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.     

A mathematical model of blood,cerebrospinal fluid and brain dynamics

Using first principles of fluid and solid mechanics a comprehensive model of human intracranial dynamics is proposed. Blood, cerebrospinal fluid (CSF) and brain parenchyma as well as the spinal canal are included. The compartmental model predicts intracranial pressure gradients, blood and CSF flows and displacements in normal and pathological conditions like communicating hydrocephalus. The system of differential equations of first principles conservation balances is discretized and solved numerically. Fluid–solid interactions of the brain parenchyma with cerebral blood and CSF are calculated. The model provides the transitions from normal dynamics to the diseased state during the onset of communicating hydrocephalus. Predicted results were compared with physiological data from Cine phase-contrast magnetic resonance imaging to verify the dynamic model. Bolus injections into the CSF are simulated in the model and found to agree with clinical measurements.

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A new mathematical model for the homeostatic effects of sleep loss on neurobehavioral performance

The two-process model of sleep regulation makes accurate predictions of sleep timing and duration for a variety of experimental sleep deprivation and nap sleep scenarios. Upon extending its application to waking neurobehavioral performance, however, the model fails to predict the effects of chronic sleep restriction. Here we show that the two-process model belongs to a broader class of models formulated in terms of coupled non-homogeneous first-order ordinary differential equations, which have a dynamic repertoire capturing waking neurobehavioral functions across a wide range of wake/sleep schedules. We examine a specific case of this new model class, and demonstrate the existence of a bifurcation: for daily amounts of wakefulness less than a critical threshold, neurobehavioral performance is predicted to converge to an asymptotically stable state of equilibrium; whereas for daily wakefulness extended beyond the critical threshold, neurobehavioral performance is predicted to diverge from an unstable state of equilibrium. Comparison of model simulations to laboratory observations of lapses of attention on a psychomotor vigilance test (PVT), in experiments on the effects of chronic sleep restriction and acute total sleep deprivation, suggests that this bifurcation is an essential feature of performance impairment due to sleep loss. We present three new predictions that may be experimentally verified to validate the model. These predictions, if confirmed, challenge conventional notions about the effects of sleep and sleep loss on neurobehavioral performance. The new model class implicates a biological system analogous to two connected compartments containing interacting compounds with time-varying concentrations as being a key mechanism for the regulation of psychomotor vigilance as a function of sleep loss. We suggest that the adenosinergic neuromodulator/receptor system may provide the underlying neurobiology.     

A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration

Bacterial lipopolysaccharide (LPS; endotoxin) is a potent immunostimulant that can induce an acute inflammatory response comparable to a bacterial infection. Experimental observations demonstrate that this biological response can be either blunted (tolerance) or augmented (potentiation) with repeated administration of endotoxin. Both phenomena are of clinical relevance. We show that a four-dimensional differential equation model of this response reproduces many scenarios involving repeated endotoxin administration. In particular, the model can display both tolerance and potentiation from a single parameter set, under different administration scenarios. The key determinants of the outcome of our simulations are the relative time-scales of model components. These findings support the hypothesis that endotoxin tolerance and other related phenomena can be considered as dynamic manifestations of a unified acute inflammatory response, and offer specific predictions related to the dynamics of this response to endotoxin.     

The growth and the fluid dynamics of protein crystals and soft organic tissues: models and simulations,similarities and differences

