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Animal-to-human extrapolation using compartmental models
Neur05c1ence• 8106ehm~10ra1Rev1ew5.V01. 15. pp. 57-61. Per9am0nPre55p1c. 1991. Pr1nted1n the U.5.A.
0149-7634/91 $3.00 + .00
An1ma1-t0-Human Extrap01at10n U51n9 C0mpartmenta1 M0de15 R 1 C H A R D L. M E D 1 N A A N D R 1 C H A R D A. A L 8 A N E 5 E
Un1ted 5tate5 A1r F0rce 5ch001 0 f Aer05pace Med1c1ne 8r00k5 AF8, 5an Ant0n10, 7X 78235-5301
MED1NA, R. L. AND R. A. AL8ANE5E. An1ma1-t0-humanextrap01at10n u51n9c0mpartmenta1 nr0de15. NEUR05C1 8108EHAV REV 15(1) 57-61, 1991.--We have 6een 5tudy1n9 h0w t0 u5e c0mpartmenta1 m0de15 t0 re11a61y extrap01ate t0x1c01091ca1 exper1menta1 re5u1t5 fr0m 0ne an1ma1 5pec1e5 t0 an0ther w1th the u1t1mate 90a1 0f u5efu1 extrap01at10n t0 man. We have taken, a5 the fundamenta1 c0re 0f 0ur ana1y515, the phy51ca11y nece55ary e4uat10n5 de5cr161n9 ma55 6a1ance. 7he5e e4uat10n5 are the c1a551ca1 F1ck1an e4uat10n5 0f the f0rm VdC/dt = Q(a - v). 7he5e e4uat10n5 are n0t a c0mp1ete 5et 51nce they are 1n5uff1c1entt0 pr0v1de e5t1mat10n 0f ven0u5, arter1a1. 0r t155ue c0ncentrat10n5 w1th t1me. 7heref0re, the ma55 6a1ance e4uat10n5 mu5t 6e au9mented w1th add1t10na1 phen0men01091ca1 re1at10n5 t0 perm1t the de51red ca1cu1at10n5. Many 1nve5t19at0r5 u5e the ven0u5 ex1t c0nd1t10n. Wh11e 1t 15 c1ear h0w t0 extrap01ate the ma55 6a1ance e4uat10n5 fr0m 0ne 512ed an1ma1 t0 an0ther 0r fr0m 0ne 5pec1e5 t0 an0ther, 1t 15 n0t c1ear h0w t0 extrap01ate the ven0u5 ex1t c0nd1t10n. 1n th15 re5earch we have eva1uated the ven0u5 ex1t c0nd1t10n 6y c0mpar1n9 1t w1th appr0x1mate ana1y5e5 0f perfu510n and 5u65tance d1ffu510n 1n the t155ue. F1ck1an e4uat10n Ven0u5ex1t c0nd1t10n Extrap01at10n Part1a1 d1fferent1a1 e4uat10n
C0mpartmenta1m0de11n9
D1ffu510n m0de1
Ma55 6a1ance
V, dC,/dt = Q,(a - v.)
7H15 rep0rt de5cr16e5 re5earch the auth0r5 have pur5ued c0ncern1n9 c0mpartmenta1 m0de11n9 1n t0x1c0109y w1th part1cu1ar re9ard t0 the pr061em 0f an1ma1-t0-man 5ca1e up. 5evera1 6enef1t5 w0u1d accrue 1f med1ca1 5c1ent15t5 c0u1d re11a61y extrap01ate 6101091ca1 effect5 f1nd1n95 fr0m 0ne 512ed an1ma1 t0 an0ther, 0r fr0m 0ne 5pec1e5 t0 an0ther. F0r examp1e, 1ma91ne that, 1n m1ce, a chem1ca1 adm1n15tered f0r 3 m0nth5 at a da11y d05e D re5u1t5 1n an 1ncrement 1n cancer C, where C = f(D), when the pre5ence 0r a65ence 0f cancer 15 determ1ned 6y aut0p5y perf0rmed when the m1ce reached 2 year5 0f a9e. 61ven the data 6a5ed e4uat10n, C = f(DL f0r m1ce, what re1at10n 6etween cancer and exp05ure m19ht ex15t f0r man•• 7he 4ue5t10n c0ncern1n9 man w0u1d u5ua11y 90 6ey0nd a re91mented da11y d05e D 91ven 0n1y f0r a 5tandard 3m0nth per10d, and c0ncern f0r cancer w0u1d extend f0r the ent1re human 11fe 5pan. 7h15 art1c1e addre55e5 the 1ntr0duced c1a55 0f pr061em5. 7he appr0ach app11ed t0 th15 pr061em 15 c0mpartmenta1 m0de11n9. We 6e11eve the pr05pect5 f0r th15 endeav0r can 6e fav0ra61e a5 15 a1ready 1nd1cated 1n. the pharmac0k1net1c5 11terature. 1n th15 art1c1e, we w111 addre55 the 9enera1 meth0d0109y 0f extrap01at10n u51n9 c0mpartmenta1 m0de11n9, and w111 n0t 6e treat1n9 a 5pec1f1c app11cat10n.
V,~d dCad/dt = Q,d(a -
vad )
Va65 dCa65/dt = Qa65(a -- va65) + ka65(X 5 -- Ca65)
V¢xc dCc~¢/dt = Q¢~¢(a - v ~ )
-
hC¢xc
Qv = Q , v , + Qad vad + Q,6.~ v,65 + Qcxc vcxc (EQUA710N 1) 7he f1ve 5ym6015, Vj, repre5ent c0mpartment v01ume5 and the f1ve 5ym6015, Q r are c0mpartment f10w5. 7he 5ym6015 Cj refer t0 5u65tance c0ncentrat10n 1n each 0r9an. 7he f0110w1n9 5u65Cr1pt C0nvent10n 15 u5ed: L de519nate5 the 1un9, n de519nate5 n0nad1p05e t155ue, ad de519nate5 ad1p05e t155ue, a65 de519nate5 the a650r61n9 0r9an (f0r examp1e, the 1nte5t1ne 0r 1nte9ument), and exc de519nate5 the excret0ry 0r9an (f0r examp1e, the k1dney). 7he 5ym601 a refer5 t0 the arter1a1 c0ncentrat10n 0f 5u65tance, v refer5 t0 the c0m61ned ven0u5 c0ncentrat10n and the 5ym6015 vj refer t0 c0ncentrat10n5 0f mater1a1 1n the ven0u5 0utf10w 0f the 1nd1v1dua1 0r9an5 (exc1ud1n9 the 1un9). 7he 5ym601 ×~ c0rre5p0nd5 t0 an externa1 5u65tance c0ncentrat10n 1mp1n91n9 0n the a650r61n9 5urface. 7he5e e4uat10n5 are a11 a d1rect ref1ect10n 0f the pr1nc1p1e 0f c0n5ervat10n 0f matter (3). Except f0r the k~65(×,~ - C,6,~) and hCc, ¢ term5, wh1ch are n0t nece55ar11y app11ca61e, the5e e4uat10n5 mu5t 6e true 1n any c0mpartmenta1 5y5tem 6ecau5e they are 6a5ed 0n phy51ca1 pr1nc1p1e5. 1n the5e e4uat10n5 the Vj, Qj, ka6~ and h term5 are c0n51dered kn0wn, 6ut the Cj, a, v and vj term5 are n0t. 7hu5, there are e1even unkn0wn var1a61e5 and 0n1y 51x e4uat10n5.7heref0re, the 4ue5t10n ar15e5 0f h0w t0 c0mp1ete the5e e4uat10n5, 0r c105e the 5et, 50 the Cj, v and vj va1ue5 can 6e ca1-
7w0 PLAu518LE APPR0ACHE570 c0MPAR7MEN7AL M0DEL1N6 F0r purp05e5 0f d15cu5510n we w111 u5e the 610ck d1a9ram 5h0wn 1n F19. 1. F19ure 1 repre5ent5 a 51mp11f1ed ver510n 0f Ram5ey and Ander5en•5 P8-PK m0de1, wh1ch de5cr16ed the pharmac0k1net1c5 0f 5tyrene 1n rat5 and human5 dur1n9 and after 1nha1at10n exp05ure (9). 1n the 5ett1n9 0f th15 610ck m0de1, the f0110w1n9 ma55 6a1ance e4uat10n5 are app11ca61e: V L dCL/dt = QL(v -- a) 57
58
MED1NA AND A L 8 A N E 5 E f
7 r"
1
LUN6
CL
•
1 . 5 - . . . . , 5 ......5 ........... ~
........... ~
..........5 -
11
.................
