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Capillary structures and O2 supply to tissue

An analysis with a digital diffusion model as applied to the skeletal muscle

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Abbreviations

a:

Arterial inflow of a capillary

A(P):

O2 consumption dependent upon PO 2

A:

Maximal O2 consumption

ai :

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

AVDO2 :

Arterio-venous O2 difference

AVDO (i)2 :

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

F (i)j :

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 h x =h y

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 PO 2

¯ Pv :

Mean venous PO 2 (of a MCU)

Pv (i) :

Capillary venous PO 2 of i-th capillary

Pm :

Maximum PO 2 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 PO 2 at the n- th step of the iteration process

P50 :

PO2 where A(P)=A/2

Pmin :

Minimum (tissue) PO 2 of a MCU

Pmin :

Minimum (tissue) PO 2 of different MCUs

Pc (i) :

Intracapillary PO 2 (i.c. PO 2)

Δ P:

P class value of a PO2 frequency distribution

R:

Radius of the Krogh tissue cylinder

rc :

Capillary radius

R (i)j :

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

RSn−1 :

Difference between approximated PO 2 of the (n−1)-th and n-th iteration step (RS n−1=P n−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

S (i)j :

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

Δ s:

Arterio-venous HbO2 saturation difference

Δ s (i)j :

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

ν:

Venous outflow of a capillary

υ:

Diffusion flow density of the O 2 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 V (i)j :

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

W ic :

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. PO 2

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 PO 2 distribution

ϕ(P):

Relative PO 2 frequency distribution

ϕh(P):

Frequency distribution of N P matrix elements (ideal PO 2 frequency distribution)

ϕ +h (P):

Frequency distribution of N + P matrix elements (statistical PO 2 frequency distribution)

ϕi(P):

PO2 distribution of i-th MCU

φ:

Capillary blood flow of a MCU

φ c(i) :

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 PO 2 frequency distribution

ψm(P):

Measured PO 2 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)

References

  • Acker, H.: Der lokale Sauerstoffdruck im Glomus caroticum der Katze und seine Bedeutung für die nervöse chemorezeptive Aktivität. Habilitationsschrift, Bochum, 1975

    Google Scholar 

  • Acker, H., Lübbers, D.W., Purves, M.: O2-Transfer in der Bindegewebskapsel des Glomus caroticum der Katze und seine funktioneile Bedeutung. Pflügers Arch. ges. Physiol. 316, R 30 (1970)

    Google Scholar 

  • Acker, H., Lübbers, D.W., Purves, M.: Local oxygen tension field in the Glomus caroticum of the cat and its change at changing arterial PO2. Pflügers Arch. ges. Physiol. 329, 136–155 (1971)

    CAS  Google Scholar 

  • Bänder, A., Kiese, M.: Die Wirkung des sauerstoffübertragenden Fermentes in Mitochondrien aus Rattenlebern bei niedrigen O2-Drucken. Nauyn-Schmiedeberg's Arch. exp. Path. Pharmak. 224, 312–321 (1955)

    Google Scholar 

  • Bailey, H.R.: Oxygen exchange between capillary and tissue: some equations describing countercurrent and non-linear transport. In: Physical bases of circulatory transport: Regulation and Exchange. Philadelphia and London: Saunders, 1967

    Google Scholar 

  • Barcroft, J., Bock, A. V., Hill, A. V., Parsons, T. R., Parsons, W., Shoji, R.: Of the hydrogenion concentration and some related properties of normal human blood. J. Physiol. (Lond.) 56, 157–178 (1922)

    CAS  Google Scholar 

  • Bartels, H., Harms, H.: Sauerstoffdissoziationskurven des Blutes von Säugetieren. Pflügers Arch. ges. Physiol. 268, 334–365 (1959)

    Article  CAS  Google Scholar 

  • Batschelet, E.: Über die numerische Auflösung von Randwert-Problemen bei elliptischen partiellen Differentialgleichungen. Z. angew. Math. Phys. 3, 165 (1952)

    Google Scholar 

  • Baumgärtl, H., Grunewald, W.A., Lübbers, D.W.: Polarographic determination of the oxygen pressure field by Pt-electrodes using the PO2-field in front of a Pt-macroelectrode as a model. Pflügers Arch. ges. Physiol. 347, 49–61 (1974)

