An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PurposeTo develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.MethodsCells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.ResultsSimulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.ConclusionsThe model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.     

Simulated weightlessness changes the cytoskeleton and extracellular matrix proteins in papillary thyroid carcinoma cells

Studies of astronauts, experimental animals, and cells have shown that, after spaceflights, the function of the thyroid is altered by low-gravity conditions. The objective of this study was to investigate the cytoskeleton and extracellular matrix (ECM) protein synthesis of papillary thyroid cancer cells grown under zero g. We investigated alterations of ONCO-DG 1 cells exposed to simulated microgravity on a three-dimensional random-positioning machine (clinostat) for 30 min, 24 h, 48 h, 72 h, and 120 h (n=6, each group). ONCO-DG 1 cells grown under microgravity exhibited early alterations of the cytoskeleton and formed multicellular spheroids. The cytoskeleton was disintegrated, and nuclei showed morphological signs of apoptosis after 30 min. At this time, vimentin was increased. Vimentin and cytokeratin were highly disorganized, and microtubules (α–tubulin) did not display their typical radial array. After 48 h, the cytoskeletal changes were nearly reversed. The formation of multicellular spheroids continued. In parallel, the accumulation of ECM components, such as collagen types I and III, fibronectin, chondroitin sulfate, osteopontin, and CD44, increased. The levels of both transforming growth factor beta-1 (TGF-β₁) and TGF-β receptor type II proteins were elevated from 24 h until 120 h clinorotation. Gene expression of TGF-β₁ was clearly enhanced during culture under zero g. The amount of E-cadherin was enhanced time-dependently. We suggest that simulated weightlessness rapidly affects the cytoskeleton of papillary thyroid carcinoma cells and increases the amount of ECM proteins in a time-dependent manner.The work of Augusto Cogoli was supported by ETH Zurich, Switzerland.     

Spatio-temporal modeling of nanoparticle delivery to multicellular tumor spheroids

The inefficiency of nanoparticle penetration in tissues limits the therapeutic efficacy of such formulations for cancer applications. Recent work has indicated that modulation of tissue architecture with enzymes such as collagenase significantly increases macromolecule delivery. In this study we developed a mathematical model of nanoparticle penetration into multicellular spheroids that accounts for radially dependent changes in tumor architecture, as represented by the volume fraction of tissue accessible to nanoparticle diffusion. Parameters such as nanoparticle binding, internalization rate constants, and accessible volume fraction were determined experimentally. Unknown parameters of nanoparticle binding sites per cell in the spheroid and pore shape factor were determined by fitting to experimental data. The model was correlated with experimental studies of the penetration of 40 nm nanoparticles in SiHa multicellular spheroids with and without collagenase treatment and was able to accurately predict concentration profiles of nanoparticles within spheroids. The model was also used to investigate the effects of nanoparticle size. This model contributes toward the understanding of the role of tumor architecture on nanoparticle delivery efficiency.     

Growth Characteristics of Human Melanoma Multicellular Spheroids In Liquid-Overlay Culture: Comparisons With the Parent Tumour Xenografts

Abstract. The growth characteristics of multicellular spheroids, derived from human melanoma xenografts and cultivated in liquid-overlay culture, were studied and compared with those of the parent tumours. Six of the seven melanomas investigated formed spheroids, which grew exponentially up to a volume of 1-2 × lo7 pm3 (a diameter of 270-340 μ) before the growth rate tapered off. the morphology of the spheroids varied considerably among the melanomas; some spheroids grew as densely packed, spherical structures of cells whereas others were loosely packed and showed an irregular shape. Central necrosis developed when the spheroids attained a diameter of 150-200, μ. the histological and cytological appearance of the spheroids was remarkably similar to that of the parent xenograft in five of the six cases. the sixth melanoma contained two subpopulations with distinctly different DNA content, one of which was predominant in the spheroids, the other in the tumours. This gave rise to clear histological and cytological differences. the volume-doubling time of the spheroids during the exponential growth phase ranged from 1.7 → 0.2 to 2.7 + 0.4 days and the fraction of cells in S from 13 → 1 to 28 → 2%. the volume-doubling time decreased with increasing fraction of cells in S, indicating that the differences in growth rate were due mainly to differences in the growth fraction or to differences in the duration of G,. the spheroid volume-doubling times did not correlate with those of the parent xenografts (T_d= 4.2-22.5 days at V= 200 mm³), possibly because the cell loss factors of the xenografts were large and varied among the melanomas. the fractions of cells in G₁/G₀, S and G₂+ M in the spheroids and the xenografts did not correlate either, but were found to be within the same narrow ranges in the spheroids and the xenografts—i.e. 50-80% (G₁/G_o), 10-30% (S) and 10-20% (G₂+ M).     

