The role of theoretical modeling in microcirculation research

Oxygen transport across tank-treading red blood cell: Individual and joint roles of flow convection and oxygen-hemoglobin reaction

Gas, especially oxygen, transport in the microcirculation is a complex phenomenon, however, of critical importance for maintaining normal biological functions, and the cytoplasm fluid in red blood cells (RBCs) is the major vehicle for transporting oxygen from lungs to tissues via the circulatory system. Existing theoretical and numerical studies have neglected the cytoplasm convection effect by treating RBCs as rigid particles undergoing a constant translation velocity. As a consequence, the influence and mechanism of the cytoplasm flow on oxygen transport are still not clear in microcirculation research. In this study, we consider a tank-treading capsule in shear flow, which is generated with two parallel plates moving in opposite directions: the top plate of a higher oxygen pressure (P_O2) representing the RBC core in the central region of a microvessel and the bottom plate of a lower P_O2 representing the microvessel wall. Numerical simulations are conducted to investigate the individual and combined effects of cytoplasm convection and oxygen-hemoglobin (O₂-Hb) reaction on the oxygen transport efficiency across the tank-treading capsule, and different P_O2 situations and shear rates are also tested. Due to the lower oxygen diffusivity in cytoplasm, the presence of the capsule reduces the oxygen transfer flux across the gap by 7.34 % in the pure diffusion system where the flow convection and O₂-Hb reaction are both neglected. Including the flow convection or the O₂-Hb reaction has little influence on the oxygen flux; however, when they act together as in real microcirculation situations, the enhancement in oxygen transport could be significant, especially in the low P_O2 and high shear rate situations. In particular, with the respective P_O2 at 60 and 30 mmHg on the top and bottom plates and a 400 s^-1 shear rate, the oxygen flux reduction is only 0.02 %, suggesting that the cytoplasm convection can improve the oxygen transport across RBCs considerably. The simulation results are scrutinized to explore the underlying mechanism for the enhancement, and a new nondimensional parameter is introduced to characterize the importance of cytoplasm convection in oxygen transport. These simulation results, discussion and analysis could be helpful for a better understanding of the complex oxygen transport process and therefor valuable for relevant studies.

Microvascular Tissue Engineering-A Review

Tissue engineering and regenerative medicine have come a long way in recent decades, but the lack of functioning vasculature is still a major obstacle preventing the development of thicker, physiologically relevant tissue constructs. A large part of this obstacle lies in the development of the vessels on a microscale-the microvasculature-that are crucial for oxygen and nutrient delivery. In this review, we present the state of the art in the field of microvascular tissue engineering and demonstrate the challenges for future research in various sections of the field. Finally, we illustrate the potential strategies for addressing some of those challenges.

Challenges for microcirculation research in developing countries

BACKGROUND

Microvascular dysfunction is a main contributor to morbidity and mortality worldwide. Sophisticated technical tools (e.g. miniaturized hardware, automated software), along with skilled personnel are the prerequisite for quantitative observations of the microvasculature.

OBJECTIVE

This review aimed to get an overview about on-going microcirculatory research in developing countries, particularly of the South-East Asia region for the last five years and to project the challenges faced in microcirculation research in developing countries.

METHODS

Original research articles originating from 194 countries were searched in PubMed database on the field of microcirculation research for the last five years.

RESULTS

Our findings showed that around 1800 articles have been published from developing countries compared to more than 5000 from developed countries on different aspects of microcirculation. The overall publication per million populations for developing countries was found to be 0.37 where for developed countries it was 3.62.

CONCLUSIONS

Initiation and execution of sophisticated research in microcirculation is a demand of the time. Such research, initially, may seem unmanageable in developing countries with limited resources and infrastructure settings. Collaborative scientific projects may aid in establishing networks for microvascular research in developing countries.

Reduced-order modeling of hemodynamics across macroscopic through mesoscopic circulation scales

We propose a hemodynamic reduced-order model bridging macroscopic and mesoscopic blood flow circulation scales from arteries to capillaries. In silico tree-like vascular geometries, mathematically described by graphs, are synthetically generated by means of stochastic growth algorithms constrained by statistical morphological and topological principles. Scale-specific pruning gradation of the tree is then proposed in order to fit computational budget requirement. Different compliant structural models with respect to pressure loads are used depending on vessel walls thicknesses and structures, which vary considerably from macroscopic to mesoscopic circulation scales. Nonlinear rheological properties of blood are also included, and microcirculation network responses are computed for different rheologies. Numerical results are in very good agreement with available experimental measurements. The computational model captures the dynamic transition between large- to small-scale flow pulsatility speeds and magnitudes and wall shear stresses, which have wide-ranging physiological influences.