X=0
X=L
N0N-AD1P05E F16. 2. 7he pr0p05ed perfu510n-d1ffu510n m0de1 5h0w1n9 5u65tance d1ffu510n 1nt0 a t155ue 5e9ment 5upp11ed 6y a centra1 6100d ve55e1.
AD1P05E
C2
1--
Vad dCad/dt = -k~d [Ca0 - (a + v~0)/2] Va65 dCa65/dt = -ka65 [Ca65 - (a + va65)/2] + ka6~ (×5 - Ca6=)
A850R81N6
C~6~
Vcx¢ dCex¢1dt = -1%~c [Cexc - (a + vcx¢)/2] - hC~x¢ •
EXCRE71N6
Qx,
, 1 1 1
~
•
Y /~ 5
V F16. 1. A f1ve c0mpartment 610ck d1a9ram 0f a phy5101091ca1 pharmac0k1net1c m0de1 5tud1ed 1n th15 paper.
CU1ated 91Ven Vj, Q.1, ka65 and h. 0ne Way t0 C0mp1ete the ma55 6a1anCe e4Uat10n 5et 15 thr0U9h U5e 0f the ven0u5 eX1t C0nd1t10n (8). 7h15 C0nd1t10n a55ert5 that 6100d f10W1n9 0Ut 0f an 0r9an a1Way5 ha5 a 5U65tanCe c0ncentrat10n pr0p0rt10na1 t0 the c0ncentrat10n 0f the 5u65tance 1n the 0r9an. 7he f0110w1n9 ven0u5 ex1t c0nd1t10n e4uat10n5 can 6e wr1tten t0 c0mp1ete the ma55 6a1ance 5et. a
=
0t L
CL
v n = 0tn Cn Vad Va65
0tad Cad
=
=
0ta65
Ca65
v~x¢ = 0t,xcC¢xc (EQUA710N 2) 7he c0eff1c1ent5 e4 may 6e th0u9ht 0f a5 51mp1y c0n5tant5 0f pr0p0rt10na11ty 0r may 6e th0u9ht 0f, m0re 5pec1f1ca11y, a5 1nver5e1y pr0p0rt10na1 t0 the part1t10n c0eff1c1ent5 (9). 7he ven0u5 ex1t c0nd1t10n 15 a 51mp1e a19e6ra1c expre5510n wh1ch may 0r may n0t h01d true 1n any 91ven ca5e. 7h15 c0nd1t10n 15 an attempt t0 appr0x1mate a c0mp1ex perfu510n-d1ffu510n 5y5tem actua11y 90verned 6y part1a1 d1fferent1a1 e4uat10n5 (8). 1t 15 p055161e t0 c0mp1ete 0r c105e the ma55 6a1ance e4uat10n5 1n a 5ec0nd way. Rec09n121n9 that tran5p0rt 1nt0 an 0r9an c0mpartment 0ccur5 thr0u9h a d1ffu510n pr0ce55, appr0x1mate d1ffu510n e4uat10n5 may 6e wr1tten (6,7). We have u5ed the f0110w1n9 d1ffu510n appr0x1mat10n c105ure e4uat10n5: V L dCL/dt = --k L [C L -- (a + v)/2] V. dC,/dt = - k n
[C n
--
(a + v,)/2]
(EQUA710N 3) 7hu5, we n0w have tw0 c0mp1ete c0mpartmenta1 m0de15 c0rre5p0nd1n9 t0 F19. 1. A part 0f each m0de1 15 phy51ca11y exact, 6e1n9 6a5ed 0n c0n5ervat10n 0f matter. H0wever, each m0de1 a150 c0nta1n5 mathemat1ca1 expre5510n5 that d0 n0t exact1y ref1ect phy51ca1 1aw5, 6ut are appr0x1mat10n5. 5pec1f1ca11y, the appr0x1mat10n5 1n the m0de15 u5e a19e6ra1c 0r 0rd1nary d1fferent1a1 e4uat10n5 when the perfu510n-d1ffu510n phen0mena 1n t155ue 15 actua11y 90verned 6y part1a1 d1fferent1a1 e4uat10n5 (8). 7w0 4ue5t10n5 1mmed1ate1y ar15e. F1r5t, wh1ch 0f the tw0-c0mpartmenta1 m0de15 w111 6e5t de5cr16e 5u65tance pharmac0k1net1c5 1n a 11v1n9 an1ma1• 5ec0nd, wh1ch 0f the5e tw0 m0de15 15 m05t accurate when u5ed t0 extrap01ate fr0m a 5ma11er t0 a 1ar9er an1ma1• 7he 6e5t way t0 an5wer the5e tw0 4ue5t10n5 15 thr0u9h the u5e 0f exten51ve data 06ta1ned fr0m d1fferent 512ed an1ma15, f0r examp1e, rat5 and human5. 51nce the 1ar9e data 5et5 needed were n0t ava11a61e t0 u5 we have pur5ued an a1ternat1ve ana1y515 0f the c0mpet1n9 c0mpartmenta1 m0de15. We have c0mpared the ven0u5 ex1t and d1ffu510n appr0x1mat10n m0de15 t0 a 51mp1e perfu510nd1ffu510n 5y5tem 90verned 6y part1a1 d1fferent1a1 e4uat10n5 f0r wh1ch ana1yt1c mathemat1ca1 501ut10n5 ex15t. 7h15 te5t perfu510nd1ffu510n 5y5tem exh161t5, 4ua11tat1ve1y at 1ea5t, much 0f the actua1 5u65tance m0vement 6ehav10r 0f t155ue. We have determ1ned h0w we11 the c0mpartmenta1 m0de15 appr0x1mate th15 exact perfu510n-d1ffu510n 5y5tem, and we have 5tud1ed h0w we11 the c0mpartmenta1 m0de15 f0110w a 5ca1e-up 0f the exact 5y5tem. AN
EXAC7
PERFU510N-D1FFU510N
5Y57EM
We pr0p05e t0 exam1ne and deve10p an exact 501ut10n 0f the 5an9ren and 5heppard m0de1 0f t155ue c1earance (4,10). We c0n51der a Kr09h cy11nder (4, 10, 11) 0f 1en9th L a5 d1a9rammed 1n F19. 2. 7he t155ue 5e9ment 15 5upp11ed 6y a centra1 6100d ve55e1 w1th v01ume •,/and the c0ncentrat10n 0f the 5u65tance 1n the 6100d 8(x,t), var1e5 a5 a funct10n 0f 11near p051t10n and t1me. Let C(x,t) 6e the c0ncentrat10n 1n the t155ue 1t5e1f. A55um1n9 51mp1e d1ffu510n acr055 the ve55e1 wa11 and n0 ax1a1 0r rad1a1 d1ffu510n 1n the t155ue, the f0110w1n9 e4uat10n5 app1y: ~/c~8(x,t)/0t = k[C(x,t) -
8(x,t)] - -y1x08(x,t)/0x
V0C(x,t)/0t = -k[C(x,t) - 8(x,t)] (EQUA710N 4) 1n the5e e4uat10n5, 8 = 8(x,t) 15 the c0ncentrat10n 0f 5u65tance
AN1MAL-70-HUMAN EX7RAP0LA710N
59
{ [61(~2 "•{- ~3) "~ 66 (~4 + 65)] 51n(¢0jt)
8(0,t)
-~
H
+ [61 ( - - ~ A.
