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On the mathematics of digesta flow kinetics
J. theor. BioL (1985) 113, 743-758
On the Mathematics of Digesta Flow Kinetics J. FRANCE, J. H. M. THORNLEY, M. S. DHANOA AND R. C. SIDDONS
The Grassland Research Institute, Hurley, Maidenhead, Berkshire, SL6 5 LR (Received 3 August 1984, and in revised form 14 November 1984) A unifying mathematical analysis of the use of compartmental models with and without time lags is given, with particular reference to the use of such models for digesta movement along the gastro-intestinal tract of the ruminant. First the generalized compartmental model without time tags is developed, and then it is shown how discrete time tags may be incorporated into this formalism. The important relationship between distributed lags, especially the gamma-distributed time lags, and the equivalent compartmental scheme, is emphasized. It is shown how distributed time lags can be included in a general compartmental model. The treatment covers, as special cases, some widely used models, and shows their relationship to each other and to other possible models. Finally a compartmental interpretation is outlined for a recently proposed double-exponential model.
I. Introduction Compartmental models are much used in biochemistry and biology (Godfrey, 1983). In considering the flow o f substances through the digestive tract of animals, the compartmental approach, mostly using sequential models, has been of especial utility (e.g. Blaxter, G r a h a m & Wainman, 1956; Ellis, Matis & Lascano, 1979). The problem has been addressed with a variety of models, sometimes blending empiricism with a more mechanistic view based on a particular compartmental scheme. However, the approaches have been somewhat fragmented, and there appears to be a lack of appreciation of the relationship of the various models to each other and to more generalized models, especially where time lags are concerned. The main objective o f this paper is to present a unifying analysis o f this area, pointing out compartmental interpretations of some o f the equations that have been fitted in practice; the mathematics used is simple, yet rigorous, and most models with analytical solutions that can be routinely fitted to data using non-linear least squares (i.e. equations with up to three shape parameters) are discussed. 743 0022-5193/85/070743 + 16 $03.00/0
O 1985 Academic Press Inc. (London) Ltd
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J. F R A N C E .
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ET
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. ..
FIG. 1. The generalized n-compartment model with no lags.
2. The Generalized Compartmental Model with N o Time Lags
Consider the n-compartmental model of passage through the gastrointestinal tract shown schematically in Fig. 1. In the model, the state variables Xt, X 2 , . . . , X , represent the amounts of a unit of non-absorbable, digestaflow marker in the compartments concerned at time t ( h o u r s ) . No attempt is made at this stage to nominate any compartment other than the final one, which is faeces. The initial and final conditions are as follows. When t = 0, then Xw = 1 and X2 = X3 = . . . = Xn = 0 as the marker is applied as a single dose into the first compartment; when t - > ~ , then X i = X 2 = . . . = X , _ , = 0 and X~ = 1. k~, k 2 , . . , kn-~ are rate constants (per hour). Assuming linear kinetics, the system is described by n linear, first-order differential equations, namely dX~ dt
dX2 - klXt dt
dXi = dt
dX._t dt
(1.1)
kjX~,
-
-
k2X2,
k,-lXi-t
- k~Xi,
-k~_2Xn_2-kn_lXn_l,
dX. = ~_,X._, dt
1.2)
1.3)
1.4)
1.5)
The faecal outflow rate (units of mass/hour) is given by equation (1.5). The generalized model (equations (1)) can be solved by mathematical induction. Consider the case n = 3. The dynamics o f the system are now described by three differential equations dXi dt
-
klXl,
(2.1)
D I G E S T A FLOW K I N E T I C S dX2 dt
(2.2)
= k i X l - k2X2,
dX3 = dt
745
(2.3)
k 2 X 2.
Equations (2) are easily solved analytically using standard procedures. For k~ ~ k2 the analytic solution is Xt = e -k't,
(3.1)
r-,_,
X2 = k, L(k2-k,) X3 = 1
+ (k,-
.I
k2)J'
k2e-k, t
k,e-k2 t
(k2- kl)
(kl - k2)"
(3.2) (3.3)
Equation (3.3) gives faecal accumulation to time t. Usingequation (3.2) to substitute for X2 in equation (2.3), the faecal outflow rate becomes dX3 - kl k2
dt
r. e-k'-' + e]--X .I L(k2- k,) (k, - k2)J"
(4)
Now consider n = 4. The differential equations are dXl dt
--=
dX2 dt
(5.1)
-klXt,
= k l X i - k2X2,
(5.2)
dX3 dt
k 2 X 2 - k3X3,
(5.3)
dX, dt
k3X3.
(5.4)
Equation (5.4) describes the faecal outflow rate. For k~ # k2 ~ k3 the analytic solution to equations (5) is (6.1)
X! = e -k,t,
[
l
e-k~t e-k2' X2 = k, L(k2- k,) ~ ( k , ~ k2)J ' [ X3 = k~k2
e -k,' ( k 2 - k,)(k3-
(6.2)
e-" k~) ÷ ( k l -
k2)(k3-
e-" ] k2) ~ ( k , - k3)(k3- k3) ' (6.3)
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J. F R A N C E
X4 = 1
ET AL.
ktk3 e -~-' (kl-k2)(k3-k2)
k2k3 e -k'r
(k2-kt)(k3-kl)
klk2 e-~' (kt-k3)(k2-k3)"
(6.4)
Equation (6.4) gives faecal accumulation to time t. Substituting equation (6.3) into equation (5.4), the faecal outflow rate becomes dX4 =
dt
kl k2 k3
[e-k, t
e-%'
÷
(k2-kj)(k3-k~)
e-k~'
}
(kt-k2)(k3-k2)
]
(k,-k3)(k2-k3)"
(7)
The analytic solution to the generalized model defined by equations (l) for k,#k2#...#k,_j may be obtained by mathematical induction using equations (3) and (6), and the faecal outflow rate, d X , / d t , is given by dX, dt
l;=~
k,
"' (kj-k,) ]} , x~E/e-k"/I-I t "=IL /j=l
(8)
j#i
(cf. equations (4) and (7)). The generalized model contains n - 1 non-linear parameters (i.e. rate constants). Equations with more than three non-linear parameters are often difficult to fit in practice. The generalized model is therefore of limited utility for n > 4. 3. The Incorporation of Discrete Time Lags Discrete time lags are now incorporated into the schemes just described. First, the model of digesta flow proposed by Blaxter et al. (1956), illustrated in Fig. 2, is considered. This model has been widely applied in the analysis
X~; tureen
]
X2; abomasum
P
delay v
X3; faeces
FIG. 2. The model of Blaxter, Graham & Wainman (1956).
of marker excretion patterns in the faeces of ruminants (Grovum & Williams, 1973). It comprises three compartments with a discrete lag between the last two pools. The state variables, parameters and initial conditions are as for the model described in the previous section with n -- 3, but with the addition of a constant z (hours) to denote the lag. Furthermore, Blaxter et al., (1956) nominate X~ as the rumen and X2 as the abomasum in their scheme. Linear kinetics are assumed and the system is described by three differential
747
DIGESTA FLOW KINETICS
equations (cf. equations (2)) dXl = klXl(t), dt dX2 dt
(9.1) (9.2)
= k t X , ( t ) - k2X2(t),
dX3
dt =0,
0~< t < r ,
= k2X2( t - r),
(9.3a) t >/~-.