A multiscale model for avascular tumor growth

The desire to understand tumor complexity has given rise to mathematical models to describe the tumor microenvironment. We present a new mathematical model for avascular tumor growth and development that spans three distinct scales. At the cellular level, a lattice Monte Carlo model describes cellular dynamics (proliferation, adhesion, and viability). At the subcellular level, a Boolean network regulates the expression of proteins that control the cell cycle. At the extracellular level, reaction-diffusion equations describe the chemical dynamics (nutrient, waste, growth promoter, and inhibitor concentrations). Data from experiments with multicellular spheroids were used to determine the parameters of the simulations. Starting with a single tumor cell, this model produces an avascular tumor that quantitatively mimics experimental measurements in multicellular spheroids. Based on the simulations, we predict: 1), the microenvironmental conditions required for tumor cell survival; and 2), growth promoters and inhibitors have diffusion coefficients in the range between 10(-6) and 10(-7) cm2/h, corresponding to molecules of size 80-90 kDa. Using the same parameters, the model also accurately predicts spheroid growth curves under different external nutrient supply conditions.     

Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients

The mathematical model developed by Riveroet al. (1989,Chem. Engng Sci. 44, 2881–2897) is applied to literature data measuring chemotactic bacterial population distributions in response to steep as well as shallow attractant gradients. This model is based on a fundamental picture of the sensing and response mechanisms of individual bacterial cells, and thus relates individual cell properties such as swimming speed and tumbling frequency to population parameters such as the random motility coefficient and the chemotactic sensitivity coefficient. Numerical solution of the model equations generates predicted bacterial density and attractant concentration profiles for any given experimental assay. We have previously validated the mathematical model from experimental work involving a step-change in the attractant gradient (Fordet al., 1991Biotechnol. Bioengng.37, 647–660; For and Lauffenburger, 1991,Biotechnol. Bioengng,37, 661–672). Within the context of this experimental assay, effects of attractant diffusion and consumption, random motility, and chemotactic sensitivity on the shape of the profiles are explored to enhance our understanding of this complex phenomenon. We have applied this model to various other types of gradients with successful intepretation of data reported by Dalquistet al. (1972,Nature New Biol. 236, 120–123) forSalmonella typhimurum validating the mathematical model and supportin the involvement of high and low affinity receptors for serine chemotaxis by these cells.     

Modelling and simulation of steady-state phenol degradation in a pulsed plate bioreactor with immobilised cells of Nocardia hydrocarbonoxydans

A novel bioreactor called pulsed plate bioreactor (PPBR) with cell immobilised glass particles in the interplate spaces was used for continuous aerobic biodegradation of phenol present in wastewater. A mathematical model consisting of mass balance equations and accounting for simultaneous external film mass transfer, internal diffusion and reaction is presented to describe the steady-state degradation of phenol by Nocardia hydrocarbonoxydans (Nch.) in this bioreactor. The growth of Nch. on phenol was found to follow Haldane substrate inhibition model. The biokinetic parameters at a temperature of 30 ± 1 °C and pH at 7.0 ± 0.1 are μ _m = 0.5397 h⁻¹, K _S = 6.445 mg/L and K _I = 855.7 mg/L. The mathematical model was able to predict the reactor performance, with a maximum error of 2% between the predicted and experimental percentage degradations of phenol. The biofilm internal diffusion rate was found to be the slowest step in biodegradation of phenol in a PPBR.     

An unstructured kinetic model to study NaCl effect on volatile ester fermentation by <Emphasis Type="Italic">Candida etchellsii</Emphasis> for soy sauce production

Salt-tolerant aromatic yeast is an important microorganism arising from the solid state fermentation of soy sauce. The fermentation kinetics of volatile esters by Candida etchellsii was studied in a batch system. The data obtained from the fermentation were used for determining the kinetic parameters of the model. Batch experimental results at four NaCl levels (180, 200, 220, and 240 g/L) were used to formulate the parameter estimation model. The kinetic parameters of the model were optimized by specifically designed Runge-Kutta Genetic Algorithms (GA). The resulting mathematical model for volatile ester production, cell growth and glucose consumption simulates the experimental data well. The resulting new model was capable of explaining the behavior of volatile ester fermentation. The optimized parameters (μ_o, X _max, K _i, α, β, Y _X/S, m, and Y _P/S) were characterized by a correlation of functions assuming salinity dependence. The kinetic models optimized by GA describe the batch fermentation process adequately, as demonstrated by our experimental results.     

