BULLETIN OF

MATHEMATICAL BIOPHYSICS

VOLUME 27, 1965 SPECIAL ISSUE

C O M P A R T M E N T A L SYSTEMS AND GENERATING F U N C T I O N S

• GIORGIO SEGBE Department of Pharmacology, University of Camerino, I ta ly

Compartmental systems can be represented by direct graphs in which each node corre- sponds to a generating function and each arm to a transfer generating function. A homomorphism is established between a compartmental system and this representation, in analogy with that obtained through the use of the Laplace transformation.

From the values obtained experimentally in a given compartment, through the solution of a difference equation, tile generating function for the corresponding node can be calculated and the graph of the system can be built up within the degrees of freedom of the model.

From the graph it is possible to calculate the transfer generating function between any two connected nodes, the mean permanence time in a given node, the mean transit time between two nodes, and their precursor-successor order.

A b lack box is a physical sys tem which t ransforms an inpu t into an ou tpu t .

The input and the o u t p u t are given by cont inuous or sect ionally cont inuous

functions. I f the funct ions X(t) and Y(t) indicate the input and the o u t p u t o f

a l inear black box, t hen there is a funct ion G(t), called the weighting funct ion of

the black box, such t h a t

Y(t) = f~ G(t - 8)X(8) dS, (1)

i.e., Y(t) is given by the convolut ion between G(t) and X(t). I f one subdivides the areas sub tended b y the curves X(t), Y(t), and G(t)

t h rough equal t ime intervals At, i.e., i f one takes the mean values of X(t), Y(t),

4---B.M.~. 49

50 G. SEGRE

and G(t) for each interval At (Xk; Yk; and Gk, with k = 0, 1 , . . . , n), then it is possible to substitute integral (1) with the summatory

n

Y~ = ~ G - ~X~ at. (2) k = 0

For simplicity one can put At = 1. The value of At comes into account when the time scale is changed and in particular when one calculates lira (70; for

At-*0

instance, i f l im Go ~ 0, then (Beck and Rescigno, 1963) At-~0

Yo lira Go = lira X - ~ t At-*0 At-*0

and, vice versa, lim Yo = lim GoX oAt. At-*0 At-~0

When equation (2) is used, the input and the output of the black box are represented by the column matrices {Xk} and {Y~} (k = 0, 1 . . . . ,n) and equation (1) can be written as the following matrical equation

{ G } = 1(Tl{x~}

where 1(71 is the square matrix

Go 0 0 0 . . . 0

G1 Go 0 0 . . . 0

G2 (71 (70 0 . . . 0

G, G,_I (7,-2 G~_3 . . . (70

This matrix is called the transfer matrix of the black box. By knowing IGI and {X~} it is possible to calculate {Yk} and, vice versa, by

knowing (Yk} and (Xk} it is possible to calculate the transfer matrix of the black box by means of the equation

{G} = IXl- l{r~} where

X o 0 0 0 . . . 0

X1 Xo 0 0 . . . 0

I X I = X2 X1 Xo o . . . o

X, X , _ I X,_~. Xn-s . . . Xo

COMPARTMENTAL SYSTEMS 51

The values X~, Y~, and G k (k = 0, 1 , . . . , n) form a sequence of numbers. Let us indicate these sequences by [Xk], [ Yk], and [Gk].

Each sequence can be represented by a generating function

x(s) = X o + X18 + X2s 2 + . ." + X~s n = ~ X~sk; kffiO

y(s) = Yo + Y l s + Y2s2 + ' ' " + Y,8 ~ = ~ Y~sk; (3) kffiO

g(8) = Go + Gls + G2s 2 + . - - + G~s ~ = ~ Gk8 k. k=0

The function x(s) may be called the input generating function, the function y(s) the output generating function, and the function g(s) the transfer generating function. The variable s itseff has no significance; it is necessary that x(s), y(8), and g(s) converge, and because [Xk] , [Yk], and [Gk] are bounded (as required in order that the black box be physically possible), then a comparison with the geometric series shows that equations (3) converge at least for Isl < 1.

