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J Physiol Sci (2018) 68:355–367 https://doi.org/10.1007/s12576-017-0538-8 ORIGINAL PAPER Regulation of the glucose supply from capillary to tissue examined by developing a capillary model 1 1 1 1 • • • • Akitoshi Maeda Yukiko Himeno Masayuki Ikebuchi Akinori Noma Akira Amano Received: 28 January 2017 / Accepted: 5 April 2017 / Published online: 17 April 2017 The Physiological Society of Japan and Springer Japan 2017 Abstract A new glucose transport model relying upon Introduction diffusion and convection across the capillary membrane was developed, and supplemented with tissue space and The difference in the glucose concentration between the lymph ﬂow. The rate of glucose utilization (J ) in the local arterial and venous blood ﬂow increases with util tissue space was described as a saturation function of increasing exercise level, indicating that glucose utilization glucose concentration in the interstitial ﬂuid (C ), and by myocytes is increased [1, 2]. Nevertheless, the micro- glu,isf was varied by applying a scaling factor f to J . With dialysis method did not show an obvious decline in glucose max f = 0, the glucose diffusion ceased within *20 min. concentration in the interstitial ﬂuid (C ) during phys- glu,isf While, with increasing f, the diffusion was accelerated ical exercise. Even an increase in C above the resting glu,isf through a decrease in C , but the convective ﬂux C was reported . It has been suggested that this glu,isf glu,isf remained close to resting level. When the glucose sup- increase in C might be attributed to an increase in glu,isf plying capacity of the capillary was measured with a cri- blood ﬂow during exercise . The increase in C was glu,isf terion of J /J = 0.5, the capacity increased in not observed when muscle contraction was evoked by util max proportion to the number of perfused capillaries. A con- neuromuscular electrical stimulation . Moreover, C glu,isf sistent proﬁle of declining C along the capillary axis measured for several hours after an exercise bout was much glu,isf was observed at the criterion of 0.5 irrespective of the lower in the exercised leg than in the control rested leg in capillary number. Increasing blood ﬂow scarcely improved human experiments [5, 6]. Meanwhile, it is generally the supplying capacity. believed that glucose transport across the capillary mem- brane is mostly carried out by diffusion, and the convective Keywords Mathematical capillary model Glucose transport is small. The driving force for substrate diffusion supplying capacity Diffusion across the capillary is the concentration gradient across the membrane. These ﬁndings raise the question of how glucose transport across membrane Convective glucose ﬂux Reﬂection coefﬁcient the capillary membrane is increased during muscle exer- cise. In order to reconcile these experimental ﬁndings, quantitative analysis of glucose transport across the capil- lary membrane is a prerequisite. Most of the key parameters for both diffusion and convection have been well documented in experimental and theoretical studies . It is now possible to calculate the transcapillary exchange of major substrates based on their permeation coefﬁcients  and reﬂection coefﬁcients & Akinori Noma [9, 10] across the capillary wall in addition to water per- firstname.lastname@example.org meability [11–13] in combination with Starling’s principle. The dependency of lymph ﬂow on tissue volume is also Department of Life Sciences, Ritsumeikan University, Shiga, well explained through variation in tissue hydrostatic Japan 123 356 J Physiol Sci (2018) 68:355–367 pressure [14, 15]. Thus, we await the development of a 60 (Nc) compartments along the axis between the arterial comprehensive system model composed of capillary, tissue and venous ends to calculate the substrate diffusion as well and lymph ﬂow to analyze glucose supply by the capillary. as the convective ﬂuxes with a constant ﬂow rate (v )of ﬂow In such models, the interactions, including the positive and 1 mm/s . CC(i) is the sequential number of capillary negative feedback mechanisms between the solute and the compartments (i = 1, 2, 3,…60). The lymphatic capillary volume exchanges are calculated by solving simultaneous only provides drainage of tissue ﬂuid at a varying ﬂow rate differential equations. However, the number of mathe- determined as a function of the tissue hydrostatic pressure. matical models that calculate the substrate exchange across Deﬁnitions, dimensions and standard magnitudes of all the capillary membrane as well as lymphatic volume functional variables are described in Tables 1, 2, 3 and 4. transport is still very limited [16–19]. The model is composed of a single or several capillaries, a In the present study, we developed a new model com- lymphatic capillary and an interstitial ﬂuid space (isf). Each posed of capillary, tissue and lymph capillary for skeletal capillary is divided into 60 compartments (CC) along the muscle tissue. This model reproduces basic functions of axis. v (mm/ms) is the blood ﬂow velocity. J ﬂow vol,lum,CC(i) capillary and glucose transport well, via convection as well (ll/ms) is the volume ﬂux from CC(i)to CC(i ? 