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Choosing an appropriate multi-exponential model
J. theor. BioL (1975) 55, 137-143
Choosing an Appropriate Multi-exponential Model J. R. GP,~N
Department of Computational and Statistical Science AND
P. E. B. REILLY
Department of Biochemistry, The University, Liverpool L69 3BX, England (Received 9 September 1974, and in revised form 11 November 1974) The inter-relations of the multi-exponential and multi-compartmental models are explored mathematically, as is also the amount of freedom there is to particularize a general multi-compartmental model in any particular case.
1. Introduction
We assume a situation where a multi-compartmental model applies, giving rise in the usual way to a multi-exponential model for the quantities of some substance of interest. We also suppose that this quantity can only be measured for one compartment. This is the usual situation where a radio-active substance is intravenously injected and the blood is sampled after given time intervals and the specific activity of the substance is determined each time. Here we firstly present the basic mathematics of the multi-compartmental model, in a way which we believe to be brief, but more informative than is found elsewhere, leading to five basic equations which connect the nine intercompartmental flow rates (for three compartments) with determinable quantities. These "determinable" quantities would in practice be the estimated parameters of the multi-exponential dependence of the measurable quantity (for example, specific activity) on time. Nine unknowns and only five equations means that there are four degrees of freedom for the flow rates. Ways in which extra restrictions may or may not be imposed on these flow rates, to accord with more specific models, are then examined. Although some particular examples of the imposition of allowable extra restrictions are given by Skinner, Clarke, Baker & Shipley (1959), the investigations of what restrictions may be imposed is new as is the deduction of ranges of allowable values for the flow rates and the inclusion of a further equation 137
138
J.R.
GREEN AND P. E. B. REILLY
appropriate when measuring the specific activity of blood samples. This reduces the number of degrees of freedom to three. 2. M a t h e m a t i c a l Theory
The literature in this field since the pioneer paper by Berman & Schoenfeld (1956) rarely makes much use of matrix notation (for an exception see Brown, 1965, though that author is not primarily concerned with biochemical applications), but we have preferred to use this for conciseness, and, we believe, added insight which it affords. Following Skinner et aL (1959), we designate the size of compartment i (labelled and unlabelled material) as Ql, and its amount of labelled material as qi, so that the specific radioactivity is qJQt; ku represents the flow rate of labelled and unlabelled material from the jth to the ith compartment of which the labelled fraction is qj[Qj, so that the rate of flow of labelled material f r o m j to i is ku(q/Qj) = ;~uqJ, where 2tj = ku/Q j. There is inflow into compartment i of unlabeUed material designated Iv Further, 2n represents the total proportional flow rate out of i and is equal to 2ot+ ~ 2i~, where 2o~ is the proportional flow rate from i out of the system, and ~ 2jt is 2~eI the total proportional flow into other compartments. We here work with three compartments for clarity, but a similar treatment will apply whatever the number of compartments (see Fig. I for a diagrammarie representation of this compartmental model). We have
ql = --/~llql +'~12q2 +'~'13q3 q2 =
22 lq I -- 222q2 "l-)]'23q3
q3 =
'~31ql +'~32q2 --~33q3
where ~ represents differentiation of q with respect to time. Although these basic differential equations appear in Skinner et ol. (1959) and elsewhere, we shall change them to condense them in the matrix form. We put = -Aq,
(1)
where An = 2n > 0; A u = - ; t ~ j _< 0, for i # ~L Berman & Schoenfeld use the name 121 for the matrix we have called A, but this has i,j element -2~j (i # j ) , contrary to usual matrix notation, to which we have here adhered. We will arrange the terms of at > e2 . . . > e,. Empirically we have found that/ltx > A12 . . . > Air then obtains. We do not know fully why
C H O O S I N G AN A P P R O P R I A T E M U L T I - E X P O N E N T I A L MODEL
139
this should be except that it is intuitively reasonable that g l should be associated with the decay of specific radioactivity in the first compartment and that this should have the largest coefficient (All) associated with it. In the same way g2 should be associated with the i value (i ~ 1) for which 2u is rather larger than any 2~j (j ~ 1, i), (supposing 9~ is appreciably larger than g3) and that the second largest Aa~ should be associated with it, and so on. However, we shah not impose this ordering of the A's as a restriction on the model, we merely note the empirical finding as an interesting point. ol
,>1 "1
,,
I
I
>~
I
~1~--"--
xa9
/
. . . . J X03
F i t . 1. A general unrestricted three compartment model. The symbols are explained in the text.
