Characterization of Toxicokinetics and Toxicodynamics with Linear Systems Theory: Application to Lead-Associated Cognitive Decline.
We present a theoretical approach to analysis of toxicokinetics and toxicodynamics using linear systems theory. In our approach, we define two impulse response In simple terms, the impulse response of a system is its output when presented with a very brief signal, an impulse. While an impulse is a difficult concept to imagine, and an impossible thing in reality, it represents the limit case of a pulse made infinitely short in time functions that characterize the kinetic kinetic /ki·net·ic/ (ki-net´ik) pertaining to or producing motion.
Of, relating to, or produced by motion.
pertaining to or producing motion. behavior of an environmental agent in the body and the dynamic time-course behavior of its effect on the body. This approach provides a formalism Formalism
or Russian Formalism
Russian school of literary criticism that flourished from 1914 to 1928. Making use of the linguistic theories of Ferdinand de Saussure, Formalists were concerned with what technical devices make a literary text literary, apart for understanding the relation among exposure, dose, and cumulative biologically effective dose and for understanding the implications of an effect time-course on cross-sectional and longitudinal data analyses. We use lead-associated cognitive decline as a specific example where the approach may be applied. Key word: bone lead, exposure assessment, linear systems model, toxicodynamics, toxicokinetics. Environ en·vi·ron
tr.v. en·vi·roned, en·vi·ron·ing, en·vi·rons
To encircle; surround. See Synonyms at surround.
[Middle English envirounen, from Old French environner Health Perspect 109:361-368 (2001). [Online 16 March 2001]
Environmental and occupational epidemiologic ep·i·de·mi·ol·o·gy
The branch of medicine that deals with the study of the causes, distribution, and control of disease in populations.
[Medieval Latin epid research involves the identification of relations between past exposures to putative Alleged; supposed; reputed.
A putative father is the individual who is alleged to be the father of an illegitimate child.
A putative marriage is one that has been contracted in Good Faith and pursuant to ignorance, by one or both parties, that certain toxicants and subsequent adverse health effects in individuals within study populations. Such relations are often hard to fully characterize because of difficulties in accurately quantifying exposure, dose, and effect. Meaningful quantification is particularly difficult because the exposure--effect relation arises from a multistage mul·ti·stage
1. Functioning in more than one stage: a multistage design project.
2. Relating to or composed of two or more propulsion units. process, often referred to as the toxicologic paradigm. As a result, the use of biomarkers in molecular epidemiology molecular epidemiology Molecular medicine An evolving field that combines the tools of standard epidemiology–case studies, questionnaires and monitoring of exposure to external factors with the tools of molecular biology–eg, restriction endonucleases, research has gained widespread attention (1).
In using biomarkers in environmental epidemiology epidemiology, field of medicine concerned with the study of epidemics, outbreaks of disease that affect large numbers of people. Epidemiologists, using sophisticated statistical analyses, field investigations, and complex laboratory techniques, investigate the cause , it is critical to consider the actual process or parameter a given biomarker biomarker /bio·mark·er/ (bi´o-mahr?ker)
1. a biological molecule used as a marker for a substance or process of interest.
2. tumor marker.
1. reflects. For example, the concepts of exposure, internal dose, and biologically effective dose are often blurred blur
v. blurred, blur·ring, blurs
1. To make indistinct and hazy in outline or appearance; obscure.
2. To smear or stain; smudge.
3. in practice, and the loss of these distinctions can influence interpretation of data. Measures of exposure or internal dose are often assumed to be surrogates of the biologically effective dose; their use implies a set of assumptions that is not usually fully articulated or considered. These assumptions involve considerations of the toxicokinetics of the agent and "exposure" biomarker, which are influenced by varying exposure intensity and duration, the residence time of the active form of the agent at the sensitive target, saturation saturation, of an organic compound
saturation, of an organic compound, condition occurring when its molecules contain no double or triple bonds and thus cannot undergo addition reactions. effects, and release from body stores. Similarly, measures of health effects depend on the time course of response to a given exposure (the toxicodynamics of the agent and "response" biomarker) and are influenced by varying response magnitude and duration, dose-dependent repair mechanisms, and multiagent synergistic effects Synergistic effect
A violation of value-additivity in that the value of a combination is greater than the sum of the individual values. .
In practice, simplifying assumptions are usually made to directly relate exposure to biologically effective dose and to relate this surrogate surrogate n. 1) a person acting on behalf of another or a substitute, including a woman who gives birth to a baby of a mother who is unable to carry the child. 2) a judge in some states (notably New York) responsible only for probates, estates, and adoptions. of dose to health effect. These simplifying assumptions generally overlook the potentially important influence of bidirectional The ability to move, transfer or transmit in both directions. transfer of the agent from one anatomic anatomic /ana·tom·ic/ (an?ah-tom´ik) anatomical.
