- :: :.. • } di . 3 in ouis L • I OFL. ORNL P2644 . * 3 . . - I : . | 11 . . : . med and W 11:25 1.14 11:5 MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS - 1963 - - ... . .. . ....... P.TAT. N . . . 1 2 MT 7715 oeyь очи, NOV 2 9 1969 . : MASTER retro H.C. $ 7.00 MN.50 A TWO-COMPARTMENT MODEL WITH A RANDOM CONNECTION* TWO-COMPARTMENT MODEL WITH A Coult 661010_/ S. R. Bernard, Ph.D. Health Paysics Division and V. R. Rao Uppuluri, Ph.D. Mathematics Division Oak Ridge National Laboratory RELEASED FOR ANNOUNCEMENT IN NUCLEAR SCIENCE ABSTRACTS LEGAL NOTICE This report was preparei us an account of Government sponsored work. Neither the United Statos, por the Commission, nor any person acting on behalf of the Commission: A. Wakes any warranty or representation, expressod or implied, with respect to the accu- racy, complotaneus, or urofalnous of the information contained in the report, or that the we i of any information, apparatus, method, or procers disclosed in this roport may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or procon disclosed in this report. As used in the above, "person acting on behalf of the Commission" includes any em- h! ploys or contractor of the Commission, or employee of such contractor, to the extent that : euch employee or contractor of "he Commission, or employee of such contractor prepares, dienominates, or provides access to any information pursuant to Me employment or contract with the Commission, or his employment with such contractor. P ET *Research sponsored by the U.S. Atomic Energy Commission under. contract with the Union Carbide Corporation At 14 ht XERO COPY XERO) CONY XERO! * I XERO 2 . COPY A . - 1.. ASS RE . it .. . . . . TAN 1 2 . AL.... TSALT *** . ....... ... * . . . CONTENTS 1. INTRODUCTION .......... 2. A ONE COMPARTMENT MODEL .... 3. A TWO COMPARTMENT MODEL .... 4. FURTHER GENERALIZATIONS .... REFERENCES . .............. ..... FIGURE 1.... ................. ....... FIGURE 2 .... FIGURE 3 FIGURE 3.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIGURE 4.... FIGURE 5.... :- EFT it. lu XERO) XPRO . . * **2.rra ******** . ., . . 2. INTRODUCTION The purpose of this paper is to consider the problem of popuniform distribution of bone-seeking radionuclides, such as the alkaline-earth elements, and the effect of age on retention of strontium by dogs via a mathematical study of a compartmental system in which the connections between the compartments are random variables. In most compartmental studies it is generally assumed that the contents of the compartments are uniformly, dis- tributed. This is not a realistic assumption for tide case of bone-seeking elements such as radium where it has been well demonstrated that hot spots of activity occur as many as 20 or 30 years after intake of <-Ra by man." Rowland states that the concentrations in the hot spots exist 11 legions of bone where new mineral was laid down at the time the xadium was acquired and that 10 this mineral the original concentration of Ra, expressed as the ratio of Ra to Ca, was essentially the same as the Ra-to-la ratio that existed in the blood plaema at the time the new mineral was formed. There is also a second type of distribution which is much lower in concentration and rather uniform. This is believed to be the result of an exchange process which continually transfers Ca and/or Ra atoms from blood to bone and back again and which is characterized by an unusually long time constant. Also, there is a trend to consider turnover rates in bone in terms of at least two "sub-compartments," trabecular bone and compact bone. The turnover rates in these two types of bone are observed to be different - faster in trabecular bone and much slower in the compact bone. In a personal communi- cation Lucas bas suggested that the biological half-life for Ra in trabecular bone 18 -7 to 14 years and twice as long in compact done. Also, the hot spots ..... . ur........ XEROI XERO COPY XERO :COPY | XERO Copy! CONY . of * - * . . V' - * . . + . t Chat S TITI T . . .. , ... ....... - . -'. .--.