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.     

Shedding of mitotic cells from the surface of multicell spheroids during growth

During the growth of EMT6/Ro mammary tumor multicell spheroids, a large number of cells are shed into the suspension medium. The rate of cell shedding was 218 cells per square millimeter of spheroid surface per hour, or up to 1.5% of the total spheroid cell content per hour. Shed cells had a clonogenic capacity equal to that of exonential monolayer cultures and were further characterized by volume distribution, mitotic index, flow cytoflurometry, and autoradiography. The results indicated that cells are released from the spheroid surface at mitosis, presumably due to a loosening of the cell-to-cell attachment during this cycle phase. These mitotic cells, when placed in monolayer culture, attached and grew synchronously with a cell cycle time of about 13 hours. Shed cells kept in suspension culture had a similar cell cycle time, but these cells reaggregated immediately after mitosis. The results indicated that cell shedding and reaggregation both occur near the time of mitosis and are intrinsic factors regulating the initiation and subsequent growth of multicell spheroids. Although these studies were done with spheroids cultured in vitro, shedding of mitotic cells may play an important role in the in vivo process of metastasis.     

Transient effects on the initial rate of oxygenation of red blood cells

The rate-controlling process in the oxygenation of red blood cells is investigated using a Roughton-like model for oxygen diffusion and reaction with hemoglobin. The mathematical equations describing the model are solved using two independent techniques, numerical inversions of the Laplace transform of the equations and numerical solutions via an implicit-explicit finite difference form of the equations. The model is used to re-examine previous theoretical models that incorporate either a red cell membrane that is resistive to oxygen diffusion or an unstirred layer of water surrounding the cell. Although both models have been postulated to be equivalent, the results of the computer simulations demonstrate significant differences between the two models in the rate of oxygenation of the red cells, depending upon the values chosen for the diffusion coefficient for O₂ in the membrane and the thickness of the water layer. The difference is apparently due to differences in the induction and transient periods of the water layer model relative to the membrane model.     

A new mathematical model for avascular tumour growth

The early development of solid tumours has been extensively studied, both experimentally via the multicellular spheroid assay, and theoretically using mathematical modelling. The vast majority of previous models apply specifically to multicell spheroids, which have a characteristic structure of a proliferating rim and a necrotic core, separated by a band of quiescent cells. Many previous models represent these as discrete layers, separated by moving boundaries. Here, the authors develop a new model, formulated in terms of continuum densities of proliferating, quiescent and necrotic cells, together with a generic nutrient/growth factor. The model is oriented towards an in vivo rather than in vitro setting, and crucially allows for nutrient supply from underlying tissue, which will arise in the two-dimensional setting of a tumour growing within an epithelium. In addition, the model involves a new representation of cell movement, which reflects contact inhibition of migration. Model solutions are able to reproduce the classic three layer structure familiar from multicellular spheroids, but also show that new behaviour can occur as a result of the nutrient supply from underlying tissue. The authors analyse these different solution types by approximate solution of the travelling wave equations, enabling a detailed classification of wave front solutions.     

Model of diffusion of oxygen to spheroids grown in stationary medium —I. Complete spherical symmetry

The use of spheroids as a tumor model has become commonplace since it was discovered that many cell lines can form spheroids when grown on a surface to which the cells cannot attach. This culture system complicates experiments which depend on oxygen supply because the oxygen concentration in the vicinity of a stationary spheroid has not been well defined. We present in this paper solutions to the oxygen diffusion equation for simple geometries: a spheroid in an infinite stationary medium and in a finite spherical stationary medium. Comparison of these solutions provides an estimate of the oxygen supply to a spheroid in a Petri dish. We show that typical spheroids can be expected to cause a substantial depletion of the oxygen in the nearby medium. Any disturbance of the medium or the spheroids will temporarily increase the oxygen supply. We provide a method for estimating the rate of return to equilibrium in the finite cases. These results indicate that the oxygen supply to stationary spheroids can be altered temporarily by small movements or changes in temperature which cause convection currents, or permanently by changes in the depth of the medium. Research supported by the Alberta Heritage Savings and Trust Fund-Applied Cancer Research. Research supported by the Natural Science and Engineering Research Council of Canada, Grant No. NSERC A 4823.     

