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Modelling extravascular drug penetration using multilayered cell cultures Kyle, Alastair Hugh 1999

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MODELLING EXTRAVASCULAR DRUG PENETRATION USING MULTILAYERED CELL CULTURES by A L A S T A I R H U G H K Y L E B . S c , M c G i l l Univers i ty A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Department of Physics) W e accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1999 © Alastair H u g h K y l e , 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ffi^SlCs The University of British Columbia Vancouver, Canada Date J > C A ^ ? . [<W\ DE-6 (2788) Abstract This thesis describes experiments using multilayered cell cultures (MCCs) to examine the ability of anticancer drugs to penetrate the extravascular compartment of tumour tissue. An MCC consists of cells grown on a permeable plastic membrane to form a disc-like, three-dimensional tissue culture 10-30 layers in thickness. MCCs are similar to the spheroidal cell culture model in that they rnirnic the cell environment of tumour tissue better than monolayer cultures in terms of cell contact effects, oxygen and nutrient gradients and the extracellular matrix content. The geometry of MCCs allows for simple experiments in which the cultures are used to separate two reservoirs of a diffusion apparatus and the flux of a drug from one reservoir, through the culture and into the second reservoir, is determined. When coupled with detailed knowledge of the tissue environment and drug-cell interactions, the analysis of flux data from such experiments provides quantitative information on the rate of diffusion and metabolism, or binding, of the drug within the tissue. Such information can be used as a comparative measure between analogues within a given family of drugs or be directly applied to predict the ability of a drug to distribute into tumour tissue. The flux of representative compounds from several classes of anticancer drugs including hypoxic cell radiosensitizers, hypoxic cell cytotoxins and anthracyclines was studied and results are reported in this thesis. In addition to drug flux studies, the cell environment of MCCs was characterised using electrical impedance spectroscopy as well as through experiments involving the use of radiolabeled inulin. ii Table of Contents A B S T R A C T II LIST O F F I G U R E S V I LIST O F T A B L E S VIII LIST O F A C R O N Y M S IX A C K N O W L E D G E M E N T S X 1. I N T R O D U C T I O N 1 1.1 D R U G DELIVERY WITHIN SOLID TUMOURS l 1.2 OBJECTIVE OF T H E STUDY : < 3 Flux experiments & Mathematical Modelling 4 Characterisation of MCC cell environment : 5 1.3 O U T L I N E OF EXPERIMENTS 7 Nitroimidazoles 7 Tirapazamine 8 Doxorubicin 9 Characterisation of the Cell Environment • 10 1.4 SUMMARY 10 2. M O D E L L I N G D R U G F L U X W I T H I N M C C S 11 2.1 INTRODUCTION 11 Fickean Diffusion 12 2.2 D R U G KINETICS IN M C C S : O N E COMPARTMENT M O D E L 14 Modelling Diffusion 15 Modelling Metabolism 15 Tirapazamine Kinetics Within MCCs • 16 2.3 D R U G KINETICS IN M C C S : TWO COMPARTMENT M O D E L 17 Tissue Tortuosity 17 Diffusion & Cellular Uptake 18 Drug Binding 18 Doxorubicin Kinetics Within MCCs 20 2.4 BOUNDARY CONDITIONS & ADDITIONAL BARRIERS TO DIFFUSION ..25 Diffusion through additional layers 25 Flux into and out of the reservoirs 25 Flux between UBLs and the tissue 26 2.5 DE-DIMENSIONALIZATION OF EQUATIONS 26 2.6 DLSCRETIZING T H E EQUATIONS 29 Discrete Tissue Equations 31 Discrete Insert/UBL Equations 32 Discrete L.H.S. Boundary Condition 33 Discrete R.H.S. Boundary Condition 34 Discrete Condition for the Tissue-UBLB Boundary 35 Discrete UBLA-Tissue Boundary Condition 38 2.7 INTEGRATING THE SYSTEM 39 Initial Conditions 39 Discretization Step Sizes 39 Solving the Equations 39 2.8 VERIFICATION OF NUMERICAL INTEGRATION M E T H O D S 40 i i i Comparison with Analytical solutions 40 Modification of discretization step size 42 3. S I M U L A T I N G D R U G D I S T R I B U T I O N IN A T U M O U R C O R D 49 3.1 DRUGKINETICSUSINGCYLINDRICAL GEOMETRY 49 Enzyme Metabolism Model 50 Doxorubicin Model 50 3.2 BOUNDARY CONDITIONS 50 Blood Vessel / Tissue Boundary 50 Boundary at the outer edge of the tissue 51 3.3 DE-DIMENSIONALIZATION OF EQUATIONS 51 3.4 DISCRETTZING THE EQUATIONS 53 Discrete Tissue Equations 54 Discrete Blood Vessel / Tissue Boundary Condition 55 Discrete Boundary Condition at the Outer Edge of the Cord 56 3.5 INTEGRATING THE SYSTEM 57 Initial Conditions 57 Discretization Step Sizes 58 Solving the Equations 58 4. E L E C T R I C A L I M P E D A N C E S P E C T R O S C O P Y T H E O R Y 59 4.1 M C C IMPEDANCE MODELLING 60 4.2 ESTIMATING TISSUE TORTUOSITY AND THE FRACTION OF EXTRACELLULAR SPACE IN MCCs 62 5. M A T E R I A L S & M E T H O D S 64 5.1 CELL CULTURE 64 Monolayers 64 MCCs 64 5.2 METHODS FOR DIFFUSION EXPERIMENTS 65 Drugs 65 Experimental Set-up and Apparatus 65 HPLC Analysis 66 Scintillation Counting 70 Fluorescence Imaging 70 Fitting Drug Flux Data • 70 Permeability of the Plastic Membrane & Unstirred Boundary Layers 70 Net Permeability of the Plastic Insert to Drugs 73 5.3 METHODS FOR E IS EXPERIMENTS 75 Impedance Measurements 75 Calibration 75 Cell Factor 76 5.4 DETERMINING M C C THICKNESS 76 5.5 DETERMINING THE FRACTION OF EXTRACELLULAR SPACE IN M C C S 78 6. EIS & R A D I O L A B E L L E D I N U L I N : E X P E R I M E N T A L R E S U L T S 79 6.1 E I S EXPERIMENTS 79 SiHa MCCs 80 V79 & V79/DOX MCCs 81 6.2 INULIN EXPERIMENTS 82 Measurement of the Fraction of Extracellular Space Within MCCs 83 Measurement of the Tortuosity of the Extracellular Space of MCCs 83 6.3 COMPARISON OF E IS & INULIN RESULTS 83 6.4 DISCUSSION 84 7. N I T R O I M I D A Z O L E S : F L U X E X P E R I M E N T S & A N A L Y S I S 87 iv 8. TIRAPAZAMINE: EXPERIMENTAL RESULTS & SIMULATIONS 90 8.1 TIRAPAZAMINE FLUX EXPERIMENTS 90 Oxic Flux Experiments 90 Hypoxic Flux Experiments 91 8.2 T U M O U R CORD SIMULATIONS 91 Boundary conditions 92 Tumour cord oxygen distribution 92 Oxygen Inhibition of Tirapazamine Metabolism 93 Relating tirapazamine exposure to cell survival 93 Simulation of tirapazamine distribution 94 8.3 DISCUSSION 96 9. DOXORUBICIN: EXPERIMENTAL RESULTS & SIMULATIONS 98 9.1 F L U X EXPERIMENTS 98 V79/DOX Analysis 99 V79 & SiHa Analysis 202 9.2 FLUORESCENCE IMAGING WITHIN M C C S 105 9.3 TUMOUR CORD SIMULATIONS 106 Blood Pharmacokinetics 107 Simulation of Doxorubicin Tumour Cord Distribution 107 9.4 DISCUSSION 110 10. FUTURE DIRECTIONS 114 BIBLIOGRAPHY 116 APPENDIX I: DIVISION OF LABOUR 121 APPENDIX II: INCUBATION SYSTEM 122 TEMPERATURE CONTROL 123 STIRRING SYSTEM 124 v List of Figures Chapter 1 Figure 1.1 H&E stained section of a SiHa mouse xenograft solid tumour 1 Figure 1.2 Overview of the process by which Potential Anticaner Drugs Are Screened 3 Chapter 2 Figure 2.1 Fickean diffusion through tissue 13 Figure 2.2 Two compartment modelling of drug kinetics 17 Figure 2.3 Modelling drug binding 19 Figure 2.4 Relation between free and bound drug described by equation (2.15) 19 Figure 2.5 Modelling of doxorubicin-tissue interactions within an MCC 20 Figure 2.6 Diagram of the barriers to diffusion encountered by a drug 25 Figure 2.7 Diagram of the spatial discretization scheme 30 Figure 2.8 Comparison of analytical and numerical solutions: one-compartment diffusion 44 Figure 2.9 Comparison of analytical and numerical solutions: one-compartment diffusion and drug breakdown in the growth medium 45 Figure 2.10 Comparison of numerical integration results: diffusion only 46 Figure 2.11 Comparison of numerical integration results using different spatial discretization: one compartment model, diffusion and metabolism 47 Figure 2.12 Comparison of numerical integration results using different spatial discretization: two-compartment doxorubicin model 48 Chapter 3 Figure 3.1 Diagram of the corded architecture used in simulating drug kinetics ....49 Figure 3.2 Diagram of the spatial discretization scheme used for the cord model 53 Figure 4.1 (a) Illustration of a simplified cell environment and (b) the equivalent electrical circuit used in the interpretation of electrical impedance measurements 59 Chapter 4 Figure 4.2 Schematic of a typical, four-electrode impedance measurement set-up used for biological material 59 Chapter 5 Figure 5.1 (a) Photo and (b) sketch of MCC growth box 64 Figure 5.2 H&E stained cryostat section of an 8 day old SiHa MCC 65 Figure 5.3 (a) photo and (b) diagram of diffusion apparatus 66 Figure 5.4 Calibration curves for HPLC response to Etanidazole and Misonidazole 67 Figure 5.5 Calibration curves for HPLC response to Etanidazole and Pimonidazole 67 Figure 5.6 Calibration curves for HPLC response to 40 pi injections of Tirapazamine 68 Figure 5.7 Calibration curves for HPLC response to Doxorubicin using 30 ul injections 69 Figure 5.8 Diagram of the barriers a drug must pass through when going from one reservoir to the other for a culture insert with no cells 71 Figure 5.9 Plot of permeability versus stirring speed for diffusion of tritiated water through a plastic insert with no cells ..71 Figure 5.10 Flux data from diffusion through blank inserts 73 Figure 5.11 Plot of Pnet versus molecular weight obtained from a series of flux experiments of molecules through blank inserts 74 Figure 5.12 Diagram of the impedance measurement cell used for EIS measurements 75 Figure 5.13 Calibration of the EIS system 75 Figure 5.14 Typical data for diffusion of tritiated water (HTO) 77 Figure 5.15 Plot of SiHa MCC thickness versus the inverse of permeability to HTO 77 Chapter 6 Figure 6.1 Typical results from measurement of impedance magnitude and phase as a function of frequency 79 Figure 6.2 Results of analysis of impedance measurements from a series of experiments using SiHa MCCs 80 vi Figure 6.3 Results of analysis of impedance measurements from a series of experiments using V79 and V79/DOX MCCs 81 Figure 6.4 Extra-cellular space as a function of MCC thickness determined via equilibrium C14-inulin measurements for SiHa, V79 and V79/ADR MCCs 82 Figure 6.5 Estimate of the tortuosity factor, X, within SiHa, V79, and V79/DOX, MCCs 83 Figure 6.6 Comparison of the combined effect of tissue tortuosity and the fraction of extracellular space determined from electrical impedance and C14-inulin diffusion measurements 84 Chapter 7 Figure 7.1 Data from flux of four nitroimidazoles through 135 um thick SiHa MCCs 89 Chapter 8 Figure 8.1 Diffusion of tirapazamine under oxic conditions (p02 = 40 rnmHg) 91 Figure 8.2 Diffusion of tirapazamine under hypoxic conditions 91 Figure 8.3 (a) Variation of p02 and the rate of tirapazamine metabolism, with distance from a blood vessel 92 Figure 8.4 Simulation of tirapazamine distribution within a tumour cord 94 Figure 8.5 Simulations in which the effect of oxygen on the inhibition of tirapazamine metabolism is varied 95 Chapter 9 Figure 9.1 Typical doxorubicin flux data from donating and receiving reservoirs for experiments using V79/DOX and V79 MCCs 98 Figure 9.2 Diffusion coefficient estimates obtained from the analysis of a series of flux experiments using V79/DOX MCCs that ranged from 90 to 140 um in thickness 100 Figure 9.3 Estimation of the extracellular binding parameter from V79/DOX M C C flux data 100 Figure 9.4 Comparison of doxorubicin fluorescence distribution within M C C cryostat sections and the predicted distribution as obtained from flux analysis results 103 Figure 9.5 Plot of the rate of diffusion of doxorubicin within V79 and SiHa MCCs as predicted by the V79/DOX analysis and knowledge of tissue tortuosity 104 Figure 9.6 Results for estimation of the rate of cellular uptake of doxorubicin within (a) V79 and (b) SiHa MCCs that ranged from 100 to 250 um in thickness 105 Figure 9.7 Results for simulation of doxorubicin penetration into a tumour cord, the sheath of cells surrounding a blood vessel 108 Appendix II Figure A . l Incubator used for growing MCCs and for the execution of drug flux experiments 123 Figure A.2 Photo of the stir bar control system 123 Figure A.3 Schematic of the current source circuit used to power the incubator heaters 124 Figure A.4 Outline of stirrer control system 125 Figure A.5 Circuit diagram of Clock 1 - the timer which controls the stir speed of the system 125 Figure A.6 Circuit diagram of the Counter module of the stir bar controller 126 Figure A.7 Overview of the method used to distribute power to the electromagnetic coils 126 Figure A.8 Diagram of a one focal point stir plate 127 Figure A.9 Diagram of a four focal point stir plate 127 Figure A.10 Diagram of the stir speed display module circuit 128 vii List of Tables Table 1.1 Outline of the experiments that are presented in this thesis 7 Table 2.1 Model variables used to describe drug flux through MCCs 28 Table 2.2 Model parameters used to describe drug flux through MCCs 29 Table 3.1 Model variables used to describe drug flux within a tumour cord 52 Table 3.2 Model parameters used to describe drug flux within a tumour cord 52 Table 7.1 Summary of results from analysis of flux of several nitroimidazoles through SiHa MCCs. .88 Table 9.1 Comparison of model parameters for 150 um thick SiHa, V79 and V79/DOX MCCs 105 Table A . l Summary of the division of labour for the work presented in this thesis 122 List of Acronyms EIS Electrical Impedance Spectroscopy HPLC High Pressure Liquid Chromatography HTO Tritiated Water (Hydrogen, Tritium, Oxygen) MCC Multilayered Cell Culture PBS Phosphate Buffered Saline PDA Photo-Diode Array PDE Partial Differential Equation UBL Unstirred Boundary Layer Acknowledgements The research presented in this thesis was carried out under the supervision of Andrew Minchinton. Numerous people made contributions to the work presented here. Carmel Chan did many of the doxorubicin flux experiments presented in Chapter 9, as well ailturing the V79 and V79/DOX MCCs used in the experiments. The design of the experimental apparati was done in conjunction with Andrew, with help during construction from machinists Michel Rocher and Colin Porter of the B.C. Cancer Agency. Karen Fryer did preliminary work on the MCC project in Andrew's lab before my arrival and introduced me to the lab. Helge Grosshans worked in the lab on a MCC cytotoxicity assay and contributed useful discussions as well as the photos of the growth box and diffusion apparatus shown in Chapter 5. Darren Sutherland, grew and sectioned the mouse tumour shown in Chapter 1. Ralph Durand (BCCRC) contributed the V79 and V79/DOX cell lines, the camera set-up used in obtaining the fluorescence images shown in Chapter 9 and edited the tirapazamine and EIS manuscripts that were submitted to journals for publication. Bill Wilson (Auckland, New Zealand), who has also developed an MCC model, contributed much advice pertaining to experimental methods. Kevin Hicks, in Bill Wilson's lab, also contributed advice on experiments and mathematical modelling. My family contributed both spiritual and economic support. In addition my step-father, Michael Mackey gave helpful scientific advice and edited parts of this thesis. This work was supported in part by grants from BCHRF and NCIC. 1. Introduction 1.1 Drug Delivery Within Solid Tumours The heterogeneous structure of solid tumour tissue poses a significant barrier to the delivery of anticancer drugs to the malignant cells that are contained within it. Deregulated growth of cancer cells results in the expansion of the extravascular compartment of tumour tissue, which acts to increase the distance between blood vessels to the point where the delivery of oxygen and nutrients by the existing vascular network is maximally taxed. Extravascular expansion may also cause elongation and compression of blood vessels, resulting in restricted or intermittent blood flow (Dewhirst, et al., 1996). As the cancer cells continue to divide and push outiying cells further away from the blood vessels, regions of necrosis are formed. This type of tissue structure is shown in Figure 1.1. Tumour cells are seen to be aligned as cords, which surround functioning blood vessels. Surrounding each cord of cells is a necrotic region, which delimits the volume of hving tissue that each blood vessel can support. An optimally tumour cords Figure 1.1 H & E stained section of a S iHa mouse xenograft solid tumour. Functioning blood vessels, visible by the red haemoglobin, are surrounded by layers of cancer cells. The necrotic regions interspersed wi th viable cells are the product of expansion of the extravascular compartment, caused by deregulated cell division. A n optimally effective anticancer drug would be able to reach therapeutic levels at all clonogenic cancer cells. 1 effective anticancer drug would be able to reach therapeutic levels at all the cancer cells that lie within the tumour. To do this it would have to overcome the effects of increased diffusion distance, geometric dilution, metabolism, binding and restricted/intermittent blood flow as it travels away from blood vessels and into the tissue (for a review see (Jain, 1994, Jain, 1989). Despite knowledge of these factors, the capacity of drugs to penetrate the extravascular compartment of tumours has received little attention, in part due to the inherent difficulty in directly evaluating drug penetration within tumour tissue. While almost forty years ago the stain lissamine green was observed to distribute poorly within tumour tissue (Goldacre and Sylven, 1962), it has only been recently that cell sorting techniques have been applied to evaluate drug induced cell kill as a function of distance from functional blood vessels within a solid tumour (Chaplin, et al., 1985, Durand and Olive, 1997). While the search for new anticancer drugs often involves extensive screening of analogues to find drugs that are effective at ]<illing cells grown as monolayers, little testing is done on how these same variations act to determine the delivery of each drug within tissue. Most commonly, analogues that have been identified as having highest cytotoxicity against monolayer cultures are advanced directly to testing of toxicity, pharmacokinetics and tumour growth delay in mice. However, the success rate in bringing compounds that have been identified as good candidates from in vitro screening to being successful therapeutic agents has been almost as poor as that of medieval alchemy. In response to this, three dimensional tissue cultures are now slowly being applied as a secondary in vitro screening stage to model drug penetration within the extravascular compartment of tumour tissue, see Figure 1.2. Spheroidal cell cultures have been shown to reproduce many in vivo characteristics of solid tumours such as the existence of oxygen and pH gradients (Acker, et al., 1987), the formation of necrotic regions (Sutherland and Durand, 1976), extracellular fraction and composition (Durand, 1980, Freyer and Sutherland, 1983, Glimelius, et al., 1988), increased drug and radioresistance relative to monolayers (Durand, 1980, Wibe, 1980), etc. Spheroids have been used to evaluate the penetration of numerous cytotoxic agents, usually through visualisation of drug fluorescence from cryostat sections (Nederman, et al., 1988, Nederman, et al., 1981) or though determination of the spatial distribution of cell kill or drug uptake from dissociated cultures (Durand, 1981, Durand, 1990, Durand and Olive, 1992, Kerr, et al., 1987, Olive and Durand, 2 1994). Recently, a multilayered cell culture (MCC) model has been developed to complement the spheroid model. The MCC model comprises a disc of cells grown on a permeable membrane. Its culture and characteristics were originally described in two recent publications (Cowan, et al., 1996, Minchinton, et al., 1997). The MCC geometry facilitates the measurement of drug flux through its tissue, thereby allowing for the quantification of drug kinetics within tumour-like tissue. In vitro models Animal models Humans 3-D tissue cultures Figure 1.2 Overview of the process by which compounds are screened for their potential as anticancer agents. Most commonly, compounds that are found to be effective at killing cells grown as monolayers are advanced directly to animal testing. Three-dimensional cell cultures such as multicellular spheroids and multilayered cell cultures, are increasingly being applied to model drug penetration within the extravascular compartment of tumour tissue and have the potential to serve as a powerful tool in the primary stage of drug screening. 1.2 Objective of the Study The objective of this study was to develop the MCC model for use in characterising the ability of anticancer drugs to penetrate the extravascular compartment of tumour tissue. This involved the development of methodology for the execution and analysis of drug flux experiments as well as the development of methodology to assess MCC tissue characteristics such as thickness, fraction of extracellular space and tortuosity. The techniques for flux experiments were applied to assess the penetrative ability of representative compounds from three classes of anticancer drugs: nitroimidazole hypoxic cell radiosensitizers (Azomycin, Misonidazole, Etanidazole and Pimonidazole), bioreductive hypoxic cell cytotoxins 3 (Tirapazamine) and anthracyclines (Doxorubicin). The techniques for characterisation of MCC tissue were applied to determine characteristics of MCCs grown from three types of cell: SiHa v- cells (Human squamous cell carcinoma), V 7 9 cells (Chinese hamster fibroblast) and V79 /DOX cells, cultivated to exhibit multidrug resistance. Flux experiments & Mathematical Modelling Drug flux through MCCs was measured using a dual-well diffusion apparatus in which the MCC was oriented to separate two well-stirred, media filled reservoirs. By introducing the drug of interest to one reservoir, the time course of change to drug concentration in each reservoir could then be followed via HPLC analysis of reservoir samples. Experiments were carried out using a custom built incubator which regulated temperature, gassing and stirring within the diffusion apparatus. Analysis of drug concentration data from flux experiments was done via mathematical modelling of the drug kinetics within the MCCs. Modelling could account for drug diffusion, cellular uptake, metabolism and binding within the tissue. Partial differential equations were used to model the relations between drug concentration and the rate of flux, uptake, metabolism and bound drug at each point within the tissue. By comparing the time course change of drug concentration within the two reservoirs with that predicted by the mathematical model, estimates of model parameters related to diffusion, uptake, etc. within the tissue were determined. Mathematically, the extra- and intracellular space of the MCCs were modelled using either one or two compartments. In the case where the drug-tissue interactions were not well characterised, an empirical approach to modelling was implemented. The extra- and intracellular spaces were treated as a single, homogeneous compartment and the model described the apparent rate of diffusion, metabolism, etc. within the tissue. While this approach allowed for a very simple analysis of drug flux through MCCs, it neglected the process of cellular uptake and the results had to be interpreted carefully. In the case where the drug-tissue interactions were well characterised, drug kinetics could be modelled using a more sophisticated approach, in which the extra- and intracellular spaces were treated as separate compartments. In this case model parameters could be directly compared to physically 4 meaningful quantities such as the drug's rate of diffusion in water and the permeability of the cell membrane to the drug. Estimates of model parameters that were obtained from analysis of flux data were used either as a relative measure in the comparison of drugs of a given family or to directly predict the penetrative ability of a drug, through simulation of drug distribution within a tumour environment. Simulations were carried out by applying parameter estimates for diffusion, cellular uptake, etc. to a mathematical model which described drug kinetics within a cord of tumour cells surrounding a blood vessel, as might typically exist within a solid tumour. To reproduce clinical conditions modelling incorporated known blood pharmacokinetics for the drug. The end goal of this modelling was to calculate the spatial distribution of the drug within the tissue as a function of time, thereby enabling an assessment of its effectiveness in penetrating tumour tissue under simulated in vivo conditions. Characterisation of MCC cell environment An obstacle to the interpretation and analysis of data obtained from flux experiments is the lack of methodology for characterising the structure of individual MCCs and the variations that exist between different cell lines grown as MCCs. Accurate assessment of culture thickness before experiments and the rate of growth during experiments, as well as knowledge of the fraction of extracellular space available for diffusion and the effect of drug toxicity on cultures, are all key factors required in interpreting experimental data. The thickness of MCCs is most simply evaluated through microscopic measurement of cryostat sections taken after experiments. However this method can be time consuming and it is often difficult to obtain an accurate measure of average thickness unless the culture is perfectly uniform. Cryostat sectioning also precludes the use of the culture for other experimental endpoints such as cell survival or accumulated drug content. Two alternative methods for the evaluation of MCC thickness were investigated in the study. The first was the simultaneous diffusion of tritiated water (HTO) with the anticancer agent of interest during flux experiments. A functional relation between a culture's permeability to HTO and its thickness measured from cryostat sections was determined and then used in subsequent flux experiments. This method allowed for accurate assessment of the average thickness over the 5 entire surface area of a culture but was still time consuming, requiring repeated measurements over several hours. The second method for evaluation of MCC thickness that was investigated in this study was the characterisation of MCCs via electrical impedance spectroscopy. The MCC geometry permits the use of electrical impedance spectroscopy (EIS) to study the cell environment of three-dimensional tissue cultures. EIS utilises the inherent electrical properties of individual cells to quantitate macroscopic parameters related to the tissue environment (Cole and Cole, 1941, Foster and Schwan, 1989, Schwan, 1963). On a conceptual level, the cell cytoplasm and extracellular space act as conductive media, which are isolated from each other by the cell membrane. A simple electrical circuit can be constructed to represent these terms, where the conductivity of the extracellular space and cell cytoplasm contribute resistive components, largely due to the presence of salt ions, and the cell membrane contributes a capacitive effect. Deterrnining experimental values for each of the terms allows for empirical comparison of impedance data with physical traits of the MCCs, such as thickness or the barrier to diffusion posed by the volume and tortuosity of the extracellular space. Unlike cryostat sectioning or HTO flux experiments, EIS measurements can be done quickly and without significant perturbation of the cells. Results from analysis of impedance data were correlated with culture thickness determined from cryostat sections. In addition to knowledge of MCC thickness, the interpretation of data from drug flux experiments requires knowledge of the fraction of extracellular space within the culture. Studies have been carried out to determine extracellular space in spheroids using equilibrium levels of molecules which do not pass the cell membrane and by cell volume measurements (Durand, 1980, Freyer and Sutherland, 1983). Results suggest the existence of large variations in extracellular space between tumour cell lines, with values ranging from 15 to 60%. Since the extracellular space may be the primary route of penetration into tumour tissue for many therapeutic drugs, it is critical that this factor be well characterised when flux data is modelled using a mechanistic approach. In this study radiolabeled inulin, C14-inulin, was employed to determine the fraction of extracellular space as well as its tortuosity. Extracellular fractions were determined through measurement of equilibrium levels of C14-inulin 6 within the tissue. Tortuosity was determined from analysis of flux of C14-inulin through the MCCs. Results were used in the analysis of drug flux data and as well were compared with the predictions made from the EIS measurements of the barrier to diffusion posed by MCCs. Cell line Drug Flux Studies Tissue Characterisation Studies Nitroimidazoles Tirapazamine Doxorubicin Azo Eta Miso Pimo HTO EIS Inulin SiHa • • • • • • • • • V79 • • • • V79/DOX • • • • Table 1.1 Outline of the experiments that are presented in this thesis. Experiments are divided into two subgroups: drug flux studies and tissue characterisation studies. MCCs used in the studies were grown from three types of cells: SiHa cells (Human squamous cell carcinoma), V79-171b cells (Chinese hamster fibroblasts) and V79/DOX cells (V79-171b cells cultivated to express multi-drug resistance). The abbreviations Azo, Eta, Miso and Pimo stand for Azomycin, Etanidazole, Misonidazole and Pimonidazole respectively. See page ix for acronym definitions. 