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.     

Mirtazapine inhibits tumor growth via immune response and serotonergic system

To study the tumor inhibition effect of mirtazapine, a drug for patients with depression, CT26/luc colon carcinoma-bearing animal model was used. BALB/c mice were randomly divided into six groups: two groups without tumors, i.e. wild-type (no drug) and drug (mirtazapine), and four groups with tumors, i.e. never (no drug), always (pre-drug, i.e. drug treatment before tumor inoculation and throughout the experiment), concurrent (simultaneously tumor inoculation and drug treatment throughout the experiment), and after (post-drug, i.e. drug treatment after tumor inoculation and throughout the experiment). The "psychiatric" conditions of mice were observed from the immobility time with tail suspension and spontaneous motor activity post tumor inoculation. Significant increase of serum interleukin-12 (sIL-12) and the inhibition of tumor growth were found in mirtazapine-treated mice (always, concurrent, and after) as compared with that of never. In addition, interferon-γ level and immunocompetent infiltrating CD4+/CD8+ T cells in the tumors of mirtazapine-treated, tumor-bearing mice were significantly higher as compared with that of never. Tumor necrosis factor-α (TNF-α) expressions, on the contrary, are decreased in the mirtazapine-treated, tumor-bearing mice as compared with that of never. Ex vivo autoradiography with [(123)I]ADAM, a radiopharmaceutical for serotonin transporter, also confirms the similar results. Notably, better survival rates and intervals were also found in mirtazapine-treated mice. These findings, however, were not observed in the immunodeficient mice. Our results suggest that tumor growth inhibition by mirtazapine in CT26/luc colon carcinoma-bearing mice may be due to the alteration of the tumor microenvironment, which involves the activation of the immune response and the recovery of serotonin level.     

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.     

Global Dormancy of Metastases Due to Systemic Inhibition of Angiogenesis

Autopsy studies of adults dying of non-cancer causes have shown that virtually all of us possess occult, cancerous lesions. This suggests that, for most individuals, cancer will become dormant and not progress, while only in some will it become symptomatic disease. Meanwhile, it was recently shown in animal models that a tumor can produce both stimulators and inhibitors of its own blood supply. To explain the autopsy findings in light of the preclinical research data, we propose a mathematical model of cancer development at the organism scale describing a growing population of metastases, which, together with the primary tumor, can exert a progressively greater level of systemic angiogenesis-inhibitory influence that eventually overcomes local angiogenesis stimulation to suppress the growth of all lesions. As a departure from modeling efforts to date, we look not just at signaling from and effects on the primary tumor, but integrate over this increasingly negative global signaling from all sources to track the development of total tumor burden. This in silico study of the dynamics of the tumor/metastasis system identifies ranges of parameter values where mutual angio-inhibitory interactions within a population of tumor lesions could yield global dormancy, i.e., an organism-level homeostatic steady state in total tumor burden. Given that mortality arises most often from metastatic disease rather than growth of the primary per se, this finding may have important therapeutic implications.     

Environmental stress induced by the tumor bed effect leads to subpopulation exclusion within heterogeneous neoplasms: modeling studies

Three nested mathematical models were used to describe the compositional stability of subpopulations within artificial heterogeneous neoplasms in a stressed environment. The first models a microecology in which each subpopulation grows independently and no competition for resources occurs. In the second model, subpopulations compete for common resources, while in the third, the subpopulations compete for resources, and an additional dynamic term describes the emergence of the second population from the first. Environmental stress is a consequence of ionizing radiation damage to the normal tissue in which the tumor grows (the tumor bed effect, TBE). Compositional data observed as a function of time from experimental assays of artificial heterogeneous colon adenocarcinoma xenografts were used for this theoretical analysis. The results show that in the stressed environment, tumor subpopulations do compete for common resources, and that the "weight of competition" (i.e., the rate at which competition can retard total growth) is significantly enhanced. In contrast to unperturbed artificial heterogeneous tumors which exhibit stable composition as a function of time, TBE stress leads to selection of the majority neoplastic population.     

