rcrit (r C + t m ) 2 ] crit + t o crit The critical radius is evaluated at a given axial location z from the equation + (1 , and D = 4D = 555.

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1 hapter 1 c) When the average bld velcity in the capillary is reduced by a factr f 10, the delivery f the slute t the capillary is liited s that the slute cncentratin after crit c is equal t er at the Krgh tissue cylinder radius. After this lcatin, the slute cncentratins are evaluated fr R () [ r crit (r + t ) ] (1.8-6) Vr r c + () t R r K [ r crit (r + t ) ] (, r ) r c + + t R [ r crit (r + t ) ] 4D R D r rcrit ln r + t (1.8-7) he critical radius is evaluated at a given axial lcatin fr the equatin R ln R 4D R ( r + t ) + (1 D 4D r K )(R 1) Vr where R rcrit r + t d) Plt the critical tissue radius as a functin f the axial distance alng the capillary. 4D Let A, B 1 R ( r + t ) D 4D, and D r K Vr R ln R A + (B D)(R 1) (1.8-8) 4D A R ( r + t ) , B 1 D 4D 555.5, D r K 5600 c -1 Vr 1 1 First slve fr crit [B (R ln R A)] D R 1 where R rcrit r + t r r + t 7.77 crit c Beynd the critical axial lcatin crit, the critical radius ust be slved nuerically fr equatin (1.8-8). Let x R, at any lcatin > crit 1-61

2 B D E cnstant he nnlinear equatin t be slved is f(x) x ln(x) E(x 1) 0 f (x) ln(x) + 1 E x can be btained using Newtn s ethd where x x f ( x) f ' ( x) ince x r r crit + t, the critical radius is then r crit (r + t )x 1/ able 1.8- lists the Matlab prgra used t plt Figure able Matlab prgra t plt Figure % Krgh issue cylinder 5; R.01; V.05; rc.5*.001; K5.75e-5; t5e-5; D8e-6; r4e-3; 0:.01:.1; rctrc+t;rctsrct*rct; 1R*(r^-rcts)*/(V*rc^); R*(r^-rcts)/(*rc*K); -1; w-; rr; 3R*(r.*r-rcts)/(4*D); 4R*r^*lg(r/rct)/(*D); w+3-4; subplt(,,1);plt(,,,w,':',,,'-.') grid n title('v.05 c/s, D8e-6 c/s') xlabel('(c)');ylabel('(micr-l/c3)') legend('_','_w','_') D8e-7; rctrc+t;rctsrct*rct; 1R*(r^-rcts)*/(V*rc^); R*(r^-rcts)/(*rc*K); -1; w-; rr; 3R*(r.*r-rcts)/(4*D); 4R*r^*lg(r/rct)/(*D); w+3-4; subplt(,,);plt(,,,w,':',,,'-.') grid n title('v.05 c/s, D8e-7 c/s') xlabel('(c)');ylabel('(micr-l/c3)') legend('_','_w','_') V.005;D8e-6; 1-6

3 % lve fr the critical radius % % lve fr psitin where 0 at r r % Rr/rct;RsR*R; A4*D*/R/rcts;B1-*D/(rc*K);4*D/(V*rc*rc); te(rs*lg(rs)-a)/(rs-1); (B-te)/; i[ ]; % % Find critical radius at psitin i % Rcri[r ]; fr i:10 i(i);eb-*; % Use previus value as the first guess xrs; fr k1:10 fxx*lg(x)-a-e*(x-1); dflg(x)+1-e; exfx/df;xx-ex; if abs(ex/x)<1e-5,break, end end Rsx;Rcri(i)rct*sqrt(x); end rctrc+t;rctsrct*rct; rr[r Rcri];cal[0 i]; fr i1: cal(i); 1R*(r^-rcts)*/(V*rc^); R*(r^-rcts)/(*rc*K); (i)-1; w(i)(i)-; rr; 3R*(r*r-rcts)/(4*D); 4R*(r^).*lg(r/rct)/(*D); (i)w(i)+3-4; end fr i3:11 cal(i);rrr(i); 1R*(r^-rcts)*/(V*rc^); R*(r^-rcts)/(*rc*K); (i)-1; w(i)(i)-; (i)0; end subplt(,,3);plt(cal,,cal,w,':',cal,,'-.') grid n title('v.005 c/s, D8e-6 c/s') xlabel('(c)');ylabel('(micr-l/c3)') legend('_','_w','_') subplt(,,4);plt(i,rcri) grid n title('ritical Radius calculatin') xlabel('(c)');ylabel('r_c_r_i_t_i(c)') 1-63

