A pharmacologically based multiscale mathematical model of angiogenesis and its use in investigating the efficacy of a new cancer treatment strategy

Tumor angiogenesis is the process by which new blood vessels are formed and enhance the oxygenation and growth of tumors. As angiogenesis is recognized as being a critical event in cancer development, considerable efforts have been made to identify inhibitors of this process. Cytostatic treatments that target the molecular events of the angiogenesis process have been developed, and have met with some success. However, it is usually difficult to preclinically assess the effectiveness of targeted therapies, and apparently promising compounds sometimes fail in clinical trials.We have developed a multiscale mathematical model of angiogenesis and tumor growth. At the molecular level, the model focuses on molecular competition between pro- and anti-angiogenic substances modeled on the basis of pharmacological laws. At the tissue scale, the model uses partial differential equations to describe the spatio-temporal changes in cancer cells during three stages of the cell cycle, as well as those of the endothelial cells that constitute the blood vessel walls.This model is used to qualitatively assess how efficient endostatin gene therapy is. Endostatin is an anti-angiogenic endogenous substance. The gene therapy entails overexpressing endostatin in the tumor and in the surrounding tissue. Simulations show that there is a critical treatment dose below which increasing the duration of treatment leads to a loss of efficacy.This theoretical model may be useful to evaluate the efficacy of therapies targeting angiogenesis, and could therefore contribute to designing prospective clinical trials.     

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.

     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

A multiscale computational comparison of the bicuspid and tricuspid aortic valves in relation to calcific aortic stenosis

Patients with bicuspid aortic valve (BAV) are more likely to develop a calcific aortic stenosis (CAS), as well as a number of other ailments, as compared to their cohorts with normal tricuspid aortic valves (TAV). It is currently unknown whether the increase in risk of CAS is caused by the geometric differences between the tricuspid and bicuspid valves or whether the increase in risk is caused by the same underlying factors that produce the geometric difference. CAS progression is understood to be a multiscale process, mediated at the cell level. In this study, we employ multiscale finite-element simulations of the valves. We isolate the effect of one geometric factor, the number of cusps, in order to explore its effect on multiscale valve mechanics, particularly in relation to CAS. The BAV and TAV are modeled by a set of simulations describing the cell, tissue, and organ length scales. These simulations are linked across the length scales to create a coherent multiscale model. At each scale, the models are three-dimensional, dynamic, and incorporate accurate nonlinear constitutive models of the valve leaflet tissue. We compare results between the TAV and BAV at each length scale. At the cell-scale, our region of interest is the location where calcification develops, near the aortic-facing surface of the leaflet. Our simulations show the observed differences between the tricuspid and bicuspid valves at the organ scale: the bicuspid valve shows greater flexure in the solid phase and stronger jet formation in the fluid phase relative to the tricuspid. At the cell-scale, however, we show that the region of interest is shielded against strain by the wrinkling of the fibrosa. Thus, the cellular deformations are not significantly different between the TAV and BAV in the calcification-prone region. This result supports the assertion that the difference in calcification observed in the BAV versus TAV may be due primarily to factors other than the simple geometric difference between the two valves.     

A mathematical model applied to the fungal colony growth of Sclerotium rolfsii

Colonies of Sclerotium rolfsii growing on solid medium showed a linear increase in radius simultaneously with exponential changes in colonial weight. The mathematical model developed by Koch (Koch, A.L. 1975. J. Gen. Microbiol. 89, 209216) to describe similar kinetics of mycelial growth was applied. Most of the parameters defined in the mentioned model were directly measured or calculated, except the values of K (maximum mycelium carrying capacity) and S (initial mycelial density) which were fitted to the experimental data of growth at 30°C and 23°C. The ratios K/S at 30°C and at 23°C were 18.8 and 1200 respectively, being consistent with two-dimensional growth. The model was predictive only during the exponential increase of the colony weight.     