(9.3b)
The analytic solution to these equations for k~ # k2 is XI = e -k't,
(10.1)
r
X 2 = k, L ( k 2 - k , ) X3=O, = 1
e-n'l
(IO.2)
{-(k,~:2J '
O<~t
(k2-k,)
(lO.3a) kle-k2lt-~-) (k,-k2)
'
t~>r.
(10.3b)
Faecal accumulation to time t is given by equations (10.3a,b). Using equation (10.2) to substitute for X2 in equation (9.3b), faecal outflow becomes
dX3 dt
=0,
(lla)
O<-<-t
= k, k2 [ e - k ' ( ' - " t_
L ( k 2 - k,)
(k, - k2)J'
t~>r,
(lib)
which may be compared with equation (4). delay rl ~ ,mo,,,o,0,,,no t"......... I
delay r 2
'ar0e'°'es"°e
FIG. 3. A four-compartment model with two lags.
We now extend the scheme of Blaxter et aL (1956) to four compartments, introduce a second lag, and designate each pool and delay as shown in Fig. 3. This scheme is described by four differential equations dX~ dt
kiXl(t),
(12.1)
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dX2 = k , X , ( t ) - k2X2(t), dt dX3 dt =0,
(12.2) (12.3a)
0 4 l < 7"1,
=k2X2(t-7-,)-k3X3(t),
dX4 dt
--=0,
t>~r,,
(12.4a)
0<~ t < 7-j + 7-2, = k3X3( t - r2),
(12.3b)
t t> r, + r2.
(12.4b)
The analytic solution for kt # k2 # k3 is X! = e -kJ,
(13.1)
X2= kl
(13.2)
L(k2-k,) i (k,~k2)J'
X3=0,
I e_kl(l_rl)
= k , k2
(k2_-~_ e-k30-L)
=l
e--k2(t--Tt)
G) + (k, - k 2 ) ( k 3 - k2) 1
-~ (k] - ka)(k2- k3) ' X4=O,
(13.3a)
0<~ t<7-,,
t ~> 7-j,
(13.3b) (13.4a)
O<<.t
kl k3 e-k2('-5-~2~
( k 2 - kl)(k3- kl)
(kl - k2)(k3 - k2)
k, k 2 e -k3(t-'rl -r2) -
(k, - k3)(k2- k3)'
t ~> rl + ~'2.
(13.4b)
Faecal accumulation to time t is given by equations (13.4a,b). Substituting equation (13.3b) into (12.4b), the faecal outflow rate becomes
dX4 dt
=0,
0~< t < r ~ + r 2 , e_k,(t_~.i_r2)
= k,k2k3 L(k2~- ~ k , )
e-k~°-"-~2)- 1 4 (k~ - k3)(k 2 -- k3) j '
(14a) e-k2(t-L-~-2)
~ ( k , - k2)(k3 - k2) t t> 7-z+ r2.
(14b)
DIGESTA FLOW KINETICS
749
This particular model is attractive on two accounts; firstly, it is biologically acceptable as the main components of ruminant gastro-intestinai tract are explicitly represented, and secondly, it is statistically and computationally acceptable as it contains only three non-linear parameters (i.e. the rate constants kt, k2, k3). Recently, Faichney & Boston (1983) postulated a modification of the scheme shown in Fig. 3 to represent the gastro-intestinal tract of sheep, in which the added refinement of a discrete lag between rumen and abomasum is included to account for time spent in the omasum. The faecal outflow rate for this modified scheme becomes
dX4 dt
=0,
0~< t < ~-t+ 7"2+ 7"3,
(15a)
e-k2( 1-7-1-'r2-'r3) I e_ kl(t_~.l_r2_.r3) t=klk2k3 ( k 2 - k l ) ( k 3 - k l ) (kt - kE)(k3- k2) e- k3(t-~'l- T2-~'3) 1 + (k-~k3)--~(k2_k-~a)j,
t>~ r~ + ~-2+ r3,
(15b)
where rl, r2, ~'3 now represent time spent in the omasum, the small intestine and the distal large intestine respectively. It is useful to note that having solved the corresponding compartmental model with no lags (equations (1) et seq.), a compartmental scheme incorporating discrete lags is easily solved by inspection; the time variable t is simply lagged appropriately in the relevant exponential terms. This may be seen by comparing equations (4) with (1 lb), (7) with (14b) and (7) with (15b). 4. The Relationship Between Compartmental Models and Distributed Lags
The gamma distribution frequently provides an appropriate distribution of the times required for the completion of complex biological events (e.g. bacterial generation times). Consider a process with g sequential stages (subprocesses) (in Fig. l, n - - g + 1). The overall process is only complete when the last stage is completed and takes time ~'. The same linear rate constant k is assumed to apply to each stage. The mean time spent in any single state, m, is
m=l/k,
(16)
and the mean time for the overall process is given by gm--g/k. Kendall (1948, 1952) has shown that T is distributed according to the gamma
750
J. F R A N C E
ET
AL.
distribution, whose probability density function is given by
Tg-I e-~'/m f(r)
mgF(g ) ,
(17)
where F denotes the complete gamma function defined by F(g)=
z~-J e-~ dz;
g>0.
(18)
The cumulative distribution function is given by
f(r) =
f(r) d r = ~,(g, ~'/rn)/F(g),
(19)
where y denotes the incomplete gamma function defined by
~,(g,x)=
zg-~ e-~ dz;
g,x>O.
(20)
The mean of the distribution is given by
E(r)=gm,
(21)
V(r) = gm 2.
(22)
and the variance by This scheme is therefore equivalent to a compartmental model comprising ( g + 1) pools with the same linear rate constant k applying to flow between each pool. Fitting the curve d F /g-I e-t/m d t - rngF(g)
(23)
to marker outflow data (measured in units of mass/hour) is thus equivalent to the c o m p a r t m e n t a l scheme given in Fig. 1 with n = g + 1 and ki = k, i = 1 , 2 , . . . , n - 1. F, faecal marker accumulation to time t, is given by
F(t) = 7(g, t/m)/F(g).
(24)
Yule's process provides another distribution having a compartmental interpretation. This consists of g parallel subprocesses (Fig. 4), and the
FIG, 4, Compartmental representation of Yule's process.
DIGESTA
FLOW
KINETICS
751
same linear rate constant k (= m -~) applies to each subprocess. The overall process is completed the instant all the subprocesses are completed and takes time r. Rahn (1932), in studying the fission of a bacterium, has shown that ~" is distributed according to Yute's distribution, whose probability density function is given by f ( t ) = g e-Y/m( 1 - e - ' / " ) s-l.
