Capillary structures and O2 supply to tissue

An analysis with a digital diffusion model as applied to the skeletal muscle
  • W. A. Grunewald
  • W. Sowa
Chapter
Part of the Reviews of Physiology, Biochemistry and Pharmacology book series (volume 77)

Keywords

Capillary Network Local Blood Flow Capillary Blood Flow Capillary Structure Longitudinal Diffusion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a

Arterial inflow of a capillary

A(P)

O2 consumption dependent upon PO2

A

Maximal O2 consumption

ai

Weighting factor of basic distribution ϕi(P) for different MCUs and different structure and supply parameters

AVDO2

Arterio-venous O2 difference

AVDO2(i)

Arterio-venous O2 difference of the i-th capillary

cj

Weighting factor of structure parameters

cHb

Hemoglobin concentration in blood

d

Capillary distance

d0

Anatomical capillary distance

dm

Functional capillary distance of step m of the mobilization and demobilization of capillary reserve

D

Tissue diffusion coefficient

Dk

Krogh diffusion constant

F

Area under a frequency distribution

Fj(i)

Supply area of the i-th capillary in the j- th section

hx

Interstice length between two grid points in x and z direction

hy

Interstice length between two grid points in y direction

h

Interstice length between two grid points if hx=hy

i,j,k

Coordinates of a grid point

K

Tissue diffusion conductivity

l

Capillary length

Δ l

Length of a capillary section (=l/2 M)

δ l

Length by which the capillary ends are displaced in order to obtain a superimposed frequency distribution

L

Number of discretization steps for directions x and z

2 · M

Number of discretization steps for direction y

MCU

Microcirculatory unit

MA(i)

Location of arterial capillary end of the i-th capillary in a MCU

m

Number of steps in mobilization (and demobilization) of the capillary reserve

N

Total number of P values in a P matrix

N+

Number of random samples of a P matrix

n

Number of iterations

PO2

Oxygen partial pressure

\(\left. {\begin{array}{*{20}c}P \\{P(x,y,z),} \\{P(r)} \\{P(i,j,k)} \\\end{array} } \right\}\)

PO2 dependent upon location; PO2 field

Pj(i)(r)

PO2 of the i-th capillary on lateral surface area of the j-th section

Pa

Arterial PO2

¯ Pv

Mean venous PO2 (of a MCU)

Pv(i)

Capillary venous PO2 of i-th capillary

Pm

Maximum PO2 of the measured PO2 distribution

P(rc)

PO2 at the capillary wall

P(r)

PO2 at distance r from the capillary center in a tissue cylinder

Pn(i, j, k)

Approximated PO2 at the n- th step of the iteration process

P50

PO2 where A(P)=A/2

Pmin

Minimum (tissue) PO2 of a MCU

Pmin

Minimum (tissue) PO2 of different MCUs

Pc(i)

Intracapillary PO2 (i.c. PO2)

Δ P

P class value of a PO2 frequency distribution

R

Radius of the Krogh tissue cylinder

rc

Capillary radius

Rj(i)

i-th supply boundary on the lateral surface area of the j-th section

RSn−1

Difference between approximated PO2 of the (n−1)-th and n-th iteration step (RSn−1=Pn−1−P(n) at point (i, j, k))

r

Coordinate of the Krogh tissue cylinder

r

Location vector

s(P)

Relative HbO2 saturation of blood (HbO2 dissociation curve)

¯Sv

Mean venous HbO2 saturation

Sa

Arterial HbO2 saturation

Sj(i)

i.c. HbO2 saturation of the j- th section

Δ s

Arterio-venous HbO2 saturation difference

Δ sj(i)

i.c. HbO2 saturation difference along the capillary section Δ l of the i-th capillary of the j-th section (Δ s j (i) =s j (i) −s j−1 (i) )

ν

Venous outflow of a capillary

υ

Diffusion flow density of the O2 molecules

Vc(i)

Capillary supply volume of the i-th capillary

V

Volume of tissue fragment of a MCU

\(V_{c_0 } ,V_{c_m } \)

Capillary supply volume of step 0 and m of mobilization and demobilization of capillary reserve

Δ V

Volume of the 2 M sections (ΔV=V/2 M)

Δ Vj(i)

Capillary supply volume of the i-th capillary of the capillary section Δ l of the j-th section

W

(Local) blood flow of a MCU

Wci

Blood flow of supply volume of the i-th capillary

x, y, z

Cartesian coordinates

w=w(P, Δ P, h)

Frequency of all P values in a P matrix

w+=w+ (P, Δ P, h)

Frequency of N+P values in a P matrix

α

Bunsen solubility coefficient

δ

Diameter of the smallest sphere which includes all grid points

ε

Truncation error of the relaxation process

εc

Truncation error of the iterative calculation of i.c. PO2

κ

Number of MCUs with a displacement of the capillary ends of δ l

κ′

Number of nonidentical MCUs

κ″

Number of different (statistically distributed) parameter values

ν

Total number of frequency distributions of an analysis of a measured PO2 distribution

ϕ(P)

Relative PO2 frequency distribution

ϕh(P)

Frequency distribution of N P matrix elements (ideal PO2 frequency distribution)

ϕh+(P)

Frequency distribution of N+P matrix elements (statistical PO2 frequency distribution)

ϕi(P)

PO2 distribution of i-th MCU

φ

Capillary blood flow of a MCU

φ(i)c

Capillary blood flow of i-th capillary in a MCU

\(\phi _{c_0 } ,\phi _{c_m } \)

Capillary blood flow of step 0 and m of mobilization and demobilization of capillary reserve

ψ(P)

Superimposed PO2 frequency distribution

ψm(P)

Measured PO2 frequency distribution

ω

Overrelaxation factor

\(\nabla ^2 \equiv \frac{{\partial ^2 }}{{\partial x^2 }} + \frac{{\partial ^2 }}{{\partial y^2 }} + \frac{{\partial ^2 }}{{\partial z^2 }}\)

Laplace operator

grad

Gradient (operator)

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Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • W. A. Grunewald
  • W. Sowa

There are no affiliations available

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