1); as diffusion. We have taken an analytical route in solving J (ll/ms) is the volume ﬂux from CC(i–1) to vol,lum,CC(i–1) the question; namely we calculate the glucose ﬂux across a CC(i); J (ll/ms) and J (plasma protein: g/ms; vol,CC(i) s,CC(i) single perfused capillary at varying glucose utilization rates glucose or NaCl: mmol/ms) are the ﬂuxes of volume and in the tissue (by myocytes). Then, the effects of increasing solutes, respectively, across the capillary membrane at the ith the number of capillaries or the blood ﬂow are examined to compartment; and J (ll/ms) is the lymphatic drainage vol,LF clarify how the glucose supply via the capillaries is of interstitial ﬂuid. J (mmol/ms) is the glucose utilization util adjusted to meet the demand of working muscle. We pro- ﬂux by hypothetical skeletal myocytes. pose a new criterion to measure the glucose supplying The parameters of a single tissue compartment are based capacity of the capillary as the basis of the muscle work on a Krogh cylinder in the skeletal muscle [21, 22], where a capacity when the number of perfused capillaries or the cylindrical envelope of muscle tissue was assumed to be blood ﬂow is varied. This new capillary model may be supported by the capillary within the Krogh cylinder. The applied to various physiological and pathophysiological number of perfused capillaries can vary, depending on the conditions when studying the balance between glucose metabolic condition of the tissue or the inﬂuence of sys- demand and supply. temic regulation. The isf was assumed to be 13% of the Krogh cylinder and the rest was assumed to be occupied by the capillary and surrounding parenchymal cells. Methods The saturation kinetics of glucose utilization Model structure as determined by the rate limiting glucose uptake into skeletal myocytes The source code of the model can be downloaded at http:// www.eheartsim.com. The present computational model, Holloszy and Narahara  showed that the uptake of schematically illustrated in Fig. 1, was developed for a sugar into stimulated muscle exhibits a saturation type of skeletal muscle tissue provided with a continuous type of kinetics, and the increase in permeability is related more to capillary. Thus, most of the parameters were adopted from a change in maximum rate of uptake (V ) than in the max experiments in skeletal muscle tissue or organs, such as a half-saturation concentration (K ). This glucose transport 0.5 hind limb as described below. The capillary space is across the cell membrane is a major rate-limiting step in deﬁned by a single or a few numbers of capillaries. The glucose utilization in skeletal muscle cells . Although a single capillary unit is 0.6 mm in length and is divided into shift of the rate-limiting step from membrane transport toward phosphorylation of glucose was observed in insulin- stimulated red muscle, Furler et al.  concluded that the membrane transport step dominates muscle glucose uti- lization. Supporting this view, Ziel et al.  found that intracellular free glucose does not accumulate in skeletal muscle. Skeletal muscle membrane glucose transport is due to facilitated diffusion via GLUT4, which is increased in number on the surface cell membrane under the inﬂuence of insulin-mediated signal transduction , or by some Fig. 1 Model compartments 123 J Physiol Sci (2018) 68:355–367 357 Table 1 Deﬁnition of variables Symbols Deﬁnitions Units C Substrate concentration mmol/l f Scaling factor of the maximum rate of consumption – I Index of saturation – Q Quantity of substrate movement mmol P Hydrostatic pressure mmHg P Effective pressure mmHg p Osmotic pressure mmHg J Flux ll/ms, g/ms or mmol/ms v Velocity mm/ms or mmol/ms Table 2 Subscripts (GLUT4) times the turnover rate of the carrier. The total number of transporters is proportional to both the density Symbols Deﬁnitions of GLUT4 on the myocytes and the number of active motor s Substrate (plasma protein, glucose, NaCl) units. Thus, in the presented simulation model, J is max pp Plasma protein increased when the muscle is activated. We tentatively glu Glucose used a value of K (=3.5 mM) of GLUT4 . To rep- 0.5 NaCl NaCl resent changes in J in the tissue space, J in Eq. (1) util max LF Lymph ﬂow was given by a product of reference J (J ) and a max max,r Vol Volume scaling factor (f). We obtained J at rest max,r -16 CC(i) ith capillary compartment (i = 1, 2, 3,…60) (J = 7.75 9 10 mmol/ms) using values of C max,r isf,r -3 isf Interstitial ﬂuid space (4.7 mM) and J (1.4 9 10 mmol glucose/min/100 g util,r pl Plasma tissue) measured in the resting muscle . a Arterial end of capillary f J max;r v Venous end of capillary J ¼ : ð2Þ util 0:5 1 þ lum Luminal side of capillary glu;isf conv Convection Then, an index of saturation (I ) was determined by diff Diffusion Eq. (3) at each f. J . max,r ﬁlt Filtration J 1 util reab Reabsorption I ¼ ¼ : ð3Þ 0:5 f J 1 þ max;r r Reference C glu;isf max Maximum To measure the glucose supplying capacity of the capillary util Utilization as the basis of the work capacity of