A Discreet Compartmental Model for Lead Metabolism in the Human Body

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1 A Discreet ompartmental Model for Lead Metabolism in the Human ody Frederika E. Steyn, Stephan V. Joubert and harlotta E. oetzee Tshwane University of Technology Abstract A real-life example of a mathematical technique, employing a socalled transfer matrix, is developed for the compartmental analysis of lead metabolism in the human body. The technique is demonstrated with the aid of ert s four-compartment biokinetic model. The results produced by this time-discrete approach correspond almost perfectly with those of a continuous-time method. The powerful calculation tools of the omputer Algebra System (AS) "Scienti c Workplace" are employed to illustrate the results using tables and graphs. Introduction Holtzman at the Argonne National Laboratory was one of the pioneers who applied numerical models of compartmental analysis to predict lead levels over time in di erent parts of the human body. In 973 Rabinowitz, Wetherill & Kopple ([4], 973) published an article in which they applied compartmental analysis to a three-compartment model for lead metabolism in the human body. This was followed by more complicated models, such as those of atchelet, rand and Steiner ([], 979), ert as reviewed in U.S. EPA ([5], 2), and Leggett ([3], 993). In this article a mathematical technique, employing a socalled transfer matrix is developed and demonstrated with the aid of ert s fourcompartment biokinetic model. Results obtained by this time-discrete approach are compared with those obtained by a continuous-time method. 2 ert s four-compartment model Dainty ([2], 2) de nes a compartment as a theoretical construct that may well combine material from several di erent physical spaces. It is an amount of material that acts as though it is well mixed and kinetically homogeneous. Further, a compartmental model consists of a nite number of compartments with speci ed interconnections between them, which represent the ux of material that is transported from one location to another. y de ning a compartmental model for lead in this way, we can reduce a complex metabolic system into a small number of pathways and compartments. The outputs from such a model include the quantity of lead stored in certain body pools, the rate and extent to which lead moves from one pool to another and how long it will remain in the body before being excreted. Figure is a schematic diagram of the systemic model, showing the compartments and directions of movement of lead among these compartments. As shown, there are four main compartments where lead distributes, namely, blood (#), cortical bone (#2), trabecular bone (#3), and

2 soft tissue (#4). Although numbered 5 and 6 in the diagram, the lungs and gastrointestinal tract are not considered as compartments as they are simply the channels by which lead enters the system. Air Hairs, Nails Sweat τ 4.292/d 4 Soft Tissue x() t 4 Food and drink α τ 4.835/d β 5 Lungs τ 4.235/d τ 64.43/d ile,saliva, Gastric secr. 6 Gastrointestinal tract p.35 lood x () t q.8 Faeces τ 2.325/d τ τ.2/d Urine τ 2.578/d τ 3.24/d 2 ortical one x () t 2 3 Trabecular one x () t 3 Figure : A four-compartment model. The rate at which lead is inhaled into the lungs is g/day. ert ([5], 2) established that a portion p :35 of this lead is absorbed into the blood. The rate at which lead enters the blood via the respiratory system is thus y 5 :35, g/day. () Lead in food and drink enters the body via the gastrointestinal tract. There is also a translocation of lead from the soft tissue to the digestive system in bile, saliva and other gastric secretions. It is assumed that the rate of transfer of lead from the soft tissue to the digestive tract is directly proportional to the amount of lead x 4 (t) in the soft tissue. That is, y x 4, g/day, (2) where y 64 is the transfer rate in g/day from compartment 4 (the soft tissue) to system 6 (the gastrointestinal tract). The symbol 64 is the transfer coe cient from soft tissue to the digestive tract, and is the fraction of lead in compartment 4 that is translocated to system 6 per day. If g is the daily intake of lead by mouth and q :8 (according to ert ([5], 2)) the portion of lead + y 64 2