A computational model for microcirculation including Fahraeus-Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium

We present a two-phase model for microcirculation that describes the interaction of plasma with red blood cells. The model takes into account of typical effects characterizing the microcirculation, such as the Fahraeus-Lindqvist effect and plasma skimming. Besides these features, the model describes the interaction of capillaries with the surrounding tissue. More precisely, the model accounts for the interaction of capillary transmural flow with the surrounding interstitial pressure. Furthermore, the capillaries are represented as one-dimensional channels with arbitrary, possibly curved configuration. The latter two features rely on the unique ability of the model to account for variations of flow rate and pressure along the axis of the capillary, according to a local differential formulation of mass and momentum conservation. Indeed, the model stands on a solid mathematical foundation, which is also addressed in this work. In particular, we present the model derivation, the variational formulation, and its approximation using the finite element method. Finally, we conclude the work with a comparative computational study of the importance of the Fahraeus-Lindqvist, plasma skimming, and capillary leakage effects on the distribution of flow in a microvascular network.

Modeling the hematocrit distribution in microcirculatory networks: A quantitative evaluation of a phase separation model

OBJECTIVE

Theoretical models are essential tools for studying microcirculatory function. Recently, the validity of a well-established phase separation model was questioned and it was claimed that it produces problematically low hematocrit predictions and lack of red cells in small diameter vessels. We conducted a quantitative evaluation of this phase separation model to establish common ground for future research.

METHODS

Model predictions were validated against a comprehensive database with measurements from 4 mesenteric networks. A Bayesian data analysis framework was used to integrate measurements and network model simulations into a combined analysis and to model uncertainties related to network boundary conditions as well as phase separation model parameters. The model evaluation was conducted within a cross-validation scheme.

RESULTS

Unlike the recently reported results, our analysis demonstrated good correspondence in global characteristics between measurements and predictions. In particular, predicted hematocrits for vessels with small diameters were consistent with measurements. Incorporating phase separation model parameter uncertainties further reduced the hematocrit validation error by 17% and led to the absence of red-cell-free segments. Corresponding model parameters are presented as alternatives to standard parameters.

CONCLUSIONS

Consistent with earlier studies, our quantitative model evaluation supports the continued use of the established phase separation model.

Model-based inference from microvascular measurements: Combining experimental measurements and model predictions using a Bayesian probabilistic approach

OBJECTIVE

In vivo imaging of the microcirculation and network-oriented modeling have emerged as powerful means of studying microvascular function and understanding its physiological significance. Network-oriented modeling may provide the means of summarizing vast amounts of data produced by high-throughput imaging techniques in terms of key, physiological indices. To estimate such indices with sufficient certainty, however, network-oriented analysis must be robust to the inevitable presence of uncertainty due to measurement errors as well as model errors.

METHODS

We propose the Bayesian probabilistic data analysis framework as a means of integrating experimental measurements and network model simulations into a combined and statistically coherent analysis. The framework naturally handles noisy measurements and provides posterior distributions of model parameters as well as physiological indices associated with uncertainty.

RESULTS

We applied the analysis framework to experimental data from three rat mesentery networks and one mouse brain cortex network. We inferred distributions for more than 500 unknown pressure and hematocrit boundary conditions. Model predictions were consistent with previous analyses, and remained robust when measurements were omitted from model calibration.

CONCLUSION

Our Bayesian probabilistic approach may be suitable for optimizing data acquisition and for analyzing and reporting large data sets acquired as part of microvascular imaging studies.