4 Evaluatin f the Overall Mass ransfer efficient K Neglecting the curvature f the capillary wall, [t /r c/( c) 0.1], the transprt rate f slute acrss the capillary can be written as N πr k ( ) πr P ( r r + ) πr K ( r c t r c + ) t apillary wall r +t r r r+t apillary Figure Radial cncentratin prfile f slute. In these expressins, k is the bld side fil ass transfer cefficient and P is the slute pereability acrss the capillary wall. We can rearrange the abve equatins N πr k ( r ) N πr r k1 N πr P ( N πr K ( r + ) r c t r c + ) t N πr P N πr K 1 r r c + t 1 r c + t Adding the first tw expressins, we btain the third expressin N πr k P r c + t N πr 1 K herefre + where,, and are the ttal ass transfer resistance f K k1 P K k1 P slute fr the capillary t the utside surface f the capillary, the ass transfer resistance in the bld side, and the ass transfer resistance thrugh the capillary wall, respectively. 1-64

5 1.9 iplified lute ransprt Mdel r c+t apillary wall 0 r c Figure A siplified del fr slute transprt in the capillary. We will cnsider a siplified del fr slute transprt in the capillary. his is the special case where the slute cncentratin in the tissue space is er and the resistance t the slute in the bld is negligible s that K P. he steady state slute balance n the cntrl vlue πr gives Vπr Vπr + πr K ( 0) (1.9-1) where K is the verall ass transfer cefficient between the slute in the capillary and the slute in the tissue space next t the capillary wall, and is the slute cncentratin in the capillary. he slute cncentratin in the surrunding tissue space is er. he verall ass transfer cefficient is related t the fil ass transfer cefficient and the pereability by the expressin K k1 P 1 1 where,, and are the ttal ass transfer resistance f slute fr the capillary t K k1 P the utside surface f the capillary, the ass transfer resistance in the bld side, and the 1 1 ass transfer resistance thrugh the capillary wall, respectively. ince << k P 1 1 K P P K Dividing equatin (1.9-1) by πr and taking the liit as 0, we btain 1-65

6 d V d r P (1.9-) d πr d Vπr P πr P (1.9-3) We have replaced the capillary bld velcity V with the vluetric flw rate. Equatin (1.9-3) can be integrated ( ) O d πr P d 0 ln ( ) πr P ( ) πr exp P (1.9-4) he fractin f slute extracted r supplied t the surrunding tissue is given by E 1 ( ) πr 1 exp P (1.9-5) ince is the aunt f slute entering the capillary, it represents the axiu aunt f slute that can be transprted acrss the capillary wall. E is the actual slute transprt rate that depends n the rati f the pereability f the capillary wall (πr P ) t the bld flw rate (). he rate f slute transprted acrss the capillary wall is flw liited if πr P πr P >> 1 and is diffusin liited if << 1. We shuld recgnie that (πr P ) has the sae unit as and (πr ) is the inside surface area f ne capillary. Fr regins f tissue cntaining any capillaries, we can replace the quantity (πr P ) by (P ), where represents the ttal surface area f the capillaries within the tissue regin f interest and represents the ttal bld flw t these regins. he grup (P ) ay be deterined fr large tissue regins using the ultiple tracer indicatr diffusin technique. A slutin cntaining equal cncentratins f pereable test slute and a nnpereable reference slute (typically labeled albuin), is rapidly injected int a ain artery leading t the regin f interest. he cncentratins f the test and reference slute are then easured in the bld saple taken fr the vein leaving the tissue regin. he cncentratin f the test slute in the venus bld will be less than that f the reference slute due t the diffusin f the test slute int the surrunding interstitiu as shwn in Figure 1.9-a. After the reference slute has been washed ut f the capillaries, its cncentratin in the venus bld will fall belw that f the test slute due t the back diffusin f test slute int the bld strea as shwn in Figure 1.9-b. 1-66

