On the completely symmetric compartmental system

Bulletin of Mathemaeical Biology, Vol. 42, pp. 481~488

0007-4985/80/0501-04gl $02.00/0

Pergamon Press Ltd. 1980. Printed in Great Britain

NOTE ON T H E C O M P L E T E L Y S Y M M E T R I C C O M P A R T M E N T A L SYSTEM

. J . Z . HEARON M a t h e m a t i c a l Research B r a n c h N a t i o n a l I n s t i t u t e of Arthritis, M e t a b o l i s m a n d Digestive Diseases N a t i o n a l Institutes of H e a l t h Bethesda, M a r y l a n d 20205, U.S.A.

The completely symmetrical system is defined as having identical transfer coefficients between pairs of compartments and the same loss coefficient for each compartment. The eigenvalues and eigenvector are explicitly found along with the inverses of the system matrix and the matrix of eigenvectors. Many properties, special instances of more general theorems, can be seen at once from the explicit analytic solution of the initial value, washout and washin problems. The system serves as a known case for testing estimation procedures, algorithms for solutions of linear systems, eigenvalue-eigenvector and inversion routines and is of considerable tutorial value.

The system discussed in what follows, and defined precisely in Section 2, exhibits several properties of interest. It illustrates (cf. Hearon 1969) conditional lumping (dependent upon initial conditions) and structural lumping (dependent upon connectivity) as well as a kind of approximate lumping, dependent upon the size of the system, which has not previously been noted. The properties of the system can be explicitly shown in that the analytical solution is possible even though the size (order) of the system is arbitrary. The eigenvalues, eigenvectors and inverse of the system matrix can be simply written down. The eigenyector expansion of the solution vector can be shown in detail, many properties rendered transparent, and the "washout" and "washin" experiments of compartmental analysis treated easily. Thus the system is an excellent teaching example, for all of the above can be achieved without resort to small (or low order) systems in order to I. Introduction.

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J.Z. HEARON

keep the linear algebra tractable. Further, though quite special, the system can serve as a known case for testing estimation procedures, algorithms for solution of linear systems and other computational procedures of linear numerical algebra. Linear donor-controlled systems have assumed widespread importance in many disparate subject-matter areas so that interest in such systems by no means any longer resides almost solely in compartmental analysis, tracer kinetics and linear chemical kinetics. Due to the high degree of connectivity and symmetry assumed, the system discussed below is not likely to be of widespread occurrence or to be an adequate model for m a n y reallife situations. Thus any claim for interest in what follows is not based primarily upon relevance in applied situations but rather upon the tutorial value of the aspects discussed in the above paragraphs.

2. The Completely Symmetrical System. We will, in the following, use the language of compartmental analysis. The transcription of the vocabulary to other physical situations will usually be obvious. We define the completely symmetric compartmental system to be a system in which (i) each compartment is directly connected to every other compartment in the system, (ii) the transfer coefficient between compartments i and j is the same for all i and j and (iii) the transfer coefficient for loss from the system is the same for each compartment. Thus the completely symmetrical system is characterized by two transfer coefficients and its size (order, or number of compartments). If A = [ a J is the system matrix we have aij =

~,

for all

i,j (1)

a u = - [(n - 1 )e + r],

for all i

for an n-compartment system with transfer coefficient ~ and loss coefficient r. If xi(t ) is the a m o u n t in the ith compartment at time t we consider the differential system

=Ax,

x(O)=x °

(2)

where x is the column vector of the x i, 2 = d x / d t and x(O) is the arbitrary initial vector with entries x[, i= 1, 2 .... ,n. We also define the mean of the compartmental amounts as:

m(t)=~xi(t)/n,

mo=~X~/n.

(3)