(25)
m
The mean of the distribution is given by E(O=m(I+½+½+...+
1).
(26)
Expressions for the other moments of the distribution and the cumulative density function are unwieldy, but Finney & Martin (1951) show that, for large g, the following are valid: E ( ~ ' ) - m(yE +In g),
(27)
V ( r ) - m2H2/6,
(28)
where y~ is Euler's constant (= 0.5572157...). 5. The Incorporation of Distributed Time Lags We illustrate the incorporation of distributed lags into compartmental models with reference to the model of Blaxter et al. (1956) (Fig. 2 and equations (9), (10) and (11)). The delay ~" is a positive constant in their model. If instead ~" is allowed to take a range of non-negative values then the differential equations become dXi dt
--=
dX2 dt
(29.1)
-k~X~(t),
= k ~ X l ( t ) - k2X2(t),
dX3 _ k2 dt
Io
f ( r ) X 2 ( t - r) dr,
(29.2) (29.3)
where f0") is the probability density function of the lag ~-. The analytic solutions to equations (29) for k~ # k2 are XI = e -k'',
(30.1)
1
X2 = k, L(k2 - k,) } (k-~- k2) J'
(30.2)
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J. F R A N C E E T A L .
Iolo Io[
X3=k, k2 =
f(r) L(k2_ k,) 4
1 k2 e-k,(,--) (k2-k,)
f(r)
k' e-k~('-~)] (-~2) J dr.
(30.3)
The gamma, Weibull, truncated normal and beta distributions are frequently used for non-negative random variables; the first three are used for example as time-to-failure distributions in the age replacement problem (Gl asset, 1967). Unfortunately these distributions lead, in general, to intractable mathematics in this particular problem. Some special cases are, however, of interest. It is interesting to note that the Blaxter model is a special case of a gamma-distributed lag; as g--,m, the probability density function of the gamma distribution (equation (17)) becomes the Dirac delta function and equations (29) reduce to equations (9). The exponential distribution, with density function f ( r ) = A e -A',
0~
(31)
where A is a scale parameter (A -I is the mean), is a special case of both the gamma (g = 1) and Weibuil distributions. Substituting equation (31) in equations (29.3) and (30.3), we get dX3 = k i k2A
dt
e-k, ' (k2-kl)(A-kl)
4"
e-~' (ki-k2)(A-k2)
e-"'
]
(kl-~2-X)J'
(32.1)
and X3=l
k2 A e-k~ t
kl A e-k2 r
k i k 2 e -xt
(k2_kl)()t_k,)
(k,-k2)(X-k2)
(k,-A)(k2-X)'
(32.2)
(cf. equations (6.4) and (7)). The rectangular distribution, whose density function is given by
f(r)=l/b,
a<-r<~a+b,
(33a)
= 0, otherwise,
(33b)
where a is a location parameter and b a scale parameter, is a special case of the beta distribution. Equation (29.3) now becomes
dX~ dt
=0, 1
- b
0<~t
(34a)
k 2 e -kl(~-a)
k I e -k2(t-a)
(k2-k,)b
(k,-k2)b'
k2(e k'b - 1) e -k,('-") ( k 2 - kl)b
+
a<~t<_a+b,
k l ( e k 2 b - 1) e -k2(t-a)
(kl - k2)b
(34b)
t>a+b
(34c)
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DIGESTA FLOW KINETICS
and equation (30.3) becomes X 3 = O,
0 ~< t < a,
k2e -k~°-a) -kl(k2-kl)b
= 1
+
(35a)
kl e -k2('-a)
k2(kl-k2)b
~
(t-a) b
(kl+k2) k~k2b '
kz(e k , b - 1) e -k,('-a)
kl(e k2b- 1) e -k2('-a)
kt(k2- kl)b
k2(kl - k2)b
"
a<~t<~a+b,
(35b)
t > a + b.
(35c)
Equations (34) and (35) are amenable to routine regression methods. We conclude this discussion of distributed lags by providing a simple derivation of the digesta-flow model of Ellis et al. (1979). This model is a derivative o f the Blaxter model, introducing the notion o f gamma time d e p e n d e n c y into the first compartment o f Fig. 2. The model follows earlier work by Matis (1972), and involves fitting the curve
"[
A2 .l+e_,,:,_~.). x ~, _,. L(--~-2---~) (A2-At)2J (A2_A,)2 j
r _,,,,_~.,rA#(t-~-) y=Co~e
"
(36)
to marker outflow data. A compartmental derivation is as follows. Consider the scheme depicted in Fig. 5, which divides the rumen into two compartments, Xi~ and X~2,
i
I
'l rumen . . . . . . . . . . . . . . . . . .
I
I t
I
X2; obomosum
1
k2 •
deloy r
X3;foeces
J
FiG. 5. A compartmental scheme for the model of Ellis, Matis & Lascano (1979).
with the same rate constant k~ applying to both pools. The initial and final conditions are as follows. When t = 0, then Xti = 1 and Xl2 = X2 = X3 = 0; when t ~ 00, then X ~ = X~2 = X2 = 0 and X3 = 1. Assuming linear kinetics we have dX~t dt dXi2
dt dX2
dt
k, X t t ( t ) ,
(37.1)
- k~X~ t(t) - ktX~2(t),
(37.2)
= k~XI 2(t) - k2X2(t),
(37.3)
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ET AL.
and dX3 ......... 0, dt
0 ~< t < r,
= k2X2(t - r),
(37.4a)
t >1 r.
(37.4b)
The faecal marker outflow rate is given by equations (37.4a,b). The analytic solution to equations (37) for k~ ¢ k2 is XI 1= e- k, ,,
(38.1)
XI2 = kit e-k, ',
(38.2) k~e-~'
X2=-
(k,-k2)"
(k~je-k"+
(38.3)
( k ! -- k2) 2'
O<-t
X3=0,
(38.4a)
k2(2kl - k2) k, k z ( t - r ) ] e_k,(,_,) = 1 + L (ki _ k2)2 ~ ( k , - ~ 2 ) J
k~ _LI,_,) ( k , - k 2 ) 2e '
t>~r.
(38.4b) Equations (38.4a,b) give faecal accumulation to time t. Using equation (38.3) to substitute in (37.4b), the facal outflow rate becomes
d~ dt
= 0,
0 ~< t < r
[
k~k2
(39a)
k ~ k 2 ( t - r)]
L(kl----~:) 2÷
~,Tk-~.)J e - k " ' - " 4
k~k2 e -k'3'-'' (kl-k2)
2
'
t/> r,
(39b)
(cf. equation (36)). Alternatively, the Ellis model may be derived using the equivalent scheme depicted in Fig. 6. The differential equations for this system are dX, - -k,X(t), dt
dX2 _ dt
(40.1)
Io
koC(rj)Xl(t - r,) dr, - k2X2(t),
where f ( r , ) = k, e -k,~,,
(40.2) (40.3)
dX3 d t = 0,
0 <~ t < %,
= k2X2(t - r2),
(40.4a) t I> r2.