skeletal muscle, we apply a new criterion level 0.5 to I and determine the magnitude of the scaling factor f (f ), which gives 0.5 intrinsic cellular mechanisms [28, 29]. Moreover, under I = 0.5 at a given number of capillaries or a blood ﬂow. It the regulation of the central nervous system, skeletal will be shown in the Results Section that the glucose muscle activity is increased by increasing the number of supplying capacity, represented by f increases when the 0.5 active fraction of motor units. Based on these ﬁndings, we number of perfused capillaries is increased. calculated the rate of glucose utilization (J ) in the util model tissue space using a saturation function [Eq. (1)] Hydrostatic pressure of capillary and tissue [23, 24]. Here, the glucose ‘utilization’ includes two steps: transmembrane transport and intracellular The hydrostatic pressure P in a CC(i) was deﬁned as a CC(i) metabolism. linear function of the axial number i in Eq. (4). A standard max arterial pressure (P ) of 25 mmHg and a venous P of a v J ¼ : ð1Þ util 0:5 1 þ 15 mmHg were assumed. glu;isf P ¼ P þðP P Þ ð4Þ Here, J represents the maximum glucose uptake, which CCðiÞ a v a max is given by the product of total number of transporters 123 358 J Physiol Sci (2018) 68:355–367 Table 3 Parameters of the model Symbols Deﬁnitions Values Units a Degree of dissociation NaCl: 1.87, Glucose: =1 – -5 2 A Area of cross section of capillary 5.027 9 10 mm CC -4 2 A Surface area of a capillary compartment 2.513 9 10 mm -4 -7 -7 2 D Diffusion coefﬁcient of substrates D : 1.40 9 10 , D : 1.54 9 10 , D : 2.47 9 l0 (9) mm /ms s pp glu NaC1 K Half saturating glucose concentration 3.5 (30) mmol/l 0.5 I Length of a capillary compartment 0.01 mm CC -9 L Hydraulic conductivity of capillary membrane 1.54 9 l0 (34) mm/ms mmHg N Number of capillary compartments 60 – -13 -10 -10 p Substrate permeability coefﬁcient p : 4.73 9 10 , p : 1.0 9 10 , p : 3.60 9 10 (8) mm/ms s pp glu NaCl -3 E Gas constant 62.4 9 10 mmHg l/K/mmol T Absolute temperature 310 K r Reﬂection coefﬁcient for substrate r : 0.973, r : 0.066, r : 0.054 (9) – s pp glu NaCl g Viscosity of the serum 0.004 N ms/mm Numbers in parenthesis indicate references Table 4 Initial values Symbols Deﬁnitions Values Units C Concentration of plasma protein in capillary 75.9 g/l pp,CC(i) C Concentration of substrates in capillary C :5 C : 150 mmol/l s,CC(i) glu NaC1 C Concentration of plasma protein in interstitial ﬂuid 17.6 g/l pp,isf C Concentration of substrates in interstitial ﬂuid C : 4.71 C :l49 mmol/l s,isf glu NaC1 -23 K Boltzmann constant 1.38 9 10 J/K P Blood pressure at arterial end 25 mmHg P Blood pressure at venous end 15 mmHg P Hydrostatic pressure in interstitial space -1.08 mmHg isf v Velocity of plasma ﬂow 1 mm/s ﬂow -7 V Volume of ith capillary segment 5.03 9 10 ll CC(i) -4 V Original volume of interstitial space 3.18 9 10 ll isf -4 V Volume of interstitial space at control 3.68 9 10 ll isf -16 J Standard rate of glucose consumption 7.75 9 10 mmol/ms max p Colloid osmotic pressure of capillary See text mmHg s,CC p Osmotic pressure of interstitial space See text mmHg s,isf The interstitial hydrostatic pressure P is subatmo- The magnitude of V was determined as 13% of a isf isf spheric in most tissues . The interstitial volume-pres- Krogh muscle cylinder of dimensions 0.6 mm in length and sure relation (compliance curve) was represented by a 0.036 mm in radius . 0 0 sigmoidal function of V =V , where V is a reference isf isf isf Lymph ﬂow volume of isf. The parameters were determined by ﬁtting an equation [Eq. (5)], which includes an exponential and a The tissue hyrostatic pressure given by Eq. (5) is the major linear terms, to the experimental data obtained in canine factor in driving lymphatic drainage of isf, J (l/ms). hind limbs , shown in Fig. 2a. vol,LF The dependency of J on the P obtained in the hind vol,LF isf ðrVisf1:07Þ=0:0953 P ¼e þð1:4 rV 2:38Þ; isf isf limb preparation  is cited in Fig. 2b, and was ﬁtted with ð5Þ isf rV ¼ : the empirical Eq. (6) isf isf 123 J Physiol Sci (2018) 68:355–367 359 Fig. 2 Model ﬁtting of the experimental volume-pressure relation (a), and the pressure-lymph ﬂow relationships (b). Dots are experimental measurements in the hindlimb preparation, and the continuous curve is the theoretical relationship of Eq. (5) and Eq. (6) We corrected diffusion coefﬁcient D (mm /ms) for T (K), 9 10 J ¼ : ð6Þ vol;LF P þ6:08 P 18:9 and viscosity (g) of the serum (=0.004 N ms/mm ) using isf isf 2:16 4:5 e þ e the Stoke–Einstein relation [Eq. (11)]. K T D ¼ ; ð11Þ Colloid and crystalloid osmotic pressures 6pgc -23 where K is the Boltzmann constant (=1.38 9 10 J/K), The colloid osmotic pressure p (mmHg) is a function of pp b p is the circular constant (=3.14) and c (mm) is the Stokes– plasma protein (pp) concentration C (g/l) as exempliﬁed pp Einstein radius of the molecule . For c, we referred to in an experimental ﬁnding , and was determined by -4 Rippe and Haraldsson . Thus, D : 1.40 9 10 , D : ﬁtting an empirical Eq. (7). Equation (7) was applied to alb glu -7 -7 1.54 9 10 , and D : 2.47 9 10 were obtained. both the plasma and