A circulatory model for human metabolism

Keilson, Julian; Kester, Adri; Waterhouse, Christine

doi:10.1016/0022-5193(78)90240-0

Cited by 13 publications

(5 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A general linear pharmacokinetic system model for drug disposition which takes drug recirculation into account has been proposed (Cutler 1979;Vaughan & Hope 1979;Weiss & Forster 1979), based on previous developments in physiology (Keilson et al 1978;Waterhouse & Keilson 1972). From the open and closed loop transfer functions of such models, mean time parameters dealing both with average recycling times (a specific form of an MTT parameter) and average residence times can be derived.…”

Section: Mean Time Parameters Relating To Recirculationmentioning

confidence: 99%

Mean Time Parameters in Pharmacokinetics

Veng‐Pedersen

1989

Clinical Pharmacokinetics

View full text Add to dashboard Cite

show abstract

Section: Mean Time Parameters Relating To Recirculationmentioning

confidence: 99%

Mean Time Parameters in Pharmacokinetics

Veng‐Pedersen

1989

Clinical Pharmacokinetics

View full text Add to dashboard Cite

show abstract

“…This new Markov chain is governed by a transition matrix a N E X T • The stationary vector e N E X T is related to eE = [ej:j E E] by e N E X T = kEeE, where k E is a renormalization constant. This is intuitively clear because e N E X T describes the relative stationary distribution of the observed set E c J and this is not influenced by non-observation of the set F. A formal proof is available in [7].…”

Section: Evaluation Of the Ergodic Probabilities On The Boundarymentioning

confidence: 89%

“…In this section a simple method for evaluating eJ and e1 is exhibited. The method is based on results for partially observable chains in a biological context [7]. A key ingredient of the method, certain ruin probabilities, of independent interest, are also obtained in this section.…”

Section: Evaluation Of the Ergodic Probabilities On The Boundarymentioning

confidence: 99%

Homogeneous row-continuous bivariate markov chains with boundaries

Keilson¹,

Zachmann²

1988

Journal of Applied Probability

Self Cite

View full text Add to dashboard Cite

In a previous paper with state space B = {(j,n) : 0 < j < J, 0 < n < N) was described as row-continuous in the sense that the marginal process N(t), indexed by row coordinate n, changed at transition epochs by at most one. In the present paper we restrict our discussion to those row-continuous chains for which the transition rate matrices, vn , vn , describing rates =n =n =n local to row n, are independent of n for each 1 s n -N -1. For n = 0, one has vO = , and for n = N, = °0. Such processes may be described as row-homogenous, row-continuous processes modified by two retaining boundaries, as for earlier similar univariate contexts [ 2].For all such processes the behavior of the bounded process is intimately related to that of the associated row-homogeneous processwith n, vn, n independent of n for all integer n. To obtain the distribution of the process B(t) defined on the state space 8 we first obtain that for the associated row-homogeneous process BH(t) defined on the infinite lattice B H . The latter distribution is the time-dependent Green's function gn(t) defined formally below.Two procedures for finding gn(t) will be developed. The ergodic Green's function -Sn ef o n (t )dt is shown to be finite for all n under simple, intuitive conditions on the row-homogeneous process. The row-homogeneous row-continuous process BH(t) = [J(t), NH(t)] isdefined on an infinite lattice B H = {(j,n) : 0 < j < J, -X < n <}. The behavior of B H(t) is governed by three transition rate matrices H', v=H'and vH which define the transitions between the states of a row and the two contiguous rows. Specifically, vH;ij is the hazard rate for transitions from (i,n) to (j, n+l), vH;ij is that from (i,n) to (j,n-l), and vH;ij is that from (i,n) to (j,n). The transition probabilities for the row-homogeneous chain, from (i,O) to (j,n) are the components of the time-dependent matrix Green's function gn(t) = [gij;n(t)]. In particular, we haveWe will call the set {gn(t): -X < n < }1 the time-dependent Green's function for B(t). Because of the row-homogeneity,The generating ij;n-m function g(u,t) = gn(t)un for the two-sided sequence (gnc(t will be of value. Questions of summability and analytic structure have been discussed at length in [6,7].The forward Kolmogorov equations for B H(t) areas the reader may verify from the component equations. Multiplying (1.4)by un and summing we get + =a H is the ergodic probability vector for J(t).Proof: The proof follows directly from (1.3).Although Proposition (1.3) is usefrl for its structural insights, it may also be used for direct calculation of the gn(t) when, for example,= H >> aH so that the cross-products converge quickly.From (1.3) we havewhere ~bkn is the matrix coefficient of u n in [a 0 + uaH + 1The following approach may be used to find _n(s) = £[gn(t)] = jf e -s t gn(t)dt when the upwards and downwards firstpassage time probability matrices are known. Formally, as in [ 8] we define the first passage times Def 1.8 (a) sH(t) = [sij(t)], _H(t) = [s.i(t)] (b) a+(s) = £[sH(t)], a (s) = £[sH(t)](c...

