Novel Formula for Cell Kinetics in Xenograft Model of Hepatocellular Carcinoma Using Histologically Calculable Parameters

The growth rate of tumors should be assessed in terms of both tumor cell proliferation and death. The former is considered to be determined by growth fraction and cell-cycle time, whereas the latter is mainly determined by apoptosis, especially in tumors with a low level of necrosis. While most hepatocellular carcinomas (HCCs) in a relatively early stage contain only a small amount of necrosis, the growth rate supposedly depends mainly on growth fraction, cell-cycle time, and apoptosis. However, their quantitative relationship remains unknown. We have derived a novel theoretical formula for determining this relationship in nonnecrotic HCC, using Ki-67-positive index, apoptotic score, and a correction factor, all calculable by histological assessment without injecting labeling agents. Furthermore, we confirmed the reliability of this formula, using a xenograft model of human HCC with less than 15% necrosis. In this model the values of cell-cycle time calculated from the formula were very close to those estimated by a conventional double-labeling method and showed high correlations. Since our novel formula can clarify the cell kinetics without cumbersome labeling procedures, it is expected to be clinically applicable to HCC with a small portion of necrosis, using the radiographically measured growth rate and the histologically assessed cell kinetic parameters.     

How well is the probability of tumor cure after fractionated irradiation described by Poisson statistics?

The probability of tumor cure in a homogeneous population of tumors exposed to fractionated radiotherapy was modeled using numerical simulations and compared with the predictions of Poisson statistics, assuming exact knowledge of the relevant tumor parameters (clonogen number, radiosensitivity, and growth kinetics). The results show that although Poisson statistics (based on exact knowledge of all parameters) accurately describes the probability of tumor cure when no proliferation occurs during treatment, it underestimates the cure rate when proliferation does occur. In practice, however, the inaccuracy is not likely to be more than about 10%. When the tumor parameters are unknown and are estimated by fitting an empirical Poisson model to tumor-cure data from a homogeneous population of proliferative tumors, the resulting estimates of tumor growth rate and radiosensitivity accurately reflect the true values, but the estimate of initial clonogen number is biased downward. A new formula that is more accurate than Poisson statistics in predicting the probability of tumor cure when proliferation occurs during treatment is discussed.     

Application of purging biotinylated liposomes from plasma to elucidate influx and efflux processes associated with accumulation of liposomes in solid tumors

Although small, 100-nm liposomes are known to selectively accumulate in solid tumors, the individual contributions of liposome influx and egress rates are not well understood. The aim of this work was to determine influx and efflux kinetics for 100-nm, 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC)/cholesterol (Chol) liposomes by inducing aggregate formation of biotinylated liposomes upon administering avidin. Injecting 50 microg of neutravidin intravenously to mice that had previously been administered 100 mg/kg DPSC/Chol liposomes containing 0.5 mol% biotin-conjugated lipid resulted in >90% elimination of the liposomes from plasma within 1 h. This rapid removal by the reticuloendothelial system (RES) permitted the determination of the tumor efflux kinetics due to negligible tumor influx after neutravidin injection. The tumor efflux rate constant (k(-1)) was determined to be 0.041 h(-1) when neutravidin was injected 4 h after liposome injection. This allowed the determination of the tumor influx rate constant (k(1)), which under these conditions was 0.022 h(-1). Therefore, DSPC/Chol liposomal accumulation, in LS180 solid tumors, is dictated primarily by plasma liposome concentrations and liposome egress is comparable or slightly faster than influx into the tumors. This method is applicable for a wide range of lipid doses, and can be used to characterize influx and efflux parameters at different time points after accumulation. The application, therefore, has the potential to be used to fully characterize the impact of different liposome parameters such as lipid composition, steric stabilization, size and dose on tumor accumulation kinetics.     