    Google Scholar 

  • Baumgärtl, H., Lübbers, D.W.: Platinum needle electrode for polarographic measurement of oxygen and hydrogen. In: Oxygen supply. Theoretical and practical aspects of oxygen supply and microcirculation of tissue. Kessler, M., Bruley, D. F., Clark, L. C., Jr., Lübbers, D. W., Silver, I. A., Strauss, J. (eds.). Munich-Berlin-Vienna: Urban and Schwarzenberg, 1973, p. 130

    Google Scholar 

  • Bicher, H.I., Bruley, D.F.: Oxygen Transport to Tissue. Advances in Experimental Medicine and Biology, 37 A and 37 B, New York and London: Plenum Press, 1973

    Google Scholar 

  • Bicher, H.I., Bruley, D., Reneau, D.W., Knisely, M.: Regulatory mechanisms of brain oxygen supply. In: Oxygen supply. Theoretical and practical aspects of oxygen supply and microcirculation of tissue. Kessler, M., Bruley, D. F., Clark, L.C., Jr., Lübbers, D.W., Silver, I.A., Strauss, J. (eds.). Munich-Berlin-Vienna: Urban and Schwarzenberg, 1973, p. 180

    Google Scholar 

  • Bicher, H.I., Knisely, M. H.: Brain tissue reoxygenation, demonstrated with a new ultramicro oxygen electrode. J. appl. Physiol. 28, 387–390 (1970)

    CAS  PubMed  Google Scholar 

  • Blum, J.J.: Concentration profiles in and around capillaries. Amer. J. Physiol. 198, 991–998 (1960)

    CAS  PubMed  Google Scholar 

  • Bork, R., Vaupel, P., Thews, G.: Atemgas-pH-Nomogramme für das Rattenblut bei 37 °C. Anaesthesist 24, 84–90 (1975)

    CAS  PubMed  Google Scholar 

  • Bruley, D. F., Knisely, M. H.: Hybrid simulation—oxygen transport in the microcirculation. Chem. Eng. Progr. 66, 22–32 (1970)

    Google Scholar 

  • Caligara, F., Rooth, G.: Measurements of the oxygen diffusion coefficient in the subcutis of man. Acta physiol. scand. 53, 114–127 (1961)

    CAS  PubMed  Google Scholar 

  • Cobourn, R.F., Mayers, L.B.: Myoglobin O2 tension determined from measurements of carboxymyoglobin in skeletal muscle. Amer. J. Physiol. 220, 66–74 (1971)

    Google Scholar 

  • Collatz, L.: Numerische Behandlung von Differentialgleichungen. Berlin-Göttingen-Heidelberg: Springer, 1955

    Google Scholar 

  • Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)

    Article  Google Scholar 

  • Diemer, K.: Eine verbesserte Modellvorstellung zur O2-Versorgung des Gehirns. Naturwissenschaften 50, 617–618 (1963)

    Article  Google Scholar 

  • Diemer, K.: Über die Sauerstoffdiffusion im Gehirn. I. Mitt.: Räumliche Vorstellung und Berechnung der Sauerstoffdiffusion. II. Mitt.: Die Sauerstoffdiffusion bei O2-Mangelzuständen. Pflügers Arch. ges. Physiol. 285, 99–108, 109–118 (1965)

    CAS  Google Scholar 

  • Doll, E.: Oxygen pressure and content in the blood during physical exercise and hypoxia. In: Limiting Factors of Physical Performance. Stuttgart: Thieme, 1973

    Google Scholar 

  • Erdmann, W., Heidenreich, J., Metzger, H.: H2-Clearance und PO2-Messung am Ratten-und Katzenhirn mit derselben Pt-Mikroelektrode. Pflügers Arch. ges. Physiol. 307, R 51 (1969)

    CAS  Google Scholar 

  • Evans, N. T., Naylor, P.F.D.: An electrical analogue for the study of tissue saturation processes with oxygen and inert gases. Phys. in Med. Biol. 9, 43 (1964)

