Coupled mathematical model of tumorigenesis and angiogenesis in vascular tumours

Objectives: Mathematical models are useful for studying vascular and avascular tumours, because these allow for more logical experimental design and provide valuable insights into the underlying mechanisms of their growth and development. The processes of avascular tumour growth and the development of capillary networks through tumour‐induced angiogenesis have already been extensively investigated, albeit separately. Despite the clinical significance of vascular tumours, few studies have combined these approaches to develop a single comprehensive growth and development model. Materials and methods: We develop a continuum‐based mathematical model of vascular tumour growth. In the model, angiogenesis is initiated through the release of angiogenic growth factors (AGFs) by cells in the hypoxic regions of the tumour. The nutrient concentration within the tumour reflects the influence of capillary growth and invasion induced by AGF. Results and conclusions: Parametric and sensitivity studies were performed to evaluate the influence of different model parameters on tumour growth and to identify the parameters with the most influence, which include the rates of proliferation, apoptosis and necrosis, as well as the diffusion of sprout tips and the size of the region affected by angiogenesis. An optimization was performed for values of the model parameters that resulted in the best agreement with published experimental data. The resulting model solution matched the experimental data with a high degree of correlation (r = 0.85).     

Targeting of the vascular system of solid tumours by photodynamic therapy (PDT).

Owing to morphological and rheological differences of the tumour vascular system as compared to the vascular system of the surrounding tissue, the efficacy of several experimental and clinical therapeutic approaches is limited. This fact has put the vascular system of solid tumours into focus and two new therapeutic strategies, anti-angiogenesis and vascular targeting, have emerged. Under the term vascular targeting various therapeutic approaches are summarized, e.g. chemoembolization, chemotherapy, hyperthermia, vascular targeting agents (VTA) and photodynamic therapy (PDT). As shown using the clinically approved photosensitiser Photofrin the irreversible destruction of the tumour vascular system is primarily responsible for an effective PDT of solid tumours. However, the clinical disadvantages of Photofrin are well known. Thus, several new photosensitisers, e.g. aminolaevulinic acid (ALA), porphycenes and indocyanine green (ICG), have been evaluated in vitro and in vivo regarding their suitability for vascular targeting of solid tumours. The promising experimental findings with the photosensitiser ICG led to first clinical results in treating Kaposi's sarcomas. In summary, systemic PDT is only effective when leading to complete ischaemia of solid tumours with subsequent necrosis. An essential prerequisite is the use of a chemically and photophysically defined photosensitiser localizing in the intravascular space due to e.g. a high molecular weight. The specific properties of such a photosensitiser are outlined.     

Tetraploid cycle in ageing solid tumours

When the mouse mammary adenocarcinoma 755 (Ca-755) reaches the plateau phase of growth, non-cycling cells with a G2-DNA content can be observed. They may belong to the diploid cell cycle but they could also be blocked in G0 or G1 of a tetraploid cycle. This hypothesis was tested in three ways: (1) non-cycling G2 nuclei were stained with a combination of Feulgen and naphthol yellow which revealed two populations, one with a low protein content and the other with a high protein content--the latter may represent nuclei ready to begin a new phase of DNA synthesis; (2) Feulgen staining and autoradiography were performed after tritiated thymidine had been administered to mice continuously: this showed that there were cells synthesizing DNA with a DNA index above 2; and (3) cells having 80 chromosomes, corresponding to the tetraploid cycle, were found almost exclusively in the plateau phase tumours. On the other hand, the use of texture and DNA parameters of the Feulgen stained nuclei showed that they were concentrated in a diploid cycle for tumours in the exponential phase of growth and were divided between a diploid and tetraploid cycle for 'plateau' cells. Neither the cause for, nor the role played by, polyploid cells is known.     

Tetraploid cycle in ageing solid tumours

Abstract. When the mouse mammary adenocarcinoma 755 (Ca-755) reaches the plateau phase of growth, non-cycling cells with a G₂-DNA content can be observed. They may belong to the diploid cell cycle but they could also be blocked in G₀ or G₁ of a tetraploid cycle. This hypothesis was tested in three ways: (1) non-cycling G₂ nuclei were stained with a combination of Feulgen and naphthol yellow which revealed two populations, one with a low protein content and the other with a high protein content– the latter may represent nuclei ready to begin a new phase of DNA synthesis; (2) Feulgen staining and autoradiography were performed after tritiated thymidine had been administered to mice continuously: this showed that there were cells synthesizing DNA with a DNA index above 2; and (3) cells having 80 chromosomes, corresponding to the tetraploid cycle, were found almost exclusively in the plateau phase tumours.
On the other hand, the use of texture and DNA parameters of the Feulgen stained nuclei showed that they were concentrated in a diploid cycle for tumours in the exponential phase of growth and were divided between a diploid and tetraploid cycle for 'plateau' cells. Neither the cause for, nor the role played by, polyploid cells is known.     