(40.4b)
DIGESTA
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755
KINETICS
t
~' I' I rumen i ........................
dislribuled clelay .~.--J X2, abomasum q, parameler X=
kl!J
~
delay r2
--,.--J [
X3; faeces
i
i
FIG. 6. An alternative scheme for the Ellis model.
6. The double-exponential model It has been observed (Dhanoa et al., 1985) that a double-exponential equation o f the type dX,, _ A e -k'' exp [ - (n - 2 ) e-~k-'-k~'],
(41)
dt
where A, kt, n and k: are parameters, is able to fit a wide range of data successfully. In this section, equation (41) is derived from the generalized compartmental scheme shown in Fig. 1. First, it is noted that equation (8) is symmetrical in the ki, and therefore for the overall behaviour o f the system, the rate constants can be ordered so that k~ < k 2 < . . . < k,_~, without loss of generality. To derive a relationship between the rate constants, it is assumed that they are related by the identity ki+t = ki + 8,
i >~ 2,
(42)
where 8 is a small positive constant. The equation for i = 3 , 4 , . . . ,
ki= k 2 + ( i - 2 ) ( k 2 - k ~ ) ,
n- 1
(43)
is equivalent to equation (42) with 8 = k 2 - k ~ . Using equation (43) to substitute for ki (i~>3) in equation (8), and making use of the series expansion o f ( 1 - x ) N, where N is a positive integer, after some algebraic manipulation equation (8) yields
dXn =
A e-kd[1
-
-
e-(k2-k')t] n-2,
(44)
dt where
A=k,k2
[k2+(i-2)(k2-k,)]
[(n-2)!(k2- k,)"-2].
i
Using the approximation [1 - e-(k2-k')'] "-2 ~ exp [ -- (n --2) e-(k2-kP'],
(45)
756
J. F R A N C E
ET AL.
which is valid for large n, equation (44) then becomes
dXn - A e -k~' exp [ - ( n - 2) e-Ck'--kO']. dt
(46)
The expressions for the faecal outflow rate given by equations (44) and (46) have just three independent, non-linear parameters k~, k2, and n, where k~, k2 are rate constants and n the number of compartments in the model. A is a scale parameter dependent on kt, k2 and n. k~ and k2 may be ascribed to the two compartments of the gastro-intestinal tract having the greatest turnover times, such as the caecum and the rumen. Equations (44) and (46) are fitted to an extensive data set in Dhanoa et al., (1985). Equation (46), in its partially linearized form obtained by logarithmic transformation, was found to be particularly effective, giving acceptable parameter estimates in all cases. 7. Discussion
By describing the mathematical derivation of different compartmental models, this paper attempts to clarify the interrelationships between them. Although dealing more specifically with the process of digesta flow along the gastro-intestinal tract of ruminants, it is also relevant to the use of compartmental models in other areas of biology and biochemistry. The generalized compartmental model, without time lags and assuming first-order kinetics, is taken as the basis for the analysis and this leads to the general expression of equation (8). In practice, equations involving more than three non-linear parameters (i.e. equations which are non-linear in their parameters) are often difficult to fit and, where possible, tend to give very high rate constants that are biologically unacceptable. Defares & Sneddon (1961) conclude that as the number of pools (i.e. rate constants) increases so their theoretical significance decreases. The generalized model is therefore of limited utility for n > 4. Discrete time lags can be incorporated into the generalized compartmental model, and the method of doing this is outlined. Blaxter et al. (1956) propose a three-compartment model comprising the rumen, abomasum and faeces with a single discrete time lag between the abomasum and faeces to describe digesta flow along the intestinal tract of sheep; it contains only two nonlinear parameters. A more sophisticated four-compartment model, consisting of rumen, abomasum, caecum and faeces with discrete time lags between abomasum and caecum and between caecum and faeces is proposed here; Faichney & Boston (1983) promulgate a similar scheme, though their equations are neither specified nor solved analytically. The model has three
DIGESTA
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KINETICS
757
non-linear parameters, which are readily amenable to estimation by nonlinear squares provided the data are of sut~cient resolution. Although it contains two discrete lags, in practice it is not possible to differentiate between them and only their sum will be produced. In addition to discrete lags, it is also possible to incorporate distributed lags. A distributed lag is in essence a series of interrelated processes which together comprise the lag. The nature of the interrelationship of the processes varies for different types of distribution. For example, the gamma distribution is equivalent to a series of pools with the same linear rate constant applying to the flux between each pool; Yule's distribution on the other hand is equivalent to a series of parallel subprocesses each having the same linear rate constant. Although any distributed lag can be replaced by a compartmental representation, it can be useful to combine distributed lags with a compartmental model and the technique for doing this is presented; this includes, as a special case, the model of Ellis et al. (1979). They propose the use of gamma time dependence in describing faecal marker outflow data because their data was not adequately described by the original model of Blaxter et al. (1956). In the Ellis model the notion of gamma time dependency is incorporated into the first compartment of the Blaxter model; they suggest that a gamma time dependence of two is most appropriate. The introduction of distributed lags, whilst in essence increasing the number of compartments, tends not to result in a significant increase in the number of non-linear parameters. Simplifying assumptions to reduce the number of non-linear parameters are made by Dhanoa et al. (1985) in deriving a double-exponential model. In this model, which appears to have a wide range of practical applications, the rate constants for the two pools with the greatest turnover times are independent, whereas the other rate constants are interrelated. The Grassland Research Institute is financed through the Agricultural and Food Research Council; the work is in part commissioned by the Ministry of Agriculture, Fisheries and Food. REFERENCES BLAXTER, K. L., GRAHAM, N. McC. & WAINMAN, F. W. (1956). Br. J. Nutr. 10, 69. DEFARES, A. G. & SNEDDON, I. N. (1961). An Introduction to the Mathematics of Medicine and Biology. pp. 582-586. Amsterdam: North Holland Publishing Company. DHANOA, M. S., SIDDONS, R. C., FRANCE, J. & GALE, D. L. (1985). Br. J. Nutr., (in press). ELLIS, W. C., MATIS, J. H. & LASCANO, C. (1979). Fedn Proc. Am. Soc. Exp. BioL 38, 2702. FAICHNEY, G. J. & BOSTON, R. C. 0983). J. Agric. Sci. 101, 575. FINNEY, D. J. & MARTIN, L. (1951). Biometrics 7, 133. GLASSER, G. J. (1967). Technometrics 9, 83.
758
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E T AL.
GODFREY, K. (1983). Compartmental Models and their Application. London: Academic Press. GROVUM, W. L. ~/. WILLIAMS, V. J. (1973). Br. J. Nutr. 30, 313. KENDALL, D. G. (1948). Biometrika 35, 316. KENDALL, D. G. (1952). Z Roy. Stat. Soc. B 14, 41. MATIS, J. H. (1972). Biometrics 28, 597. RAHN, O. (1932). J. gen. Physio. 15, 257.