Inactivation of chymotrypsin and human skin chymase: kinetics of time-dependent inhibition in the presence of substrate

The reaction of chymase, a chymotryptic proteinase from human skin, and bovine pancreatic chymotrypsin with a number of time-dependent inhibitors has been studied. An integrated equation, relating product formation with time, has been derived for the reaction of enzymes with time-dependent inhibitors in the presence of substrate. This is based on a two-step model in which a rapidly reversible, non-covalent complex (EI) is formed prior to a tighter, less readily reversible complex (EI)*). The equation depends on the simplifying assumption [I] much greater than [E], but is applicable to reversible and irreversible slow-binding and tight-binding inhibitors whether or not they show saturation kinetics. The method has been applied to the reaction of chymase and chymotrypsin with the tetrapeptide aldehyde, chymostatin, basic pancreatic trypsin inhibitor and Ala-Ala-Phe-chloromethylketone (AAPCK). The irreversible inhibitor, AAPCK, showed the expected saturation kinetics for both enzymes and the apparent first-order rate constants (k2) and dissociation constants (Ki) for the non-covalent complexes were determined. Chymostatin was a much more potent inhibitor which failed to show a saturation effect. The second-order rate constant of inactivation (k2/Ki), the first-order reactivation rate constant (k-2), and the dissociation constant of the covalent complex (Ki*) were determined. Basic pancreatic trypsin inhibitor, a potent inhibitor of chymotrypsin, had similar kinetics to chymostatin but failed to inhibit chymase. The applicability of the two-step model and the integrated equation to slow- and tight-binding inhibitors is discussed in relation to a number of examples from the literature.     

Distinguishing effects on tumor multiplicity and growth rate in chemoprevention experiments

In some types of cancer chemoprevention experiments and short-term carcinogenicity bioassays, the data consist of the number of observed tumors per animal and the times at which these tumors were first detected. In such studies, there is interest in distinguishing between treatment effects on the number of tumors induced by a known carcinogen and treatment effects on the tumor growth rate. Since animals may die before all induced tumors reach a detectable size, separation of these effects can be difficult. This paper describes a flexible parametric model for data of this type. Under our model, the tumor detection times are realizations of a delayed Poisson process that is characterized by the age-specific tumor induction rate and a random latency interval between tumor induction and detection. The model accommodates distinct treatment and animal-specific effects on the number of induced tumors (multiplicity) and the time to tumor detection (growth rate). A Gibbs sampler is developed for estimation of the posterior distributions of the parameters. The methods are illustrated through application to data from a breast cancer chemoprevention experiment.     

On the analysis of oxygen diffusion and reaction in biological systems

An analysis of oxygen diffusion and reaction in multiregion biological systems is presented. This analysis considers a time-dependent flux boundary condition and oxygen consumption governed by Michaelis-Menten kinetics. The mathematical problem is developed in a uniform fashion, so as to include both the single cell and anisotropic systems with distinct regions which are characteristic of either a multicell spheroid or a tumor mass. Both transient and steady-state solutions are obtained, based on orthogonal collocation. Literature results on single-cell analysis are corroborated, and detailed transient solutions are presented for the oxygenation of a multicell spheroid, and for systemic oxygenation of both small and large tumors.     

Origins and evolution of cell phenotypes in breast tumors

This study presents a stochastic model that correlates genomic instability with tumor formation. The model describes the time- and space-variant volumetric concentrations of cancer cells of various phenotypes in a breast tumor. The cells of epithelial origin in the cancerous breast tissue are classified into four different phenotypes, normal epithelial cells and the grade 1, grade 2 and grade 3 cancer cell types with increasing potential for growth and invasion. Equations governing the time course of volumetric concentrations of cell phenotypes are derived by using the principle of conservation of mass. Cell migration into and from the stroma is taken into account. The transformations between cell phenotypes are due to genetic inheritance and chromosome aberrations. These transformations are assumed to be stochastic functions of the local cell concentration. The simulations of the model for planar geometry replicate the shapes of human breast tumors and capture the time history of tumor growth in animal models. Simulations point to transformation of tumor cell population from heterogeneous compositions to a single phenotype at advanced stages of invasive tumors. Systematic variations of model parameters in the computations indicate the important roles the migration capacity, proliferation rate, and phenotype transition probability play in tumor growth. The model developed provides realistic simulations for standard breast cancer therapies and can be used in the optimization studies of chemotherapy, radiotherapy, hormone therapy and emerging individualized therapies for cancer.     

A two-mutation model for radon-induced lung tumors in rats

The recessive oncogenesis model, according to which inactivation of both alleles of specific genes leads to cancer, has received much recent attention. A mathematical formulation of a two-mutation model for carcinogenesis, which includes the recessive oncogenesis model as a special case, was fitted to data from a large experimental study in which rats exposed to radon daughters developed malignant lung tumors. The model described the data well. The results indicate that fractionation of exposure increased the lifetime probability of tumor. Examination of the parameters of the model suggests that the effect of fractionation can be explained by the relative effects of radon daughters on the mutation rates and on the kinetics of growth of initiated cells. The first mutation rate is very strongly dependent upon the rate of exposure to radon daughters, the second mutation rate much less so, suggesting that the nature of the two mutational events is different. The model makes predictions which are testable in future experiments.     