I f with the sequences [Gk] and [Xk] one carries out the following operation

G o X 1 -~" G1Xt_ 1 ~- G2X]_ 2 -~-...-~- G I _ I X 1 --~ GyX 0 (4)

for eachj (j ~ n), then one obtains the sequence [Yk] which corresponds to the convolution between [G~] and [Xk]. I t is easy to see that the above convolution corresponds to the matrical equation lal(X,} which gives {r,}. Moreover, if [G,] and [X1c] are the sequences with generating functions g(s) and x(s), and [Y~] is their convolution, then the generating function y(s) is given by the product (Feller, 1959, p. 251)

y ( s ) = g ( s ) . z ( s ) . (5)

From equation (5) there results that the transfer generating function g(s) is a rational function in s

g(s) = y(s)/x(s).

t i t h e degree in s ely(s) is lower than that of x(s) and equation x(s) -- 0 has m distinct (real or imaginary) roots (poles) sl, s2,. •., sin, then

x(8) = (8 - 81)(8 - 8 2 ) . . . (8 - 8m),

and g(8) can be decomposed into partial fractions

g(8) = C~/(81 - 8) + c 2 / ( 8 2 - 8) + . . . + c ~ / ( 8 ~ - 8). (6)

I t can be shown (Feller, 1959, p. 258) that

Y('~) I c ~ = d x ~ d 8 J . . 8~ " (7)

52 G. SEGRE

If sk is a root of x ( s ) = 0 of multiplicity p, then in the partial fraction expansion (6), p terms of the form C~/(s k - s ) j ( j = 1, 2 . . . . . p) will appear. Once expansion (6) is obtained, i t i s easy to calculate Gr, the coefficient of s T in g(s). In fact

1 1 1 8 k - - s s k 1 - - 8 / 8 k '

and, for Is] < sk

1 - = I + + + + . . . .

By introducing this expression in equation (6), one obtains for Gr

01 8~2 +2 Cra (~, = ~ + - - ~ + . . . + s~+~"

As shown by W. Feller (1959, p. 259), ff*l is a root ofx(a) = 0 which is smaller in absolute value than all other roots, then, as r - ~ ~ , Gr = C1/8~ +1

Because the representation of the input and of the output by means of generating functions belongs to a linear space and because the commutative, the associative, and the distributive (with respect to addition) properties are valid for the transfer generating functions, this representation can be employed in drug and tracer kinetics (Stephenson and Jones, 1963). Therefore the representation by means of generating functions is equivalent, in this domain, to the representation by means of the Laplace transformation. In effect, all the considerations which lead to equation (6) are analogous to those applied to the compartmental analysis through the Laplace transformation, and expansion (6) corresponds, in compartmental systems, to the multiexponential function to which, following the theory, the experimental data can be fitted.

I f the Laplace transform of a function F(0 is denoted by f(p), then one sees immediately the similarity between equation (6) and the equation

f(p) = ~ Mk/(p + "k) k

which corresponds to a transfer function in a compartmental system (Rescigno and Segre, 1962), and for which

L - l{f(p)} = ~ Mk e - % t. k

In effect each fraction C k / ( s k - s ) in equation (6) is the generating function of a sequence of values corresponding to a single exponential function.

C O M P A R T M E N T A L S Y S T E M S 53

Let us take the sequence [AI.~] = 1, A1, A~, A~ . . . . . A~ of the generating function

n

g ( s ) = kffi0

I f 0 < A l s < 1, then for n --> ov one can write

g(s) = ~ A~8 ~ = 1/(1 - J l s ) ; k = 0

n O W

Ck = Ck 1 8 k - - 8 S~ 1 - - s / 8 k '

and, if one writes 1/s k = A k , there follows

1 g(8) = C~,A~, 1 - A ~ (8)

I t can be noticed that the sequence [A~.k] is the solution of the difference equation

Wk = - a W ~ , (9)

i . e . ,

where 1 - a = A e obtains

W~+I = (1 - a)Wk,

I f the first value of [W~], i.e., W 0 is known, then one

Wk = AI{Wo •

For [W~] one obtains the sequence [A~.k]W o. I t can be observed that 1 > a > 0 and therefore 0 < At < 1.*

I t is known that it is possible to use generating functions in solving difference equations (Jordan, 1947). The following table shows the transformation between a term of the sequence [Wk] and the generating function (see Goldberg, 1961).

Equation (9) corresponds therefore to the equation y(s) = W0/(1 - A l s ) .

Because the Laplace transformation and the generating function trans- formation in compartmental analysis belong to the same algebra, it is possible to m a p a system of compartments into a direct graph by means of both transformations. In analogy to the graph mapping by Laplace transformation (Rescign0 and Segre, 1961a) it is therefore possible to represent a compartmental system by means of a directed graph formed by nodes and oriented arms: a

• Note : I n t h e different ial f o rm of equa t i on (9) ~one ha s Wk+1 ---- e -~ ; 1 -- a is t h e approxi - m a t i o n o f e - a t h r o u g h Tay lo r series w h e n one t ake s t he first two t e r m s only.