On the Mathematics of Digesta Flow Kinetics J. FRANCE, J. H. M. THORNLEY, M. S. DHANOA AND R. C. SIDDONS
The Grassland Research Institute, Hurley, Maidenhead, Berkshire, SL6 5 LR (Received 3 August 1984, and in revised form 14 November 1984) A unifying mathematical analysis of the use of compartmental models with and without time lags is given, with particular reference to the use of such models for digesta movement along the gastro-intestinal tract of the ruminant. First the generalized compartmental model without time tags is developed, and then it is shown how discrete time tags may be incorporated into this formalism. The important relationship between distributed lags, especially the gamma-distributed time lags, and the equivalent compartmental scheme, is emphasized. It is shown how distributed time lags can be included in a general compartmental model. The treatment covers, as special cases, some widely used models, and shows their relationship to each other and to other possible models. Finally a compartmental interpretation is outlined for a recently proposed double-exponential model.
I. Introduction Compartmental models are much used in biochemistry and biology (Godfrey, 1983). In considering the flow o f substances through the digestive tract of animals, the compartmental approach, mostly using sequential models, has been of especial utility (e.g. Blaxter, G r a h a m & Wainman, 1956; Ellis, Matis & Lascano, 1979). The problem has been addressed with a variety of models, sometimes blending empiricism with a more mechanistic view based on a particular compartmental scheme. However, the approaches have been somewhat fragmented, and there appears to be a lack of appreciation of the relationship of the various models to each other and to more generalized models, especially where time lags are concerned. The main objective o f this paper is to present a unifying analysis o f this area, pointing out compartmental interpretations of some o f the equations that have been fitted in practice; the mathematics used is simple, yet rigorous, and most models with analytical solutions that can be routinely fitted to data using non-linear least squares (i.e. equations with up to three shape parameters) are discussed. 743 0022-5193/85/070743 + 16 $03.00/0
O 1985 Academic Press Inc. (London) Ltd
744
J. F R A N C E .
.
ET
.
AL,
. ..
FIG. 1. The generalized n-compartment model with no lags.
2. The Generalized Compartmental Model with N o Time Lags
Consider the n-compartmental model of passage through the gastrointestinal tract shown schematically in Fig. 1. In the model, the state variables Xt, X 2 , . . . , X , represent the amounts of a unit of non-absorbable, digestaflow marker in the compartments concerned at time t ( h o u r s ) . No attempt is made at this stage to nominate any compartment other than the final one, which is faeces. The initial and final conditions are as follows. When t = 0, then Xw = 1 and X2 = X3 = . . . = Xn = 0 as the marker is applied as a single dose into the first compartment; when t - > ~ , then X i = X 2 = . . . = X , _ , = 0 and X~ = 1. k~, k 2 , . . , kn-~ are rate constants (per hour). Assuming linear kinetics, the system is described by n linear, first-order differential equations, namely dX~ dt
dX2 - klXt dt
dXi = dt
dX._t dt
(1.1)
kjX~,
-
-
k2X2,
k,-lXi-t
- k~Xi,
-k~_2Xn_2-kn_lXn_l,
dX. = ~_,X._, dt
1.2)
1.3)
1.4)
1.5)
The faecal outflow rate (units of mass/hour) is given by equation (1.5). The generalized model (equations (1)) can be solved by mathematical induction. Consider the case n = 3. The dynamics o f the system are now described by three differential equations dXi dt
-
klXl,
(2.1)
D I G E S T A FLOW K I N E T I C S dX2 dt
(2.2)
= k i X l - k2X2,
dX3 = dt
745
(2.3)
k 2 X 2.
Equations (2) are easily solved analytically using standard procedures. For k~ ~ k2 the analytic solution is Xt = e -k't,
(3.1)
r-,_,
X2 = k, L(k2-k,) X3 = 1
+ (k,-
.I
k2)J'
k2e-k, t
k,e-k2 t
(k2- kl)
(kl - k2)"
(3.2) (3.3)
Equation (3.3) gives faecal accumulation to time t. Usingequation (3.2) to substitute for X2 in equation (2.3), the faecal outflow rate becomes dX3 - kl k2
dt
r. e-k'-' + e]--X .I L(k2- k,) (k, - k2)J"
(4)
Now consider n = 4. The differential equations are dXl dt
--=
dX2 dt
(5.1)
-klXt,
= k l X i - k2X2,
(5.2)
dX3 dt
k 2 X 2 - k3X3,
(5.3)
dX, dt
k3X3.
(5.4)
Equation (5.4) describes the faecal outflow rate. For k~ # k2 ~ k3 the analytic solution to equations (5) is (6.1)
X! = e -k,t,
[
l
e-k~t e-k2' X2 = k, L(k2- k,) ~ ( k , ~ k2)J ' [ X3 = k~k2
e -k,' ( k 2 - k,)(k3-
(6.2)
e-" k~) ÷ ( k l -
k2)(k3-
e-" ] k2) ~ ( k , - k3)(k3- k3) ' (6.3)
746
J. F R A N C E
X4 = 1
ET AL.
ktk3 e -~-' (kl-k2)(k3-k2)
k2k3 e -k'r
(k2-kt)(k3-kl)
klk2 e-~' (kt-k3)(k2-k3)"
(6.4)
Equation (6.4) gives faecal accumulation to time t. Substituting equation (6.3) into equation (5.4), the faecal outflow rate becomes dX4 =
dt
kl k2 k3
[e-k, t
e-%'
÷
(k2-kj)(k3-k~)
e-k~'
}
(kt-k2)(k3-k2)
]
(k,-k3)(k2-k3)"
(7)
The analytic solution to the generalized model defined by equations (l) for k,#k2#...#k,_j may be obtained by mathematical induction using equations (3) and (6), and the faecal outflow rate, d X , / d t , is given by dX, dt
l;=~
k,
"' (kj-k,) ]} , x~E/e-k"/I-I t "=IL /j=l
(8)
j#i
(cf. equations (4) and (7)). The generalized model contains n - 1 non-linear parameters (i.e. rate constants). Equations with more than three non-linear parameters are often difficult to fit in practice. The generalized model is therefore of limited utility for n > 4. 3. The Incorporation of Discrete Time Lags Discrete time lags are now incorporated into the schemes just described. First, the model of digesta flow proposed by Blaxter et al. (1956), illustrated in Fig. 2, is considered. This model has been widely applied in the analysis
X~; tureen
]
X2; abomasum
P
delay v
X3; faeces
FIG. 2. The model of Blaxter, Graham & Wainman (1956).
of marker excretion patterns in the faeces of ruminants (Grovum & Williams, 1973). It comprises three compartments with a discrete lag between the last two pools. The state variables, parameters and initial conditions are as for the model described in the previous section with n -- 3, but with the addition of a constant z (hours) to denote the lag. Furthermore, Blaxter et al., (1956) nominate X~ as the rumen and X2 as the abomasum in their scheme. Linear kinetics are assumed and the system is described by three differential
747
DIGESTA FLOW KINETICS
equations (cf. equations (2)) dXl = klXl(t), dt dX2 dt
(9.1) (9.2)
= k t X , ( t ) - k2X2(t),
dX3
dt =0,
0~< t < r ,
= k2X2( t - r),
(9.3a) t >/~-.