An advanced discrete state-discrete event multiscale simulation model of the response of a solid tumor to chemotherapy: Mimicking a clinical study

In this paper an advanced, clinically oriented multiscale cancer model of breast tumor response to chemotherapy is presented. The paradigm of early breast cancer treated by epirubicin according to a branch of an actual clinical trial (the Trial of Principle, TOP trial) has been addressed. The model, stemming from previous work of the In Silico Oncology Group, National Technical University of Athens, is characterized by several crucial new features, such as the explicit distinction of proliferating cells into stem cells of infinite mitotic potential and cells of limited proliferative capacity, an advanced generic cytokinetic model and an improved tumor constitution initialization technique. A sensitivity analysis regarding critical parameters of the model has revealed their effect on the behavior of the biological system. The favorable outcome of an initial step towards the clinical adaptation and validation of the simulation model, based on the use of anonymized data from the TOP clinical trial, is presented and discussed. Two real clinical cases from the TOP trial with variable molecular profile have been simulated. A realistic time course of the tumor diameter and a reduction in tumor size in agreement with the clinical data has been achieved for both cases by selection of reasonable model parameter values, thus demonstrating a possible adaptation process of the model to real clinical trial data. Available imaging, histological, molecular and treatment data are exploited by the model in order to strengthen patient individualization modeling. The expected use of the model following thorough clinical adaptation, optimization and validation is to simulate either several candidate treatment schemes for a particular patient and support the selection of the optimal one or to simulate the expected extent of tumor shrinkage for a given time instant and decide on the adequacy or not of the simulated scheme.     

A mathematical model for the growth of mycelial pellet populations

In liquid culture, filamentous organisms often grow in the form of pellets. Growth result in an increase in radius, whereas shear forces result in release of hyphal fragments which act as centers for further pellet growth and development. A previously published model for pellet growth of filamentous microorganisms has been examined and is found to be unstable for certain parameter values. This instability has been identified as being due to inaccuracies in estimating the numbers of fragments which seed the pellet population. A revised model has been formulated, based on similar premises, but adopting a finite element approach. This considers the population of pellets to be distributed in a range of size classes. Growth results in movement to classes of increasing pellet size, while fragments enter the smallest size class, from which they grow to form further pellets. The revised model is stable and predicts changes in the distribution of pellet sizes within a population growing in liquid batch culture. It considers pellet growth and death, with fragmentation providing new centers of growth within the pellet population, and predicts the effects of shear forces on pellet growth and size distribution. Predictions of pellet size distributions are tested using previously published data on the growth of fungal pellets and further predictions are generated which are suitable for experimental testing using cultures of filamentous fungi or actinomycetes. (c) 1995 John Wiley & Sons, Inc.     

A mechanistic mathematical model for the catalytic action of glutathione peroxidase

Abstract

Glutathione peroxidase (GPx) is a well-known seleno-enzyme that protects cells from oxidative stress (e.g., lipid peroxidation and oxidation of other cellular proteins and macromolecules), by catalyzing the reduction of harmful peroxides (e.g., hydrogen peroxide: H₂O₂) with reduced glutathione (GSH). However, the catalytic mechanism of GPx kinetics is not well characterized in terms of a mathematical model. We developed here a mechanistic mathematical model of GPx kinetics by considering a unified catalytic scheme and estimated the unknown model parameters based on different experimental data from the literature on the kinetics of the enzyme. The model predictions are consistent with the consensus that GPx operates via a ping-pong mechanism. The unified catalytic scheme proposed here for GPx kinetics clarifies various anomalies, such as what are the individual steps in the catalytic scheme by estimating their associated rate constant values and a plausible rationale for the contradicting experimental results. The developed model presents a unique opportunity to understand the effects of pH and product GSSG on the GPx activity under both physiological and pathophysiological conditions. Although model parameters related to the product GSSG were not identifiable due to lack of product-inhibition data, the preliminary model simulations with the assumed range of parameters show that the inhibition by the product GSSG is negligible, consistent with what is known in the literature. In addition, the model is able to simulate the bi-modal behavior of the GPx activity with respect to pH with the pH-range for maximal GPx activity decreasing significantly as the GSH levels decrease and H₂O₂ levels increase (characteristics of oxidative stress). The model provides a key component for an integrated model of H₂O₂ balance under normal and oxidative stress conditions.     

