Oxygen consumption and diffusion effects in photodynamic therapy

Effects of oxygen consumption in photodynamic therapy (PDT) are considered theoretically and experimentally. A mathematical model of the Type II mechanism of photooxidation is used to compute estimates of the rate of therapy-dependent in vivo oxygen depletion resulting from reactions of singlet oxygen (1O2) with intracellular substrate. Calculations indicate that PDT carried out at incident light intensities of 50 mW/cm2 may consume 3O2 at rates as high as 6-9 microM s-1. An approximate model of oxygen diffusion shows that these consumption rates are large enough to decrease the radius of oxygenated cells around an isolated capillary. Thus, during photoirradiation, cells sufficiently remote from the capillary wall may reside at oxygen tensions that are low enough to preclude or minimize 1O2-mediated damage. This effect is more pronounced at higher power densities and accounts for an enhanced therapeutic response in tumors treated with 360 J/cm2 delivered at 50 mW/cm2 compared to the same light dose delivered at 200 mW/cm2. The analysis further suggests that the oxygen depletion could be partially overcome by fractionating the light delivery. In a transplanted mammary tumor model, a regimen of 30-s exposures followed by 30-s dark periods produced significantly longer delays in tumor growth when compared to the continuous delivery of the same total fluence.     

Method for the determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids

A method has been developed for the quantitative evaluation of oxygen tension (PO2) distributions in multicellular spheroids measured with O2-sensitive microelectrodes. The experimental data showed that multicellular tumor spheroids in stirred growth media were characterized by a diffusion-depleted zone surrounding the spheroids. This zone was elicited by an unstirred layer of medium next to the spheroid leading to a continuous decrease in the PO2 values from the bulk medium towards the spheroid surface. Theoretical considerations demonstrate that the volume-related O2 consumption rate, Q, in the spheroids can be assessed by measuring the PO2 gradient in the diffusion-depleted zone outside the spheroids. Accordingly, Krogh's diffusion constant, KS, in the spheroids can be determined through measuring the PO2 gradient within the spheroids. The results obtained suggest that multicellular spheroids represent useful in vitro tumor models for the experimental and theoretical analysis of the interrelationship among O2 supply to tumor cells, O2 metabolism in tumors tissue, and the responsiveness of cancer cells to treatment.     

Oxygen diffusion and consumption in active aerobic granules of heterogeneous structure

The interior structure of aerobic granules is highly heterogeneous, hence, affecting the transport and reaction processes in the granules. The granule structure and the dissolved oxygen profiles were probed at the same granule in the current work for possible estimation of transport and kinetic parameters in the granule. With the tested granules fed by phenol or acetate as carbon source, most inflow oxygen was consumed by an active layer thickness of less than 125 μm on the granule surface. The confocal laser scanning microscopy scans also revealed a surface layer thickness of approximately 100 μm consisting of cells. The diffusivities of oxygen transport and the kinetic constant of oxygen consumption in the active layers only were evaluated. The theoretical models adopted in literature that ignored the contributions of the layered structure of aerobic granule could have overlooked the possible limitations on oxygen transport.     

Oxygen enhancement ratio in V79 spheroids

Chinese hamster V79 spheroids were stained with a nontoxic fluorescent stain, Hoechst 33342, which penetrates slowly into the spheroids. Single cells from these spheroids were then sorted by a fluorescence-activated cell sorter according to staining intensity (and therefore position in the spheroids). Flow cytometry characterization of the various cell subpopulations indicated that the innermost cells were more radiosensitive than expected on the basis of cell cycle position or cell thiol content. However, comparison of the radiosensitivities of cells sorted from equivalent depths from completely aerobic or anoxic V79 spheroids indicated that the oxygen enhancement ratio remained remarkably constant at 2.7 +/- 0.2 through the spheroid.     

Oscillating growth patterns of multicellular tumour spheroids

The growth kinetics of 9L (rat glioblastoma cell line) and U118 (human glioblastoma cell line) multicellular tumour spheroids (MTS) have been investigated by non-linear least square fitting of individual growth curves with the Gompertz growth equation and power spectrum analysis of residuals. Residuals were not randomly distributed around calculated growth trajectories. At least one main frequency was found for all analysed MTS growth curves, demonstrating the existence of time-dependent periodic fluctuations of MTS volume dimensions. Similar periodic oscillations of MTS volume dimensions were also observed for MTS generated using cloned 9L cells. However, we found significant differences in the growth kinetics of MTS obtained with cloned cells if compared to the growth kinetics of MTS obtained with polyclonal cells. Our findings demonstrate that the growth patterns of three-dimensional tumour cell cultures are more complex than has been previously predicted using traditional continuous growth models.     