54 G. S E G R E

TABLE I

General Term of the Sequence Generating Function

W , + I

W~+2

w(8)

(1/s)(wO)- Wo)

( 1 / 8 2 ) ( W ( 8 ) - W 0 - W18)

(1/s")(w(s) -- Wo - W18 . . . . . W . s " - l )

node corresponds to the sequence of the generating function which results from determinations carried out in that compartment, and an arm between two nodes corresponds to the transfer generating function between the two nodes.

The transfer generating function between two adjacent nodes i and j takes the form of equation (8), i.e.,

C~j/(1 - A f ) = l q j A j / ( 1 - Af) ,

where b~j is the transfer constant between i and j, Aj = 1 - %, and % (for At equal to the t ime scale unit) indicates the fractional outflow from compartment j. Therefore, once the A / s and the Ct/s of the fractions of the expansion of a transfer generating function are known, it is possible to calculate the %'s and the various k~j = C~j[Aj.

The product of two generating functions corresponds to the convolution of their sequence; therefore the same rules (Mason's rules) applied to the graph in the Laplace transform representation can be applied in this case, too:

(a) Two tandem arms can be substituted by an arm equal to their product; (b) Two parallel arms can be substituted by an arm equal to their sum; (c) The arms entering a node which belongs to a cycle whose value is given by

g(s) can be multiplied by 1/(1 - g(~)) and the cycle eliminated. The initial dose D introduced as an impulse in compartment 1 must be

represented by a node 0 of value D, connected to node 1 by an arm of value

Co I(I - A 8) = & l ( 1 -

I t is easily shown that every input and output of a linear black box can be represented by the sequences of generating functions. The various inputs can be formed by constants, exponentials, sinusoids, positive integer powers of independent variable, or any product or sum of these, and each of these functions can be represented by a generating function.

COMPARTMENTAL SYSTEMS 55

For instance, the unit impulse is represented by the sequence 1, 0, 0 , . . . . A step function of height D, beg~nnlug at time t = 0, corresponds to the generat- ing function D](1 - 8) = D(1 + s + s 2 + . . . ) , with a sequence [D] = D, D, D . . . . ; in this case node 0 has the value D/(1 - s). A multiexponential input is represented by the generating function indicated in equation (6). In order to obtain a trigonometric function one can recall that the difference equation W~+2 + Wk = 0 gives the following generating function (see Table I): w(s) = (Wo + Wls ) / (1 + s2); if Wo = 1 and W1 = O, then the generating function becomes

w(8) = 1 / 0 + 8 2) with the sequence

1,0, - 1 ,0 ,1 . . . . .

i.e., the sequence given by cos krr/2 (k = O, 1 , . . . ). I t can be observed that the generating function 1/(1 + s 9') corresponds to the convolution between 1/(1 - is) and 1/(1 + is) and that

1/(1 - i8) = 1 + is + i2s 2 + i383 + . . . ;

1/(1 + is) = 1 - is + i2s 2 - i383 + . . . .

In general the difference equation corresponding to a cosinusoid function is Wk + n + Wk = O, which affords the following generating function

w(s) = (Wo + W l s + ' " + W n - l S n - 1 ) / ( 1 ÷ S n)

with a sequence given by cos (k/n)(Ir/2).

The sequence given by sin krr/2 corresponds to the generating function

W(S) = --8/(1 + s2),

etc. A delay of a time interval unit is equivalent to multiplying by s a given

generating function; for example

1/(1 - s) gives the sequence: 1, 1, 1 , . . . ;

s/(1 - s) gives the sequence: 0, 1, 1 , . . . .

The multiplication of a generating function by s k corresponds to a translation of its sequence by k time interval units (k At).

I t can be observed that multiplying a generating function by 1/(1 - s ) corresponds to the integration (by sum) of the sequence of that generating function; the derivation (by difference) is, on the other hand, obtained by multiplying a given generating function by (1 - s) and by subtracting the first value of the sequence.