1.3 Outline of Experiments An overview of the experiments that are presented in this thesis is shown in Table 1.1. Experiments can be divided into drug flux and tissue characterisation studies conducted on MCCs grown from three types of cells: SiHa, V79 and V79/DOX. Nitroimidazoles The flux of several nitroimidazoles through SiHa MCCs was measured and analysed using a one compartment model describing diffusion within the MCCs. Nitroimidazoles studied included Azomycin, as well as three compounds with differing side chains replacing the hydrogen atom at the 1-position of the 2-nitroimidazole ring: Misonidazole, Etanidazole and Pimonidazole. All four compounds act as hypoxic cell radiosensitising agents, which can substitute for oxygen in the process of fixation of radiation damage to DNA (Dische, et al., 1980). While these molecules are not currently used in the clinical treatment of cancer, they are representatives of a class of well-characterised molecules that served as a starting point for 7 the development of the experimental and analytical techniques used in drug flux experiments. Considerable data exist regarding their physicochemical characteristics such as size, partition coefficient, pK, etc. (Wardman, 1980), as well as HPLC analytical methods. Data for flux of the nitroimidazoles through MCCs were used to determine the apparent rate of diffusion of each molecule within the tissue. Results for determination of the rate of diffusion were tabulated along with other physicochemical characteristics of each molecule. In addition, the apparent rate of diffusion of each molecule through the MCCs was compared with the rate predicted when one assumes that the molecules are confined to the extracellular space of the tissue and diffuse at a rate equal to their rate of diffusion in water. Tirapazamine Tirapazamine (3-amino-l,2,4-benzotriazine-l,4-di-N-oxide; Triazone; SR 259075; formerly SR 4233) is a bioreductive cytotoxin exhibiting preferential toxicity to cells at reduced oxygen tension. It is believed that under hypoxic conditions tirapazamine is reductively metabolised to a cytotoxic free radical intermediate which causes DNA strand breaks and cell death (Laderoute, et al., 1988). Hypoxic cells in suspension culture are about 40 times more sensitive to tirapazamine than corresponding well-oxygenated cells (Zeman, et al., 1986), though a much smaller differential in sensitivity has been observed in multicellular spheroids and tumours (Durand, 1994, Durand and Olive, 1992, Durand and Olive, 1997). Tirapazamine appears to enhance radiation and chemotherapy based anti-tumour activity in experimental animals (Brown and Lemmon, 1990, Dorie and Brown, 1993, Zeman, et al., 1988), and is the lead compound in this new class of anticancer agents. It is presently undergoing clinical evaluation (Brown, 1993, Brown and Lemmon, 1991, Graham, et al., 1997, Senan, et al., 1994). Tirapazamine could complement clinical radiotherapy and chemotherapy by sterilising cells at reduced oxygen tension that can be located distal to functional tumour blood vessels. However a prerequisite for its effectiveness is that it be able to reach therapeutic levels in all target cells. Data from tirapazamine flux experiments through SiHa MCCs were analysed using a one compartment mathematical model based on diffusion and enzymatic metabolism. Experimental estimates for the rate of diffusion and metabolism within MCCs were then 8 applied to a second mathematical model describing a corded tumour architecture. Using this model, simulations were carried out to examine the extent of tirapazamine penetration into solid tumour tissue. Doxorubicin The cytotoxic anthracycline doxorubicin is among the three most widely used anticancer chemotherapy agents. It is used clinically to treat a wide array of tumours despite the common emergence of drug resistance after repeated administration. In vitro evidence from multicellular spheroids suggests that doxorubicin penetrates tumour tissue slowly and it is our hypothesis that while doxorubicin may be effective against cells that exist proximally to blood vessels, cells located more distally may be exposed to sub-lethal levels. Repeated exposure of cells to sub-lethal doses of doxorubicin may then provide an environment that encourages the development of cellular expression of multidrug resistance. The multicellular spheroid model has allowed for studies which investigate cell kill and fluorescence of doxorubicin as a function of depth into tumour-like tissue (Durand, 1981, Durand, 1990, Kerr, et al., 1988). However, these studies have not provided an understanding of the relative importance of factors involved in limiting doxorubicin penetration. There exists considerable information regarding doxorubicin binding to DNA (Frezard and Garnier-Suillerot, 1990, Tarasiuk, et al., 1989, Zunino, et al., 1980), cellular membranes (de Wolf, et al., 1993, Demant and Friche, 1998, Gallois, et al., 1996) and serum proteins (Chassany, et al., 1996, Demant and Friche, 1998), as well as cellular uptake and efflux (Frezard and Garnier-Suillerot, 1991, Speelmans, et al., 1994). While this information has been recently used to mathematically model the uptake and binding of doxorubicin in cell suspensions (Demant and Friche, 1998), there has been no straightforward way of seeing how these effects combine with diffusion limited penetration to determine doxorubicin distribution within tumour tissue. Experimental flux data were used to determine the rate of diffusion of doxorubicin through the extracellular space as well as the rate of uptake by cells and the extent of extracellular binding. Flux experiments were conducted using MCCs grown from SiHa and V79 cells, as well as from V79/DOX cells, cultured to exhibit multidrug resistance. Results from the analysis of flux data using a two-compartment model were used to predict the distribution of 9 doxorubicin in tumour tissue after typical clinical administration and to investigate the relative importance of the factors involved in limiting its penetration. Characterisation of the Cell Environment The cell environment of MCCs grown from SiHa, V79 and V79/DOX cells were characterised using electrical impedance spectroscopy and through radiolabeled inulin experiments. Measurement of extracellular fraction and tortuosity as well as EIS parameters were made on cultures that ranged from about 80 to 250 pm in thickness. Results were characterised in terms of their variation with culture thickness. 1.4 Summary The chapters of this thesis can be grouped into three major sections: theory, methods and results. Chapters 2, 3 and 4 describe the mathematical methods used to model drug kinetics within the cultures, simulate drug distribution within tumour tissue and model the electrical characteristics of the tissue respectively. In Chapter 5, the experimental methods used for the flux and tissue characterisation experiments are described. In Chapters 6, 7, 8 and 9 results and analysis of the experiments, including simulations and discussion, are presented. In addition to the three major sections, Chapter 10 provides an outline of the future directions that may be taken based on these studies, Appendix I lists the people who contributed to work presented here and Appendix II describes the incubation system used for MCC growth and flux experiments. 10 2. Modelling Drug Flux Within MCCs 2.1 Introduction The data from experiments in which the flux of a drug through MCCs was measured over time were used to deterrnine values of the parameters of a mathematical model that described drug kinetics within tissue. Mathematical modelling of drug kinetics was carried out using partial differential equations to account for drug diffusion, metabolism, binding and cellular uptake within the tissue. By comparing experimental data with the predictions made by the mathematical model, estimates of undetermined model parameters were obtained. Results of analysis were used to satisfy three objectives: to provide a relative measure of the penetrative ability of drugs of a given family, to compare the barrier to penetration posed by MCCs grown from different cell lines and to predict a drug's effectiveness via simulation of its distribution within tissue under clinical conditions. The modelling approach that was used to describe drug kinetics within the MCCs was similar to that used in recent publications to describe drug diffusion and reaction in MCCs (Hicks, et al., 1997), tissue (Rippley and Stokes, 1995) and cellular uptake in cell suspension (Demant, et al., 1990). Mathematical modelling of drug kinetics within tissue was based on simple Fickean diffusion, where the flux of a drug through a section of tissue is taken to be linearly proportional to the concentration difference across it. In order to account for the effects of metabolism, binding and cellular uptake, Fickean diffusion was described using partial differential equations, which modelled drug concentration as a function of position within the tissue. Tissue was modelled as a homogeneous material using either one compartment, in which case the extra- and intracellular space were not distinguished; or two compartments, one for the extracellular space and one for the intracellular space. Modelling the tissue using only one compartment allowed for simple analysis of flux data in situations where a drug was known to enter cells easily, and hence did not necessitate the discrimination between extra-and intracellular drug levels, or where not enough was known of the drug-tissue interactions to allow more complicated modelling. Two compartment models were used in situations where the rate of a drug's entry into cells posed an important rate lirniting step in the process of 11 metabolism or binding within cells. In this case, the contribution to the net flux through the tissue from drug diffusion occurring within cells was assumed to be negligible compared to the contribution from extracellular diffusion and drug flux was modelled as being a purely extracellular process. Analysis of drug flux data also required that the model account for the effects of unstirred boundary layers and of the permeable plastic growth membrane. These barriers to diffusion were incorporated into the model as additional layers through which the drug must diffuse. Boundary conditions were then used to relate drug flux into and out of each layer as the drug travelled from the donating reservoir and towards the receiving reservoir. The complete set of equations describing drug kinetics within the MCCs, along with the relevant boundary conditions, were discretized and numerical solutions obtained using the Crank-Nicolson methodology (Crank, 1975). Estimates of model parameters were made using standard, non-linear chi-squared rnimrnisation technique (Press, 1992), from the comparison of drug flux data with model predictions. Fickean Diffusion The partial differential equations used to model drug diffusion during flux experiments were based on simple Fickean diffusion, where the flux of a molecule through a section of material is taken to be linearly proportional to the concentration difference across it. This relation can be written in algebraic form as: Flux = m±Zhlf (2.1) i 1000 where Flux is in mol/s, A (cm2) is the cross-sectional area of the material, £ (cm) is the thickness of the material, cA and cB (M) are the concentrations of the drug on either side of the material and D (cm2/s) is the drug diffusion coefficient within the material examined, see Figure 2.1 (a). The factor of 1000 appearing in the denominator converts cm3 to litres. Equation (2.1) describes steady state drug flux through a section of tissue where no binding, metabolism or uptake occur. The condition for steady state flux is that a stable concentration gradient exist within the tissue, which changes only as the concentration of drug on either side 12 of the tissue changes. For a slab/MCC geometry, the shape of the stable gradient will be in the form of a linear decrease in concentration with distance from the high concentration side, see Figure 2.1 (b). The period of time required to establish this gradient, given initial the condition of zero drug concentration throughout the material, can be approximated as (Crank, 1975): 2-D (2.2) Hence, for a drug with a diffusion coefficient of 5xl0"7 cm2/s and a 150pm thick tissue section, the time to reach steady state diffusion will be about 4 minutes. For diffusion of a drug from reservoir A through an MCC and into reservoir B, the time course change of drug concentration in the two reservoirs can be derived from equation (2.1) as: 2 1-e D A A ve and C A = C o - C B (2.3) (2.4) (a) Reservoirs A B (b) [Drug] 0 /drug RUXJ * " (c) 0 t Position Within Material 1 .-c/cD 0.5 h Time Figure 2.1 Fickean diffusion through tissue, (a) Diagram of M C C orientation during a flux experiment. Drug passes from reservoir A, through the M C C of thickness t, to reservoir B. (b) Sketch of the initial build up of drug within an MCC. Equation 2.1 is only applicable after a stable gradient is reached within the MCC. (c) Sketch of the time course change to drug concentrations in reservoirs A - solid line, and B -dashed line, that is predicted by equations (2.4) and (2.3). where cA and cB are the drug concentrations in the donating and receiving reservoirs and cG is the initial drug concentration in the donating reservoir. Figure 2.1 (c) shows the time course change for reservoir drug concentrations predicted by equations (2.3) and (2.4). In the derivation of equations (2.3) and (2.4) the volume of the tissue is assumed to be negligible relative to the volume of the reservoirs. This ensures that the amount of drug within the tissue at any given time, which is neglected in the derivation, will not subtract appreciably from the reservoir concentrations. The time to reach steady state diffusion is also taken as being negligible in this derivation. 13 2.2 Drug Kinetics in M C C s : One Compartment Model Models based on Fickean diffusion can be developed for more complicated drug/tissue dynamics than described by equation (2.1). Terms which account for metabolism and binding can be incorporated into diffusion based models by rewriting the relation of equation (2.1) in terms of partial differential equations (PDEs) which relate drug fluxes to concentration gradients across infinitesimal slices of the tissue. One-compartment models treat the tissue through which the drug diffuses as a homogeneous material in which the extra- and intracellular spaces are grouped as a single substrate. There are two situations where one-compartment models may be useful. The first situation is when knowledge concerning a drug's interactions with the tissue is limited and a mechanistic modelling approach is not possible. In this case, the one compartment model serves as an empirical model and care must be taken in comparing or interpreting results obtained from the analysis of experimental data. The second situation where the use of a one compartment model may be applicable is when the drug being modelled is known to enter cells rapidly enough that the cell membrane does not pose a barrier to drug diffusion. For most molecules, the rate of diffusion in water will be several orders of magnitude greater than the apparent rate of diffusion within cellular membranes (Stein, 1986), hence the requirement for the second situation will usually only be approximately satisfied. The effect of approximating tissue to a one compartment model on the analysis of flux data can be understood through consideration of the situation where a drug which does not enter cells is modelled using this approach. The apparent diffusion coefficient of the drug, that is deterrnined when it is assumed to pass through the entire tissue, will differ from the true diffusion coefficient, through the extracellular space alone, by a factor equal to the ratio of the total cross sectional area of the tissue to the cross sectional area of the extracellular space alone. The ratio of cross sectional areas will be equal to the fraction of extracellular space within the tissue, fex. The flux relation described by equation (2.1) can be applied to account for drug flux using either approach: FlUX= D«™fe»'A) ( ^ - C a ) = P a p p A(CA-CB) ( 2 5 ) I 1000 I 1000 ' 14 here the true rate of diffusion and the apparent rate of diffusion are related through D a p p = f e xD l r u e. Both approaches predict the same rate of drug flux through the tissue since they are effectively trading off the rate of diffusion with the available volume. However they will have different implications when applied, for example, to equation (2.2) which predicts the speed at which drug distributions will form within tissue. Modelling Diffusion The PDE form of equation (2.1), for diffusion through a slab of tissue is written as (Crank, 1975): *=D — dt dx ^7 = D ^ . (2.6) The variable for drug concentration, c (M), is now a function of time, t (s), and position, x (cm), within the tissue. Equation (2.6) serves in essence as a book keeping equation which relates changes to drug concentration at each position within the tissue to drug flux through the tissue due to intra-tissue drug gradients. Because drug flux is now accounted for on a microscopic level, as opposed to the macroscopic approach of equation (2.1), drug flux during the initial build up period can be modelled, as well as stable state diffusion. The effect of MCC area and thickness as well as the reservoir volumes are all accounted for in the boundary conditions, addressed in Section 2.3. In addition, experimental effects such as non-zero reservoir sample sizes and unstirred boundary layers can all be incorporated into this model. Modelling Metabolism The incorporation of a tissue metabolism term requires only a simple modification to equation (2.6). To describe a situation where the rate of change of drug concentration at each infinitesimal layer within the tissue depends on drug loss/gain through diffusion and drug loss through concentration dependent metabolism, we use: where the parameter y (s1) is used in describing a linear relation between the rate of drug loss through metabolism and parent drug concentration. 15 The tissue metabolism term can be interpreted by looking at the case where drug diffusion is set to zero. When D=0, equation (2.7) can be integrated to obtain drug concentration as a function of time within the tissue: c(t) = c0e-». (2.8) In this case c0 refers to an initial drug concentration that is taken to be uniformly distributed throughout the tissue. Inspection of equation (2.8) reveals that y determines the half-life of the drug within the tissue, tV2 = ln(2)/y. Tirapazamine Kinetics Within MCCs Tirapazamine kinetics within MCCs were modelled using a one-compartment model which included terms to describe diffusion and oxygen inhibited metabolism. Modelling of intracellular metabolism using a one-compartment model will yield accurate results as long as the rate of drug entry into cells is considerably greater than the rate of drug metabolism within the cells. It was assumed that tirapazamine could enter cells quickly because of its small molecular weight (M.W. = 178), neutral charge at physiological pH and high partition coefficient, P o c t a n oi / w a t e r = 0.48 (Minchinton, et al., 1992). Cellular metabolism of tirapazamine has been shown to follow Michaelis-Menten type kinetics and modelling within the tissue was carried out by replacing the non-specific metabolism term of equation (2.7) with a concentration dependant enzymatic metabolism term: dc d2c V -c f = D ! H ^ ' (2'9) here Vm a x (M/s) and K'm (M) are the maximum rate of drug metabolism and the drug concentration at which metabolism is half saturated. If K'm is much greater than c, the metabolism term of equation (2.9) can be approximated to the term used in equation (2.7). Oxygen inhibition of tirapazamine metabolism was described using a competitive inhibition model, where oxygen dependence of K'm was modelled as: K' =K„ 1 + Po 2 v K p o 2 ; (2.10) 16 Here pOz (irtmHg) is the oxygen partial pressure, KPo2 (rrrrnHg) is the oxygen partial pressure where tirapazamine is 50% inhibited and Km is the value of K'm under fully hypoxic conditions. Km was taken as 75 uM (Wang, et al., 1993) and KPo2 was estimated to be 4 mmHg (Koch, 1993). 2.3 Drug Kinetics in M C C s : Two Compartment Model Two compartment models, which differentiate between extra- (a) and intracellular drug concentrations, allow for a mechanistic approach to modelling drug kinetics within MCCs. Modelling using two compartments requires that the tissue environment in which the drug is diffusing be characterised in terms of its fraction of extra- and intracellular space. Figure 2.2 (a) illustrates typical drug-tissue kinetics that can be described using a two-compartment model. Drug diffusion is confined to the extracellular space and the rate constant k, which depends on membrane surface area and permeability to the drug, will determine the rate of cellular drug uptake. Tissue Tortuosity Mathematically, the tissue environment illustrated in Figure 2.2 (a) can simplified to that shown in Figure 2.2 (b), where average cell size and extracellular fraction are conserved but the effect of tissue tortuosity is absorbed into the effective diffusion coefficient D, through the relation (Nicholson and Phillips, 1981): D* D = A2 (2.11) (b) D It* D. Figure 2.2 Two compartment modelling of drug kinetics, (a) Sketch of tissue kinetics where a drug diffuses through extra-cellular space, with diffusion coefficient D" the rate of diffusion in water, and is also taken up into cells, with rate constant k. (b) Diagram of a two compartment model of the tissue. Extra- and intracellular spaces are equal to those in panel (a) but the effect of tortuosity is now incorporated into the effective diffusion coefficient, D, as defined by equation (2.12). where D* is the rate of diffusion of the drug in water and X is the relative increase in path length that the drug must travel as it passes through the tissue due to the degree of tortuosity. X is defined by the relation £'=X-i, where t is the thickness of the 17 tissue and I' is the effective path length that a molecule must travel. Unlike the parameters D a p p and D t r u e that were introduced for the one compartment model, using either D or D* to predict the rate of drug build up within tissue, through equation (2.2), will produce the same results when the appropriate thickness £ or £' is used. Diffusion & Cellular Uptake Equations describing the extracellular drug diffusion and cellular uptake as illustrated in Figure 2.2 (b) can be written as: ^ i = p ^ £ i - F A ^ ' (c - c ) (2 12) and ^ = ™ ^ ( ( 2 1 3 ) dt v2 Here cx and c2 (M) are the extra- and intracellular drug concentrations, which will vary with position within the tissue and time, V i and v2 (cm3) are the extra- and intracellular volumes per cell, A c e l l (cm2) is the surface area per cell and P (cm/s) is the membrane permeability to the drug. The term involving P appears in both equations which reflects the fact that the net drug loss from the extracellular space will be matched by a drug gain within the cells and vice versa. The rate constant, k, which appears in Figure 2.2 is defined as k = PA c e l l/v 2. The effect of the fraction of extracellular space on drug flux is accounted in the boundary conditions (see Section 2.3) which relate drug flux from the reservoirs and into the tissue. Drug Binding In the case where intracellular drug binding occurs, a further embellishment to the above model involving an additional rate constant must be made. Figure 2.3 (a) shows a possible model where one rate coefficient, ku controls the rate of drug influx and efflux from the cell and a second rate coefficient, k2, is used for the rate at which drug is bound once within the cell. Such a model requires additional variables, c2f and c2b, to discern between free and bound drug within the cell, with the total intracellular drug concentration, c2, now defined as: c2=cw + c2rb. (2.14) 18 In the case where intracellular binding occurs at a rate which is much faster than the rate at which the drug enters cells, the model can be simplified back to a single rate constant model if one assumes that c2b is always in equilibrium with c2f such that they are proportional to each other, as depicted in Figure 2.3 (b). For binding that follows a reversible equilibrium relation, the concentration of bound drug as a function of free drug can be written as: Cmax-^bC2,f l+kbc2,f (2.15) where cmax (M) is the concentration of binding sites within each cell and kb (M1) is the drug-bmding site affinity constant. Figure 2.4 shows the relation between bound and free drug that is described by equation (2.15). Combining equations (2.14) and (2.15) then allows the total intracellular drug concentration to be expressed as a function of the free drug concentration: f c k ^ V l + k b C 2 f j (2.16) Modification of equations (2.12) and (2.13) to accommodate the drug binding effect leads to the following set of relations: and dt dx2 dcn P A „ (C!-C2,f>> P A cell dt (2.17) (2.18) Figure 2.3 Modelling drug binding, (a) Modelling cellular uptake and intracellular binding requires two rate constants, k, and k2 which control the rate of drug entry into cells and the rate of intracellular drug binding respectively, (b) When k2 is much greater than k[ the model can be simplified by assuming that free and bound drug are always in equilibrium. Figure 2.4 Relation between free and bound drug described by equation (2.15). As the free drug concentration is increased, the concentration of bound drug approaches saturation. The key differences between equations (2.17) and (2.18), and the previous equations, (2.12) and (2.13), are that total intracellular drug concentration, c2, has now been replaced with free 19 intracellular drug concentration, c2 f, in the term describing the drug concentration gradient across the cell membrane. Doxorubicin Kinetics Within MCCs Doxorubicin kinetics within MCCs were described using a two-compartment model. Modelling addressed the following key aspects regarding doxorubicin kinetics within the cultures; diffusion through the extracellular space, uptake by cells, DNA binding, membrane binding and extracellular binding. Figure 2.5, shows a diagrammatic representation of these interactions and Table 2.1 summarises the variables Figure 2.5 Modelling of doxorubicin-that were used to describe them mathematically. The tissue interactions within an MCC. mathematical relations that were used to model each of the processes depicted in Figure 2.5 are described in the following paragraphs. Extracellular Diffusion Diffusion of doxorubicin through the extracellular space of MCCs was modelled mathematically starting with the diffusion equation: Five key interactions are accounted for in the modelling: 1, diffusion of doxorubicin through the extracellular space; 2, passive uptake of doxorubicin through cell membranes; 3, D N A intercalation; 4, extracellular binding; 5, intracellular binding. dt dx2 (2.19) where c, (M) is the total extracellular concentration of doxorubicin and c l f (M) is the free extracellular concentration of doxorubicin, both of which will vary with time and position within the culture. Cellular Uptake It has been shown (Frezard and Garnier-Suillerot, 1991) that doxorubicin enters cells via passive uptake of molecules in the uncharged state. Flux of doxorubicin into cells was written in terms of the concentration gradient of uncharged drug across the cell membrane using: r)r P A ^ - = £ ^ ( f 1 - c u ~f2 • cu) = k• (A • cu -/2 • cu), at v, (2.20) 20 where P (cm/s) is the permeability of the cell membrane, A c e l l (cm2) is the cell membrane surface area per cell, v2 (cm3) is the volume of intracellular space per cell, fx and / 2 are the fractions of doxorubicin in the uncharged state in the extra- and intracellular space, which are pH dependent, c2 (M) is the total intracellular doxorubicin concentration, c2f (M) is the intracellular concentration of free doxorubicin and k (s1) is the effective rate of uptake, defined as k=PAcell/v2. Active Efflux by Resistant Cells lh the case of the doxorubicin resistant V79/DOX cells, modelling must also account for the active efflux of doxorubicin from cells. In cell suspensions at low doxorubicin concentrations the effect of the active efflux of doxorubicin can be described as a reduction in the initial rate of cellular influx (Frezard and Garnier-Suillerot, 1991). That is, during the period when intracellular free doxorubicin levels are negligibly low, the effect of the active efflux can be described using a simplified version of equation (2.20): where k' is the reduced apparent rate of cellular uptake. Analysis of the V79/DOX MCC flux data using equation (2.20) to describe cellular uptake indicated the concentration of free intracellular doxorubicin to always be less than 0.5% of the extracellular levels. Hence, under the experimental conditions used in this study, flux data from drug resistant MCCs could be modelled using the uptake relation described by equation (2.21). Doxorubicin Ionisation The fraction of uncharged doxorubicin, as a function of pH, is written as: 1 •^ 1/2 = 1 + 1 Q P K a - p H 1 / 2 ' (2.22) where pH 1 / 2 is the pH of the extra- or intracellular space and the pKa value for doxorubicin is taken as 8.4 (Frezard and Garnier-Suillerot, 1990), hence only -9% of molecules are uncharged at biological pH (pH=7.4). Intracellular pH was set at 7.2 throughout the MCCs. Extracellular pH was modelled based on results obtained from the outer layers of large spheroids (Acker, et 21 al., 1987), where gradients are similar to MCCs, using a decrease of 0.2 units for every 100 um depth into the MCCs. DNA Intercalation The binding of doxorubicin to DNA has been shown exhibit a Scatchard type relation between free and bound drug (Zunino, et al., 1980). The equilibrium relation between free and DNA bound doxorubicin is: nBdkdc2,f 1 A 1 + kx ' K ' where Bd (M) is concentration of DNA base pairs per cell, kd (M"1) is the doxorubicin-DNA affinity coefficient, n is the exclusion parameter and c14 (M) is the concentration of doxorubicin bound to DNA within the cell. Values for kd and n were set to 4300 mM"1 and 0.24 according to published data obtained from naked DNA at 37°C in PBS solution (Tarasiuk, et al., 1989). Bd was estimated assuming a DNA content of 6xl09 base pairs per cell, which yielded values for Bd of -5.6 mM and 12.5 mM for SiHa and V79 cells respectively. Cell volumes were taken as 1800 pm3 for the SiHa cells, based on cell diameter of 15 pm, and 800 um3 for the V79 cells (Freyer, et al, 1984). Intracellular Membrane Binding The bmding of doxorubicin to membrane phospholipids has been characterised by several investigators (de Wolf, et al., 1993, Demant and Friche, 1998, Gallois, et al, 1996). The equilibrium relation between free and membrane bound drug is taken to be: c ^ = T r r ^ ' ( 2 . 