H12
.
.
.
.
.
D
~
t
•
2D-H/2
,,t
2D
F16. 3. 7he 5u65tance temp0ra1 pr0f11e 1n the 6100d enter1n9 the Kr09h cy11nder.
-- 65) + ~6 (62 + 63)] C05(t0jt)}
where ~, = t0j/[% 2 + (k/V) 2] ~J2 = [1/(°tj 2 + (¢°j[3j)2)] [ - a j c05(t0j[3jL) + (mj[3j)51n(t0j13jL)]exp( - ctjL)
63 = %/[%2 + (%13j)2] 1n the 6100d, C = C(x,t) 15 the c0ncentrat10n 0f 5u65tance 1n the t155ue, "y 15 the v01ume 0f 6100d 1n the t155ue, 1x 15 the ve10c1ty 0f 6100d f10w 1n the t155ue, V 15 the t155ue v01ume, and k 15 the permea6111ty c0n5tant f0r the ve55e1 wa1f. We have taken 8(0,t) t0 6e the trape201da1 funct10n 0f t1me t 5h0wn 1n F19. 3. 7h15 funct10n ha5 the f0110w1n9 F0ur1er 5er1e5 repre5entat10n (1):
~.. j=
~5 = (%13j)11%2 + (%[3j)=1 = (k/v)/[% 2 + (k/v)-•]
a~
:t:
8(0,t) = (1/2) - 4/[•rr2(1-2p)]
= [1/(aj 2 + (t~j13j)2)] [ - a j 51n(t0j[3jL) - (t0j13j) c05(%[3jL)]exp( - ajL)
=
%/p.
[3j = (11p.) {t0j2 + (k2/V) [(1/V) + (1•)]}/(% 2 + (k/V) 2)
.,
(11j 2) c05(jrrp) c05 (t0jt1 where t0j = j~r/D and p = H/2D. U51n9 th15 F0ur1er 5er1e5 repre5entat10n, e4uat10n 4 can 6e read11y 501ved t0 f1nd
1n the exact perfu510n-d1ffu510n m0de1, ave[C(t)] 15 the va1ue c0rre5p0nd1n9 t0 the Cj term5 1n the c0mpartmenta1 appr0x1mat10n5. A150 8(0,t) c0rre5p0nd5 t0 5u65tance c0ncentrat10n 1n 1n-f10w1n9 6100d, wh11e 8(L,t) c0rre5p0nd5 t0 the c0ncentrat10n 1n t155ue ex1t1n9 6100d.
8(x,t) = (1/2) - 4/[w2(1-2p)] j=
C0MPAR1N6 7HE EXAC7 PERFU510N-D1FFU510N 5Y57EM 70 7HE VEN0U5 EX17 AND APPR0X1MA7E D1FFU510N C0MPAR7MEN7AL M0DEL5
.,
(102) e x p ( - ~jx/p.) c05(j1rp) c05[~j(t-0j)]
F0r a 5pec1f1c c0mpartment, the ven0u5 ex1t m0de1 15
and
VdC/dt + Q c t C C(x,t) = (1/2) - 4(k/V)/[•rr2(1 - 2p)] j= ~
(1/j 2) exp ( - $ j {[0~J(t0j 2 +
(k/V)2)] 51n[t0j(t
v=CtC
..
x/p.) c05(j~rp). -
0j)] +
= Qa
[(k/V)1(t02+(k/V)2)]
c05[%(t- 0j)]} where
1n th15 m0de1, we take a = 8(0,t) and 501ve f0r C and v t0 6e c0mpared w1th ave[C(t)] and 8(L,t) re5pect1ve1y fr0m the exact perfu510n-d1ffu510n 5y5tem. 51m11ar1y, f0r a 5pec1f1c 51n91e c0mpartment, the appr0x1mate d1ffu510n m0de1 15 VdC/dt = Q(a - v)
~ j = %- (k/-,D/(~0j2 + (k/V) 2)
VdC/dt = - k [ C
0j = (x/p.){t0j- + (k2/V)[(11V) + (1/•)•)]}1(t0j 2 + (k/V):•) Mean t155ue 5u65tance c0ncentrat10n ave[C(t)] can 6e ca1cu1ated
- (a + v)/2]
7he5e e4uat10n5 are e4u1va1ent t0 VdC/dt + [2kQ1(2Q + k)1C = [2kQ/(2Q + k)1a
a5:
v = [2k/(2Q+ k)]C + [(2Q - k)/(2Q + k)1a
ave[C(t)] = (1/L) f~ C(x,t)dx whence
ave[C(t)] = (1/2) - 4(k/V)/[~2L(1-2p)]
~ j=
(1/j 2) c05(j•rrp).
..
A9a1n, we take a = 8(0,t) and 501ve f0r C and v t0 c0mpare w1th ave[C(t)] and 8(L,t) re5pect1ve1y fr0m the exact perfu510n-d1ffu510n 5y5tem. We have c0ntra5ted the perf0rmance 0f the ven0u5 ex1t c0mpartmenta1 m0de1 and the appr0x1mate d1ffu510n c0mpartment m0de1 t0 the exact perfu510n-d1ffu510n 5y5tem 6y treat1n9 rat and human 11ver. F0r the rat 11ver V = 12 cm 3 and Q = 35.2 cm3/
60
MED1NA A N D A L 8 A N E 5 E
/
v E N 0 u 5 Ex17 APPR0x. - - E x A c 7 P-0 M0DEL 1• - - A P P R 0 x .
V E N 0 U 5 EX17 APPR0X." - - EXAC7 P-0 M 0 D E L
/
D1FFu510N
/
--
A P P R 0 X . D1FFU510N
/
.1
/
.4
1
30
60
45
/
/
//
15
15
f 1
50
71ME(M1NU7E5)
71ME(M1NU7E5)
F16. 4. F1tted data 5h0w1n9 510w chan9e 1n rat ven0u5 6100d 1n a11 three m0de15. m1n. F0r the human 11ver V = 3320 cm 3 and Q = 1800 cm31m1n (9). We have u5ed V = 3.14 x 10 - 5 cm 3 and 3• = 7.85 x 10 - 8 cm 3 1n the exact perfu510n-d1ffu510n m0de1, a55um1n9, f0r te5t purp05e5 0n1y, that 60th rat and human 11ver are made up 0f cy11ndr1ca1 re910n5 1-mm 10n9, and 100 p,m 1n rad1u5 w1th a centra1 6100d ve55e1 hav1n9 a 5 1-tm rad1u5 (2, 4, 5, 1 1). F0r the exact m0de1, 6100d ve10c1ty wa5 ch05en t0 6e 1 17 cm/m1n 1n the rat 50 the exact m0de1 w0u1d have a Q/V rat10 0f 35.2/12, and 6100d ve10c1ty wa5 ch05en t0 6e 21.7 cm/m1n 1n the human t0 c0rre5p0nd t0 a Q/V rat10 0f 1800/3320. F19ure5 4 thr0u9h 7 5h0w the re5u1t5 0f 0ur numer1ca1 exper1mentat10n. 1n the5e f19ure5 6e5t f1t (~ and k va1ue5 were f0und 50 that the ven0u5 ex1t and appr0x1mate d1ffu510n c0mpartment5 6e5t m1m1cked the rat exact m0de1. Extrap01at1n9 the human 11ver 51tuat10n, (x wa5 1eft unchan9ed wh11e k wa5 extrap01ated u51n9 the re1at10n5h1p
[(2~..~ Q,~t)1(2Q,~. + k,at) V~,] = [(2kma. Qm~.)1
F16. 6. Extrap01at10n re5u1t5 f0r man re9ard1n9 the e5t1mat10n 0f ven0u5 c0ncentrat10n.
