to drug uptake and diffusion.

A model to simulate the oxygen distribution in hypoxic tumors for different vascular architectures

PURPOSE

As hypoxic cells are more resistant to photon radiation, it is desirable to obtain information about the oxygen distribution in tumors prior to the radiation treatment. Noninvasive techniques are currently not able to provide reliable oxygenation maps with sufficient spatial resolution; therefore mathematical models may help to simulate microvascular architectures and the resulting oxygen distributions in the surrounding tissue. Here, the authors present a new computer model, which uses the vascular fraction of tumor voxels, in principle measurable noninvasively in vivo, as input parameter for simulating realistic PO2 histograms in tumors, assuming certain 3D vascular architectures.

METHODS

Oxygen distributions were calculated by solving a reaction-diffusion equation in a reference volume using the particle strength exchange method. Different types of vessel architectures as well as different degrees of vascular heterogeneities are considered. Two types of acute hypoxia (ischemic and hypoxemic) occurring additionally to diffusion-limited (chronic) hypoxia were implemented as well.

RESULTS

No statistically significant differences were observed when comparing 2D- and 3D-vessel architectures (p>0.79 in all cases) and highly heterogeneously distributed linear vessels show good agreement, when comparing with published experimental intervessel distance distributions and PO2 histograms. It could be shown that, if information about additional acute hypoxia is available, its contribution to the hypoxic fraction (HF) can be simulated as well. Increases of 128% and 168% in the HF were obtained when representative cases of ischemic and hypoxemic acute hypoxia, respectively, were considered in the simulations.

CONCLUSIONS

The presented model is able to simulate realistic microscopic oxygen distributions in tumors assuming reasonable vessel architectures and using the vascular fraction as macroscopic input parameter. The model may be used to generate PO2 histograms, which are needed as input in models predicting the radiation response of hypoxic tumors.

Modelling tumour oxygenation, reoxygenation and implications on treatment outcome

Oxygenation is an important component of the tumour microenvironment, having a significant impact on the progression and management of cancer. Theoretical determination of tissue oxygenation through simulations of the oxygen transport process is a powerful tool to characterise the spatial distribution of oxygen on the microscopic scale and its dynamics and to study its impact on the response to radiation. Accurate modelling of tumour oxygenation must take into account important aspects that are specific to tumours, making the quantitative characterisation of oxygenation rather difficult. This paper aims to discuss the important aspects of modelling tumour oxygenation, reoxygenation, and implications for treatment.

Advection, diffusion, and delivery over a network

Many biological, geophysical, and technological systems involve the transport of a resource over a network. In this paper, we present an efficient method for calculating the exact quantity of the resource in each part of an arbitrary network, where the resource is lost or delivered out of the network at a given rate, while being subject to advection and diffusion. The key conceptual step is to partition the resource into material that does or does not reach a node over a given time step. As an example application, we consider resource allocation within fungal networks, and analyze the spatial distribution of the resource that emerges as such networks grow over time. Fungal growth involves the expansion of fluid filled vessels, and such growth necessarily involves the movement of fluid. We develop a model of delivery in growing fungal networks, and find good empirical agreement between our model and experimental data gathered using radio-labeled tracers. Our results lead us to suggest that in foraging fungi, growth-induced mass flow is sufficient to account for long-distance transport, if the system is well insulated. We conclude that active transport mechanisms may only be required at the very end of the transport pathway, near the growing tips.