tissue compartments. NaCl These values are nearly comparable or slightly smaller than p ¼ 0:21 C þ 0:0016 C : ð7Þ pp pp pp those used by Kellen et al. , which were D : alb -4 -7 -6 1.53 9 10 , D : 7.0 9 10 , D : 2.0 9 10 .On sucrose NaCl The crystalloid osmotic pressure p caused by a substrate s the other hand, the parameters in Kellen et al.  are (glucose or NaCl) was determined by van ‘t Hoff’s law rather similar to the diffusion coefﬁcient in water. [Eq. (8)], where R is the gas constant, T is the absolute temperature, C is concentration, and a is the degree of Fluid volume and solute ﬂuxes across the capillary dissociation. The values of a are 1.0 and 1.87 for glucose membrane and NaCl, respectively . p ¼ R T C a s;fg NaCl, glucose : ð8Þ s s The volume ﬂux across the capillary membrane J vol,CC(i) for a single capillary compartment was calculated by the Starling principle extended for crystalloid osmotic pres- Fluid volume and solute ﬂuxes between successive sures [Eq. (12)]. Note that the third factor gives the capillary compartments effective ﬁltration pressure P deﬁned in Starling’s principle. The volume ﬂux J between the (i–1)th and the vol,lum,CC(i) ith compartments is described using Eq. (9). J ¼ L A p s vol;CCðiÞ "# flow J ¼ V ; ð9Þ vol;lum;CCðiÞ CCði1Þ P P r ðp p Þ ; isf s s;isf CCðiÞ s;CCðiÞ cc where V is the volume of capillary (i–1)th compart- CC(i–1) ð12Þ ment, l is the length of a capillary compartment and v cc ﬂow where L is the hydraulic conductivity of the capillary is velocity of blood ﬂow. The substrate ﬂux across the membrane, A is the capillary surface area, and P and boundary between two successive capillary compartments s CC(i) P are the hydrostatic pressures in the capillary compart- isf J (i–1) and i was calculated as a sum of diffusion s,lum,CC(i) ment CC(i) and interstitial space, respectively. The and convection terms in Eq. (10). reﬂection coefﬁcient for a solute s is r (r = 0.973, s pp J ¼ D A C C s cc s;lum;CCðiÞ s;CCði1Þ s;CCðiÞ r = 0.066, r = 0.054) , and p and p are glu NaCl s;CCðÞ i s;isf þ J C ð10Þ vol;lum;CCði1Þ s;CCði1Þ osmotic pressures in CC(i) and interstitial space, s;fg pp, glucose, NaCl : respectively. 123 360 J Physiol Sci (2018) 68:355–367 We determined the water permeability (L ) of the cap- dQ p s;CCðiÞ ¼ J J J s;lum;CCði1Þ s;lum;CCðiÞ s;CCðiÞ illary membrane from the experimental data described in ð16Þ dt s;fg pp, NaCl, glucose ‘Textbook of Medical Physiology’ , in which Guyton wrote that ‘13 mmHg ﬁltration pressure causes, on aver- Nc dQ s;isf age, about 1/200 of the plasma in the ﬂowing blood to ﬁlter ¼ J J C s;fg pp, NaCl s;CCðiÞ vol;LF s;isf dt out of the arterial ends of the capillaries into the interstitial i¼1 spaces each time the blood passes through the capillaries’. ð17Þ Assuming an arterial end of 0.3 mm in length, blood Nc dQ glu;isf passing through the arterial end within 300 ms, and a ¼ J J C J : ð18Þ glu;CCðiÞ vol;LF glu;isf util -9 dt hematocrit of 0.4, a L of 1.54 9 10 (mm/ms/mmHg) is p i¼1 obtained. In the frog mesentery, values for L of -8 -9 1.36 9 10  and 4.06 9 10  were reported. In -10 the human forearm, a L of 1.66 9 10 , and in rat Results -10 skeletal muscle a L of 9.97 9 10 , and -10 3.32 9 10  have been reported. Thus, the magni- Glucose supply to isf space via convection tude of L in the present model is within the range of and diffusion variation. The solute ﬂux J (g/ms or mmol/ms) across a It has been suggested that the glucose supply to the skeletal s,CC(i) capillary membrane of a compartment CC(i) was deter- myocytes is largely carried out by diffusion [22, 39]. Our mined as a sum of diffusion J and convection J new capillary model examines the amplitude of glucose s,diff,CC(i) s,- : diffusion ﬂux (J ¼ J ) and the convective conv,CC(i) glu;diff glu;diff;CCðiÞ P P ﬂux (J ¼ J ; J ¼ J ; glu;conv glu;filt glu;conv;CCðiÞ glu;filt;CCðiÞ J ¼ J þ J s;CCðtÞ s;diff;CCðiÞ s;conv;CCðtÞ J ¼ J ) across the capillary membrane glu;reab glu;reab;CCðiÞ ¼ p A ðC C Þþ J C ð1 s s s;CCðiÞ s;isf vol;CCðiÞ s;x as shown in Fig. 3. The four factors of Starling’s principle r Þ x are demonstrated in Fig. 3a against sequential capillary : fCC(iÞ; isfg; compartment number at the resting level of glucose uti- ð13Þ lization in the tissue space. p (blue) is quite uniform pp;CCðÞ i where A is the surface area for diffusion, and C or s s,CC(i) in all segments of the capillary because of the large C is concentration of substrate s in the volume ﬂux s,isf reﬂection coefﬁcient for pp. In Fig. 3b, J was ll/ vol,CC(32) J across the capillary wall. vol,CC(i) ms and the net volume ﬂux J was slightly positive (not vol For the permeability p (mm/ms) across the capillary s shown), and was balanced by the negative lymph ﬂow membrane, we referred to Renkin  and Levick , and (black line in Fig. 3b). -13 -10 deﬁned a p of 4.73 9 10 , p of 1.0 9 10 , and pp glu Figure 3c1 shows the convective glucose ﬂux J glu,- -10 p of 3.60 9 10 . These values are similar to those NaCl by ﬁltration (red), and absorption (blue) in each conv,CC(i) -12 used in the Kellen model; p : 4.3 9 10 , p : alb sucrose compartment of the capillary for