Related to the physical structure of an organ or organism. or physiologic compartment compartment
a part of the body as a whole and divided from the rest by a physical partition.
that liquid part of the body excluded by cell membranes. Includes intravascular and intercellular compartments. to another. Less frequently, more sophisticated approaches to relating exposure to internal dose and internal dose to biologically effective dose at the sensitive target have been used. In general, these approaches are based on multicompartmental pharmacokinetic modeling (2-4). Even with sophisticated approaches to characterizing the toxicokinetics of a given agent--effect paradigm, simplified models of response are usually used in epidemiologic investigations (e.g., the assumption of an irreversible irreversible (ir´ēvur´sebl),
adj incapable of being reversed or returned to the original state. , static response to a given exposure). To help understand these issues, we present a conceptual framework For the concept in aesthetics and art criticism, see .
A conceptual framework is used in research to outline possible courses of action or to present a preferred approach to a system analysis project. based on linear systems theory and its application to the analysis of lead-associated neurocognitive decline. We specifically consider issues of residence time of the agent at the sensitive target, later release of the agent from body stores (with corresponding re-residence at the sensitive target), and the time course of response.
A fundamental assumption in dose--response studies of toxic agents is that the active form of the agent at the sensitive target site causes the effect (4). Thus, to characterize the relation between exposure to an environmental agent and subsequent development of an adverse health effect, two types of processes must be modeled: the toxicokinetics that describe the relation between environmental exposure and ultimate cumulative dose at the sensitive target, and the toxicodynamics that describe the relation between this cumulative dose at the sensitive target and the adverse effects. In practice, it is frequently assumed that the total adverse effect is proportional to the area under the curve ([AUC AUC
area under curve .sub.T]) of the time-concentration relation for the active form of the agent at the target site, (4):
 [AUC.sub.T] = [integral of] [C.sub.T] (t)dt
Thus, it is highly desirable in environmental epidemiologic investigations to be able to estimate [AUC.sub.T] and to be able to characterize the relation between [AUC.sub.T] and the observed effect at any given measurement time.
Toxicokinetics: Measures of [AUC.sub.T]
A common surrogate for [AUC.sub.T] is cumulative exposure, E, defined as the integral exposure to a certain time-dependent environmental concentration of a toxic agent, [C.sub.E](t):
 E = [integral of] [C.sub.E](t)dt
Another common surrogate for [AUC.sub.T] is the cumulative dose, D, which is frequently based on the assumption of a linear relation between exposure and dose:
 D(t) = kE(t) D = k [integral of] [C.sub.E](t)dt
The use of either cumulative exposure or cumulative dose as a surrogate for [AUC.sub.T] implies the assumption of a linear relation between [AUC.sub.T] and either E or D, with y-intercept equal to zero:
 [AUC.sub.T] =[k.sub.E]E = [k.sub.D]D
In other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently , Equation 4 implies that the toxicokinetics are strictly linear; that is, the transfer of a toxic agent from the environment to the sensitive target (including bioactivation, if relevant) follows a linear dependence. Of perhaps greater importance, Equation 4 implies that the residence time of the agent at the sensitive target is relatively brief, and that, if the agent is concentrated in body stores, it is not subsequently released and made bioavailable to the sensitive target. This conclusion follows directly from characterization of the transfer of an environmental agent to the body [i.e., from exposure, [C.sub.E](t), to internal dose, [C.sub.D](t)] and from the initial biodistribution in the body to the sensitive target [i.e., biologically effective dose, [C.sub.T](t)] with linear systems theory (5), as discussed below. If these assumptions are incorrect, [C.sub.T](t) will be significantly different in shape than either [C.sub.E](t) or [C.sub.D](t) [i.e., [C.sub.T](t) will be spread out compared with either [C.sub.E](t) or [C.sub.D](t)], even in the case of a truly linear relation between exposure and internal dose, and the use of Equation 4 will lead to an underestimation of the actual [AUC.sub.T]. Thus, the common use of the cumulative exposure index [i.e., integrated exposure over time (Equation 2)] (4) may underestimate the [AUC.sub.T].
For the sake of our discussion, a system is considered linear if the relation between the input [e.g., [C.sub.E](t)] and the output [e.g., [C.sub.T](t)] has the properties of additivity and scaling. A system has the property of additivity if the sum of the outputs from two independent inputs equals the unified output from the sum of the inputs. That is, if the function f that describes a system is linear, then:
f[[x.sub.1](t) + [x.sub.2](t)] = f[[x.sub.1](t)] + f[[x.sub.2](t)].