The fluid-dynamic environment within typical growth reactors as well as the interaction of such flow with the intrinsic kinetics of the growth process are investigated in the frame of the new fields of protein crystal and tissue engineering. The paper uses available data to introduce a set of novel growth models. The surface conditions are coupled to the exchange mass flux at the specimen/culture-medium interface and lead to the introduction of a group of differential equations for the nutrient concentration around the sample and for the evolution of the construct mass displacement. These models take into account the sensitivity of the construct/liquid interface to the level of supersaturation in the case of macromolecular crystal growth and to the "direct" effect of the fluid-dynamic shear stress in the case of biological tissue growth. They then are used to show how the proposed surface kinetic laws can predict (through sophisticated numerical simulations) many of the known characteristics of protein crystals and biological tissues produced using well-known and widely used reactors. This procedure provides validation of the models and associated numerical method and at the same time gives insights into the mechanisms of the phenomena. The onset of morphological instabilities is discussed and investigated in detail. The interplay between the increasing size of the sample and the structure of the convective field established inside the reactor is analysed. It is shown that this interaction is essential in determining the time evolution of the specimen shape. Analogies about growing macromolecular crystals and growing biological tissues are pointed out in terms of behaviours and cause-and-effect relationships. These aspects lead to a common source (in terms of original mathematical models, ideas and results) made available for the scientific community under the optimistic idea that the contacts established between the "two fields of engineering" will develop into an ongoing, mutually beneficial dialogue.     

A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis

A dynamic population model of Ixodes scapularis, the vector of a number of tick-borne zoonoses in North America, was developed to simulate effects of temperature on tick survival and seasonality. Tick development rates were modelled as temperature-dependent time delays, calculated using mean monthly normal temperature data from specific meteorological stations. Temperature also influenced host-finding success in the model. Using data from stations near endemic populations of I. scapularis, the model reached repeatable, stable, cyclical equilibria with seasonal activity of different instars being very close to that observed in the field. In simulations run using data from meteorological stations in central and eastern Canada, the maximum equilibrium numbers of ticks declined the further north was the station location, and simulated populations died out at more northerly stations. Tick die-out at northern latitudes was due to a steady increase in mortality of all life stages with decreasing temperature rather than a specific threshold event in phenology of one life stage. By linear regression we investigated mean annual numbers of degree-days >0 degrees C (DD>0 degrees C) as a readily mapped index of the temperature conditions at the meteorological stations providing temperature data for the model. Maximum numbers of ticks at equilibrium were strongly associated with the mean DD>0 degrees C (r2>0.96, P<0.001), when the Province of origin of the meteorological station was accounted for (Quebec>Ontario, beta=103, P<0.001). The intercepts of the regression models provided theoretical limits for the establishment of I. scapularis in Canada. Maps of these limits suggested that the range of southeast Canada where temperature conditions are currently suitable for the tick, is much wider than the existing distribution of I. scapularis, implying that there is potential for spread. Future applications of the model in investigating climate change effects on I. scapularis are discussed.     

A mathematical model of the methionine cycle

Building on the work of Martinov et al. (2000), a mathematical model is developed for the methionine cycle. A large amount of information is available about the enzymes that catalyse individual reaction steps in the cycle, from methionine to S-adenosylmethionine to S-adenosylhomocysteine to homocysteine, and the removal of mass from the cycle by the conversion of homocysteine to cystathionine. Nevertheless, the behavior of the cycle is very complicated since many substrates alter the activities of the enzymes in the reactions that produce them, and some can also alter the activities of other enzymes in the cycle. The model consists of four differential equations, based on known reaction kinetics, that can be solved to give the time course of the concentrations of the four main substrates in the cycle under various circumstances. We show that the behavior of the model in response to genetic abnormalities and dietary deficiencies is similar to the changes seen in a wide variety of experimental studies. We conduct computational "experiments" that give understanding of the regulatory behavior of the methionine cycle under normal conditions and the behavior in the presence of genetic variation and dietary deficiencies.     

A mathematical model of human heart including the effects of heart contractility varying with heart rate changes

Incorporating the intrinsic variability of heart contractility varying with heart rate into the mathematical model of human heart would be useful for addressing the dynamical behaviors of human cardiovascular system, but models with such features were rarely reported. This study focused on the development and evaluation of a mathematical model of the whole heart, including the effects of heart contractility varying with heart rate changes. This model was developed based on a paradigm and model presented by Ottesen and Densielsen, which was used to model ventricular contraction. A piece-wise function together with expressions for time-related parameters were constructed for modeling atrial contraction. Atrial and ventricular parts of the whole heart model were evaluated by comparing with models from literature, and then the whole heart model were assessed through coupling with a simple model of the systemic circulation system and the pulmonary circulation system. The results indicated that both atrial and ventricular parts of the whole heart model could reasonably reflect their contractility varying with heart rate changes, and the whole heart model could exhibit major features of human heart. Results of the parameters variation studies revealed the correlations between the parameters in the whole heart model and performances (including the maximum pressure and the stroke volume) of every chamber. These results would be useful for helping users to adjust parameters in special applications.     