56 G. SEGRE

For example [1/(1 - As)] .[1~(1 - 8)] can be calculated to be equal to

A A 2 A s

1 A A 2

1 A

1

° ° °

& • Q

• O •

O I $

1 I + A I + A + A 2 I + A + A 2 + A 8 . . .

and [1/(1 - As)] /[1/ (1 - s)] can be calculated to be equal to

1 A A 2 A s - 1 --1 - 1

1 - A 1 - A A - A 2

1 A - 1 A 2 - A A s - A 2

Q O O

g Q O

° ° °

° ° o

The first difference is obtained by the following operation on function g(8)

(1 -s)g(s)s - G° I $ = 0

The product ( 1 - s)- [ 1/(1 - A s ) ] can be also calculated in the following way: the generating function s/(1 - A s ) is equal to: a + A s 2 + A2s 8 + . - - , correspond- ing to the sequence: 0, 1, A, A 2 . . . . ; this sequence subtracted from the sequence generated by 1/(1 - A s ) gives: 1, (A - 1), (A 2 - A) . . . . .

The operation 1/(1 - As) : 1/(1 - s), and in general the division between two generating functions, y(s) /x (s ) , corresponds to the inverse operation of the convolution and can be called "deconvolution." The calculation of a given transfer generating function corresponds in fact to such a deconvolution between the output generating function and the input generating function.

I t is therefore clear that the application of the generating functions to a black box brings forth the same results as the analysis by means of the transfer functions (Rescigno and Segre, 1961a). The analogy covers also other func- tional relationships. For instance, ff the generating function of the sequence [X~] (k = 0, 1, 2 . . . . ) is multiplied by s and then derived with respect to s, the sequence [(k + 1)Xk] is obtained, which corresponds to the multiplication of each value of the sequence by the index of the time interval. This operation can be writ ten as d s x ( s ) / ~ . I f the summatory ~ X~. (/~ + 1) is divided by ~ Xk, one obtains the approximate value of the m e a n permanence tim#. in the

COMPARTMENTAL SYSTEMS 57

compartment that corresponds to the sequence [Xk]; this value can be indicated a s

d ~X(8)/& . X(8) .ffil'

this expression corresponds to the expression

d x(p)/dp x(p) p = o

in the Laplace transform representation (Rescigno and Segre, 1961a). The two values obtained by the two expressions are equal for At --> 0.

I f g(s) is the transfer generating function between compartment i and j, the above expressions represent the mean tranait t ime from compartment i to compartment j, and are valid provided no recycling occurs in compartment i and in compartment j.

In the same way it is evident that the area under the curve passing through the experimental points is approximated by

k

this expression is equivalent to

g(p)[,=o in the Laplace transform representation (Rescigno and Segre, 1961a).

In the graph representation of a system of compartments one can write

y~(s)= ~. Cj~ D~ ( i ~ j ; i = l , 2, ,n) (10) j=z 1 ---A~s yj(s) + 1 - Af'----8 "'"

where D~ indicates the dose introduced into compartment i at the beginning of the experiment.

In order to solve system (10) one has to solve the determinant

I 1 Als -C21 - -C31 . . . - C . 1 I !

A = --C12 1 -- A28 - 0 3 2 . . . -On2 1"

I -C1 . - 0 2 . - 0 3 . . . . 1 - A.sj

If the material has been introduced in compartment 1 only, then the solution for y~(s) is given by

yt(s) = DI(- 1) t + l Al.t A

58 G. S E G R E

where AI.~ is the minor of A obtained by suppressing the 1st row and the i th column.

Because of the validity of the superposition principle, if the material has been introduced into several compartments, the system is equal to the sum of the systems in which the material has been introduced into one compartment only.

The function y~(s) has the form

crns m + C,n-lS m-1 + " " + Co (m < n). (11) yt(s) = s~ + dn_ls ~-1 + . . . + d o

By partial fraction expansion one obtains an equation like equation (6), if y~(s) has n distinct poles; for a pole of multiplicity p, p terms of the form

( , k - s ) J ( j = 1 , 2 . . . . , p )

will appear. I t is known that a system of n simultaneous linear differential equations of 1st

order can be reduced to one linear differential equation of nth order; in the same way a system of n simultaneous linear difference equations can be reduced to one linear difference equation of n th order. Therefore, for a compartmental system one can write

a, Yk+,t + an- lYk+,~- i + " " + a lY~+l + Yk = O. (12)

The values of [Yk+z] (1 = 0, 1 , . . . , n) correspond to the observed values in that compartment or to the areas subtended by the curve formed by the experimental values through equal intervals At. For At = 1, there is a good correspondence between the value of Y(t~) and the area subtended by the curve Y(t) in the interval between t~ + At~2 and t~ - At/2, provided At is sufficiently small.

