Mathematical modeling tissues

Rubinsky, B. Pegg, D.E. (1988). A mathematical model for the freezing process in biological tissue. Proc. Roy. Soc. London B 234, 343-358. 

Zaharko DS, Dedrick RL, Bischoff KB, Longstreth JA, Oliverio VT. Methotrexate tissue distribution prediction by a mathematical model. / Natl Cancer Inst 1971 Apr 46(4) 775-84. 

So far, we have reviewed the various ways in which complex dose-response curves in intact-tissue bioassays can be the result, the pharmacological resultant, of two or more interacting activities. Now, if all that these bioassays achieved was to blur and obscure the underlying activities, they would have to give way to the newer, analytically simpler assays based on chemistry and biochemistry. However, the beauty of intact-tissue bioassays is that they are analytically tractable by using families of dose-response curves and appropriate mathematical models, the complexity of intact hormone-receptor systems can, indeed, be interpreted. Bioassay allows them to be studied as systems in ways denied to simple biochemical assays. 

Based on this concept of correlation between high replication rate/high persistent mutation risk, Pike et al. (1983) formulated the hypothesis of breast tissue age and developed a mathematical model to predict the effects of exposure to ovarian hormones. This model incorporates reproductive and endocrine items related to breast cancer and is able to predict the relative risk of individual situations with results that are very close to those observed in clinical trials. According to this hypothesis, both the years of exposure and the circulating serum levels of estrogens are associated to short-term breast cancer risk in postmenopausal women (Toniolo et al. 1995). 

Belousova IM, Mironova NG, Yur ev MS (2005) A mathematical model of the photodynamic fullerene-oxygen action on biological tissues. Optic Spectrosc 98 349-356. 

The effectiveness of the internal O2 transport by diffusion or convection depends on the physical resistance to movement and on the O2 demand. The physical resistance is a function of the cross-sectional area for transport, the tortuosity of the pore space, and the path length. The O2 demand is a function of rates of respiration in root tissues and rates of loss of O2 to the soil where it is consumed in chemical and microbial reactions. The O2 budget of the root therefore depends on the simultaneous operation of several linked processes and these have been analysed by mathematical modelling (reviewed by Armstrong... 

This chapter first reviews and discusses selected research on local dose aspects of ozone toxicity, the morphology of the respiratoty tract and mucus layer, air and mucus flow, and the gas, liquid, and tissue components of mathematical models. Next, it discusses the approaches and results of the few models that exist. A similar review was recently done to defme an analytic framework for collating experiments on the effects of sulfur oxides on the lung. Pollutant gas concentrations are generally stated in parts per million in this chapter, because experimental uptake studies are generally quoted only to illustrate behavior predicted by theoretical models. Chapter 5 contains a detailed discussion of the conversion from one set of units to another. 

In the physiological sense, one can divide the body into compartments that represent discrete parts of the whole-blood, liver, urine, and so on, or use a mathematical model describing the process as a composite that pools together parts of tissues involved in distribution and bioactivation. Usually pharmacokinetic compartments have no anatomical or physiological identity they represent all locations within the body that have similar characteristics relative to the transport rates of the particular toxicant. Simple first-order kinetics is usually accepted to describe individual... 

Toxicokinetics studies are designed to measure the amount and rate of the absorption, distribution, metabolism, and excretion of a xenobiotic. These data are used to construct predictive mathematical models so that the distribution and excretion of other doses can be simulated. Such studies are carried out using radiolabeled compounds to facilitate measurement and total recovery of the administered dose. This can be done entirely in vivo by measuring levels in blood, expired air, feces, and urine these procedures can be done relatively noninvasively and continuously in the same animal. Tissue levels can be measured by sequential killing and analysis of organ levels. It is important to measure not only the compound administered but also its metabolites, because simple radioactivity counting does not differentiate among them. 

Patterson MS, Pogue BW. Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissue. Applied Optics 1994, 33, 1963-1974. 

These results led us to analyze the relationship between carrier-wave frequency and power density. We developed a mathematical model (6) which takes into account the changes in complex permittivity of brain tissue with frequency. This model predicted that a given electric-field intensity within a brain-tissue sample occurred at different exposure levels for 50-, 147-, and 450-MHz radiation. Using the calculated electric-field intensities in the sample as the independent variable, the model demonstrated that the RF-induced calcium-ion efflux results at one carrier frequency corresponded to those at the other frequencies for both positive and negative findings. In this paper, we present two additional experiments using 147-MHz radiation which further test both negative and positive predictions of this model. 

Outline of Mathematical Analysis. We have described the basis for the mathematical model in detail elsewhere (6). The brain tissue plus buffer was assumed to approximate a spherical object in order to simplify the mathematical development. 

Table I and Figure 3 highlight another intriguing feature of the results obtained to date. The experimental results at 50 and 147 MHz have demonstrated two effective power density ranges separated and bracketed by regions of no effect. The mathematical model, applied to this data, predicts an additional effective power density range at each carrier frequency. Moreover, if the experimental results of Adey (1 1) are evaluated in this manner, a fourth effective power density window is predicted for each of the carrier frequencies. Confirmation of the predictions of the mathematical model resulting from these data is necessary in order to ensure that these results reflect a true response of the biological tissue.

Alternative methods include (1) computer-based methods (mathematical models and expert systems) (2) physicochemical methods, in which physical or chemical effects are assessed in systems lacking cells and, most typically, (3) in vitro methods, in which biological effects are observed in cell cultures, tissues, or organs. 

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