-.. of Ra would disappear from the bone after perhaps 200 years since by then the whole bone would be remodeled and turned over. How day we more edequately represent these phenomena in terms of a compartmental approach? One way to gain more insight into this problem 13 to consider the bone.. blood system as a set of randomly connected compartments. We need to determine the behavior of the system--what are the concentiations in a compartment as a : function of timew-bow long must one wait until a compartment releases trapped material--how many randomly connected compartments are required to represent the available data on retention in bone? We find that these are difficult .... questions and that before we can get some answers to the general case we . -en have to initiate our studies on simpler systems namely, that of only a single in'. irti compartment witia random flow and that of two compartments with random 'lows. Here we gain some insight into the behavior of these systems and we find that i the two-compartment system gives some information on the age effect on retention of Sr by beagles. Thus we present the results of the study of these simpler models in this paper. Also, generalizations of these models are discussed. . models wat XEROT COPY XERO XEROI OPY. . .- Y ... most converte*** ************************ sort A . 47 4 ** . ." 4. *** . .. . . .. - .. ... . * * *** * ************** 2. A CNE COMPARTMENT MODEL In order to Lllustrate in a concrete manner the nature of this . and 01:ċ of which water flows at a constant rate of v cc's per minute. Suppose also we have a stirrer in the beaker. When an ordinary drop of a dye is placed in the beaker then the concentration denoted by c(t) will dilute exponentially and at any time t after injecting the drop ... . . c(t) = c(0) e-(v/v)t. (1) .. ... . . . . . . i -.- . . . - .. . .. -... .--. - Now split the time axis t into m.At intervals of time and in each of these At time intervals flip a coin to decide whether or not to stop or continue the dilution process. If a head appears let the water flow into (and out of) the beaker during that interval of time to dilute the dye. If a ted. I appears stop the flow and maintain the concentration established as the end of the preceding interval of time. This results in a random process as opposed to the deterministic process which occurs when the water is permitted to continually flow into and out of the beaker. We can study the behavior of: this random process in terms of a random sequence which describes the conc- entration at time a At in terms vf the concentration at tiue (m-1) At as *. follows. Let C depote the concentration at time tm = m At, n = 1,2, ..., given Co = 1 and let X denote a sequence of independent Bernoulli random variables with Prob[x = 1) = p and Prob[x = 0) = 1-p = q. A1.80 let d = e-lv/st. At the mth interval of time if a coin is flipped and a head appears, we let - XERO CONYI XIRO, CY XERO COPY ..... . . . -40 the concentration at the nth stage be equal to d times the concentration at the (n-1 )tika stage, 1.e.; - 6-2 198,- 1. It a tail appears then C = Cm-1 1.e., If x= 0. Le . . A compact expression for the above is C = dC-1 + 0-1 (1-x), n = 1,2,... It can easlly be seen that eq. (2) describes a branching process where the possible values of Chave the associated probabilities of a 'binonial distribution. More precisely Probic = dkg = (.pcm-k, k = 0,1,...,.. The expected concentration, E(C) at the mth interval of time is just given by : E(Cn) = (q + pa). It is also of interest to derive the expected concentration in the beaker at any time t from this random model. Let t n At be fixed and substitute into (4) for d to obtain ..ECC) = (a + pe-(v/v)t/mym which upon taking limits as m + and At +0 yields . e-P(v/v)t .. .. XEROT COYOY XRRO, COPY t: -I-,4 _ . . . 4 ....... . . ... ... . "'".. . 11 L L L . . 1 . T . R . . . .... . -:- .. .. : in: im. - - -- . . L . I -*... . : P T R : - As can be seen when p = 1 we have a deterministic process, the expression (5) is ldentical to that in eq (1), when c(0) = 1. Only when p * 1 we have a stochastic penomenon and the expected dilution does not occur as rapidly as in the deterministic case. . The 'nehavior of this simple-one compartnent model is completely described in terms of the closed form for the distribution of the concentrations. We shall see in the two compartment model that the distributional problems become difficult. XERO COPY XERO COPY XERO! XCRO COPY! -------- -- KUZ U NGU . T . 