    Article  Google Scholar 

  • Fatt, J.: An ultramicro oxygen electrode. J. appl. Physiol. 19, 326–329 (1964)

    CAS  PubMed  Google Scholar 

  • Forsythe, G.I., Wasow, W.R.: Finite difference methods for partial differential equations. New York: John Wiley and Sons, 1967

    Google Scholar 

  • Gonzales-Fernandez, J.M., Atta, S.E.: Transport and consumption of oxygen in capillary-tissue structures. Math. Biosci. 2, 225–262 (1968)

    Google Scholar 

  • Grote, J.: Der Einfluß der Temperatur und des CO2-Druckes auf die O2-Affinität und den pH-Wert des menschlichen Blutes. Pflügers Arch. ges. Physiol. 300, R 5 (1968)

    Google Scholar 

  • Grote, J., Thews, G.: Die Bedingungen für die O2-Versorgung des Herzmuskelgewebes. Pflügers Arch. ges. Physiol. 276, 142–165 (1962)

    CAS  Google Scholar 

  • Grunewald, W.A.: Theoretical analysis of the oxygen supply in tissue. In: Oxygen transport in blood and tissue. Stuttgart: Thieme, 1968, pp. 100–114

    Google Scholar 

  • Grunewald, W.A.: Die Beeinflussung der PO2-Verteilung im Gewebe durch die PO2-Messung mit der Pt-Elektrode. Pflügers Arch. ges. Physiol. 312, R 144–145 (1969a)

    CAS  Google Scholar 

  • Grunewald, W.A.: Digitale Simulation eines räumlichen Diffusionsmodelles der O2-Versorgung biologischer Gewebe. Pflügers Arch. ges. Physiol. 309, 266–284 (1969b)

    CAS  Google Scholar 

  • Grunewald, W.A.: Diffusionsfehler und Eigenverbrauch der Pt-Elektrode bei PO2-Messungen im steady-state. Pflügers Arch. ges. Physiol. 320, 24–44 (1970)

    CAS  Google Scholar 

  • Grunewald, W.A.: Bedeutung der Kapillarstrukturen für die O2-Versorgung der Organe und ihre Analyse anhand digital simulierter Modelle. Habilitationsschrift, Bochum, 1971

    Google Scholar 

  • Grunewald, W.A., Lübbers, D. W.: Die Bedeutung asymmetrischer Capillarstrukturen für die Sauerstoffversorgung der Organe. Pflügers Arch. ges. Physiol. 289, R 98 (1966)

    Google Scholar 

  • Grunewald, W.A.: Lübbers, D. W.: Quantitative Beurteilung der “O2-Diffusions-Kurzschlußgefährdung” bei der O2-Versorgung des Gewebes. Pflügers Arch. ges. Physiol. 300, R 20 (1968)

    Google Scholar 

  • Grunewald, W.A., Sowa, W.: The influence of capillary structures, O2-supply parameters and O2-shunt on the frequency distribution of intracapillary HbO2-saturation. Pflügers Arch. ges. Physiol. 355, R 3 (1975)

    Google Scholar 

  • Günther, H., Aumüller, G., Kunke, S., Vaupel, P., Thews, G.: Die Sauerstoffversorgung der Niere. I. Verteilung der O2-Drucke in der Rattenniere unter Normbedingungen. Res. exp. Med. 163, 251–264 (1974)

    Article  Google Scholar 

  • Günther, H., Vaupel, P., Metzger, H., Thews, G.: Stationäre Verteilung der O2-Drucke im Tumorgewebe (DS-Carcinosarkom). II. Messungen in vivo unter Verwendung von Goldmikroelektroden. Z. Krebsforsch. 77, 26–39 (1972)

    Article  Google Scholar 

  • Heidenreich, J., Metzger, H., Erdmann, W.: Änderung des regionalen Sauerstoffpartialdruckes nach plötzlichem Wechsel der inspiratorischen O2-Konzentration. Pflügers Arch. ges. Physiol. 312, R 64 (1969)

    CAS  Google Scholar 

  • Henquell, L., Honig, C.R.: Intercapillary distance and capillary reserve in right and left ventricles. Significance for control of tissue PO2. Microvasc. Res. 12, 35–41 (1976)