(9.3b)
The analytic solution to these equations for k~ # k2 is XI = e -k't,
(10.1)
r
X 2 = k, L ( k 2 - k , ) X3=O, = 1
e-n'l
(IO.2)
{-(k,~:2J '
O<~t
(k2-k,)
(lO.3a) kle-k2lt-~-) (k,-k2)
'
t~>r.
(10.3b)
Faecal accumulation to time t is given by equations (10.3a,b). Using equation (10.2) to substitute for X2 in equation (9.3b), faecal outflow becomes
dX3 dt
=0,
(lla)
O<-<-t
= k, k2 [ e - k ' ( ' - " t_
L ( k 2 - k,)
(k, - k2)J'
t~>r,
(lib)
which may be compared with equation (4). delay rl ~ ,mo,,,o,0,,,no t"......... I
delay r 2
'ar0e'°'es"°e
FIG. 3. A four-compartment model with two lags.
We now extend the scheme of Blaxter et aL (1956) to four compartments, introduce a second lag, and designate each pool and delay as shown in Fig. 3. This scheme is described by four differential equations dX~ dt
kiXl(t),
(12.1)
748
j. F R A N C E
E T AL.
dX2 = k , X , ( t ) - k2X2(t), dt dX3 dt =0,
(12.2) (12.3a)
0 4 l < 7"1,
=k2X2(t-7-,)-k3X3(t),
dX4 dt
--=0,
t>~r,,
(12.4a)
0<~ t < 7-j + 7-2, = k3X3( t - r2),
(12.3b)
t t> r, + r2.
(12.4b)
The analytic solution for kt # k2 # k3 is X! = e -kJ,
(13.1)
X2= kl
(13.2)
L(k2-k,) i (k,~k2)J'
X3=0,
I e_kl(l_rl)
= k , k2
(k2_-~_ e-k30-L)
=l
e--k2(t--Tt)
G) + (k, - k 2 ) ( k 3 - k2) 1
-~ (k] - ka)(k2- k3) ' X4=O,
(13.3a)
0<~ t<7-,,
t ~> 7-j,
(13.3b) (13.4a)
O<<.t
kl k3 e-k2('-5-~2~
( k 2 - kl)(k3- kl)
(kl - k2)(k3 - k2)
k, k 2 e -k3(t-'rl -r2) -
(k, - k3)(k2- k3)'
t ~> rl + ~'2.
(13.4b)
Faecal accumulation to time t is given by equations (13.4a,b). Substituting equation (13.3b) into (12.4b), the faecal outflow rate becomes
dX4 dt
=0,
0~< t < r ~ + r 2 , e_k,(t_~.i_r2)
= k,k2k3 L(k2~- ~ k , )
e-k~°-"-~2)- 1 4 (k~ - k3)(k 2 -- k3) j '
(14a) e-k2(t-L-~-2)
~ ( k , - k2)(k3 - k2) t t> 7-z+ r2.
(14b)
DIGESTA FLOW KINETICS
749
This particular model is attractive on two accounts; firstly, it is biologically acceptable as the main components of ruminant gastro-intestinai tract are explicitly represented, and secondly, it is statistically and computationally acceptable as it contains only three non-linear parameters (i.e. the rate constants kt, k2, k3). Recently, Faichney & Boston (1983) postulated a modification of the scheme shown in Fig. 3 to represent the gastro-intestinal tract of sheep, in which the added refinement of a discrete lag between rumen and abomasum is included to account for time spent in the omasum. The faecal outflow rate for this modified scheme becomes
dX4 dt
=0,
0~< t < ~-t+ 7"2+ 7"3,
(15a)
e-k2( 1-7-1-'r2-'r3) I e_ kl(t_~.l_r2_.r3) t=klk2k3 ( k 2 - k l ) ( k 3 - k l ) (kt - kE)(k3- k2) e- k3(t-~'l- T2-~'3) 1 + (k-~k3)--~(k2_k-~a)j,
t>~ r~ + ~-2+ r3,
(15b)
where rl, r2, ~'3 now represent time spent in the omasum, the small intestine and the distal large intestine respectively. It is useful to note that having solved the corresponding compartmental model with no lags (equations (1) et seq.), a compartmental scheme incorporating discrete lags is easily solved by inspection; the time variable t is simply lagged appropriately in the relevant exponential terms. This may be seen by comparing equations (4) with (1 lb), (7) with (14b) and (7) with (15b). 4. The Relationship Between Compartmental Models and Distributed Lags
The gamma distribution frequently provides an appropriate distribution of the times required for the completion of complex biological events (e.g. bacterial generation times). Consider a process with g sequential stages (subprocesses) (in Fig. l, n - - g + 1). The overall process is only complete when the last stage is completed and takes time ~'. The same linear rate constant k is assumed to apply to each stage. The mean time spent in any single state, m, is
m=l/k,
(16)
and the mean time for the overall process is given by gm--g/k. Kendall (1948, 1952) has shown that T is distributed according to the gamma
750
J. F R A N C E
ET
AL.
distribution, whose probability density function is given by
Tg-I e-~'/m f(r)
mgF(g ) ,
(17)
where F denotes the complete gamma function defined by F(g)=
z~-J e-~ dz;
g>0.
(18)
The cumulative distribution function is given by
f(r) =
f(r) d r = ~,(g, ~'/rn)/F(g),
(19)
where y denotes the incomplete gamma function defined by
~,(g,x)=
zg-~ e-~ dz;
g,x>O.
(20)
The mean of the distribution is given by
E(r)=gm,
(21)
V(r) = gm 2.
(22)
and the variance by This scheme is therefore equivalent to a compartmental model comprising ( g + 1) pools with the same linear rate constant k applying to flow between each pool. Fitting the curve d F /g-I e-t/m d t - rngF(g)
(23)
to marker outflow data (measured in units of mass/hour) is thus equivalent to the c o m p a r t m e n t a l scheme given in Fig. 1 with n = g + 1 and ki = k, i = 1 , 2 , . . . , n - 1. F, faecal marker accumulation to time t, is given by
F(t) = 7(g, t/m)/F(g).
(24)
Yule's process provides another distribution having a compartmental interpretation. This consists of g parallel subprocesses (Fig. 4), and the
FIG, 4, Compartmental representation of Yule's process.
DIGESTA
FLOW
KINETICS
751
same linear rate constant k (= m -~) applies to each subprocess. The overall process is completed the instant all the subprocesses are completed and takes time r. Rahn (1932), in studying the fission of a bacterium, has shown that ~" is distributed according to Yute's distribution, whose probability density function is given by f ( t ) = g e-Y/m( 1 - e - ' / " ) s-l.