Bioreactor for Blood Product Production

The feasibility of ex vivo blood production is limited by both biological and engineering challenges. From an engineering perspective, these challenges include the significant volumes required to generate even a single unit of a blood product, as well as the correspondingly high protein consumption required for such large volume cultures. Membrane bioreactors, such as hollow fiber bioreactors (HFBRs), enable cell densities approximately 100-fold greater than traditional culture systems and therefore may enable a significant reduction in culture working volumes. As cultured cells, and larger molecules, are retained within a fraction of the system volume, via a semipermeable membrane it may be possible to reduce protein consumption by limiting supplementation to only this fraction. Typically, HFBRs are complex perfusion systems having total volumes incompatible with bench scale screening and optimization of stem cell-based cultures. In this article we describe the use of a simplified HFBR system to assess the feasibility of this technology to produce blood products from umbilical cord blood-derived CD34+ hematopoietic stem progenitor cells (HSPCs). Unlike conventional HFBR systems used for protein manufacture, where cells are cultured in the extracapillary space, we have cultured cells in the intracapillary space, which is likely more compatible with the large-scale production of blood cell suspension cultures. Using this platform we direct HSPCs down the myeloid lineage, while targeting a 100-fold increase in cell density and the use of protein-free bulk medium. Our results demonstrate the potential of this system to deliver high cell densities, even in the absence of protein supplementation of the bulk medium.

Tumour thermotolerance, a physiological phenomenon involving vessel normalisation

The purpose of this study was to delineate the mechanisms by which stromal components of cancer may induce tumour thermotolerance and exploit alterations in stromal and tumour physiology to enhance radiation therapy. The vascular thermoresponse was monitored by daily one-hour 41.5°C heatings in two murine solid tumour models, SCK murine mammary carcinoma and B16F10 melanoma. A transient increase was seen in overall tumour oxygenation for 2-3 days, followed by a progressive decline in tumour pO(2) upon continued daily heatings. Vascular thermotolerance was further studied by treating tumours with different heating strategies, i.e. (1) a single 60 min 41.5°C treatment; (2) two consecutive daily treatments of 41.5°C for 60 min; (3) a single 60 min 43°C treatment or (4) two days of 41.5°C for 60 min followed by treatment with 43°C for 60 min on the third day. Pre-heating tumours with mild temperature hyperthermia induced vascular thermotolerance, which was accompanied by evidence of vessel normalisation, i.e. a decrease in microvessel density and an increase in pericyte coverage. Rational scheduling of fractionated radiation during heat-induced increases in tumour oxygen levels rendered a significantly greater, synergistic, tumour growth inhibition. In vitro clonogenic survival responses of the individual cell types associated (endothelial cells, fibroblasts, pericytes and tumour cells) indicated only a direct cellular thermotolerance in endothelial cells. Overall, this suggests that tumour thermotolerance is a physiological phenomenon mediated through improvement of functional vasculature.

Membrane bioreactors enhance microenvironmental conditioning and tissue development

In membrane bioreactors the cells are isolated from the bulk medium through a semipermeable membrane. This concept, which is analogous to how the circulatory system supplies solid tissues with nutrients, allows the maintenance of cells at much higher densities than is possible in traditional cultures. The membrane-based microbioreactor described herein is easy to operate, requiring only a pipette to load and harvest cells. A 10 microL culture volume was isolated from 1 mL of bulk medium through a semipermeable membrane having a molecular weight cutoff of 10 kDa. Here we describe the benefits regarding the retention of both cells and their secretions within this small culture volume using the following two model systems: hematopoietic stem cell expansion and mesenchymal stem cell-derived cartilage matrix accumulation.

Mild temperature hyperthermia and radiation therapy: role of tumour vascular thermotolerance and relevant physiological factors

Here we review the significance of changes in vascular thermotolerance on tumour physiology and the effects of multiple clinically relevant mild temperature hyperthermia (MTH) treatments on tumour oxygenation and corresponding radiation response. Thus far vascular thermotolerance referred to the observation of significantly greater blood flow response by the tumour to a second hyperthermia exposure than in response to a single thermal dose, even at temperatures that would normally cause vascular damage. New information suggests that although hyperthermia is a powerful modifier of tumour blood flow and oxygenation, sequencing and frequency are central parameters in the success of MTH enhancement of radiation therapy. We hypothesise that heat treatments every 2 to 3 days combined with traditional or accelerated radiation fractionation may be maximally effective in exploiting the improved perfusion and oxygenation induced by typical thermal doses given in the clinic.

