Respiratory flow in a realistic tracheostenosis model

The possible mechanism of wheeze generation in tracheostenosis was identified by measuring inspiratory and expiratory flow in a "morphological and distensible" realistic tracheostenosis model. The shape of the model was based on CT (Computed Tomography) images of a patient that had tracheostenosis. A trachea consists of tracheal cartilage rings and smooth muscle. Spatial variation of wall distensibility was achieved in the model by varying the wall thickness based on the elastic modulus measured in pig airways. The spatial variation influenced the flow in the airway and the turbulence production rate decreased faster at smooth muscles. Using the model, we investigated the mechanism of wheeze generation by focusing on the turbulence intensity. The turbulence intensity in expiratory flow was about twice that in inspiratory flow, and larger vortices existed in post-stenosis in expiratory flow, and thus might contribute to wheeze generation.     

A mathematical model for the axial migration of suspended particles in tube flow

A plausible model for the decrease in particle concentration in a fluid suspension flowing through a tube has been developed. It involves only reasonable postulations of the concentration and viscosity profiles. With proper mathematical manipulations, it is developed into a simple expression relating the Reynolds and Kármán numbers and two empirical parameters to the concentration reduction. All correlations are reasonable. However, the two empirical parameters are inseparable in the final mathematical expression, thus making the plotting of the concentration and viscosity profile virtually impossible. Other models based on the same idea but evolved into a different form should be sought. Although calculations and correlations were attempted, missing information behind the data has made them at best only good indicators. Future experimental work aimed at obtaining all variables called for in this model is highly recommended.     

The threshold bias model: a mathematical model for the nomothetic approach of suicide

Background

Comparative and predictive analyses of suicide data from different countries are difficult to perform due to varying approaches and the lack of comparative parameters.

Methodology/Principal Findings

A simple model (the Threshold Bias Model) was tested for comparative and predictive analyses of suicide rates by age. The model comprises of a six parameter distribution that was applied to the USA suicide rates by age for the years 2001 and 2002. Posteriorly, linear extrapolations are performed of the parameter values previously obtained for these years in order to estimate the values corresponding to the year 2003. The calculated distributions agreed reasonably well with the aggregate data. The model was also used to determine the age above which suicide rates become statistically observable in USA, Brazil and Sri Lanka.

Conclusions/Significance

The Threshold Bias Model has considerable potential applications in demographic studies of suicide. Moreover, since the model can be used to predict the evolution of suicide rates based on information extracted from past data, it will be of great interest to suicidologists and other researchers in the field of mental health.     

A mathematical model for the dissolution of non-occlusive blood clots in fast tangential blood flow

Our aim was to study the effect of an axially directed blood plasma flow on the dissolution rate of cylindrical non-occlusive blood clots in an in vitro flow system and to derive a mathematical model for the process. The model was based on the hypothesis that clot dissolution dynamics is proportional not only to the biochemical proteolysis of fibrin but also to the power of the flowing blood plasma dissipated along the clot. The predicted rate of thrombolysis is then proportional to the square of the average blood plasma velocity for laminar flow and to the third power of the average velocity for turbulent flow. To verify the model, the time dependence of the clot cross-sectional area was measured by dynamic magnetic resonance microscopy during fast (turbulent) and slow (laminar) flow of plasma through an axially directed channel along the clot. The flowing plasma contained a magnetic resonance imaging contrast agent (Gd-DTPA) and a thrombolytic agent (recombinant tissue-type plasminogen activator). The experimental data fitted well to the model, and confirmed the predicted increase in the dissolution rate when blood flow changed from a laminar to a turbulent flow regime.     

Identifying non-sperm particles during flow cytometric physiological assessment: a simple approach

Flow cytometry is now being used more frequently to determine sperm functional characteristics during semen assessment for artificial insemination. With this methodology, viable and potentially functional cells are detected as unstained events differentiated from non-sperm events through their light-scattering characteristics. However, it can be shown mathematically that identification of sperm on the basis of light scatter leads to significant overestimation of unstained viable cells and underestimation of responding cells in tests of sperm function (subpopulations expressing different fluorescence patterns). We have developed a simple and cost-efficient flow cytometric approach for identifying non-sperm particles that can be carried out in parallel with functional assessments. Our method is based on the sperm's osmotic intolerance. Diluted in water, lethal osmotic shock causes major damage to the cell membranes, and all sperm will stain with propidium iodide (PI). Particulate material which is not PI-positive can then be quantitatively evaluated by FACS analysis and the results substituted in mathematical equations to provide true values for sperm counts and subpopulations. In practical tests, the percentage of non-sperm particles determined by this technique was closely comparable to the figure obtained either by SYBR14^®/PI staining or by PI/CFDA staining. As well as being valuable with respect to tests of sperm function, the procedure is also suitable for obtaining accurate sperm counts during routine semen evaluation.     