1n F19.4 we 5ee that a11 three m0de15 can 6e made t0 a9ree a5 re9ard5 a 510w chan9e 1n rat ven0u5 6100d. H0wever, 1n F19. 5 we 5ee that the 5ame parameter5 wh1ch 9ave a9reement 1n rat ven0u5 6100d d0 n0t w0rk a5 we11 f0r rat t155ue c0ncentrat10n5.1n part1cu1ar, the ven0u5 ex1t c0nd1t10n pr0v1de5 a p00rer 4ua11ty f1t t0 rat t155ue c0ncentrat10n5. 70 c0rre5p0nd t0 man, the exact perfu510n-d1ffu510n 5y5tem 6100d ve10c1ty wa5 chan9ed fr0m 117 cm/m1n t0 21.7 cm/m1n. 7he parameter e~ wa5 1eft unchan9ed f0r the ven0u5 ex1t m0de1 and k wa5 adju5ted a5 de5cr16ed 6ef0re. 1n F19.6 we 5ee that the ven0u5 ex1t m0de1 and d1ffu510n appr0x1mat10n m0de1 60th extrap01ate we11 t0 man a5 re9ard5 the e5t1mat10n 0f ven0u5 c0ncentrat10n5. H0wever, a5 5een 1n F19. 7, the ven0u5 ex1t c0nd1t10n a9a1n d0e5 n0t perf0rm we11 re9ard1n9 t155ue c0ncentrat10n5. 7he pattern5 f0und 1n F195. 4 thr0u9h 7 are amp11f1ed when fa5ter chan9e5 1n 6100d 1eve15 are 5tud1ed. 1n the5e 5ett1n95 a9a1n the d1ffu510n appr0x1mat10n m0de1 exce15 the ven0u5 ex1t m0de1.
(2Qm~. + km~.) Vm~.]
/,
VEN0U5 EX17 APPR0X. ~ - --EXAC7 P-0 M0DEL --
APPR0X. D1FFU510N
--
APPf10X. D1FFU510N
J~
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F16. 5. Rat t155ue c0ncentrat10n f0r a11 three m0de15. 5ame parameter5 u5ed wh1ch 9ave a9reement 1n rat ven0u5 6100d.
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60
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F16. 7. Extrap01at10n re5u1t5 f0r man re9ard1n9 the e5t1mat10n 0f t155ue c0ncentrat10n.
AN1MAL-70-HUMAN EX7RAP0LA710N
61
c0NcLu510N C0mpartmenta1 m0de15 have a c0re 5et 0f mathemat1ca1 re1at10n5 wh1ch are der1ved fr0m the phy51ca1 pr1nc1p1e 0f c0n5ervat10n 0f matter. 7h15 c0re 5et 0f e4uat10n5 15 n0t 5uff1c1ent f0r m0de11n9, 6ut mu5t 6e au9mented 6y aux111ary c0nd1t10n5. 7he ven0u5 ex1t c0nd1t10n 15 c0mm0n1y u5ed, and a d1ffu510n appr0x1mat10n a150 15 p055161e. 7he5e appr0x1mat10n5 u5e a19e6ra1c e4uat10n5 0r 0rd1nary d1fferent1a1 e4uat10n5 t0 de5cr16e perfu510nd1ffu510n phen0mena 1n t155ue 90verned 6y part1a1 d1fferent1a1 e4uat10n5. We c0mpared a ven0u5 ex1t m0de1 and a d1ffu510n appr0x1mat10n m0de1 t0 an exact perfu510n-d1ffu510n m0de1 0f t155ue 90verned 6y part1a1 d1fferent1a1 e4uat10n5. We have f0und
that the d1ffu510n appr0x1mat10n m0de1 perf0rm5 6etter than the ven0u5 ex1t m0de1 1n m1m1ck1n9 the perfu510n-d1ffu510n 5y5tem, part1cu1ar1y a5 re9ard5 the pred1ct10n 0f t155ue c0ncentrat10n5. We were 5urpr15ed t0 065erve that a m0de1 that f1t5 6100d c0ncentrat10n chan9e5 we11 may n0t f1t t155ue c0ncentrat10n event5. 0n1y m0re exper1ence w1th a var1ety 0f 5u65tance5 and rea1 w0r1d an1ma1-t0-human extrap01at10n eff0rt5 w111 pr0v1de 5ecure 9u1dance c0ncem1n9 ch01ce am0n9 c0mpet1n9 c0mpartmenta1 m0de15. 7h15 rep0rt 5u99e5t5 that c0mpar150n 0f c0mpet1n9 c0mpartmenta1 m0de15 t0 perfu510n-d1ffu510n m0de15 90vemed 6y part1a1 d1fferent1a1 e4uat10n5 may 6e an add1t10na1 1mp0rtant 9u1de t0 c0mpartmenta1 m0de1 5e1ect10n.
REFERENCE5 1. 8eyer, W. H., ed. CRC 5tandard mathemat1ca1 ta61e5. 28th ed. 80ca Rat0n, FL: CRC Pre55, 1nc.; 1987:405---406. 2. C0penhaver, W. M.; Ke11y, D. E.; W00d, R. L. 8a11ey•5 text600k 0f h15t0109y. 17th ed. 8a1t1m0re: 7he W1111am5• W11k1n5C0mpany; 1978. 3. Dedr1ck, R. L. An1ma1 5ca1e-up. J. Pharmac0k1net. 810pharm. 1: 435-461; 1973. 4. 6055e11n, R. P.; 6055e11n, R. E. 7he k1net1c5 0f 501ute c1earance 6y a 51n91e t155ue cap111ary 0f 11m1tedpermea6111ty: Exp11c1t 501ut10n50f tw0 m0de15 and the1r 60und5. 8u11. Math. 8101. 49:329-349; 1987. 5. Henne55y, 7. R. 7he 1nteract10n 0f d1ffu510n and perfu510n 1n h0m09ene0u5 t155ue. 8u11. Math. 8101. 36:505-526; 1974. 6. Jac4ue2, J. A. C0mpartmenta1 ana1y5151n 610109y and med1c1ne. 2nd ed. Ann Ar60r: 7he Un1ver51ty0f M1ch19an Pre55; 1985:11-13. 7. Landah1, H. D. An appr0x1mat10n meth0d f0r the 501ut10n 0f d1ffu510n and re1ated pr061em5. 8u11. Math. 810phy5. 15:49-61; 1953.
8. Per1, W. An exten510n 0f the d1ffu510n e4uat10n t0 1nc1ude c1earance 6y cap111ary 6100d f10w. 1n: Wh1pp1e, H. E., et a1., ed5. Mu1t1-c0mpartment ana1y5150f tracer exper1ment5. New Y0rk: New Y0rk Academy 0f 5c1ence5; 1972:92-105. (Ann. NY Acad. 5c1., v01. 108, art. 1.) 9. Ram5ey, J. C.; Ander5en, M. E. A phy5101091ca11y6a5ed de5cr1pt10n 0f the 1nha1at10n pharmac0k1net1c5 0f 5tyrene 1n rat5 and human5. 70x1c01. App1. Pharmac01. 73:159-175; 1984. 10. 5an9ren, W. C.; 5heppard, C. W. A mathemat1ca1 der1vat10n 0f the exchan9e 0f a 1a6e11ed 5u65tance 6etween a 114u1d f10w1n9 1n a ve55e1 and an externa1 c0mpartment. 8u11. Math. 810phy5. 15:387-394; 1953. 11. 7hew5, 6.; Hutten, H. 6a5 exchan9e 1n t155ue. 1n: H0ppe, W., et a1., ed5. 810phy51c5 0f re5p1rat0ry 9a5 tran5p0rt. New Y0rk: 5pr1n9er-Ver1a9; 1983.