Enzyme deactivation

Enzyme deactivation kinetics is often first-order. Different examples of first-order deactivation kinetics exhibited by different enzymes under a wide variety of conditions are presented. Examples of both soluble and immobilized enzymes are presented. The influence of different parameters, chemical modification of specific residues, inhibitors, inactivators, protecting agents, induced conformational changes by external agents, enzyme concentration, and different substrates on the first-order inactivation kinetics of different enzymes is analyzed. The different examples presented from a variety of different areas provides a judicious framework and collection demonstrating the wide applicability of first-order deactivation kinetics. Examples of reversible first-order deactivation kinetics and deactivation-disguise kinetics are also presented.Different mechanisms are also presented to model complex enzyme deactivations. The non-series type mechanisms are emphasized and these involve the substrate and chemical modifiers. Substrate-dependent deactivation rate expressions that are of "separable" and "non-separable" type are presented. Rate expressions involving time-dependent rate constants along with their corresponding mechanisms are presented. Examples of enzymes that exhibit a deactivation-free grace period are also given. An interesting case of enzyme inactivation is the loss of activity in the presence of an auto-decaying reagent. The method is presented by which the intrinsic inactivation rate constants may be obtained. Examples of pH-dependent enzyme inactivation are presented that may be modelled by a five-step (or a simplified two-step) mechanism, and also by a single-step mechanism involving residual activity for the final state. Appropriate examples of enzyme inactivation are presented in each case to highlight the different mechanisms involved.     

Noninvasive monitoring of therapy-induced microvascular changes in a pancreatic cancer model using dynamic contrast-enhanced magnetic resonance imaging with P846, a new low-diffusible gadolinium-based contrast agent

A predictive technique in the management of patients with cancer could improve the therapeutic index by allowing better individualization of treatment. Dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is a noninvasive technique that can provide anatomical and physiological information on the tumor and its microenvironment. We studied the effect of chemotherapy (gemcitabine), anti-angiogenesis therapy (sunitinib) and radiotherapy on the kinetics of DCE-MRI parameters in a preclinical model of pancreatic cancer using P846, a new low-diffusible contrast agent. Mice underwent DCE-MRI before treatment (MRI1), after 1 week of treatment (MRI2), and after 1 additional week (MRI3). Combined treatment with radiotherapy and sunitinib had a synergistic effect on tumor growth. In radiotherapy/sunitinib-treated mice, a decrease in K(trans) at MRI2 predicted its superior antivascular and antitumor effect at an early time. An increased K(trans) at MRI2, as seen in gemcitabine- and gemcitabine/sunitinib-treated mice, reflects increased permeability for P846 and might predict a smaller therapeutic effect at this early time. This study shows that the kinetics of DCE-MRI parameters depends on the contrast agent used. P846 appears to be a promising low-diffusible agent to monitor therapeutic effects in this preclinical cancer model, but further studies are needed to compare its behavior with Gd-DTPA and macromolecular-weight contrast agents. Sunitinib as a radiosensitizer is promising for future clinical trials in human pancreatic cancer.     

A Biased Competition Theory of Cytotoxic T Lymphocyte Interaction with Tumor Nodules

The dynamics of the interaction between Cytotoxic T Lymphocytes (CTL) and tumor cells has been addressed in depth, in particular using numerical simulations. However, stochastic mathematical models that take into account the competitive interaction between CTL and tumors undergoing immunoediting, a process of tumor cell escape from immunesurveillance, are presently missing. Here, we introduce a stochastic dynamical particle interaction model based on experimentally measured parameters that allows to describe CTL function during immunoediting. The model describes the competitive interaction between CTL and melanoma cell nodules and allows temporal and two-dimensional spatial progression. The model is designed to provide probabilistic estimates of tumor eradication through numerical simulations in which tunable parameters influencing CTL efficacy against a tumor nodule undergoing immunoediting are tested. Our model shows that the rate of CTL/tumor nodule productive collisions during the initial time of interaction determines the success of CTL in tumor eradication. It allows efficient cytotoxic function before the tumor cells acquire a substantial resistance to CTL attack, due to mutations stochastically occurring during cell division. Interestingly, a bias in CTL motility inducing a progressive attraction towards a few scout CTL, which have detected the nodule enhances early productive collisions and tumor eradication. Taken together, our results are compatible with a biased competition theory of CTL function in which CTL efficacy against a tumor nodule undergoing immunoediting is strongly dependent on guidance of CTL trajectories by scout siblings. They highlight unprecedented aspects of immune cell behavior that might inspire new CTL-based therapeutic strategies against tumors.     