Initially (injecting into compartment 1 at t = 0) ql = ql0 (known), q2 = q3 = 0. Here, we will take r = 3, to be specific, but the theory of the present section is dearly true more generally. If total flow into a compartment = total flow out (non-isotopic steady state),
If + ~ 21jQj = 2i~Q~, for all i. That is, in matrix form, I =
AQ.
(2)
The solution of equation (1) using Laplace transforms yields q = Ale-C], where A is a 3 x 3 matrix with
(3)
(i,j) element A o, l'e - ~ ] is a 3 x I vector with
140
J.R.
AND
GREEN
P. E. B . R E I L L Y
ith element e -g~t. We write g = ( g l g 2 g 3 ) ' , where prime stands for transpose. Also
0 :]
G=
g2
0
g3
The elements of A1 and g are all necessarily positive, where A t is the first row of A. Substituting equation (3) into (1) yields
AE-ge-"] --AAEe-"],
(4)
for all t, where [ge -g*] has ith element g,e -*'t. That is - A G [ e - " ] ----AAEe-gt]. Since this is an identity, we have AG = A A
(5)
A = AGA -1
(6)
or
Since G is diagonal, equation (6) tells us that the gl's are the eigenvalues of A, and corresponding eigenvectors are given by the columns of A. This equation and some of the following consequences of it were derived by Berman & Schoenfeld (1956). This can be used to derive all of the properties and equations connecting A, A and G (but some relationships may be obtained in other ways). For example, from equations (6), we see that trace (A') = ~ 0~, for any positive integer r, and when r = 1 this becomes A1x+A22+A33 = gl+g~+g3. Also IAI = glg2ga, and each Atl is a linear combination of the gl's with coefficients summing to 1, and so lies between the smallest and largest g~, whereas each A~j, for i ~ j is a contrast of the Or's, that is a linear combination of them with coefficients summing to 0, and it can easily be shown that the sum of squares of the coefficients is less than 1, so that each will lie between 0 and (93 -gl)/.,/2, being all negative. S o m e other relations follow by considering the zero-time situation. Initially
A3d where 1 is the unit vector.
C H O O S I N G AN A P P R O P R I A T E
MULTI-EXPONENTIAL
MODEL
141
Postmuttiply equation (5) by 1, giving
(8)
A9 = ql0A1, where A1 is the first column of A. That is
At =
TO, where T = A / q t o =
H1
H2
Ha I
K1
K2
K3
L1
L2
La
of Skinner et al. Expressions for the elements of T are given by Skinner et aL, but we do not need these here. The first row of equation (8) yields Ald~ = qloAll, which gives us All when the Ali's, g~'s and ql0 are known (or estimated) ---equivalently to this equation we may write Axt = ~. H~#~. One may ask whether equation (3) (that is with three observed q's at each of a series of times) implies equation (1) which is the mathematical representation of the multicompartmental model. In fact it does, for we can easily derive equation (1) from (3), putting A = A G A -1.
However, in practice we may only have the first row of equation (3) as a model satisfactorily fitting our data. All we can say is that this is consistent with the whole of equation (3) applying, where q2 and q3 could represent the quantities of labelled material in two unknown compartments. This full equation (3) would then yield equation (I) for the multi-compartmental model. 3. Choice of Particular Models within the Multi-exponential type
The particular models considered in the literature are distinguished by the fact that different combinations of the constants A~j, /~, 2o~, for various i and j, have been set equal to zero. We now consider some of the restrictions which prevent us from arbitrarily equating these terms to zero. We have already seen that A~ = Au > 0, for all i, A~ = - ~ u -< 0, for all i and j (i ¢ j), and certain other relations about the A~j's being linear combinations of the g~'s, resulting from equation (6).