Characterization of a carbocyanine derivative as a fluorescent penetration probe

The fluorescent carbocyanine dye 3,3-diheptyloxycarbocyanine [DiOC7(3)], originally described as a membrane potential probe, penetrates poorly into multicell spheroids. Since the dye is retained in the cells following spheroid disaggregation, cells can be selected from different depths within the spheroid using fluorescence-activated cell sorting. Characterization of the binding kinetics, stability, and toxicity of this probe were undertaken, and intercompared with Hoechst 33342. The optimum drug dose for achieving good separation of internal and external cells of spheroids is about tenfold lower than for Hoechst 33342, and like Hoechst, DiOC7(3) is toxic at concentrations at least tenfold higher than those required to produce a good gradient for cell separation. When cells are removed from the stain, cellular fluorescence decreases to half the initial intensity within 2 hours; however, unlike Hoechst, the carbocyanine dye does not transfer between cells.     

Use of Hoechst 33342 for cell selection from multicell systems

Hoechst 33342 staining of multicell spheroids, three-dimensional cell clusters grown in vitro, results in a marked gradient of cellular fluorescent intensities inward from the spheroid periphery. The penetration of the dye is concentration and time dependent, so staining can be coupled with fluorescence activated cell sorting techniques to allow disaggregated single cells to be sorted or selected according to their degree of staining and therefore their depth within the spheroid. We have found the staining procedure to be highly reproducible, and to result in minimal toxicity even to the more brightly staining external cells. Comparison of this technique with others for cell selection suggests that increased resolution is available with the Hoechst technique.     

Analytical solutions for transient thermal behavior in biological systems

Analytical solutions are presented for transient heat conduction in biological media. General boundary conditions and internal sources varied in both spatial and time variables are considered, thus, solutions for many special cases can be obtained with ease from the general solutions presented in this analysis.     

The role of stress in the growth of a multicell spheroid

Rather recent experimental results demonstrate the non–negligible role of mechanical stress in the growth of a multicell spheroid. In this paper we discuss a theoretical framework for volumetric growth suitable for modeling the growth of soft tissues exhibiting the properties of a solid. After a proper kinematic decomposition, balance equations for mass, momentum and energy are discussed together with constitutive relationships. The mathematical model is then applied to avascular tumor growth. We show by numerical simulation that, under assumption of spherical symmetry, the mathematical model is able to reproduce the experimental data with a satisfying qualitative agreement.     

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.     

Closed loop control of oxgenation and ventilation

Closed loop control of oxygenation and ventilation during mechanical ventilatory support is essential for remote medical care in an austere environment. Closed loop control allows for expert systems to provide the current standard of care in the absence of on-site expertise. Ventilation may be controlled by simple systems incorporating patient height or by advanced systems incorporating measurements of end-tidal carbon dioxide (ETCO2) and pulmonary impedance. Oxygenation may be controlled by adjustments of inspired oxygen concentrations (FIO2) and positive end-expiratory pressure (PEEP) using pulse oximetry (SpO2) as the input. Control of oxygenation can prevent hypoxemia and has the potential to reduce oxygen requirements. A double closed loop system of oxygenation control including control of FIO2 via SpO2 and control of oxygen generation by a portable oxygen generator (POG) based on FIO2 and minute ventilation (VE) promises safety and efficiency. Remote control of ventilation and oxygenation is possible using existing technology.     

Is there a relationship between hypoxia,contact resistance,and intercellular communication?

Summary This investigation addresses the shape of radiation survival curves of cells cultured as multicell spheroids. It is shown that spheroids of cells capable of intercellular communication by gap-junctions display survival curves lacking a radioresistant fraction of hypoxic cells. Compared to the corresponding monolayers, these spheroid survival curves exhibit a uniform increase in radioresistance due to the contact effect. In contrast, biphasic survival curves indicative of hypoxic cells are obtained with non-communicating spheroids, however, without indication of a contact effect. Evidence is presented that this relationship between intercellular communication, hypoxia, and contact effect may possibly also hold for survival curves of solid tumors.     