A mathematical description of blood spiral flow in vessels: application to a numerical study of flow in arterial bending

Local arterial haemodynamics has been associated with the pathophysiology of several cardiovascular diseases. The stable spiral blood-flows that were observed in vivo in several vessels, may play a dual role in vascular haemodynamics, beneficial since it induces stability, reducing turbulence in the arterial tree, and accounts for normal organ perfusion, but detrimental in view of the imparted tangential velocities that are involved in plaque formation and development. Being a spiral flow considered representative of the local blood dynamics in certain vessels, a method is proposed to quantify the spiral structure of blood flow. The proposed function, computed along a cluster of particle trajectories, has been tested for the quantitative determination of the spiral blood flow in a three-dimensional, s-shaped femoral artery numerical model in which three degrees of stenosis were simulated in a site prone to atherosclerotic development. Our results confirm the efficacy of the Lagrangian analysis as a tool for vascular blood dynamics investigation. The proposed method quantified spiral motion, and revealed the progression in the degree of stenosis, in the presented case study. In the future, it could be used as a synthetic tool to approach specific clinical complications.     

The linear cable theory as a model of gill blood flow

The vascular organization of the teleost gill suggests that blood flow distribution from the filamental artery to the respiratory lamellae is governed by relationships analogous to the cable conduction properties of a nerve axon. The space constant (λ) by definition is the distance along the gill filament at which the in-series resistance of the afferent filament artery equals the in-parallel resistance of the afferent lamellar arteriolar, lamellar, efferent lamellar arteriolar (ALA-L-ELA) segments. Constriction of the afferent filamental artery or uniform dilation of the ALA-L-ELA will decrease λ. As λ decreases, flow through the proximal (basal) lamellae greatly increases at the expense of distal lamellar perfusion. When λ increases in a system of finite length the flow profile must account for reflected pressures within the main vessel. The λ calculated from corrosion casts of gill vasculature is $\text{1}\text{4}$ to $\text{1}\text{2}$ the filament length. This favors blood flow through the proximal lamellae and when cardiac output increases, the proportion of cardiac output perfusing the proximal areas increases at the expense of distal lamellar blood flow. To offset these changes it is proposed that increased distal lamellar perfusion is achieved by simultaneous vasodilatation of distal and constriction of proximal ALA-L-ELA segments and dilation of the afferent filamental artery.     

The "protocell": a mathematical model of self-maintenance

The concepts of self-generation, autonomous boundary and self-maintenance are explained briefly. The "protocell" is presented as a model of self-maintenance which is based on simple physical mechanisms of diffusion and reaction. The time evolution of the surface of the protocell is taken into account explicitly in the form of a Stefan condition giving rise to a non-linear feedback of the surface motion to the reaction and diffusion processes inside the protocell. The spatio-temporal dynamics are investigated, particularly in the neighbourhood of the stationary states, showing a self-maintaining behaviour under a certain range of nutritional conditions. Under another set of conditions we find an instability leading to a division process so that the population of protocells becomes self-maintaining instead of the single individual. The presented formulation of the protocell model is crucially improved compared with a previous version which required boundary conditions at infinity. The previous version was not strictly self-maintaining since dynamics outside the cell were essential for its behaviour.     

A mathematical model of flow through a collapsible tube--I. Model and steady flow results

A one-dimensional model is presented to describe the flow through a collapsible tube whose ends are tethered to rigid tubes and which is enclosed in a pressurized chamber. Results are presented for the special case of steady flow. Predicted pressure drop versus flow rate (delta P-Q) characteristics agree qualitatively with available experimental data. The significance of the model and of various physical parameters, in regard to the shape of these characteristics, is discussed.     

Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress

A mathematical model of the arterial baroreflex was developed and used to assess the stability of the reflex and its potential role in producing the low-frequency arterial blood pressure oscillations called Mayer waves that are commonly seen in humans and animals in response to decreased central blood volume. The model consists of an arrangement of discrete-time filters derived from published physiological studies, which is reduced to a numerical expression for the baroreflex open-loop frequency response. Model stability was assessed for two states: normal and decreased central blood volume. The state of decreased central blood volume was simulated by decreasing baroreflex parasympathetic heart rate gain and by increasing baroreflex sympathetic vaso/venomotor gains as occurs with the unloading of cardiopulmonary baroreceptors. For the normal state, the feedback system was stable by the Nyquist criterion (gain margin = 0.6), but in the hypovolemic state, the gain margin was small (0.07), and the closed-loop frequency response exhibited a sharp peak (gain of 11) at 0.07 Hz, the same frequency as that observed for arterial pressure fluctuations in a group of healthy standing subjects. These findings support the theory that stresses affecting central blood volume, including upright posture, can reduce the stability of the normally stable arterial baroreflex feedback, leading to resonance and low-frequency blood pressure waves.     