An application of Lacker's mathematical model for the prediction of ovarian response to superstimulation

Introduction. A mathematical model of ovarian follicular growth is applied to the problem of predicting ovarian response in a superstimulation protocol. Methods. Fifty-four women enrolled in an ovarian superstimulation program of therapy for the amelioration of idiopathic infertility had their ovarian cycles synchronized by taking Demulen 30 for two weeks prior to the study. Daily ultrasonographic imaging, measurements of serum estradiol and doses of hMG began on day 5 after the patients stopped taking Demulen. The diameters of individual follicles were measured and followed daily. When the largest follicle attained a diameter of 19 mm, hCG was given to induce ovulation. Individual follicle growth data were fit to a mathematical model of ovarian follicle maturation and the resulting parameters were used to classify patients into low and high ovarian response groups. Results. The parameters computed from the mathematical model fit were found to be predictive of ovarian response with a sensitivity of 71% and a specificity of 70%. The parameters were also meaningful within the context of the original mathematical model and have value for determining how doses of hMG may be adjusted during the course therapy to increase the ovarian response in individuals. Conclusion. Mathematical modeling of ultrasonographically derived follicular growth data has significant potential for clinical application in ovarian superstimulation protocols. The method of fitting follicular growth data to a mathematical follicle maturation surface furthermore provides a straightforward approach for the characterization of ovarian follicular dynamics in general.     

A mathematical model for pH patterns in the rhizospheres of growth zones

In the classical model by Nye (1981), the main process for the change in pH across the rhizosphere is assumed to be diffusion. The classical model focuses on the non-growing part of the root and assumes that the distribution of ion fluxes along the root is spatially uniform. We consider the rhizosphere of the growth zone and take into account the root growth rate and spatially varying flux along the root surface. We present both analytical (dimensional analysis) and experimental (computational) evidence of the importance of taking into account the root growth rate. We describe a conceptual and mathematical model to analyse the pH field around the root tip over time. The model is used with published data to show that, for typical growth rates in sandy soil, the pH field becomes steady (independent of time) after 6 h. Dimensional analysis reveals that a version of the Péclet number, related to the quotient of root elongation rate and proton diffusivity, can be used to predict the extent of the rhizosphere and the time required for it to become steady. For Péclet numbers much greater than 1 (soils), the root influences soil pH for distances on the millimetre scale. In contrast, for Péclet numbers much less than one (agar, aqueous solution), the root influences substrate pH for radial distances on the scale of centimetres. We also present some evidence that agar-contact techniques to measure the soil pH may not be appropriate for measuring the millimetre-scale gradients in soil pH.     

A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation

The acute inflammatory response, triggered by a variety of biological or physical stresses on an organism, is a delicate system of checks and balances that, although aimed at promoting healing and restoring homeostasis, can result in undesired and occasionally lethal physiological responses. In this work, we derive a reduced conceptual model for the acute inflammatory response to infection, built up from consideration of direct interactions of fundamental effectors. We harness this model to explore the importance of dynamic anti-inflammation in promoting resolution of infection and homeostasis. Further, we offer a clinical correlation between model predictions and potential therapeutic interventions based on modulation of immunity by anti-inflammatory agents.     

A conceptual mathematical model of the aquatic communities of lakes Naroch and Myastro (Belarus)

A conceptual mathematical model of the dynamics of fish and zooplankton (rotifer) populations of connected lakes Naroch and Myastro (Belarus) is built and examined with parameters based on field data. It is shown that community coupling and trophic interactions give rise to both regular and irregular oscillations in population numbers.     

A mathematical model for pattern formation of glioma cells outside the tumor spheroid core

Glioblastoma is the most common and the most aggressive type of brain cancer. The median survival time from the time of diagnosis is approximately one year. Invasion of glioma cells from the core tumor into the surrounding brain tissue is a major reason for treatment failure: these migrating cells are not eliminated in surgical resection and cause tumor recurrence. Variations are seen in number of invading cells, and in the extent and patterns of migration. Cells can migrate diffusely and can also be seen as clusters of cells distinct from the main tumor mass. This kind of clustering is also evident in vitro using 3D spheroid models of glioma invasion. This has been reported for U87 cells stably expressing the constitutively active EGFRVIII mutant receptor, often seen expressed in glioblastoma. In this case the cells migrate as clusters rather than as single cells migrating in a radial pattern seen in control wild type U87 cells. Several models have been suggested to explain the different modes of migration, but none of them, so far, has explored the important role of cell–cell adhesion. The present paper develops a mathematical model which includes the role of adhesion and provides an explanation for the various patterns of cell migration. It is shown that, depending on adhesion, haptotactic, and chemotactic parameters, the migration patterns exhibit a gradual shift from branching to dispersion, as has been reported experimentally.     