2 4 ) l + kc, f where B2 m (M) is the concentration of doxorubicin membrane binding sites within the cell, km (M'1) is the doxorubicin-membrane affinity coefficient and c2m (M) is the intracellular concentration of membrane bound doxorubicin. The values of km and B 2 m were set to 0.62 mM"1 (Demant and Friche, 1998) and 55 mM (Demant and Friche, 1998) respectively. Doxorubicin Serum Binding Recent reports have shown doxorubicin to bind to human and bovine serum albumin with affinity coefficients of 2.6 mM"1 (Chassany, et al., 1996, Demant and Friche, 1998) and 1.4 22 mM'1 (Demant and Friche, 1998) respectively. The relation between free and serum bound drug within the diffusion reservoirs is assumed to be: ^=7*1^ ' ( 2 - 2 5 ) 1 + k.c. C p C r , f where C r b and C r f (M) are the concentrations of serum bound and free doxorubicin in either of the reservoirs, Bp (M) is the reservoir concentration of serum albumin and kp (M"1) is the doxorubicin-albumin affinity coefficient. Bp was set at 0.03 mM for analysis of diffusion experiment data, as calculated assuming 2 mg/ml bovine serum albumin within the growth medium (Boone, et al., 1972) and kp was set to the value determined from bovine serum albunriin. Doxorubicin Hydrolysis in Media Doxorubicin exhibited instability in the growth medium, which was detected through loss of peak area response from HPLC analysis. This effect was characterised using a single exponential decay term, i.e. assuming an irreversible process where the rate of doxorubicin disappearance was proportional to doxorubicin concentration: ^ = - P C r , (2.26) where P (s"1) is related to the media half-life of doxorubicin by t1/2=ln(2)/p. The effect of doxorubicin hydrolysis during diffusion experiments was accounted for by combining equation (2.26) with the boundary relations, see equations (2.32) and (2.33). The half-life of doxorubicin, under conditions of the flux experiments, was approximately 32 hours. Extracellular Binding The effect of an array of extracellular binding sites, including membrane phospholipids and proteins as well as components of the extracellular matrix, was modelled by approximating their combined effect to an apparent binding relation given by: where B^ (M) and kj (M1) are the concentration of extracellular binding sites and the doxorubicin-receptor affinity coefficients for each of the i classes of binding sites and c l b (M) is 23 the equihbrium concentration of extracellular bound doxorubicin given a free doxorubicin concentration of clf. The approximation to the simple linear relation involving the term Bk requires that kj-c1/f be small compared to unity for each of the i classes of binding sites. This should hold true given the typical values of kt for phospholipids and serum albumin of ~1 mM'1 and experimental doxorubicin concentrations generally less than 0.05 mM. Complete Model Combining equations (2.19)-(2.27) leads to the following set of equations, which describe doxorubicin kinetics within the MCCs: ^•^(fi-Cu-A-Cu), (2.29) Ci=c 1 > f+c 1 ( b=c 1 ( f(l + Bk) (2.30) (2.31) a n d c2=c2ii+c2im+c2id^c2il \ { B2,mkm | Bdkd A l + kmc2,f l + kdc2f j Here equations (2.28) and (2.29) relate changes to extra- and intracellular doxorubicin concentrations produced by diffusion and cellular uptake within the cultures. Equations (2.30) and (2.31) express total extra- and intracellular doxorubicin concentrations as functions of the free doxorubicin concentrations. In writing c2 and c2 in terms of c l f and c2f one implicitly assumes that equilibrium between free and bound drug is reached at a rate much greater than the rate of change of drug concentrations within the culture. For this assumption to hold true drug binding within cells must occur much faster than the rate of cellular drug uptake and within the extracellular space binding must occur much faster than the rate of change of the extracellular drug concentration. Numerous studies have shown that DNA intercalation occurs on a time scale that is much more rapid than the rate at which doxorubicin enters cells, e.g. (Tarasiuk, et al., 1989). Hence, intracellular doxorubicin will always be in a state of near equilibrium between free and bound drug. For extracellular binding, the tortuosity of the tissue and the high concentration of binding sites relative to drug concentration (see results), should ensure that a rapid equilibrium occurs between free and bound drug. 24 2.4 Boundary Conditions & Additional Barriers to Diffusion Figure 2.6 shows a diagram of the individual layers that a drug must diffuse through in order to pass from one reservoir to the other. They consist of two unstirred boundary layers (UBLs), the tissue and a permeable plastic membrane. Boundary conditions are required to relate flux through the M C C to concentration changes in the reservoirs and, as well, to relate flux out of one layer and into the next. A Reservoirs • • • Cells mm 1 M lip* D i -Plastic membrane h \ Figure 2.6 Diagram of the barriers to diffusion encountered by a drug as it passes between reservoirs during a diffusion experiment. Drug added to reservoir A must first pass U B L A followed by the M C C and then the plastic insert and UBL B in order to arrive in reservoir B. Reservoirs have drug concentrations cA / cb and volumes VA, VB. Diffusion through additional layers Drug flux through the UBLs and plastic insert was described using equation (2.6), where the diffusion coefficient was set to that of the drug in water and the thickness of each layer was deterrnined experimentally. To simplify matters, the effect of the right side UBL and the plastic insert were described mathematically as a single layer. Flux into and out of the reservoirs The boundary condition at the border between the reservoirs and the UBLs must relate flux into and out of the UBLs to drug concentration change in the reservoirs. Equation (2.32) indicates this condition for the border between reservoir A and U B L A (Crank, 1975): dc dt A _ D*A m r r dcu mcc i,i dx (2.32) here V A (cm3) is the volume of reservoir A , cA is the total concentration of reservoir A , c A f is the free concentration of reservoir A , D* is the diffusion coefficient within U B L A , p is the rate of breakdown of free drug within the reservoir and dcu/dx I U B L a is the concentration gradient of free drug at the border of the UBL. There is no negative sign in front of the gradient term since a decreasing concentration from left to right will a produce a negative gradient term which 25 results in a negative value for rate of change of the concentration in reservoir A. The last term of equation (2.32) is the non-saturable reaction term used to model drug loss through breakdown within the reservoirs. The condition for the boundary between reservoir B and UBLB is similar to equation (2.32) but with a negative sign in front of the gradient term to reflect increases to the drug concentration in reservoir B for negative gradients (Crank, 1975): dc, _-D'A m c c dcu\ dt V B dx - P c w . (2-33) Flux between UBLs and the tissue At the border between UBLA and the tissue, the flux from one layer into the other must be equal. This constraint is written as (Crank, 1975): F = A M C C D ^ = (4 x"AM C C)D.^f ,» dx left (2.34) right where F (mol/s) is flux and 'left' and 'right' indicate concentration gradients evaluated to the left and the right of the UBL/tissue border. The factor fex that appears on the tissue side of the border balances diffusion through the entire cross sectional area of the MCC within the UBL to diffusion constrained through the extracellular space within the tissue. A similar relation holds at the border between the tissue and the layer combining the plastic membrane and UBL„: F = ( f . -A M C C )D| i - A D * ^ left (2.35) right 2.5 De-dimensionalization of Equations In order to facilitate integration of the system, the equations describing the drug kinetics within the tissue and the boundary layers were rewritten in dimensionless form, using the normalizations outlined in Table 2.1 and 2.2. The tissue equations for both one and two compartment models are generalised to the following set of dimensionless equations: 26 a r ~ <?x2 V l l / l L w h u> K m +c 1 / f Y l w <9C2 - k(/j • Q f / 2 • C2 f) y2 • C2 f. (2.36) (2.37) With the dimensionless total extra- and intracellular drug concentrations within the tissue written as: C =C +C = C ( , B bk b ^ 1 + ^ — S -l + k b C w , v c = c + c + c = c ^2 *-2,f T ^2,111 T *-2,d V"2,f B 2 m k m B dk d l + kmC2, f l + kdC2 < f (2.38) (2.39) To describe a one compartment model k is set to zero in equations (2.36) and (2.37), and fe set to 1 in equations (2.45) and (2.46). The dimensionless equation for diffusion through the UBLs is written as: dCx _ D* d2Cu ar D dx2 p Clf" With the dimensionless total drug concentration within the UBLs written as: (2.40) c = c + c = c 1 + -—p-^-1 + K P C u y (2.41) Dimensionless versions of the boundary conditions describing flux into and out of the reservoirs, equations (2.32)-(2.33), are written as: dC £ D* dCu\ dT L 4 D dX dCB _ / D* dCu dT L B D dX ~ P C A , f P C B , f (2.42) (2.43) With the dimensionless total drug concentrations within either of the reservoirs written as: ( D i . ^ c r = c r , f +c r , b = c r, f B„k„ 1 + - p -p (2.44) 27 Dimensionless versions of the boundary conditions describing flux from the UBLs and into the tissue, equations (2.34)-(2.35), are written as: F —= AD dX = (4,'A)D Ueft F f = ( f . - ' A ) D % dX Ueft = AD dX dX right \right (2.45) (2.46) Name Units Definition Value Non-dimensional c i .f M free extracellular drug concentration, cu(x,t) C u = c 1 ( / c 0 c i , b M bound extracellular drug concentration, f(cw) Ci,b=c1,b/c0 C i M total extracellular drug concentration, C 1 =c 1 / c„ C2,f M free intracellular drug concentration, c2l(x,t) C2.f=c2,,/c0 C2,m M membrane bound intracellular drug cone, f(cv) C 2 , m = C 2 , m / C 0 C2,d M DNA bound intracellular drug cone, f(cw) C2,d=C2,d/Co Cl M total intracellular free drug cone, C2= C2,f + C2,m + C2,d C 2=c 2/c 0 CA/B M total drug cone, in reservoir A or B CA/B(0 . C A =c A /c 0 CA/B,f M free drug cone, in reservoir A or B CA/B,f=CA/B~CA/B,b £-A,f=CA,t/Co CA/B,b M bound drug cone, in reservoir A or B F(CA/B.f) ^"A,p=CA,p/Co X cm position within MCC x=x/e t s time T=tD/e Table 2.1 Variables used by the mathematical model describing drug flux through the MCCs. The notation 'CA/B' indicates a contraction of two distinct variables ' C / and 'C B ' . 28 Name Units Definition Value Note Non-dimensional • D cm2/s diffusion coefficient through extracellular space K m M Michaelis-Menten half saturation level 75 uM 1 • V * max M/s maximum rate of enzyme metabolism 1 • k s"1 effective tissue uptake rate 2 B d M concentration of cellular DNA base pairs 6-12 mM 2 n - DNA intercalation exclusion parameter 0.24 2 kd M" 1 DNA intercalation affinity constant 4300 mM' 1 2 B 2 ,m M cone, of intracellular membrane binding sites 55 mM 2 K M" 1 membrane binding affinity constant 0.6 mM' 1 2 M concentration of extracellular binding sites 2 K M" 1 extracellular binding affinity constant 2 • B k — effective extracellular binding term ~ B U • kb 2 Bp M media concentration of FBS 0.03 mM 2 M" 1 FBS binding affinity constant 1.4 mM' 1 2 P s"1 non-saturable drug loss in reservoirs 8.2 xlO'6 s'1 2 Yi s"1 non-saturable drug loss in extracellular space Y2 s"1 non-saturable drug loss in intracellular space pK a — acid dissociation constant 8.4 2 PHl/2 - pH of extra- or intracellular space see text 2 /l/2 — fraction of doxorubicin that is uncharged see text 2 v 2 cm3/cell volume of intracellular space per cell see text 2 V i cm3/cell volume of extracellular space per cell v 2 • f „ / d - f J 2 v, cm3/cell volume occupied per cell v 2/(l-f e x) 2 A l l ] cm2 / cell surface area per cell 2 p cm/s cell membrane permeability k • v2/AceH 2 — fraction of extracellular space 0.2-0.4 2 V cm2 MCC cross-sectional area 0.64 cm2 A m c c cm2 effective extracellular space cross sectional area f • A ex mcc 2 I D* un MCC thickness 80-250 pm cm2/s rate of diffusion in water see text v A / B cm volume of reservoir A or B 5-10 ml cm effective length of each reservoir V A / B / A Km = K r a / c 0 V m „ = V m a x £2/D/c0 k = k / v 2 f/D. B d = B d / c „ kd=k dc 0 B2,m = B 2,m /c„ K= k m c„ By, = B,,b /c„ K= k b c 0 B P = B P kp = kp c„ B=p f/D Yi=Y, <7D Y2=Y2 Table 2.2 Parameters used by the mathematical model describing drug flux through MCCs. The • indicates parameters that were estimated from analysis of drug flux through MCCs. The notation ' X 1 / 2 ' indicates a contraction of two distinct parameters 'X,' and 'X 2', where 1 and 2 refer to the extra- and intracellular compartments. Notes: 1 -parameter used specifically for modelling tirapazamine, 2 - parameter used specifically for modelling doxorubicin flux. 2.6 Discretizing the Equations Equations (2.36)-(2.46) were discretized and integrated numerically using the Crank-Nicolson implicit integration scheme (Crank, 1975). This technique involves discretizing the equations so that they relate flux to concentration gradients over finite steps in time and space. The approximations used in discretizing the equations were chosen so that the final equations formed a system of linear equations. 29 Figure 2.7 shows the spatial discretization that was used for the system. The tissue and UBLs are divided into a total of N layers. The first Sx layers describe UBLA, which has thickness £A. The next SJ-SJ layers describe the tissue, of thickness t. The last N - S 2 slices describe the effect of the plastic insert and UBLB of total thickness £B. The thickness of the individual layers within the UBLs and the tissue are taken as; 8xA=^A/(S!) for UBLA, bx=£/(S^S^ for the tissue and 5xB-£B/(N-S2) for the plastic insert and UBLB. 8xA 8x 5xB V A = L A A Si S i + 1 S 2 - 1 S 2 + 1 0 V B =L B A -»•*-Donating Reservoir UBLA Tissue Insert+UBLB J * e c e ™ S Reservoir Figure 2.7 Diagram of the spatial discretization scheme that is used in describing drug flux through the tissue, insert and UBLs. The tissue, thickness i, is divided into S2-S, layers. UBLA, thickness £A, is divided into S, layers. The plastic insert and UBL„ are described as single barrier of thickness (B, which divided into N-S2 layers. Layer thicknesses are: 8x=£/(S2-S,) for the tissue, 5xA=£A/(S,) for UBLA and 5xB=^B/(N-S2) for the insert and UBLB. The tissue has a cross sectional area A and reservoirs have volumes V A B which can be written as V A B =L A B A. Reservoirs are modelled as having drug concentrations equal to the middle of the outer layers. Following the Crank-Nicolson implicit integration scheme (Crank, 1975), the partial derivative of drug concentration with respect to time is discretized for small a time step, ST, as: dC_(Cl+1-C!) • + 0{ST), (2.47) dT <5T where the subscript i indicates spatial positioning within the culture, with i an integer falling between 0 and N. The temporal variable is represented by the integer superscript j. The spatial derivative is discretized, using a spatial step 8X, by the approximation: d2Cu _ dX2 2(6X)2 + 0(SX)2, (2.48) which is averaged over the/h and;+lth time intervals (Crank, 1975). 30 Discrete Tissue Equations. Substituting the relations of equations (2.47) and (2.48) into equation (2.36) yields the following linear equation relating extracellular drug concentrations at the / h time to concentrations at the ; + l t h time: 2{bXf V_. 1 + ^ ST Y.+ ^ /V / + r 7+1 ^  *—i,f, T v ~ i , f , -K m + C 7 , m l,f,- J y'+i / i C 7 + C 7 C 7 + c 7 S!<f<S2 (2.49) Here the binding/metabolism terms are also averaged over the two time intervals. The first term of the R.H.S. of this equation describes the relation between drug flux into and out of the I t h tissue slice and the drug concentrations of the z-l l h and z + l t h slices. The second two terms describe drug loss from the z'th tissue slice due to reactions that depend only on drug concentration in the i* layer. Equation (2.49) can be rearranged so as to separate drug concentrations at the y + l t h time step and those at the;* time step: (C 7 + 1 + C 7+1V ST 1 + -(SXf B b k b + 2-81 (6Xf • + m l,f, J ST+fk^SI v. v, i+-B,k„ l + k b Q , f 7 y _ 5 J _ (<5X)2 V v i max K m + C l f 7 m l,t v, • J ST-^k^ST v. + C 2 , f 7 . / 2 k^ .«ST for S!<i<S2 (2.50) In a similar fashion the application of equation (2.47) to discretize intracellular drug concentrations within the tissue, equation (2.37), yields: 31 +2/2k SI + y2SI -Cj^-fkSI -2f2kSI-y2SI for S^^S, (2.51) Equations (2.50) and (2.51) describe a set of linear relations between extra- and intracellular drug concentrations within the tissue at the;* and;'+lth time steps. Discrete Insert/UBL Equations Substituting equation (2.47) and (2.48) into equation (2.40), for diffusion within the UBLs, leads to the following discrete relation: D* ( C i , - 2C ' + C '', + C f\ - 2Q f''+1 + Q >*\ V 1 Mi+i M i i.t i - l I.t i+l M i M i - l BDkD 1 + - p-^-P M i ' ( c u ; + 1 - c u ; ) _ 81 D P 2(<5XA / B) 2 for 0</<S1 & S2<z<N (2.52) 8X A / B indicates spatial steps will be either 8XA , for UBLA, or 8XB, for UBLB. Separation of equation (2.52) in terms of drug concentrations at the ;* and ;'+lth time steps leads to: (C >+1+C i+1V SI D 1 + -B„k J p p P M i +2-ST D' + P5T (CMLI+CWLI)' SI D f f («J 2 D Bk. 1+ ~ p p ,. P M i _ 2 3 ^ - | i * r M D for O ^ S i & S2<z'<N (2.53) which describes a set of linear relations between drug concentration at each position within the UBLs at the;* and;'+lth time steps. 32 Discrete L.H.S. Boundary Condition Using the Crank-Nicolson discretization scheme for the interface between the reservoir A and UBLA; equation (2.42) is approximated to: BDk„ 1 + - 81 L. D 2SK, (2.54) Here the -1th spatial position refers a ghost layer which facilitates the derivation but does not produce a physical effect (Crank, 1975). To remove the ghost layer, equation (2.40) is also applied to the boundary thereby obtaining an additional relation: BDk„ 1 + - p-^- 81 D (<5XA) (2.55) By rewriting equation (2.54) as: Cu>Cj+i-28XA^ B k 1 + 1 + KPCi/oy 81 (2.56) and substituting it into equation (2.55) to remove the ghost layer, we obtain: Bk >+1-c > ( * A ) ' 1+- p p 1 + K P C u i ; ' l , f 0 '-l.fQ _ 81 2^c1,fll-2^c1;o-(fixA)2pc1,f -28X^ B„k„ 1 + -(2.57) which relates drug fluxes at the boundary of reservoir A and UBLA This can be regrouped to become: ( 2L V BDkD 1 + -\ p j+i _ p ; V 2 D* 81 (8XAY ^ ( C ^ - Q ^ - p f t ^ V i (2.58) and then by setting: h D* "5" A / D' ^XA/2 D (dXA/2 + LJ£) (2.59) We can write equation (2.57) as: 33 B X 1+-—2-2-*-i,f0 *-i,fo _ . z 51 (SXA) Taking the average of equation (2.60) for the/* and;'+lth time step we can obtain P (2.60) 1 + e_E_ 1 + *"i,fo *-uo _ ^  ±_ (r i+l _ r 1+1 + r ; - r ; ^  ST (5xA)2 (2.61) This can be regrouped as a linear combination of the ;+lth time drug concentration as a function of the fh time drug concentrations: c i+1 B k 1 + - 2_JL 1 + k Q ' p'-i-fo J + b. -y + P— 1 (axj 2 - C ;+1-b ST ( «A) Z C ; B k 1 + - 2—2- b 5 1 i+k p c u j ' (ag 2 p 2 (^XA)2 (2.62) Discrete R.H.S. Boundary Condition Similarly the R.H.S. boundary conditions of equation (2.43) can be discretized as: B„kD 1 + - 2 -2-1 + k p C w ! w D L„ 2 < 5 X „ (2.63) and rewritten as: B„kD 1 + - S - 8 -1 + W N ; U N , / (2.64) Applying the relation for diffusion within the UBL, equation (2.40), yields: B„k„ 1 + -C 1 + 1 -C 1 n' C ' -2C ' + C <5T D Combining equation (2.64) with equation (2.65) yields: (2.65) ( « - ) S BDkD 1 + -1 + K P C ^ y r )+1 — C 1 ST 2 T J c « - " 2 r J c ^ - ( ^ B ) 2 p c ^ - 2 5 X B — BDkD 1 + - 2 -2-c 7+1—c ; U N U N , o p / "'"M'-l.fN ST (2.66) which can be rewritten as: 34 ( L V 1 + -^-v £5X B k 1 + -• A D* 81 • (Cuir - i -Cui ) -P 1 + -£8XC C ' H f N (2.67) Equation (2.67) can be rewritten as: 1 + - p—s-1 + k C > ^l.fN * - l , f N P ^ U N J 81 = b„ • ( c i , f L i - C i , / N ) - P c v (2.68) where bB is defined as: 8 D i 2LB 1 + — B -D' B y D (dXj2 + LJt) (2.69) Taking the time average of the/* and;'+lth time steps, we obtain a similar relation to equation (2.61) but now for the R.H.S.: BDk„ 1 + - —— V i+1 P >+1 —C > *-l,fw *-l,fw -c'r+c. l , f N ^ - U N _ I ^ oT " B(«9X B) 2 i+1. UN-I i^,f/v T ^ w - i M N C, - P - l , fN ^ l , f N 2 (2.70) This can be regrouped as a linear combination of the ;'+l* time drug concentration as a function of the;'* time drug concentrations: 1 + --C i+1 b *-i,fN_i De 1 + +b B _ * I , + I « r B(«9XB)2 2 (<?xB) C ; BpkD 1 + - — , ST B -b„ - -v--oT +ci,fLrb-1 + W J 2 ( ^ B ) 2 (2.71) Discrete Condition for the Tissue-UBLB Boundary The boundary condition relating flux between the tissue and UBLB/ equation (2.46), can be discretized (Crank, 1975) to the following two relations: f ADc„ 2<5X (2.72) and ¥1 Cj -Cj AD*c„ 2<5X„ (2.73) 35 For the left/tissue side of the boundary, equation (2.49) can be applied, using a ghost layer, to obtain: 1 + - b-A ^  7+1 _ (- j l + k b C / b l,fs, j ST r i -2C ' +C *-l,fS,-l ^i,fs, fs,+i Yi V_ (SKY -k^^. C l ,4-A-c 2 ,4) K +C c ; (2.74) Substituting equation (2.72) into equation (2.74) yields the relation for the tissue to the left of the boundary: B b k b 1 + b Ms, J C i+1 — C ' ST V„ Km+C ' C ; 2F>. <3XDfexAc0 Similarly, applying equation (2.49) to the right/UBL side one obtains: BDkD 1 + - 2_P_ 1+kpC ' C ; + 1 - C ; T V C ; - 2C ' + C fs,+i •2 y ST D Which is then combined with equation (2.74) to yield: BDk„ 1 + -1 + k.C D 2FA. Combining equations (2.75) and (2.77), one then obtains: LD , r Bk i ls> SX„DAc„ +SX-, B„k„ 1 + 2 - 2 — 2 y D p ; + 1 — c > *-i,fs2 i^,fs2 <5T <5X LSXD Yi + V„ cs2 fs2 f 5XD, v k^(/iq4+/2c2,4) (2.75) (2.76) (2.77) (2.78) Which can be rewritten as: 36 ' B b k b ^ 1 + l + k b Q 7 i J +<5X, 1+-1 + k P c * U ST ^ C ' -C ' T V C ; - C ; ^ <5X D <5X f v_ c ; -fex5xk^(/lCl,4+/2c2,4) (2.79) Finally, taking the average of the;* and;+l t h time steps one obtains: L6X \ Bbkb ^ 1 + —^^  V +<5Xt 1 + -i+kbcw; B_k_ '2 J C 7 + 1 — C > v-i,fs2 *-i,fs2 _ ST + r *--i,fs2-i ^his2 " r v-i,fs z-i ^Us 2 D <5XH - ( f ^ Y . H A P ) ^ C / + 1 + C ' -L5X v „ V c > +c 1+1 ^ -U>xk^ C ' +C / + 1 C y' +C (2.80) Rewriting equation (2.80) to separate the;* and;'+l t h time drug concentrations yields: 37 D* ST D SK B y +C i + 1 fs 2 f ax +«5X ' i Bbkb ^ 1 + b b . v 1 + W s 2 y p p 1 + -v 1 + k p C ^ 2 ; + a r f 4 + D * 1 ' <5X D SK B y ST +—iex(SKyl + SK^) v V /,k-^ + :—^JWL-V V i K m +C ' +C / + 1 ST - f » t J - c - : * f f « ^ k ^ c ; D* ST } D ax s y +C Vs. f 5X 1 + b—b—-+axt BDkD 1 + -—— -ST ST 1 + k p C ^ y L D* 1 • + -ax D SK B y - f t * V_ K n + C W s 2 y +C ' If — ''ax +c2,4-f fex5X/2k^ (2.81) Discrete UBLA-Tissue Boundary Condition The boundary condition of equation (2.45) can be discretized in a similar fashion as used for the tissue-UBLB boundary of equation (2.46) to obtain the final boundary relation: c >+i . |-f USl+l [ ex ^ +c 1 + 1 LSK \ Bbkb ^ 1 + b—b-+8K, +81 1 + -Bpkp , 1 + k p c 4 ; SK D SK A y ST +yf e x(5X Y l +5K AP) +c ; + 1 D" ar D SK -Cj^-^SXfr ST C ' • f — ar ax f e x < * +axA -ar ' i Bhkb ^ 1 + b —^ BDk„ 1 + - p - p -f„ D* 1 ax D ax A y -^fex(axYl+<5xAB) +CJs D* ar fs,-i D ax A y v, K m + C 4 j +c 24-^f e xax/ 2k: (2.82) 38 2.7 Integrating the System Equations (2.50), (2.51), (2.53), (2.62), (2.71), (2.81) and (2.82) combine to relate changes to drug concentrations throughout the system between the ;-th and ;'+lth time intervals. Together they form a set of linear equations that can be solved analytically to determine the j+lth drug concentrations as a function of known, ;'* drug concentrations. Hence, given some set of initial drug concentrations for j-0, drug concentrations at any later time can be calculated iteratively from time zero. Initial Conditions Initial conditions for diffusion experiments are set as follows: Q, f°=l i = 0 C lf° = 0, 0<f<N . (2.83) C2>f°=0, S X </<S 2 Discretization Step Sizes The tissue was divided into slices of ~5pm by setting S^Sj to the nearest integer to £/(5 um) and then defining 8x=^/(S2-S1). Each UBL was divided in three slices, Si=3 and N-S2=3. Giving step sizes of 8xA= A^/3 and 5xB= B^/3, where £A and £B were determined experimentally. Time steps were set to 8t=15 s. However, the first 30 seconds were integrated using 200 steps with 8t=0.00015 s and then 999 steps with 8t=0.03 s. This was done to avoid integration errors which occur when large concentration gradients are combined with large time steps. Initially, there is a step-function for concentration as a function of position within the tissue, which quickly changes to a curved decrease in concentration going from left to right within the tissue. Solving the Equations The linear equations (2.50), (2.51), (2.53), (2.62), (2.71), (2.81), (2.82) and (2.83) describe a system of (N+1)+(S2-S!