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Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

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PER protein circadian oscillations in Drosophila have been described by Goldbeter according to a five-dimensional model that includes the possibility of genetic mutation described by changing one parameter, the maximum degradation rate of the PER protein. Assuming that, in a mutant Drosophila this parameter is unreachable, we modify another parameter, the translation rate between the mRNA and the nonphosphorylated form of PER protein, by periodic intermittent activation or inhibition. We show how such a modification, simulated in the model by a periodic, on/off, piecewise constant stimulation (which increases or decreases this parameter) allows the entrainment of oscillations exactly at, or close to, a desired period. In a different context, this suggests that some diseases may be corrected using pharmacological agents according to specific periodic delivery schedules.

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Heart rate varies with respiration, blood pressure, emotion, etc., and heart rate variability (HRV) is presently one of the best indices to predict fatal issues in cardiac failure and after myocardial infarction. HRV depends on various reflexes. In addition, parallel studies of HRV and the myocardial adrenergic and muscarinic transduction system in experimental models of cardiac hypertrophy (CH) have suggested that the myocardial phenotype at the sinus-node level may also play a role. A transgenic strain of mice with atrial overexpression of the beta 1-adrenergic receptors was generated with attenuated HRV, which demonstrates that the phenotype itself is a determinant of HRV. HRV is explored by noninvasive techniques, including simple determination of the standard error of the mean, time-domain analysis, and Fourier transformation. We recently developed a time and frequency domain method of analysis, the smoothed pseudo-Wigner-Ville transformation, which allows better exploration of nonstationarity. Nonlinear methods have also been applied due to the extreme complexity of the biological determinants, and have provided evidence of a chaotic attractor in certain conditions. It is proposed that in steady state a very simple process, which is not completely deterministic, could better explain intermit interval regulations than chaotic behavior. In contrast, under extreme circumstances the regulation proceeds using chaotic behavior. Arrhythmias and HRV can be quantitated in 16-month-old unanesthetized spontaneously hypertensive rats (SHR). Ventricular premature beats are more frequent in SHR than in age-matched controls; they disappear after converting enzyme inhibition (CEI) relative to the reduction of both cardiac hypertrophy and ventricular fibrosis. HRV is attenuated in SHR, as it is in compensatory CH in humans. When CH is prevented, HRV returns to normal. CEI is therefore antiarrhythmic. Another pharmacological application of this concept concerns the bradycardic agents that may improve HRV.

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Heart rate is a function of at least three factors located in the sinus node, including the pacemaker and the activity of the sympathetic and vagal pathways. Heart rate varies during breathing and exercising. The is far from being a purely academic question because, after myocardial infarction or in cardiac insufficiency, reduced heart rate variability (HRV) represents the most valuable prognostic factor. HRV is usually considered index of the sympathovagal balance and is explored using time domain analysis, such as spectral analysis. Nevertheless, methods such as the Fast Fourier Transformation are not applicable to small rodents which have an unstable heart rate with asymmetric oscillations. Nonlinear methods show chaotic behavior under some conditions. A time and frequency domain method of analysis, the Wigner-Villé Transform, has been proposed for the study of HRV in both humans and small rodents, as a compromise between linear and nonlinear methods. We developed a method to quantify both arrhythmias and HRV in unanesthetized rodents. Such a method allows study of the relationship between the physiological parameters and the myocardial phenotype. Ventricular premature beats are more frequent in 16-month-old spontaneously hypertensive rats than in age-matched controls. In addition, HRV is attenuated in spontaneously hypertensive rats, as in compensatory cardiac hypertrophy in humans, and such attenuation is considered a prognostic index. Converting enzyme inhibition reduces in parallel arterial hypertension, cardiac hypertrophy, and ventricular fibrosis; it prevents ventricular premature beats and normalizes heart rate variability. It can be demonstrated that the incidence of ventricular premature beats is linked to the myocardial phenotype in terms of both cardiac hypertrophy and fibrosis. The two factors act as independent variables. HRV is correlated with the incidence of arrhythmias, suggesting that the beneficial effects of converting enzyme inhibition are related to prevention of arrhythmias.

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Heart rate variability (HRV) depends on various reflexes, including the baroreflex or respiratory reflex. Experimental studies have suggested that the sinoatrial node density in G protein-linked receptors may be involved. Transgenic mice, with a specific eightfold atrial overexpression of human beta 1-adrenoceptor (beta 1-AR), have been generated to evaluate the role of the atrial beta 1-AR density on HRV. The heart rate was monitored using telemetry, and the signal was analyzed using a quantitative time-frequency domain analysis, the smoothed pseudo-Wigner-Ville method, and phase portrait maps. 1) Heart rate was unchanged, but the two normal components of HRV were decreased in transgenic mice. Transgenic mice have an unshortened life span and no arrhythmias. 2) Challenge of the animals by propranolol showed no modulation of the HRV in transgenic mice compared with controls. 3) In isolated atrial strips from transgenic mice, basal contractility was increased and there was no isoproterenol-induced inotropic effect. 4) The basal level of adenosine 3',5'-cyclic monophosphate production was lowered in transgenic mice, suggesting a shift in adenylate cyclase isoforms.