3 that is absorbed by the blood, then the rate at which lead is translocated from the digestive system to the blood is y 6 :8 ( + y 64 ) :8 ( + 64 x 4 ), g/day. (3) The pathways with directions of transfer of lead in Figure indicate the interconnection between compartments. Associated with each pathway and direction is a constant, ij, which denotes the transfer rate of lead from compartment j to compartment i. The constant rate, ij, is called the transfer coe cient of lead from compartment j to compartment i and is the fraction of the amount x j (t) of lead in compartment j that is translocated to compartment i per day. So, the rate of lead transfer from compartment j to compartment i is given by y ij ij x j, g/day. (4) Denoting the environment by and using values found in the aforementioned publication, U.S. EPA ([5], 2), we construct the following table of values for ij for a normal healthy human adult male. Table : Transfer coe cients ij of lead (/day) j j 2 j 3 j 4 i :2 :292 i :325 :229 :235 i 2 :578 i 3 :24 i 4 :835 i 6 :43 Note that it is assumed that the transfer rate for lead from compartment j to i is directly proportional to the amount x j (t) of lead in compartment j and that the constant of proportionality, ij, is a daily fraction of x j. 3 The transfer matrix (a discrete model) In this article the custom in many textbooks is followed, namely to denote a column vector (n matrix) by a small letter in boldface and an mn matrix by APITALS (not in boldface). Let the entry x i (t) in the 4 column vector x, x (t) x (t) x 2 (t) x 3 (t) x 4 (t) A, x (t + t) x (t + t) x 2 (t + t) x 3 (t + t) x 4 (t + t) represent the amount of lead (in g) in compartment i at time t. The column vector x (t) speci es the state of the system at time t. The 4 column vector x (t + t) is the state of the system t units of time later. Assume that a xed fraction ij of the lead in compartment j is passed to compartment i every t units of time. Note that it is assumed that the system changes only A, 3

4 at times t + t; t + 2t; : : : ; t + nt; : : :. The calculations of x i (t + t) for the four compartments are as follows: x (t + t) x (t) + [ 2 x 2 (t) + 3 x 3 (t) + 4 x 4 (t) + y 5 + y 6 ] [ x (t) + 2 x (t) + 3 x (t) + 4 x (t)]. Use Equations () and (3) to obtain [ x (t + t) ( )] x (t) + 2 x 2 (t) + 3 x 3 (t) + ( 4 + :8 64 ) x 4 (t) + :35 + :8. If ( ), then is merely the fraction of the lead content of compartment that remains in, and so x (t + t) x (t) + 2 x 2 (t) + 3 x 3 (t) + ( 4 + :8 64 ) x 4 (t) + :35 + :8. (5) x 2 (t + t) x 2 (t) + 2 x (t) 2 x 2 (t) ( 2 ) x 2 (t) + 2 x (t) 22 x 2 (t) + 2 x (t), (6) where 22 remains in 2. where 33 remains in 3. 2 is the fraction of the lead content of compartment 2 that x 3 (t + t) 33 x 3 (t) + 3 x (t), (7) 3 is the fraction of the lead content of compartment 3 that x 4 (t + t) x 4 (t) + 4 x (t) [ 4 x 4 (t) + 4 x 4 (t) + 64 x 4 (t)] [ ( )] x 4 (t) + 4 x (t) 44 x 4 (t) + 4 x (t), (8) where 44 ( ) is the fraction of the lead content of compartment 4 that remains in 4. The last four equations, Equations (5) to (8), can be written as follows in matrix form: x (t + t) T x (t) + b, (9) where T : A : :325 :229 : :578 : :24 :9977 :835 :983 3 A () and b :35 + :8 T. () 4

5 The matrix T is referred to as the transfer matrix. It follows from Equation (9) that x (t + 2t) T x (t + t) + b T 2 x (t) + T b + b, x (t + 3t) T x (t + 2t) + b T 3 x (t) + T 2 b + T b + b.. x (t + nt) T n x (t) + T n b + T n 2 b + : : : + T 2 b + T b + b. Rearranging and grouping terms gives x (t + nt) T n x (t) + I + T + T 2 + : : : + T n 2 + T n b, (2) where I is the 4 4 identity (unit) matrix. Let Premultiply by T to obtain S I + T + T 2 + : : : + T n 2 + T n : (3) T S T + T 2 + T 3 + : : : + T n 2 + T n + T n : (4) Subtract Equation (4) from Equation (3) to obtain (I T ) S I T n. If (I T ) is nonsingular, then (I T ) exists and premultiplication by (I T ) is possible. So, S (I T ) (I T n ). Using this result, Equation (2) can be re-expressed as Let t and t day, then x (t + nt) T n x (t) + (I T ) (I T n ) b. x (n) T n x () + (I T ) (I T n ) b, (5) where x () is the initial state of the system at time t and n is the number of days, starting to count from t. Since the determinant ji T j 6, (I T ) can be calculated as follows: (I T ) 44: : : : : : : : 7 46: : : : : : : : A system of ODEs (continuous model) A. (6) The ow of lead through the human body is based on the basic mass balance law x i (t) dx i dt X X y ij y ji (g/day), (7) j j 5