An algorithm for molecular dissection of tumor progression

The volumetric growth of tumor cells as a function of time is most often likely to be a complex trait, controlled by the combined influences of multiple genes and environmental influences. Genetic mapping has proven to be a powerful tool for detecting and identifying specific genes affecting complex traits, i.e., quantitative trait loci (QTL), based on polymorphic markers. In this article, we present a novel statistical model for genetic mapping of QTL governing tumor growth trajectories in humans. In principle, this model is a combination of functional mapping proposed to map function-valued traits and linkage disequilibrium mapping designed to provide high resolution mapping of QTL by making use of recombination events created at a historic time. We implement an EM-simplex hybrid algorithm for parameter estimation, in which a closed-form solution for the EM algorithm is derived to estimate the population genetic parameters of QTL including the allele frequencies and the coefficient of linkage disequilibrium, and the simplex algorithm incorporated to estimate the curve parameters describing the dynamic changes of cancer cells for different QTL genotypes. Extensive simulations are performed to investigate the statistical properties of our model. Through a number of hypothesis tests, our model allows for cutting-edge studies aimed to decipher the genetic mechanisms underlying cancer growth, development and differentiation. The implications of our model in gene therapy for cancer research are discussed.     

Positive feedback and angiogenesis in tumor growth control

In vivo tumor growth data from experiments performed in our laboratory suggest that basic fibroblast growth factor (bFGF) and vascular endothelial growth factor (VEGF) are angiogenic signals emerging from an up-regulated genetic message in the proliferating rim of a solid tumor in response to tumor-wide hypoxia. If these signals are generated in response to unfavorable environmental conditions, i.e. a decrease in oxygen tension, then the tumor may play an active role in manipulating its own environment. We have idealized this type of adaptive behavior in our mathematical model via a parameter which represents the carrying capacity of the host for the tumor. If that model parameter is held constant, then environmental control is limited to tumor shape and mitogenic signal processing. However, if we assume that the response of the local stroma to these signals is an increase in the host's ability to support an ever larger tumor, then our models describe a positive feedback control system. In this paper, we generalize our previous results to a model including a carrying capacity which depends on the size of the proliferating compartment in the tumor. Specific functional forms for the carrying capacity are discussed. Stability criteria of the system and steady state conditions for these candidate functions are analyzed. The dynamics needed to generate stable tumor growth, including countervailing negative feedback signals, are discussed in detail with respect to both their mathematical and biological properties.     

Hybrids between normal cells,communication-deficient and communication-competent tumor cells

Summary Communication competent rat cells, communication incompetent (i.e., gap junction defective) rat tumor cells and communication competent rat tumor cells were fused in different combinations and the resultant hybrid cells were tested with regard to their growth properties and communication-competence. The ability to communicate via gap junctions was strictly inherited in a dominant fashion. Hybrids of normal cells and non-communicating tumor cells exhibited normal growth properties,i.e., the transformed phenotype was not expressed. This adds evidence to the hypothesis that the transformed growth of communication-incompetent tumor cells may be due to the loss or closure of their gap junctions.Communication-competent tumor cells behaved different when fused with normal cells in that their transformed growth was observed in each of the hybrids examined. No complementation was observed when different communication-incompetent cells or different tumor cells were fused.     

Antiangiogenic effects of the novel camptothecin ST1481 (gimatecan) in human tumor xenografts

ST1481 (gimatecan) is a novel lipophilic camptothecin with a promising preclinical pharmacological profile. On the basis of its high antitumor efficacy when delivered by the oral route, the compound is suitable for prolonged administration. This schedule of treatment has been reported as the most appropriate to exploit the antiangiogenic effects of cytotoxic drugs. The aim of the study was to investigate the antiangiogenic and antitumor effects of oral ST1481 in human tumor xenografts. In spite of a marginal drug effect against the s.c. growing A549 lung carcinoma following administration with an intermittent schedule (q4dx4 times, maximum tolerated dose: 2 mg/kg), tumor growth was strongly inhibited by a daily low-dose (0.5 mg/kg) prolonged administration. Immunohistochemical analysis showed a reduced number of microvessels in tumors of both treated groups versus controls and a significantly higher reduction in the daily versus the q4dx4-treated tumors (P < 0.0001, by Student's t test). In our experimental model, the relation between microvessel density and tumor size (r = 0.738, by the Spearman rank test) suggests a role of inhibition of tumor vasculature in tumor response. Significant inhibition of tumor angiogenesis (P < 0.0001 versus control tumors) was observed even with a very low drug dose (0.06 mg/kg) in the orthotopically implanted (i.d.) MeWo melanoma, under conditions causing minimal tumor growth inhibition. Additional evidences of the antiangiogenic activity of ST1481 were provided by antimotility effects on endothelial cells, in vivo inhibition of vascularization in the Matrigel assay, and down-regulation of the expression of the proangiogenic basic fibroblast growth factor in A549 tumor cells associated with inhibition of the pathway involving Akt. In conclusion, the available results support the possibility that the antiangiogenic properties of ST1481 contribute to its antitumor potential and that this effect might be enhanced by the continuous low-dose treatment.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

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.     