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To complete traditional time- and frequency-domain analyses, new methods derived from non-linear systems analysis have recently been developed for time series studies. A panel of the most widely used methods of heart rate analysis is given with computations on mouse data, before and after a single atropine injection.

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To assess the influence of intrauterine growth retardation on heart rate (HR) and HR variability during sleep, we performed polygraphic recordings in 10 small-for-gestational age (SGA) and 16 appropriate-for-gestational age (AGA) newborns. Both groups were clinically and neurologically normal and were at 37 to 41 wk conceptional age. RR intervals were analyzed using the short-time Fourier transform in three frequency bands: 1) high frequency, with a period 3-8 heartbeat; 2) mid frequency, with a period 10-25 heartbeat; and 3) low frequency, with a period 30-100 heartbeat. In both active and quiet sleep, SGA newborns significantly differed from AGA newborns by having a shorter RR interval (p < 0.01) and lower amplitude of HR variability in all bands (p < 0.05) except low frequency in quiet sleep. Quiet sleep differed from active sleep by having a longer RR interval (p < 0.05), higher high-frequency variability (p < 0.02) in both SGA and AGA newborns, and lower low-frequency variability (p < 0.005 for AGA newborns). Our data give evidence of clear modifications of both sympathetic and parasympathetic HR control in the at-risk SGA population. Similarity of between-state characteristics suggests maintained CNS control of HR in SGA as well as in AGA newborns. We speculate that between-group HR and HR variability differences may be related to augmented metabolic rate in SGA compared with AGA newborns.

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To investigate the influence of prematurity and postnatal age on the maturation of the autonomic nervous system function, we analysed heart-rate and heart-rate variability in twelve prematurely born infants (< 37 weeks gestational age) reaching the conceptional age of 37-41 weeks. These neonates were compared with sixteen 37-41 week conceptional age newborns (< 10 days postnatal age). Heart-rate variability was analysed by spectral analysis of interbeat intervals using Short-Time Fourier Transform. We found that during both active and quiet sleep, the durations of RR-intervals were shorter and the amplitude of heart-rate variability in different frequency bands was lower in prematures reaching term than in newborns of the same conceptional age (P < 0.001). Between-state comparison showed differences in both groups. In both groups, low-frequency heart-rate variability was higher in active sleep than in quiet sleep. Between-state differences of RR-intervals and high-frequency heart-rate variability were present only in newborns (P < 0.01). Discrimination between newborns and prematures reaching term, based on RR-intervals and heart-rate variability, was correct in both sleep states with errors between 7 to 16%. However, in both newborns and prematures reaching term, between-state discrimination showed less reliable results, especially for quiet sleep discrimination with 24% (in PRT) and 20% (in NB) of errors. Our results, especially information given by factor analysis, suggest that the differences between newborns and prematures reaching term, concerning RR-interval and heart-rate variability, may be related to a changed balance between the sympathetic and parasympathetic nervous systems with a diminished parasympathetic component of heart rate control in prematures reaching term, as compared to newborns.

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To assess maturation of the Autonomic Nervous System (ANS) and sleep states, Heart Rate Variability (HRV) was studied in 24 healthy sleeping newborns, aged from 31 to 41 weeks, conceptional age (CA). Spectral analysis of the interbeat interval (RR) signal, was performed by Short-Time Fourier Transform, in three frequency bands: high (HF), of purely vagal origin, mid (MF), and low (LF), vagal and sympathetic, thus allowing evaluation of both branches of the ANS, observed in Active Sleep (AS = REM Sleep) and in Quiet Sleep (QS = nREM Sleep). Principal Component Analysis, Discriminant Analysis, and hypothesis tests were used to investigate the evolution of spectral variables and their relation with sleep states. HF, MF, LF, and mean RR all increased with age; the differences from the premature to the full-term group, were more marked, as a whole, in AS than in QS. HF showed the highest increase from the premature (31-36 weeks CA) to the intermediate (37-38) group, whereas LF showed equal differences from the premature to the intermediate, and from the intermediate to the full-term (39-41) groups. These results suggest a steep increase in vagal tone at 37-38 weeks CA, with stability afterwards, and a more regular increase in sympathetic tone from 31 to 41 weeks CA.

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