I f 2n values of Y (i.e., Yo, Y1 . . . . , Y2,-~) are known, then it is possible to calculate the coefficients a~, a2 . . . . , a,, of (12) through the system

~ Y~+jaz= 0 (ao = 1 ; j = 0 , 1 , . . . , n - 1). (13) t = 0

I f m + n values of Y (m > n) are known, then one can write the n normal equations (Rescigno and Segre, 1962)

a, ~ = 0 (h = 1, 2 , . . . , n ) . i = 0 / '=0

I f 2rn values of Y are known, then one can form the 2n partial totals q r - 1

Sq = ~ Y~ (q = 1,2 . . . . . 2n) i = ( q - 1 ) r

COMPARTMENTAL SYSTEMS 59

and treat the 2n partial totals Sq in the same way as the 2n values of Yt in (13) (see Cornell, 1 9 6 2 ) .

Once the at's are known, one can proceed b y solving the auxiliary equation

F(A) = anA ~ + a ,_l~ n- i + . . . + ai)t + 1 = 0.

With the n roots of this equation, the n functions

(P I , (P2, • • • , (Pn

are formed as follows: to each simple real root whose natural logarithm is - a t there corresponds the generating function Ct/(1 - A t s ) , where At = 1 - a t (ff At = 1, otherwise at must be divided by At); to each real root of multiplicity p, whose natural logarithm is - fl there corresponds the lo generating functions

cl cl c; • .

1 - Bs' ( 1 - B s ) : ; " " ( 1 - Bs) ~'

w h e r e B = 1 - f l ( f f A t = 1). The generating function is

y(s) = ~1 + ~2 + "'" + Tn.

The constants C1, C2 . . . . . C n are determined by expanding each generating function

~t = C d ( 1 - A t s ) = Ct[1, At, A 2 . . . . . At]

and by solving the system

n

Yk+r = ~ , C , A ~ (r = 0 , 1 . . . . . n - 1), i l l

where the n values of Yk+T (r = 0, 1 , . . . , n - 1) are known experimentally (see equation (12)).

On the other hand, when the at's are known, by applying to equation (12) the rules of Table I, one obtains

(a,/s'~){Y(s) - Yo - Y i s . . . . . Y,~_i s '~-l} + (a,~_i/s È - i ) x

( y ( s ) - Y o - Yls . . . . . Y,-2s n-2} + " " +

(a l / s ) {y ( s ) - Yo}]+ y(s) = 0

which gives

b , _ l s n -1 + ba_2 8n-2 + ' " "4" bib + be (14) y(s) = s" + a l s '~-I + . . . + a , ,_ l s + an

6 0 G . S E G R E

where

I b,~_l = a l Y o + a2Y1 + ' " + anYn-1

bn_2 = a2Yo + a3Y1 + + anYn-2

[ bl = an- 1 Y o + an Y1

b o = anY o

(15)

Equation (15) corresponds to the matrical equation

If°} ° °Iant bl = Y1 Yo . . . 0 a ,_ l

• ' i

n-1 Y,~- I Y n - 2 . . . Yo [ i, al J

and therefore the square matrix ] YI may be considered as a transfer matrix of a black box with input given by the sequence a,, am-1 . . . . , al and with output given by the sequence bo, b l , . . . , b._ 1, and y(s) can be considered as the transfer generating function whose output generating function and input generating function are given by the numerator and by the denominator of (14).

Equation (14) can be decomposed into partial fractions like equation (6) and therefore y(8) can be determined from the experimental values. I f b,_ 1 = 0 ( Yo ~ 0), then the numerator of y(s) = p(s)/q(s) will be of two degrees lower in s than the denominator•

I f

Ibn-~ (Yo ~ 0), 0 b,_ 2 O,

then p(s) will be of three degrees lower in 8 than q(s), etc. The number of degrees by which p(s) is lower than q(s) corresponds to the

precursor's order of the input in respect to the output in compartmental systems, that is it corresponds to the number of arms connecting the precursor (input) to the successor (output) (Rescigno and Segre, 1961b).