comparison with the dif- -10 -10 2.6 9 10 , p : 8.1 9 10 . NaCl fusion ﬂux J (black line), which is virtually glu,diff,CC(i) uniform for all capillary components. The J is glu,conv,CC(i) quite proportional to the ﬂuid ﬂux J (Fig. 3b, c1), vol,CC(i) Ordinary differential equations to determine rate since the difference in C is relatively small between glu of volume and concentration changes capillary and tissue. In this respect, the proﬁle of J glu,- (Fig. 3c1) is similar to J (Fig. 3c3), conv,CC(i) NaCl,conv,CC(i) Changes in the interstitial and capillary volumes and the but in strong contrast to J , showing a marked pp,conv,CC(i) quantity of substrate (Q) were calculated by the numerical asymmetry between ﬁltration and reabsorption, because of time integration of the transcapillary exchange of ﬂuid (dV/ the large difference between C and C (Fig. 3c2). pp,CC(i) pp,isf dt) and solute (dQ /dt) using the Euler method. The capillary factors p and P remain virtually pp;CCðÞ i cc(i) dV CCðiÞ constant during various simulation conditions in the pre- ¼ J J J ð14Þ vol;lum;CCði1Þ vol;lum;CCðiÞ vol;CCðiÞ dt sent study. On the other hand, the tissue parameters p and isf Nc P change in a dynamic manner, and are mainly respon- isf dV isf ¼ J J ð15Þ vol;CCðiÞ vol;LF sible for adjusting the convective glucose ﬂux in response dt i¼1 to various experimental conditions. 123 J Physiol Sci (2018) 68:355–367 361 Fig. 3 Main factors involved in the glucose ﬂux under the control lymphatic drainage of interstitial ﬂuid J is indicated by a black vol,LF -12 (resting) condition. a Capillary colloid osmotic pressure p line, all in a dimension of 10 ll/ms. c1, c2, c3 The solute pp,CC(i) (blue), capillary hydrostatic pressure P (red), tissue colloid convective ﬂux J by ﬁltration (red) and reabsorption CC(i) s,conv,CC(i) -17 osmotic pressure p (green), and tissue hydrostatic pressure P J (blue). c1 Glucose ﬂux (10 mmol/ms), c2 plasma pp,isf isf s,conv,CC(i) -18 -16 (black). b The volume ﬂux across the capillary membrane: the protein ﬂux (10 g/ms), and c3 NaCl ﬂux (10 mmol/ms). The ﬁltration J (red) and reabsorption J (blue). The diffusion ﬂuxes J are indicated by a black line in each graph vol,ﬁlt,CC(i) vol,reab,CC(i) s,diff,CC(i) Balance between diffusional glucose supply events occurred to recover the control levels of each vari- and cellular glucose utilization during exercise able within 15 min. The decrease in C during the period of accelerated glu,isf If glucose utilization (J ) by the tissue was nulliﬁed, the J largely lowered p and decreased V , as shown in util util isf isf glucose gradient across the capillary membrane dissipated, Fig. 4b3, and thereby C was secondarily increased as pp,isf and J totally disappeared within *20 min, indicating shown by the red trace in Fig. 4b2. On the other hand, the glu,diff that glucose diffusion was indirectly driven by J C remained at the control level (blue trace in util NaCl,isf (Fig. 4a). On the other hand, an increase in tissue J Fig. 4b2), because of its high conductivity across the util during exercise greatly increased J . In Fig. 4b, J capillary membrane. The slow process of re-equilibration glu,diff util was increased by augmenting J by 20-fold [i.e., of pp, however, caused a drift of C and caused max pp,isf changing f to 20 from 1 in Eq. (2)] at 50 min. This inter- delayed recovery of the V as shown in Fig. 4b3. These isf vention caused an immediate rise of J (blue curve in changes in the tissue parameters were completely util Fig. 4b1) close to the target J level (black line), which reversible. util was determined by f (=20) and the initial C (=4.7 mM) The magnitude of J (red curve) is compared with glu,isf glu,diff (Eq. 2). Then, J started to decrease to a new steady level J (ﬁltration, yellow and absorption purple) in more util glu,conv (57% of the initial level) by the decrease in C to less detail in Fig. 4b1. During the control, the magnitude of glu,isf than 2 mM as indicated in Fig. 4b2 (black curve). This glucose supply to isf due to ﬁltration J was nearly glu,ﬁlt decrease in C largely magniﬁed J until the net comparable (*70%) to J . This magnitude of J glu,isf glu,diff glu,diff glu,ﬁlt glucose ﬂux across the capillary membrane, subtracted by remained almost unchanged during the high J period util the glucose removal via lymph ﬂow (red curve in Fig. 4b1) because the ﬂuid ﬂux was not signiﬁcantly modulated by matched the augmented J (blue curve) well at around the increase in J . On the other hand, J (purple) was util util glu,reab 50 mmol/ms *15 min after switching f to 20. decreased by the depletion of C , thereby the net con- glu,isf When f.J was returned to the control level by vective ﬂux of glucose was increased during exercise, max switching f to 1 from 20 at 150 min, a reverse sequence of although the magnitude was much smaller than the 123 362 J Physiol Sci (2018) 68:355–367 Fig. 4 Time course of the glucose transport response evoked by (blue), the net glucose transport J (red), glucose ﬁltration glu,diff varying the metabolic activity (a f = 0, b f = 20) in the tissue space. transport J (yellow), glucose reabsorption transport