0149-7634/91 $3.00 + .00
An1ma1-t0-Human Extrap01at10n U51n9 C0mpartmenta1 M0de15 R 1 C H A R D L. M E D 1 N A A N D R 1 C H A R D A. A L 8 A N E 5 E
Un1ted 5tate5 A1r F0rce 5ch001 0 f Aer05pace Med1c1ne 8r00k5 AF8, 5an Ant0n10, 7X 78235-5301
MED1NA, R. L. AND R. A. AL8ANE5E. An1ma1-t0-humanextrap01at10n u51n9c0mpartmenta1 nr0de15. NEUR05C1 8108EHAV REV 15(1) 57-61, 1991.--We have 6een 5tudy1n9 h0w t0 u5e c0mpartmenta1 m0de15 t0 re11a61y extrap01ate t0x1c01091ca1 exper1menta1 re5u1t5 fr0m 0ne an1ma1 5pec1e5 t0 an0ther w1th the u1t1mate 90a1 0f u5efu1 extrap01at10n t0 man. We have taken, a5 the fundamenta1 c0re 0f 0ur ana1y515, the phy51ca11y nece55ary e4uat10n5 de5cr161n9 ma55 6a1ance. 7he5e e4uat10n5 are the c1a551ca1 F1ck1an e4uat10n5 0f the f0rm VdC/dt = Q(a - v). 7he5e e4uat10n5 are n0t a c0mp1ete 5et 51nce they are 1n5uff1c1entt0 pr0v1de e5t1mat10n 0f ven0u5, arter1a1. 0r t155ue c0ncentrat10n5 w1th t1me. 7heref0re, the ma55 6a1ance e4uat10n5 mu5t 6e au9mented w1th add1t10na1 phen0men01091ca1 re1at10n5 t0 perm1t the de51red ca1cu1at10n5. Many 1nve5t19at0r5 u5e the ven0u5 ex1t c0nd1t10n. Wh11e 1t 15 c1ear h0w t0 extrap01ate the ma55 6a1ance e4uat10n5 fr0m 0ne 512ed an1ma1 t0 an0ther 0r fr0m 0ne 5pec1e5 t0 an0ther, 1t 15 n0t c1ear h0w t0 extrap01ate the ven0u5 ex1t c0nd1t10n. 1n th15 re5earch we have eva1uated the ven0u5 ex1t c0nd1t10n 6y c0mpar1n9 1t w1th appr0x1mate ana1y5e5 0f perfu510n and 5u65tance d1ffu510n 1n the t155ue. F1ck1an e4uat10n Ven0u5ex1t c0nd1t10n Extrap01at10n Part1a1 d1fferent1a1 e4uat10n
C0mpartmenta1m0de11n9
D1ffu510n m0de1
Ma55 6a1ance
V, dC,/dt = Q,(a - v.)
7H15 rep0rt de5cr16e5 re5earch the auth0r5 have pur5ued c0ncern1n9 c0mpartmenta1 m0de11n9 1n t0x1c0109y w1th part1cu1ar re9ard t0 the pr061em 0f an1ma1-t0-man 5ca1e up. 5evera1 6enef1t5 w0u1d accrue 1f med1ca1 5c1ent15t5 c0u1d re11a61y extrap01ate 6101091ca1 effect5 f1nd1n95 fr0m 0ne 512ed an1ma1 t0 an0ther, 0r fr0m 0ne 5pec1e5 t0 an0ther. F0r examp1e, 1ma91ne that, 1n m1ce, a chem1ca1 adm1n15tered f0r 3 m0nth5 at a da11y d05e D re5u1t5 1n an 1ncrement 1n cancer C, where C = f(D), when the pre5ence 0r a65ence 0f cancer 15 determ1ned 6y aut0p5y perf0rmed when the m1ce reached 2 year5 0f a9e. 61ven the data 6a5ed e4uat10n, C = f(DL f0r m1ce, what re1at10n 6etween cancer and exp05ure m19ht ex15t f0r man•• 7he 4ue5t10n c0ncern1n9 man w0u1d u5ua11y 90 6ey0nd a re91mented da11y d05e D 91ven 0n1y f0r a 5tandard 3m0nth per10d, and c0ncern f0r cancer w0u1d extend f0r the ent1re human 11fe 5pan. 7h15 art1c1e addre55e5 the 1ntr0duced c1a55 0f pr061em5. 7he appr0ach app11ed t0 th15 pr061em 15 c0mpartmenta1 m0de11n9. We 6e11eve the pr05pect5 f0r th15 endeav0r can 6e fav0ra61e a5 15 a1ready 1nd1cated 1n. the pharmac0k1net1c5 11terature. 1n th15 art1c1e, we w111 addre55 the 9enera1 meth0d0109y 0f extrap01at10n u51n9 c0mpartmenta1 m0de11n9, and w111 n0t 6e treat1n9 a 5pec1f1c app11cat10n.
V,~d dCad/dt = Q,d(a -
vad )
Va65 dCa65/dt = Qa65(a -- va65) + ka65(X 5 -- Ca65)
V¢xc dCc~¢/dt = Q¢~¢(a - v ~ )
-
hC¢xc
Qv = Q , v , + Qad vad + Q,6.~ v,65 + Qcxc vcxc (EQUA710N 1) 7he f1ve 5ym6015, Vj, repre5ent c0mpartment v01ume5 and the f1ve 5ym6015, Q r are c0mpartment f10w5. 7he 5ym6015 Cj refer t0 5u65tance c0ncentrat10n 1n each 0r9an. 7he f0110w1n9 5u65Cr1pt C0nvent10n 15 u5ed: L de519nate5 the 1un9, n de519nate5 n0nad1p05e t155ue, ad de519nate5 ad1p05e t155ue, a65 de519nate5 the a650r61n9 0r9an (f0r examp1e, the 1nte5t1ne 0r 1nte9ument), and exc de519nate5 the excret0ry 0r9an (f0r examp1e, the k1dney). 7he 5ym601 a refer5 t0 the arter1a1 c0ncentrat10n 0f 5u65tance, v refer5 t0 the c0m61ned ven0u5 c0ncentrat10n and the 5ym6015 vj refer t0 c0ncentrat10n5 0f mater1a1 1n the ven0u5 0utf10w 0f the 1nd1v1dua1 0r9an5 (exc1ud1n9 the 1un9). 7he 5ym601 ×~ c0rre5p0nd5 t0 an externa1 5u65tance c0ncentrat10n 1mp1n91n9 0n the a650r61n9 5urface. 7he5e e4uat10n5 are a11 a d1rect ref1ect10n 0f the pr1nc1p1e 0f c0n5ervat10n 0f matter (3). Except f0r the k~65(×,~ - C,6,~) and hCc, ¢ term5, wh1ch are n0t nece55ar11y app11ca61e, the5e e4uat10n5 mu5t 6e true 1n any c0mpartmenta1 5y5tem 6ecau5e they are 6a5ed 0n phy51ca1 pr1nc1p1e5. 1n the5e e4uat10n5 the Vj, Qj, ka6~ and h term5 are c0n51dered kn0wn, 6ut the Cj, a, v and vj term5 are n0t. 7hu5, there are e1even unkn0wn var1a61e5 and 0n1y 51x e4uat10n5.7heref0re, the 4ue5t10n ar15e5 0f h0w t0 c0mp1ete the5e e4uat10n5, 0r c105e the 5et, 50 the Cj, v and vj va1ue5 can 6e ca1-
7w0 PLAu518LE APPR0ACHE570 c0MPAR7MEN7AL M0DEL1N6 F0r purp05e5 0f d15cu5510n we w111 u5e the 610ck d1a9ram 5h0wn 1n F19. 1. F19ure 1 repre5ent5 a 51mp11f1ed ver510n 0f Ram5ey and Ander5en•5 P8-PK m0de1, wh1ch de5cr16ed the pharmac0k1net1c5 0f 5tyrene 1n rat5 and human5 dur1n9 and after 1nha1at10n exp05ure (9). 1n the 5ett1n9 0f th15 610ck m0de1, the f0110w1n9 ma55 6a1ance e4uat10n5 are app11ca61e: V L dCL/dt = QL(v -- a) 57
58
MED1NA AND A L 8 A N E 5 E f
7 r"
1
LUN6
CL
•
1 . 5 - . . . . , 5 ......5 ........... ~
........... ~
..........5 -
11
.................