142
J. R. GREEN AND P. E. B. REILLY
It is apparent that the 21o'S are determined by the Aij's, when these are all known. QI may be determined from the initial specific activity, which may be estimated from the fitted relationships for compartment I, and the known value of qlo. However, equation (2) shows that the li values still depend on two unknowns (Q2, Q3) even when the Aij's are all known. Thus up to two of the/~ could be put equal to zero. We will now concentrate on the elements of A alone. Here we have nine unknowns, restricted by five equations [for example, equations (9), (10), (11) of Skinner et aL--there are three equations in (11)]. This leaves four degrees of freedom, so that at most four further restrictions can be imposed on A, and/or on the 2o{S. The first element of equation (8) gives equation (9) below, equating the traces of A and G, and using (9), gives (10). Equating the traces of A -1 and G -1, and using IA-11 = IGt - I = (glg2g3) -I, and equation (9) and (10) gives (12); also equating the leading element of A-1A1 and AG-I1 [using equations (8) and (7)also] gives equation (11). Finally putting IAI = 010203, and using (9) and (I1) yields (13). These equations are as follows [as given by Skinner et aL, equations (8)-(11), in their notation] = = = =
Z as,g,/Z Ali = Z nigi = c3 Z g,-21x = Z g,-c = c1 Hig2g3+H2glg3+H3glg2 = C2 C z + C 3 C l - g l g z - g 2 g s - g 3 g l = C4
(9) + (lO) ,~.22,~.3~-22s2s2 (11) ;tls231 +;t12221 (12) ,~.I3(232221 "1-422431) -~ 412(423431 +433421 ) -~- C 5 = C2C3--.ql,q2g 3 (13) It can be shown that the constants, C1, C2, C3, C4 and Cs are all necessarily I> 0, but we wish to examine what combinations of the Aii's these equations will allow to be zero. The diagonal elements cannot be zero. We can show mathematically, using these equations (that is, assuming that the three-exponential model applies, but not assuming any particular observed values for the quantities involved) that the following apply. We cannot have two non-diagonal A's in a row both zero. Nor can we have both (at least one ofAla, A31 = 0) and (at least one of At2, A21 = 0). Nor can we have both Ala and A23 zero, nor both A~2 and A3z. We have thus seen that 8 of the possible 6C2 = 15 pairs of non-diagonal A's cannot both be zero. Any one of the remaining seven pairs is possible. These are (A12, A23), (A13, A32), (A21 , A32), (A23 , ha1), (A12 , A21), (A13 , Aal), (A23,A32)---the last three being symmetrically opposite pairs, which are intuitively plausible. The impossibility of any of the eight excluded pairs of A's being both zero applies quite generally. The reasoning leading to the excluded pairs of zero A's was verified with a particular set of appropriate data. The excluded pairs include all cases of two elements of A in
CHOOSING
AN APPROPRIATE MULTI-EXPONENTIAL MODEL 143
the same row, or in the same column. This agrees with physical sense, for two zero elements in the same row virtually rules out the corresponding compartment, since no material passes into it from any other. Two zero dements in a column would mean that we had a compartment passing its material completely out of the system with none to another compartment. A pair of zero elements like A13, A2:, would mean in this case that the two compartments, 2 and 3, would effectively act as a single compartment. Clearly we cannot have three or more zero non-diagonal elements in A, since this would necessarily give two such in the same row or the same column. Any one non-diagonal element may be zero, from the mathematical viewpoint, though it may be physically or chemically unlikely to have Ao = 0 and Ai~ # 0. When the first compartment is the blood plasma (which is usually the case), it is reasonable to conclude on physical grounds that 2or = 0. This imposes a further condition on the A's, leaving now only three degrees of freedom, but not otherwise affecting the above reasoning. In this case, we have I ' A i = 0, which in equation (8) yields 1A'g = O.
(14)
I f we could impose three conditions on the model, then these could, with equations (9)-(14), give a complete solution of the system. The authors are grateful to Dr D. Y. Downham for helpful discussion. REFERENCES BEP.MAN,M. & SCHOENFELD,F. (1956). d. appl. Phys. 27, 1361. BROWN,B. M. (1965). The Mathematical Theory o f Linear Systems. London: Chapman and Hall. SKINNER,S. M., CLARKE,R. E., BAKER,N. & SHIPLEY,R. A. (1959). Am. d. Physiol. 196, 238.