Spatio-Temporal Dynamics of Hypoxia during Radiotherapy

Tumour hypoxia plays a pivotal role in cancer therapy for most therapeutic approaches from radiotherapy to immunotherapy. The detailed and accurate knowledge of the oxygen distribution in a tumour is necessary in order to determine the right treatment strategy. Still, due to the limited spatial and temporal resolution of imaging methods as well as lacking fundamental understanding of internal oxygenation dynamics in tumours, the precise oxygen distribution map is rarely available for treatment planing. We employ an agent-based in silico tumour spheroid model in order to study the complex, localized and fast oxygen dynamics in tumour micro-regions which are induced by radiotherapy. A lattice-free, 3D, agent-based approach for cell representation is coupled with a high-resolution diffusion solver that includes a tissue density-dependent diffusion coefficient. This allows us to assess the space- and time-resolved reoxygenation response of a small subvolume of tumour tissue in response to radiotherapy. In response to irradiation the tumour nodule exhibits characteristic reoxygenation and re-depletion dynamics which we resolve with high spatio-temporal resolution. The reoxygenation follows specific timings, which should be respected in treatment in order to maximise the use of the oxygen enhancement effects. Oxygen dynamics within the tumour create windows of opportunity for the use of adjuvant chemotherapeutica and hypoxia-activated drugs. Overall, we show that by using modelling it is possible to follow the oxygenation dynamics beyond common resolution limits and predict beneficial strategies for therapy and in vitro verification. Models of cell cycle and oxygen dynamics in tumours should in the future be combined with imaging techniques, to allow for a systematic experimental study of possible improved schedules and to ultimately extend the reach of oxygenation monitoring available in clinical treatment.     

A reduction in the in situ rates of oxygen and glucose consumption of cells in EMT6/Ro spheroids during growth

The rates of consumption of oxygen and glucose by EMT6/Ro cells in multicellular spheroids were measured at various times during normal growth. In situ spheroid cellular consumption rates were similar to those of exponentially growing single cells up to a spheroid diameter of 150 micron. Further growth resulted in decreases in the rates of both oxygen and glucose consumption which were correlated with the increase in spheroid diameter and cell number. At a diameter of 1300 micron, both rates of cellular consumption had decreased by a factor of 2.5. The rates of consumption per unit of nonnecrotic spheroid volume decreased in a similar manner. Measurements with single cells demonstrated that the rate of oxygen consumption was coupled with glucose concentration, and vice versa. The rates of consumption for cells dissociated from small spheroids indicated that there was some effect of the spheroid environment. As the spheroids grew, however, association in the spheroid structure accounted for a smaller proportion of the total observed reduction in the rates of nutrient consumption. The presence of central necrosis also appeared to have no effect on the rates of consumption of these nutrients. Spheroid-derived cells showed a decrease in cell volume with growth as the cells accumulated in a quiescent state. Measurements with single cells demonstrated that oxygen and glucose consumption were correlated with cell volume and with the development of nonproliferating cells. We conclude that the observed decrease in oxygen and glucose consumption with growth in spheroids is largely due to the progressive accumulation of cells in a quiescent state characterized by an inherently lower cellular rate of nutrient utilization.     

Quantitative description of blood oxygenation process in oxygenator for extracorporal circulation

Based on conceptions and assumptions concerning the blood oxygenation process, some fundamental quantitative relations for red blood corpuscle oxygenation and blood oxygenation kinetics are presented. A distribution function is introduced expressing the probability density for the occurrence of a red blood cell with a specific oxygen content. By means of a kinetic equation deduced the distribution function is connected with spatial distribution of oxygen pressure and with blood flow rate. For the given initial conditions the kinetic equation is solved for a one-dimensional case, and this solution is applied to a generalized oxygenator in a stationary case. The generalized oxygenator presents a system of through-flow elements in which blood flows and contacts oxygen. Each of the through-flow elements is characterized by length, blood flow rate, probability of red blood corpuscle entry and by a quantity depending on oxygen pressure. Results obtained for the generalized oxygenator are then applied to a disc oxygenator with certain presumptions concerning blood oxygen saturation at the system's output expressed in dependence on geometry and performance conditions. Stress is laid upon the influence of blood flow in the oxygenator, on oxygenation; and two extreme cases are compared—series and parallel types of disc oxygenator.     

Recovery from potentially lethal damage and recruitment time of noncycling clonogenic cells in 9L confluent monolayers and spheroids