Incorporating time postinoculation into a dose–response model of Yersinia pestis in mice

Aims:  To develop a time-dependent dose–response model for describing the survival of animals exposed to Yersinia pestis.
Methods and Results:  Candidate time-dependent dose–response models were fitted to a survival data set for mice intraperitoneally exposed to graded doses of Y. pestis using the maximum likelihood estimation method. An exponential dose–response model with the model parameter modified by an inverse-power dependency of time postinoculation provided a statistically adequate fit to the experimental survival data. This modified model was verified by comparison with prior studies.
Conclusions:  The incorporated time dependency quantifies the expected temporal effect of in vivo bacteria growth in the dose–response relationship. The modified model describes the development of animal infectious response over time and represents observed responses accurately.
Significance and Impact of the Study:  This is the first study to incorporate time in a dose–response model for Y. pestis infection. The outcome may be used for the improved understanding of in vivo bacterial dynamics, improved postexposure decision making or as a component to better assist epidemiological investigations.     

Time course of bioluminescent signal in orthotopic and heterotopic brain tumors in nude mice

In vivo bioluminescence imaging is becoming increasingly popular. Quantification of bioluminescence signals requires knowledge of the variability and reproducibility of this technique. The objective of this study was to analyze the time course of luminescent signal emitted from firefly luciferase-expressing tumors in two locations, following luciferin injection and at different times after tumor cell implantation. Knowledge of the kinetics of the bioluminescent signals is required for the reliable quantification and comparison of signal during longitudinal studies. The kinetics of bioluminescence was evaluated in orthotopic and heterotopic brain tumors in mice using a human brain tumor cell line constitutively expressing luciferase. Tumor cells were implanted in the brains and flanks of the animals, and whole-body images revealing tumor location were obtained. Tumor burden was monitored over time by the quantitation of photon emission. The magnitude of bioluminescence measured in vivo varied with time after the injection of luciferin, as well as with dose, which necessitated that the comparison of the quantitative results take into consideration the time after injection. Heterotopic and orthotopic tumors exhibited significantly different time courses; however, time after implantation as characterized by kinetic studies performed on days 4 and 14 after cell implantation revealed no significant differences in orthotopic tumors. Future quantitative longitudinal studies must take into account the differences in the kinetics of different models.     

Modeling and computer simulations of tumor growth and tumor response to radiotherapy

A model of tumor growth and tumor response to radiation is introduced in which each tumor cell is taken into account individually. Each cell is assigned a set of radiobiological parameters, and the status of each cell is checked in discrete intervals. Tumor proliferation is governed by the cell cycle times of tumor cells, the growth fraction, the apoptotic capacity of the tumor, and the degree of tumor angiogenesis. The response of tumor cells to radiation is determined by the radiosensitivities and the oxygenation status. Computer simulation is performed on a 3D rigid cubic lattice, starting out from a single tumor cell. Random processes are simulated by Monte Carlo methods. Short cell cycle time, high growth fraction, and tumor angiogenesis all increase tumor proliferation rates. Accelerated time-dose patterns result in lower total doses needed for tumor control, but the extent of dose reduction depends on the kinetics and the radiosensitivities of tumor cells. Tumor angiogenesis alters fully oxygenated and hypoxic fractions within the tumor and subsequently affects the radiation response. It is demonstrated for selected radiobiological parameters that the simulation tools are suitable to quantitatively assess the total doses needed for tumor control. Using the simulation tools, it is feasible to simulate time-dependent effects during fractionated radiotherapy and to compare different time-dose patterns in terms of their tumor control.     

Efficient analysis of Weibull survival data from experiments on heterogeneous patient populations

An efficient method is presented for analyses of death rated in one-way or cross-classified experiments where expected survival time for a patient at time of entry on trial is a function of observable covariates. The survival-time distribution used is a Weibull form of Cox's (1972) model. The analysis proceeds in two steps. In the first, goodness of fit of the model is checked, inefficient estimates of the parameters are obtained, and survival times adjusted for the entry covariates are calculated. In the second, efficient estimates and tests for the rate parameters are obtained. These can easily be calculated using hand or desk equipment. Reorganized data sets can be analyzed without repetition of step one, thereby reducing the computational load to hand level and facilitating exploratory data analysis.     