A validated model of passive muscle in compression

A better characterisation of soft tissues is required to improve the accuracy of human body models used, amongst other applications, for virtual crash modelling. This paper presents a theoretical model and the results of an experimental procedure to characterise the quasi-static, compressive behaviour of skeletal muscle in three dimensions. Uniaxial, unconstrained compression experiments have been conducted on aged and fresh animal muscle samples oriented at various angles from the fibre direction. A transversely isotropic hyperelastic model and a model using the theory of transverse isotropy and strain dependent Young's moduli (SYM) have been fitted to the experimental data. Results show that the hyperelastic model does not adequately fit the data in all directions of testing. In contrast, the SYM gives a good fit to the experimental data in both the fibre and cross-fibre direction, up to 30% strain for aged samples. The model also yields good prediction of muscle behaviour at 45° from the fibre direction. Fresh samples show a different behaviour than aged tissues at 45° from the fibre direction. However, the SYM is able to capture this difference and gives a good fit to the experimental data in the fibre, the cross-fibre and at 45° from the fibre direction. The model also yields good prediction of muscle behaviour when compressed at 30° and 60° from the fibre direction. The effect of the time of test after death has also been investigated. Significant stiffening of muscle behaviour is noted a few hours after death of the subject.     

Testicular germ cell tumours: the paradigm of chemo-sensitive solid tumours

Testicular germ cell tumours (TGCTs) are the most frequent solid malignant tumour in men 20–40 years of age and the most frequent cause of death from solid tumours in this age group. Up to 50% of the patients suffer from metastatic disease at diagnosis. The majority of metastatic testicular cancer patients, in contrast to most other metastatic solid tumours, can be cured with highly effective cisplatin-based chemotherapy. From a genetic point of view, almost all TGCTs in contrast to solid tumours are characterised by the presence of wild type p53. High p53 expression levels are associated with elevated Mdm2 levels and a loss of p21^Waf1/Cip1 expression suggesting a changed functionality of p53. Expression levels of other proteins involved in the regulation of cell cycle progression indicate a deregulated G1–S phase checkpoint in TGCTs. After cisplatin-induced DNA damage, the increasing levels of p53 lead to the trans-activation of a number of genes but not of p21^Waf1/Cip1, preferentially directing TGCT cells into apoptosis or programmed cell death, both via the mitochondrial and the death receptor apoptosis pathways. The sensitivity of TGCTs to chemotherapeutic drugs may lay in the susceptibility of germ cells to apoptosis. Taken together, this provides TGCT as a tumour type model to investigate and understand the molecular determinants of chemotherapy sensitivity of solid tumours. This review aims to summarise the current knowledge on the biological basis of cisplatin-induced apoptosis and response to chemotherapy in TGCTs.     

A technique to analyse the emerging zonality of heterogeneous solid tumours

A technique to analyse the time-dependent emergence of homogeneous regions composed of cells from a single clone within an artificial (clonally) heterogeneous tumour is described. Neoplasms were grown in vivo as xenografts made from varying proportions of the dichotomous subpopulations (Clones A and D). They were sampled frequently for volume and composition. A variable number of tumour cross-sections were taken as part of the sampling technique. The random subsamples obtained from each cross-section were enzymatically disaggregated into single cells. From the single cell disaggregates the composition of the tumours was estimated. An estimator for the global proportion of cells was then calculated from all the single cell disaggregates. Time-dependent changes in the overall composition of the tumour requires that a time-dependent estimate of the global proportions of the subpopulations be calculated from each sample. Analysis of the sample proportions results in a statistic which can be tested for goodness-of-fit against a standardized normal variate as a test of emerging zonality. Data from three artificial admixtures were examined. The results show that 'zonality', i.e. regions composed primarily of single subpopulations, emerges in all cases. However, the rate at which the zones emerge appears to depend on the 'compositional stability'. Robustness studies show that the technique is robust with respect to the global estimator of the proportion.     

A technique to analyse the emerging zonality of heterogeneous solid tumours

Abstract A technique to analyse the time-dependent emergence of homogeneous regions composed of cells from a single clone within an artificial (clonally) heterogeneous tumour is described. Neoplasms were grown in vivo as xenografts made from varying proportions of the dichotomous subpopulations (Clones A and D). They were sampled frequently for volume and composition. A variable number of tumour cross-sections were taken as part of the sampling technique. The random subsamples obtained from each cross-section were enzymatically disaggregated into single cells. From the single cell disaggregates the composition of the tumours was estimated. An estimator for the global proportion of cells was then calculated from all the single cell disaggregates. Time-dependent changes in the overall composition of the tumour requires that a time-dependent estimate of the global proportions of the subpopulations be calculated from each sample. Analysis of the sample proportions results in a statistic which can be tested for goodness-of-fit against a standardized normal variate as a test of emerging zonality.
Data from three artificial admixtures were examined. The results show that 'zonality', i.e. regions composed primarily of single subpopulations, emerges in all cases. However, the rate at which the zones emerge appears to depend on the 'compositional stability'.
Robustness studies show that the technique is robust with respect to the global estimator of the proportion.     