    CAS  PubMed  Google Scholar 

  • Henquell, L., LaCelle, P. L., Honig, C. R.: Capillary diameter in rat heart in situ, relation to erythrocyte deformability, 02-transport, and transmural O2-gradient. Microvasc. Res. In press (1976)

    Google Scholar 

  • Hill, A.V.: The diffusion of oxygen and lactic acid through tissue. Proc. roy. Soc. B 104, 39–96 (1929)

    Google Scholar 

  • Honig, C.R., Bourdeau-Martini, J.: O2 and the number and arrangement of coronary capillaries; effect on calculated tissue PO2. In: Oxygen transport to tissue. Instrumentation, methods, and physiology. Advance exp. Med. Biol. Bicher, H.I., Bruley, D.F. (eds.). New York and London, Plenum Press, 1973, Vol. XXXVII A, pp. 519–524

    Google Scholar 

  • Hudson, J. A., Cater, D.B.: An analysis of factors affecting tissue oxygen tension. Proc. roy. Soc. B 101, 247–274 (1964)

    Google Scholar 

  • Iwanow, K. P., Kisliakow, Yn. Ya.: The oxygen available in a neuron and surrounding tissue. Sechenow Physiol. J. USSR, LX, 6, 900–905 (1974)

    Google Scholar 

  • Jacobs, M.H.: Diffusion processes. Ergebn. Biol. 12, 1 (1935)

    Google Scholar 

  • Kadatz, R.: Sauerstoffdruck und Durchblutung im gesunden und coronarinsuffizierten Myocard des Hundes und ihre Beeinflussung durch coronarerweiternde Pharmaka. Habilitationsschrift, Tübingen, 1967

    Google Scholar 

  • Kantorowitsch, L.W., Krylow, W.J.: Näherungsmethoden der höheren Analyse. Berlin: Deutscher Verlag der Wissenschaften, 1956

    Google Scholar 

  • Kessler, M.: Normale und kritische O2-Versorgung der Leber bei Normo-und Hypothermie. Habilitationsschrift, Marburg, 1967

    Google Scholar 

  • Kessler, M., Bruley, D. F., Clark, L. C., Jr., Lübbers, D. W., Silver, I.A.. Strauss, J.: Oxygen supply. Theoretical and practical aspects of oxygen transport and microcirculation of tissue. Munich-Berlin-Vienna: Urban and Schwarzenberg, 1973

    Google Scholar 

  • Keul, J.: Die Wirkung des sportlichen Trainings auf Substratumsatz, Sauerstoffverbrauch und Durchblutung des Skelettmuskels. In: Probleme der Skelettmuskeldurchblutung. Arzneimittel-Forsch. 21, 366–376 (1970)

    Google Scholar 

  • Kisliakow, Yn. Ya., Iwanow, K. P.: PO2-distribution within neurons and cerebral capillaries with regard to the blood flow speed in normal and in hypoxia. Sechenow Physiol. J. USSR, LX, 8, 1216–1222 (1974)

    Google Scholar 

  • Knisely, M. H., Reneau, D. D., Bruley, D. F.: The development and use of equations for predicting the limits on the rates of oxygen supply to the cells of living tissues and organs. Angiology (Suppl.) 20, 1–56 (1969)

    Google Scholar 

  • Krogh, A.: The number and the distribution of capillaries in muscles with the calculation of the oxygen pressure head necessary for supplying the tissue. J. Physiol. (Lond.) 52, 409–415 (1918/19)a

    Google Scholar 

  • Krogh, A.: The rate of diffusion of gases through animal tissue, with some remarks on the coefficient of invasion. J. Physiol. (Lond.) 52, 391–409 (1918/19)b

    Google Scholar 

  • Kunze, K.: Das Sauerstoffdruckfeld im normalen und pathologisch veränderten Muskel. Schriftenreihe “Neurologie”. Berlin-Heidelberg-New York: Springer, 1969, Vol. III

    Google Scholar 

  • Kunze, K.: Die lokale kontinuierliche Sauerstoffdruckmessung in der menschlichen Muskulatur. Pflügers Arch. ges. Physiol. 282, 151–160 (1966)

