Regulation of the glucose supply from capillary to tissue examined by developing a capillary model

Abstract

A new glucose transport model relying upon diffusion and convection across the capillary membrane was developed, and supplemented with tissue space and lymph flow. The rate of glucose utilization (J _util) in the tissue space was described as a saturation function of glucose concentration in the interstitial fluid (C _glu,isf), and was varied by applying a scaling factor f to J _max. With f = 0, the glucose diffusion ceased within ~20 min. While, with increasing f, the diffusion was accelerated through a decrease in C _glu,isf, but the convective flux remained close to resting level. When the glucose supplying capacity of the capillary was measured with a criterion of J _util /J _max = 0.5, the capacity increased in proportion to the number of perfused capillaries. A consistent profile of declining C _glu,isf along the capillary axis was observed at the criterion of 0.5 irrespective of the capillary number. Increasing blood flow scarcely improved the supplying capacity.

Keywords: Convective glucose flux; Diffusion across the capillary membrane; Glucose supplying capacity; Mathematical capillary model; Reflection coefficient.

Fig. 1

Model compartments

Fig. 2

Model fitting of the experimental volume-pressure relation (a) [32], and the pressure-lymph flow relationships (b) [14]. Dots are experimental measurements in the hindlimb preparation, and the continuous curve is the theoretical relationship of Eq. (5) and Eq. (6)

Fig. 3

Main factors involved in the glucose flux under the control (resting) condition. a Capillary colloid osmotic pressure π _pp,CC(i) (blue), capillary hydrostatic pressure P _CC(i) (red), tissue colloid osmotic pressure π _pp,isf (green), and tissue hydrostatic pressure P _isf (black). b The volume flux across the capillary membrane: the filtration J _{vol,filt,CC(i)} (red) and reabsorption J _{vol,reab,CC(i)} (blue). The lymphatic drainage of interstitial fluid J _vol,LF is indicated by a black line, all in a dimension of 10⁻¹² μl/ms. c1, c2, c3 The solute convective flux J _s,conv,CC(i) by filtration (red) and reabsorption J _s,conv,CC(i) (blue). c1 Glucose flux (10⁻¹⁷ mmol/ms), c2 plasma protein flux (10⁻¹⁸ g/ms), and c3 NaCl flux (10⁻¹⁶ mmol/ms). The diffusion fluxes J _s,diff,CC(i) are indicated by a black line in each graph

Fig. 4

Time course of the glucose transport response evoked by varying the metabolic activity (a f  = 0, b f = 20) in the tissue space. a Cessation of glucose diffusion (J _glu,diff) after nullifying J _util through equilibration of glucose concentration between capillary and isf. The glucose utilization J _util (blue) and the net glucose transport J _{glu, diff} (red) are shown. b1 Glucose flux J _glu across the capillary membrane given in 10⁻¹⁶ mmol/ms. The target J _util determined by f (=20) and the initial C _glu,isf (=4.7 mM) (Eq. 2) (black), the evolution of J _util (blue), the net glucose transport J _glu,diff (red), glucose filtration transport J _glu,filt (yellow), glucose reabsorption transport J _{glu, reab} (purple), and lymphatic glucose drainage of interstitial fluid J _{glu, LF} (green) are shown. b2 Interstitial concentration of C _glu,isf (mmol/l, black), C _pp,isf (10 g/l, red) and C _NaCl,isf (10² mmol/l, blue). b3 Relative tissue volume (rV _isf, green). On returning to the control condition, rV _isf showed a rebound because of an accumulation of pp during the test period of f = 20

Fig. 5

The relation between J _util and the scaling factor f of J _max.r at different numbers of capillaries indicated at the right side

Fig. 6

The J _util-f relations (a) and the profiles of C _glu,CC(i) along the capillary axis when the numbers of capillaries were one (b) and three (c), respectively. a v _flow was varied by fourfold (green), onefold (control, red), 0.1-fold (blue) and 0.01-fold (black). The J _util-f relations were nearly superimposed. b Profiles of C _glu,CC(i) along the capillary axis when the capillary number was one. c The same profiles of C _glu,CC(i) obtained when the capillary number was three

Fig. 7

The steady-state relationship between the I _s (a) and f (b) as revealed by the horizontal line drawn at the criterion level I _s = 0.5

Fig. 8

The profile of C _glu.CC(i) along the capillary axis at I _s = 0.5. The values of f were 8.74 at one (black), 17.5 at two (red) and 26.2 at three capillaries (green). For better visibility, the three curves, nearly identical, were plotted in an alternating way with different colors