+1) equations and (N+1)+(S2-S1+1) unknowns. The equations were organised in a matrix and, using a matrix inversion method (Press, 1992), were solved to calculate drug concentrations at the;'+lth time step as a function of the previous;"1 time step. 39 2.8 Verification of Numerical Integration Methods The numerical integration scheme used to compute drug kinetics within the tissue and UBLs was verified through comparison with analytical solutions for a reduced version of the system and through self-consistency tests, where integration results using finer discretization steps were compared with original results. In the first set of tests, analytical solutions were derived for a limited version of the system, involving diffusion and media breakdown only. Results from the analytical solutions were directly compared with results from numerical integration. In the second set of tests, the integration error attributable to discretization step size was estimated by repeating integration using finer spatial and temporal steps. Comparison with Analytical solutions An analytical solution for diffusion through the tissue and UBLs and the relevant boundary conditions, equations (2.6) and (2.32)-(2.33), can be derived which is valid for highly permeable cultures where the ratio of reservoir volume to culture volume is large. The requirement of high permeability of the cultures ensures that the time required to reach steady diffusion within the culture is negligibly small. The requirement of small culture volume ensures that the amount of drug taken up in the culture at any time is also negligible. An analytic solution for the case of equal reservoir volumes, V=VA=VB, and a starting concentration of cD in reservoir A and zero concentration in reservoir B can be written as: 'B cA=e ( 2£A\ 1-e fTV v J f 2 D ' M \ 1- 1- e f'v 2 V V ) / (2.84) (2.85) where cA and cB are the drug concentrations in reservoirs A and B, which are functions of time t, £j is the total thickness of the tissue and UBLs, £T=£+£A+£B, P is the rate of drug loss in media and De is the effective diffusion coefficient, defined as De = Pnet • £r. Pnet can be calculated using the inverse addition rule: J ~ - L + I + - 1 , (2.86) P P P P 1 net A 1 B 40 where PA=D'/£A, P=D/£, and PB=D'/£B. Using equations (2.84)-(2.86)/ a comparative study of the time rate of change of reservoir concentrations was carried out between the analytical solutions and the numerical integration method. Results were calculated using values for D which spanned the expected experimental range, 1x10s to lxlO^cmVs. Tissue thickness, £, was set to 150 pm. UBLs were modelled as having thickness' of ^ =40 pm and 4=20 M-™/ with D*=5xl0"6 cm2/s. The cross-sectional area of the culture, A M C C , was set to 7i(0.45)2 cm2, reservoir volumes were set to 5 ml, starting drug concentration, c0, was set to 100 uM and the fraction of extracellular space, fex, was set to 1. Figure 2.8 shows results for drug concentration as a function of time using the above values with (3 set to zero. The tissue diffusion coefficient is set to (a) lxlO"5, (b) 3xl0"7 and (c) lxlO"8 cm2/s. Solid lines show numerical results and dashed lines show analytical results. Comparison of analytical and numerical results confirm that the numerical model is behaving as expected and is able to reproduce the analytical results. In panel (a) there is no visible difference between the two methods. In panel (b) there is a slight offset between analytical and numerical results and in panel (c), where the diffusion coefficient is reduced by another factor of thirty, the offset becomes larger. The increase in offset between the two methods with decrease in the rate of diffusion is consistent with the breakdown of the assumption used in the derivation of the analytical solutions, i.e. that the culture be highly permeable so that the time to reach steady state diffusion be negligible. The offset between the methods is due to the time required to reach conditions of steady state diffusion within the culture, which is neglected in the analytical derivation but not in the numerical solutions. Using equation (2.2), a rough estimate of the time to reach steady state diffusion for the conditions of panel (c) can be calculated as ~3 hours, which is consistent with the offset that is observed. Figure 2.9 shows results where the drug decay rate in media, (3, is set to lxlO"4 s"1, resulting in a half life of approximately 2 hours for the drug within the reservoirs. Once again the tissue diffusion coefficient is set to (a) 1x10s, (b) 3xl0"7 and (c) lxlO"8 cm2/s. In panel (a) results for the two methods show excellent consistency. In panels (b) and (c) the time delay observed in Figure 2.8 (b) and (c) comes into play again. The reason why the donating reservoir data sets 41 do not show the offset is because its effect on drug concentration is now negligible compared to the effect of drug decay within the reservoir. Modification of discretization step size Figures 2.10-2.12 show results for self-consistency tests where discretization step sizes were varied. The objective of such a test is to determine an estimate of the error that is introduced by the finite size of the spatial and temporal steps. Comparison of the difference between simulations made using increasingly fine steps will yield an estimate of the error that is incurred by using discrete equations to model a continuous system. In each figure, panels show an overlay of three curves, each made using a different number of culture slices: dotted lines show results when 3 slices are used for each UBL and 10 slices are used for the tissue, dashed lines show results when 3 slices are used for each UBL and 30 slices are used for the tissue, solid lines show results when 20 slices are used for each UBL and 100 slices are used for the tissue. The second choice of discretization step size corresponds to that used for data analysis in the following chapters. For the simulations, parameters were kept the same as those used in the previous section with the exceptions of p\ which was kept at zero, and fex, which was set to 0.25 for the two compartment modelling results shown in Figure 2.12. Figure 2.10 shows results for simple diffusion, with D set equal to (a) 1x10s, (b) 3xl0"7 and (c) lxl0"8cm2/s. The simulations show similar results irrespective of the number of slices used. In the case of the simulations using finest slices, results were changed' by less than 0.01% from those done using the moderate slices. When these results were repeated using a time step of one fifteenth that used initially, no noticeable change was observed in the curves, indicating that the temporal step size was set to a small enough value. Figure 2.11 shows results where the effect of drug loss through tissue metabolism was modelled in addition to drug diffusion. The metabolic terms Vm a x and Km were set to 5 uM/s and 75 uM respectively, roughly what was observed experimentally for tirapazamine. Results indicate that for diffusion coefficients above 3xl0"7 cm2/s, the numerical results using 30 tissue slices will be accurate to at least 1% of those with S=100. However, when modelling at lower 42 diffusion rates, the number of tissue slices must be further increased in order to achieve accurate results. In all cases switching to a time step that was fifteen times smaller than that used for the simulation produced only negligible changes to the results. Figure 2.12 shows results for a two-compartment model, using parameters similar those used in the doxorubicin flux analysis. Parameters were set as; fex=0.25, v2=1000 pm3, k=0.5 s"1, km=20, Bp=0.03 mM, kp=1.4 mM"1, Bm=55 mM, km=0.62 mM"1, n=0.24, kd=4300 mM"1, Bd=10 mM, pKa=8.4, pHj=7.2 and pHe=7.4 (minus 0.2 for every 100 um away from the edges of the tissue). Again, panels show results when D set equal to (a) lxlO"5, (b) 3xl0"7 and (c) lxlO"8 cm2/s. For diffusion rates above 3xl0"7 cm2/s the results show good self-consistency when the number of tissue slices is greater than 30. However, for lower rates of diffusion finer discretization steps are required. Again, switching to a time step that was fifteen times smaller than that used in the simulations produced only negligible changes to the results. 43 (a) 0 1 2 3 4 5 0 1 2 3 4 5 Time (hours) Time (hours) (b) Tjme (hours) Time (hours) Figure 2.8 Comparison of analytical and numerical solutions for a limited version of the diffusion model: one-compartment diffusion. Solid lines show results from the numerical method and dashed lines show results from the analytical method. The diffusion coefficient was set to (a) lxlO"5, (b) 3xl0"7 and (c) lxlO" 8 cm 2/s. A comparison of the results shown in (a) and (b) indicates that the numerical method is consistent with the analytical method. In (c) the numerical and analytical results diverge due to the breakdown in the assumption, required for application of the analytical solutions, that the time to reach steady state diffusion is negligible. 44 (a) Time (hours) Time (hours) Figure 2.9 Comparison of analytical and numerical solutions of a limited version of the diffusion model: one-compartment diffusion and drug breakdown in the growth media. The rate of drug breakdown, P, is set to lxlO"4 s"1 (t1/2 = 2 hours). Solid lines show results from the numerical method and dashed lines show results from the analytical method. The diffusion coefficient is set to: (a) lxlO' 5, (b) 3xl0'7 and (c) lxlO" 8 cm2/s. Results of (a) and (b) validate the numerical method, while (c) indicates that the time delay required to set up a continuous concentration gradient across the tissue culture is no longer negligible and hence the analytical solutions break down. 45 (a) o u u (b) 2 3 4 Time (hours) 0.05 0.04 1 1 1 r Receiving Reservoir 1 2 3 4 Time (hours) ~l 1 1 Donating Reservoir! 1 2 3 4 Time (hours) H 1 1 Donating Reservoir 1 2 3 4 Time (hours) (c) 0.0015 o U u 0.001 U 0.0005 r -2 3 Time (hours) lh-o 0.999 U 0.998 r -0.997 2 3 Time (hours) Figure 2.10 Comparison of numerical integration results using different spatial discretization: one compartment model, diffusion only. Each panel shows results where the number of discrete layers used to describe the tissue and each UBL were set to: (10, 3) - dotted lines, (30, 3) - dashed lines and (100, 20) -solid lines, where the notation (nun2) refers to the number of layers, and n2, that the tissue and UBLs are respectively divided into. The diffusion coefficient is set to (a) lxlO"5, (b) 3xl0"7 and (c) lxlO^cmVs. 46 (a) o U u 0 (k) 0.0006 0.0005 Q 0.0004 -u u Receiving Reservoir" _L o U i u 1 2 3 4 Time (hours) 2 3 4 Time (hours) n i i Donating Reservoir J 2 3 4 Time (hours) 1 2 3 4 Time (hours) (c) 1.5E-17 o u u 1E-17U 5E-18 \-n 1 n r Receiving Reservoir J L 1 2 3 4 Time (hours) 1.02 1 Q 0.98 U 0.96 U 0.94 | -0.92 0 —I 1 1 Donating Reservoir -I _L 2 3 4 Time (hours) Figure 2.11 Comparison of numerical integration results using different spatial discretization: one compartment model, diffusion and metabolism. Each panel shows results where the number of discrete layers used to describe the tissue and each UBL were set to: (10, 3) - dotted lines, (30, 3) - dashed lines and (100, 20) - solid lines, where the notation (n„n 2) refers to the number of layers, nr and n2, that the tissue and UBLs are respectively divided into. The diffusion coefficient is set to: (a) lxlO"5, (b) 3xl0"7 and (c) lxl0' 8 cm 2/s. Tissue metabolism terms V m a x and K m were set to 5 uM/s and 75 uM. 47 (a) Time (hours) T i m e (hours) Figure 2.12 Comparison of numerical integration results using different spatial discretization: two-compartment doxorubicin model. Each panel shows results where the number of discrete layers used to describe the tissue and each UBL were set to: (10, 3) - dotted lines, (30, 3) - dashed lines and (100, 20) -solid lines, where the notation (n„n2) refers to the number of layers, n, and n2, that the tissue and UBLs are respectively divided into. The diffusion coefficient is set to (a) lxlO"5, (b) 3xl0"7 and (c) lxlO"8 cm2/s. Values of the parameters describing drug binding and uptake are stated in the text. 48 3. Simulating Drug Distribution in a Tumour Cord Experimental estimates of the parameters governing drug kinetics within the MCC model can be applied to simulate drug penetration within a solid tumour. Tumour architecture was modelled assuming a repeating cord motif, in which blood vessels are surrounded by sheaths of tumour cells, see Figure 3.1. Modelling of drug distribution within this system is then done by examining drug penetration into a typical cord structure. This chapter describes the reformulation of the equations that were introduced in Chapter 2 to model drug kinetics within MCCs, to model drug kinetics within the cylindrical geometry of a tumour cord. Application of experimental estimates of the MCC model parameters, describing diffusion and reaction, to the cord model then allows for simulation of the spatial distribution of drug within a tumour cord as a function of time. Figure 3.1 Diagram of the corded architecture used in modelling drug kinetics within a solid tumour environment. A cord of cells with thickness b surrounds a blood vessel of radius a. As a drug diffuses away from the blood vessel it is subject to geometric dilution. 3.1 Drug Kinetics Using Cylindrical Geometry Diffusion based models that describe drug kinetics within a cylindrical geometry can be derived from equation (2.1) by switching to a radial co-ordinate system. The radial form of equation (2.6) is written as: dc 1 d frdc^ dt r dry dr (3.1) here the variable for drug concentration, c (M), is a function of time, t (s), and radial position within the tissue, r (cm) and D (cm2/s) is the drug's diffusion coefficient within the tissue. 4 9 Enzyme Metabolism Model The tissue model describing diffusion with Michaelis-Menten type metabolism that was used to describe tirapazamine kinetics within MCCs can be rewritten to suit a radial geometry. Using equation (3.1), the radial geometry analogue of equation (2.9) is: dc d (rdc^ dt r dr\dr j V -c max K ' +c' (3.2) Doxorubicin Model The two-compartment model used to describe doxorubicin kinetics within MCCs can also be reformulated to satisfy a radial geometry. Using equation (3.1), the radial geometry analogue of equations (2.28)-(2.31) is; dc, = p l d (rdclf dt r dry dr j k 2 C/i' c i , f f i ' c 2 , i ) > v dc2 IF : k ( f r c i , f - / 2 - c 2 , f ) and Ci=cw+c1 < b=cw(l + Bk), C 2 — C 2 , f C 2 ,m C 2 , d — C 2 , f B 2 mkm + • Bdkd 1 + k m C 2 , f l+kdC2,f (3.3) (3.4) (3.5) (3.6) 3.2 Boundary Conditions Boundary conditions satisfying the radial geometry must describe drug flux into the cord from the blood vessel as well as flux out of the cord and into neighbouring cords. Blood Vessel / Tissue Boundary The boundary condition at the blood vessel/tissue border, relates flux out of the blood vessel to the drug within the tissue (Crank, 1975); - D -dr -P v ( C v , f C l , f | r = a ) ' (3.7) here cV/f (M) is the concentration of free drug in the blood vessel, Pv (cm/s) is permeability of the blood vessel to the drug and 'a' (cm) is the blood vessel radius. The drug concentration in 50 the blood vessel is an external variable and can be assigned any desired functional dependence on time. Boundary at the outer edge of the tissue The condition at the outer layer of cells of the cord is chosen to describe the situation where efflux from the cord is equal to influx from other cords. Hence the net flux at the outer layer of cells will be zero. This condition is written as; - D -dr = 0, (3.8) where, a+b is the distance from the centre of the blood vessel to the outer layer of cells. 3.3 De-dimensionalization of Equations Using the transformations outlined in Table 3.1 and 3.2, a generalised form of equations (3.2)-(3.8) can be written in non-dimensional form. The tissue equations for extra- and intracellular drug concentration become: dCl=l d (RdCu^ dT RdR{ dR v V C ' k ~ (/i ' Q,f ~ f i ' C 2 , f) _ a , n ' ~ Y i ' Q,f / and dC2 Hr - k(/j • C1(f f2 • C2 f) y 2 • C2 f. The dimensionless versions of the relations between total and bound drug are: C =C + C =C 1 + — ^ v l + k b C f and C = C. +C +C =C 2 B2,mkm • + - Bdkd l + kmC 2, l + k dC 2, while the dimensionless boundary conditions for the tumour cord are: and dR ' dR = ^L(C -C I ) = 0. (3.9) (3.10) (3.11) (3.12) (3.13) (3.14) lR=l+b/a 51 Name Units Definition Value Non-dimensional cl,b M M M free extracellular drug concentration, bound extracellular drug concentration, total extracellular drug concentration, cu(x,t) f(c u ) Cl— C U + C l , m C,,,=c,, f/c 0 Ci,b=Ci / b/c 0 C,=c , /c 0 C2,f c 2,m C2,d c 2 M M M M free intracellular drug concentration, membrane bound intracellular drug cone, D N A bound intracellular drug cone, total intracellular free drug cone, cu(x,t) f(c2,f) f(c2,r) C 2 — C 2,f + c 2 m + c2 d C 2 , f = C 2 , f / C o ^ - 2 , m = C 2 , m / C o C 2 l d = C 2 . d / C o C 2 = c 2 / c 0 c v C v,b M M M total drug cone in blood vessel free drug cone in blood vessel bound drug cone blood vessel Cv,l~Cv-C b f(cv,f) C v =c v / c Cv,f=c v , , /c 0 C v , p =c v , p /c 0 r t cm s radial position within the rumour cord time R=r / a T=t D / a 2 Table 3.1 Variables used by the mathematical model describing drug flux wi thin a tumour cord. Name Units Definition Value Note Non-dimensional D cm 2 / s diffusion coefficient through extracellular space e n k d B2,m K B..b k„ Bk Bn M M/s rrrriHg M M " 1 M M'1 M M 1 M M " 1 p H 1 / 2 /l/2 fex V , v, P a b cm 3 / ce l l cm 3 / ce l l cm 3 / ce l l cm 2 / ce l l cm/s u n i m Michaelis-Menten half saturation level 75 u M 1 maximum rate of enzyme metabolism 1 oxygen partial pressure within tumour cord f(r) 1 Michaelis-Menten oxygen inhibition term 4 mmHg effective tissue uptake rate 2 concentration of cellular DNA base pairs 6-12 m M 2 D N A intercalation exclusion parameter 0.24 2 D N A intercalation affinity constant 4300 m M " 1 2 cone of intracellular membrane binding sites 55 m M 2 membrane binding affinity constant 0.6 mM" 1 2 concentration of extracellular binding sites 2 extracellular binding affinity constant 2 effective extracellular binding term ~Bib • k b 2 media concentration of FBS 0.03 m M 2 FBS binding affinity constant 1.4 m M ' 1 2 acid dissociation constant 8.4 2 p H of extra- or intracellular space see text 2 fraction of doxorubicin that is uncharged see text 2 fraction of extracellular space 0.2-0.4 2 volume of intracellular space per cell see text 2 volume of extracellular space per cell v 2 • f e x / ( l - f e x ) 2 volume occupied per cell v 2 / ( l - f e x ) 2 surface area per cell 2 cell membrane permeability k - v 2 / A c c l l 2 blood vessel radius 5 um tumour cord thickness 150 u m KB = K m / c 0 V m „ = V m a x a 2 / D / c n k = k / v 2 a 2 / D B d = B d / c c B2,„ = B2,m / c G K= Blib_= B u / c c k b — k b c c B p = Bp / q , kp= k p c 0 Table 3.2 Parameters used by the mathematical model describing drug flux wi th in a tumour cord. The notation ' X 1 / 2 ' indicates a contraction of two distinct parameters ' X / and ' X 2 ' , where 1 and 2 refer to the extra- and intracellular compartments. Notes: 1 - parameter used specifically for modelling tirapazamine, 2 - parameter used specifically for modelling doxorubicin flux. 52 3.4 Discretizing the Equations Equations (3.9)-(3.14) can be discretized and integrated numerically using the Crank-Nicolson integration scheme (Crank, 1975). This technique involves discretizing the equations using finite steps of time and space. The discretized equations are then approximated to a system of linear equations that relate drug concentrations at all positions at a given time to the drug concentrations at a previous time step. Figure 3.2 shows the spatial discretizing used for the tumour cord. The tissue of radial thickness b is divided into N cylindrical shells. Figure 3.2 Diagram of the spatial discretization scheme used for the tumour cord model. Tumour tissue of thickness b surrounds a blood vessel of radius a, r is the spatial co-ordinate within the tissue. The tissue is divided into N shells and drug concentration is modelled assuming radial symmetry. Following the Crank-Nicolson scheme, the partial derivative of drug concentration with respect to time is discretized for small a time step, 8T, as being: dC (Cj+1-Cl) dr ar ' (3.15) where the integer subscript i indicates spatial positioning within the tissue, with i nirvning between 0 and N. The temporal variable is represented by the integer superscript;'. The spatial positioning variable r can be written in discrete form as; r^a + i&c (3.16) which can then be written in non-dimensional form as; R: =1 + iSR -r z + -SR <5R = z"<5R, (3.17) where z'=z+l/8R. The spatial derivative is then discretized as; 53 1 d (RdC)=d2C | 1 dC R <9R I oR (9R2 R oR ffCL-2C/+C>.1+Cig-2Cr+Cff>i1 2(<5R)2 i (cL-cL+cg-cK) i'SR 45R (3.18) Which can be rewritten as; <92C 1 <3C 1 1 • + — dR2 R dR 2 (SR)2 C ^ ( 1 + 2 7 ] " 2 Q + 1 + C - f 1 + 2 7 +q+1\i+— \-2q+cu 2V (3.19) Discrete Tissue Equations The relation for extracellular drug flux within the tissue cord, equation (3.9), can be discretized by substituting in the discrete relations of equations (3.15), (3.17) and (3.19), yielding; B b k b 1 + - 6 - j L -i+kbQ,f; Vc ; + 1-c ;) SI 1 1 2(<5R)2 i+—l-2C1/+1+c1 H1+— + C ' I 1 + — ] -2C/+C/' fl + — 2i"' J_ 2z" Yi + — v „ K ' + C / m 1, j : •fij f >+C >+1^ Jl' O /2 ' T for 0<z'<N (3.20) Which can be regrouped as; 54 _c M . J L f i + J ^ ^ ' ( S R f l 2i' ' ' Bbkb ^ 1 + b— V i+k bcw; j (SR) + 2-ST Yi + — ST -c 2, fr/ 2k^sr M , + 1 (SR) 2 l 2i' M (SR) 2 l 2f ' ' Bbkb A 1 + b—b-V i+k„c1 ( f; , ST (SR)2 - / . k ^ S T - Yi + — V_ m l , t | ST +c 2, f;-/ 2k^ST for 0<z'<N (3.21) Similarly, discretization of equation (3.10) yields: 1 + - B_k. Bdkd i+kmc2,f; i+kdc2,f; i J ST f r >+i + r ' c i+1 + c VI r ^l,fj T *--l,f,- _ r * " 2 , f i T * "2,f; 7 i 2 2 -Y2 C >+1+C ;' for 0<i<N (3.22) Which can be regrouped as: i+1 Bmkm Bdkd i+kmc 2 / f; i+kdc2, f; +/2ksr+/3sr [-c uf 1 k sr c ; -Bmkm Bdkd ^ 1 _j_ m m | a a v i+kmc2,f; i+kdc2,f; -/2kST-/3ST [+cw; -/x k sr for 0<z'<N (3.23) Equations (3.21) and (3.23) define a set of linear equations, which relate drug concentrations within the tissue at the ;'+lth time interval to concentrations at the ;-th time interval concentrations. Discrete Blood Vessel / Tissue Boundary Condition A discrete form of the boundary condition for the border between the blood vessel and tissue, equation (3.13) can be written using a ghost layer as (Crank, 1975); 55 C ' -C 1 A P 2SR (3.24) Which can be rewritten as; C 1 =C ' +2SR^-(C 1-C ') (3.25) To remove reference to the ghost layer, equation (3.25) is substituted into equation (3.21), with i set to zero, yielding; ST -C 1+1-2 SI (SRJ ( r i +c 1+1 1 + - b—b-1 + kbCM;o + 2-SI (SR)2 +f1k^ST + v. Yi + — v„ m l.fn ST D SRl 2V C ; -2 (SR)2 1 + - b—b- , ST (SR)2 -flk-±8I- Y,+17T V_ aP„ SI -2=-*-D SR 1-— 2i" ST + c 2 ; . / 2 k ^ s r + v^,f +cv,f j D SRv 2Z" for f = 0 (3.26) Equation (3.26) describes a linear relation between ;'+lth time interval drug concentrations and the /* time interval concentrations at the boundary between the blood vessel and the tissue. Discrete Boundary Condition at the Outer Edge of the Cord A discrete version of equation (3.14), the boundary condition at the outer edge of the cord, can be written using a ghost layer as (Crank, 1975); C 1 -C > 1-fN+l l.^N-l _ Q 2SR (3.27) which simplifies to; C ' =C 1 ^ i , f N + i ' - M N - r (3.28) 56 To remove reference to the ghost layer equation (3.28) is substituted into equation (3.21) yielding the following linear relation between drug concentrations for the ;'+lth time interval at the N* layer: -C ; + 1 -2 SI (SR)2 +C i + 1 f \ Bbkb A 1 + v 1 + k b C i , f N y + 2-SI '(SR)2 +/1k-^ 2-<5T + • 1 Ts< v_ ST - C ^ k ^ S T v, C ' -1 (<5R)2 Bbkb ^ 1 + fe—^ V ST (SR)2 -/.k^ST-v, Yi + V„ K ' m + C l , / N SI +cJN-f2^si v (3.29) 3.5 Integrating the System Equations (3.21), (3.23), (3.26) and (3.29) combine to relate changes to drug concentrations throughout the system between the f1 and /+lth time intervals. Together they form a set of 2(N+1) linear equations and 2(N+1) unknowns that can be solved algebraically to determine they+l* drug concentrations as a function of the/"1 drug concentrations. Hence, given some set of initial drug concentrations at ;=0, drug concentrations at any later time ; can be calculated iteratively from the starting time. Initial Conditions Initial drug concentrations within the tissue are set to zero at time zero; c °=0 2,f ,• U , 0<z'<N. (3.30) Since the blood vessel drug concentration is modelled as an independent variable, it can be assigned any functional dependence on time. Whatever average value it has during each time step is then substituted into the equations. 5 7 Discretization Step Sizes The tissue section was divided using steps of 8R=5 pm. Time steps were set to 8t=15 sec, with the exception of the first 200 steps were made with 8t=0.00015 s and the following 199 were made with 8t=0.03 sec. This was done to avoid computation errors that occur when the large concentration gradients initially present within the cord are combined with large time steps. Solving the Equations The system of linear equations that described the complete tumour cord model, equations (3.21), (3.23), (3.26) and (3.29), were solved in matrix form using a matrix inversion method described in Numerical Recipes in C (Press, 1992) to solve for drug concentrations for the ;+lth time step as a function of the concentrations from the;'"1 time step. 58 4. Electrical Impedance Spectroscopy Theory Electrical impedance spectroscopy (EIS) (a) (b) utilises the inherent electrical properties of individual cells to quantitate macroscopic parameters related to the tissue environment (Cole and Cole, 1941, Foster and Schwan, 1989, Schwan, 1963). On a conceptual level, 1 Figure 4.1 (a) Illustration of a simplified cell the cell cytoplasm and extracellular space act environment and (b) the equivalent electrical circuit used in the interpretation of electrical impedance measurements. Key impedance model parameters include: extra- and intracellular resistance, R. and R,., and membrane capacitance, C m . as conductive media, which are isolated from each other by the cell membrane, Figure 4.1 (a). A simple electrical circuit, Figure 4.1 (b), can be constructed to represent these terms; where the conductivity of the extracellular space and cell cytoplasm, contribute the resistive components, Re and R; (largely due to the presence of salt ions) and the cell membrane contributes the capacitive effect, Cm. Deterrnining experimental values for these parameters allows for empirical comparison of impedance data with physical traits such as MCC thickness. Impedance results can also be used to estimate the relative barrier to diffusion posed by MCCs grown using different types of cells. Impedance spectroscopy involves passing an alternating current over a range of frequencies through an object to determine the functional form of its electrical impedance. The application of this technique to biological material has been recently described by numerous investigators (Ackmann, 1993, Bao, et al., 1993, McRae, et al., 1997). The basic premise of the four terminal system used to make measurements is illustrated in Figure 4.2. Two large area stainless steel electrodes are used to pass a sine wave current of known 1=30 uA r.m.s. j current electrodes voltage electrodes to gain/phase detector Figure 4.2 Schematic of a typical, four-electrode impedance measurement set-up used for biological material. Current is passed through two large area electrodes by a constant current source driven by a sine wave generator. The voltage drop across the material is detected by two smaller electrodes connected to a high impedance differential amplifier that leads to a gain/phase detection system. 59 amplitude and frequency through the material under study. The voltage drop and phase shift generated by the material is then measured by two small area chloridized silver electrodes. How the voltage drop and phase shift vary with frequency is used to determine the functional form of the magnitude, I Z I (Q), and phase, <|) (°), of impedance of the culture. Where, an arbitrary impedance can be written in the form: Z=A+z'B, (4.1) with IZ12=A2+B2 (4.2) and (j>=arctan(B/A), (4.3) here A (Q) and B (Q) are the real and imaginary components of the impedance and z'=(-l)1/2. 4.1 M C C Impedance Modell ing The electrical properties of conduction and capacitance of the tumour cells are modelled empirically using a variation of the Cole-Cole relaxation equation (Foster and Schwan, 1989) in which the membrane capacitive term is replaced with a constant phase element term (CPE): Z c e U - R i + T T ^ T n - ^ (4-4) where R; (Q.) is the net intracellular resistance, Y0 (Q "1/n-s/rad) and n are the CPE terms which model membrane capacitance, co (rad/s) is the angular frequency defined as &*=2jtf, with f (s"1) the sine wave frequency at which the measurements are made and fn=cos(nJt/2) + i sin(n7t/2). The CPE term is thought to better model the existence of a distribution of relaxation times often seen in biological material, see Rigaud for a recent review (Rigaud, et al., 1996). When n is set to 1, the second term in equation (4.4) describes a purely capacitive effect. Generally, for biological tissue, n is found to be between 0.5 and 0.8 (Foster and Schwan, 1989). The total impedance for the MCC can then written as a combination of Z c e l l and net extracellular resistance, Re (Q): 7 = fj_ + J_ V1 R. V ^ e Zcell J Ri(ifl)Y0)n+l (Re+Ri)(za)Y0)n+l (4-5) 60 This relation can be expanded and then written in the form of equation (4.1), such that A and B are equated to functions of R e , R i 7 Y0, n and co. Experimental estimates of the parameters R,, Rj, Y„ and n can then be determined, using equations (4.2) and (4.3), through measurement of the voltage drop and phase shift generated by the MCC over a range of frequencies. In order to relate equation (4.5) to A and B from equations (4.1) the zn term must be expanded using zn=cos(n7i/2) + i sin(n7t/2) and reference to i must be removed from the denominator. This leads to: R, R; (fl)Y„)" (cos(n7r/2) + isin(nw/2)) + i l l + (R, + Re )(fl>Y„ )n (cos(n7r/2) - zsin(n;r/2)) — M l _ e r -|2r -|2 l + (R1+Re)(fl)Y0) cos(n^ /2)j [(R, +Re)(aY0)nsin(n;r/2)] which can then be expanded as: Re[(2R, +Re)(ft)Y0)"cos(n^ /2) + Ri(Ri +Re)(q)Y0)2n +l-fRe(a;Yo)nSin(n^/2)] ,(4.6) •"MCC L + (R; +Re)(«Y0)ncos(n^/2)] [(R, .+ Rj(©Y0)nsin(wr/2)' (4.7) Then, writing the total impedance as the sum of the media resistance, R^ , and the MCC impedance leads to: ^ total — ^ M C C • (4.8) The two parameters, A and B, of equation (4.1) can then be written as: Re[(2R; +Re)(6jY0)ncos(n /^2) + Ri(Ri + Re)(ft>Y0)2n +1 A = R +-1 + (R, +Re)(«Y0)ncos(n7r/2)] [(Ri + Re)(fl>Y0)nsin(n;r/2)] and B = Re -Re(^Y0)nsin(n^/2)] l + (Ri+Re)(coY0)nc :os(n^/2)]2[(Ri+Re)(fl)Y0)nsin(n^/2)]2 (4.9) (4.10) A and B can then be directly compared with experimentally determined values for I Z I and <|> through equations (4.2) and (4.3). This in turn allows for estimates of the parameters Re, Rir Y0 and n. 61 4.2 Estimating Tissue Tortuosity and the Fraction of Extracellular Space i n M C C s If the conductivity of the growth media and the culture thickness are known, then an estimate of the combined effect of extracellular space and tissue tortuosity can be obtained through measurement of Re. In general, the resistance of a cylindrical piece of purely conductive material can be written as: £ 1 R = T - - ' ( 4- n) A a where £ (cm) is the thickness, A (cm2) is the surface area and a (Q^ -cm"1) is the conductivity of the material. Applying this to the low frequency impedance of an MCC, where the cell membrane acts as an insulator and one is effectively measuring the resistance of the extracellular space, yields the relation: £' 1 R.=4T-—' ( 4 - 1 2 ) A <je where ae (Q -^crn1) is the conductivity of the extracellular medium and £' (cm) and A' (cm2) are the effective thickness and cross-sectional area of the extracellular space. Taking £' as equal to £-X, where X is the tortuosity factor (Nicholson and Phillips, 1981) then leads to A' being equal to fex -VT/(£-X) - fex -A/X. Here fex is the fraction of extracellular space which is defined as fex=Ve/VT, where Ve and V x are the volumes of the extracellular space and total MCC space respectively. Substituting these results into equation (4.12) leads to the following relation: £-X2 1 Af e x cre Measurement of Re and £ and knowledge of A and oe, then allows for an estimate of the combined effect of tortuosity and fraction of extracellular space to be made. Rearranging equation (4.13) and defining 9 as the ratio of f e x A 2 , the factor by which tortuosity and extracellular space will modify measurement of Re, leads to: ^ 1 1 <PE,S = — • — • — . (4.14) 62 In a similar fashion, the effect of tissue tortuosity and the fraction extracellular space on the diffusion of small molecules through the extracellular space of MCCs can described using the Fickean diffusion relation of equation (2.1). If the molecule is constrained to the extracellular space then A and £ can be substituted with A' and £', as was done in equation (4.12), which leads to the following modified version of equation (2.1): Flux = D'A' — = ^ -D*A— = <pdiffD*A—, (4.15) £' A £ d l f f £ here D* is the diffusion coefficient of the molecule in water and (pdiff is the diffusion analogue of the impedance parameter, (pEIS. The parameter, (pdiff, will vary between cell lines and serves as a relative measure of the barrier presented by MCCs to the passage of a molecule which is constrained to diffuse through the extracellular space. 