ON

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' - 483

We are going to demonstrate the following properties: P1. Each x~(t) is a linear combination of at most two exponentials. P2. If x ° = m o for some i, then for that i, xi(t ) is a single exponential. This condition can be achieved for each compartment of any chosen subset of the compartments. P3- There are n linear combinations of the x~(t), i = 1, 2, ...,n which are single exponentials. They are re(t) and m ( t ) - x i ( t ) , i > 2 . P4. The washout curve is a single exponential. Ps- U n d e r constant inputs.f~, to the ith. compartment, i = 1 , 2,...,n the asymptotic- values x i (oo) satisfy xl (°c) = . . . = x, (ce) if and only if fl =f2 . . . . . f.. P6- Let e, r and the initial vector x ° be arbitrary but fixed. Then for any to and g > 0 there exists an N such that for n > N we have, for each i, ]xi(t ) - g(t)] t o, where g(t) is a single exponential. Some Algebraic Preliminaries. We begin with a LEMMA. Let U be an n-square-matrix each entry of which-is unity. Then U has two distinct eigenvalues #1---n, which is simple,-and #2 = O, which is of multiplicity n - 1 . The vectors p 1 = [ 1 , 1,...,1]*, p 2 = [ 1 , - 1 , 0,...,0]*, P3 =[1, 0, - 1 , 0 , . . . , 0 ] * , . . . . , p , = [ 1 , 0, 0,...,0, - 1 ] * are a complete set of • eigenvectors, where * denotes transposition and p j, j > 2 , has 1 in the first position, - 1 in the jth position and zeroes elsewhere. P r o o f Since each row sum of U is n it is clear that Upa = npl. Since U is of rank 1, 0 is a root of multiplicity n - 1 . Hence the toots of U are-as claimed. Any vector p such that the sum of its entries is 0 satisfies Up =0. The pj, j = 2 , 3,...,n satisfy this condition and are obviously linearly independent. Since each p j, j > 2 , is orthogonal to Pl, the entire set is linearly independent and constitutes a complete set of eigenvectors. Another simple proof is to observe that U = p l p ~ . Since P'P1 = n we have Up1 = n p l . The null space of U is the set of all vectors orthogonal to PlThe p j, j > 2, shown above are a basis for this subspace. There are m a n y approaches to writing down the solution of the system (2). In case A is diagonable with eigenvalues 21, )~2," ',)~n, distinct or not, then •one f o r m of the solution is 3.

x(t)=~

aipie~i~,

(4)

i

where the Pi are eigenvectors corresponding to the 2i and the a~ are scalar coefficients to be determined so that x ° = ~ alp i. i

(5)

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J. Z. H E A R O N

This has -been discussed in Hearon (1963) from x=exp(At)x °, by substituting x ° exp(At)pi=exp(2~t)pv We further observe and P is the matrix whose columns are Pl,

in detail. It can be seen at once f r o m (5) and observing that that if a is the vector of the ai, P2,..., P,, then (5) can be written

Pa=x °.

(6)

4. Properties of the System.

It is clear that A as defined in (1) can be

written

A = e U - (an + r)I,

(7)

where I is the identity matrix. The eigenvalues of A are ~ # i - (en + r)-where the #~ are the eigenvalues of U. By the lemma then, 21 = - r and 2 2 = - - ( a n + r ) are the distinct eigenvalues o f A with multiplicities 1 and n - 1 respectively. In what follows the special case r = 0 , in which case A is singular occasions no special comment. We note however that in any washout or washin problem (P4, P5 and all-of Section 5) we require r ¢ 0 of necessity. Further, Upi = #~p~ implies Api =~.~Pv Thus the Pi, i--1, 2,..., n shown in the lemma are a complete set of eigenvectors for A, with Pl corresponding to 21 and P2,-.-,P, corresponding to 22. For (4) we have

x(t)=alPlezlt +(i=~ aiPi) e;tzt and from the structure of the Pi, it follows that x 1(t) = alealt + (a 2 ~ - a 3 + . . . + a , ) e xzt

(8) x~(t)=aleXlt-aie ~2~, i>2. N o w (6) takes the form '1 1

1

1

-1

...

0

1

0

1

0

-1

0

)(i1)(xtO

0

...

_

0

a3

--1

n

x °

x°

x,~0

ON THE C O M P L E T E L Y S Y M M E T R I C C O M P A R T M E N T A L SYSTEM

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from which we read at once that at+a2+...

+ a , = x °,

a t - a i = x °,

i>2.

Upon adding these equations we obtain a 1 = rno. Thus (8) can be written as

x~(t)=moeZ~ +(x°-rno)e ~ ,

l
(9)

F r o m (9), Pt and the first statement in P2 are obvious. Choose any subset of k compartments. It is no restriction to consider them to be first k. By assigning X k0' + t , " " Z~ X no arbitrarily, defining q (x°+l+ ... +x°)/n, and choosing x lo- - x 2 o = . . . . xk=nq/(n--k), o we see that x ° = m o , for l<_i<_k.' This is the second statement of P2. Since m(t)=mo e_N the quantity Ix°-mo[exp(-(~n+r)to) is smaller than any preassigned e > 0 . This assertion is P6-

5. Washout and Washin Curves.

If f is a constant, but otherwise arbitrary, input vector, the-asymptotic solution vector x(oo)-obeys A x ( o o ) = - f . If we express x ( o o ) = P b = E b ~ p i, then A P b = P A b = - f , since p - t A P = A = d i a g (21, 22,..-, 2,). Thus the quantities 2/bi are determined in terms of the .f~ in the same way that the a i are-determined by-(6) in terms of the x °. Thus we write at once that )~lba= - f and 2 i b m = - ( f - f ) , i>2, where f = ~ + f z + . . - + f , ) / n . Then, with due respect for the fact that 2 2 = 2 3 = ....