(25)
m
The mean of the distribution is given by E(O=m(I+½+½+...+
1).
(26)
Expressions for the other moments of the distribution and the cumulative density function are unwieldy, but Finney & Martin (1951) show that, for large g, the following are valid: E ( ~ ' ) - m(yE +In g),
(27)
V ( r ) - m2H2/6,
(28)
where y~ is Euler's constant (= 0.5572157...). 5. The Incorporation of Distributed Time Lags We illustrate the incorporation of distributed lags into compartmental models with reference to the model of Blaxter et al. (1956) (Fig. 2 and equations (9), (10) and (11)). The delay ~" is a positive constant in their model. If instead ~" is allowed to take a range of non-negative values then the differential equations become dXi dt
--=
dX2 dt
(29.1)
-k~X~(t),
= k ~ X l ( t ) - k2X2(t),
dX3 _ k2 dt
Io
f ( r ) X 2 ( t - r) dr,
(29.2) (29.3)
where f0") is the probability density function of the lag ~-. The analytic solutions to equations (29) for k~ # k2 are XI = e -k'',
(30.1)
1
X2 = k, L(k2 - k,) } (k-~- k2) J'
(30.2)
752
J. F R A N C E E T A L .
Iolo Io[
X3=k, k2 =
f(r) L(k2_ k,) 4
1 k2 e-k,(,--) (k2-k,)
f(r)
k' e-k~('-~)] (-~2) J dr.
(30.3)
The gamma, Weibull, truncated normal and beta distributions are frequently used for non-negative random variables; the first three are used for example as time-to-failure distributions in the age replacement problem (Gl asset, 1967). Unfortunately these distributions lead, in general, to intractable mathematics in this particular problem. Some special cases are, however, of interest. It is interesting to note that the Blaxter model is a special case of a gamma-distributed lag; as g--,m, the probability density function of the gamma distribution (equation (17)) becomes the Dirac delta function and equations (29) reduce to equations (9). The exponential distribution, with density function f ( r ) = A e -A',
0~
(31)
where A is a scale parameter (A -I is the mean), is a special case of both the gamma (g = 1) and Weibuil distributions. Substituting equation (31) in equations (29.3) and (30.3), we get dX3 = k i k2A
dt
e-k, ' (k2-kl)(A-kl)
4"
e-~' (ki-k2)(A-k2)
e-"'
]
(kl-~2-X)J'
(32.1)
and X3=l
k2 A e-k~ t
kl A e-k2 r
k i k 2 e -xt
(k2_kl)()t_k,)
(k,-k2)(X-k2)
(k,-A)(k2-X)'
(32.2)
(cf. equations (6.4) and (7)). The rectangular distribution, whose density function is given by
f(r)=l/b,
a<-r<~a+b,
(33a)
= 0, otherwise,
(33b)
where a is a location parameter and b a scale parameter, is a special case of the beta distribution. Equation (29.3) now becomes
dX~ dt
=0, 1
- b
0<~t
(34a)
k 2 e -kl(~-a)
k I e -k2(t-a)
(k2-k,)b
(k,-k2)b'
k2(e k'b - 1) e -k,('-") ( k 2 - kl)b
+
a<~t<_a+b,
k l ( e k 2 b - 1) e -k2(t-a)
(kl - k2)b
(34b)
t>a+b
(34c)
753
DIGESTA FLOW KINETICS
and equation (30.3) becomes X 3 = O,
0 ~< t < a,
k2e -k~°-a) -kl(k2-kl)b
= 1
+
(35a)
kl e -k2('-a)
k2(kl-k2)b
~
(t-a) b
(kl+k2) k~k2b '
kz(e k , b - 1) e -k,('-a)
kl(e k2b- 1) e -k2('-a)
kt(k2- kl)b
k2(kl - k2)b
"
a<~t<~a+b,
(35b)
t > a + b.
(35c)
Equations (34) and (35) are amenable to routine regression methods. We conclude this discussion of distributed lags by providing a simple derivation of the digesta-flow model of Ellis et al. (1979). This model is a derivative o f the Blaxter model, introducing the notion o f gamma time d e p e n d e n c y into the first compartment o f Fig. 2. The model follows earlier work by Matis (1972), and involves fitting the curve
"[
A2 .l+e_,,:,_~.). x ~, _,. L(--~-2---~) (A2-At)2J (A2_A,)2 j
r _,,,,_~.,rA#(t-~-) y=Co~e
"
(36)
to marker outflow data. A compartmental derivation is as follows. Consider the scheme depicted in Fig. 5, which divides the rumen into two compartments, Xi~ and X~2,
i
I
'l rumen . . . . . . . . . . . . . . . . . .
I
I t
I
X2; obomosum
1
k2 •
deloy r
X3;foeces
J
FiG. 5. A compartmental scheme for the model of Ellis, Matis & Lascano (1979).
with the same rate constant k~ applying to both pools. The initial and final conditions are as follows. When t = 0, then Xti = 1 and Xl2 = X2 = X3 = 0; when t ~ 00, then X ~ = X~2 = X2 = 0 and X3 = 1. Assuming linear kinetics we have dX~t dt dXi2
dt dX2
dt
k, X t t ( t ) ,
(37.1)
- k~X~ t(t) - ktX~2(t),
(37.2)
= k~XI 2(t) - k2X2(t),
(37.3)
754
J. F R A N C E
ET AL.
and dX3 ......... 0, dt
0 ~< t < r,
= k2X2(t - r),
(37.4a)
t >1 r.
(37.4b)
The faecal marker outflow rate is given by equations (37.4a,b). The analytic solution to equations (37) for k~ ¢ k2 is XI 1= e- k, ,,
(38.1)
XI2 = kit e-k, ',
(38.2) k~e-~'
X2=-
(k,-k2)"
(k~je-k"+
(38.3)
( k ! -- k2) 2'
O<-t
X3=0,
(38.4a)
k2(2kl - k2) k, k z ( t - r ) ] e_k,(,_,) = 1 + L (ki _ k2)2 ~ ( k , - ~ 2 ) J
k~ _LI,_,) ( k , - k 2 ) 2e '
t>~r.
(38.4b) Equations (38.4a,b) give faecal accumulation to time t. Using equation (38.3) to substitute in (37.4b), the facal outflow rate becomes
d~ dt
= 0,
0 ~< t < r
[
k~k2
(39a)
k ~ k 2 ( t - r)]
L(kl----~:) 2÷
~,Tk-~.)J e - k " ' - " 4
k~k2 e -k'3'-'' (kl-k2)
2
'
t/> r,
(39b)
(cf. equation (36)). Alternatively, the Ellis model may be derived using the equivalent scheme depicted in Fig. 6. The differential equations for this system are dX, - -k,X(t), dt
dX2 _ dt
(40.1)
Io
koC(rj)Xl(t - r,) dr, - k2X2(t),
where f ( r , ) = k, e -k,~,,
(40.2) (40.3)
dX3 d t = 0,
0 <~ t < %,
= k2X2(t - r2),
(40.4a) t I> r2.