show abstract

“…The one-step transition probability matrix al see Keilson [10], Kemeny et al [14], Schassberger [23], and Vantilborgh [33]. It is known (see e.g., Courtois and Semal [5], Keilson and Kester [11], Meyer [17], Schassberger [23], Sumita and Rieders [27], and Vantilborgh [33]) that J m {k) governed by gl im)L{m) has the ergodic probability vector which is proportional to i£( m ), i.e., _ r…”

Section: Construction Of Replacement Processes For Rank Reductionmentioning

confidence: 99%

“…In the study of a circulatory model for human metabolism, Keilson and Kester [11] developed a Markov chain model where the state space S is decomposed into two sets E and F = S \E. The Markov chain describes the sequence of metabolites and distributions of intervening times.…”

Section: Introductionmentioning

confidence: 99%

A New Algorithm for Computing the Ergodic Probability Vector for Large Markov Chains

Sumita

Rieders

1990

Prob. Eng. Inf. Sci.

View full text Add to dashboard Cite

A novel algorithm is developed which computes the ergodic probability vector for large Markov chains. Decomposing the state space into lumps, the algorithm generates a replacement process on each lump, where any exit from a lump is instantaneously replaced at some state in that lump. The replacement distributions are constructed recursively in such a way that, in the limit, the ergodic probability vector for a replacement process on one lump will be proportional to the ergodic probability vector of the original Markov chain restricted to that lump. Inverse matrices computed in the algorithm are of size (M -1), where M is the number of lumps, thereby providing a substantial rank reduction. When a special structure is present, the procedure for generating the replacement distributions can be simplified. The relevance of the new algorithm to the aggregation-disaggregation algorithm of Takahashi [29] is also discussed.Working Paper Series No. QM88-08. This paper is dedicated with respect to Tomoharu Sekine in honor of his 60th birthday. This paper has been partially supported by the IBM Program of Support for Education in the Management of Information Systems. © 1990 Cambridge University Press 0269-9648/90 $5.00 + .00 89 90 U. Sumita and M. Rieders J(k) on S: the original Markov chain Partial aggregation (L(r) -• r * for r e M, L(m) intact) J (m) (k) on Van der Heyden's algorithm Replacement distributions Total aggregation (L(r) -» r * for all r e M) J(k) on M: Aggregation procedure with output Partial aggregation ( m* -• m* based on n T J m V ) on L(m): J (m) (k) on L(m) u{m**}: Replacement process Disaggregation procedure procedure FIGURE 6.1. Replacement algorithm and aggregation-disaggregation algorithm.

show abstract

A circulatory model for human metabolism

Cited by 13 publications

References 10 publications

Mean Time Parameters in Pharmacokinetics

Mean Time Parameters in Pharmacokinetics

Homogeneous row-continuous bivariate markov chains with boundaries

A New Algorithm for Computing the Ergodic Probability Vector for Large Markov Chains

Contact Info

Product

Resources

About