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J . Z . HEARON

=2,, we obtain from P b = x ( o o ) iv f-f,. x,(ao ) = - - ~ - -~ A1-- --'~'2

(10)

Now the washout vector is y(t), which obeys S'=Ay with y(0)=x(oo). Thus the yi(t) are given by (9) when the xi(oo ) from (10) are substituted for the initial values. We have then

y~(t) = @ 1

e ~ ' + ( f - f i ) ca2,,

l
(11)

and the washout curve is Z y ~ = e x p ( - r t ) n f / r , since 21 = - r . With (9), (10) and (11) the washin vector is easily written down. From 2 = A x + f x ( 0 ) = x ° we have

x (t) = eAtx ° + A 1(e* - I ) f which, recalling that - A - i f = x (oo) we write as

x(t)=eatx ° --eAtx(oO )+ x(oO ) The components of exp(At)x ° are given by (9), while exp(At)x(oo) is just y(t) with components given by (11) and x(oo) is given component wise by (10). -The asymptotic vector x(oo) is the steady state solution of the system = A x + f The sum, S, of its components is the total steady state amount-in the system under constant inputs fl, fz,--.,f,, or under total inflow F = f l +f2 + . . - +f,- F r o m (10) we can easily see a well known fact:

S= Z x,( oo )= C r = F/r= rF for it is an easy exercise to verify that ?-= l/r, is the mean life of particles in the system.

6. Discussion. Some of the properties P1-P6 can be deduced from more general theorems. For example P1 follows from the fact that each compartment of the completely symmetric system obeys the M a n n Gurpide condition (Mann and Gurpide, 1969. This is most simply seen from Theorem IV in Hearon, 1969). We have already noted that P5 is a special case of a more general proposition. We saw, in the proof of P2 that

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x°=mo implies xi(t)=m(t ) for all t. This remarkable fact seems to be peculiar to the system at hand. Similarly P6, [which states that for a sufficiently large system after any elapsed time to, however small, we have, with as small error as we please, Xl(t)=x2(t ) . . . . = x , ( t ) ] is highly dependent upon the complete symmetry involved. It is of interest to observe that the linear combinations noted in P3 are those which "decouple" the system. Let P be as defined in (6) and z = P x - 1 Then, since P diagonalizes A, we have ~ = A z or zi(t)=zi(O)exp(2it), i = 1 , 2.... ,n. It is easily seen that the columns of P - a are pl/n, pl/n-e2, pl/n -e3,...,pa/n-e,, where ei is the ith column of the identity matrix. Thus

Px-l=z=[m, m-x2, m-x3,...,m-x,]*. Finally, by using an eigenvector expansion we have evaded the direct computation of A -~ in Section 5 in determining x(oe). However, that inverse is easily written down from a known theorem (Householder, 1964, p. 3). -THEOREM (Householder). Let u and v-be given column vectors and E

=I-~ruv* where ~ is a scalar. Then E-1 =I-'cuv*, where z is determined by v*u = 1/G + 1/z. We apply the theorem as follows: A can be written as A=22(I+op~p~), where G~---~/)C 2. Then, by the theorem A-l=(I-'cplp~)/)~2, where z = o/(na

-- 1 ) =

-- o~/r = (x/21 ,

Thus

.

A-' -(I-~.plp?)I22 --(~U+l)/(an+r). Observe that - A - 1 is a strictly positive matrix. It is known that the negative of a nonsingular compartmental matrix is an M-matrix (see Poole and Boullion 1974 for definitions and listing of equivalent properties of such matrices). The inverse of an M-matrix is nonnegative matrix (e.g. Poole and Boullion, p. 420) and if the M-matrix is irreducible the inverse is strictly positive (Hearon, 1968, p. 33). One definition of an M-matr-ix (e.g. Poole and Boullion, p. 420) is that the matrix can be written C = p I - B where B is nonnegative and p exceeds the spectral radius of B. Note that - A = ( ~ n + r ) I - o ~ U , from (7), is already in this form, since ~n+r>~n = spectral radius of eU.

LITERATURE Hearon, J. Z. 1963. "Theorems on Linear Systems." Ann. N . Y Acad_ Sci. 108, 36-68. . 1968. "The W a s h o u t Curve in Tracer Kinetics." Math. Biosci. 3, 31 39. . 1969. "Interpretation of Tracer Data." Biophys. J. 9, 1363-1370.

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Householder, A. S. 1964. The Theory of Matrices in Numerical Analysis. New York: Blaisdell. Mann, J, and E. Gurpide. 1969. "Interpretation of Tracer Data: Significance of the Number of Terms in Specific Activity Functions." Biophys_ J. 9, 810-821. Poole, G. and T. Boullion. 1974. "A survey on M-Matrices." S I A M Rev_ 16, 419 427.