Supplementing glycosylation: A review of applying nucleotide-sugar precursors to growth medium to affect therapeutic recombinant protein glycoform distributions

Glycosylation is a critical quality attribute (CQA) of many therapeutic proteins, particularly monoclonal antibodies (mAbs), and is a major consideration in the approval of biosimilar biologics due to its effects to therapeutic efficacy. Glycosylation generates a distribution of glycoforms, resulting in glycoproteins with inherent molecule-to-molecule heterogeneity, capable of activating (or failing to activate) different effector functions of the immune system. Glycoforms can be affected by the supplementation of nucleotide-sugar precursors, and related components, to culture growth medium, affecting the metabolism of glycosylation. These supplementations has been demonstrated to increase nucleotide-sugar intracellular pools, and impact glycoform distributions, but with varied results. These variations can be attributed to five key factors: Differences between cell platforms (enzyme/transporter expression levels); differences between recombinant proteins produced (glycan-site accessibility); the fermentation and sampling timeline (glucose availability and exoglycosidase accumulation); glutamine levels (affecting ammonia levels, which impact Golgi pH, as well as UDP-GlcNAc pools); and finally, a lack of standardized metrics for observing shifts in glycoform distributions (glycosylation indices) across different experiments. The purpose of this review is to provide detail and clarity on the state of the art of supplementation strategies for nucleotide-sugar precursors for affecting glycosylation in cell culture processes, and to apply glycosylation indices for standardized comparisons across the field.     

A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration

Bacterial lipopolysaccharide (LPS; endotoxin) is a potent immunostimulant that can induce an acute inflammatory response comparable to a bacterial infection. Experimental observations demonstrate that this biological response can be either blunted (tolerance) or augmented (potentiation) with repeated administration of endotoxin. Both phenomena are of clinical relevance. We show that a four-dimensional differential equation model of this response reproduces many scenarios involving repeated endotoxin administration. In particular, the model can display both tolerance and potentiation from a single parameter set, under different administration scenarios. The key determinants of the outcome of our simulations are the relative time-scales of model components. These findings support the hypothesis that endotoxin tolerance and other related phenomena can be considered as dynamic manifestations of a unified acute inflammatory response, and offer specific predictions related to the dynamics of this response to endotoxin.     

A novel approach to investigate the effect of methionine oxidation on pharmacokinetic properties of therapeutic antibodies

Preserving the chemical and structural integrity of therapeutic antibodies during manufacturing and storage is a major challenge during pharmaceutical development. Oxidation of Fc methionines Met252 and Met428 is frequently observed, which leads to reduced affinity to FcRn and faster plasma clearance if present at high levels. Because oxidation occurs in both positions simultaneously, their individual contribution to the concomitant changes in pharmacokinetic properties has not been clearly established. A novel pH-gradient FcRn affinity chromatography method was applied to isolate three antibody oxidation variants from an oxidized IgG1 preparation based on their FcRn binding properties. Physico-chemical characterization revealed that the three oxidation variants differed predominantly in the number of oxMet252 per IgG (0, 1, or 2), but not significantly in the content of oxMet428. Corresponding to the increase in oxMet252 content, stepwise reduction of FcRn affinity in vitro, as well as faster clearance and shorter terminal half-life, in huFcRn-transgenic mice were observed. A single Met252 oxidation per antibody had no significant effect on pharmacokinetics (PK) compared with unmodified IgG. Importantly, only molecules with both heavy chains oxidized at Met252 exhibited significantly faster clearance. In contrast, Met428 oxidation had no apparent negative effect on PK and even led to somewhat improved FcRn binding and slower clearance. This minor effect, however, seemed to be abrogated by the dominant effect of Met252 oxidation. The novel approach of functional chromatographic separation of IgG oxidation variants followed by physico-chemical and biological characterization has yielded the first experimentally-backed explanation for the unaltered PK properties of antibody preparations containing relatively high Met252 and Met428 oxidation levels.     