A mathematical model of tumor growth. III. comparison with experiment

In this paper a model of nonuniform inhibitor production is used to discuss some experimental results obtained by Folkman and Hochberg for multicellular spheroids (of V-79 Chinese hamster lung tissue). Built into the mathematical model is a parameter b which is a measure of the degree of nonuniformity of the inhibitor production rate. The “inverse problem” is solved by finding the value of b necessary to account for results with which the uniform model is incompatible [6]. The resulting value of b enables a good estimate to be made for the observed width of peripheral mitotic zones in the V-79 spheroids, and the resulting stability parameters lie in appropriate ranges for the model to be a significant improvement over uniform models. Further improvements are discussed, including a heuristic model for estimating the destabilizing effect of vascularization on tissue growth.     

Growth of Saccharomycopsis fibuligera in a continuous-stirred-tank fermentor

Growth rates and amylase production rates were determined for the yeast Saccharomycopsis fibuligera grown on a simulated potato processing waste in a continuous-stirred-tank fermentor. S. fibuligera formed large multicellular flocs in the fermentor, and cell growth was reduced at low dilution rates because of mass-transfer resistance. The average Thiele modulus, which is the measure of extent of substrate diffusion, had a value ranging from ?_av = 2.2 for D = 0.10 to 0.3 for D = 0.40. Growth rates were described by the Monod equation modified to include mass-transfer effects. This modified Monod equation was used to predict growth rates from measured floc-size distribution. Experimentally determined growth rates were in close agreement with these predicted values.     

Uptake of lactose and continuous lactic acid fermentation by entrapped non-growing Lactobacillus helveticus in whey permeate

 Continuous production of lactic acid from lactose has been carried out in a stirred-tank reactor with non-growing Lactobacillus helveticus entrapped in calcium alginate beads. A considerably longer operation half-life was obtained in a continuously operated reactor than in a batch-operated reactor. It is possible to simulate the action of entrapped non-growing cells on the basis of information from diffusion and kinetic experiments with suspended free cells. The simulation fit the experimental data over a broad range of substrate concentrations if the specific lactic acid production rate, q _P, was used as a variable parameter in the model. The dynamic mathematical model used is divided into three parts: the reactor model, which describes the mass balance in a continuously operated stirred-tank reactor with immobilized biomass, the mass-transfer model including both external diffusion and internal mass transfer, and the kinetic model for uptake of substrate on the basis of a Michaelis-Menten-type mechanism. From kinetic data obtained for free biomass experiments it was found, with the use of non-linear parameter estimation techniques, that the conversion rate of lactose by L. helveticus followed a Michaelis-Menten-type mechanism with K _S at half-saturation=0.22±0.01 g/l. The maximum specific lactose uptake rate for growing cells, q _S,max, varied between 4.32±0.02 g lactose g cells^-1 h^-1 and 4.89 ±0.02 g lactose g cells^-1 h^-1. The initial specific lactose uptake rate for non-growing cells, q _S,0, was found to be approximately 40% of the maximum specific lactose uptake rate for growing cells. Received: 4 October 1995/Received last revision: 23 April 1996/Accepted: 29 April 1996     

Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seenin vivo, including immunostimulation of tumor growth, “sneaking through” of the tumor, and formation of a tumor “dormant state”. The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL₁ in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measuredin vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.     

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

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

A model for the growth of multicellular spheroids

Abstract. Based on biological observations and the basic physical properties of tri-dimensional structures, a mathematical expression is derived to relate the growth rate of multicellular spheroids to some easily measurable parameters. This model involves properties both of the individual cells and of the spheroid structure, such as the cell doubling time in monolayer, the rate of cell shedding from the spheroid and the depth of the external rim of cycling cells. The derived growth equation predicts a linear expansion of the spheroid diameter with time. The calculated growth rate for a number of spheroid cell types is in good agreement with experimental data. The model provides a simple and practical view of growth control in spheroids, and is further adapted to include parameters presumably responsible for the growth saturation in large spheroids.     