On the role of endothelial progenitor cells in tumor neovascularization

The exact role that bone marrow (BM)-derived endothelial progenitor cells (EPCs) play in tumor neovascularization is heavily debated. We develop a quantitative three-compartment model with predictive power regarding the dynamics of tumorigenesis. There are two distinct processes by which tumor neovasculature can be built: angiogenesis is the formation of new blood vessels from preexisting vessels; vasculogenesis is the formation of new vessels by recruiting circulating EPCs. We show that vasculogenesis-driven and angiogenesis-driven tumors grow in different ways. (i) If angiogenesis is the prevailing process, then the tumor mass (and volume) will grow as a cubic power of time, and BM-derived EPCs will stay at a constant level. (ii) If vasculogenesis is the dominant process, then the tumor mass will be characterized by a linear growth in time, and the number of circulating EPCs (after possibly increasing to a maximum) will decrease to low levels. With this information, one can identify the "signature" of each of the processes in the observations of tumor growth and the dynamics of the relevant characteristics, such as the level of BM-derived EPCs. We show how our results can help explain some apparently contradictory experimental data. We also propose ways to couple this study with directed experiments to identify the exact role of vasculogenesis in tumor progression.     

Anti-cancer therapies targeting the tumor stroma

For anti-tumor therapy different strategies have been employed, e.g., radiotherapy, chemotherapy, or immunotherapy. Notably, these approaches do not only address the tumor cells themselves, but also the tumor stroma cells, e.g., endothelial cells, fibroblasts, and macrophages. This is of advantage, since these cells actively contribute to the proliferative and invasive behavior of the tumor cells via secretion of growth factors, angiogenic factors, cytokines, and proteolytic enzymes. In addition, tumor stroma cells take part in immune evasion mechanisms of cancer. Thus, approaches targeting the tumor stroma attract increasing attention as anti-cancer therapy. Several molecules including growth factors (e.g., VEGF, CTGF), growth factor receptors (CD105, VEGFRs), adhesion molecules (alphavbeta3 integrin), and enzymes (CAIX, FAPalpha, MMPs, PSMA, uPA) are induced or upregulated in the tumor microenvironment which are otherwise characterized by a restricted expression pattern in differentiated tissues. Consequently, these molecules can be targeted by inhibitors as well as by active and passive immunotherapy to treat cancer. Here we discuss the results of these approaches tested in preclinical models and clinical trials.     

Inhibition of human renal cancer by monoclonal-anti-body-linked methotrexate in an ascites tumor model

Summary The monoclonal antibody DAL K29 against a cell-surface antigen associated with a human renal cell carcinoma was covalently linked to the antifolate methotrexate with full retention of antibody activity and partial retention of drug activity. Using an ascites tumor model, developed after intraperitoneal (i.p.) inoculation of 5 × 10⁶ cells of the human kidney cancer line Caki-1 per pristane-primed nude mouse, we showed that the methotrexate-Dal-K29 conjugate was a more potent tumor inhibitor (P <0.0005) of human renal cell carcinoma (which is resistant to currently available modalities including chemotherapy) than the drug or mAb alone, the drug linked to an isotype-matched nontumor-specific IgG or a mixture of the drug and the mAb. Only the conjugate could produce tumor-free survival in a proportion of the mice during the period of observation (i.e. 150 days after tumor inoculation).     

In vivo imaging in experimental preclinical tumor research--a review.

The multiparametric molecular cell and tissue analysis in vitro and in vivo is characterized by rapid progress in the field of image generation technologies, sensor biotechnology, and computational modeling. Fascinating new potentials in unraveling the detailed functions of single cells, organs, and whole organisms are presently emerging and permit the close monitoring i.e. tumor development or basic cell development processes with an unprecedented multiplicity of promising investigative possibilities. To answer basic questions of in vivo tumor development and progression fluorescence based imaging techniques provide new insights into molecular pathways and targets. Genetic reporter systems (eGFP, DsRED) are available and high sensitive detection systems are on hand. These techniques could be used for in vitro assays and quantified e.g. by microscopy and CCD based readouts. The introduction of novel fluorescent dyes emitting in the near infrared range (NIR) combined with the development of sensitive detector systems and monochromatic powerful NIR-lasers for the first time permits the quantification and imaging of fluorescence and/or bioluminescence in deeper tissues. Laser based techniques particularly in the NIR-range (like two-photon microscopy) offer superb signal to noise ratios, and thus the potential to detect molecular targets in vivo. In combination with flat panel volumetric computed tomography (fpVCT), questions dealing e.g. with tumor size, tumor growth, and angiogenesis/vascularization could be answered noninvasively using the same animal. The resolution of down to 150 microm/each direction can be achieved using fpVCT. It is demonstrated by many groups that submillimeter resolutions can be achieved in small animal imaging at high sensitivity and molecular specificity. Since the resolution in preclinical small animal imaging is down to approximately 10 microm by the use of microCT and to subcellular resolutions using ( approximately 1 microm) microscope based systems, the advances of different techniques can now be combined to "multimodal" preclinical imaging and the possibilities for in vivo intravital cytometry now become within one's reach.     