Choosing an Appropriate Multi-exponential Model J. R. GP,~N
Department of Computational and Statistical Science AND
P. E. B. REILLY
Department of Biochemistry, The University, Liverpool L69 3BX, England (Received 9 September 1974, and in revised form 11 November 1974) The inter-relations of the multi-exponential and multi-compartmental models are explored mathematically, as is also the amount of freedom there is to particularize a general multi-compartmental model in any particular case.
1. Introduction
We assume a situation where a multi-compartmental model applies, giving rise in the usual way to a multi-exponential model for the quantities of some substance of interest. We also suppose that this quantity can only be measured for one compartment. This is the usual situation where a radio-active substance is intravenously injected and the blood is sampled after given time intervals and the specific activity of the substance is determined each time. Here we firstly present the basic mathematics of the multi-compartmental model, in a way which we believe to be brief, but more informative than is found elsewhere, leading to five basic equations which connect the nine intercompartmental flow rates (for three compartments) with determinable quantities. These "determinable" quantities would in practice be the estimated parameters of the multi-exponential dependence of the measurable quantity (for example, specific activity) on time. Nine unknowns and only five equations means that there are four degrees of freedom for the flow rates. Ways in which extra restrictions may or may not be imposed on these flow rates, to accord with more specific models, are then examined. Although some particular examples of the imposition of allowable extra restrictions are given by Skinner, Clarke, Baker & Shipley (1959), the investigations of what restrictions may be imposed is new as is the deduction of ranges of allowable values for the flow rates and the inclusion of a further equation 137
138
J.R.
GREEN AND P. E. B. REILLY
appropriate when measuring the specific activity of blood samples. This reduces the number of degrees of freedom to three. 2. M a t h e m a t i c a l Theory
The literature in this field since the pioneer paper by Berman & Schoenfeld (1956) rarely makes much use of matrix notation (for an exception see Brown, 1965, though that author is not primarily concerned with biochemical applications), but we have preferred to use this for conciseness, and, we believe, added insight which it affords. Following Skinner et aL (1959), we designate the size of compartment i (labelled and unlabelled material) as Ql, and its amount of labelled material as qi, so that the specific radioactivity is qJQt; ku represents the flow rate of labelled and unlabelled material from the jth to the ith compartment of which the labelled fraction is qj[Qj, so that the rate of flow of labelled material f r o m j to i is ku(q/Qj) = ;~uqJ, where 2tj = ku/Q j. There is inflow into compartment i of unlabeUed material designated Iv Further, 2n represents the total proportional flow rate out of i and is equal to 2ot+ ~ 2i~, where 2o~ is the proportional flow rate from i out of the system, and ~ 2jt is 2~eI the total proportional flow into other compartments. We here work with three compartments for clarity, but a similar treatment will apply whatever the number of compartments (see Fig. I for a diagrammarie representation of this compartmental model). We have
ql = --/~llql +'~12q2 +'~'13q3 q2 =
22 lq I -- 222q2 "l-)]'23q3
q3 =
'~31ql +'~32q2 --~33q3
where ~ represents differentiation of q with respect to time. Although these basic differential equations appear in Skinner et ol. (1959) and elsewhere, we shall change them to condense them in the matrix form. We put = -Aq,
(1)
where An = 2n > 0; A u = - ; t ~ j _< 0, for i # ~L Berman & Schoenfeld use the name 121 for the matrix we have called A, but this has i,j element -2~j (i # j ) , contrary to usual matrix notation, to which we have here adhered. We will arrange the terms of at > e2 . . . > e,. Empirically we have found that/ltx > A12 . . . > Air then obtains. We do not know fully why
C H O O S I N G AN A P P R O P R I A T E M U L T I - E X P O N E N T I A L MODEL
139
this should be except that it is intuitively reasonable that g l should be associated with the decay of specific radioactivity in the first compartment and that this should have the largest coefficient (All) associated with it. In the same way g2 should be associated with the i value (i ~ 1) for which 2u is rather larger than any 2~j (j ~ 1, i), (supposing 9~ is appreciably larger than g3) and that the second largest Aa~ should be associated with it, and so on. However, we shah not impose this ordering of the A's as a restriction on the model, we merely note the empirical finding as an interesting point. ol
,>1 "1
,,
I
I
>~
I
~1~--"--
xa9
/
. . . . J X03
F i t . 1. A general unrestricted three compartment model. The symbols are explained in the text.