63 5. Materials & Methods 5.1 Cel l Culture Monolayers SiHa (Human cervix squamous cell carcinoma) cells were purchased from American Type Culture Collection. V79-171b cells (Chinese hamster fibroblasts) and V79/ADR cells, cultured to exhibit multi-drug resistance, were obtained from Dr. Ralph Durand (B.C. Cancer Research Centre, Canada). Cells were grown in monolayers using minimum essential media (10437-028, GIBCO BRL, Burlington, ON) supplemented with 10% foetal bovine serum (15140-122, GIBCO BRL) and passaged every 3 to 5 days upon reaching confluence. On every fourth passage, the V79/ADR cells were exposed to 5 pg/ml doxorubicin (Faulding; Vaudreuil, QC) in order to maintain their drug resistance. Cells were maintained as monolayer cultures on tissue culture plates (Falcon, Franklin Lake, NJ). Incubator temperature was kept at 37°C and gassing was at 5% 0 2 , 5% C 0 2 and 90% N 2 . (a) During passaging, cell suspensions were obtained from monolayer cultures by mcubating with 1.5 ml trypsin-EDTA (Sigma, Oakville, ON, 1 g/1 porcine trypsin, 0.4 g/1 EDTA) at 37 °C for 3-5 minutes. Doubling times were determined to be approximately 21 hours for SiHa cells, 14 hours (Chan, 1998) for V79 and 23 hours (Chan, 1998) for V79/ADR. (b) gas in Media MCCs MCCs were grown by seeding sub-monolayer densities of cells onto the permeable membrane culture insert M C C plastic membrane Magnetic induction of tissue culture inserts (Millipore, Nepean, ON, Figure 5.1 (a) Photo and (b) sketch of MCC growth box. Up to 6 MCCs are supported in a C M 12 mm, pore size 0.4 pm). Prior to adding to plastic cage immersed in 120ml stirred media. Gassing is provided at ~5ml/min (5% 0 2, 5% cells, the surface of the tissue culture insert co2 and 90% N 2) and the entire box is kept at 37 membrane was coated with 50 pi of 0.75 mg/ml C u n d e r f o r c e d a i r h e a h n 8 -64 collagen (type I, Sigma Chemical, Oakville, ON) dissolved in 60% Ethanol with 50 pi of 1 M HC1 and allowed to dry overnight. Approximately 4xl0 5 cells, in 0.5 ml growth media, were then inoculated onto the coated surface of the membrane and incubated for 6-12 hours to allow the cells to attach. Silicone o-rings placed around the inserts were used to support them in a frame with slots for six inserts. The frame was then completely immersed in 120 ml of stirred media, see Figure 5.1. Stirring was carried out using one-inch magnetic stir bars that were spun at 500 r.p.m. Gassing was provided using 5 foot sections of 0.05 mm Peek tubing held at 7 p.s.i., which provided a flow rate of ~5ml/min. Cultures were incubated for up to 8 days with continual gassing (5% 02, 5% C02 and 90% N2) at 37.5° C, medium was changed daily after the first three days. For a description of the incubator in which the growth boxes were kept see Appendix II. Figure 5.2 shows a section of an 8 day old SiHa MCC. It is approximately 200 pm thick and rests on the -25 pm thick permeable plastic growth membrane. 5.2 Methods for Diffusion Experiments Drugs Doxorubicin was purchased from Faulding (Vaudreuil, QC). Tirapazamine was obtained from Sterling Winthrop (Collegeville, Pa). Azomycin was obtained from Sigma (Oakville, ON). The nitroimidazoles Misonidazole, Etanidazole and Pimonidazole were a gift from Dr. Peter Wardman (Gray Laboratory, UK). Radiolabelled inulin and tritiated water were purchased from Amersham Life Science (Oakville ON). Experimental Set-up and Apparatus Diffusion experiments were performed using the dual reservoir diffusion apparatus shown in Figure 5.3. The apparatus consists of two dual-well reservoirs made from machined plexiglass. Reservoir lids contained sampling ports as well as fittings for gassing via Peek tubing. Prior to the experiment, MCCs were selected and visually assessed for uniformity. The cultures were 65 Figure 5.2 H&E stained cryostat section of an 8 day old SiHa MCC. The culture, approximately 20 layers thick, lies on the permeable, plastic growth membrane. then placed in the diffusion apparatus with 6-10 ml medium per reservoir. Each reservoir was individually gassed, 10 ml/min, and maintained at 37°C via forced air heating. Each reservoir was stirred using a magnetic stir bar (15 mm x 4 mm) at 450 r.p.m.; higher stir speeds caused excessive disturbance of the culture. For a description of the magnetic induction system used to spin the stir bars see Appendix n. Experiments were carried out over time periods ranging from several hours to several days. Reservoirs were sampled periodically using 50 ill or 100 ul Hamilton syringes (Hamilton Inc., Reno, NA) and samples were placed in 1.5 ml plastic Eppendorf microcentrifuge vials (Hamburg, Germany). HPLC Analysis (b) Reservoirs C A I C B J ! Stir bars M C C HPLC analysis of samples taken from the diffusion 5 3 (a) P h o t o a n d (b) d i a & a m o f diffusion apparatus. MCCs are used to apparatus reservoirs was generally begun midway separate two stirred reservoirs, which are gassed and maintained at 37°C. through drug flux experiments. Analysis was made using a Waters HPLC system (Mississauga, ON) which included a model 510 pump, model 712 WISP Injector, a model 996 Photo-Diode Array (PDA) Detector and a 474 Scanning Fluorescence Detector. All eluents were made using HPLC grade products and distilled and de-ionised water was used for all dilutions. Before running sample analysis, HPLC eluents were degassed via vacuum filtration (0.5 um FHLP filter, Millipore, Mississauga, ON). Sample Preparation Reservoir medium samples were prepared for HPLC analysis immediately following their acquisition. Samples were deproteinated by adding an aliquot of a 40% aqueous solution of ZnS0 4 (volume equal to 10% of sample volume) following which samples were shaken for a few seconds and methanol, of equal or greater volume than the original sample, was added. After shaking, the protein precipitate was pelleted by centrifugation for 10 minutes at 7 x 103 r.p.m. (radial distance of ~5 cm). Sample supernatant was then removed and placed in glass 66 HPLC vials. Use of ZnS04 allowed up to a 1:1 ratio of medium to methanol while mamtaining effective protein precipitation. Nitroimidazoles Reservoir medium samples were 100 pi and an equal volume of methanol was used to pellet protein. After preparation, samples were placed into low volume (10-200 pi), re-useable glass HPLC vials. Simultaneous flux experiments with etanidazole and misonidazole were quantitated using a Kromasil Eka Noble HPLC column (4.6 x 250 mm, KR100-5C18, Bohus, Sweden) and a two eluent HPLC method for molecule separation. Molecule concentration was quantitated via absorbance at 340 nm using the PDA detector. The two eluents used for molecule separation were: A - buffered H20 (50 mM ammonium dihydrogen orthophosphate, pH set to 3.9 with" HCl) and B - 90% Acetonitrile. The eluents were run through the HPLC column, flow rate 1.5 ml/min, using the following gradient: Time (min) % A % B 0 100 0 0.5 100 0 12.5 55 45 15 100 0 22 100 0. Using this method, sample injections of 100 ul were made whereby etanidazole eluted after 3.7 minutes and misonidazole after 7.5 minutes. Figure 5.4 shows a plot of peak area response, 1.5E+06 3 < s o (X cn <y 1.0E+06 5.0E+05L 1 1 1 1 A Misonidazole i i v Etanidazole A ' , - ' ' -- r ' • t i i 1 1 20 40 60 80 Concentration (]iM) 100 Figure 5.4 Calibration curves for HPLC response to Etanidazole and Misonidazole using lOOul injections. Data are fitted to a straight line; Etanidazole, slope = 1.07 x 10" Au-s/uM, intercept = 2.3 x 104 Au-s. Misonidazole slope = 1.27 x 104 Au-s/uM, intercept = 0.7 x 104 Au-s. Concentration is that of the sample before processing, which dilutes the drug by a factor of 2.1. 7.5E+05 3 < C o O H CO 5.0E+05U 2.5E+05L 1 1 1 1 A Pimonidazole 1 1 1 _ * Etanidazole , 0 ' ' 1 i 0 20 40 60 80 100 Concentration (pM) Figure 5.5 Calibration curves for HPLC response to Etanidazole and Pimonidazole for 75ul injections. Data is fitted to a straight line through zero; Etanidazole, slope = 8.01 x 103 Au-s/uM, intercept = -2.2 x 103 Au-s. Pimonidazole slope = 8.02 x 103 Au-s/uM, intercept = -5.7 x 103 Au-s. Concentration is that of the sample before processing which dilutes the drug by a factor of 2.1. 67 the integral of absorbance units (Au) over time, versus concentration of etanidazole and misonidazole that was obtained using this procedure. Quantitation of samples from flux experiments where etanidazole and pimonidazole were simultaneously diffused through MCCs was carried out using the Kromasil Noble HPLC column. A two eluent HPLC gradient was used, with: eluent A - buffered H20 (100 mM glycine, 5 mM hydro-sulphuric acid, pH adjusted to 3.9 with HC1) and eluent B - 90% Acetonitrile. The gradient ran at 1.5 ml/min and was set as follows; Time (min) %A %B 0 95 5 16 75 25 17 95 5 23 95 5 Using this method, 75 pi injections of samples were made whereby etanidazole eluted after 3.7 minutes and pimonidazole after 16.7 minutes. Absorbance detection was carried out at 340 nm. Figure 5.5 shows a plot of peak area response versus concentration of etanidazole and pimonidazole that was obtained using this procedure. Tirapazamine Reservoir medium samples were 50 pi in volume and 100 pi of methanol was used to pellet protein. After preparation, samples were placed into low volume (10-200 pi), re-useable glass HPLC vials. A Symmetry C18 HPLC column (3.9 x 150 mm, Waters, Mississauga, ON) was used for sample separation. Samples of 40 pi were injected using a mobile phase mixture of acetonitrile and H20 (0.13:0.87) flowing at 1.5 rnl/min, whereby tirapazamine eluted after 1.8 rninutes. Absorbance detection 3 1 • was carried out at 460 nm using the PDA detector. Metabolites SR 4330 and SR 4317 were separated from the tirapazamine peak with retention times of 5.3 and 6.2 minutes 1 o 3 1.0E+06 7.5E+05 5.0E+05 2.5E+05 0.0E+00 U v Tirapazamine y _l_ _L _1_ _L 0 20 40 60 80 100 Concentration (uM) Figure 5.6 Calibration curves for HPLC response to 40 pi injections of Tirapazamine. Data are fitted to a straight line, slope = 9.89 x 103 Au-s/uM, intercept = -2.1 x 103 Aus. Concentration is that of the sample before processing which dilutes the drug by a factor of 68 respectively. Figure 5.6 shows the plot of peak area response versus tirapazamine concentration that was obtained using this method. Doxorubicin Reservoir medium samples were 50 pi in volume. Due to large differences in the magnitude of drug concentration present in donating and receiving reservoirs, samples from the donating reservoir were diluted relative to receiving reservoir samples and disposable glass vials were used to avoid sample contamination. To meet the rrunimum volume of the disposable vials (400 pi), methanol aliquots were of 395 ul for receiving reservoirs and of 995 pi for donating reservoirs. To circumvent the effect of high methanol concentration on peak retention time and height, methanol aliquots were diluted with water such that final concentration of methanol was always 50%. A Symmetry C18 column (3.9 x 150 mm, Waters, Mississauga, ON) was used for sample separation. Samples of 30 pi were injected using 20 40 Concentration (uM) 60 (b) Ol c o a, OS 7.5E+05 1 1 1 ~ o Doxorubicin o -o 5.0E+05 -2.5E+05 X -0.0E+00 i i i 1 0 1 2 3 Concentration (uM) Figure 5.7 Calibration curves for HPLC response to Doxorubicin using 30 pi injections, (a) High concentration data, fitted to a straight line slope = 2.93 x 105 mV-s/uM, intercept = -1.6 x 105 mVs. (b) Low concentration data fitted to a quadratic a mobile phase of 90% acetonitrile/buffered H20 equation, quadratic term = 5.34 x 104 m V s / u M 2 , linear term = 6.58 x 104 mV-s/uM, constant term =0.1 mixture (0.3:0.7) flowing at 1 rm/min, whereby x 10" mV-s. Concentration is that of the sample before processing according to donating reservoir doxorubicin eluted after 4.1 minutes. The protocol, where samples are diluted by a factor of 21 prior to injection. buffered H 20 eluent consisted of 0.1 M ammonium acetate, pH adjusted to 4 with HC1. Detection was carried out using the 474 Scanning Fluorescence Detector using an excitation wavelength of 480nm and an emission wavelength of 560 nm (bandwidth 40 nm). 69 Scintillation Counting Scintillation counting was carried out using a LKB/Wallac 1214 Rackbeta liquid scintillation counter (Turku, Finland). Medium samples were 50 pi in volume and were added to 1 ml of 30% Scintisafe™ (Fisher Scientific, Nepean, ON) in clear plastic 1.5 ml Eppendorf (Hamburg, Germany) centrifuge vials. Vials were shaken vigorously and then left for 2 hours before performing scintillation counting over a 5 minute time interval. Counting efficiency was in the range of 40%. Fluorescence Imaging Digital images of the distribution of doxorubicin fluorescence in cryostat sections of MCCs were obtained using an ITT camera (Fort Wayne, IN), mounted on a Zeiss epifluorescence microscope. Excitation was at 546 nm and emission detected at 590 nm. Captured images were 482x752 pixels (-600x900 pm field of view) using 8-bit greyscale. Images were captured the same day cryostat sections were made. Fitting Drug Flux Data Estimates of the model parameters listed in Table 2.2, for drug diffusion, metabolism and binding, were obtained from drug flux experiment data via standard, non-linear chi-square minimisation fitting techniques (Press, 1992). Flux data from both reservoirs were fitted to the mathematical model simultaneously in order ensure a mass balance in terms of flux into and out of the tissue and drug lost through metabolism or binding. Calculations were done using Ansi-C programming language on a power Macintosh computer. Division of the Total Insert Permeability between the Plastic Membrane & Unstirred Boundary Layers The barriers to diffusion presented by the plastic insert and unstirred boundary layers (UBLs) were modelled as additional layers through which the drug had to diffuse. Figure 5.8 shows a diagram of the physical orientation of these barriers. Their effect on drug flux was modelled mathematically in Chapter 2 by treating UBLA as a layer with permeability PA, located to the 70 left of the tissue, and treating the net effect of UBLB and the plastic insert as a single layer located to the right of the tissue, with net permeability PB. The permeabilities, PA and PB, of the two layers were estimated by performing flux experiments at different stirring speeds using blank plastic inserts (no cells). Figure 5.9 shows data from a series of experiments, carried out over a range of stirring speeds, that measured the flux of tritiated water through the plastic insert and UBLs. The net permeability that was measured experimentally can be expressed as the combined permeability of the UBLs and plastic insert, which add inversely as; _ L - _ L + J _ + J L - J _ + 1 (a) (b) M y / \ PA PBI PB2 PA PB P P A 1 B l P P A B Figure 5.8 (a) Diagram of the barriers a drug must pass through when going from one reservoir to the other for a culture insert with no cells. From left to right they are U B L A / plastic inset and UBL B . (b) The effect of the plastic insert and UBL B can be modelled as a single barrier by combining permeabilities PB 1 and PB 2. (5.1) 3E-03|-s 2E-03 8 IE-03U 01 ca-using equation (5.1), the values of PA and PB, at a given stirring speed f, can be written as; OE+00 i 1 1 r As stirring speed is increased, the effect of the unstirred boundary layers will become less important and Pnet will approach PB1 the permeability of the plastic insert alone. From the data shown in Figure 5.9, net permeability approaches (3.5±0.5)xl0"3cm/s when stir speed is extrapolated beyond to infinity. At the stirring speed of typical flux experiments, 450 r.p.m., the net permeability is seen to be (1.65±0.05)xl0"3cm/s. 8 J I I L Pnet(0 l - P „ e t ( f ) / P n e t W (1 + r) P n e t (f) "(1 + r) (5.2) (5.3) r + P n e , ( f ) / P n e t H where, f (r.p.m.) is stirring speed, Pnet(f) is the net 0 200 400 600 800 1000 Stir Speed (r.p.m.) Figure 5.9 Plot of permeability versus stirring speed for diffusion of tritiated water through a plastic insert with no cells. As stirring speed increases the effect of the UBLs diminishes and permeability increases towards a maximal value equal to the permeability of the plastic insert alone. From the data the permeability of the plastic insert is estimated to be (3.5±0.5)xl0 3cm/s. 71 permeability at stirring speed f, Pnet(°°) is the net permeability extrapolated to infinite stirring speed and 'r' is the ratio of the permeability of UBLA to UBLB, P A / P B 2 . A first order approximation for the relation between the permeabilities of the UBLs would be that P A =P B 2 , however from the geometry of the diffusion set up we know that UBLA will be greater than UBLB, due to obstruction by the walls of the plastic insert, hence P A <P B 2 . A reasonable second order approximation would be to take P A =0.5P B 2 , i.e. r=0.5. Using this one then obtains: PA=4.7xlO" 3 cm/s and PB=2.5xlO" 3 cm/s for diffusion of tritiated water at 450 r.p.m.. The maximum error for these two values can be obtained by calculating P A and P B using the lirmting values for of r: r=l when the UBLs have equal permeabilities and r=0 when UBLB is negligible when compared with UBLA. Using r=l one obtains P A =6xl0" 3 cm/s and P B=2.2xl0" 3 cm/s. When r=0, P A = 3 . 0 x l 0 3 cm/s and PB=3.5xl0" 3 cm/s. Hence values for the two permeabilities can be taken as PA=(4.7±1.5)xlO"3cm/s and PB=(2.5±l)xl0" 3cm/s. While the mdeterrninacy in these values appears quite large, it is important to note that for any value of r, P A and P B always add up to the true permeability of the layers, P n e t / and hence variation in Y only changes how the net permeability is divided between P A and P B . In the case of a drug that solely undergoes diffusion as it passes through the tissue culture, the order of the diffusive barriers is not important and choice of r will not affect the end results. It is only when drug reactions are allowed, and the actual drug concentration at each cell layer need be known, that it becomes important to know if a diffusive barrier comes before or after the tissue. In this case, the resultant error in analysis of drug reaction within the tissue due to indeterminacy of P A and P B , will still not be significant because the barrier to diffusion posed by the plastic insert and UBLs is generally much smaller than that of the tissue itself and hence the concentration drop over the UBLs will be negligible when compared to the drop across the tissue. Experimentally, the net permeability of the plastic insert and UBLs is more than 20 times that of the tissue; i.e. it comprises less than 5% of the total barrier to diffusion. Hence a 20% uncertainty in the division of P n e t between P A and P B reduces to less than a 1% overall error in uncertainty of concentration within the tissue. These results have been obtained using tritiated water, however they can be carried over to other drugs by measuring Pn e t(450 r.p.m.) for each individual drug and then using the values for 72 the ratio of Pnet(450 r.p.m.)/Pnet(°°) and for 'r' that were determined from tritiated water. Net Permeability of the Plastic Insert to Drugs The net permeability of the plastic insert to each of the drugs studied in this report was determined from flux experiments using blank inserts. Results for Pnet were used to calculate values for PA and PB/ using equations (5.2) and (5.3), which were in turn used to account for the effects of the plastic insert and UBLs on data for drug flux through MCCs. Figure 5.10 shows data for flux of misonidazole, etanidazole and doxorubicin through blank inserts. Each panel shows drug concentration data from both the receiving and donating reservoirs, normalised to the starting concentration of the donating reservoir: 150 uM, 150 uM, 30 uM respectively. Experiments were carried out using the standard conditions of temperature, gassing and media 0 10 20 30 40 50 Time (hours) that were used for regular MCC flux experiments. Flux F i g u r e 5 1 0 F l u x d a t a from d i f f u s i o n , , , . , , . t h r o u g h blank inserts. Each panel shows data were fitted using the one compartment diffusion , . , , , ° r data from the donating and receiving model described in Section 2.2, to determine the net permeability of the plastic insert and UBLs. In the case reservoirs, normalised to the starting concentration of the donating reservoir. Data were fitted using the one compartment model for diffusion. The of the misonidazole and etanidazole data, no drug loss doxorubicin data required modelling of drug instability in addition to diffusion. occurred during the experiments and the two reservoirs equilibrated to C0/2. Pnet for each of the drugs was determined as (5.25±0.25)xl0"4 cm/s and (5.18±0.25)xl0"4 cm/s respectively. For the doxorubicin flux data, drug instability in the growth medium led to drug decay for both reservoirs. In this case data were fitted to determine the rate of drug breakdown, P, as well as the permeability of the insert. Pnet and p were determined as (3.24+0.16)xl0"4 cm/s and (1.07±0.04)xl0"5 s"1 respectively. 73 Figure 5.11 shows the results of a series of flux experiments, where the permeability of the insert to each of the drugs/molecules examined in this study was determined. Results for Pnet were: HTO, 16.5xl0'4 cm/s; Azomycin, 6.1xl0"4 cm/s; Tirapazamine, 7.1xl0"4 cm/s; Misonidazole, 4.9xl0"4 cm/s; Etanidazole, 4.85xl0"4 cm/s; Pimonidazole, 5.4xl0"4 cm/s; Doxorubicin, 3.4xl0'4 cm/s; Inulin, 2.1xl0"3 cm/s. Data were fitted to the relation Pnet-h-(M.W.)m, yielding a value for m, the power dependency of permeability on molecular weight, of -0.40 ± 0.04, which compared favourably with the expected value of -0.418, from published data for diffusion of molecules through pure water (Boag, 1969). By combining the data for Pnet with the rate of diffusion of each molecule in water, as predicted from consideration of M.W. (Boag, 1969), the effective thickness of the UBLs and porosity of the plastic membrane were calculated. UBL thicknesses were determined to be 46 um (s.d. 8 um), 23 um (s.d. 4 um) for UBLA and UBLB respectively and the porosity of the plastic membrane, defined as the ratio of the molecule's diffusion coefficient within the membrane to its diffusion coefficient in pure water, was determined to be 0.51 (s.d. 0.09). 1E-03 h (30 o 1E-04 HTO 10 Tira B Pimo Azp Miso Eta Dox • • •1 • 100 1000 Molecular Weight • I I I Inulin 10000 Figure 5.11 Plot of P„et versus molecular weight obtained from a series of flux experiments of molecules through blank inserts. When fitted assuming a power dependency of permeability on molecular weight, i.e. Pnet=b-(M.W.)m, the value for m was determined as -0.40±0.04, which was consistent with the predicted value of -0.42 (Boag, 1969) from diffusion through pure water. The value for b was determined as 4.4xl0"3. 74 5.3 M e t h o d s for EIS Exper iments Impedance Measurements Impedance measurements were made using the EndOhm-12 impedance cell (World Precision Instruments, Sarasota, FI), see Figure 5.12, and a stripped down version of an impedance measurement system described by J.J. Ackmann (Ackmann, 1993), the basic premise of which is illustrated in Figure 4.2. A sine wave generator (HP 3312A, Hewlett Packard, Palo Alto, CA) EndOhm-12 tissue culture \ insert electrodes Figure 5.12 Diagram of the impedance measurement cell used for EIS measurements, manufactured by World Precision Instruments. Current and voltage electrodes have a circular form and lie in planes above and below the cell culture. The top and bottom of the cells are electrically insulated from each other by the plastic tissue culture insert. is used to drive a 30 pA r.m.s. current source connected to the current electrodes in the measurement cell. Voltage and phase measurements across the voltage electrodes are then made using a buffered differential amplifier connected to an oscilloscope (V-1085, Hitachi, Japan). For a given frequency, measurement of the voltage drop, I VI, across the culture yields IZI = IVI / 111 where III is fixed. The phase shift, <|), produced by the culture was then determined through comparison of V i n versus V o u t using the oscilloscope in the X-Y mode. Measurements were carried out at room temperature over frequencies from 0.1 kHz to 1 MHz and took approximately 10 minutes per culture. Calibration Calibration of the measurement system was made by detennining the gain of the system as a function of frequency over the range of 0.1 kHz to 1 MHz. Figure 5.13 shows results of measurements of gain made using the measurement cell with medium and a blank growth insert. Data were fit empirically using the function: 10 h c 3 5 TTTT] 1 I I 1111 E| 1 I IT1111| 1 I I mill e o ooo—e—e-o—QQO <~>&& uL 0.1 1 10 100 Frequency (kHz) 1000 Figure 5.13 Calibration of the EIS system. Plot of gain - O and phase shift - • measured across purely conductive medium and permeable plastic membrane. Data were fitted to determine gain and phase shift of the system as a function of frequency. 75 gain(f) = a-e ^+b-f + a0, (5.4) where a=0.30, 30=8.9, b=4.1xl0"4 kHz"1 and x-11 kHz and f is the measurement frequency in kHz. The magnitude of the impedance for subsequent measurements of IVI at a given frequency was calculated using the relation IZ I = IVI / 111 xl/gain, where 111 was fixed at 30 uA r.m.s. Phase shift data were fitted using a linear relation between phase shift, and frequency, f. This effect was attributed to a constant time delay internal to the oscilloscope when used in the X-Y mode. Fitting the data to <|>=27ifAt, resulted in At=(8.15±0.2)xl0"8 s. This shift was then subtracted from subsequent phase measurements on cultures. Cell Factor The resistance of an electrically conductive medium is related to its conductivity by equation (4.11), where / is taken as the distance between the voltage measuring electrodes, A is the cross sectional area of the measurement cell and a is the conductivity of the medium. In practice, the effective value of / / A, referred to as the cell factor, is determined experimentally for an impedance cell by measuring the resistivity of a medium of known conductivity. Using our measurement cell we determined the resistivity of a 150 mM saline solution (cj=1.83 Q^ m"1) to be 16.7±0.4 Q, which leads to a cell factor of 0.30±0.01 cm"1. This result compares favourably with an approximate value for the cell factor of 0.35 cm"1 obtained using /=0.275 cm and A=0.79 cm2 for the dimensions of the measurement cell. Using this result, we determined the conductivity of our growth medium to be 1.95±0.05 QW 1 from the measurement of its resistivity. 5.4 Determining M C C thickness Accurate measurement of MCC thickness was required for analysis of data from drug flux experiments. Measurement of MCC thickness from cryostat sections was possible, however obtaining a measure of average thickness over the entire culture area required multiple measurements using multiple sections. Rather than repeating this process for each flux experiment, the relation between MCC permeability to tritiated water (HTO) and MCC thickness was determined from a series of HTO flux experiments and results were then 76 applied to determine MCC thickness via the simultaneous diffusion of HTO and the drug of interest during flux experiments. Figure 5.14 shows data from a typical experiment where HTO diffuses through a SiHa MCC, the results of which are used to determine the tissue culture thickness. Both curves were fitted simultaneously to the mathematical diffusion model to obtain the permeability of the MCC to HTO. Figure 5.15 shows a plot of SiHa MCC thickness versus the inverse of permeability to HTO. Permeability was determined using diffusion experiments such as shown in Figure 5.14 and thickness was determined through cryostat sectioning. The data indicate a linear relationship over the range of 70-300 um, which confirms that the diffusion coefficient for HTO is effectively constant over this range. The data are fitted to a line with the y-intercept forced through zero. Results yield D H T O = slope = 3.45 ± 0.1 x 10"6 cm2/s with r=0.95. If the data were not linear over the region it would indicate the diffusion coefficient varied with changes in the tissue environment due to increased thickness or age. However, this was not observed. Similar experiments using MCCs grown from V79 and V79/ADR cells yielded diffusion coefficients of (2.92 ± 0.1) xlO"6 cm2/s and (2.17 ± 0.1) xlO"6 0 2 4 6 8 Time (hours) Figure 5.14 Typical data for diffusion of tritiated water (HTO) through a SiHa M C C approximately 200 um in thickness. Panel (a) shows concentration in the donating reservoir. Panel (b) is for the receiving reservoir. Both data sets are fit simultaneously to the diffusion model, P=1.7 x 10'4 cm/s. Error bars indicate the estimated measurement error, 3% fractional error and a 0.003 constant error. 400 300 $ 200 c o i — i o o h I I I T / - T ; 7 -m i i i 0 0.25 0.5 0.75 1 1/Px10-4s/cm Figure 5.15 Plot of SiHa M C C thickness versus the inverse of permeability to HTO. Data show a linear relation, which indicates that the diffusion coefficient remains constant over the range of MCC thickness used here. Data fitted with a straight line forced through zero. 77 cm2/s respectively (Chan, 1998). The results for were used to determine culture thickness using I = D H X O /P H T O where PH T O was measured for each culture. This yielded a more accurate measure of the effective thickness of each culture since cryostat sectioning gave uncertainties of 10-20% while the uncertainty in determining of P was less than 5%. 