For instance, ff two compartments i and j are adjacent (i.e., if they are connected by one arm), the transfer generating function has the form Ctj/(1 " .4f); that is, the difference in degree in 8 between the numerator and the denominator of their transfer generating function is one. In the system

1 3

COMPARTMENTAL SYSTEMS 61

one has X3(8) : C12 C23

X1(8 ) 1 -- A28 1 -- A38

and the difference in degree of numerator and denominator is equal to two. The value of p(s)],=o in a transfer generating function g(s) - -p(s) /q(s)

corresponds to the sum of the products of all the Co's of the elementary paths (i.e., of the paths which do not pass twice through the same node) which connect the precursor node to the successor node (Rescigno and Segre, 1964), because the recyelings for s = 0 do not contribute to the numerator (see rule (c) on page 54). I f each Ctj is divided by Aj then one obtains the precursor

principal term (Rescigno and Segre, 1961b). In the previous example

X3(8 ) p(8) C12G23 X1(8 ) q(8) (1 -- A28)(1 - A3s)

andp(s)ls= o = C12C2a. If in equation (15) Yo = 0, then b o = 0; therefore equation (14)becomes

s(bn_lS '~-2 + b . _ 2 8 n - 3 - } - . . . -]- 51)

y(s) = sn + alsn-1 + . . . + am _is + am '

that is y{s) corresponds to the generating function

bn_lsn-2 + . . . + bl 8n + a 1 8 n - 1 .~_ . . . + am

with a delay of one time interval unit.

In the analysis of a compartmental system one has to build up the model (the directed graph); from the model and the values of the arms and of node O one can obtain the transfer generating function from outside (node 0) to any compartment as well as the generating function between any two connected compartments. The transfer generating function between any two connected nodes is equal to the sum of all the paths which connect the two nodes; the value of each path is equal to the product of the values of all the arms forming that path, and the cycles must be accounted for by the use of rule (c), page 54.

A transfer generating function has the form of equation (11). On the other hand, from the experimental values the equation (14) is obtained by the method previously outlined. Therefore, the following system of equations is derived by equating the coefficients in equation (11) and (14):

I! n : 1 -~" Cm

n 2 Cm- 1 I! 1 = d n - 1

2 dn - 9.

6 2 G . S E G R E

where the b's and the a's are calculated from the experimental values. By solving this system, if the model is determined and unique, it is possible

to obtain the values of each arm. System (10) can also be solved by taking into consideration the following

determinant:

1 -T21 -T31 . . . - T ~ I

-- T12 1 - Tz2 . . . - T . 2

- T I , ~ - T 2 , ~ - T s , ~ . . . 1

= I n + D

where I , is the n x n identity matrix and D is an invertebrate determinant whose element - T,j is equal to - C l j / ( 1 - A j s ) (Rescigno and Segre, 1965).

In this case the transfer generating function g(s) between any two connected nodes can be obtained by analysing the strong subgraphs (Rescigno and Segre, 1964) of the associated graph. The denominator of g(s) is equal to the product of the determinants of the strong subgraphs entered by the elementary paths which connect the two nodes; each of these determinants has the form: one minus the sum of the elementary cycles, plus the product of the elementary cycles two by two, three by three, etc., without nodes in common, each cycle in the products having a negative sign.

The numerator of g(s) is formed by the sum of the values of the elementary paths between the two nodes plus the sum of the products of each elementary path times the elementary cycles untouched by the said path, the cycles being taken one by one, two by two, etc., without nodes in common, each with a negative sign.

LITERATURE

Beck, J. and A. Rescigno. 1963. "Determination of Precursor Order and Particular Weighting Function from Kinetic Data." Jaur. The.or. Biol., 6, 1.

Cornell, R. G. 1962. "A Method for Fitting Linear Combination of Exponentials.'" Biome~r/c~, 18, 104.

Feller, W. 1959. A n InSroduction of Probability Theory and it~ Applications. NewYork: Wiley.

Goldberg, S. 1961. Introduction to Difference Equations, p. 192. New York: Science Editions, Inc.

Jordan, C. 1947. Calculus of FiniZe Differences, second edition. New York: Chelsea Publ. Co.

COMPARTMENTAL SYSTEMS 63

Rescigno, A. and G. Segre. 1961a. La cinetica dei farmaci c dei traccianti radioattivi. Torino: Boringhieri. (English translation: 1965. Drug and Tracer Kinetics. New York: Blaisdell.)

1961b. "The Precursor-Product Relationship." Jour. Theor. Biol., 1, 498. 1962. "Analysis of Multicompartmen~ed Biological Systems." Jour. Thor .

Biol., 8, 149. 1964. "On Some Topological Properties of the Systems of Compartments."

Bull. Math. Biophysics, 26, 31. 1965. "On Some Metric Properties of the Systems of Compartments." Ibid.,

in press. S~ephenson, J. L. and A. P. Jones. 1963. "Application of Linear Analysis to Tracer

Kinetics." Ann. N.Y. Acad. Sci., 108, 15.