6 where x i (t) is the rate at which the mass of the lead in compartment i changes with time; y ij is the time rate at which lead is transferred from compartment j to compartment i; and y ji is the time rate at which lead is transferred from compartment i to compartment j. Applying this law to ert s four-compartment model yields the following four coupled linear rst-order ordinary di erential equations: x 2 x 2 (t) + 3 x 3 (t) + [ 4 + :8 64 ] x 4 (t) + :35 + :8 [ ] x (t). (8) x 2 (t) 2 x (t) 2 x 2 (t). (9) x 3 (t) 3 x (t) 3 x 3 (t). (2) x 4 (t) 4 x (t) [ ] x 4 (t). (2) If the values for the transfer coe cients, ij, in Table are substituted into Equations (8) to (2), these equations can be written in matrix form as where x (t) and A x (t) x 2 (t) x 3 (t) x 4 (t) A ; x (t) x (t) Ax (t) + b, (22) x (t) x 2 (t) x 3 (t) x 4 (t) A ; b :3 5 :325 :229 : :578 :325 :24 :229 :835 :6 7 :35 + :8 A ; A : (23) Matrix A with constant entries is referred to as the coe cient matrix and column vector b as the input. 5 A lead environment Rabinowitz et al. ([4], 973) tested a healthy 53-year-old man, weighing 7 kg, whose diet contained, on average, 367 g/day Pb. It was estimated that the testee inhaled, on average, 49 g/day Pb. Substituting into Equation (), gives 49 g/day and 367 g/day b 46:5 T. (24) onsider a healthy adult male who, at time t, had no lead stored in his body, that is, x () : 6

7 With this initial condition, Equation (5) becomes x (n) (I T ) (I T n ) b. (25) Equation (25) predicts the lead content of each compartment, after n days of lead intake. Now, consider the nonhomogeneous system (22). Using standard methods for the solution of the system, the following equations are obtained: x (t) lo o d x 2 (t) o r t ic a l x 3 (t) Trab ecular x 4 (t) T is s u e 8 < : 8 < : 8 < : 8 < : 423:633e :365t 72:832 7e :63369t 47:924e :29788t 424:9 6e : t +269:29 (26) 26:583e :365t + 25:872 7e :63369t +43:98 e :29788t 3: e : t + 3: (27) 6:535e :36 5t + 2:4733e :63369t 847:3e :29788t 45:3925e : t +268:688 75:273e :365t 337:23 9e :63369t 8:589 4e :29788t 46:76 49e : t + 227:374 The functions (26) to (29) are plotted in Figure 2 below. (28) (29) 2 5 lood x(t) ( µ g) ortical bone 5 Trabecular bone 5 t (days) 5 Soft tissue Figure 2: Lead levels during lead intake (continuous model) The graph for blood in Figure 2 shows that during lead intake, the rate at which lead in the blood increases is rapid at the beginning but slows down with time, and that the lead level in the blood has practically reached equilibrium after 2 days. In other words, after 2 days of lead intake, the rate at which lead is passed from the other compartments to the blood nearly equals the rate at which lead is passed from the blood to the other compartments. During lead intake, the lead level in the soft tissue increases at a much lower rate than that in the blood and also reaches an equilibrium state after 2 days. After 2 days, 2 7

8 blood and tissue levels approach equilibrium, but bone lead soars, especially in cortical bone. These trends of lead levels in the blood, the soft tissue and the skeleton are con rmed by published results. Equation (25) which has been derived for the discreet model produces Table 2 below: Table 2: Distribution of lead at intervals of days t (days) x (g) x 2 (g) x 3 (g) x 4 (g) () () 45 (4) (2) 5 (5) 3 (4) 2 7 (694) 43 (44) 8 (8) 2 (3) 3 98 (9) 89 (9) 36 (37) 24 (24) 4 77 (69) 47 (48) 59 (6) 37 (37) 5 94 (86) 22 (23) 85 (85) 5 (5) 6 28 (274) 283 (85) 2 (2) 64 (64) (338) 359 (36) 4 (4) 76 (76) (387) 438 (439) 7 (7) 88 (87) (423) 59 (52) 99 (2) 98 (98) 454 (45) 62 (63) 229 (229) 8 (7) 475 (472) 687 (687) 259 (259) 6 (5) 2 49 (488) 772 (772) 288 (288) 23 (23) 3 53 (5) 858 (859) 37 (37) 29 (29) 4 52 (5) 945 (945) 346 (345) 35 (35) 5 52 (59) 32 (33) 374 (374) 4 (4) (526) 2 (2) 42 (4) 44 (44) (532) 28 (28) 429 (428) 48 (48) (536) 296 (296) 456 (455) 5 (5) 9 54 (54) 385 (385) 482 (48) 54 (54) (544) 474 (474) 57 (57) 56 (56) The gures in Table 2 are rounded o to the nearest g and those within parentheses are the calculations from Equations (26) to (29). lood x(t) ( µ g) ortical bone Trabecular bone Soft tissue t (days) Figure 3: Lead levels during lead intake (discrete model) The graphs in Figure 3 depict the results in Table 2, generated by Equation 8