Multiscale modelling of fluid and drug transport in vascular tumours

A model for fluid and drug transport through the leaky neovasculature and porous interstitium of a solid tumour is developed. The transport problems are posed on a micro-scale characterized by the inter-capillary distance, and the method of multiple scales is used to derive the continuum equations describing fluid and drug transport on the length scale of the tumour (under the assumption of a spatially periodic microstructure). The fluid equations comprise a double porous medium, with coupled Darcy flow through the interstitium and vasculature, whereas the drug equations comprise advection-reaction equations; in each case the dependence of the transport coefficients on the vascular geometry is determined by solving micro-scale cell problems.

Modeling corneal metabolism and oxygen transport during contact lens wear

PURPOSE

A metabolic model is developed for cornea-contact-lens system to elucidate the role of glucose metabolism in oxygenation of the cornea and to gauge the role that contact lens oxygen transmissibility plays in avoiding hypoxia-induced corneal abnormalities for extended wear applications.

METHODS

Oxygen transport through the cornea and contact lens system is typically described by oxygen diffusion with reactive loss. Oxygen in the cornea, however, interacts with other metabolic species, specifically glucose, lactate ion, bicarbonate ion, hydrogen ion, and carbon dioxide via aerobic glycolysis (Krebs or tricarboxylic acid cycle) and anaerobic glycolysis. Here, corneal aerobic and anaerobic metabolic reactions are incorporated into a six-layer (endothelium, stroma, epithelium, postlens tear film, contact lens, and prelens tear film) steady-state continuum reaction-diffusion model to quantify oxygen transport. We also define a new index, the oxygen deficiency factor (ODF), for gauging corneal oxygenation. As opposed to other current gauges of hypoxia, ODF is a local and sensitive measure of both the extent and severity of corneal oxygen deprivation.

RESULTS

We calculate not only oxygenation of the cornea but also its coupled glucose, lactate, and acidosis behavior. For the first time, the metabolic shift from aerobic to anaerobic glycolysis is explicitly incorporated into the transport and consumption of oxygen in the cornea on closed-eye contact lens wear. Adoption of enzymatic Monod kinetics for the metabolic reactions permits realistic assessment of local species concentrations throughout the cornea. We find that anerobic-produced lactate transports out of the cornea into the anterior chamber, whereas buffering bicarbonate ion transports into the comea from the anterior chamber.

CONCLUSIONS

The coupling of oxygen with other reactive species in corneal metabolism provides useful insight into the transport of oxygen in cornea-contact-lens system. Specifically, we find that in addition to oxygen depletion and acidosis in the cornea, lactate concentration increases while glucose and bicarbonate concentrations decrease from the endothelium toward the epithelium. Unlike other indices of corneal oxygenation, ODF is sensitive specifically to regions of cornea with local oxygen deficiency. Accordingly, ODF is a useful physiologic index to assess the extent and severity of hypoxia in the cornea.

Theoretical models of microvascular oxygen transport to tissue

To improve understanding of microvascular O(2) transport, theoretical modeling has been pursued for many years. The large number of studies in this area attests to the complexities (i.e., biochemical, structural, and hemodynamic) involved. This article focuses on theoretical studies from the last two decades and, in particular, on models of O(2) transport to tissue by discrete microvessels. A brief discussion of intravascular O(2) transport is first given, highlighting the physiological importance of intravascular resistance to blood-tissue O(2) transfer. This is followed by a description of the Krogh tissue cylinder model of O(2) transport by a single capillary, which is shown to remain relevant in modified forms that relax many of the original biophysical assumptions. However, there are many geometric and hemodynamic complexities that require the consideration of microvascular arrays and networks. Multivessel models are discussed that have shown the physiological importance of heterogeneities in vessel spacing, O(2) supply, red blood cell flow path, as well as interactions between capillaries and arterioles. These realistic models require sophisticated methods for solving the governing partial differential equations, and a range of solution techniques are described. Finally, the issue of experimental validation of microvascular O(2) delivery models is discussed, and new directions in O(2) transport modeling are outlined.