12. .. U... A r t. trad .. . " --- - 3. À TWO-COMPARTMENT MODEL Let us suppose that we have two compartments of tize same volume, with a gate between them which can be opened or closed at will (Figure 1). :) Initially, say we have wait concentration in compartment 1 and zero concentratio in compartment 2. Also, suppose that there 18 provision for an Iniet and outlet through compartment I which will enable the concentratia (in 1 only If the gate is closed or in both if the gate is open) to reduce by a factor of est in a time interval At. (Here 18 a constant.) Take a fala coin, that is, with P(H) = 1/2 = P(T), and toss 15 once. If a head appears, open the gate and assume that instantly the material in both the compartments becomes homogeneous, that is, of concentration 1/2 in each. Wait for a time intervai At and suppose that during the at the inlet and outlet through compartment 1 are functioning. After time at, the concen- tration in 1 and 2 will become e-Ast/2. eachi Denvie en dat by d. If a tall appears, keep the gate between compartments 1 and 2 closed, and after time at (during which the inlet and outlet through compartment 1. are functioning) the concentration in 1 will become e Not and of course the concentration in compartment 2 is still zero. :: IH . A Two-Compartment Model Figure 1. XERO ...ico'Y XERO Cox'Y XEROI COPY -;-... .......--------- --------- ------ --- dones bas----- *** ata de la comme*** The equations for this system are set up as follows: Let Co, denote the initial concentration in compartment 1 and Co,2 denote the initial concentration in compartment 2, that 18, This corresponds to an experiment where initially the gate is always closed. More generally, let Connt denote the concentratica at time tos = 1,2,3,..., in compartment i = 1,2. iet X denote the Bernoulli random variable, corresponding to the fair coin, that 18, j ſi 1f I appears (with probability 1/2) LO 11 T appears (with probability 1/2) Thea, it is easily seen that (60,2 +00, 220/2 CO2/ where Xz denotes the Bernoulli random variable at the first stage of the experiment, that 18, more simply, (0) C:6) which is a compact way of representing all the four possible outcomes, ..y . T: KERO OPY XERO COPY XERO XEROL icony) E it 7 ' . . . . EM 2 . . . ! - - - " .. Mi1 . ETTETY . ". T 21 .. ...... ... 1. : . L . . namely, 1f heads, that 18, xy = 1, C2,2 = 8/2 = 67,2 if talls, that 18, X = 0, 09,1 = d; C2,2 = 0. Next, we shall obtain a recursive relation between connected through the raudou variable X: 0.(.) se or X \zare 2-412 -4126) \0,2) • - + f 4 Li ệ u XEROI XERO! CO'YI XERO! COPY Copy where we recall that By following the same reasoning we obtain, 1f Xm denotes the Bernoulli random variable at the mth stage, Cm, 21 to read ). 1x m = 1,2,.. where J is the above matrix. We have not been able to obtain the probability distri?ution in a closed form since Jme Juny are noncommunicative random matrices. In the one compartment Case this problem does not arise since the do -2 were scalars. XERO CONY XERO COPY XERO COPY t"" ""1 " = + = ter - -- t------- --< 1 - - - - - - - -:":"Erri'CT' **** = ',a, A. . . . . . . . - - - - "" TIE..……. ... .1 .....14 1. T 2, 4-MALT. 1-4 "'-.- " " ." . ". :* . .. N R N * . . P : . . . ..,. . . 1 . . . . . . . . . . . . . . - i. : :.. ... ?". 10- . If we assume that Xy,x2,..., X are independent, identically distributed, ... random variables and E denotes the expectation operator, then (on, 2) alt alt ) (2) Pn, al /4 1/2+0/4/ because E(X) = 1/2; m = 1,2,...,1; At = t/n. Let un ae^t/a and denote the eigenvalues of the matrix It turns out that . 1,4 na ten + 2 + Veeb-29+3/ Hy + de 9+2 V22 - 2 + 21,8, este (1) COD) . . -. XERO COPY XKNO XCRO: " .. * V . , therefore .. . .. .. . .. . . ... * * (22 [i9 (, $22) 0 (43 -8,2)) It 18 of interest to determine the solution as a , At - 0. Now, as n + , we heve: en 1, 811 + 3/4, 612 * 1/4, My * 1, Mx + 1/2, .. (y) - -1/2, 11-2-3ht/", and wito. Therefore lim gled 2 + 0 In equation see that the expected concentrations are functions of two parameters, 1 and At, and the initial conditions. We investi- gate the effect of At on the concentrations by graphical techniques. Figure 2 presents a plot of the sum of the concentrations in both compartments for . . At = 0+,1,2,4, and N = 0.693. As can be seen, when the interval width . .. . Synonim y 1st - . . ... - XERO) COPY XERO COPY XORO CONY XERO }сору: 01 mengine UNCLASSYRO Now 4-7034 min . ... . . 10 12 : 8 TIME (days). : orio Fig. 227%;/ Plot of the Sum of Concontrations in Como partmanis 1 and 2 ve Time for Cose of Equally Spaced... Subintervale of Time. :: ...! . . XIERO (xeno i -13- increases, the total concentrations decrease more rapidly, initially, but then it decreases more slowly along a second exponential component. Note also the intercept of this second exponential decreases with increasing At. It may be that the most appropriate value of at varies with the age of the animal, and thus the model might suggest an effect of age on the biol.ogical turnover time. It has been noted that the biological half-life for retention of Cs 18 larger for young than for old dogs.? Now, if a second exponential component would occur, does it have a longer half-life and a smaller inter- cept in older dogs? In the