J glu,ﬁlt glu, reab a Cessation of glucose diffusion (J ) after nullifying J through (purple), and lymphatic glucose drainage of interstitial ﬂuid J glu,diff util glu, LF equilibration of glucose concentration between capillary and isf. The (green) are shown. b2 Interstitial concentration of C (mmol/l, glu,isf glucose utilization J (blue) and the net glucose transport J black), C (10 g/l, red) and C (10 mmol/l, blue). b3 util glu, diff pp,isf NaCl,isf (red) are shown. b1 Glucose ﬂux J across the capillary membrane Relative tissue volume (rV green). On returning to the control glu isf, -16 given in 10 mmol/ms. The target J determined by f (=20) and condition, rV showed a rebound because of an accumulation of pp util isf the initial C (=4.7 mM) (Eq. 2)(black), the evolution of J during the test period of f = 20 glu,isf util diffusional ﬂux. It is concluded that the increase in J util during heavy exercise is largely compensated for by J . glu,diff Adjustments of glucose supply by increasing the number of capillaries and velocity of blood ﬂow The increase in J in response to heavy exercise as util demonstrated in Fig. 4b was examined over a wide range of f (Fig. 5). Within a range of approximately 0 * 5 times f, it seems that J increased in proportion to f. However, util the relationship is evidently a saturating function; the ratio J /(fJ ) gradually decreased with increasing f (red util max,r curve in Fig. 5). This ﬁnding might suggest that the glu- cose supply to the skeletal muscle by capillary becomes Fig. 5 The relation between J and the scaling factor f of J at util max.r deﬁcient and the work capacity is decreased, when the different numbers of capillaries indicated at the right side exercise level is raised. Under the condition in situ this shortage of glucose supply is compensated for by the autonomous increase in the number of perfused capillaries. Fig. 5, the J was much increased with increasing number util of capillaries, with an obvious trend of saturation in the This situation was simulated by increasing the number of capillaries for the model tissue compartment. As shown in effect. 123 J Physiol Sci (2018) 68:355–367 363 An alternative way of increasing the glucose supply proportion to the capillary number. This proportional might be applied by accelerating the velocity of the blood increase in the glucose supplying capacity was obtained ﬂow through the capillary. This possibility was examined when the criterion level of I was varied within a range of as shown in Fig. 6. Surprisingly, however, effects of 0.3 * 0.57. Most probably, the work capacity of muscle increasing the ﬂow rate were very limited. On the other might be simultaneously augmented in proportion to the hand, if the ﬂow rate was decreased to 0.1 mm/s, C capillary number. glu,CC(i) gradually declined with increasing compartment number It is also evident that the steady-state level of C was glu,isf by *0.14 mM at the venous end. A marked decay in the raised with increasing number of capillaries. This ﬁnding is C proﬁle was obtained when the ﬂow rate was consistent with the measurements of C , which glu,CC(i) glu,isf further decreased to 0.01 to simulate profound ischemia increased with increasing workload of the muscle , . The decrease in the C proﬁle was largely provided that the number of perfused capillaries was glu,CC(i) compensated for by increasing the number of capillaries increased by the local as well as systemic regulations. It threefold (Fig. 6c). should be noted that glucose diffusion across the capillary membrane is still well driven by the difference between The capacity of capillaries for supplying glucose C and C in Fig. 7b. glu,CC(i) glu,isf measured by applying a criterion of 0.5 The I (Fig. 7a) or f (Fig. 7b) were plotted on each to the saturation index I abscissa against the common ordinate of C . Different s glu,isf. colors in Fig. 7b indicate the different number of capil- Using the criterion of I = 0.5, the glucose supplying laries assumed within the tissue space, as indicated on the capacity was measured from the magnitude of f at a right side of the ﬁgure. In Fig. 7b, the C at f = 0is 0.5 glu,isf given number of capillaries. Figure 7a shows the rela- equal to the C . glu,CC(i) tionship between I on the abscissa and C on the When the blood ﬂow was increased to 0.01, 0.1, 1 and 4 s glu,isf ordinate. The ﬁve relationships between the scaling factor (mm/s) in the single capillary, the capacity of supplying f (on the abscissa) and the steady-state C (on the glucose (f ), was increased to 1.22, 4.72, 5.96 and 6.03, glu,isf 0.5 ordinate) were determined at different numbers of capil- respectively (data not shown). Obviously, the glucose laries as indicated with different colors. Thus, f is indi- supplying capacity was saturated at v [ 1 mm/s. 0.5 ﬂow cated by the arrowheads drawn from each intersection of The ﬁndings described so far indicate that the C glu,CC(i) the horizontal line of I = 0.5 with individual curves of proﬁle reﬂects the glucose supplying capacity of the cap- C . It is evident that the f increased with increasing illary well. This notion was further