    Google Scholar 

  • Leichtweiss, H.P., Lübbers, D. W., Weiss, Ch., Baumgärtl, H., Reschke, W.: The oxygen supply of the rat kidney: measurements of intrarenal PO 2. Pflügers Arch. ges. Physiol. 309, 328–349 (1969)

    CAS  Google Scholar 

  • Leonard, E. F., Jørgensen, S. B.: The analysis of convection and diffusion in capillary beds. Ann. Rev. Biophys. Bioeng. 3, 293–339 (1974)

    CAS  Google Scholar 

  • Liebmann, H.: Die angenäherte Ermittlung harmonischer Funktionen und konformer Abbildungen. S. math.-phys. Kl. Bayer. Akad. Wiss., Munich, 1918

    Google Scholar 

  • Longmuir, J. S.: Respiration rate of the rat-liver cells at low oxygen concentrations. Biochem. J. 65, 378–382 (1957)

    CAS  PubMed  Google Scholar 

  • Lübbers, D. W.: Kritische O2-Versorgung und MikroZirkulation. Marburger Jahrb. Marburg: Elwerth, 1967, pp. 305–319

    Google Scholar 

  • Lübbers, D. W., Baumgärtl, H.: Herstellungstechnik von palladinierten Pt-Stichelektroden (1–5 μ Außendurchmesser) zur polarographischen Messung des Wasserstoffdruckes für die Bestimmung der MikroZirkulation. Pflügers Arch. ges. Physiol. 294, R 39 (1967)

    Google Scholar 

  • Lübbers, D. W., Baumgärtl, H., Fabel, H., Huch, A., Kessler, M., Kunze, K., Riemann, H., Seiler, D., Schuchhardt, S.: Principle of construction and application of various platinum electrodes. In: Oxygen pressure recording in gases, fluids, and tissue. Progr. resp. Res. Kreuzer, F., Herzog, H. (eds.). Basel: Karger, 1969, Vol. III, pp. 136–146

    Google Scholar 

  • Lübbers, D. W., Luft, U. C., Thews, G., Witzleb, E.: Oxygen transport in blood and tissue. Stuttgart: Thieme, 1968

    Google Scholar 

  • Martini, J., Honig, C. R.: Direct measurement of intercapillary distance in beating rat heart in situ under various conditions of O2-supply. Microvasc. Res. 1, 244–256 (1969)

    Article  CAS  PubMed  Google Scholar 

  • Mendier, N., Schuchhardt, S., Sebening, F.: Measurements of intromyocardial oxygen tension during cardiac surgery in man. Res. exp. Med. 159, 231–238 (1973)

    Google Scholar 

  • Metzger, H.: Der Sauerstoffaustausch zwischen Blutkapillaren und Gewebe. Dissertation, Darmstadt, 1967

    Google Scholar 

  • Metzger, H.: Verteilung des O2-Partialdruckes im Mikrobereich des Gehirngewebes. Polarographische Messung und mathematische Analyse. Habilitationsschrift, Mainz, 1972

    Google Scholar 

  • Metzger, H., Erdmann, W., Thews, G.: Vergleichende Untersuchung der O2-und H2-Übergangsfunktionen im Gehirngewebe nach inspiratorischem Konzentrationswechsel. Pflügers Arch. ges. Physiol. 316, R 4 (1970)

    Google Scholar 

  • Niesel, W., Thews, G.: Ein elektronisches Analogrechenverfahren zur Lösung physiologischer Diffusionsprobleme. 1. Mitt.: Die allgemeinen Analogiebeziehungen und ihre Anwendung auf Diffusionsprozesse mit konstanten Nebenbedingungen. Pflügers Arch. ges. Physiol. 269, 262–305 (1959)

    Article  Google Scholar 

  • Niesel, W., Thews, G.: Ein elektronisches Analogrechenverfahren zur Lösung physiologischer Diffusionsprobleme. 2. Mitt.: Anwendung des Verfahrens auf Diffusionsprozesse mit konstanten und periodisch veränderlichen Verbrauchsgrößen. Pflügers Arch. ges. Physiol. 276, 182–191 (1962)