Improved decision making for prioritizing tumor targeting antibodies in human xenografts: Utility of fluorescence imaging to verify tumor target expression,antibody binding and optimization of dosage and application schedule

Preclinical efficacy studies of antibodies targeting a tumor-associated antigen are only justified when the expression of the relevant antigen has been demonstrated. Conventionally, antigen expression level is examined by immunohistochemistry of formalin-fixed paraffin-embedded tumor tissue section. This method represents the diagnostic “gold standard” for tumor target evaluation, but is affected by a number of factors, such as epitope masking and insufficient antigen retrieval. As a consequence, variances and discrepancies in histological staining results can occur, which may influence decision-making and therapeutic outcome. To overcome these problems, we have used different fluorescence-labeled therapeutic antibodies targeting human epidermal growth factor receptor (HER) family members and insulin-like growth factor-1 receptor (IGF1R) in combination with fluorescence imaging modalities to determine tumor antigen expression, drug-target interaction, and biodistribution and tumor saturation kinetics in non-small cell lung cancer xenografts. For this, whole-body fluorescence intensities of labeled antibodies, applied as a single compound or antibody mixture, were measured in Calu-1 and Calu-3 tumor-bearing mice, then ex vivo multispectral tumor tissue analysis at microscopic resolution was performed. With the aid of this simple and fast imaging method, we were able to analyze the tumor cell receptor status of HER1–3 and IGF1R, monitor the antibody-target interaction and evaluate the receptor binding sites of anti-HER2-targeting antibodies. Based on this, the most suitable tumor model, best therapeutic antibody, and optimal treatment dosage and application schedule was selected. Predictions drawn from obtained imaging data were in excellent concordance with outcome of conducted preclinical efficacy studies. Our results clearly demonstrate the great potential of combined in vivo and ex vivo fluorescence imaging for the preclinical development and characterization of monoclonal antibodies.     

Modeling tumor growth with random onset

The longitudinal assessment of tumor volume is commonly used as an endpoint in small animal studies in cancer research. Groups of genetically identical mice are injected with mutant cells from clones developed with different mutations. The interest is on comparing tumor onset (i.e., the time of tumor detection) and tumor growth after onset, between mutation groups. This article proposes a class of linear and nonlinear growth models for jointly modeling tumor onset and growth in this situation. Our approach allows for interval-censored time of onset and missing-at-random dropout due to early sacrifice, which are common situations in animal research. We show that our approach has good small-sample properties for testing and is robust to some key unverifiable modeling assumptions. We illustrate this methodology with an application examining the effect of different mutations on tumorigenesis.     

Bakuchiol inhibits lung cancer by modulating tumor microenvironment and the expression of PD-L1

Immune checkpoint therapy is an emerging frontier in cancer therapy. With the aim to develop an efficient herb derived compound to facilitate immune checkpoint therapy, here we investigate if a herb-derived compound, Bakuchiol (BAK), can be used to treat lung cancer and elucidate if BAK could serve as a PD-L1 regulator. To this end, a murine lung cancer model was established by subcutaneously inoculating murine Lewis lung carcinoma (LLC) cells. BAK of 5 to 40 mg/kg was used for treatment in vivo for 15 days. On Day 15, the population of CD4+ and CD8+ T cells, Treg cells. BAK could effectively inhibit tumor growth by starting treatment either on Day 0 or 6 after tumor inoculation at doses of 5−40 mg/kg. BAK treatment increased the population of cytotoxic immune cells (i.e., CD8+ T cells, and M1 macrophages), meanwhile decreasing pro-tumor immune cells (i.e., CD3+ T cells, Treg cells, and M2 macrophages). Anti-inflammatory cytokines, including IL1β, IL2, IFNγ, TNF-α, IL4 and IL10 were upregulated by BAK. PD-L1 expression in the tumor was also lowered by BAK. AKT and STAT3 signaling were inhibited by BAK. BAK is an efficient agent in reducing LLC tumor growth. These data support the potential of BAK as a new drug for treating lung cancer by serving as a PD-L1 inhibitor that suppresses the activation of AKT and STAT3.