Residual stress generation and necrosis formation in multi-cell tumour spheroids

We consider how cell proliferation and death generate residual stresses within a multi-cell tumour spheroid (MCTS). Previous work by Jones and co-workers [8] has shown that isotropic growth in a purely elastic MCTS produces growth induced stresses which eventually become unbounded, and hence are physically unrealistic. Since viscoelastic materials show stress relaxation under a fixed deformation we consider the effect of the addition of a small amount of viscosity to the elastic system by examining formation of equilibrium stress profiles within a Maxwell type viscoelastic MCTS. A model of necrosis formation based upon that proposed by Please and co-workers (see [16] [17] [18]) is then presented in which necrosis forms under conditions of adverse mechanical stress rather than in regions of extreme chemical stress as is usually assumed. The influence of rheology on necrosis formation is then investigated, and it is shown that the excessive stress generated in the purely elastic tumour can be relieved either by the addition of some viscosity to the system or by accounting for an inner necrotic interface with an appropriate stress boundary condition.     

Modelling the internalization of labelled cells in tumour spheroids

In this paper a mathematical model is developed to describe the migration of labelled particles within a multicell spheroid. In the model, spatial variations in cell proliferation and death create an internal velocity field which leads to redistribution of the labelled and unlabelled cells. By applying a range of numerical and analytical techniques to the model equations, it is possible to show that, whilst the speed with which the labelled cells migrate through the tumour is independent of the type of cells that are labelled, their limiting distribution depends crucially on whether inert polystyrene microspheres or live tumour cells are labelled. These predictions are shown to be in good qualitative agreement with independent experimental results.     

Multicellular tumour spheroids derived from human brain tumours

Multicellular tumour spheroids were prepared from a total of 46 human brain tumour biopsies by collagenase digestion and plating into agar coated flasks. Both primary malignant and secondary tumours formed spheroids with some correlation between the malignancy of tumour and the ability to undergo spheroid formation. The spheroids were capable of progressive growth, the rate of which was dependent, to some extent, on environmental conditions and was reflected by an increase in cell number within the spheroids. Spheroids prepared in this way may prove to be useful models for in vitro chemosensitivity and the general biology of brain tumours.     

Mathematical modelling of drug transport in tumour multicell spheroids and monolayer cultures

In this paper we adapt an avascular tumour growth model to compare the effects of drug application on multicell spheroids and on monolayer cultures. The model for the tumour is based on nutrient driven growth of a continuum of live cells, whose birth and death generates volume changes described by a velocity field. The drug is modelled as an externally applied, diffusible material capable of killing cells, both linear and Michaelis-Menten kinetics for drug action on cells being studied. Numerical solutions of the resulting system of partial differential equations for the multicell spheroid case are compared with closed form solutions of the monolayer case, particularly with respect to the effects on the cell kill of the drug dosage and of the duration of its application. The results show an enhanced survival rate in multicell spheroids compared to monolayer cultures, consistent with experimental observations, and indicate that the key factor determining this is drug penetration. An analysis of the large time tumour spheroid response to a continuously applied drug at fixed concentration reveals up to three stable large time solutions, namely the trivial solution (i.e. a dead tumour), a travelling wave (continuously growing tumour) and a sublinear growth case in which cells reach a pseudo-steady-state in the core. Each of these possibilities is formulated and studied, with the bifurcations between them being discussed. Numerical solutions reveal that the pseudo-steady-state solutions persist to a significantly higher drug dose than travelling wave solutions.     

Oxygen consumption in diapausing blastocysts

In vitro hybridization of seven pairs of genetically different murine cell has been demonstrated by the use of karyological markers, and pure cultures of these hybrids have been isolated. All somatic hybrids showed a progressive loss of chromosomes during this proliferation in vitro.     