A rapid bioassay to determine stabilities of anticancer agents under conditions of the clonogenic assay

Summary We developed a rapid bioassay to determine in vitro drug stabilities in the clonogenic assay. The in vitro half-lives of 11 standard antitumor agents, actinomycin D, Adriamycin, bleomycin, cis-Platinum, dacarbazine, 5-fluorouracil, melphalan, mitomycin C, 1,3-bis(2-chloroethyl)-1-nitrosourea (BCNU), vinblastine, and vincristine, and four investigational drugs, Aziridinylbenzoquinone (AZQ) (NSC 182986), diamminecyclobutane-dicarboxylatoplatinum (CBDCA) (NSC 241240), Dihydroxyanthracenedione dihydrochloride (DHAD) (NSC 301739), and 1-(2-chloroethyl)-3-(2,6-dioxo-3-piperidyl-1-nitrosourea) (PCNU) (NSC 95466), were determined under conditions of the clonogenic assay as well as under storage conditions at −40° and−196°C. BCNU and AZQ at−40°C had t1/2 of 4.7 d and 2.5 d, respectively. All other drugs were stable at −40° and −196°C with t1/2>6 wk. Under assay conditions at 37°C, actinomycin D, bleomycin, dacarbazine, 5-fluorouracil, mitomycin C, vinblastine, vincristine, and DHAD were stable, with t1/2>14 d. CBDCA, AZQ, Adriamycin, cis-Platinum, melphalan, BCNU, and PCNU had t1/2 of 94,72,29,18.5,1.8,1, and 0.5 h, respectively. This work was supported by Veterans Administration Medical Research Service and by Contract CM57710 of the National Cancer Institute.     

A novel approach to investigate the effect of methionine oxidation on pharmacokinetic properties of therapeutic antibodies

Preserving the chemical and structural integrity of therapeutic antibodies during manufacturing and storage is a major challenge during pharmaceutical development. Oxidation of Fc methionines Met252 and Met428 is frequently observed, which leads to reduced affinity to FcRn and faster plasma clearance if present at high levels. Because oxidation occurs in both positions simultaneously, their individual contribution to the concomitant changes in pharmacokinetic properties has not been clearly established. A novel pH-gradient FcRn affinity chromatography method was applied to isolate three antibody oxidation variants from an oxidized IgG1 preparation based on their FcRn binding properties. Physico-chemical characterization revealed that the three oxidation variants differed predominantly in the number of oxMet252 per IgG (0, 1, or 2), but not significantly in the content of oxMet428. Corresponding to the increase in oxMet252 content, stepwise reduction of FcRn affinity in vitro, as well as faster clearance and shorter terminal half-life, in huFcRn-transgenic mice were observed. A single Met252 oxidation per antibody had no significant effect on pharmacokinetics (PK) compared with unmodified IgG. Importantly, only molecules with both heavy chains oxidized at Met252 exhibited significantly faster clearance. In contrast, Met428 oxidation had no apparent negative effect on PK and even led to somewhat improved FcRn binding and slower clearance. This minor effect, however, seemed to be abrogated by the dominant effect of Met252 oxidation. The novel approach of functional chromatographic separation of IgG oxidation variants followed by physico-chemical and biological characterization has yielded the first experimentally-backed explanation for the unaltered PK properties of antibody preparations containing relatively high Met252 and Met428 oxidation levels.     

A New Modeling Approach to the Effect of Antimicrobial Agents on Heterogeneous Microbial Populations

We develop a mathematical framework to model the dynamics of the effect of antimicrobial agents on heterogeneous microbial populations of distributed antimicrobial resistance. Our framework uses the concept of cumulants of a distribution. Simplifications that result in easily usable approximation tools are presented. A case study on experimental data exemplifies shortcomings of standard methods and the usefulness of the proposed approach. Suggestions for future development are made. Author to whom all correspondence should be addressed