Multiscale modeling of fluid transport in tumors

A model for fluid flow through the leaky neovasculature and porous interstitium of a solid tumor is developed. A network of isolated capillaries is analyzed in the limit of small capillary radius, and analytical expressions for the hydraulic conductivities and fractional leakage coefficients derived. This model is then homogenized to give a continuum description in terms of the vascular density. The resulting equations comprise a double porous medium with coupled Darcy flow through the interstitium and vasculature.

Analytic solution to steady-state radial diffusion of a substrate with first-order reaction kinetics in the tissue of a Krogh's cylinder

It is often useful to calculate the concentration profile for a substrate undergoing reaction in the tissue surrounding a capillary. In this paper, we consider a model geometry consisting of a long straight cylinder of tissue surrounding a capillary. Substrate diffuses radially out of the capillary through the tissue, with consumption of substrate in the tissue directly proportional to substrate concentration (i.e., first-order reaction kinetics). The model is extended to include the case where a cylinder of necrotic tissue surrounds a metabolically active inner tissue cylinder. A simple analytic solution is derived, and concentration profiles are generated for various combinations of parameters. Compared to the case where substrate consumption is independent of concentration, this model predicts much more rapid depletion of substrate near the capillary interface. This can have significant implications for the calculation of the hypoxic fraction (e.g., tissue with pO(2)<0.5-5 mmHg) when tumor oxygenation is modeled. The model also permits calculation of the limiting substrate concentration for cell viability when the reaction rate constant is known and vice versa.

Hypoxia and radiotherapy: opportunities for improved outcomes in cancer treatment

A large body of clinical evidence exists to suggest that tumor hypoxia negatively impacts radiotherapy. As a result, there has been longstanding active research into novel methods of improving tumor oxygenation, targeting hypoxic tumor cells, and otherwise modulating the effect hypoxia has on how tumors respond to radiation. Over time, as more has been learned about the many ways hypoxia affects tumors, our understanding of the mechanisms connecting hypoxia to radiosensitivity has become increasingly broad and complicated. This has opened up new potential avenues for interrupting hypoxia's negative effects on tumor radiosensitivity. Here, we will review what is currently known about the spectrum of influence hypoxia has over the way tumors respond to radiation. Particular focus will be placed on recent discoveries suggesting that hypoxia-inducible factor-1 (HIF-1), a transcription factor that upregulates its target genes under hypoxic conditions, plays a major role in determining tumor radiosensitivity. HIF-1 and/or its target genes may represent therapeutic targets which could be manipulated to influence hypoxia's impact on tumor radiosensitivity.

A geometrical model of dermal capillary clearance

A new microscopic model is developed to describe the dermal capillary clearance process of skin permeants. The physiological structure is represented in terms of a doubly periodic array of absorbing capillaries. Convection-dominated transport in the blood flow within the capillaries is coupled with interstitial diffusion, the latter process being quantified via a slender-body-theory approach. Convection across the capillary wall and in the interstitial phase is treated as a perturbation which may be added to the diffusive transport. The model accounts for the finite permeability of the capillary wall as well as for the geometry of the capillary array, based on realistic values of physiological parameters. Calculated dermal concentration profiles for permeants having the size and lipophilicity of salicylic acid and glucose illustrate the power and general applicability of the model. Furthermore, validation of the model with published in vivo experimental results pertaining to human skin permeation of hydrocortisone is presented. The model offers the possibility for in-depth theoretical understanding and prediction of subsurface drug distribution in the human skin following topical application, as well as rates of capillary clearance into the systemic circulation. A simpler approach that treats the capillary bed as a homogeneously absorbing zone is also employed. The latter may be used in conjunction with the capillary exchange model to estimate measurable dermal transport and clearance parameters in a straightforward manner.