A generalized compartmental model to estimate the fibre mass in the ruminoreticulum: 1. Estimating parameters of digestion

Parameters related to the microbial digestion of nutrients in the ruminoreticulum have been estimated by fitting mathematical models to degradation profiles generated from kinetic studies. In the present paper, we propose a generalized compartmental model of digestion (GCMD) based on implicit theoretical concepts and the gamma probability density function to estimate fibre digestion parameters. The proposed model is consistent to a broader compartmental model presented in a companion paper that integrates aspects of fibre digestion and passage. Different versions of the GCMD were generated by increasing the integer order of time dependency of the gamma function. These versions were fitted to 192 published fibre degradation profiles that were obtained using an in vitro fermentation technique. The quality of fit was evaluated based on the frequency of minimum sum of squares of errors (SSE), the number of runs of signs of residuals, and its likelihood probability calculated according to the Akaike's Information Criterion. The likelihood of the proposed model was also compared to a discrete lag time model (DLT), which is commonly used to interpret fibre degradation profiles. The GCMD had superior quality of fit compared to the DLT and was considered more likely in describing 68.75% of the profiles evaluated. Only 9.38% of the degradation profiles that were fitted to the DLT model had a lower SSE. Even though the degradation profiles studied were generated by incubating feed samples up to 96 h, the true asymptotic limit of fibre degradation can only be achieved by long-term fermentations. This fact leads to questioning the uniformity of the potentially digestible fibre fraction and a further approach based on GCMD-type model was used to account for its heterogeneous nature.     

水分胁迫与光照条件对棉花干物质和产量形成影响的数学模型

从作物冠层净同化速率入手,通过引入对CO2浓度、空气湿度、光照强度和土壤含水量反映较敏感的光能利用系数（β）,建立了考虑水分胁迫和光照条件对作物干物质积累与产量形成影响的数学模型,模型考虑了水分胁迫与低光照下冠层阻力增加的设定,将反映作物冠层水分状况的功能叶水势（Ψl）作为参数纳入本模型,通过对土壤相对含水量（Aw)、气温（Ta）、水汽压差（VPD）的多元回归估算出Ψl,并将空气动力学阻力（Ra）简化为风速（u）的函数,盆载试验应用实例和敏感性分析表明,该模型在诊断环境因子特别是土壤水分与光照因子对作物生长和产量构成的影响具有一定的实用性。     

Development of a stable low-dose aglycosylated antibody formulation to minimize protein loss during intravenous administration

The physical and chemical integrity of a biopharmaceutical must be maintained not only during long-term storage but also during administration. Specifically for the intravenous (i.v.) delivery of a protein drug, loss of stability can occur when the protein formulation is compounded with i.v. bag diluents, thus modifying the original composition of the drug product. Here we present the challenges associated with the delivery of a low-dose, highly potent monoclonal antibody (mAb) via the i.v. route. Through parallel in-use stability studies and conventional formulation development, a drug product was developed in which adsorptive losses and critical oxidative degradation pathways were effectively controlled. This development approach enabled the i.v. administration of clinical doses in the range of 0.1 to 0.5 mg total protein, while ensuring liquid drug product storage stability under refrigerated conditions.     