(40.4b)
DIGESTA
FLOW
755
KINETICS
t
~' I' I rumen i ........................
dislribuled clelay .~.--J X2, abomasum q, parameler X=
kl!J
~
delay r2
--,.--J [
X3; faeces
i
i
FIG. 6. An alternative scheme for the Ellis model.
6. The double-exponential model It has been observed (Dhanoa et al., 1985) that a double-exponential equation o f the type dX,, _ A e -k'' exp [ - (n - 2 ) e-~k-'-k~'],
(41)
dt
where A, kt, n and k: are parameters, is able to fit a wide range of data successfully. In this section, equation (41) is derived from the generalized compartmental scheme shown in Fig. 1. First, it is noted that equation (8) is symmetrical in the ki, and therefore for the overall behaviour o f the system, the rate constants can be ordered so that k~ < k 2 < . . . < k,_~, without loss of generality. To derive a relationship between the rate constants, it is assumed that they are related by the identity ki+t = ki + 8,
i >~ 2,
(42)
where 8 is a small positive constant. The equation for i = 3 , 4 , . . . ,
ki= k 2 + ( i - 2 ) ( k 2 - k ~ ) ,
n- 1
(43)
is equivalent to equation (42) with 8 = k 2 - k ~ . Using equation (43) to substitute for ki (i~>3) in equation (8), and making use of the series expansion o f ( 1 - x ) N, where N is a positive integer, after some algebraic manipulation equation (8) yields
dXn =
A e-kd[1
-
-
e-(k2-k')t] n-2,
(44)
dt where
A=k,k2
[k2+(i-2)(k2-k,)]
[(n-2)!(k2- k,)"-2].
i
Using the approximation [1 - e-(k2-k')'] "-2 ~ exp [ -- (n --2) e-(k2-kP'],
(45)
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which is valid for large n, equation (44) then becomes
dXn - A e -k~' exp [ - ( n - 2) e-Ck'--kO']. dt
(46)
The expressions for the faecal outflow rate given by equations (44) and (46) have just three independent, non-linear parameters k~, k2, and n, where k~, k2 are rate constants and n the number of compartments in the model. A is a scale parameter dependent on kt, k2 and n. k~ and k2 may be ascribed to the two compartments of the gastro-intestinal tract having the greatest turnover times, such as the caecum and the rumen. Equations (44) and (46) are fitted to an extensive data set in Dhanoa et al., (1985). Equation (46), in its partially linearized form obtained by logarithmic transformation, was found to be particularly effective, giving acceptable parameter estimates in all cases. 7. Discussion
By describing the mathematical derivation of different compartmental models, this paper attempts to clarify the interrelationships between them. Although dealing more specifically with the process of digesta flow along the gastro-intestinal tract of ruminants, it is also relevant to the use of compartmental models in other areas of biology and biochemistry. The generalized compartmental model, without time lags and assuming first-order kinetics, is taken as the basis for the analysis and this leads to the general expression of equation (8). In practice, equations involving more than three non-linear parameters (i.e. equations which are non-linear in their parameters) are often difficult to fit and, where possible, tend to give very high rate constants that are biologically unacceptable. Defares & Sneddon (1961) conclude that as the number of pools (i.e. rate constants) increases so their theoretical significance decreases. The generalized model is therefore of limited utility for n > 4. Discrete time lags can be incorporated into the generalized compartmental model, and the method of doing this is outlined. Blaxter et al. (1956) propose a three-compartment model comprising the rumen, abomasum and faeces with a single discrete time lag between the abomasum and faeces to describe digesta flow along the intestinal tract of sheep; it contains only two nonlinear parameters. A more sophisticated four-compartment model, consisting of rumen, abomasum, caecum and faeces with discrete time lags between abomasum and caecum and between caecum and faeces is proposed here; Faichney & Boston (1983) promulgate a similar scheme, though their equations are neither specified nor solved analytically. The model has three
DIGESTA
FLOW
KINETICS
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non-linear parameters, which are readily amenable to estimation by nonlinear squares provided the data are of sut~cient resolution. Although it contains two discrete lags, in practice it is not possible to differentiate between them and only their sum will be produced. In addition to discrete lags, it is also possible to incorporate distributed lags. A distributed lag is in essence a series of interrelated processes which together comprise the lag. The nature of the interrelationship of the processes varies for different types of distribution. For example, the gamma distribution is equivalent to a series of pools with the same linear rate constant applying to the flux between each pool; Yule's distribution on the other hand is equivalent to a series of parallel subprocesses each having the same linear rate constant. Although any distributed lag can be replaced by a compartmental representation, it can be useful to combine distributed lags with a compartmental model and the technique for doing this is presented; this includes, as a special case, the model of Ellis et al. (1979). They propose the use of gamma time dependence in describing faecal marker outflow data because their data was not adequately described by the original model of Blaxter et al. (1956). In the Ellis model the notion of gamma time dependency is incorporated into the first compartment of the Blaxter model; they suggest that a gamma time dependence of two is most appropriate. The introduction of distributed lags, whilst in essence increasing the number of compartments, tends not to result in a significant increase in the number of non-linear parameters. Simplifying assumptions to reduce the number of non-linear parameters are made by Dhanoa et al. (1985) in deriving a double-exponential model. In this model, which appears to have a wide range of practical applications, the rate constants for the two pools with the greatest turnover times are independent, whereas the other rate constants are interrelated. The Grassland Research Institute is financed through the Agricultural and Food Research Council; the work is in part commissioned by the Ministry of Agriculture, Fisheries and Food. REFERENCES BLAXTER, K. L., GRAHAM, N. McC. & WAINMAN, F. W. (1956). Br. J. Nutr. 10, 69. DEFARES, A. G. & SNEDDON, I. N. (1961). An Introduction to the Mathematics of Medicine and Biology. pp. 582-586. Amsterdam: North Holland Publishing Company. DHANOA, M. S., SIDDONS, R. C., FRANCE, J. & GALE, D. L. (1985). Br. J. Nutr., (in press). ELLIS, W. C., MATIS, J. H. & LASCANO, C. (1979). Fedn Proc. Am. Soc. Exp. BioL 38, 2702. FAICHNEY, G. J. & BOSTON, R. C. 0983). J. Agric. Sci. 101, 575. FINNEY, D. J. & MARTIN, L. (1951). Biometrics 7, 133. GLASSER, G. J. (1967). Technometrics 9, 83.
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GODFREY, K. (1983). Compartmental Models and their Application. London: Academic Press. GROVUM, W. L. ~/. WILLIAMS, V. J. (1973). Br. J. Nutr. 30, 313. KENDALL, D. G. (1948). Biometrika 35, 316. KENDALL, D. G. (1952). Z Roy. Stat. Soc. B 14, 41. MATIS, J. H. (1972). Biometrics 28, 597. RAHN, O. (1932). J. gen. Physio. 15, 257.