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Register a new userMathematical modeling of the metastatic process
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Mathematical modeling in cancer has been growing in popularity and impact since its inception in 1932. The first theoretical mathematical modeling in cancer research was focused on understanding tumor growth laws and has grown to include the competition between healthy and normal tissue, carcinogenesis, therapy and metastasis. It is the latter topic, metastasis, on which we will focus this short review, specifically discussing various computational and mathematical models of different portions of the metastatic process, including: the emergence of the metastatic phenotype, the timing and size distribution of metastases, the factors that influence the dormancy of micrometastases and patterns of spread from a given primary tumor.
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In this article, we review the mathematical modeling for the vascular system.
To meet the current need for skeletal tumor-load estimation in prostate cancer (mCRPC), we developed a novel approach, based on adaptive bone segmentation. In this study, we compared the program output with existing estimates and with the radiological outcome. Seventy-six whole-body 99mTc-DPD-SPECT/CT from mCRPC patients were analyzed. The software identified the whole skeletal volume (SVol) and classified it voxels metastases (MVol) or normal bone (BVol). SVol was compared with the estimation of a commercial software. MVol was compared with manual assessment and with PSA-level. Counts/voxel were extracted from MVol and BVol. After six cycles of 223RaCl2-therapy every patient was re-evaluated as progressing (PD), stabilized (SD) or responsive (PR). SVol correlated with the one of the commercial software (R=0,99, p<0,001). MVol correlated with manually-counted lesions (R=0,61, p<0,001) and PSA (R=0,46, p<0.01). PD had a lower counts/voxel in MVol than PR/SD (715 pm 190 Vs. 975 pm 215 and 1058 pm 255, p<0,05 and p<0,01) and in BVol (PD 275 pm 60, PR 515 pm 188 and SD 528 pm 162 counts/voxel, p<0,001). Segmentation-based tumor load correlated with radiological/laboratory indices. Uptake was linked with the clinical outcome, suggesting that metastases in PD patients have a lower affinity for bone-seeking radionuclides and might benefit less from bone-targeted radioisotope therapies.
The CVS is composed of numerous interacting and dynamically regulated physiological subsystems which each generate measurable periodic components such that the CVS can itself be presented as a system of weakly coupled oscillators. The interactions between these oscillators generate a chaotic blood pressure waveform signal, where periods of apparent rhythmicity are punctuated by asynchronous behaviour. It is this variability which seems to characterise the normal state. We used a standard experimental data set for the purposes of analysis and modelling. Arterial blood pressure waveform data was collected from conscious mice instrumented with radiotelemetry devices over $24$ hours, at a $100$ Hz and $1$ kHz time base. During a $24$ hour period, these mice display diurnal variation leading to changes in the cardiovascular waveform. We undertook preliminary analysis of our data using Fourier transforms and subsequently applied a series of both linear and nonlinear mathematical approaches in parallel. We provide a minimalistic linear and nonlinear coupled oscillator model and employed spectral and Hilbert analysis as well as a phase plane analysis. This provides a route to a three way synergistic investigation of the original blood pressure data by a combination of physiological experiments, data analysis viz. Fourier and Hilbert transforms and attractor reconstructions, and numerical solutions of linear and nonlinear coupled oscillator models. We believe that a minimal model of coupled oscillator models that quantitatively describes the complex physiological data could be developed via such a method. Further investigations of each of these techniques will be explored in separate publications.
Simple ideas, endowed from the mathematical theory of control, are used in order to analyze in general grounds the human immune system. The general principles are minimization of the pathogen load and economy of resources. They should constrain the parameters describing the immune system. In the simplest linear model, for example, where the response is proportional to the load, the annihilation rate of pathogens in any tissue should be greater than the pathogens average rate of growth. When nonlinearities are added, a reference value for the number of pathogens is set, and a stability condition emerges, which relates strength of regular threats, barrier height and annihilation rate. The stability condition allows a qualitative comparison between tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response.
Until recently many studies of bone remodeling at the cellular level have focused on the behavior of mature osteoblasts and osteoclasts, and their respective precursor cells, with the role of osteocytes and bone lining cells left largely unexplored. This is particularly true with respect to the mathematical modeling of bone remodeling. However, there is increasing evidence that osteocytes play important roles in the cycle of targeted bone remodeling, in serving as a significant source of RANKL to support osteoclastogenesis, and in secreting the bone formation inhibitor sclerostin. Moreover, there is also increasing interest in sclerostin, an osteocyte-secreted bone formation inhibitor, and its role in regulating local response to changes in the bone microenvironment. Here we develop a cell population model of bone remodeling that includes the role of osteocytes, sclerostin, and allows for the possibility of RANKL expression by osteocyte cell populations. This model extends and complements many of the existing mathematical models for bone remodeling but can be used to explore aspects of the process of bone remodeling that were previously beyond the scope of prior modeling work. Through numerical simulations we demonstrate that our model can be used to theoretically explore many of the most recent experimental results for bone remodeling, and can be utilized to assess the effects of novel bone-targeting agents on the bone remodeling process.
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