Cancer screening: a mathematical model relating secreted blood biomarker levels to tumor sizes

Background

Increasing efforts and financial resources are being invested in early cancer detection research. Blood assays detecting tumor biomarkers promise noninvasive and financially reasonable screening for early cancer with high potential of positive impact on patients'' survival and quality of life. For novel tumor biomarkers, the actual tumor detection limits are usually unknown and there have been no studies exploring the tumor burden detection limits of blood tumor biomarkers using mathematical models. Therefore, the purpose of this study was to develop a mathematical model relating blood biomarker levels to tumor burden.

Methods and Findings

Using a linear one-compartment model, the steady state between tumor biomarker secretion into and removal out of the intravascular space was calculated. Two conditions were assumed: (1) the compartment (plasma) is well-mixed and kinetically homogenous; (2) the tumor biomarker consists of a protein that is secreted by tumor cells into the extracellular fluid compartment, and a certain percentage of the secreted protein enters the intravascular space at a continuous rate. The model was applied to two pathophysiologic conditions: tumor biomarker is secreted (1) exclusively by the tumor cells or (2) by both tumor cells and healthy normal cells. To test the model, a sensitivity analysis was performed assuming variable conditions of the model parameters. The model parameters were primed on the basis of literature data for two established and well-studied tumor biomarkers (CA125 and prostate-specific antigen [PSA]). Assuming biomarker secretion by tumor cells only and 10% of the secreted tumor biomarker reaching the plasma, the calculated minimally detectable tumor sizes ranged between 0.11 mm³ and 3,610.14 mm³ for CA125 and between 0.21 mm³ and 131.51 mm³ for PSA. When biomarker secretion by healthy cells and tumor cells was assumed, the calculated tumor sizes leading to positive test results ranged between 116.7 mm³ and 1.52 × 10⁶ mm³ for CA125 and between 27 mm³ and 3.45 × 10⁵ mm³ for PSA. One of the limitations of the study is the absence of quantitative data available in the literature on the secreted tumor biomarker amount per cancer cell in intact whole body animal tumor models or in cancer patients. Additionally, the fraction of secreted tumor biomarkers actually reaching the plasma is unknown. Therefore, we used data from published cell culture experiments to estimate tumor cell biomarker secretion rates and assumed a wide range of secretion rates to account for their potential changes due to field effects of the tumor environment.

Conclusions

This study introduced a linear one-compartment mathematical model that allows estimation of minimal detectable tumor sizes based on blood tumor biomarker assays. Assuming physiological data on CA125 and PSA from the literature, the model predicted detection limits of tumors that were in qualitative agreement with the actual clinical performance of both biomarkers. The model may be helpful in future estimation of minimal detectable tumor sizes for novel proteomic biomarker assays if sufficient physiologic data for the biomarker are available. The model may address the potential and limitations of tumor biomarkers, help prioritize biomarkers, and guide investments into early cancer detection research efforts.     

Genomic mapping by single copy landmark detection: a predictive model with a discrete mathematical approach

One of the goals of the Human Genome Project is to produce libraries of largely contiguous, ordered sets of molecular clones for use in sequencing and gene mapping projects. This is planned to be done for human and many model organisms. Theory and practice have shown that long-range contiguity and the degree to which the entire genome is covered by ordered clones can be affected by many biological variables. Many laboratories are currently experimenting with different experimental strategies and theoretical models to help plan strategies for accomplishing longrange molecular mapping of genomes. Here we describe a new mathematical model and formulas for helping to plan genome mapping projects, using various single-copy landmark (SCL) detection, or anchoring, strategies. We derive formulas that allow us to examine the effects of interactions among the following variables: average insert size of the cloning vector, average size of SCL, the number of SCL, and the redundancy in coverage of the clone library. We also examine and compare three different ways in which anchoring can be implemented: (1) anchors are selected independently of the library to be ordered (random anchoring); (2) anchors are made from end probes from both ends of clones in the library to be ordered (nonrandom anchoring); and (3) anchors are made from one end or the other, randomly, from clones in the library to be ordered (nonrandom anchoring). Our results show that, for biologically realistic conditions, nonrandom anchoring is always more effective than random anchoring for contig building, and there is little to be gained from making SCL from both ends of clones vs. only one end of clones. We compare and contrast our results with other similar mathematical models.     

A mathematical model of the flow in the circle of Willis

A mathematical model of the flow in the circle of Willis has been designed and the effects of (a) the large anatomical variation of the communicating arteries and (b) physiological changes of the resistances of the vertebral arteries have been studied. The influence of the posterior perforating arteries on the flow in the posterior communicating arteries has been investigated as well, with special attention being paid to the possible occurrence of a 'dead point'. In the model, the influence of diameters of the communicating arteries on the flow in the afferent vessels and the segments of the circle turns out to be considerable, especially in the range of the anatomical variation of the diameters. Within this range flow reductions due to an increased resistance of the vertebral artery will be compensated for by the system. Assuming that the values and ratios of the peripheral resistances are within the physiological range, a dead point is not to be expected in the flow in the posterior communicating arteries.