    Google Scholar 

  • Opitz, E.: Über die Sauerstoffversorgung des Zentralnervensystems. Naturwissenschaften 35, A 80 (1948)

    Article  Google Scholar 

  • Opitz, E., Bartels, H.: Gasanalyse. In: Hoppe-Seyler/Thierfelder Hdb. d. Physiolog. und Patholog. Chemie. Analyse, 10th ed. Berlin-Göttingen-Heidelberg: Springer, 1955, Vol. II, pp. 183–311

    Google Scholar 

  • Opitz, E., Schneider, M.: Über die Sauerstoffversorgung des Gehirns und den Mechanismus von Mangelwirkungen. Ergebn. Physiol. 46, 126–260 (1950)

    Google Scholar 

  • Opitz, E., Thews, G.: Einfluß von Frequenz und Faserdicke auf die Sauerstoffversorgung des menschlichen Herzmuskels. Arch. Kreisl.-Forsch. 18, 137–152 (1952)

    Google Scholar 

  • Reneau, D. D., Jr., Bicher, H.I., Bruley, D.F., Knisely, M. H.: A mathematical analysis predicting cerebral tissue reoxygenation time as a function of the rate of change of effective cerebral blood flow. In: Blood Oxygenation. New York: Plenum Press, 1970

    Google Scholar 

  • Reneau, D. D., Jr., Bruley, D. F., Knisely, M. H.: A digital simulation of transient oxygen transport in capillary tissue systems (cerebral grey matter). Aiche J. 15, 916–925 (1969)

    Article  CAS  Google Scholar 

  • Reneau, D. D., Jr., Bruley, D. F., Knisely, M. H.: A mathematical simulation of oxygen release, diffusion, and consumption in the capillaries and tissue of the human brain. In: Chemical engineering in medicine and biology. New York: Plenum Press, 1967

    Google Scholar 

  • Reneau, D. D., Knisely, M. H.: A mathematical simulation of oxygen transport in the human brain under conditions of countercurrent capillary blood flow. Chem. Engng. Progr. 67, 18–27 (1971)

    Google Scholar 

  • Renkin, E.M.: Transcapillary exchange in skeletal muscle. In: Probleme der Skelettmuskeldurchblutung. Arzneimittel-Forsch. 21, 366–376(1970)

    Google Scholar 

  • Rodenhäuser, J. H., Baumgärtl, H., Lubbers, D.W., Briggs, D.: Behavior of the oxygen partial pressure in the vitreous body under various oxygen conditions. Experimental studies in cats. Exc. Med. Intern. Congr. Ser. 222, Ophthalmology, 1624–1628 (1970)

    Google Scholar 

  • Roughton, F. R. S.: Diffusion and chemical reaction in cylindrical and spherical systems of physiological interest. Proc. roy. Soc. B. 140, 203–230 (1952)

    CAS  Google Scholar 

  • Schuchhardt, S.: Die Sauerstoffdruckverteilung im hämoglobinfrei perfundierten Meerschweinchenherzen bei Ruhe und Tätigkeit. Pflügers Arch. ges. Physiol. 322, 131–151 (1971a)

    CAS  Google Scholar 

  • Schuchhardt, S.: PO2-Messung im Myocard des schlagenden Herzens. Pflügers Arch. ges. Physiol. 322, 83–94 (1971b)

    CAS  Google Scholar 

  • Schuchhardt, S.: Statik und Dynamik der Sauerstoffversorgung des Herzens. Habilitationsschrift, Bochum, 1971c

    Google Scholar 

  • Silver, J. A.: Measurement of oxygen tension in tissues. In: Oxygen measurements in blood and tissues. London: Churchill, 1966, p. 135

    Google Scholar 

  • Silver, J. A.: Some observations on the cerebral cortex with an ultramicro membrane-covered oxygen electrode. Med. Electron, biol. Eng. 3, 377–387 (1965)

    CAS  Google Scholar 

  • Stainsby, W.N., Otis, A. B.: Blood flow, blood oxygen tension, oxygen uptake, and oxygen transport in skeletal muscle. Amer. J. Physiol. 206, 858–866 (1964)