A model of growing vascular structures

Increasing attention is being paid to the configuration and development of vascular structures and their possible correlations with physiological events. The study of angiogenesis in normal and pathological states as well as in the embryo and adult has provided new insights into the mechanism of vessel growth and organization of the vasculature. Various mathematical branching models have been developed. These constructions are mainly geometrical and only involve a branching phenomenon. We propose the use of a deterministic non-linear model based on physiological laws and hydrodynamics. Growth, branching and anastomosis, the three actual main events occurring in vascular growth, are included in this model. Space growth, including cells and vessels, is defined by a decreasing transformation. Space density and the length of new sprouts are controlled by a set of parameters. The conditions on these parameters are well established, which allows the production of realistic patterns.     

Diversity of penetration of anti-cancer agents into solid tumours

Failure of anti-cancer agents to reach all clonogenic cells at cytotoxic concentrations is recognized as an important form of resistance in solid tumours. Subcutaneously implanted mammary adenocarcinoma 16/C was used to evaluate the intratumour distribution of five alkylating, bioreductive alkylating and intercalating agents and two radiation sensitizers. The agents were classified according to their in vivo distribution in well- and poorly-perfused tumour regions, as delineated by lissamine green. The classifications were: (1) distribution in direct proportion to the vascular supply; (2) uniform distribution to well- and poorly-perfused tumour regions; and (3) preferential retention in the poorly-perfused tumour regions. Our current state of knowledge did not allow reliable prediction of the classification based on chemical structure or mechanism of action.     

A model of the compression of acoustic information in neuronal networks

A model for processing of acoustic information in neuron networks has been proposed. The model takes into account the final rate of transfer of electric signal along the nerve fibers. The model indicates a possibility for the existence of a mechanism of compression of the input wide frequency band signal that retains its energetic parameters (in a certain limit). The model allowed to create a very efficient synthesizer of repellent acoustic signals that may be used to manage the bird behavior and regulate their numbers.     

A nonlocal constitutive model for trabecular bone softening in compression

Using the three-dimensional morphological data provided by computed tomography, finite element (FE) models can be generated and used to compute the stiffness and strength of whole bones. Three-dimensional constitutive laws capturing the main features of bone mechanical behavior can be developed and implemented into FE software to enable simulations on complex bone structures. For this purpose, a constitutive law is proposed, which captures the compressive behavior of trabecular bone as a porous material with accumulation of irreversible strain and loss of stiffness beyond its yield point and softening beyond its ultimate point. To account for these features, a constitutive law based on damage coupled with hardening anisotropic elastoplasticity is formulated using density and fabric-based tensors. To prevent mesh dependence of the solution, a nonlocal averaging technique is adopted. The law has been implemented into a FE software and some simple simulations are first presented to illustrate its behavior. Finally, examples dealing with compression of vertebral bodies clearly show the impact of softening on the localization of the inelastic process.     

Serum metallothionein in newly diagnosed patients with childhood solid tumours

Tumour markers are substances produced by malignant cells or by the organism as a response to cancer development. Determination of their levels can, therefore, be used to monitor the risk, presence and prognosis of a cancer disease or to monitor the therapeutic response or early detection of residual disease. Time-consuming imaging methods, examination of cerebrospinal fluid or tumour tissue and assays for hormones and tumour markers have been used for cancer diagnosis. However, no specific marker for diagnosis of childhood solid tumours has been discovered yet. In this study, metallothionein (MT) was evaluated as a prospective marker for such diseases. Serum metallothionein levels of patients with childhood solid tumours were determined using differential pulse voltammetry - Brdicka reaction. A more than 5-fold increase in the amount of metallothionein was found in sera of patients suffering from cancer disease, compared with those in sera of healthy donors. The average metallothionein level in the sera of healthy volunteers was 0.5 ± 0.2 μmol ? dm?3 and was significantly different (p<0.05, determined using the Schefe test) from the average MT level found in serum samples of patients suffering from childhood solid tumours (3.4 ± 0.8 μmol ? dm?3). Results found in this work indicate that the MT level in blood serum can be considered as a promising marker for diagnostics, prognosis and estimation of therapy efficiency of childhood tumours.