case of sr retention in dogs of different ages, as observed by Glad, Mays, and Fisher, it appears that this model gives approximately the same quantitative picture as far as the intercept of the long-term component 18 concerned, but not the biological half-life, I. Figure 3 shows a plot of the data of Glad et al. Note the half-life of the second exponential component indicated in these figures 18 generally larger for young dogs that are injected than is the I, for the second component fitting retention in old dogs. Note also that the intercept, of that second component decreases with age. We find that we can get a better interpretation of these data in terms of increasing, subinterval widths. Also, Glad and Mays have shown that these data can also be represented with a power function. It is possible to obtain something akin to a power function from this two-compartment model with random flows simply by taking a partitioning of time intervals which increase monotonically. In this case we have to have a more general notation. Let 1- . , . - ..st, a to - t2_2 which are not necessarily equal intervals. Then, instead of Equation (6), '. ..! .h. 0722 RO XERO COPY . XERO OY ! XCRO COPY Line YI .. .14 Om Die nu WECTION AGE 1452 dopo :2.: TON AGE 19 deyo. A N ECTION AGE 364 doya NECTION AGE 58 do you and-for-tatoto 7, +0.437, 38, 19 +028 i mate 3 RENENTION . 1 LO Emapomonweabe safe porn es manténor afectiva on 240 NWECTION AGE 1923 ore INJECTION AGF. 2410 days. isoon one sapo tunato NJECTIWY AGE 3520 6. DO 160 TIME (daya) : • . : asos, 805 T # 0195 116 2 KTION 2: :.: .. : 1 0 160 240 0 .80 150 240 0 .80 160 . 240 ::. ia ..TIME" (days) ... ,, i Fig. 27.3. Strontium-35 Retention in Dogs. :: . . t .... Xino COLY XAMO XERO conyi سس ممسنجهنمنلمنشينمسلسلسه مهم "" we have • T s(at) where 8 is the matrix vaich has already been defined in Eq. (7). When all At's are equal, we recover Eq. (8) again. Now, taking a specific case, that 18, at, = 1 and 1 = 1,2,...,7, and letting 1 = 0.693, we insert into Eq. (9) and calculate the expected concentrations. Figure 4 presents a plot of the total concentrations, that 18, E(C+,2 # C+2), as a function of time. . As can be seen, the slope of this function is monotonically decreasing. It can be seen from inspection of Eqs. (7) and (9) that as At, gets larger s(at) 1/2 . - and hence the concentration in the second compartment decreases by 1/2, but the time interval for this reduction to occur 18 larger the ferther : out we go. This behavior 18 similar to a power function. To gain some further insight into the effect of age on retention from this model, consider the case where we let at, = 1 + T, ato = 0, and 1 = 0,1,2,3,... Here we let I correspond to the age of an animal at the time of injection. This corresponds to a simulation of aging. The gate opens and closes in a shorter time interval in a younger animal and in a longer time interval in an older animal. For example, in a newborn animal the gate first opens. (or closes) after only one unit of time, while in a 3-year-old animal tihe gate first opens (or closes) after three units of time. We calculate the sum of the expected concentrations in compartments 1 and 2 for this case, and the results ... . - XERO XERO CONY XERO! COPY + + + - - . . -. . -16- are presented graphically in Figure 5. As can be seen, the curves are of the same qualitative shape as Glad and Mays observed in their experiments on 85sr retention in dogs of different ages. It is to be pointeri out here that the trend with age observed in Figure 5 could be reversed by simply changing the initial conditions on the column vector. Recall that we dealt with the specific case where the gate between compartments 1 and 2 18 initially closed. If we were to decide whether this gate was initially opened or closed by a random toss of the coin, then the expected value of the initial concentrations would be 1 Similar store If we used this vector of initial concentrations, then the sum of the expected concentrations in compartments 1 and 2 would behave differently for different interval widths of time. One would expect that as the interval was increased, then, since compartment 2 initially starts out with material in it, it would keep it longer the longer the intervals. The chances for release would be greater the shorter the intervals. Before closing, we list the equation for the two-compartment model, that is, the case where flows between compartments 1 and 2 are random and flow out of compartment 1 also is random. Let ^denote one rate of flow and g. the other rate of flow out of compartment 1. Assume these have equally probable occurrences. Then it can be shown that ... ti # 1. . XERO COPY XERO: XERO COPY! -- ....... . 