tested by plotting the glu,isf 0.5 number of perfused capillaries. Namely, the magnitude of C proﬁle along the axis of the capillary obtained glu,CC(i) f was increased to 5.96, 11.9 and 17.8 with the number of using the criterion of I = 0.5 with various numbers of 0.5 s capillaries = 1, 2 and 3, respectively (Fig. 7b), indicating perfused capillaries. As shown in Fig. 8, the C glu,CC(i) that the f increases in proportion to the number of cap- proﬁles at I = 0.5 were superposable at one (black points), 0.5 s illaries within a single tissue space. That is, the glucose two (red) and three (green) capillaries. Thus, the value of supplying capacity of the capillaries was magniﬁed in f deﬁnes a unique set of C and C . These 0.5 glu,CC(i) glu,isf Fig. 6 The J -f relations util (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 was varied ﬂow 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 along the capillary glu,CC(i) axis when the capillary number was one. c The same proﬁles of C obtained when the glu,CC(i) capillary number was three 123 364 J Physiol Sci (2018) 68:355–367 Fig. 7 The steady-state relationship between the I (a) and f (b) as revealed by the horizontal line drawn at the criterion level I = 0.5 has also been well measured in the canine hind limb . The presented model based on those experimental mea- surements simulated the supply of glucose to the skeletal muscle well at varying levels of muscle activities scaled by f. The present model study conﬁrmed that the glucose supply to tissue cells largely depends on glucose diffusion across the endothelial wall. When the glucose supplying capacity of the capillary is measured with the criterion f 0.5 deﬁned by the saturation kinetics, the capacity was increased in proportion to the number of perfused capil- laries; in other words, in proportion to the glucose diffusion area of the capillary. This simulation model substantiated by the criterion of f might be relevant to evaluation of the 0.5 capillary capacity for supplying glucose in the intact tissues Fig. 8 The proﬁle of C along the capillary axis at I = 0.5. glu.CC(i) s or most probably to evaluate the increase in the work 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 capacity of the muscle, induced by increasing the number identical, were plotted in an alternating way with different colors of perfused capillaries. results prove the relevance of using the criterion of f in 0.5 evaluating the capillary capability to provide glucose. It The convection ﬂux of glucose across the capillary should be noted that the overlap of the C proﬁles wall glu,CC(i) only occurs when the diffusion ﬂux dominates over the convective glucose ﬂux, since the convective ﬂux is almost The evaluation of the convection ﬂux of glucose in the independent from C . present study should be thoroughly discussed in respect to glu,isf the water ﬂux, which was calculated by the classic Starling principle. It has been suggested that the ﬂuid balance is not Discussion directly determined by the plasma protein concentration in the interstitial ﬂuid, but that the concentration just below Relevance of the presented capillary model the ﬁltration structure (underneath space), glycocalyx sheet, should be used in calculating the ﬂuid balance The histological composition of the model was determined [41–43]. The detailed quantitative studies strongly sug- according to a Krogh cylinder in skeletal muscle. The gested that the concentration of protein in a local space just biophysical parameters, such as the reﬂection coefﬁcients, beneath the glycocalyx sheet should be lower than that in the colloid, as well as the crystal osmotic pressures, the the bulk interstitial space by about 30% . This is diffusion conductivity of various substrates and conduc- because ultraﬁltrate through the glycocalyx toward the tivity of water across the capillary membrane are all based tissue side inhibits the back diffusion of protein molecule on experimental data obtained in skeletal muscle tissue or from the bulk tissue space to the underneath space. This organs, as described in the Method. The magnitude of the effect should be dependent on the ﬂow rate of ﬁltrate lymph ﬂow as a function of the tissue hydrostatic pressure through the narrow gap (*4 nm, ) of tight junctional 123 J Physiol Sci (2018) 68:355–367 365 strand, where the ﬂow rate is much accelerated. They Participation of large and small pores in the water measured the effective oncotic pressure across rat mesen- ﬂux in the model teric microvessels with and without albumin in the inter- stitial ﬂuid. They found the effective oncotic pressure was Morphological and physiological studies indicated mul- tiple routes for substrate transport across capillary near 70% of the luminal oncotic pressure when the albumin concentrations were equal across the endothelial wall. endothelium. However, the assignment of speciﬁc transport roles to morphologically identiﬁable pathways Thus, the ratio of J and J might be only mar- glu,conv glu,diff ginally modiﬁed, but the major role of diffusional ﬂux in has been only partly achieved. The contributions of junctions, single