Initially (injecting into compartment 1 at t = 0) ql = ql0 (known), q2 = q3 = 0. Here, we will take r = 3, to be specific, but the theory of the present section is dearly true more generally. If total flow into a compartment = total flow out (non-isotopic steady state),
If + ~ 21jQj = 2i~Q~, for all i. That is, in matrix form, I =
AQ.
(2)
The solution of equation (1) using Laplace transforms yields q = Ale-C], where A is a 3 x 3 matrix with
(3)
(i,j) element A o, l'e - ~ ] is a 3 x I vector with
140
J.R.
AND
GREEN
P. E. B . R E I L L Y
ith element e -g~t. We write g = ( g l g 2 g 3 ) ' , where prime stands for transpose. Also
0 :]
G=
g2
0
g3
The elements of A1 and g are all necessarily positive, where A t is the first row of A. Substituting equation (3) into (1) yields
AE-ge-"] --AAEe-"],
(4)
for all t, where [ge -g*] has ith element g,e -*'t. That is - A G [ e - " ] ----AAEe-gt]. Since this is an identity, we have AG = A A
(5)
A = AGA -1
(6)
or
Since G is diagonal, equation (6) tells us that the gl's are the eigenvalues of A, and corresponding eigenvectors are given by the columns of A. This equation and some of the following consequences of it were derived by Berman & Schoenfeld (1956). This can be used to derive all of the properties and equations connecting A, A and G (but some relationships may be obtained in other ways). For example, from equations (6), we see that trace (A') = ~ 0~, for any positive integer r, and when r = 1 this becomes A1x+A22+A33 = gl+g~+g3. Also IAI = glg2ga, and each Atl is a linear combination of the gl's with coefficients summing to 1, and so lies between the smallest and largest g~, whereas each A~j, for i ~ j is a contrast of the Or's, that is a linear combination of them with coefficients summing to 0, and it can easily be shown that the sum of squares of the coefficients is less than 1, so that each will lie between 0 and (93 -gl)/.,/2, being all negative. S o m e other relations follow by considering the zero-time situation. Initially
A3d where 1 is the unit vector.
C H O O S I N G AN A P P R O P R I A T E
MULTI-EXPONENTIAL
MODEL
141
Postmuttiply equation (5) by 1, giving
(8)
A9 = ql0A1, where A1 is the first column of A. That is
At =
TO, where T = A / q t o =
H1
H2
Ha I
K1
K2
K3
L1
L2
La
of Skinner et al. Expressions for the elements of T are given by Skinner et aL, but we do not need these here. The first row of equation (8) yields Ald~ = qloAll, which gives us All when the Ali's, g~'s and ql0 are known (or estimated) ---equivalently to this equation we may write Axt = ~. H~#~. One may ask whether equation (3) (that is with three observed q's at each of a series of times) implies equation (1) which is the mathematical representation of the multicompartmental model. In fact it does, for we can easily derive equation (1) from (3), putting A = A G A -1.
However, in practice we may only have the first row of equation (3) as a model satisfactorily fitting our data. All we can say is that this is consistent with the whole of equation (3) applying, where q2 and q3 could represent the quantities of labelled material in two unknown compartments. This full equation (3) would then yield equation (I) for the multi-compartmental model. 3. Choice of Particular Models within the Multi-exponential type
The particular models considered in the literature are distinguished by the fact that different combinations of the constants A~j, /~, 2o~, for various i and j, have been set equal to zero. We now consider some of the restrictions which prevent us from arbitrarily equating these terms to zero. We have already seen that A~ = Au > 0, for all i, A~ = - ~ u -< 0, for all i and j (i ¢ j), and certain other relations about the A~j's being linear combinations of the g~'s, resulting from equation (6).