7 Pereable test slute Nnpereable reference slute Artery apillary Vein Figure 1.9-a Diffusin f test slute fr capillary t interstitiu. Pereable test slute Nnpereable reference slute Artery apillary Vein Figure 1.9-b Diffusin f test slute fr interstitiu t capillary. able lists a typical test and reference slutes cncentratin in the venus bld with a knwn bld vluetric flw rate. he slute cncentratins are in % dse/0. c 3 f venus saple fr varius perids f tie fllwing their injectin int the artery. 1-67

8 able Data fr the deterinatin f (P ) ies (secnds) est slute Reference slute E ince the ultiple tracer indicatr diffusin technique is a transient test while the equatin E 1 ( ) P 1 exp (1.9-5) is a steady state del. herefre, we need t lk fr a perid f tie ver which the fractin f slute extractin E is a cnstant as shwn with the bld values in able A tepral plt f E can als indicate a perid f steady state diffusin f slute thrugh the capillaries as shwn in Figure Fr the data E 0.39, P can then be slved fr equatin (1.9-5) P ln(1 E) P If the slute transprt is flw liited, P >> 1, we have E 1 and exp 0. his cnditin will prduce large errr fr P.. herefre the test shuld be cnducted at high bld flw rate () s that E < 0.5 t insure that the slute transprt is nt flw liited. he ter clearance is defined by the fllwing expressin L() E Under flw liited cnditins, dependent n the flw rate. P >> 1, E 1 we have L(). he clearance is linearly P Under diffusin liited cnditins, P << 1, E 1 exp P 1 P 1 We have L() E P. he clearance is independent f the flw rate and is dependent n the capillary wall pereability. 1-68

9 E 1-/ t(sec) Figure he perid fr 0.5 t 1.0 sec can be used t evaluate E A parisn f nvectin and Diffusin Effects he Peclet (Pe) nuber represents the rati f axial cnvectin t axial diffusin. When Pe >> 1, axial diffusin can be neglected in cparisn t axial cnvectin. We will use the values given in able t cpare the effects f cnvectin and diffusin fr the transprt within the capillary. We als cpare the effect f radial and axial diffusin within the tissue space. Prperties Inside diaeter (D c ) Length (L) Average bld velcity (V) Krgh tissue cylinder radius (r ) Plasa slute diffusivity (D ) able apillary characteristics Value c 0.1 c 0.05 c/sec c c /sec 1-69

10 r D L r O V D D ransprt Within the apillary Figure Paraeters used fr transprt cparisn. lute transprt by axial cnvectin 4 π D V π D lute transprt by axial diffusin D 4 L Pe cnvectin diffusin transprt transprt VL (0.05)(0.1) D 550 >> 1 Axial diffusin f the slute within the capillary can be neglected in cparisn t axial cnvectin f slute. ransprt Within the issue pace lute transprt by radial diffusin πr L D r r lute transprt by axial diffusin π(r r ) D L radial axial diffusin diffusin ( r r L r )( r r ) ( (0.001)(0.1 ) )[(4 10 ) ( ) ] radial axial diffusin diffusin 181 >> 1 Axial diffusin f the slute within the tissue space can be neglected in cparisn t radial diffusin f slute. 1-70

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