vesicles, chains of vesicles and fenes- supplying glucose at enhanced tissue activity remains as estimated in the present study. trae to total macromolecular transport have been sug- At the venous end of the capillary, reabsorption of ﬂuid gested, but their quantitative contributions are not yet might also be modiﬁed if pp is accumulated in the under- known precisely . Identiﬁcation of the ‘‘small-pore’’ neath space by the ﬂow directed from the bulk to the pathway for water and lipid-insoluble molecules still underneath space . At present, however, quantiﬁcation largely remains questionable. A theoretical approach of this effect is beyond the scope of the present study. from physics may be indispensable for understanding the mechanism of solute permeation through the glycocalyx Involvement of the gel structure in the tissue colloid sheet. Indeed, some of the theoretical calculations of parameters, optimized by ﬁtting the experimental mea- pressure surement of substrate ﬂux, are explicitly related to the One of the major comprehensive microcirculation models coefﬁcients determined by Kedem and Katchalsky . was published by Kellen et al. [16, 17] for the cardiac When the total ﬂux through the capillary membrane is muscle. Most of the model parameters are similar to the reconstructed, it might be necessary to adjust the ratio of presented model, except those parameters which are different types of pores according to the experimental speciﬁc for the cardiac tissue; such as the hydraulic pres- data. Rippe and Haraldsson suggested that 86–87% of the hydraulic conductivity (L sure of the tissue. In the cardiac muscle there is a huge ) was accounted for by the gradient of hydraulic pressure in the tissue between the small-pore pathway and 3.0–4.1% of L by the endothelial pathway, while the remaining fraction (10%) epicardial and endocardial sides. In the endocardial muscle layer, the tissue hydrostatic pressure is similar to the would be accounted for by a large-pore pathway . On the other hand, Kellen and Bassingthwaighte (2003) pressure in the ventricular cavity during systole, and this effect of systolic pressure is minimal on the epicardial side , based on the osmotic weight transient data in iso- lated rabbit hearts, estimated that the endothelial path- of the muscle tissue. The colloid osmotic pressure in the interstitial ﬂuid is way for transcapillary water-only exchange accounts for much higher (*25 mmHg) in the Kellen model than in our 28% of total transcapillary hydraulic conductivity, the skeletal muscle model. This high colloid osmotic pressure in large-pore pathway accounts for 5% of L ,and the the Kellen model was attributed to the matrix protein. majority 67% is via a small-pore pathway . In the However, this colloid osmotic pressure is largely different to present study, we simply referred directly to the exper- the detailed analysis based on the glycocalyx theory descri- imental permeability coefﬁcient determined for the whole area of capillary membrane (surface) without bed above, or the hypothesis of Guyton . Note that the ﬂuid balance across the glycocalyx layer was determined discriminating large and small pores. only by assuming the freely diffusive protein in the under- neath space bordered by the tight junction strand underneath Limitations of the present model the glycocalyx. The involvement of the matrix proteins, such as proteoglycan was not considered in this space. Guyton To get a deeper insight into the balance between glucose assumed a 2 mmHg oncotic pressure at most, when the gel is supply and glucose utilization by skeletal muscle, it might included in a bag made by a ﬁltration membrane, where the be important to replace the simple saturation kinetics glycosaminoglycan is not freely diffusive, but is restrained [Eq. (1)] by a detailed metabolic pathway for glucose by the cross-linkage within the gel structure. However, it is consumption in the skeletal muscle. We await the addition unlikely that the gel structure extends to the glycocalyx of mechanisms of physiologically active substances, such as serotonin and histamine  to the model, to further membrane through the gap of the tight junction strand. If the matrix protein is anchored to a long chain of hyaluronan and develop a physiological capillary model which can be applied to a variety of capillary functions in different is totally separated from the ﬁltration sheet, no inﬂuence is expected for the effective oncotic pressure. tissues. 123 366 J Physiol Sci (2018) 68:355–367 Acknowledgements We thank colleagues in the laboratory of Reg- 17. Kellen MR, Bassingthwaighte JB (2003) Transient transcapillary ulation of Tissue Functions at the Department of life Sciences, Rit- exchange of water driven by osmotic forces in the heart. Am J sumeikan University, for very fruitful discussions. Physiol Heart Circ Physiol 285:H1317–H1331 18. Bassingthwaighte JB, Raymond GM, Ploger JD, Schwartz LM, Bukowski TR (2006) GENTEX, a general multiscale model for Compliance with ethical standards in vivo tissue exchanges and intraorgan metabolism. 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