A multiscale mathematical model of avascular tumor growth to investigate the therapeutic benefit of anti-invasive agents

With the aim of inhibiting cancer growth and reducing the risk of metastasis, pharmaceutical companies in the early 1990s developed anti-metastatic agents called inhibitors of metalloproteinases (MMPi). Despite the promising results obtained in pre-clinical studies, results of Phase III trials have been somewhat disappointing for late stage cancer patients. With the aim of mathematically investigating this therapeutic failure, we developed a mechanistically based model which integrates cell cycle regulation and macroscopic tumor dynamics. By simulating the model, we evaluated the efficacy of MMPi therapy. Simulation results predict the lack of efficacy of MMPi in advanced cancer patients. The theoretical model may aid in evaluating the efficacy of anti-metastatic therapies, thus benefiting the design of prospective clinical trials.     

A theory of drug tolerance and dependence II: the mathematical model

The preceding paper presented a model of drug tolerance and dependence. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process. The oral detection and analysis of exogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model considered secondary stimuli, becoming primary in dependence and addiction or when the drug administration bypasses the natural-oral-route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking autonomous tolerance processes. Simulations with the mathematical model demonstrate the model's behaviour to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence. The present paper discusses the mathematical model in terms of its design. The model is a nonlinear, learning feedback system, fully satisfying control theoretical principles. It accepts any form of the stimulus-the drug intake-and describes how the physiological processes involved affect the distribution of the drug through the body and the stability of the regulation loop. The mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes.     

A mathematical model on the optimal timing of offspring desertion

We consider the offspring desertion as the optimal strategy for the deserter parent, analyzing a mathematical model for its expected reproductive success. It is shown that the optimality of the offspring desertion significantly depends on the offsprings' birth timing in the mating season, and on the other ecological parameters characterizing the innate nature of considered animals. Especially, the desertion is less likely to occur for the offsprings born in the later period of mating season. It is also implied that the offspring desertion after a partially biparental care would be observable only with a specific condition.     

A quasi-static three-dimensional, mathematical, three-body segment model of the canine knee

A mathematical, three-dimensional, anatomically accurate model of the canine knee was created to determine the forces in the knee ligaments and the knee joint reaction forces during the stance phase of a slow walk. This quasi-static model considered both the tibio-femoral and patello-femoral articulations. The geometric and morphometric data of the hind limb were obtained from cadaver data. Muscle forces acting on the femur and the hip joint reaction force were determined by numerical optimization. Ligaments were modeled as non-linear-springs. Ligament material properties were obtained from the literature pertaining to the human knee. The model consists of-28 non-linear algebraic equations describing equilibrium of the femur and the patella, and geometric constraints. This system of equations was solved by a non-linear least-squares method. Results are presented for a knee with an intact cranial cruciate ligament (CCL) and for a knee with a ruptured CCL. Forces predicted to occur in the CCL by analysis of the model were found to be very similar to reported results of CCL forces measured in vivo in goats.     

A generalized compartmental model to estimate the fibre mass in the ruminoreticulum: 2. Integrating digestion and passage

Models used to predict digestibility and fill of the dietary insoluble fibre (NDF) treat the ruminoreticular particulate mass as a single pool. The underlying assumption is that escape of particles follows first-order kinetics. In this paper, we proposed and evaluated a model of two ruminoreticular sequential NDF pools. The first pool is formed by buoyant particles (raft pool) and the second one by fluid dispersed particles (escapable pool) ventrally to the raft. The transference of particles between these two pools results from several processes that reduce particles buoyancy, assuming the gamma distribution. The exit of escapable pool particles from the ruminoreticulum is exponentially distributed. These concepts were evaluated by comparing ruminoreticular NDF masses as 43 and 27 means from cattle and sheep, respectively, to the same predicted variable using single- and two-pools models. Predictions of the single-pool model were based on lignin turnover and the turnover associated to the descending phase of the elimination of Yb-labelled forage particles in the faeces of sheep. Predictions of the two-pool model were obtained by estimating fractional passage rates associated to the ascending and descending phases of the same Yb excretion profiles in sheep faeces. All turnovers were scaled to the power 0.25 of body mass for interspecies comparisons. Predictions based on lignin turnover (single pool) and the two-pool model presented similar trends, accuracies and precisions. The single-pool approach based solely on the descending phase of the marker yielded biased estimates of the ruminoreticular NDF mass.