Distributions of oxygen,nutrient, and metabolic waste concentrations in multicellular spheroids and their dependence on spheroid parameters

The distribution of oxygen, nutrients and metabolic wastes in multicellular tumor spheroids and its dependence on the parameters characterizing the spheroid (i.e., spheroid geometry, diffusivity, and consumption/ production rates of biological substances) have been investigated by a theoretical analysis: 1. Parameter dependence is qualitatively demonstrated and visualized. 2. Reduction of the number of variables by specific coordinate transformations made it possible to generate nomograms from which concentration distributions for any choice of parameter values may easily be obtained. In particular, these nomograms may also be used for estimating concentration profiles of metabolic waste products, e.g. of lactate, which are expected to accumulate in the tumor spheroids. 3. An additional set of nomograms is given which is more convenient for determining time courses of these concentrations during spheroid growth. 4. A quantitative sensitivity analysis of parameter dependencies is performed to identify those parameters upon which a concentration of interest depends most critically in a given experimental situation. Offprint requests to: W Mueller-Klieser     

Human Neuroendocrine Tumor Cell Lines as a Three-Dimensional Model for the Study of Human Neuroendocrine Tumor Therapy

Neuroendocrine tumors (NETs) are rare tumors, with an incidence of two per 100, 000 individuals per year, and they account for 0.5% of all human malignancies.¹ Other than surgery for the minority of patients who present with localized disease, there is little or no survival benefit of systemic therapy. Therefore, there is a great need to better understand the biology of NETs, and in particular define new therapeutic targets for patients with nonresectable or metastatic neuroendocrine tumors. 3D cell culture is becoming a popular method for drug screening due to its relevance in modeling the in vivo tumor tissue organization and microenvironment.^2,3 The 3D multicellular spheroids could provide valuable information in a more timely and less expensive manner than directly proceeding from 2D cell culture experiments to animal (murine) models.To facilitate the discovery of new therapeutics for NET patients, we have developed an in vitro 3D multicellular spheroids model using the human NET cell lines. The NET cells are plated in a non-adhesive agarose-coated 24-well plate and incubated under physiological conditions (5% CO₂, 37 °C) with a very slow agitation for 16-24 hr after plating. The cells form multicellular spheroids starting on the 3^rd or 4^th day. The spheroids become more spherical by the 6^th day, at which point the drug treatments are initiated. The efficacy of the drug treatments on the NET spheroids is monitored based on the morphology, shape and size of the spheroids with a phase-contrast light microscope. The size of the spheroids is estimated automatically using a custom-developed MATLAB program based on an active contour algorithm. Further, we demonstrate a simple method to process the HistoGel embedding on these 3D spheroids, allowing the use of standard histological and immunohistochemical techniques. This is the first report on generating 3D spheroids using NET cell lines to examine the effect of therapeutic drugs. We have also performed histology on these 3D spheroids, and displayed an example of a single drug''s effect on growth and proliferation of the NET spheroids. Our results support that the NET spheroids are valuable for further studies of NET biology and drug development.     

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.     

Substrate consumption and excess sludge reduction of activated sludge in the presence of uncouplers: a modeling approach

A mathematical model with a consideration of energy spilling is developed to describe the activated sludge in the presence of different levels of metabolic uncouplers. The consumption of substrate and oxygen via energy spilling process is modeled with a Monod term, which is dependent on substrate and inhibitor. The sensitivity of the developed model is analyzed. Three parameters, maximum specific growth rate (μ _max), energy spilling coefficient (q _max), and sludge yield coefficient (Y _H) are estimated with experimental data of different studies. The values of μ _max, q _max, and Y _H are found to be 6.72 day^-1, 5.52 day^-1, and 0.60 mg COD mg^-1 COD for 2, 4-dinitrophenol and 7.20 day^-1, 1.58 day^-1, and 0.62 mg COD mg^-1 COD for 2, 4-dichlorophenol. Substrate degradation and sludge yield could be predicted with this model. The activated sludge process in the presence of uncouplers that is described more reasonably by the new model with a consideration of energy spilling. The effects of uncouplers on substrate consumption inhibition and excess sludge reduction in activated sludge are quantified with this model.     

Study of the Chemotactic Response of Multicellular Spheroids in a Microfluidic Device

We report the first application of a microfluidic device to observe chemotactic migration in multicellular spheroids. A microfluidic device was designed comprising a central microchamber and two lateral channels through which reagents can be introduced. Multicellular spheroids were embedded in collagen and introduced to the microchamber. A gradient of fetal bovine serum (FBS) was established across the central chamber by addition of growth media containing serum into one of the lateral channels. We observe that spheroids of oral squamous carcinoma cells OSC–19 invade collectively in the direction of the gradient of FBS. This invasion is more directional and aggressive than that observed for individual cells in the same experimental setup. In contrast to spheroids of OSC–19, U87-MG multicellular spheroids migrate as individual cells. A study of the exposure of spheroids to the chemoattractant shows that the rate of diffusion into the spheroid is slow and thus, the chemoattractant wave engulfs the spheroid before diffusing through it.