    CAS  PubMed  Google Scholar 

  • Starlinger, H., Lübbers, D. W.: Polarographic measurement of oxygen pressure performed simultaneously with optical measurements of the redox state of the respiratory chain in suspensions of mitochondria under steady-state conditions at low oxygen pressure. Pflügers Arch. ges. Physiol. 341, 15–22 (1973)

    CAS  Google Scholar 

  • Tang, P.S.: Quart Rev. Biol 8, 260 (1958)

    Google Scholar 

  • Thews, G.: Über die mathematische Behandlung physiologischer Diffusionsprozesse in cylinderförmigen Objekten. Acta biotheor. (Leiden) 10, 105–136 (1953)

    Google Scholar 

  • Thews, G.: Ein Verfahren zur Berechnung des O2-Diffusionskoeffizienten aus Messungen der Sauerstoffdiffusion in Hämoglobin-und Myoglobinlösungen. Pflügers Arch. ges. Physiol. 265, 138–153 (1957)

    CAS  Google Scholar 

  • Thews, G.: Die Sauerstoffdiffusion im Gehirn. Pflügers Arch. ges. Physiol. 271, 197–226 (1960)

    CAS  Google Scholar 

  • Thews, G.: Die Sauerstoffdrucke im Herzmuskelgewebe. Pflügers Arch. ges. Physiol. 276, 166–181 (1962)

    CAS  Google Scholar 

  • Thews, G., Niesei, W.: Zur Theorie der Sauerstoffdiffusion im Erythrozyten. Pflügers Arch. ges. Physiol. 268, 318–333 (1959)

    CAS  Google Scholar 

  • Thomas, L.J., Jr.: Algorithmus for selected blood acid-base and blood gas calculations. J. appl. Physiol. 33, 154–158 (1972)

    PubMed  Google Scholar 

  • Vaupel, P.: Atemgaswechsel und Glucosestoffwechsel von Implantationstumoren (DS-Carcinosarkom) in vivo. Funktionsanalyse Biologischer Systeme 1, 1–138 (1974)

    Google Scholar 

  • Vaupel, P., Braunbeck, W., Thews, G.: Respiratory gas exchange and PO2-distribution in splenic tissue. In: Oxygen transport to tissue. Instrumentation, methods, and physiology. Advance exp. Med. Biol. Bicher, H.I., Bruley, D.F. (eds.). New York and London: Plenum Press, 1973, Vol. XXXVII A, pp. 401–406

    Google Scholar 

  • Vaupel, P., Günther, H., Erdmann, W., Kunke, S., Thews, G.: Dreidimensionale Registrierung stationärer PO2-Werte im Tumorgewebe unter in vivo-Bedingungen bei Verwendung von Multi-Goldmikroelectroden. Verh. dtsch. Ges. inn. Med. 78, 133–136 (1972)

    CAS  PubMed  Google Scholar 

  • Warburg, O.: Versuche an überlebendem Carcinomgewebe. Biochem. Z. 142, 317–333 (1923)

    CAS  Google Scholar 

  • Whalen, W. J., Nair, P.: Microelectrode measurements of tissue PO2 in the carotid body of the cat. In: Oxygen Supply. Theoretical and Practical Aspects of Oxygen Supply and Microcirculation of Tissue. Munich-Berlin-Vienna: Urban and Schwarzenberg, 1973

    Google Scholar 

  • Whalen, W.J., Riley, J., Nair, P.: A microelectrode for measuring intracellular PO 2. J. appl. Physiol. 23, 798–801 (1967)

    CAS  PubMed  Google Scholar 

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Authors
  1. W. A. Grunewald
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  2. W. Sowa
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Dedicated to: Prof. Dr. D.W. Lübbers, director of the Max-Planck-Institut für Systemphysiologie in Dortmund in honor of his 60th birthday.

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Grunewald, W.A., Sowa, W. (1977). Capillary structures and O2 supply to tissue. In: Reviews of Physiology, Biochemistry and Pharmacology. Reviews of Physiology, Biochemistry and Pharmacology, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0050159

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  • DOI: https://doi.org/10.1007/BFb0050159

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