Oxygen consumption rate in rotifers

The oxygen consumption rate (OCR) is a cumulative index of metabolic losses during aerobic metabolism. The generalized relationship of oxygen consumption rate (R, n1 O₂ ind^–1 h^–1) and dry body mass (M, µg) for rotifers is described by the equation: R = 9.15M^0.716. The level of rotifer metabolism is slightly lower than that of multicellular poikilothermic animals. Differences of OCR values in ontogenesis are substantial. Embryos and senile individuals are characterized by minimal OCR values. The OCR of oviparous females in the beginning of reproduction exceeds 2–3 times OCR values of juveniles. Differences in oxygen consumption intensity (OCI) are not so essential. OCR depends on food concentration. An increase of food concentration from 1.4 to 7.0 µg dry mass m1^–1 resulted in Brachionus calyciflorus in an OCR escalation of 2.5 times at 30°C, and 0.5 times at 25°C. Maximal OCR values occur at food concentration close to the saturation concentration for population growth rate. An exponential equation is adequate to describe R-t dependence for animals, long-term adapted to different constant temperatures (2 < Q₁₀ < 3). Acclimation effects observed during sharp temperature changes are determined by peculiarities of compensation reactions in species and separate populations. The formation of a zone of relative temperature independence of OCR (Q¹⁰ 1) at fluctuating temperature is observed. It is necessary to study enzymatic activities parallel to OCR and OCI measurements.     

The effect of combined heat and ultrasound on multicellular tumour spheroids

The effect of combined ultrasound and heat treatments on Chinese hamster multicellular spheroids of varying size was investigated using growth rate, single cell survival and ultrastructural damage as endpoints. Ultrasonic irradiation at 37 degrees C had no effect on the growth rate of 200-730 microns spheroids. Similarly there was no effect on the growth rate of 350 microns spheroids when irradiated during a 60 min exposure to 41.5 degrees C. However, spheroids of 200-700 mm diameter showed growth delay when held at 43 degrees C for 1 h. The effect was enhanced with concomitant ultrasound irradiation but was not dependent on spheroid size. When 200 and 400 microns spheroids held at 43 degrees C for 60 min were irradiated with different ultrasonic intensities a dose-dependent decrease in surviving fraction and a dose-dependent increase in growth delay was obtained. When surviving fraction was plotted as a function of growth delay a good correlation was obtained, suggesting that the combination of heat and ultrasound irradiation does not produce cytostasis in the surviving cells of either 200 or 400 microns spheroids. At the ultrastructural level increased cytoplasmic vacuolation was the only result of ultrasonic irradiation at 37 degrees C. Exposure to 43 degrees C for 60 min was required to elicit thermal damage. This took the form of membrane evagination at the spheroid surface, vacuolation of the cytoplasm, grouping of organelles around the periphery of the nucleus, and fragmentation of the nucleolus. These effects were enhanced with concomitant ultrasonic irradiation but other features were also noted, viz. disaggregation of polyribosomes, dilation of the rough endoplasmic reticulum and blebbing of the nuclear membrane. Damage was independent of spheroid size. These results are in agreement with previous data obtained from single-cell studies. Indicating that there is a non-thermal, non-cavitational component to the cell killing in multicellular spheroids resulting from combined heat and ultrasound treatment.     

Imaging growth dynamics of tumour spheroids using optical coherence tomography

Optical coherence tomography (OCT) was used to monitor the dynamics of tumour spheroid formation by the hanging drop method. In contrast to microscopy, the estimates obtained using OCT for the volume of the spheroid, were consistent with the measured changes in cell number as a function of time. The OCT images also revealed heterogeneous structures in the spheroids of ∼200 μm diameter. These corresponded to the necrotic regions identified by fluorescence of propidium iodide stained cells.     

Oscillations in growth of multicellular tumour spheroids: a revisited quantitative analysis

Objectives: Multicellular tumour spheroids (MTS) provide an important tool for study of the microscopic properties of solid tumours and their responses to therapy. Thus, observation of large‐scale volume oscillations in MTS, reported several years ago by two independent groups ( 1 , 2 ), in our opinion represent a remarkable discovery, particularly if this could promote careful investigation of the possible occurrence of volume oscillations of tumours ‘in vivo’. Materials and methods: Because of high background noise, quantitative analysis of properties of observed oscillations has not been possible in previous studies. Such an analysis can be now performed, thanks to a recently proposed approach, based on formalism of phenomenological universalities (PUN). Results: Results have provided unambiguous confirmation of the existence of MTS volume oscillations, and quantitative evaluation of their properties, for two tumour cell lines. Proof is based not only on quality of fitting of the experimental datasets, but also on determination of well‐defined values of frequency and amplitude of the oscillations for each line investigated, which would not be consistent with random fluctuation. Conclusions: Biological mechanisms, which can be directly responsible for observed oscillations, are proposed, which relates also to recent work on related topics. Further investigations, both at experimental and at modelling levels, are also suggested. Finally, from a methodological point of view, results obtained represent further confirmation of applicability and usefulness of the PUN approach.