9 (25). The graphs displayed in Figure 3 are almost identical to those displayed in Figure 2. 6 A nonlead environment Suppose that after 2 days of lead intake the testee moves to a nonlead environment, that is, and. The question now is: How will the lead levels in the compartments change with time in a lead-free environment after the 2 days of lead intake? The problem can be stated as follows: Determine x (t) if x () 544: 4 473: 5 57: 5 56: 3 T : For this situation, Equations (5) and (22) respectively become and x (n) T n x () (3) x (t) Ax (t) (3) since b. Using Equation (3), lead levels in the four compartments are calculated for every th day, starting from the moment the testee stops taking in lead. Table 3 summarizes the calculations to the nearest g, and Figure 4 is a graphic representation of the data. The gures within parentheses in Table 3 are the calculations from Equations (32) to (35) below. Table 3: Lead levels during isolation from lead t (days) x (g) x 2 (g) x 3 (g) x 4 (g) 544 (544) 474 (474) 58 (58) 56 (56) 42 (47) 55 (55) 528 (527) 55 (54) (857) 69 (67) 54 (539) 48 (47) (643) 65 (65) 545 (544) 38 (37) (487) 683 (682) 546 (545) 26 (26) (37) 77 (76) 544 (543) 4 (3) 6 28 (286) 725 (724) 539 (538) 2 (2) 7 28 (224) 739 (738) 533 (532) 9 (9) 8 73 (77) 75 (749) 526 (525) 79 (8) 9 39 (43) 758 (758) 58 (57) 7 (7) 4 (7) 765 (765) 59 (58) 6 (62) 95 (98) 77 (77) 5 (499) 53 (54) 2 8 (83) 775 (775) 49 (49) 47 (47) 3 7 (73) 779 (779) 48 (48) 4 (4) 4 63 (64) 782 (782) 472 (472) 36 (36) 5 57 (58) 785 (785) 463 (462) 3 (3) 6 52 (53) 788 (788) 454 (453) 27 (28) 7 48 (49) 79 (79) 444 (444) 24 (24) 8 45 (45) 792 (792) 435 (435) 2 (2) 9 42 (43) 794 (794) 427 (426) 8 (9) 2 4 (4) 796 (796) 48 (48) 6 (7) 2 39 (39) 798 (798) 49 (49) 4 (5) (37) 799 (799) 4 (4) 3 (3) 9

10 is The exact solution to the system of linear rst-order ODEs in Equation (3) x (t) lo o d x 2 (t) o r t ic a l x 3 (t) Trab ecular x 4 (t) T is s u e 42:76e :36t + 7:38 22e :634t +5:688 59e :297 8t + 2:9 83e :25839t (32) 26:5 e :36t 24:879 6e :634t 4:857 e :297 8t + 9:487e :25839t (33) 6:322 e :36t :994 82e :634t +633:7 8e :297 8t + 2:32 477e :25839t (34) 74:892 6e :36t + 324: 295 6e :634t +6:369 84e :297 8t + :24 9 6e :25839t (35) 8 ortical bone 6 4 x(t) ( µ g) Trabecular bone 2 Tissue lood t (days) Figure 4: Lead levels in a nonlead environment (discrete model) Figure 4 shows that as soon as the testee stops taking in lead, his blood lead immediately decreases rapidly, while the lead in his soft tissue decreases slowly. Lead in trabecular bone still builds up during the rst 4 days because of the presence of lead in his blood. After about 4 days, the lead in trabecular bone starts to decrease slowly. Even when lead levels in the other compartments are decreasing, the lead in cortical bone continues to increase for a long time. Equation (33) calculates that only after three years does the lead level in cortical bone start to drop. After years there is still a considerable amount of lead in cortical bone. one lead acts as an endogenous source of lead, and by continuous release to other tissues it can represent a health risk long after exposure has ended

11 REFERENES [] ATHELET, E., RAND, L. & STEINER, A On the kinetics of lead in the human body. Journal of Mathematical iology, 8, July: [2] DAINTY, J.R. 2. Use of stable isotopes and mathematical modelling to investigate human mineral metabolism. Nutrition Research Reviews. 4: Online, available at: < /2/4/2>. Accessed: 6//8. [3] LEGGETT, R.W An age-speci c kinetic model of lead metabolism in humans. Environmental Health Perspectives, (7), December: [4] RAINOWITZ, M.., WETHERILL, G.W. & KOPPLE, J.D Lead metabolism in the normal human: stable isotope studies. Science, 82(3), November: [5] U.S. ENVIRONMENTAL PROTETION AGENY (U.S. EPA). 2. Review of adult lead models: Evaluation of models for assessing human health risks associated with lead exposures at non-residential areas of Superfund and other hazardous waste sites. Document EPA, OSWER # , August: -85. Online, available at: < contaminants/lead/index.htm>. Accessed: 6//8.