X=0
X=L
N0N-AD1P05E F16. 2. 7he pr0p05ed perfu510n-d1ffu510n m0de1 5h0w1n9 5u65tance d1ffu510n 1nt0 a t155ue 5e9ment 5upp11ed 6y a centra1 6100d ve55e1.
AD1P05E
C2
1--
Vad dCad/dt = -k~d [Ca0 - (a + v~0)/2] Va65 dCa65/dt = -ka65 [Ca65 - (a + va65)/2] + ka6~ (×5 - Ca6=)
A850R81N6
C~6~
Vcx¢ dCex¢1dt = -1%~c [Cexc - (a + vcx¢)/2] - hC~x¢ •
EXCRE71N6
Qx,
, 1 1 1
~
•
Y /~ 5
V F16. 1. A f1ve c0mpartment 610ck d1a9ram 0f a phy5101091ca1 pharmac0k1net1c m0de1 5tud1ed 1n th15 paper.
CU1ated 91Ven Vj, Q.1, ka65 and h. 0ne Way t0 C0mp1ete the ma55 6a1anCe e4Uat10n 5et 15 thr0U9h U5e 0f the ven0u5 eX1t C0nd1t10n (8). 7h15 C0nd1t10n a55ert5 that 6100d f10W1n9 0Ut 0f an 0r9an a1Way5 ha5 a 5U65tanCe c0ncentrat10n pr0p0rt10na1 t0 the c0ncentrat10n 0f the 5u65tance 1n the 0r9an. 7he f0110w1n9 ven0u5 ex1t c0nd1t10n e4uat10n5 can 6e wr1tten t0 c0mp1ete the ma55 6a1ance 5et. a
=
0t L
CL
v n = 0tn Cn Vad Va65
0tad Cad
=
=
0ta65
Ca65
v~x¢ = 0t,xcC¢xc (EQUA710N 2) 7he c0eff1c1ent5 e4 may 6e th0u9ht 0f a5 51mp1y c0n5tant5 0f pr0p0rt10na11ty 0r may 6e th0u9ht 0f, m0re 5pec1f1ca11y, a5 1nver5e1y pr0p0rt10na1 t0 the part1t10n c0eff1c1ent5 (9). 7he ven0u5 ex1t c0nd1t10n 15 a 51mp1e a19e6ra1c expre5510n wh1ch may 0r may n0t h01d true 1n any 91ven ca5e. 7h15 c0nd1t10n 15 an attempt t0 appr0x1mate a c0mp1ex perfu510n-d1ffu510n 5y5tem actua11y 90verned 6y part1a1 d1fferent1a1 e4uat10n5 (8). 1t 15 p055161e t0 c0mp1ete 0r c105e the ma55 6a1ance e4uat10n5 1n a 5ec0nd way. Rec09n121n9 that tran5p0rt 1nt0 an 0r9an c0mpartment 0ccur5 thr0u9h a d1ffu510n pr0ce55, appr0x1mate d1ffu510n e4uat10n5 may 6e wr1tten (6,7). We have u5ed the f0110w1n9 d1ffu510n appr0x1mat10n c105ure e4uat10n5: V L dCL/dt = --k L [C L -- (a + v)/2] V. dC,/dt = - k n
[C n
--
(a + v,)/2]
(EQUA710N 3) 7hu5, we n0w have tw0 c0mp1ete c0mpartmenta1 m0de15 c0rre5p0nd1n9 t0 F19. 1. A part 0f each m0de1 15 phy51ca11y exact, 6e1n9 6a5ed 0n c0n5ervat10n 0f matter. H0wever, each m0de1 a150 c0nta1n5 mathemat1ca1 expre5510n5 that d0 n0t exact1y ref1ect phy51ca1 1aw5, 6ut are appr0x1mat10n5. 5pec1f1ca11y, the appr0x1mat10n5 1n the m0de15 u5e a19e6ra1c 0r 0rd1nary d1fferent1a1 e4uat10n5 when the perfu510n-d1ffu510n phen0mena 1n t155ue 15 actua11y 90verned 6y part1a1 d1fferent1a1 e4uat10n5 (8). 7w0 4ue5t10n5 1mmed1ate1y ar15e. F1r5t, wh1ch 0f the tw0-c0mpartmenta1 m0de15 w111 6e5t de5cr16e 5u65tance pharmac0k1net1c5 1n a 11v1n9 an1ma1• 5ec0nd, wh1ch 0f the5e tw0 m0de15 15 m05t accurate when u5ed t0 extrap01ate fr0m a 5ma11er t0 a 1ar9er an1ma1• 7he 6e5t way t0 an5wer the5e tw0 4ue5t10n5 15 thr0u9h the u5e 0f exten51ve data 06ta1ned fr0m d1fferent 512ed an1ma15, f0r examp1e, rat5 and human5. 51nce the 1ar9e data 5et5 needed were n0t ava11a61e t0 u5 we have pur5ued an a1ternat1ve ana1y515 0f the c0mpet1n9 c0mpartmenta1 m0de15. We have c0mpared the ven0u5 ex1t and d1ffu510n appr0x1mat10n m0de15 t0 a 51mp1e perfu510nd1ffu510n 5y5tem 90verned 6y part1a1 d1fferent1a1 e4uat10n5 f0r wh1ch ana1yt1c mathemat1ca1 501ut10n5 ex15t. 7h15 te5t perfu510nd1ffu510n 5y5tem exh161t5, 4ua11tat1ve1y at 1ea5t, much 0f the actua1 5u65tance m0vement 6ehav10r 0f t155ue. We have determ1ned h0w we11 the c0mpartmenta1 m0de15 appr0x1mate th15 exact perfu510n-d1ffu510n 5y5tem, and we have 5tud1ed h0w we11 the c0mpartmenta1 m0de15 f0110w a 5ca1e-up 0f the exact 5y5tem. AN
EXAC7
PERFU510N-D1FFU510N
5Y57EM
We pr0p05e t0 exam1ne and deve10p an exact 501ut10n 0f the 5an9ren and 5heppard m0de1 0f t155ue c1earance (4,10). We c0n51der a Kr09h cy11nder (4, 10, 11) 0f 1en9th L a5 d1a9rammed 1n F19. 2. 7he t155ue 5e9ment 15 5upp11ed 6y a centra1 6100d ve55e1 w1th v01ume •,/and the c0ncentrat10n 0f the 5u65tance 1n the 6100d 8(x,t), var1e5 a5 a funct10n 0f 11near p051t10n and t1me. Let C(x,t) 6e the c0ncentrat10n 1n the t155ue 1t5e1f. A55um1n9 51mp1e d1ffu510n acr055 the ve55e1 wa11 and n0 ax1a1 0r rad1a1 d1ffu510n 1n the t155ue, the f0110w1n9 e4uat10n5 app1y: ~/c~8(x,t)/0t = k[C(x,t) -
8(x,t)] - -y1x08(x,t)/0x
V0C(x,t)/0t = -k[C(x,t) - 8(x,t)] (EQUA710N 4) 1n the5e e4uat10n5, 8 = 8(x,t) 15 the c0ncentrat10n 0f 5u65tance
AN1MAL-70-HUMAN EX7RAP0LA710N
59
{ [61(~2 "•{- ~3) "~ 66 (~4 + 65)] 51n(¢0jt)
8(0,t)
-~
H
+ [61 ( - - ~ A.