1- T htt .. L A A TTO RIT M I . . .w . : : . . -27- -------... --- ----- ......-- UNCLASS CD ORHOO 84-7936 UNCLASSICO OROWO €4-17 . . ." , .:. . ::: ... Eig,, + Equita ..: :::. . - ::: - - - - - 1 0. | 0 11 4 i 3 1 2 1 28 1. 8 18 TIME (doya). . : como . . 9. . 20 8 • • 8 2 18 TIME (days) tº 24 : 28 . " ::: ::: : / Fig. 221. Expected Sum of Concentrations in Como ; Fig. (22.5. Expected Sum of Concentrations for Cose partments for Cose of increasing Subinterval Width. :::: of Increasing Subinterval Width and for Difforont Agos. . XERO XIMO Selena .. i EURO - *'aime las t med barn how members who. * Poker 2 Tartu . . C : . . . . -18- 1 (32-246-24/8 (2124440 324 3/8 , Veyat , what i at te? )/8 1/2+1^245 1/2 of (e " + e. 1/8/ 2,2 This solution 18 only for the case where one flips two fair coins at the same time, that 18, it 18 assumed that 'ooth gates are opened (or closed) in the equally spaced intervals of time according as the respective coins fall heads or talls. ... ANA 191. 1 th M . apo XERO XERO LY. copy . 4. FURTHER GENERALIZATIONS Tae two-compartment' model just described can be generalized in several ways. We can conceive of multicompartment systems connected in several possible ways. "Two such systems which can be studied systematically would be (1) the so-called mammillary system where the system consists of peripheral compartments which are connected only to a central compartment or (2) the so-called 'Catenary System' in which the compartments are connected in series with each other. · When once we are in a multicompartment system we can also increase the complexity by considering different kinds of probability schemes. For instance, in a memmillary system with compartments we can consider the case where at each instant of time, only one peripheral compartment 18 in communication with the central compartment with a certain probability or perhaps only a subset of the peripaeral compartments are in communication with the central compartment or one could also consider the case where one might have 23-2 possible events. at each instant of time; viz., none of the peripheral compartments are in communication with the central compartment or exactly one is in communication, or exactly two are in communication and so on. Another possible generalization concerns the principle of 'mixing' which we have lovoked for the apportionment of material in compartments that happen to be connected. . Some of these generalizations have been considered by Bernard, Shenton and Uppulurit. In the case of 3-compartments, the mammillary system is discussed in detail in this paper, and expressions have been derivea Ior the mead ad expressions have been derived for the mean amounts in the compartments of the system. In general, covariances for the amounts in XENO) CONY XE.RO: XERO Copyi : on? TA i -20 the system are nore complicated. For example, for a manillary system, It turns out that the covariance matrix (uncorrected second-order moments) Qm at the mth stage is given in terms of that at the previous stage by the relation j-l Quote P4 A 2-1 Azn m = 1,2, ... where the A;'s are certain matrix operators related to the probability structure of the communicating valves. In the same paper, a three compartment model of a mammillary system is discussed in the appendix, and the feasibility of the application of these models to the experimental data mentioned in this report is studied in detail. .- .. XERO PY .. xr.no C01.Y IXCRO COPY . an21 RHIERENCES 2. . 0.1 1960. 2. R. E. Rowland, Iate Observations of the Distribution of Radium in the Human Skeleton, p. 339 in A Symposium on Redioisotopes in the Blosphere, ed. by R. S. Caldecott and L. A. Snyder, University of Minnesota, Minneapolis, 1960. W. P. Norris, late Effects of Radiation Exposure, presented at Radiation Research Society Meeting, Miami Beach, Fla., May 18-20, 1964. B. W. Glad, C. W. Mays, and W. Fisher, "Strontium Studies in Beagles," Radiation Res. 12, 672-81 (1960). S. R. Bernard, L. R. Shenton and V. R. R. Uppuluri, (1965) Stochastic 4. Models for the Distribution of Radioactive Material in a connected System of Compartments. To appear in the Proceedings of the 5th Berkeley Syraposium (Part IV), University of California Press. Also ORNL Report No. 3809 (1965), Oak Ridge National Laboratory Report. . XERO) CONY XERO XER Cop ------- 1 - - . .. . . . . . . . . . . . 1 .1- . . . 1 END .. DATE FILMED 12/ 23 / 66 **- . 1 . . * S 1 4 :17 . : : P . .. 4T4 .25 '. W P. - . M . ig . I . . ST . . . 5 . . VS .