Tumor Growth Rate Determines the Timing of Optimal Chronomodulated Treatment Schedules

In host and cancer tissues, drug metabolism and susceptibility to drugs vary in a circadian (24 h) manner. In particular, the efficacy of a cell cycle specific (CCS) cytotoxic agent is affected by the daily modulation of cell cycle activity in the target tissues. Anti-cancer chronotherapy, in which treatments are administered at a particular time each day, aims at exploiting these biological rhythms to reduce toxicity and improve efficacy of the treatment. The circadian status, which is the timing of physiological and behavioral activity relative to daily environmental cues, largely determines the best timing of treatments. However, the influence of variations in tumor kinetics has not been considered in determining appropriate treatment schedules. We used a simple model for cell populations under chronomodulated treatment to identify which biological parameters are important for the successful design of a chronotherapy strategy. We show that the duration of the phase of the cell cycle targeted by the treatment and the cell proliferation rate are crucial in determining the best times to administer CCS drugs. Thus, optimal treatment times depend not only on the circadian status of the patient but also on the cell cycle kinetics of the tumor. Then, we developed a theoretical analysis of treatment outcome (TATO) to relate the circadian status and cell cycle kinetic parameters to the treatment outcomes. We show that the best and the worst CCS drug administration schedules are those with 24 h intervals, implying that 24 h chronomodulated treatments can be ineffective or even harmful if administered at wrong circadian times. We show that for certain tumors, administration times at intervals different from 24 h may reduce these risks without compromising overall efficacy.     

A mathematical model of tumor growth and its response to single irradiation

Background

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

Methods

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

Results

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

Conclusions

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

     

Modeling age-specific incidence of colon cancer via niche competition

Cancer development is a multistep process often starting with a single cell in which a number of epigenetic and genetic alterations have accumulated thus transforming it into a tumor cell. The progeny of such a single benign tumor cell expands in the tissue and can at some point progress to malignant tumor cells until a detectable tumor is formed. The dynamics from the early phase of a single cell to a detectable tumor with billions of tumor cells are complex and still not fully resolved, not even for the well-known prototype of multistage carcinogenesis, the adenoma-adenocarcinoma sequence of colorectal cancer. Mathematical models of such carcinogenesis are frequently tested and calibrated based on reported age-specific incidence rates of cancer, but they usually require calibration of four or more parameters due to the wide range of processes these models aim to reflect. We present a cell-based model, which focuses on the competition between wild-type and tumor cells in colonic crypts, with which we are able reproduce epidemiological incidence rates of colon cancer. Additionally, the fraction of cancerous tumors with precancerous lesions predicted by the model agree with clinical estimates. The correspondence between model and reported data suggests that the fate of tumor development is majorly determined by the early phase of tumor growth and progression long before a tumor becomes detectable. Due to the focus on the early phase of tumor development, the model has only a single fit parameter, the time scale set by an effective replacement rate of stem cells in the crypt. We find this effective rate to be considerable smaller than the actual replacement rate, which implies that the time scale is limited by the processes succeeding clonal conversion of crypts.     

Effect of interbreath fluctuations on characterizing exercise gas exchange kinetics

Breathing has inherent irregularities that produce breath-to-breath fluctuations ("noise") in pulmonary gas exchange. These impair the precision of characterizing nonsteady-state gas exchange kinetics during exercise. We quantified the effects of this noise on the confidence of estimating kinetic parameters of the underlying physiological responses and hence of model discrimination. Five subjects each performed eight transitions from 0 to 100 W on a cycle ergometer. Ventilation, CO2 output, and O2 uptake were computed breath by breath. The eight responses were interpolated uniformly, time aligned, and averaged for each subject; and the kinetic parameters of a first-order model (i.e., the time constant and time delay) were then estimated using three methods: linear least squares, nonlinear least squares, and maximum likelihood. The breath-by-breath noise approximated an uncorrelated Gaussian stochastic process, with a standard deviation that was largely independent of metabolic rate. An expression has therefore been derived for the number of square-wave repetitions required for a specified parameter confidence using methods b and c; method a being less appropriate for parameter estimation of noisy gas exchange kinetics.