Cells that have been grown as multicell tumor spheroids exhibit radioresistance compared to the same cells grown in monolayers. Comparison of potentially lethal damage (PLD) repair and its kinetics was made between 9L cells grown as spheroids and confluent monolayers. Survival curves of cells plated immediately after irradiation showed the typical radioresistance associated with spheroid culture compared to plateau-phase monolayers. The dose-modification factor for spheroid cell survival is 1.44. Postirradiation incubations in normal phosphate-buffered saline (PBS), conditioned media, or 0.5 M NaCl in PBS reduced the differences in radiosensitivity between the two culture conditions. Postirradiation treatment in PBS or conditioned medium promoted repair of potentially lethal damage, and 0.5 M NaCl prevented the removal of PLD and allowed the fixation of damage resulting in lower survival. Survival of spheroid and monolayer cells after hypertonic NaCl treatment was identical. NaCl treatment reduced Do more than it did the shoulder (Dq) of the survival curve. PLD repair kinetics measured after postirradiation incubation in PBS followed by hypertonic NaCl treatment was the same for spheroids and for plateau-phase monolayers. The kinetics of PLD repair indicates a biphasic phenomenon. There is an initial fast component with a repair half-time of 7.9 min and a slow component with a repair half-time of 56.6 min. Most of the damage (59%) is repaired slowly. Since the repair capacity and kinetics are the same for spheroids and monolayers, the radioresistance of spheroids cannot be explained on this basis. Evidence indicates that the time to return from a Go (noncycling G1 cells) state to a proliferative state (recruitment) for cells from confluent monolayers and from spheroids after dissociation by protease treatment may be the most important determinant of the degree of PLD repair that occurs. Growth curves and flow cytometry cell cycle analysis indicate that spheroid cells have a lag period for reentry into a proliferative state. Since plating efficiency remains high and unchanging during this period, one cannot account for the delay on the basis of the existence of a large fraction of Go cells which are not potentially clonogenic. The cell cycle progression begins in 6-8 h for monolayer cells and in 14-15 h for spheroids. It is hypothesized that the slower reentry of spheroid cells into a cycling phase allows more time for repair than for the rapidly proliferating monolayer cells.     

Biofilm history and oxygen availability interact to affect habitat selection in a marine invertebrate

In marine systems, oxygen availability varies at small temporal and spatial scales, such that current oxygen levels may not reflect conditions of the past. Different studies have shown that marine invertebrate larvae can select settlement sites based on local oxygen levels and oxygenation history of the biofilm, but no study has examined the interaction of both. The influence of normoxic and hypoxic water and oxygenation history of biofilms on pre-settlement behavior and settlement of the bryozoan Bugula neritina was tested. Larvae used cues in a hierarchical way: the oxygen levels in the water prime larvae to respond, the response to different biofilms is contingent on oxygen levels in the water. When oxygen levels varied throughout biofilm formation, larvae responded differently depending on the history of the biofilm. It appears that B. neritina larvae integrate cues about current and historical oxygen levels to select the appropriate microhabitat and maximize their fitness.     

In situ oxygen consumption rates of cells in V-79 multicellular spheroids during growth

The rate of consumption of oxygen by V-79 cells in multicellular spheroids was measured as a function of the spheroid diameter. In situ consumption was equal to that of exponentially growing cells for spheroids less than 200 micron in diameter. The rate of oxygen consumption decreased for cells in spheroids between 200 and 400 micron diameter to a value one-fourth the initial, then remained constant with further spheroid growth. Comparison of consumption rates for spheroid-derived cells before and after dissociation from the spheroid structure indicated that the spheroid microenvironment accounted for only 20% of the change in oxygen consumption rate. Cell-cell contact, cell packing, and cell volume were not critical parameters. Plateau-phase cells had a fivefold lower rate of oxygen consumption than exponential cells, and it is postulated that the spheroid quiescent cell population accounts for a large part of the intrinsic alteration in oxygen consumption of cells in spheroids. Some other mechanism must be involved in the regulation of cellular oxygen consumption in V-79 spheroids to account for the remainder of the reduction observed in this system.     

The interaction of salts, amides, and water.

Infrared and Raman spectra are presented for salt solutions in N-methyl formamide and N,N'-dimethylformamide. Viscosities are reported for many of these solutions. Spectroscopic and viscosity data are also given for amide-salt solutions containing water. The three-component systems exhibit a hydrogen-bonding strength of water proton greater than amide proton, and an acceptor strength of Cl- greater than amide carbonyl oxygen greater than water oxygen. The salt cation is deduced by means of viscosity measurements to interact strongly with the amide carbonyl oxygen, even in the presence of appreciable quantities of water. Association of amides by hydrogen bonding through halide ions is also indicated.     

Closing and opening of gap junction pores between two- and threedimensionally cultured tumor cells

Intercellular signal transfer via gap junction pores in cultured multicell spheroids of BICR/M1R-K cells decreases with increasing spheroid age. In two days old spheroids the pores allow passage of Lucifer yellow molecules. Two days later, this fluorescent dye is retained in the injected cell even though the cells are still electrically coupled. Gap junction plaques of considerable size are still found in 9 days old spheroids, when the cells are completely uncoupled. The same cells growing as monolayer cultures do not exhibit such a gradual closing of their gap junction pores: Their coupling is established at first cell contact, probably by a gradual opening of the pores, which remain open even up to 9 days in culture.Based on material presented at the Symposium Intercellular Communication Stuttgart, September 16–17, 1982