142
J. R. GREEN AND P. E. B. REILLY
It is apparent that the 21o'S are determined by the Aij's, when these are all known. QI may be determined from the initial specific activity, which may be estimated from the fitted relationships for compartment I, and the known value of qlo. However, equation (2) shows that the li values still depend on two unknowns (Q2, Q3) even when the Aij's are all known. Thus up to two of the/~ could be put equal to zero. We will now concentrate on the elements of A alone. Here we have nine unknowns, restricted by five equations [for example, equations (9), (10), (11) of Skinner et aL--there are three equations in (11)]. This leaves four degrees of freedom, so that at most four further restrictions can be imposed on A, and/or on the 2o{S. The first element of equation (8) gives equation (9) below, equating the traces of A and G, and using (9), gives (10). Equating the traces of A -1 and G -1, and using IA-11 = IGt - I = (glg2g3) -I, and equation (9) and (10) gives (12); also equating the leading element of A-1A1 and AG-I1 [using equations (8) and (7)also] gives equation (11). Finally putting IAI = 010203, and using (9) and (I1) yields (13). These equations are as follows [as given by Skinner et aL, equations (8)-(11), in their notation] = = = =
Z as,g,/Z Ali = Z nigi = c3 Z g,-21x = Z g,-c = c1 Hig2g3+H2glg3+H3glg2 = C2 C z + C 3 C l - g l g z - g 2 g s - g 3 g l = C4
(9) + (lO) ,~.22,~.3~-22s2s2 (11) ;tls231 +;t12221 (12) ,~.I3(232221 "1-422431) -~ 412(423431 +433421 ) -~- C 5 = C2C3--.ql,q2g 3 (13) It can be shown that the constants, C1, C2, C3, C4 and Cs are all necessarily I> 0, but we wish to examine what combinations of the Aii's these equations will allow to be zero. The diagonal elements cannot be zero. We can show mathematically, using these equations (that is, assuming that the three-exponential model applies, but not assuming any particular observed values for the quantities involved) that the following apply. We cannot have two non-diagonal A's in a row both zero. Nor can we have both (at least one ofAla, A31 = 0) and (at least one of At2, A21 = 0). Nor can we have both Ala and A23 zero, nor both A~2 and A3z. We have thus seen that 8 of the possible 6C2 = 15 pairs of non-diagonal A's cannot both be zero. Any one of the remaining seven pairs is possible. These are (A12, A23), (A13, A32), (A21 , A32), (A23 , ha1), (A12 , A21), (A13 , Aal), (A23,A32)---the last three being symmetrically opposite pairs, which are intuitively plausible. The impossibility of any of the eight excluded pairs of A's being both zero applies quite generally. The reasoning leading to the excluded pairs of zero A's was verified with a particular set of appropriate data. The excluded pairs include all cases of two elements of A in
CHOOSING
AN APPROPRIATE MULTI-EXPONENTIAL MODEL 143
the same row, or in the same column. This agrees with physical sense, for two zero elements in the same row virtually rules out the corresponding compartment, since no material passes into it from any other. Two zero dements in a column would mean that we had a compartment passing its material completely out of the system with none to another compartment. A pair of zero elements like A13, A2:, would mean in this case that the two compartments, 2 and 3, would effectively act as a single compartment. Clearly we cannot have three or more zero non-diagonal elements in A, since this would necessarily give two such in the same row or the same column. Any one non-diagonal element may be zero, from the mathematical viewpoint, though it may be physically or chemically unlikely to have Ao = 0 and Ai~ # 0. When the first compartment is the blood plasma (which is usually the case), it is reasonable to conclude on physical grounds that 2or = 0. This imposes a further condition on the A's, leaving now only three degrees of freedom, but not otherwise affecting the above reasoning. In this case, we have I ' A i = 0, which in equation (8) yields 1A'g = O.
(14)
I f we could impose three conditions on the model, then these could, with equations (9)-(14), give a complete solution of the system. The authors are grateful to Dr D. Y. Downham for helpful discussion. REFERENCES BEP.MAN,M. & SCHOENFELD,F. (1956). d. appl. Phys. 27, 1361. BROWN,B. M. (1965). The Mathematical Theory o f Linear Systems. London: Chapman and Hall. SKINNER,S. M., CLARKE,R. E., BAKER,N. & SHIPLEY,R. A. (1959). Am. d. Physiol. 196, 238.