5.5 Determining the Fraction of Extracellular Space in M C C s Equilibrium levels of radiolabelled inulin (C14-inulin; Amersham Life Science, Oakville, ON., Canada) within MCCs were used to deteiTnine the extracellular water fraction. MCCs of known thickness, determined via permeability to tritiated water, were incubated with 0.2 pCi/ml C14-inulin for 4 hours, under controlled gassing, stirring and temperature, to allow equilibration of C14-inulin throughout the extracellular space of the cultures. After incubation, the cultures were quickly rinsed in fresh media for 10 seconds and then wicked with a tissue to remove excess liquid from the surface of culture. Each culture was then placed in a 20 ml scintillation vial with 500 pi of tissue solubilising agent, Scintigest (Fisher Scientific; Nepean ON., Canada), and left for 1 hour. Following dissolution of the tissue, 10 ml of scintillation liquid, Scintisafe Econo-1 (Fisher Scientific; Nepean ON., Canada), was added to each vial which was then vigorously shaken for 1 minute. Vials were left in darkness for one hour and disintegrations were then measured using a RackBeta Scintillation counter (Turku, Finland). Scintillation decays per minute (DPM) from the MCCs were compared with DPM from 100 ul samples of the original incubation medium containing the C14-inulin to calculate the volume of space occupied by the C14-inulin within the MCCs. The effect of the excess C14-inulin contained within the permeable plastic membrane and boundary layers was determined by separate experiments done using inserts with no cells. This effect was then subtracted from the results of experiments done with cells. 78 6. EIS & Radiolabelled Inulin: Experimental Results The structure of MCCs grown from SiHa, V79 and V79/DOX cells was characterised through electrical impedance spectroscopy and C14-inulin experiments. EIS measurements were made on MCCs of each cell type, using cultures that ranged from 80 to 250 um in thickness. EIS data were analysed, using the electrical model of the cell environment presented in Section 4.1, to determine estimates of the model parameters related to extra- and intracellular resistance and membrane capacitance. Empirical relations between model parameters and MCC thickness were determined for each type of MCC. EIS results were also used to determine the combined effect of tissue tortuosity and extracellular fraction on diffusion through the cultures. Measurement of the equilibrium level of C14-inulin within MCCs was used to determine the fraction of extracellular space within the cultures and flux of C14-inulin through the cultures was used to determine the tortuosity of their extracellular space. Estimation of the combined effect of tortuosity and extracellular fraction on diffusion that was determined from the C14-inulin results were directly compared with that determined from the EIS results. 6.1 EIS E x p e r i m e n t s Typical data from electrical impedance measurements using a 3 day old V79 MCC are shown in Figure 6.1. The data were fitted using the impedance model described by equations (4.2), (4.3), (4.9) and (4.10). Impedance magnitude and phase data were fitted simultaneously using the Levenberg-Marquardt non-linear, chi-squared rnimrnisation technique (Press, 1992). Model parameters were determined as Re = 34.4 ± 0.8 a R i = 1-6 ± 0.8 Q , Y D = (3.3 ± 0.8)xl0"8 0.1 1 10 100 1000 Frequency (kHz) Figure 6.1 Typical results from measurement of impedance magnitude and phase as a function of frequency (IZ I - O, (()-•). A 3 day old, 145 um thick V79 M C C was used for measurements. The two data sets were fitted simultaneously to the impedance model. 79 Q"1 -s/rad and n = 0.73 ± 0.03, with RD = 17.5 Q. Standard errors were calculated by the fitting routine and were based on predetermined measurement error of 3%. Culture thickness was determined to be 145 ± 5 pm via permeability to HTO. SiHa MCCs Figure 6.2 (a)-(d) shows results for estimates of model parameters Re, R;, Y0 and n as a function of culture thickness from a series of experiments using SiHa MCCs. The parameter representing extracellular resistance, Re, Figure 6.2 (a), shows a curved increase with MCC thickness. If the structure of the MCC extracellular space were to remain unchanged with increasing MCC thickness, then the parameter Rg would be expected to increase linearly with MCC thickness. In this case, the curved relation between Re and thickness suggests an increase to the barrier posed by the MCC, via either increased tortuosity or decreased fraction of extracellular space, with increasing MCC thickness. Data for R e fitted using a second order polynomial, with the constant term forced to zero, to match the physical situation of zero impedance for an MCC of zero 10 50 100 150 Thickness (um) 250 a of 5 h 1 (b) I 1 1° rx --^ i 1 T T p T . -L _L 1 1r 50 100 150 Thickness (um) 200 250 Figure 6.2 Results of analysis of impedance measurements from a series of experiments using SiHa MCCs. (a) R^ (b) R,, (c) Y 0 and (d) n as a function of culture thickness. 80 thickness, yielded the following fitting terms; linear term of 0.14 ± 0.01 Q/um and quadratic term of (9.5 ± 0.5)xl0"4 Q/um2. Determination of Rir the cytoplasm resistance, Figure 6.2 (b), was more difficult due to its small value relative to the system measurement error. It shows an upward trend as expected with increased culture thickness. Data for Rs fitted using a linear relation, with the constant term forced to zero, yielded a slope of 0.024 ± 0.001 Q. /um. The parameter YD, Figure 6.2 (c), which models the membrane capacitance effect, shows a downward trend which is consistent with the expected decrease in net MCC capacitance with an increase in the number of cell layers. Data for Y0 fitted using a second order polynomial yielded the following fitting terms: constant term (3.7 ± 0.04)xl0"7 CJ 1 / n. s/rad, linear term of (-2.8 ± 0.5)xl0"9 Q'1/n-s/rad/um and quadratic term of (5.9 ± 1.4)xl0"12 Q"1/n-s/rad/um2. The parameter n, Figure 6.2 (d), also decreases with increasing thickness. When fitted to a linear relation the following fitting terms were obtained: constant term 0.91 ± 0.03 and linear term (-2 ± 1) xlO"4 um"1. V79 & V79/DOX MCCs Figure 6.3 (a)-(c) shows Re, Y0 and n plotted as a function of thickness as obtained from a series of experiments using MCCs comprised of V79 and V79/DOX cells. Comparing panel (a) from Figure 6.2 and 6.3, the SiHa and V79 MCCs yielded similar R, values. However, the V79/DOX MCCs produced higher values at each thickness. 100 a 50 100 150 Thickness (um) Figure 6.3 Results of analysis of impedance measurements from a series of experiments using V79 - O and V79/DOX - • MCCs. (a) Pv (b) Y0 and (c) n as a function of MCC thickness. 81 The data for Re from the V79 MCCs was fitted using a second order polynomial, with the constant term forced to zero, yielding the following terms; linear term of 0.15 ± 0.01 Q/um and quadratic term of (5.9 ± l)xl0"4 ft/pm2. The V79/DOX data for Re could be fitted to a simple linear relation and still satisfy the requirement of zero impedance at zero thickness; a slope of 0.35 ± 0.01 Q./\im was obtained from the fit. Results for the parameter R; for the V79 and V79/DOX MCCs are not shown because the measurement error (~1 Q) was generally as large as the values themselves. Rj ranged from 0 to 2 Q for the V79 MCCs and from 0 to 1 for the V79/DOX MCCs. The difference in Y0 and n between the two cell types, Figure 6.3 (b), (c), may be related to differences in cell density, caused by differences in the extracellular fraction or average cell volume, which would affect the total membrane capacitance of an MCC. 6.2 Inulin Experiments Measurement of the Fraction of Extracellular Space Within MCCs The fraction of extracellular space within SiHa, V79 and V79/DOX MCCs was estimated via measurement of equilibrium levels of C14-inulin. Figure 6.4 shows the estimated fraction of extracellular space as a function of MCC thickness. The results indicate a general trend of decreasing extracellular space with MCC thickness, with the exception of the SiHa MCCs where this is not detected. These results are of similar magnitude to those reported in the literature for V79 spheroid cultures (Durand, 1980, Freyer and Sutherland, 1983). Fitting the results as a linear function of MCC thickness yielded constant terms of 0.22, 0.50 and 0.39 and slopes of 0.2xl0"4 pm"1, -8.2x10 pm"1 and -7.1xl0"4 pm"1 for SiHa, V79 and V79/DOX MCCs respectively. 250 Thickness (um) Figure 6.4 Extra-cellular space as a function of MCC thickness determined via equilibrium C14-inulin measurements for SiHa - •, V79 - O and V79/ADR -• MCCs. 82 Measurement of the Tortuosity of the Extracellular Space of MCCs 'In O 3 O H 50 100 150 200 Thickness (um) 250 Figure 6.5 shows experimental estimates of the tortuosity factor, X, within the extracellular space of SiHa, V79 and V79/DOX MCCs as determined from a series of flux experiments using C14-inulin. Data from each flux experiment were analysed using the two compartment drug kinetic model of Section 2.3, with the cellular uptake rate set to zero and the fraction of extracellular space set according to the results from the previous section. Using this Figure 6.5 Estimate of the tortuosity factor, X, within model, flux of C14-inulin through the MCCs was SiHa - • , V79 - O, and V79/DOX - • , MCCs. Results were obtained from analysis of C 1 4-inulin flux used to determine an estimate of the model experiments. Each data point is from a separate experiment where the rate of diffusion of C 1 4-inulin parameter D, the rate of diffusion of C -inulin m e e x t r a c e l l u l a r s p a c e o f a n M C C was used xi. u L L i. i i i c IA A itr^r^ to determine an estimate of X. through the extracellular space of the MCCs. Tissue tortuosity was then calculated using the relation, ?i2=D*/D, where D* is the rate of diffusion of C14-inulin in water. The value of D* for C14-inulin was taken as 3.0xl0"6 cm2/s (Lanman, et al., 1971). The results for X that are shown in Figure 6.5 indicate that there were large variations in tortuosity between the three cell lines. These differences can be attributed to a variety of factors mduding cell size, -800 um3 for V79 cells (Freyer, et al., 1984) versus -1800 um3 for SiHa cells, as well as differences in the composition of the extracellular space and cell-cell interactions. Fitting the results as a linear function of MCC thickness yielded constant terms of 1.75,1.55 and 2.66 and slopes of 0.4xl0"3 um"1, 7.4xl0"3 um"1 and 3.0xl0"3 um"1 for SiHa, V79 and V79/DOX cells respectively. 6.3 Comparison of EIS & Inulin Results Both EIS and Inulin experimental results can be used to predict the barrier to diffusion posed by the MCCs. Results from either method can be used to estimate the parameter cp, which quantifies the combined effect of tissue tortuosity and the extracellular fraction. Figure 6.6 shows a comparison of cp calculated using each method. Panel (a) shows results for (pEIS, 83 obtained using equation (4.14) and the impedance data shown in Figures 6.2 (a) and 6.3 (a). Panel (b) shows similar results for (Pinuiin/ obtained using equation (4.15) and the C14-inulin experimental results shown in Figure 6.4 and 6.5. Comparison of the two graphs indicates the extent to which impedance measurements can predict the effect of the MCC environment on C14-inulin flux. The values for <pEIS and (p ,^, differ roughly by a factor of two. In both cases SiHa and V79 MCCs are observed to rank higher than V79/DOX MCCs, however the trends for the rate of change of the parameter (pwith MCC thickness were not conserved between the two methods. 6.4 D i s c u s s i o n The data presented here provides an assessment of the ability of electrical impedance spectroscopy to characterise the cell 0.05 0.04 0.03 0.02 0.01 1 (a) 1 1 1 -— • u — i 1 1 | 0.06 50 100 150 200 Thickness (pm) 250 Figure 6.6 Comparison of the combined effect of tissue tortuosity and the fraction of extracellular space determined from electrical impedance and C 1 4-inulin diffusion measurements, (a) Results for calculation of (pE,s from the data of Figures 6.2 (a) and 6.3 (a) and equation (4.14). (b) Similar results for (ft^m/ as calculated using the data of Figure 6.4 and 6.5 and environment of MCCs. Results show good equation (4.15). Data points show results for SiHa - • , V79 - O, and V79/DOX - • , MCCs. correlation of impedance measurements with culture growth and indicate sensitivity to increasing culture thickness and net membrane capacitance. In addition, the differences between impedance characteristics of the three cell lines studied here, manifested by the parameter (pEIS, indicates the ability of impedance spectroscopy to serve as a method of comparison of in vitro tissue structure. Plots of the parameter for extracellular resistance, Re, as a function of culture thickness show that impedance measurements can be used to determine culture thickness and monitor growth. Such measurements can be done quickly and non-destructively. The net cell membrane capacitance, related to the parameter Y0, is seen to change with culture thickness. Y0 shows a downward 84 trend with increasing culture thickness, which is consistent with what is expected to occur if the cells behave according to the simple electrical model of Figure 4.1. The C14-inulin measurements of extracellular space and the effect of tissue tortuosity on diffusion were consistent with previous studies done using spheroids (Casciari, et al., 1988, Freyer and Sutherland, 1983) that compared a range of cell lines. In that study there was no direct correlation between rates of diffusion for the two cell lines once the extracellular space had been accounted for, indicating that the degree of extracellular tortuosity plays an important role in determining the barrier to diffusion posed by the culture. The general trend of decreasing extracellular space with increasing MCC thickness that was observed for both V79 and V79/DOX MCCs, was not seen in the earlier spheroid study (Freyer and Sutherland, 1983). The possibility that the decrease was caused by incomplete removal of surface liquid during the wicking process was rejected because the effect is not observed for the SiHa MCCs which have the smallest amount of extracellular space and would be the most prone to such measurement artefact. The combined effect of tortuosity and extracellular fraction calculated from impedance measurements did not match exactly with the results determined via C14-inulin experiments. While the estimates of (p differed by a roughly a factor of two, between impedance and diffusion methods, the order of ranking of the three cell lines was reasonably similar between the two methods. The increase in the penetrative barrier posed by V79/DOX relative to V79 MCCs, observed in both impedance and diffusion measurements, was consistent with reported data of wild type versus drug resistant EMT-6 cells grown as spheroids (Kobayashi, et al., 1993). In the Kobayashi report, visual assessment of histological sections found that spheroids grown from the drug resistant cells formed a more compact structure than the wild type cells. There are many possible explanations for the disagreement between the absolute value of (p determined using the two methods. The impedance results were obtained using a measurement jig that did not produce a uniform field over the entire surface area of the MCC. This is not necessarily a problem when experimental results, such as the parameter Re, are empirically compared with MCC thickness. However, it does pose a problem when the data are applied to equations (4.14), which implicitly require a uniform electrical field. The source of the discrepancy may also derive from the fact that the impedance measurements were based on the 85 conductivity of salt ions such as Na+, Cl", K+, etc., which are all small molecules (M.W. <100), while the diffusion data were derived from measurements made using the large C14-inulin molecule (M.W. ~5175). However, it is not clear why the much larger mulin molecule would experience less resistance to diffusion within the extracellular space of the MCCs than the small ions would. Despite this disparity between the two methods, the results do indicate that this simple impedance spectroscopy technique is reasonably successful at comparing the relative barriers to diffusion presented by the different cells types when grown as MCCs. In addition, impedance measurements made before and after experiments using MCCs could serve as a toxicity assay to determine the effect of a drug on the integrity of the cultures and cell membranes. In conclusion, the results presented here indicate that electrical impedance spectroscopy can serve as a useful tool to characterise and monitor the cell environment of MCCs. The impedance system that is presented here is simple and requires little developmental labour since a commercially available measurement jig was used for these experiments. In addition there are numerous automated impedance measurement devices that could be used to monitor MCCs in real time. 86 7. Nitroimidazoles: Flux Experiments & Analysis The flux of several nitroimidazoles through SiHa MCCs was measured and the data analysed using the one compartment model for diffusion within the tissue. Results for estimation of the apparent rate of diffusion through the tissue were used as a comparative measure of the penetrative ability of each of the compounds. Figure 7.1 shows the time rate of change of drug concentration within the donating and receiving reservoirs of the diffusion apparatus from flux experiments for Azomycin, Misonidazole, Etanidazole and Pimonidazole. Data were normalised to the starting concentration of the donating reservoir, Cc, which was equal to 100 uM, 140 uM, 140 uM and 100 uM for Azomycin, Misonidazole, Etanidazole and Pimonidazole respectively. MCCs were all approximately 135 um thick, as determined via permeability to HTO. Solid lines show results of simultaneously fitting donating and receiving reservoir data using the one compartment diffusion model, to determine the apparent rate of diffusion through the MCCs. In general the data were well fitted using the simple model for diffusion within the tissue. However, data for both misonidazole and etanidazole showed slightly skewed fits for the donating reservoir data, indicating the possibility of a small amount of drug loss within the cultures. Table 7.1 summarises results from the analysis of flux data to determine the apparent rate of diffusion through the cultures. From comparison of the ranking of the diffusion coefficient and molecular weight of each drug, it is clear that the simple relation that was shown to exist for drug flux through blank inserts in Figure 5.11 does not hold for flux through the tissue. For drug diffusion through tissue, factors such as drug ionisation and ability to pass the cell membrane will play important roles in determining the rate of flux through the tissue. In order to determine an estimate of the degree by which each drug is constrained to the extracellular space of the tissue, the experimental values for D a p p the apparent rate of diffusion that is determined when the tissue is modelled as a single compartment were compared with the value of D a p p that was predicted when each drug was assumed to be constrained solely to the extracellular space, in which case a theoretical value for the apparent 87 rate of diffusion can be calculated using Dapp=D*-fex/A,2. Here, D* is the rate of diffusion of the molecule in water, which can be predicted from consideration of molecular weight (Boag, 1969), and fex and X are the extracellular fraction and tortuosity of the tissue, which are taken from Figure 6.6 (b) of Chapter 6 (iex/X2~ 0.07). Comparison of the experimental and predicted values for D a p p suggests that drug diffusion is predominantly constrained to the extracellular space of the tissue for the four nitroimidazoles that are studied here. Notably, diffusion of Azomycin is well predicted by this approach. Misonidazole and Pimonidazole both have experimental values of D a p p that are moderately higher than the predicted values, suggesting that some para-cellular flux occurs. Etanidazole, which possess the lowest octanol/water partition coefficient, appears to experience a larger barrier to diffusion than predicted. For comparison, diffusion of HTO is also listed in Table 7.1, high value of D a p p that is observed experimentally for HTO relative to the predicted value, indicates that it can access the intracellular compartment of the tissue much more effectively than the nitroimidazoles. Experimental Predicted Drug M.W. P o c t / w a t e r pKa D a p p (xlO-7 cmVs) D'-fexA2 (xlO 7 cm'/s) Azomycin 113.2 1.4 7.04 7.1 ± 0.2 7.20 Misonidazole 201.2 0.43 - 6.2 ± 0.2 5.65 Etanidazole 214 0.046 - 3.7 ± 0.3 5.51 Pimonidazole 290.8 8.5 8.71 5.7 ± 0.3 4.84 HTO 19 - - 34.5 ± 1 15.3 Table 7.1 Summary of results from analysis of flux of several nitroimidazoles through SiHa MCCs. Data were analysed using a one compartment model to determine the apparent rate of diffusion of each drug within the MCCs. Data for molecular weight, octanol-water partition coefficient and pK a value were obtained from Dr. Peter Wardman (Gray Laboratory, UK). The predicted value of D a p p was calculated based on the rate of diffusion of the molecule in water D", calculated from consideration of molecular weight (Boag, 1969), and properties of the extracellular space, fex and X, which are characterised in Chapter 6. 88 Time (hours) Time (hours) Figure 7.1 Data from flux of four nitroimidazoles through 135 um thick SiHa MCCs. Left and right panels show drug concentration data from the receiving and donating reservoirs, normalised to the starting concentration of the donating reservoir. In each case, data from both reservoirs were fitted simultaneously, using the one compartment diffusion model, to determine the apparent rate of diffusion of each drug though the tissue. 89 8. Tirapazamine: Experimental Results & Simulations 8.1 Tirapazamine flux experiments Tirapazamine flux experiments were carried out to determine the rate of diffusion and metabolism of tirapazamine within SiHa MCCs. Experimental conditions were controlled during flux experiments so that MCCs could be considered as either uniformly well oxygenated or uniformly poorly oxygenated throughout. To attain well oxygenated conditions, flux experiments were conducted using thin MCCs, approximately 100 um in thickness, with gassing maintained at 5% 02, 5% C02, 90% N2. Under these conditions, metabolism of tirapazamine within the MCCs was expected to have negligible effect on drug flux between reservoirs. To attain uniformly low oxygenation conditions, flux experiments were conducted with gassing maintained at 5% C02, 95% N2. Prior to begiruiing flux experiments, MCCs were allowed to equilibrate for 2.5 hours, at which point oxygen levels in the growth media, measured using the Eppendorf oxygen sensor, were less than 0.5%. Under these conditions, tirapazamine flux through the MCCs was modelled with both diffusion and metabolism allowed. Data from flux experiments were analysed using the mathematical model describing tirapazamine kinetics within MCCs to determine estimates of the rate of diffusion and metabolism within the tissue (see Section 2.2). Flux data from oxic experiments were analysed with the metabolic term fixed at zero. The simultaneous analysis of data from donating and receiving reservoirs allowed for verification that drug loss through metabolism was negligible. Flux data from hypoxic experiments were analysed according to the complete diffusion-metabolism model, with the oxygen partial pressure set to zero throughout the tissue. Oxic Flux Experiments Figure 8.1 shows typical experimental results in which tirapazamine is allowed to diffuse through an 85 um thick SiHa MCC under oxic conditions. Analysis of results from a series of such experiments, with cultures ranging in thickness from 80-110 um, yielded a value for DT i r a = 6.7 x 10'7 cm2/s, standard deviation 0.4 x 10"7cm2/s (n=4). For these experiments, using thin, well-oxygenated cultures, the metabolic rate parameter, Vm a x, was set to zero and the results of the fit indicated no detectable loss of tirapazamine to metabolism. A substantial level of drug 90 metabolism would have skewed the fit and it would not have been possible to satisfy both data sets simultaneously. In addition, neither of the two major tirapazamine metabolites, SR 4330 and SR 4317, were detected during HPLC measurement of tirapazamine levels. Hypoxic Flux Experiments The diffusion of tirapazamine through a fully hypoxic, 140 pm thick MCC is shown in Figure 8.2. Results from several experiments, with cultures ranging in thickness from 80-180 pm, gave values for DT i r a = 7.3 x 10"7 cm2/s, standard deviation 0.5 x 10"7 cm2/s (n=7) and Vm a x =1.5 uM/s, standard deviation 0.4 uM/s (n=7). For these experiments, there was a large amount of drug loss through tissue metabolism, the donating reservoir concentration was seen to decrease by 12% after 5 hours while the receiving reservoir concentration increases by just 3% of C c over the same time period. 8.2 Tumour Cord Simulations Results from the analysis of tirapazamine flux through MCCs were used to model drug distribution within a cord of cells surrounding a blood vessel. The experimental estimates for the diffusion coefficient, D, and the metabolic rate, Vm a x, were applied to the mathematical model describing drug kinetics within the geometry of a tumour cord, as described in Chapter 3. To 0 2 4 6 8 10 Time (hours) Figure 8.1 Diffusion of tirapazamine under oxic conditions (p02 = 38 mmHg). Panels (a) and (b) show drug concentrations in donating and receiving reservoirs from one experiment. Both data sets are fit simultaneously, with V m a x set to zero. Error bars indicate the estimated measurement, 3% fractional error plus 0.003 constant error. (b) I 1 1 1 r 0 1 2 3 4 5 Time (hours) Figure 8.2 Diffusion of tirapazamine under hypoxic conditions. Panels (a) and (b) show donating and receiving reservoir concentrations from one experiment. Data are fit simultaneously to determine D and V m a x Error bars indicate the estimated measurement error, plus 0.003 constant error. 91 complete the model the effect of the variation of oxygen partial pressure on the rate of cellular tirapazamine metabolism was also accounted for. Boundary conditions The condition for the blood vessel-tissue boundary was chosen to model the situation where the barrier to diffusion posed by the endothelial wall was negligible, so that the first layer of tumour cells was exposed directly to the blood tirapazamine level. The blood concentration was held constant to reflect the clinical practice of adrninistering tirapazamine by intravenous infusion over several hours (Graham, et al., 1997). Diffusion of tirapazamine beyond the outer layer of cells was modelled using a mirror blood vessel scheme, where drug diffusing out of the cord was matched by drug diffusing in from a mirror vessel. Tumour cord oxygen distribution Oxygen partial pressure as a function of depth into the cord was modelled using a diffusion-reaction model for cylindrical geometry (Boag, 1969): P02(r) = p02 2-ln(r/rmax) + l - r 2 / r m 2-ln(a/rmx) + l -a 2 /r i r (8.1) where p02(r) is the oxygen partial pressure as a function of radial distance from the centre of the blood vessel, p02 is the blood oxygen partial pressure (r=0), a is the radius of the blood vessel and rmn is the distance where oxygen partial pressure falls to zero. For blood p02 values of 40, 20 and 10 mmHg, rmjx was set to 150, 110 and 80 pm respectively (Boag, 1969). Figure 8.3 (a), broken Une, shows the oxygen profile that is predicted within a cord 150 pm thick when blood vessel p02 is 40 (a) O OH 0 25 50 75 100 125 150 Distance from blood vessel (um) 0.00 0 25 50 75 100 125 150 Distance from blood vessel (um) Figure 8.3 (a) Variation of p02 and the rate of tirapazamine metabolism, with distance from a blood vessel. The functional dependence of p02 is obtained using equation (8.1). Variation of the metabolic rate follows a competitive inhibition model, equation (2.10), with K p 0 2 set to 4 mmHg Tirapazamine concentration, C, is set uniform throughout the cord at 1 pM. (b) Simulated cell survival, as calculated using equation (8.2), after a 30 hour exposure using the conditions of panel (a). 