Physiological models of the lateral superior olive

In computational biology, modeling is a fundamental tool for formulating, analyzing and predicting complex phenomena. Most neuron models, however, are designed to reproduce certain small sets of empirical data. Hence their outcome is usually not compatible or comparable with other models or datasets, making it unclear how widely applicable such models are. In this study, we investigate these aspects of modeling, namely credibility and generalizability, with a specific focus on auditory neurons involved in the localization of sound sources. The primary cues for binaural sound localization are comprised of interaural time and level differences (ITD/ILD), which are the timing and intensity differences of the sound waves arriving at the two ears. The lateral superior olive (LSO) in the auditory brainstem is one of the locations where such acoustic information is first computed. An LSO neuron receives temporally structured excitatory and inhibitory synaptic inputs that are driven by ipsi- and contralateral sound stimuli, respectively, and changes its spike rate according to binaural acoustic differences. Here we examine seven contemporary models of LSO neurons with different levels of biophysical complexity, from predominantly functional ones (‘shot-noise’ models) to those with more detailed physiological components (variations of integrate-and-fire and Hodgkin-Huxley-type). These models, calibrated to reproduce known monaural and binaural characteristics of LSO, generate largely similar results to each other in simulating ITD and ILD coding. Our comparisons of physiological detail, computational efficiency, predictive performances, and further expandability of the models demonstrate (1) that the simplistic, functional LSO models are suitable for applications where low computational costs and mathematical transparency are needed, (2) that more complex models with detailed membrane potential dynamics are necessary for simulation studies where sub-neuronal nonlinear processes play important roles, and (3) that, for general purposes, intermediate models might be a reasonable compromise between simplicity and biological plausibility.

Computational models help our understanding of complex biological systems, by identifying their key elements and revealing their operational principles. Close comparisons between model predictions and empirical observations ensure our confidence in a model as a building block for further applications. Most current neuronal models, however, are constructed to replicate only a small specific set of experimental data. Thus, it is usually unclear how these models can be generalized to different datasets and how they compare with each other. In this paper, seven neuronal models are examined that are designed to reproduce known physiological characteristics of auditory neurons involved in the detection of sound source location. Despite their different levels of complexity, the models generate largely similar results when their parameters are tuned with common criteria. Comparisons show that simple models are computationally more efficient and theoretically transparent, and therefore suitable for rigorous mathematical analyses and engineering applications including real-time simulations. In contrast, complex models are necessary for investigating the relationship between underlying biophysical processes and sub- and suprathreshold spiking properties, although they have a large number of unconstrained, unverified parameters. Having identified their advantages and drawbacks, these auditory neuron models may readily be used for future studies and applications.

Knife-edge scanning microscopy for in silico study of cerebral blood flow: From biological imaging data to flow simulations

Knife-edge scanning microscopy provides the capability to image whole-brain cerebral microvasculature of small organisms, such as mice, at sub-micron resolution, providing a feasible foundation for the reconstruction of circulatory pathways from the systemic to cellular scale. In this paper, we illustrate the feasibility of using this data to model cerebral blood flow using numerical simulations. Starting with a small vascular element in microcirculation of interest, we present its segmentation from the imaging-data volume, construction of its triangular surface mesh, assembly of its tetrahedral volumetric mesh from the surface, and then conclude with Stokes flow simulation of plasma through the microvascular vessel.

Making microvascular networks work: angiogenesis, remodeling, and pruning

The adequate and efficient functioning of the microcirculation requires not only numerous vessels providing a large surface area for transport but also a structure that provides short diffusion distances from capillaries to tissue and efficient distribution of convective blood flow. Theoretical models show how a combination of angiogenesis, remodeling, and pruning in response to hemodynamic and metabolic stimuli, termed "angioadaptation," generates well organized, functional networks.

A one-dimensional mathematical model for studying the pulsatile flow in microvascular networks

Techniques that model microvascular hemodynamics have been developed for decades. While the physiological significance of pressure pulsatility is acknowledged, most of the microcirculatory models use steady flow approaches. To theoretically study the extent and transmission of pulsatility in microcirculation, dynamic models need to be developed. In this paper, we present a one-dimensional model to describe the dynamic behavior of microvascular blood flow. The model is applied to a microvascular network from a rat mesentery. Intravital microscopy was used to record the morphology and flow velocities in individual vessel segments, and boundaries are defined according to the experimental data. The system of governing equations constituting the model is solved numerically using the discontinuous Galerkin method. An implicit integration scheme is adopted to increase computing efficiency. The model allows the simulation of the dynamic properties of blood flow in microcirculatory networks, including the pressure pulsatility (quantified by a pulsatility index) and pulse wave velocity (PWV). From the main input arteriole to the main output venule, the pulsatility index decreases by 66.7%. PWV obtained along arterioles declines with decreasing diameters, with mean values of 77.16, 25.31, and 8.30 cm/s for diameters of 26.84, 17.46, and 13.33 μm, respectively. These results suggest that the 1D model developed is able to simulate the characteristics of pressure pulsatility and wave propagation in complex microvascular networks.