H12
.
.
.
.
.
D
~
t
•
2D-H/2
,,t
2D
F16. 3. 7he 5u65tance temp0ra1 pr0f11e 1n the 6100d enter1n9 the Kr09h cy11nder.
-- 65) + ~6 (62 + 63)] C05(t0jt)}
where ~, = t0j/[% 2 + (k/V) 2] ~J2 = [1/(°tj 2 + (¢°j[3j)2)] [ - a j c05(t0j[3jL) + (mj[3j)51n(t0j13jL)]exp( - ctjL)
63 = %/[%2 + (%13j)2] 1n the 6100d, C = C(x,t) 15 the c0ncentrat10n 0f 5u65tance 1n the t155ue, "y 15 the v01ume 0f 6100d 1n the t155ue, 1x 15 the ve10c1ty 0f 6100d f10w 1n the t155ue, V 15 the t155ue v01ume, and k 15 the permea6111ty c0n5tant f0r the ve55e1 wa1f. We have taken 8(0,t) t0 6e the trape201da1 funct10n 0f t1me t 5h0wn 1n F19. 3. 7h15 funct10n ha5 the f0110w1n9 F0ur1er 5er1e5 repre5entat10n (1):
~.. j=
~5 = (%13j)11%2 + (%[3j)=1 = (k/v)/[% 2 + (k/v)-•]
a~
:t:
8(0,t) = (1/2) - 4/[•rr2(1-2p)]
= [1/(aj 2 + (t~j13j)2)] [ - a j 51n(t0j[3jL) - (t0j13j) c05(%[3jL)]exp( - ajL)
=
%/p.
[3j = (11p.) {t0j2 + (k2/V) [(1/V) + (1•)]}/(% 2 + (k/V) 2)
.,
(11j 2) c05(jrrp) c05 (t0jt1 where t0j = j~r/D and p = H/2D. U51n9 th15 F0ur1er 5er1e5 repre5entat10n, e4uat10n 4 can 6e read11y 501ved t0 f1nd
1n the exact perfu510n-d1ffu510n m0de1, ave[C(t)] 15 the va1ue c0rre5p0nd1n9 t0 the Cj term5 1n the c0mpartmenta1 appr0x1mat10n5. A150 8(0,t) c0rre5p0nd5 t0 5u65tance c0ncentrat10n 1n 1n-f10w1n9 6100d, wh11e 8(L,t) c0rre5p0nd5 t0 the c0ncentrat10n 1n t155ue ex1t1n9 6100d.
8(x,t) = (1/2) - 4/[w2(1-2p)] j=
C0MPAR1N6 7HE EXAC7 PERFU510N-D1FFU510N 5Y57EM 70 7HE VEN0U5 EX17 AND APPR0X1MA7E D1FFU510N C0MPAR7MEN7AL M0DEL5
.,
(102) e x p ( - ~jx/p.) c05(j1rp) c05[~j(t-0j)]
F0r a 5pec1f1c c0mpartment, the ven0u5 ex1t m0de1 15
and
VdC/dt + Q c t C C(x,t) = (1/2) - 4(k/V)/[•rr2(1 - 2p)] j= ~
(1/j 2) exp ( - $ j {[0~J(t0j 2 +
(k/V)2)] 51n[t0j(t
v=CtC
..
x/p.) c05(j~rp). -
0j)] +
= Qa
[(k/V)1(t02+(k/V)2)]
c05[%(t- 0j)]} where
1n th15 m0de1, we take a = 8(0,t) and 501ve f0r C and v t0 6e c0mpared w1th ave[C(t)] and 8(L,t) re5pect1ve1y fr0m the exact perfu510n-d1ffu510n 5y5tem. 51m11ar1y, f0r a 5pec1f1c 51n91e c0mpartment, the appr0x1mate d1ffu510n m0de1 15 VdC/dt = Q(a - v)
~ j = %- (k/-,D/(~0j2 + (k/V) 2)
VdC/dt = - k [ C
0j = (x/p.){t0j- + (k2/V)[(11V) + (1/•)•)]}1(t0j 2 + (k/V):•) Mean t155ue 5u65tance c0ncentrat10n ave[C(t)] can 6e ca1cu1ated
- (a + v)/2]
7he5e e4uat10n5 are e4u1va1ent t0 VdC/dt + [2kQ1(2Q + k)1C = [2kQ/(2Q + k)1a
a5:
v = [2k/(2Q+ k)]C + [(2Q - k)/(2Q + k)1a
ave[C(t)] = (1/L) f~ C(x,t)dx whence
ave[C(t)] = (1/2) - 4(k/V)/[~2L(1-2p)]
~ j=
(1/j 2) c05(j•rrp).
..
A9a1n, we take a = 8(0,t) and 501ve f0r C and v t0 c0mpare w1th ave[C(t)] and 8(L,t) re5pect1ve1y fr0m the exact perfu510n-d1ffu510n 5y5tem. We have c0ntra5ted the perf0rmance 0f the ven0u5 ex1t c0mpartmenta1 m0de1 and the appr0x1mate d1ffu510n c0mpartment m0de1 t0 the exact perfu510n-d1ffu510n 5y5tem 6y treat1n9 rat and human 11ver. F0r the rat 11ver V = 12 cm 3 and Q = 35.2 cm3/
60
MED1NA A N D A L 8 A N E 5 E
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v E N 0 u 5 Ex17 APPR0x. - - E x A c 7 P-0 M0DEL 1• - - A P P R 0 x .
V E N 0 U 5 EX17 APPR0X." - - EXAC7 P-0 M 0 D E L
/
D1FFu510N
/
--
A P P R 0 X . D1FFU510N
/
.1
/
.4
1
30
60
45
/
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15
15
f 1
50
71ME(M1NU7E5)
71ME(M1NU7E5)
F16. 4. F1tted data 5h0w1n9 510w chan9e 1n rat ven0u5 6100d 1n a11 three m0de15. m1n. F0r the human 11ver V = 3320 cm 3 and Q = 1800 cm31m1n (9). We have u5ed V = 3.14 x 10 - 5 cm 3 and 3• = 7.85 x 10 - 8 cm 3 1n the exact perfu510n-d1ffu510n m0de1, a55um1n9, f0r te5t purp05e5 0n1y, that 60th rat and human 11ver are made up 0f cy11ndr1ca1 re910n5 1-mm 10n9, and 100 p,m 1n rad1u5 w1th a centra1 6100d ve55e1 hav1n9 a 5 1-tm rad1u5 (2, 4, 5, 1 1). F0r the exact m0de1, 6100d ve10c1ty wa5 ch05en t0 6e 1 17 cm/m1n 1n the rat 50 the exact m0de1 w0u1d have a Q/V rat10 0f 35.2/12, and 6100d ve10c1ty wa5 ch05en t0 6e 21.7 cm/m1n 1n the human t0 c0rre5p0nd t0 a Q/V rat10 0f 1800/3320. F19ure5 4 thr0u9h 7 5h0w the re5u1t5 0f 0ur numer1ca1 exper1mentat10n. 1n the5e f19ure5 6e5t f1t (~ and k va1ue5 were f0und 50 that the ven0u5 ex1t and appr0x1mate d1ffu510n c0mpartment5 6e5t m1m1cked the rat exact m0de1. Extrap01at1n9 the human 11ver 51tuat10n, (x wa5 1eft unchan9ed wh11e k wa5 extrap01ated u51n9 the re1at10n5h1p
[(2~..~ Q,~t)1(2Q,~. + k,at) V~,] = [(2kma. Qm~.)1
F16. 6. Extrap01at10n re5u1t5 f0r man re9ard1n9 the e5t1mat10n 0f ven0u5 c0ncentrat10n.