92 mmHg. Oxygen Inhibition of Tirapazamine Metabolism Oxygen inhibition of tirapazamine metabolism within the tumour cord was described using a competitive inhibition model, in which the dependency of the Michaelis-Menten parameter K'm on oxygen partial pressure was described using equation (2.10). The application of equation (2.10) to model the rate of tirapazamine metabolism within the cord requires specification of the parameters Km, the value of K'm under conditions of zero oxygen, and K p D 2, the oxygen partial pressure at which low concentration tirapazamine metabolism is 50% inhibited. The value of Km was taken as 75 pM (Wang, et al., 1993). The value for K p 0 2 was estimated to be 4 ±1 mmHg using published data for the exposure to tirapazamine, as a function of oxygen partial pressure required to achieve a 1% cell survival using V79 cells (Koch, 1993). From the Michaelis-Menten competitive inhibition relation, a value of 4 rnrnHg for Kp G, will imply a ratio of -10 for the concentration of tirapazamine required to achieve equal cytotoxicity under oxic and hypoxic conditions (p02 is set equal to 40 mmHg for oxic conditions, and 0 rnrnHg for hypoxic conditions). If oxic levels are chosen to be 160 mmHg (20% oxygen) then the oxic/hypoxic cytotoxicity ratio becomes -40 which is consistent with the reported value of -38 for experiments done in air using human tumour cell lines (Zeman, et al., 1986). Figure 8.3 (a), solid line, shows the rate of tirapazamine metabolism that is calculated based on the oxygen profile shown by the broken line and assuming a uniform distribution of tirapazamine throughout the cord, using metabolic rate = Vm a x-C/ (K'm +C) with C fixed at 1 uM. Relating tirapazamine exposure to cell survival A relationship between the amount of tirapazamine metabolised and cell survival was estimated using published data for cell survival as a function of exposure to parent drug under hypoxic conditions (Siim, et al., 1996). The published data showed a general trend for human cell lines, which was averaged. In order to relate these results for exposure to tirapazamine under hypoxic conditions to exposure at any oxygen partial pressure, the amount of drug metabolised under the hypoxic conditions was calculated, yielding a direct link between amount of drug metabolised and cell kill. The data for surviving fraction, now as a function of total metabolised drug (exposure), E, were then fitted to a sigmoidal function of the form: 93 SF - S F SF= ° ,S;+SF„ 1 + e A E where the model parameters were determined as follows; SF0=1, SF„=-0.06, E0=1.4 mM and EM=0.32 mM. This relation produced a good fit to the data for the range of accumulated metabolised drug exposures from 0 mM to 2.25 mM. Figure 8.3 (b) shows simulated cell kill predicted using this relation. Simulation of tirapazamine distribution By combining the relations described by equations (8.1) and (8.2), along with the experimental results and mathematical model for tirapazamine kinetics within tissue, the spatial distribution of tirapazamine within a tumour cord was simulated as a function of exposure time. Blood vessel tirapazamine levels were chosen to match the reported clinical level of ~5 pg/ml (30 uM) which was achieved through intravenous infusion over several hours (Graham, et al., 1997). Figure 8.4 (a)-(c) shows results of simulation of the distribution of parent drug, metabolised drug and surviving fraction respectively, within the tumour cord. Each panel shows distributions at three different blood vessel partial pressures 10, 20 and 40 mrnHg after 4 hours exposure to tirapazamine. In each case, a stable drug (8.2) (a) (b) (c) 0 25 50 75 100 125 150 Distance from blood vessel (pm) Figure 8.4 Simulation of tirapazamine distribution within a tumour cord. Each Figure shows results of simulations where the concentration of tirapazamine within the blood is kept at 30 uM and blood p02 is set to 40 mrnHg (—), 20 mrnHg (- -) and 10 mrnHg (•••). Panel (a) shows the stable parent drug distribution which forms within 15 minutes. Panel (b) shows the distribution of metabolised drug after a four-hour exposure under the conditions of panel (a). Panel (c) shows simulated cell kill as a function of distance from blood vessel, calculated from the data of panel (b) using equation 8.2. The shaded regions shown for the simulations at 40 mrnHg shows the extent to which measurement error for D and V m a x will modify simulation results. 94 gradient is seen to form within the tissue within approximately 15 minutes, at which point a balance was reached between drug influx from the blood vessel and drug loss through metabolism within the cord. The simulations of Figure 8.4 (a) indicate that the cells furthest from the blood vessel will see a maximum parent drug concentration of only -10% that of the proximal cells. Figures 8.4 (b) and 8.4 (c) show the net metabolised drug and resultant surviving fraction after the four-hour exposure. At blood p02 concentrations of 40 mrnHg very little cell kill occurs throughout the cord of cells, but when the blood p02 is reduced to 20 and 10 mrnHg there is a significant increase in cytotoxicity towards cells proximal to the vessel but little toward the cells distal to blood vessels. The shaded regions, shown for the simulations at 40 mrnHg, indicate the range of behaviour that the experimental uncertainty in D and Vm a x can produce. The boundaries of the shaded regions were determined by carrying out additional simulations, first with both D and Vm a x increased by one standard deviation and then with both parameters reduced by one standard deviation, consistent with what occurred experimentally. The effect of an increase in the degree of oxygen inhibition on drug distribution within the cord was examined by varying the parameter K^. Figure 8.5 (a)-(c) shows results where K p G 2 was set to 1, 2 and 4 mrnHg, with p02 fixed at 40 mrnHg. Results indicate that if oxygen inhibition of tirapazamine metabolism were increased, the distribution of 0 25 50 75 100 125 150 Distance from blood vessel (um) Figure 8.5 Simulations in which the effect of oxygen on the inhibition of tirapazamine metabolism is varied. Distributions are shown after a four hour exposure at 30 p M blood tirapazamine levels with p02 set to 40 mrnHg and K p 0 2 set to 4 mrnHg (—), 2 mrnHg(- -) and 1 mrnHg (•••). Panel (a) shows the stable parent drug distribution which forms within 15 minutes. Panel (b) shows the distribution of metabolised drug after a four-hour exposure under the conditions of panel (a). Panel C shows simulated cell kill as a function of distance from blood vessel, calculated from the data of panel (b) using equation 8.2. 95 metabolised drug and cell kill would become much more favourable in terms of achieving the desired goal of highly specific cytotoxicity towards cells existing at low oxygen tension far from blood vessels. 8.3 D i s c u s s i o n The measurement of tirapazamine flux through MCCs under oxic and hypoxic conditions was well fitted using the mathematical diffusion-metabolism model. The diffusion coefficient, D is a measure of the apparent rate of diffusion of tirapazamine within MCCs comprised of SiHa cells. It incorporates the net effect of drug diffusion through and around cells. While it would be expected to vary with changes of the cell environment, such as the fraction of extracellular space, no systematic variations with culture thickness were detected. The value of DX i r a was determined to be 7.0 ± 0.5 x 10~7 cm2/s by taking the average of all results. The maximal metabolic rate, Vm a x, deterrrtines the rate that a fully hypoxic MCC metabolises tirapazamine and was found to be 1.5 ± 0.4 uM/s. Measurements of the diffusion of tirapazamine under oxic and hypoxic conditions allowed us to simulate perivascular penetration of tirapazamine in a tumour cord. The simulations that were carried out using our estimates of the rate of diffusion and metabolism of tirapazamine uluminate the possible range of drug distributions expected to occur in a solid tumour. The microregional distribution of tirapazamine is governed by a balance between drug influx and oxygen dependent drug metabolism. As tirapazamine diffuses away from blood vessels it is geometrically diluted and, as the oxygen tension decreases, metabolism increases. This metabolism generates a cytotoxic species that contributes to the antitumour effect, but also acts to consume drug and therefore hinder drug penetration. Overall, our simulations predict that cells peripheral to blood vessels are exposed to only 10% of blood tirapazamine levels. As a result of this, little activity against hypoxic tumour cells residing distal to tumour blood vessels is predicted. Only when hypoxic cells reside close to the vasculature, for example as a result of depleted blood p02 at the end of tumour capillaries, does tirapazamine become significantly cytotoxic. This finding is consistent with experimental data from spheroids (Durand and Olive, 1992) and human xenografts and murine tumours (Durand and Olive, 1997) using flow cytometric sorting techniques. In the case of spheroids, under low oxygen tension, tirapazamine 96 was found to have minimal penetration and in animal systems employing SiHa xenografts or SCCVII tumours only a small differential in cell kill between the cells close to and distant from vasculature was observed. One possible method for improving drug penetration and cytotoxic exposure to peripheral cells would be to select a tirapazamine analogue that possesses a lower Kp 0 2 value, the oxygen partial pressure at which the rate of low concentration tirapazamine metabolism drops to half its hypoxic rate. Figure 8.5 (a) shows results where KpG2 is set to 1, 2 and 4 mrnHg, other parameters are as in the simulations of Figure 8.4 (a) with p02= 40 mrnHg. These chosen values for KpD2 translate to oxic versus hypoxic cytotoxicity ratios of -40, 20 and 10 respectively. The simulations show a moderate increase in drug distribution throughout the cord, when KpD2 is lowered from 4 mrnHg to 1 mrnHg, along with a substantial redistribution of metabolised drug and cytotoxic exposure. Hence a drug with a lower Kp 0 2 value would cause increased cell kill at the peripheral cells while at the same time sparing cells close to blood vessels. 97 9. Doxorubicin: Experimental Results & Simulations The kinetics of doxorubicin distribution within SiHa, V79 and V79/DOX MCCs were examined through the measurement and analysis of the flux of doxorubicin through MCCs and by imaging the distribution of doxorubicin fluorescence within the MCCs. Results from the analysis of the flux experiments were used to predict the fluorescence data as well as to simulate doxorubicin distribution within a tumour cord under clinical conditions. 9.1 F l u x Exper iments Typical data for diffusion of doxorubicin through V79 and V79/DOX MCCs are shown in Figure 9.1. Data points show the time rate of change of doxorubicin concentration as measured in the donating and receiving reservoirs, normalised to the starting concentration of the donating reservoir. Comparison of the receiving reservoir data for the V79 and V79/DOX 135umV79/DOXMCC 145 um V79 MCC 180umV79MCC Time (hours) Figure 9.1 Typical doxorubicin flux data from donating and receiving reservoirs for experiments using V79/DOX and V79 MCCs. (a)-(b) 135 pm thick V79/DOX MCC; (c)-(d) 145 pm thick V79 MCC; (e)-(f) 180 um thick V79 MCC. Solid lines indicate fitting results as described in the text. Broken lines show the effect of setting f3, dashed lines, or Bk, dotted lines, to zero while keeping other parameters fixed at the values determined from the fit. 98 MCCs of similar thickness indicated drug flux through the MCC composed of drug resistant cells to be considerably greater than that observed for the MCC composed of the parent cells. This difference was attributed to the higher rate of doxorubicin cellular uptake that was expected for the sensitive V79 cells in comparison to the resistant V79/DOX cells. For the thick V79 MCC, the rate of appearance of doxorubicin in the receiving reservoir was reduced to less than half that seen from the thin V79 MCC data. This decrease in flux was attributed to the combined effect of the increased barrier to diffusion posed by the thicker MCC and by the increased number of cells taking up doxorubicin. All three data sets showed a high rate of drug loss in the donating reservoir compared to the rate of drug appearance in the receiving reservoir. The concentration in the donating reservoirs was seen to be reduced by 10-15% over the course of the experiment, while only a fraction of this, 1% or less, was seen to enter the receiving reservoirs. Doxorubicin uptake by cells as well as its instability in the growth medium were both found to play important roles in determining the rate of drug loss from the donating reservoir. V79/DOX Analysis Analysis of doxorubicin flux data, such as shown in Figure 9.1, was done using a multi-step process. Data from V79/DOX flux experiments were analysed first and results were then carried over for use in the analysis of V79 and SiHa data. Flux data were fitted using the mathematical model for doxorubicin kinetics within tissue to obtain estimates of the parameters for diffusion, D, rate of cellular uptake, k, and extracellular bmcling, Bk. Flux data from both reservoirs were fitted simultaneously so as to force consistency when accounting for doxorubicin concentration changes in the two reservoirs. Despite this, two reasons made it impossible to fit for all model parameters simultaneously. First, it was found that the rate of drug breakdown, P, in the growth medium, determined in separate experiments as t1/2~32 hours, was more important in determining the rate of drug loss in the donating chamber than was the rate of flux into the culture. The difference between the dashed and solid lines in panels (a), (c) and (e) of Figure 9.1 indicate the contribution of drug breakdown to the rate of drug loss in the donating chamber. In the case of the V79/DOX data, this difference represents more than 75% of the total drug loss. Because of moderate variation in the rate of drug 99 breakdown between experiments, p had to be made a parameter of the fit, hence the donating chamber flux data did not allow for a precise estimate of the rate of doxorubicin flux into the MCC. The second factor causing difficulty in parameter estimation was a degree of redundancy presented by the parameters of diffusion, uptake and breakdown and the role they played in deterrnining the rate of drug loss and gain in the two chambers. In the case of the V79/DOX data, it was found that when the cellular uptake parameter, k, was fixed at a value from 0 up to -0.06 s"1, the fitting routine could produce equally good fits to the experimental data, through modification of the values of D and p. In order to determine experimental values for D, k and Bk within the V79/DOX MCCs, flux data were fitted with k fixed at a range of values and results for D were then compared with those predicted from knowledge of tissue tortuosity, using equation 2.11. Figures 9.2 and 9.3 show results for the estimation of D and Bk as obtained by fitting doxorubicin flux data from a series of experiments using V79/DOX MCCs that ranged from 90 to 140 um in thickness. The flux data from each experiment were fitted with the parameter k fixed at 0, 0.03 and 0.06 s"1, panels (a), (b) and (c) respectively, and the results for "(a) ' I I k = 0 s"1 i i X Q -(b) 1 i 1 1 k = 0.03 s -1 * * ' ' • 1 8 r i 1 . 1 1 * I * £ i 6 - -4 k = 0.06 s -1 i i 100 120 140 MCC Thickness (um) Figure 9.2 Diffusion coefficient estimates obtained from the analysis of a series of flux experiments using V79/DOX MCCs that ranged from 90 to 140 umin thickness. Panels (a)-(c) show results when the rate of cellular uptake, k, is fixed at 0, 0.03 and 0.06 s"1 respectively. Solid lines show the linear best fit for the data and dashed lines show the slope predicted from knowledge of tissue tortuosity. 50 40 30 20 10 0 50 40 30 3 20 3 10 u 2 0 2 50 40 30 20 10 0 .5 T 3 C ta -(a) 1 , 1 k = 0 s-1 _ T T • T • T ± • • -j- T — » -1 • 1 | -(b) T r T • T • T • ± • k = 0.03 s"1. 1 \ -• - 1 • 1 1 1 i i -t-T • 1 1 -(c) 1 T • T • • i • ' k = 0.06 s'"1 - - ~T-• - 1 • i i — I l I 1 ^ _ T • 1 , 100 120 140 MCC Thickness (pm) Figure 9.3 Estimation of the extracellular binding parameter, Bk, from V79/DOX MCC flux data. Panels (a)-(c) show results when the rate of cellular uptake, k, is fixed at 0, 0.03 and 0.06 s'1 respectively. Solid lines show the best linear fit for the data and dashed lines show data fit to a constant value. 100 D and Bk are plotted as a function of culture thickness. The decrease in the rate of diffusion with increasing culture thickness, shown in Figure 9.2, can be explained by the increase in the tortuosity of the cultures that was previously characterised by the C14-inulin flux experiment results, Section 6.2. The dashed lines in Figure 9.2 show the change to the rate of diffusion of doxorubicin through V79/DOX MCCs that is predicted from knowledge of the tissue tortuosity parameter X that was characterised in Section 6.2, using equation 2.11. Because a large fraction of the doxorubicin molecules will carry a charge at physiological pH, the direct application of equation 2.11 is not expected to produce exact results. To correct for this, the value of D*doxo, the rate of doxorubicin diffusion in water, appearing in equation 2.11 was varied to fit the V79/DOX data shown in each panel of Figure 9.2. Hence, the dashed lines of Figure 9.2 show the slope to the diffusion versus thickness plots that is predicted from the C 1 4-inulin results, through equation 2.11, with the constant term D*doxo varied to fit the data. In contrast, the solid lines in Figure 9.2 show results of best linear fits, where both the slope and intercept were allowed to vary. It is evident from comparison of the dashed and solid lines shown in each panel that a cellular uptake rate of ~0.03 s"1, panel (b), produces the closest match between the predicted and actual change to the rate of diffusion with culture thickness. With k set to 0.03 s"1, the value of D*doxo was determined as (5.6 ± 0.1) x 10"6 cm2/s. This result compares favourably with the expected rate of diffusion of doxorubicin in water of ~(5 ± 1) x 10"6 cm2/s, as obtained from consideration of its molecular weight (Boag, 1969). The fact that the effective rate of diffusion of doxorubicin within MCCs can, through consideration of tissue structure and drug interactions, be related to its rate of diffusion in water confirms that the method of analysis used here reasonably describes the major aspects of doxorubicin flux within the MCCs. Figure 9.3 shows results for Bk that were obtained in conjunction with the results for D shown in Figure 9.2. In each panel the data are fitted to a constant value (dashed line) and to a linear relation (solid line). Comparison of the linear and constant fits indicates that when the rate of cellular uptake is set to -0.03 s"1, panel (b), results most closely match the situation where Bk is independent of culture thickness. Since incorrect choice of the rate of cellular uptake will skew the Bk estimates, the results of panel (b) can be considered the most reasonable if one expects the actual concentration of extracellular binding sites not to vary with culture thickness. With k set to 0.03 s"1, an average value for Bk was determined as 19 101 (s.d. = 8). The results of Figures 9.2 and 9.3 can really only be considered as indicating the range within which the parameters D, k and Bk fall. However, if the predictions made by equation 2.11 are considered as reasonable, then the parameters can be further constrained to the results shown in panel (b) of each figure. The results of analysis of the V79/DOX MCC flux data shown by the solid lines in Figure 9.1 (a)-(b) were obtained with the parameter k fixed at 0.03 s"1. Analysis determined D as 5.7 ± 0.2 x 10"7 cm2/s, Bk as 22 ± 6 and P as (6.0 ± 2) x 10"6 s"1. In order to indicate the relative importance of the model parameters in determining flux, P and Bk were individually set to zero while leaving other parameters unchanged, results are shown by the dashed and dotted lines respectively. In the case where P is set to zero, one finds that the majority of the drug loss from the donating reservoir can be attributed to drug breakdown rather than flux into the MCC. Conversely in the receiving reservoir, setting P to zero only moderately affected the predicted change to drug concentration, indicating that the rate of drug breakdown in the receiving reservoir is secondary when compared with the rate of drug efflux from the MCC. Setting the parameter Bk to zero had a negligible effect on the donating reservoir concentration, producing no visible change from the best-fit line. For the receiving reservoir, setting Bk to zero had the effect of removing the initial slow rate of change of drug concentration (compare the solid and dotted lines in Figure 9.1 (b)), indicating the effect of extracellular binding only plays an important role in deterrniriing drug flux during the transient period when the gradient of doxorubicin across the MCC is first developed. The extracellular binding sites act to slow the build up of free doxorubicin within the culture during the initial stage when the concentration gradient is first developed within the culture. After this initial stage only gradual changes in extracellular drug concentration gradient occur within the culture, due to the slow rate of change of concentration in the two reservoirs. At this point the extracellular binding sites play only a minor role in limiting drug flux out of the culture, since an equilibrium between free and bound drug has already been reached. V79 & SiHa Analysis Analysis of V79 and SiHa MCC flux data was carried out in two stages in order to circumvent the same problems that were encountered with the V79/DOX MCC flux data when 102 simultaneously estimating the parameters for diffusion and cellular uptake. In the first stage, an estimate of the diffusion coefficient for doxorubicin in the V79 and SiHa MCCs was calculated from the V79/DOX results. Then, with the rate of diffusion fixed, data from V79 and SiHa MCC flux experiments were analysed to determine an estimate of k and Bk for each cell line. The rate of diffusion of doxorubicin within V79 and SiHa MCCs was calculated using equation 2.11, where the value of X for each cell line was taken from the C14-inulin results of Section 6.2 and the value of D*doxo was taken from the V79/DOX data analysis section. Figure 9.4 shows the calculated rate of diffusion plotted as a function of culture thickness, dotted and dashed lines for SiHa and V79 MCCs respectively. For comparison, the V79/DOX results from Figure 9.2 (b) are shown in this figure as well. The data points for V79 and SiHa MCCs shown in the figure are the result of preliminary analysis of the doxorubicin flux data. In this case, receiving reservoir flux data were fitted to the diffusion model with the rate of cellular uptake set to zero to determine an apparent value for Dd o x o. The difference between apparent and predicted values for D d o x o, data points versus lines, is attributed to the unaccounted cellular uptake of doxorubicin. When cellular uptake is negligible relative to flux through the MCC, the apparent value of D d o x o should approach the actual rate of diffusion within the MCC. Hence, as culture thickness becomes small, and doxorubicin flux increases while net cellular uptake is reduced, the apparent value for D d o x o should tend towards the predicted value. Comparison of the predicted values for D d o x o and the trends shown by the grey lines, which extrapolate the data points to zero thickness, indicates that equation 2.11 serves as a i i 1 —i 1 r 0 50 100 150 200 250 M C C Thickness (um) Figure 9.4 Plot of the rate of diffusion of doxorubicin within V79 and SiHa MCCs as predicted by the V79/DOX analysis and knowledge of tissue tortuosity. Dashed and dotted lines show the predicted rate of diffusion for V79 and SiHa MCCs respectively. Data points for V79 - O and SiHa - • MCCs, indicate the apparent rate of diffusion when data from a series of flux experiments were fitted with the rate of cellular uptake fixed at zero. Grey lines show crude extrapolation of data points to zero thickness. Comparison of the predicted curves and the data points indicates that as M C C thickness becomes small, and cellular uptake becomes negligible compared to flux, the apparent rate of diffusion tends towards the predicted rate. For comparison the V79/DOX data from Figure 9.1 (b) is included here, as shown by the solid line and closed circles. 103 reasonable method for estimation of the actual rate of doxorubicin diffusion within the cultures. Using the predicted values for Dd o x o, V79 and SiHa MCC flux data were analysed to determine k and Bk. Examples of typical fitting results are shown for the two sets of V79 MCC flux data shown in Figure 9.1, panels (c)-(f). Solid lines indicate the results of fitting. For the analysis the parameter D d o x o was fixed at its calculated value, using equation 2.11. Results for the thin MCC, panels (c)-(d), were k = 0.38 ± 0.02 s"1, Bk = 52 ± 20 and (3 = (5.0 ± 2) x 10"6 s"1. For the thick MCC, panels (e)-(f), the results were k = 0.25 ± 0.02 s"1, Bk = 16 ± 27 and p = (6.9 ± 2) x 10"6 s1. Figure 9.5 shows results for determination of the rate cellular uptake, k, plotted as a function of culture thickness from a series of flux experiments using V79 and SiHa MCCs, panels (a) and (b) respectively. It is difficult to predict what the relation between k and MCC thickness, and hence age, should be. Since k is defined through the relation k=P-Acell/v2, it can be expected to be independent of MCC thickness only if the average cell volume and surface area, as well as any membrane composition effects on cell permeability, remain unchanged. Conversely, k will vary with MCC thickness if a systematic change occurs with any of these factors. In addition to actual changes in the rate of cellular uptake, poor estimation of the rate of diffusion of doxorubicin within the MCCs will also skew the relation between k and MCC thickness. Given this degree of indeterminacy, the results shown in Figure 9.5 were tentatively interpreted as indicating the rate of cellular uptake to be reasonably independent of culture thickness for cultures greater than 125 pm. The solid lines show the average value for k from each data set; ky^ = 0.27 s"1 (s.d. = 0.07), excluding results from the two thinnest cultures, and kSiHa = 0.18 sl (s.d. = 0.03). Cell membrane permeabilities were calculated from the cellular uptake rates to be PSiHa = 4.6 x 10"5 cm/s, PV79 = 5.3 x 10s cm/s and PV79/Dox = 0.6 x 10"5 cm/s, assuming 104 0.6 0.4 (a) o o CO * 0.2 h ca £ o 2 0.6 u n | 0.4 h U O Q rO . . . . 3 ° o o (b) 0.2r-^urn-D_ rV1-_L _L _L 100 150 200 250 MCC Thickness (um) Figure 9.5 Results for estimation of the rate of cellular uptake of doxorubicin within (a) V79 and (b) SiHa MCCs that ranged from 100 to 250 um in thickness. Flux data from each experiment were analysed with the rate of diffusion fixed at a value predicted from the V79/DOX results, using equation 2.11. Solid lines show the average value of k for each data set. cell surface area based on a spherical geometry. Table 9.1 summarises the results from the analysis of doxorubicin flux experiments through SiHa, V79 and V 7 9 / D O X M C C s . Cell Type D fex k P Bk xlO" 7 xlO" 5 cm/s 2 ~ § j cm/s --SiHa 17.3 0.22 0.18 (0.03) 4.6 26 (13) V79 7.9 0.38 0.27 (0.07) 5.3 34 (14) V79/DOX 5.8 0.28 0.03 ( - ) 0.6 19 (8) Table 9.1 Comparison of model parameters for 150 pm thick SiHa, V79 and V79/DOX MCCs. Values shown in brackets indicate standard deviations. The rate of flux of doxorubicin through the MCCs will depend on both the diffusion coefficient and the fraction of extracellular space available for diffusion. Cell permeability to the neutral form of doxorubicin, P, was calculated from the rate of cellular uptake, k, and consideration of cell volume and surface area (see section 2.3). 9.2 Fluorescence Imaging Within M C C s Figure 9.6 shows a comparison of the relative distribution of doxorubicin fluorescence within SiHa, V79 and V 7 9 / D O X M C C s and that predicted using the results of analysis of doxorubicin flux data. Fluorescence distributions were obtained through imaging of doxorubicin fluorescence within cryostat sections of M C C s . The predicted distribution of doxorubicin fluorescence within each of the M C C s was made using the mathematical model and the experimental estimates of the model parameters summarised in Table 9.1. Panels (a)-(c) show typical fluorescence images of M C C sections that were obtained for the measurement of the distribution of doxorubicin. The M C C s were exposed to 20 u M doxorubicin for 2 hours under conditions Depth into M C C (vim) Figure 9.6 Comparison of doxorubicin fluorescence distribution within MCC cryostat sections and the predicted distribution as obtained from flux analysis results. Panels (a), (b) and (c) show fluorescence images from SiHa, V79 and V79/DOX MCCs respectively. Panels (e), (f) and (g) show plots of fluorescence brightness as a function of depth in the MCCs from three separate cryostat sections, dashed lines. Grey lines show the predicted distribution of doxorubicin using parameter estimates from Table 9.1. Exposures were made over 2 hours using 20 u M doxorubicin under conditions matching the flux experiments. 