Capillary recruitment in a theoretical model for blood flow regulation in heterogeneous microvessel networks

In striated muscle, the number of capillaries containing moving red blood cells increases with increasing metabolic demand. This phenomenon, termed capillary recruitment, has long been recognized, but its mechanism has been unclear. Here, a theoretical model for metabolic blood flow regulation in a heterogeneous network is used to test the hypothesis that capillary recruitment occurs as a result of active control of arteriolar diameters, combined with unequal partition of hematocrit at diverging microvascular bifurcations. The network structure is derived from published observations of hamster cremaster muscle in control and dilated states. The model for modulation of arteriolar diameters includes length-tension characteristics of vascular smooth muscle and responses of smooth muscle tone to myogenic, shear-dependent, and metabolic stimuli. Blood flow is simulated including nonuniform hematocrit distribution. Convective and diffusive oxygen transport in the network is simulated. Oxygen-dependent metabolic signals are assumed to be conducted upstream from distal vessels to arterioles. With increasing oxygen demand, arterioles dilate, blood flow increases, and the numbers of flowing arterioles and capillaries, as defined by red blood cell flux above a small threshold value, increase. Unequal hematocrit partition at diverging bifurcations contributes to recruitment and enhances tissue oxygenation. The results imply that capillary recruitment, as observed in the hamster cremaster preparations, can occur as a consequence of local control of arteriolar tone and the resulting nonuniform changes in red blood cell fluxes, and provide an explanation for observations of sequential recruitment of individual capillaries in response to modulation of terminal arteriolar diameter.

Oxygen transport in the microcirculation and its regulation

OBJECTIVE

Cells require energy to carry out their functions and they typically use oxidative phosphorylation to generate the needed ATP. Thus, cells have a continuous need for oxygen, which they receive by diffusion from the blood through the interstitial fluid. The circulatory system pumps oxygen-rich blood through a network of increasingly minute vessels, the microcirculation. The structure of the microcirculation is such that all cells have at least one nearby capillary for diffusive exchange of oxygen and red blood cells release the oxygen bound to hemoglobin as they traverse capillaries.

METHODS

This review focuses first on the historical development of techniques to measure oxygen at various sites in the microcirculation, including the blood, interstitium, and cells.

RESULTS

Next, approaches are described as to how these techniques have been employed to make discoveries about different aspects of oxygen transport. Finally, ways in which oxygen might participate in the regulation of blood flow toward matching oxygen supply to oxygen demand is discussed.

CONCLUSIONS

Overall, the transport of oxygen to the cells of the body is one of the most critical functions of the cardiovascular system and it is in the microcirculation where the final local determinants of oxygen supply, oxygen demand, and their regulation are decided.

Study of tumor growth under hyperthermia condition

The new concept of keeping primary tumor under control in situ to suppress distant foci sheds light on the treatment of metastatic tumor. Hyperthermia is considered as one of the means for controlling tumor growth. To simulate the tumor growth, a continuum mathematical model has been introduced. The newest understanding of the Warburg effect on the cellular metabolism and diffusion of the nutrients in the tissue has been taken into consideration. The numerical results are compared with the in vivo experimental data by fitting the tumor cell doubling time/tumor cell growth rate under different thermal conditions. Both the tumor growth curve and corresponding average glucose concentration have been predicted. The numerical results have quantitatively illustrated the controlling effect on tumor growth under hyperthermia condition in the initial stage.

