The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue

Krogh, August

doi:10.1113/jphysiol.1919.sp001839

Cited by 1,285 publications

(665 citation statements)

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“…However, average indices cannot capture the spatial heterogeneity in capillary supply generated by the local metabolic environment and the distribution of fibre size (Ahmed et al, 1997), and the scale-dependency inherent in such indices may explain why data on capillarity is variable, if not conflicting (Egginton, 1990). Second, area-based methods, such as those based on Krogh cylinders (Krogh, 1919) or Voronoi polygons (equivalently known as capillary domains; Hoofd et al, 1985), are used to avoid the spatial limitations of models based on global indices. A capillary domain is defined to be the area of a muscle cross section surrounding an individual capillary and closer to its centre than all neighbouring capillaries.…”

Section: Introductionmentioning

confidence: 99%

Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited

Alshammari

Gaffney

Egginton

2014

Journal of Theoretical Biology

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Developing effective therapeutic interventions for pathological conditions associated with abnormal oxygen transport to muscle fibres critically depends on the objective characterisation of capillarity. Local indices of capillary supply have the potential to identify the onset of fine-scale tissue pathologies and dysregulation. Detailed tissue geometry, such as muscle fibre size, has been incorporated into such measures by considering the distribution of Voronoi polygons (VP) generated from planar capillary locations as a representation of capillary supply regions. Previously, detailed simulations have predicted that this is generally accurate for muscle tissue with uniform oxygen uptake. Here we extend this modelling framework to heterogeneous muscle for the assessment of capillary supply capacity under maximal sustainable oxygen consumption. We demonstrate for muscle with heterogeneous fibre properties that VP theoretically provide a computationally simple but often accurate representation of trapping regions (TR), which are predicted from biophysical transport models to represent the areas of tissue supplied by individual capillaries. However, this use of VP may become less accurate around large fibres, and at the interface of fibres of largely different oxidative capacities. In such cases, TR may provide a more robust representation of capillary supply regions. Additionally, given VP can only approximate oxygen delivery by capillaries, we show that their generally close relationship to TR suggests that (1) fibre type distribution may be tightly regulated to avoid large fibres with high oxidative capacities, (2) the anatomical fibre distribution is also tightly regulated to prevent large surface area of interaction between metabolically dissimilar fibres, and (3) in chronically hypoxic tissues capillary distribution is more important in determining oxygen supply than the spatial heterogeneity of fibre demand.

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Section: Introductionmentioning

confidence: 99%

Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited

Alshammari

Gaffney

Egginton

2014

Journal of Theoretical Biology

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“…The classical Krogh tissue cylinder model, 50 a single cylindrical tissue unit supplied by a single capillary, has been the basis for most of the theoretical studies on microcirculatory oxygen transport to tissue over the past 80 years. Reviews by Hellums et al 37 and Popel 54 and recent studies oriented toward PET 15 O-oxygen image analysis by Beyer et al, 20 Deussen and Bassingthwaighte, 27 and Li et al 51 illustrate the applicability of the geometry that Krogh modeled.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Blood–Tissue Exchange of Oxygen, Carbon Dioxide, Bicarbonate, and Hydrogen Ion

Dash

Bassingthwaighte

2006

Ann Biomed Eng

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A detailed nonlinear four-region (red blood cell, plasma, interstitial fluid, and parenchymal cell) axially distributed convection-diffusion-permeation-reaction-binding computational model is developed to study the simultaneous transport and exchange of oxygen (O 2 ) and carbon dioxide (CO 2 ) in the blood-tissue exchange system of the heart. Since the pH variation in blood and tissue influences the transport and exchange of O 2 and CO 2 (Bohr and Haldane effects), and since most CO 2 is transported as (bicarbonate) via the CO 2 hydration (buffering) reaction, the transport and exchange of and H + are also simulated along with that of O 2 and CO 2 . Furthermore, the model accounts for the competitive nonlinear binding of O 2 and CO 2 with the hemoglobin inside the red blood cells (nonlinear O 2 -CO 2 interactions, Bohr and Haldane effects), and myoglobin-facilitated transport of O 2 inside the parenchymal cells. The consumption of O 2 through cytochrome-c oxidase reaction inside the parenchymal cells is based on MichaelisMenten kinetics. The corresponding production of CO 2 is determined by respiratory quotient (RQ), depending on the relative consumption of carbohydrate, protein, and fat. The model gives a physiologically realistic description of O 2 transport and metabolism in the microcirculation of the heart. Furthermore, because model solutions for tracer transients and steady states can be computed highly efficiently, this model may be the preferred vehicle for routine data analysis where repetitive solutions and parameter optimization are required, as is the case in PET imaging for estimating myocardial O 2 consumption.

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“…On the microscale, several models based on the Krogh Cylinder model (Krogh, 1919) have been proposed (Baish et al, 1996;Maseide and Rofstad, 2000) whilst Breward et al (2001) followed the approach of Ward and King (1997). These papers focus on the interactions between a blood vessel supplying oxygen to a region of tissue within a tumour; the latter paper incorporating the effect of blood vessel compliance on the subsequent remodelling of the tumour tissue.…”

Section: Introductionmentioning

confidence: 99%

A Multiphase Model Describing Vascular Tumour Growth

Breward¹,

Byrne

Lewis

2003

Bulletin of Mathematical Biology

145

126

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In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modifed to incorporate the effect of other phases, or to include the effect of drugs.

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The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue

Cited by 1,285 publications

References 0 publications

Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited

Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited

Simultaneous Blood–Tissue Exchange of Oxygen, Carbon Dioxide, Bicarbonate, and Hydrogen Ion

A Multiphase Model Describing Vascular Tumour Growth

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