105 similar to the flux experiments, i.e. using the dual reservoir diffusion apparatus with one reservoir starting at 20 uM and the other at zero. The relative fluorescence, as obtained from these images, plotted as a function of depth into the MCCs is shown by the dashed lines in panels (d)-(f). In each panel, each line represents measurements made using a separate section from the same MCC. Individual fluorescence measurements for each MCC were normalised to remove differences in cryostat sectioning thickness; hence only a comparison of the relative distribution of fluorescence within each MCC is made here, i.e. this is not a comparison of the absolute fluorescence within SiHa versus V79 MCCs. For many of the cryostat sections, the cells on the edge of the MCC that was exposed directly to the reservoir containing the doxorubicin showed substantially more uptake than cells a few layers further in. This effect was attributed to a loss of tissue homogeneity at the border between tissue and growth medium. While not accounted for in the modelling, its effect on the parameter estimates made in the results section was expected to be negligible due to the thickness of the MCCs used for experiments. The solid grey lines in panels (d)-(f) show the distribution of doxorubicin within the MCCs that was predicted from the mathematical simulations. Conditions for the simulations were chosen to match the experimental situation outlined above. A comparison of actual and simulated distributions indicates a good correlation between measurement and prediction. The V79 MCCs show a slightly steeper drop with depth into the MCCs than do the SiHa MCCs. This difference can be attributed to the higher rate of cellular uptake in the V79 MCCs, which has the effect of limiting doxorubicin penetration. The V79/DOX distributions show a lower signal to noise ratio than seen for the SiHa and V79 results, due to the lower total amount of doxorubicin within the cultures. The more linear drop of fluorescence with depth that was seen within the V79/DOX MCCs can be explained by the very low rate of cellular uptake within these MCCs. In the case where the rate of cellular uptake is zero, a linear gradient would form within the tissue due to the planar geometry of the MCCs. 9.3 Tumour Cord Simulations In order to aid the interpretation of the experimental results, mathematical simulations were carried out to determine the spatial distribution of doxorubicin within a tumour cord, the sheath of cells surrounding a blood vessel, under clinically relevant conditions. Modelling of 106 doxorubicin kinetics within a cord of cells was carried out using the mathematical model for doxorubicin kinetics within a tumour cord described in Section 3.1. In addition to modelling the tissue-drug interactions, the pharmacokinetics for doxorubicin within blood plasma were also incorporated into the modelling. Blood Pharmacokinetics In order to reproduce clinical conditions, the tumour cord model incorporated a time dependence for blood plasma doxorubicin concentration, matdiing that produced by intravenous administration of doxorubicin using: c w = C0(OJ-eH"T°ln2 +0.2-e-("T2)ln2 +0.1-eH"T')ln2), (9.1) where Cvessd is the blood plasma concentration of doxorubicin (uM), C 0 is the initial blood plasma doxorubicin concentration (uM), t is time in hours and Tlr x2 and x3 are the parameters that determine the doxorubicin rate of decay. For the simulations C 0 was set to 1.4 uM and xu x2 and T 3 were set to 5 minutes, 1 hour and 24 hours respectively, matching typical clinical conditions (Ames, et al., 1983, Dobbs and James, 1987). The fraction of blood plasma doxorubicin that was bound to serum albumin and lipoproteins was taken to be 0.7 (Chassany, et al., 1996). Simulation of Doxorubicin Distribution in a Tumour Cord Using the model for doxorubicin kinetics within a tumour cord from Section 3.1, the plasma pharmacokinetics of doxorubicin described by equation (1.1) and the parameter estimates shown in Table 9.1, simulations for doxorubicin penetration into the tumour cord model were carried out. Results are summarised in Figure 9.7, which shows results of simulations made using parameter data for SiHa, V79 and V79/DOX MCCs. For each cell type, the distribution of extra- and intracellular free doxorubicin as well as the percent of occupied D N A binding sites is shown at times of 1, 12 and 48 hours. For all three cell types, results showed poor penetration of doxorubicin away from the blood vessel. Comparison of the simulations made using SiHa and V79 MCC data allows several observations to be made. Firstly, the extracellular penetration of doxorubicin is slightly better in the simulations using the SiHa data. This is consistent with the MCC results from Figure 9.6. In both cases the improved 107 SiHa data V79 data V79/DOX data Distance from Blood Vessel (urn) Figure 9.7 Results for simulation of doxorubicin penetration into a tumour cord, the sheath of cells surrounding a blood vessel. Blood pharmacokinetics were set to match clinically relevant conditions. Distributions are shown at three times after the start of exposure: 1 hour (•••), 12 hours (- -) and 48 hours (—). Rows show results for extracellular free doxorubicin as a percent of the initial blood plasma doxorubicin concentration; intracellular free doxorubicin as a percent of the blood plasma doxorubicin concentration; the percent of occupied DNA binding sites. Columns show results using the parameters from Table 9.1 for SiHa, V79 and V79/DOX MCCs respectively. Note the y-axis scale used for the V79/DOX simulations shown in panels (h) and (i) is one tenth of that used for the SiHa and V79 simulations. penetration can be attributed to the lower rate of cellular uptake that was observed in the SiHa MCCs relative to the V79 MCCs (the effect of the higher rate of diffusion for doxorubicin in the SiHa MCCs is offset by a lower fraction of extracellular space available for diffusion). Secondly, the level of intracellular doxorubicin and the percent of occupied DNA binding sites appears to be higher for the SiHa MCCs, despite having a lower rate of cellular uptake. This can be explained by the higher density of DNA per unit volume within the V79 MCCs, due to the smaller size of the V79 cells relative to SiHa cells, hence the total amount of doxorubicin within the V79 cells is in fact higher than the SiHa cells. In the case of the simulations done using the V79/DOX data, drug distribution gradients are substantially less steep than those observed using the V79 data, resulting in much better extracellular penetration. In contrast, penetration of doxorubicin as measured by intracellular levels or the percent of occupied DNA 108 binding sites was roughly one tenth that seen for the V79 results, as was expected from comparison of the rates of cellular uptake used for the V79 and V79/DOX simulations. Additional simulations were carried out to determine the relative importance of the model parameters in limiting doxorubicin penetration. When the cellular uptake rate, all bmding terms and the plasma decay rate were set to zero, a uniform distribution of extracellular doxorubicin was achieved throughout the cord (within 5% of blood vessel levels) in ~10 minutes using SiHa data and -20 minutes using V79 data. The difference between the SiHa and V79 results can be attributed to their doxorubicin diffusion coefficients, which differ by roughly a factor of two. Repeating these simulations with the extracellular binding term set back to experimental levels had the effect of extending the time to reach a uniform distribution to 4 and 12 hours, using SiHa and V79 data respectively. The effect of the extracellular binding term on the net exposure to doxorubicin within the cord was investigated by repeating the original simulations shown Figure 9.7 with B k set to zero. Increases of 20% and 70% were observed in cellular doxorubicin levels at 150 pm after the first hour of exposure, using SiHa and V79 data respectively, but by twelve hours the increase had fallen to just over 1% and 3% respectively. This indicated that the effect of the delay in the build up of extracellular doxorubicin on cellular uptake, in the original simulations, was compensated for by retention of the drug as doxorubicin blood plasma levels decreased. Setting the intracellular membrane binding parameters to zero had negligible effect on simulations, due to the much higher affinity of doxorubicin for DNA. Removing the effect of the geometric dilution caused by the radial geometry of the tumour cord, by setting the blood vessel radius to a value much greater than the thickness of the cord and thereby making the cord geometry similar to an MCC, resulted in improved penetration. In this case, simulations showed a much gentler decrease in doxorubicin concentration with distance into the tissue; after 48 hours the ratio between intracellular doxorubicin at 0 pm versus 150 pm went from 20:1 (SiHa data) and 38:1 (V79 data) using the corded geometry to 7:1 (SiHa data) and 13:1 (V79 data) for the MCC-like geometry. The effect that the number of DNA binding sites had on drug penetration was investigated by reducing its value, the When the number of DNA bmding sites was halved, both the 109 intracellular concentration of free drug and the percent occupied binding sites were almost doubled. However, the effect on the relative distribution of bound doxorubicin was more moderate; the ratio between intracellular doxorubicin at 0 um versus 150 um after 48 hours became 15:1 (SiHa data) and 30:1 (V79 data). 9.4 Discussion This study has used the measurement and analysis of doxorubicin flux through MCCs to model the extravascular distribution of doxorubicin in tumour tissue after clinical administration. Flux data were analysed, using a mathematical model of doxorubicin kinetics within the three-dimensional tissue environment pf the MCCs to determine experimental estimates for doxorubicin diffusion, cellular uptake and extracellular binding. The diffusion of doxorubicin through the extracellular space of the MCCs was described assuming a Fickean relation, i.e. linear relation, between concentration gradient and resultant drug flux (Crank, 1975). The key factors that determined the flux of doxorubicin through the tissue were the fraction of extracellular space and the effective rate of diffusion of doxorubicin, as determined by tissue tortuosity. The ionisation of doxorubicin at biological pH was assumed to have a negligible effect on diffusion due to the low concentrations used in the experiments (Robinson and Stokes, 1955). Analysis of flux data using the full diffusion/uptake/binding model yielded estimates of the rate of diffusion of doxorubicin within the cultures which, upon consideration of tissue tortuosity, could be directly compared with the rate of diffusion of doxorubicin in water. Experimental results yielded an apparent rate of diffusion of doxorubicin in water of (5.6 ± 0.1) x 10"6 cm2/s compared to an expected value of ~(5 ± 1) x 10"6 cm2/s. The reasonable correspondence between these two values indicates that the method of analysis used here was able to correctly describe the major aspects of doxorubicin kinetics within the MCCs. Cellular uptake of doxorubicin was modelled assuming passive uptake of the molecule in its uncharged state (Frezard and Garnier-Suillerot, 1991). The key factors that determined the rate of flux through the cell membranes were permeability, surface area and the concentration gradient of uncharged doxorubicin. The slow rate of cellular uptake of doxorubicin and its high 110 affinity for the DNA resulted in very low levels of unbound drug within the cells, hence the concentration gradient across the cell membranes was determined almost solely by the extracellular free drug concentration. In the case of the V79/DOX MCCs, modelling indicated intracellular free doxorubicin never reached more than 0.5% of the free extracellular concentration. This allowed the effect on cellular uptake of active drug efflux to be described, without further adjustments to the model, by taking it to simply reduce the apparent permeability of the cells (Frezard and Garnier-Suillerot, 1991). Analysis of doxorubicin flux data yielded estimates for cell permeability to the uncharged form of doxorubicin of 4.6 x 10"5 cm/s, 5.3 x 10"5 cm/s, 0.6 x 10"5 cm/s for MCCs grown from SiHa, V79 and V79/DOX cells respectively. These results fall within the range of previously reported uptake rates of 2 x 10"6 to 7 x 10"4 cm/s for a variety of cell types (Stein, 1997). They are also comparable with permeabilities obtained using single cell suspensions of parental and resistant K562 cells (Frezard and Garnier-Suillerot, 1991). In the Frezard report, a ratio of ~5 between parental and resistant cell permeability was found. The binding of doxorubicin to DNA, intracellular membranes and serum proteins was modelled using equiubrium relations between free and bound drug that depended on drug-binding site affinity and the concentration of unoccupied binding sites. The assumption that the system was always in a state of near equilibrium relied on the high concentration of binding sites relative to the concentration of doxorubicin and the slow rate of change of drug concentrations relative to movement of the doxorubicin molecules. These two requirements were thought to be satisfied by the slow rate of cellular uptake and the high tortuosity of the extracellular space. The effect of doxorubicin DNA-binding on its own was great enough to maintain the simulated intracellular free drug concentration at levels that were small compared to extracellular free drug concentration. Hence the introduction of additional classes of intracellular binding sites to the model, including intracellular membrane binding, had a limited effect on the spatial distribution of doxorubicin within the cultures. The high affinity of doxorubicin for DNA had a profound effect on the limitation of simulated drug penetration into the tissue. I l l Because of the undetermined variety and extent of extracellular binding sites, extracellular bmding of doxorubicin was modelled using a non-saturable equilibrium relation. This was deemed acceptable because of the low binding affinity of doxorubicin for cell membranes, 0.6-0.9 mM"1, (Demant and Friche, 1998, Gallois, et al., 1996) and proteins, 1.3-2.7 mM"1, (Chassany, et al., 1996, Demant and Friche, 1998). Experimental analysis yielded values for the extracellular binding parameter, Bk, that fell between 19 and 34. By assuming a typical affinity coefficient of 1 mM"1 for extracellular binding, the concentration of extracellular binding sites can be estimated as ranging from 19 to 34 mM. From our simulations, extracellular binding was observed to have the effect of slowing the build up of free doxorubicin within the extracellular space of the tissue. However, its effect on intracellular levels was offset by prolonged retention of doxorubicin at later periods when blood plasma levels dropped off. While the source(s) of these binding sites remains undetermined, the contributions from several types of sites can be speculatively considered. Major sources of extracellular binding sites may include membrane phospholipids, glycosaminoglycans and extracellular proteins. The concentration of membrane phospholipids exposed to the extracellular space can be estimated to lie between 5-10 mM, assuming -2.5 x 106 phospholipids/pm2 and using the data for V79 and SiHa cell size, surface area and fraction of extracellular space described in this report. Doxorubicin has also been shown to interact with negatively charged sulphate groups of glycosaminoglycans with affinities similar those of phospholipids (Garnier-Suillerot and Gattegno, 1988), though in their report the affinity of doxorubicin for both phospholipids and sulphate groups appeared to be 100 times higher than those from the more recent reports. Unfortunately, little quantitative data exists concerning glycosaminoglycan content and sulphate density within the extracellular space of tumour tissue, though it has been estimated to range from 0.1 up to 1 g/kg tumour depending on the tumour type (Swabb, et al., 1974). Assuming that the glycosaminoglycan's disaccharide constituents have average molecular weights of 250 daltons and contain, on average, one sulphate group, their contribution to the concentration of extracellular binding sites would be of the order of several millimolar. While the multicellular spheroid model has been used to investigate variation in the relative level of glycosaminoglycans (GUmelius, et al., 1988), no absolute measure of glycosaminoglycan levels has been reported in any three-dimensional 112 culture system. In addition to phospholipid and glycosaminoglycan binding, doxorubicin will also be subject to binding to extracellular matrix and membrane proteins of undetermined variety and content. While the exact composition and nature of the doxorubicin binding sites remains undetermined, our experimental results, which suggest extracellular binding sites of the order of several tens of millimolar, appear to be consistent with what is known of the variety and concentration of the potential binding sites. In conclusion, MCC based modelling of doxorubicin distribution in tissue suggests that the extravascular penetration of doxorubicin is poor and may be a major factor limiting the effectiveness of this drug in vivo. A fundamental assumption required for the simple mathematical modelling approach used here was that intracellular doxorubicin was always in a state of near equilibrium between free and DNA-bound drug. A breakdown in this assumption, caused by high intracellular concentration gradients, would lead to higher than predicted intracellular free doxorubicin levels and would force an increase in the cell permeability estimates that were obtained from analysis of MCC flux data. This would be because the rate of cellular uptake depends on the product of the concentration gradient across the cell membrane and the permeability of the membrane to the drug. In a situation were the resultant change was great enough to allow saturation of cellular uptake of drug, this would then lead to an increase in the penetration of doxorubicin that is predicted in the simulations. Validation of the results presented here could be done through further comparison between predicted doxorubicin distribution and that measured via fluorescence imaging in MCCs and tumour cords. 113 10. Future Directions The experimental methodology and results presented in this thesis serve as the basis for future studies that employ MCCs to model drug penetration into tumour tissue. The methodology presented here describes the application of flux experiments to characterise the ability of anticancer drugs to penetrate tumour tissue. Results from the analysis of such experiments may be useful in the predicting the effectiveness of anticancer drugs as well as in understanding how the multitude of pharmacokinetic and physicochemical factors of a drug combine to determine its penetrative ability. The results that are presented here indicate that MCCs could be employed as a primary screening tool in the search for new anticancer drugs. With this in mind, future work might focus on the advancement of two experimental methods: the EIS system for tissue characterisation and a cell survival assay for determination of the spatial distribution of cytotoxicity within MCCs, similar to those that exist for the spheroid model. The experimental EIS results that were presented in this thesis indicate that EIS can potentially serve as a rapid and effective method for the assessment of MCC thickness and possibly for the characterisation of tissue structure. The temporary system that was employed to obtain the results shown here was fairly basic, requiring time consuming measurements using an oscilloscope. An automated system could be developed employing a computer based data acquisition system, which would allow for faster measurements at a greater number of frequencies. Once developed, such a system would greatly facilitate any MCC based study in which knowledge of culture thickness was required. A cell survival assay that could determine the spatial distribution of cell kill within MCCs after exposure to an anticancer agent would complement the existing technique for the assessment of drug penetration via drug flux experiments. Methodology could be based on sequential dissociation techniques that already exists for the spheroid model (Freyer and Schor, 1989, Kunz-Schughart and Freyer, 1997). The assay would serve to link drug penetration predicted from flux experiments to the actual distribution of drug cytotoxicity. As such it would serve as a valuable tool for verification of experimental flux results. 114 The primary objective of the development of the MCC model is to provide a simple method for evaluating a drug's ability to penetrate the extravascular compartment of a solid tumour. Future studies involving the use of MCCs for evaluation of drug penetration, either in the search for new anticancer drugs or for explaining the success or failure of existing drugs, could be based on a combination of drug flux and cell survival experiments. Analysis of drug flux data using the mathematical modelling approach presented in this report allows for an understanding of the relative importance of the multitude of factors which act to limit penetration, as well as the ability to predict penetration under clinical conditions. Combining this with the simple cell survival assay, for measurement of the distribution of cell kill within the MCCs, would create a robust system, capable of high throughput drug screening and elucidating why a drug is or is not a successful therapeutic agent. Achievement of these goals will also require a higher level of sophistication when carrying over results obtained from the MCC tissue environment to the highly variable in vivo tumour environment. 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Biochim Biophys Acta. 607:206-14. 120 Appendix I: Division of Labour Apparatus Alastair Andrew Carmel Machinists Growth Boxes • Diffusion apparatus • • • Incubator case • • • Heating system • • Gassing system • • • Stirring system • EIS system • Cell Culture S i H a • V79 • V79/DOX • Experiments Nitroimidazoles Tirapazamine • Doxorubicin • • EIS • Inulin • HTO • • Fluorescence Imaging • Data Analysis • Table A . l Summary of the division of labour for the work presented in this thesis. Contributors included the author - Alastair Kyle, supervisor - Andrew Minchinton, undergraduate student - Carmel Chan and machinists - Michel Rocher and Colin Potter. 1 2 1 Appendix II: Incubation System MCCs were maintained under physiological conditions both during growth and flux experiments using a custom built incubation system. MCC growth boxes and the diffusion apparatus were kept within the incubator under conditions of controlled temperature, gassing and stir speed. The incubator consisted of a plexiglass case that was maintained at 37 °C via forced air heating, see Figure A.l (a). Gassing within the growth boxes and the diffusion apparatus was provided via peek tubing (described in Chapter 5). Stir bars within the growth boxes and the diffusion apparatus were spun using an electromagnetic drive system, see Figure A.2. The incubation system satisfied several requirements that a conventional laboratory incubator could not meet. For growth of MCCs these included: the ability to maintain individual growth boxes under different gassing mixtures and the ability to regularly monitor MCCs from the side or from above without disturbing growth conditions. For flux experiments the custom incubator allowed for frequent sampling of diffusion apparatus reservoirs without disturbing the incubator. Temperature control unit Figure A . l Incubator used for growing MCCs and for the execution of drug flux experiments, (a) Photo of the incubator, (b) Diagram of temperature control system used to maintain the incubator at 37 °C. The programmable temperature control unit measures the temperature of the incubator via a thermocouple and sends a graded signal to the current source which in turn powers the heaters. Figure A.2 Photo of the stir bar control system. Stir bars are spun using electromagnets, using either a single focal point unit (right) - for growing MCCs, or a four focal point unit (left) - for flux experiments. Stirring speed can be varied between 0 to 900 r.p.m and is displayed on a digital readout. 122 The custom-built stir system was necessitated by the small separation distance between the four reservoirs of the diffusion apparatus, which was much smaller than the separation distance of commercially available stir plates. In the development of the stirring controller an electromagnetic drive system was chosen over a simpler, magnetic drive for several reasons, including the ability to function submerged or in a humid environment and that the digital electronics of the control circuit facilitated the incorporation of an exact stir speed readout. Temperature Control Figure A.l (b) shows a diagram of the temperature control system. A Love (model 15160, Wheeling, IL) control unit was used to measure temperature within the plexiglass case, via a copper-constantine thermocouple, and send a 0-20 mA signal to the current source which powered the incubator heaters. Figure A.3 shows a schematic of the electrical circuit used for the current source. The circuit is designed to pass current through the heating coils in proportion to the signal at the control input. The circuit was calibrated to pass 0-1.5 A in response to the 0 to 20 mA control signal. Calibration was carried out by the following procedure: 1 - set the input signal to zero and set 10 kQ pot on the positive arm of the op-amp to hold the voltage low, 2 - adjust the 10 kQ pot on the negative arm of the op-amp until it is just below the level where it will cause the output of the op-amp to go high, 3 - do the same for the 10 kQ pot on the positive arm of the op-amp (the circuit is now set to allow zero current for zero input and will not require a minimum input current before allowing the output current to flow), 4 - now set the input signal to maximum and adjust the 500 Q pot on the Control Figure A.3 Schematic of the current source circuit used to power the incubator heaters. The input signal from the temperature control unit (0-20 mA) acts to control the current through the heating coils (0-1.5 A). 123 positive arm of the op-amp so that the desired maximum output current passes through the heating coils. The heating coils were obtained from consumer grade hair dryers. The elements within the hair dryers were rewired so as to produce a combined resistance of 12 Q., hence the maximum power output of the heating system was 27 W. The high power fans within the hair dryers were removed and replaced with quiet 12 V fans. Stirring System Stir bars within the growth boxes and diffusion apparatus were spun using ,a custom built electromagnetic induction system, see Figure A.2. Figure A.4 (a) shows the basic principle upon which the system functioned and Figure A.4 (b) shows an outline of the electronic circuit designed to run it. Diagrams of the individual segments of the circuit outlined in Figure A.4 (b) are shown in Figures A.5 through A.10. (a) Stage 1 2 3 4 Coil 1 N - S -2 - N - S 3 S - N -4 - S - N Coils Figure A.4 Outline of stirrer control system, (a) Magnetic stir bars are spun via electromagnetic induction using wire coils that are sequentially powered in a four stage repetitive pattern, (b) Schematic of the electronic circuit used to control the power to the coils. output +9 V grid. i!i k JIL-Q R out trig gnd 555 / 1= cv dis/ thr +v T J - t r Ri 4.7 k£2 R2 100 ktipot R 3 1 5 kQ Ci 0.005 |xF C 2 0.01 nF +9 V Figure A.5 Circuit diagram of Clock 1 - the timer which controls the stir speed of the system. The circuit produces repeating pulses at a rate that can be varied by changing the value of R2. 124 A N D gate for computer on/off control n f\ Ji n n v- ai ai ao b< ao 4081 • . ._ ci co do di di c d t t v+ 4 ground to disable sequences through pins 0-3 =1-4017 Mm LL T divides frequency by 5 4017 £1 18 4 9 cocleCl RII U U U U LI El LL divides clock frequency by 3 v-L Output: pins a,b,c,d control the mosfets L L _ u L L q3 ql q2 j2 jl a 4018 a q4 i5 a5 Cl R U U U U 1 • v+ 4018: divide by N cmos chip 4017: decade counter, cmos 4081: quad A N D gate, cmos Input: from clock 1 Figure A.6 Circuit diagram of the Counter module of the stir bar controller. Clock 1 drives a sequence of counters that are used to lower the clock rate and sequence through four output stages, which in turn activate the mosfet power amplifiers. :a) Stage 1 2 3 4 Coil la,3b N - S -2a,4b - N - S lb,3a S - N -2b,4a - S - N (b) ! n-channel enhancement mosfet: M-TP5N05 Figure A.7 Overview of the method used to distribute power to the electromagnetic coils of the stirrer system, (a) Outline of the wiring of the electromagnetic coils. Each coil has a split solenoid so that current can be drawn through either section a or section b of the coil, thereby producing either a North or a South pole at the top of the coil, (b) Circuit used to power the individual section of the four coils. Each mosfet leads to one of the four output stages of the Counter module, which sequentially enables current flow through them. 125 (a) Figure A.8 Diagram of a one focal point stir plate, (a) The stir plate consists of four electromagnetic coils fixed on a metal plate. Each coil consists of two 125-foot segments of 34 gauge magnetic wire, (b) Outline of the wiring used to connect the coils to the mosfets. (b) 6 c m -top view iron base (0.15 cm thick) teflon bobbin - » - 2 . 5 4 c m - * , t 0.2 c m side view of coil ** iron field guide, fits on top of each bobbin (0.15 cm thick) +12 V stage 1 stage 2 stage 3 stage 4 (a) -7 cm-r«-2.4cm-»| .ks^J. hid L £ 4 -2 cm j l . 8 cm side view of coil specifications: -34 gauge magnet wire -coils 1,3,7,9 :two 66 feet 2,4,6,8 :two 100 feet 5 :two 133 feet (b) 5b stage 1 la 3a 7a 9a 2a 8a +12 V I stage 2 4b 6b 5a stage 3 lb 3b 7b 9b stage 4 Figure A.9 Diagram of a four focal point stir plate, (a) The stir plate consists of nine electromagnetic coils fixed on a metal plate, (b) Outline of the wiring used to connect the coils to the mosfets. 126 Clock 2 out Ri 100 kQ. R 2 4.7 kQ Rs 1 MQt r im C i 1 pF C 2 0.01 pF +9V to display counter (b) .n /MTII II JLO. v- ai ai ao bo bi bi 4011 • ci ci co do di di v-t to display reset (C) in from Clock 1 p n n n n n n n |v- 3 7 6 2 0 1 5 4017 q h h v+ in from Clock 2 4011: quad N A N D gate 4017: decade counter display reset display count reset counter gnd omronH7EC v+ __r -TJ D " +9v -all resistors 4.7kQ Figure A.10 Diagram of the Display Module that is used to display the stir speed of the stirring system, (a) Clock 2 acts to gate the count period during which the display/counter chip (c) counts pulses from Clock 1. (b) Circuit used to control the display/counter, (c) Display/counter chip that shows the stir speed in r.p.m. 

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