Modeling of angioadaptation: insights for vascular development

Vascular beds are generated by vasculogenesis and sprouting angiogenesis, and these processes have strong stochastic components. As a result, vascular patterns exhibit significant heterogeneity with respect to the topological arrangement of the individual vessel segments and the characteristics (length, number of segments) of different arterio-venous pathways. This structural heterogeneity tends to cause heterogeneous distributions of flow and oxygen availability in tissue. However, these quantities must be maintained within tolerable ranges to allow normal tissue function. This is achieved largely through adjustment of vascular flow resistance by control of vessel diameters. While short-term diameter control by changes in vascular tone in arterioles and small arteries plays an important role, in the long term an even more important role is played by structural adaptation (angioadaptation), occurring in response to metabolic and hemodynamic signals. The effectiveness, stability and robustness of this angioadaptation depend sensitively on the nature and strength of the vascular responses involved and their interactions with the network structure. Mathematical models are helpful in understanding these complex interactions, and can be used to simulate the consequences of failures in sensing or signal transmission mechanisms. For the tumor microcirculation, this strategy of combining experimental observations with theoretical models, has led to the hypothesis that dysfunctional information transport via vascular connexins is a major cause of the observed vascular pathology and increased heterogeneity in oxygen distribution.

The microcirculation: physiology at the mesoscale

The microcirculation exemplifies the mesoscale in physiological systems, bridging larger and smaller scale phenomena. Microcirculatory research represents an example of a 'middle-out,' rather than 'top-down' or 'bottom-up,' approach to the study of biological function. Computational and mathematical approaches can be used to analyse the functioning of the microcirculation and to establish quantitative relationships between microvascular processes and phenomena occurring on larger and smaller scales, leading to insights which could not be obtained solely by reductionist biological experiments. Given its integrative approach to processes occurring on disparate scales and its emphasis on theoretical as well as experimental approaches, microcirculatory research belongs within current definitions of systems biology.

Cerebral blood flow modeling in primate cortex

We report new results on blood flow modeling over large volumes of cortical gray matter of primate brain. We propose a network method for computing the blood flow, which handles realistic boundary conditions, complex vessel shapes, and complex nonlinear blood rheology. From a detailed comparison of the available models for the blood flow rheology and the phase separation effect, we are able to derive important new results on the impact of network structure on blood pressure, hematocrit, and flow distributions. Our findings show that the network geometry (vessel shapes and diameters), the boundary conditions associated with the arterial inputs and venous outputs, and the effective viscosity of the blood are essential components in the flow distribution. In contrast, we show that the phase separation effect has a minor function in the global microvascular hemodynamic behavior. The behavior of the pressure, hematocrit, and blood flow distributions within the network are described through the depth of the primate cerebral cortex and are discussed.

Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell

A two-dimensional computer simulation model was proposed for tank-treading and tumbling motions of an elastic biconcave red blood cell (RBC) under steady shear flow. The RBC model consisted of an outer cellular membrane and an inner fluid; the membrane's elastic properties were modeled by springs for stretch/compression and bending to consider the membrane's natural state in a practical manner. Membrane deformation was coupled with incompressible viscous flow of the inner and outer fluids of the RBC using a particle method. The proposed simulation model was capable of reproducing tank-treading and tumbling motions of an RBC along with rotational oscillation, which is the transition between the two motions. In simulations using the same initial RBC shape with different natural states of the RBC membrane, only tank-treading motion was exhibited in the case of a uniform natural state of the membrane, and a nonuniform natural state was necessary to generate the rotational oscillation and tumbling motion. Simulation results corresponded to published data from experimental and computational studies. In the range of simulation parameters considered, the relative membrane elastic force versus fluid viscous force was approximately 1 at the transition when the natural state nonuniformity was taken into account in estimating the membrane elastic force. A combination of natural state nonuniformity and elastic spring constant determined that change in the RBC deformation at the transition is that from a large compressive deformation to no deformation, such as rigid body.

Microcirculation and the physiome projects

The Physiome projects comprise a loosely knit worldwide effort to define the Physiome through databases and theoretical models, with the goal of better understanding the integrative functions of cells, organs, and organisms. The projects involve developing and archiving models, providing centralized databases, and linking experimental information and models from many laboratories into self-consistent frameworks. Increasingly accurate and complete models that embody quantitative biological hypotheses, adhere to high standards, and are publicly available and reproducible, together with refined and curated data, will enable biological scientists to advance integrative, analytical, and predictive approaches to the study of medicine and physiology. This review discusses the rationale and history of the Physiome projects, the role of theoretical models in the development of the Physiome, and the current status of efforts in this area addressing the microcirculation.