1n F19.4 we 5ee that a11 three m0de15 can 6e made t0 a9ree a5 re9ard5 a 510w chan9e 1n rat ven0u5 6100d. H0wever, 1n F19. 5 we 5ee that the 5ame parameter5 wh1ch 9ave a9reement 1n rat ven0u5 6100d d0 n0t w0rk a5 we11 f0r rat t155ue c0ncentrat10n5.1n part1cu1ar, the ven0u5 ex1t c0nd1t10n pr0v1de5 a p00rer 4ua11ty f1t t0 rat t155ue c0ncentrat10n5. 70 c0rre5p0nd t0 man, the exact perfu510n-d1ffu510n 5y5tem 6100d ve10c1ty wa5 chan9ed fr0m 117 cm/m1n t0 21.7 cm/m1n. 7he parameter e~ wa5 1eft unchan9ed f0r the ven0u5 ex1t m0de1 and k wa5 adju5ted a5 de5cr16ed 6ef0re. 1n F19.6 we 5ee that the ven0u5 ex1t m0de1 and d1ffu510n appr0x1mat10n m0de1 60th extrap01ate we11 t0 man a5 re9ard5 the e5t1mat10n 0f ven0u5 c0ncentrat10n5. H0wever, a5 5een 1n F19. 7, the ven0u5 ex1t c0nd1t10n a9a1n d0e5 n0t perf0rm we11 re9ard1n9 t155ue c0ncentrat10n5. 7he pattern5 f0und 1n F195. 4 thr0u9h 7 are amp11f1ed when fa5ter chan9e5 1n 6100d 1eve15 are 5tud1ed. 1n the5e 5ett1n95 a9a1n the d1ffu510n appr0x1mat10n m0de1 exce15 the ven0u5 ex1t m0de1.
(2Qm~. + km~.) Vm~.]
/,
VEN0U5 EX17 APPR0X. ~ - --EXAC7 P-0 M0DEL --
APPR0X. D1FFU510N
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APPf10X. D1FFU510N
J~
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,/
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71ME(M1NU7E5)
F16. 5. Rat t155ue c0ncentrat10n f0r a11 three m0de15. 5ame parameter5 u5ed wh1ch 9ave a9reement 1n rat ven0u5 6100d.
30
/ J a5 71ME(M11~U7E5)
60
75
90
F16. 7. Extrap01at10n re5u1t5 f0r man re9ard1n9 the e5t1mat10n 0f t155ue c0ncentrat10n.
AN1MAL-70-HUMAN EX7RAP0LA710N
61
c0NcLu510N C0mpartmenta1 m0de15 have a c0re 5et 0f mathemat1ca1 re1at10n5 wh1ch are der1ved fr0m the phy51ca1 pr1nc1p1e 0f c0n5ervat10n 0f matter. 7h15 c0re 5et 0f e4uat10n5 15 n0t 5uff1c1ent f0r m0de11n9, 6ut mu5t 6e au9mented 6y aux111ary c0nd1t10n5. 7he ven0u5 ex1t c0nd1t10n 15 c0mm0n1y u5ed, and a d1ffu510n appr0x1mat10n a150 15 p055161e. 7he5e appr0x1mat10n5 u5e a19e6ra1c e4uat10n5 0r 0rd1nary d1fferent1a1 e4uat10n5 t0 de5cr16e perfu510nd1ffu510n phen0mena 1n t155ue 90verned 6y part1a1 d1fferent1a1 e4uat10n5. We c0mpared a ven0u5 ex1t m0de1 and a d1ffu510n appr0x1mat10n m0de1 t0 an exact perfu510n-d1ffu510n m0de1 0f t155ue 90verned 6y part1a1 d1fferent1a1 e4uat10n5. We have f0und
that the d1ffu510n appr0x1mat10n m0de1 perf0rm5 6etter than the ven0u5 ex1t m0de1 1n m1m1ck1n9 the perfu510n-d1ffu510n 5y5tem, part1cu1ar1y a5 re9ard5 the pred1ct10n 0f t155ue c0ncentrat10n5. We were 5urpr15ed t0 065erve that a m0de1 that f1t5 6100d c0ncentrat10n chan9e5 we11 may n0t f1t t155ue c0ncentrat10n event5. 0n1y m0re exper1ence w1th a var1ety 0f 5u65tance5 and rea1 w0r1d an1ma1-t0-human extrap01at10n eff0rt5 w111 pr0v1de 5ecure 9u1dance c0ncem1n9 ch01ce am0n9 c0mpet1n9 c0mpartmenta1 m0de15. 7h15 rep0rt 5u99e5t5 that c0mpar150n 0f c0mpet1n9 c0mpartmenta1 m0de15 t0 perfu510n-d1ffu510n m0de15 90vemed 6y part1a1 d1fferent1a1 e4uat10n5 may 6e an add1t10na1 1mp0rtant 9u1de t0 c0mpartmenta1 m0de1 5e1ect10n.
REFERENCE5 1. 8eyer, W. H., ed. CRC 5tandard mathemat1ca1 ta61e5. 28th ed. 80ca Rat0n, FL: CRC Pre55, 1nc.; 1987:405---406. 2. C0penhaver, W. M.; Ke11y, D. E.; W00d, R. L. 8a11ey•5 text600k 0f h15t0109y. 17th ed. 8a1t1m0re: 7he W1111am5• W11k1n5C0mpany; 1978. 3. Dedr1ck, R. L. An1ma1 5ca1e-up. J. Pharmac0k1net. 810pharm. 1: 435-461; 1973. 4. 6055e11n, R. P.; 6055e11n, R. E. 7he k1net1c5 0f 501ute c1earance 6y a 51n91e t155ue cap111ary 0f 11m1tedpermea6111ty: Exp11c1t 501ut10n50f tw0 m0de15 and the1r 60und5. 8u11. Math. 8101. 49:329-349; 1987. 5. Henne55y, 7. R. 7he 1nteract10n 0f d1ffu510n and perfu510n 1n h0m09ene0u5 t155ue. 8u11. Math. 8101. 36:505-526; 1974. 6. Jac4ue2, J. A. C0mpartmenta1 ana1y5151n 610109y and med1c1ne. 2nd ed. Ann Ar60r: 7he Un1ver51ty0f M1ch19an Pre55; 1985:11-13. 7. Landah1, H. D. An appr0x1mat10n meth0d f0r the 501ut10n 0f d1ffu510n and re1ated pr061em5. 8u11. Math. 810phy5. 15:49-61; 1953.
8. Per1, W. An exten510n 0f the d1ffu510n e4uat10n t0 1nc1ude c1earance 6y cap111ary 6100d f10w. 1n: Wh1pp1e, H. E., et a1., ed5. Mu1t1-c0mpartment ana1y5150f tracer exper1ment5. New Y0rk: New Y0rk Academy 0f 5c1ence5; 1972:92-105. (Ann. NY Acad. 5c1., v01. 108, art. 1.) 9. Ram5ey, J. C.; Ander5en, M. E. A phy5101091ca11y6a5ed de5cr1pt10n 0f the 1nha1at10n pharmac0k1net1c5 0f 5tyrene 1n rat5 and human5. 70x1c01. App1. Pharmac01. 73:159-175; 1984. 10. 5an9ren, W. C.; 5heppard, C. W. A mathemat1ca1 der1vat10n 0f the exchan9e 0f a 1a6e11ed 5u65tance 6etween a 114u1d f10w1n9 1n a ve55e1 and an externa1 c0mpartment. 8u11. Math. 810phy5. 15:387-394; 1953. 11. 7hew5, 6.; Hutten, H. 6a5 exchan9e 1n t155ue. 1n: H0ppe, W., et a1., ed5. 810phy51c5 0f re5p1rat0ry 9a5 tran5p0rt. New Y0rk: 5pr1n9er-Ver1a9; 1983.