Hypoxia-inducible factor-1alpha and the glycolytic phenotype in tumors

Metastatic tumors generally exhibit aerobic glycolysis (the Warburg effect). The advent of [18F]fluorodeoxyglucose positron emission tomography imaging, coupled with recent findings linking hypoxia-inducible factor (HIF-1alpha) overexpression to aggressive cancers, has rekindled an interest in this aspect of tumor metabolism. These studies explore the role of HIF-1alpha in human breast cancer lines and its relationship to glycolytic regulation. Here we demonstrate that, under normal oxygen conditions, nonmetastatic cells consume less glucose and express low HIF-1alpha, whereas metastatic cells constitutively express high glycolysis and HIF-1alpha, suggesting that dysregulation of HIF-1alpha may induce the Warburg effect. This hypothesis was tested by renormalizing HIF-1alpha levels in renal carcinoma cells, leading to inhibition of aerobic glycolysis.

Modeling Intercellular Interactions during Carcinogenesis

By modulating the microenvironment of malignant or premalignant cells, inhibitory or stimulatory signals from nearby cells can play a key role in carcinogenesis. However, current commonly used quantitative models for induction of cancers by ionizing radiation focus on single cells and their progeny. Intercellular interactions are neglected or assumed to be confined to unidirectional radiation bystander effect signals from cells of the same tissue type. We here formulate a parsimoniously parameterized two-stage logistic (TSL) carcinogenesis model that incorporates some effects of intercellular interactions during the growth of premalignant cells. We show that for baseline tumor rates, involving no radiation apart from background radiation, this TSL model gives acceptable fits to a number of data sets. Specifically, it gives the same baseline hazard function, using the same number of adjustable parameters, as does the commonly used two-stage clonal expansion (TSCE) model, so it is automatically applicable to the many data sets on baseline cancer that have been analyzed using the TSCE model. For perturbations of baseline rates due to radiation, the models differ. We argue from epidemiological and laboratory evidence, especially results for the atomic bomb survivors, that for radiation carcinogenesis the TSL model gives results at least as realistic as the TSCE or similar models, despite involving fewer adjustable parameters in many cases.

Predicting the effect of temporal variations in PO2 on tumor radiosensitivity

PURPOSE

Tumor hypoxia is associated with less effective radiation-mediated cell killing, increased metastatic potential, and poorer prognosis. Transient variations in hypoxia, with characteristic periodicity on the order of 1 to 10 min, have been observed in animal models. This article explores the effect of these temporal variations in PO(2) on the oxygen enhancement ratio, effective radiation dose to the tumor, and tumor control probability.

METHODS AND MATERIALS

PO(2) over a 50-60 min period was determined at multiple sites in rat fibrosarcomas, 9L gliomas, and R3230Ac mammary adenocarcinomas. Using a correlation derived from the data of Elkind et al. (1965), PO(2) data are converted into oxygen enhancement ratios (OERs.) A tumor is assumed to consist of 10(3)-10(4) independent oxygenation subvolumes, each with a randomly chosen starting point on the OER-time curve. The effect of temporal variations in OER is examined for three cases: conventionally fractionated external beam radiotherapy (EBRT), stereotactic radiosurgery (SRS) and intraoperative radiotherapy (IORT). The oxygen effective dose (OED) for a subvolume is calculated from the dose to that subvolume modified by the OER. In turn, the distribution of OED for a tumor is analyzed for each treatment case and representative tumor control probabilities (TCPs) calculated.

RESULTS

Oxygen enhancement ratio varied from 1 to 3 over the range of PO(2) measured in this study. Mean OER ranged from 1.6 to 2.6, and the variation in OER vs. time was greater with decreasing PO(2). In EBRT, the standard deviation in OED was small, <2%. In contrast, the standard deviation in OED was much higher for both SRS and IORT, typically ranging from 3 to 6%, with the greatest variation at the lowest PO(2)s. Compared with a tumor with equal mean OED and uniform PO(2), TCP was minimally poorer for either EBRT or well-oxygenated tumors. However, for both SRS and IORT, temporal variations in more hypoxic tumors can produce a significant decrease in TCP.

CONCLUSION

Temporal variations in tumor PO(2) can produce significant variations OER, particularly at low PO(2), resulting in decreased TCP for hypofractionated treatment regimens.