In our case, this implies that the sum of the two actual time--concentration curves at the sensitive target resulting from two separate exposures would be the same as the single time--concentration curve resulting from the sum of the exposures. That is, we assume that a single, complex time-varying exposure may be conceptualized as a series of intensity-scaled instantaneous in·stan·ta·ne·ous
1. Occurring or completed without perceptible delay: Relief was instantaneous.
2. exposures, and that the actual observed time--concentration curves resulting from the single, complex time-varying exposure can be modeled as the sum of individual time--concentration curves from the series of intensity-scaled instantaneous exposures.
A system has the property of scaling if the output is scalable by the input:
f[ax(t)] = af[x(t)].
In our case, this implies that the time--concentration curve at the sensitive target multiplicatively mul·ti·pli·ca·tive
1. Tending to multiply or capable of multiplying or increasing.
2. Having to do with multiplication.
mul scales with a multiplicative mul·ti·pli·ca·tive
1. Tending to multiply or capable of multiplying or increasing.
2. Having to do with multiplication.
mul change in exposure. For example, if the exposure intensity doubles, we assume that the time--concentration curve will double (i.e., the concentration will double, and the curve will retain the same shape).
Additivity and scaling are usually combined into the principle of superposition Noun 1. principle of superposition - (geology) the principle that in a series of stratified sedimentary rocks the lowest stratum is the oldest
superposition principle, superposition . In general, biological systems follow this super-position principle, up to the point of saturation or other mass effects at very high concentrations.
An equally important consideration for a linear system is time invariance in·var·i·ant
1. Not varying; constant.
2. Mathematics Unaffected by a designated operation, as a transformation of coordinates.
An invariant quantity, function, configuration, or system. . In a time-invariant system A time-invariant system is one whose output does not depend explicitly on time.
Causing irritation, especially physical irritation.
A source of irritation.
n 1. an agent that causes an irritation or stimulation.
2. may influence airway airway /air·way/ (-wa)
1. the passage by which air enters and leaves the lungs.
2. a device for securing unobstructed respiration. caliber or mucociliary clearance, thus changing the toxicokinetics of subsequent exposures. For the example given below (lead-associated neurocognitive decline), we assume that the system is time invariant (programming) invariant - A rule, such as the ordering of an ordered list or heap, that applies throughout the life of a data structure or procedure. Each change to the data structure must maintain the correctness of the invariant. .
Given a linear, time-invariant system, the output [e.g., [C.sub.T](t)] for any given arbitrary input [e.g., [C.sub.E](t)] can be directly predicted from knowledge of the input and the system's impulse response function (IRF IRF Interferon Regulatory Factor
IRF International Religious Freedom
IRF Institut for Rationel Farmakoterapi (German)
IRF Inherited Rights Filter (Novell)
IRF Inherited Rights Filter ). This IRF characterizes the system's response (i.e., the output) to an infinitely short duration input (mathematically equivalent to a delta or Dirac function in time, denoted as [Delta]) (5):
 [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]
From a toxicokinetic perspective, [IRF.sub.TK] is the [C.sub.T](t) curve that would be observed from a single, infinitely short duration exposure. Because [IRF.sub.TK] is a time-varying curve and is conceptually derived from a single pulse exposure, its use implies that the ongoing toxicokinetics of even a single molecule of a toxicant toxicant /tox·i·cant/ (tok´si-kant)
1. A poison or poisonous agent.
2. An intoxicant.
adj. could be captured by this formalism. For example, [IRF.sub.TK] could represent the behavior of a single lead molecule that enters the brain (and produces an effect), is cleared and stored in bone, is then released back to the blood, and reenters the brain (and produces a second effect).
Assuming that the toxicokinetics can be described by a linear, time-invariant system, the observed target site concentration--time curve from an arbitrary exposure time course is given by the mathematical convolution convolution /con·vo·lu·tion/ (-loo´shun) a tortuous irregularity or elevation caused by the infolding of a structure upon itself. (denoted by [cross product]) of the actual exposure time course with the target site [IRF.sub.TK](5):
 [C.sub.T](t) = [k.sub.E][C.sub.E](t) [cross product] [IRF.sub.TK](t),
where [C.sub.E](t) is the exposure time--concentration curve, and [k.sub.E] is a constant relating units of exposure to units of biologically effective dose. Equation 6 may be recast re·cast
tr.v. re·cast, re·cast·ing, re·casts
1. To mold again: recast a bell.
2. as the standard convolution integral:
 [C.sub.T](t) = [k.sub.E] [integral of] [C.sub.E]([Tau])[IRF.sub.TK](t-[Tau])d[Tau],
where [Tau] is a dummy variable This article is not about "dummy variables" as that term is usually understood in mathematics. See free variables and bound variables.
In regression analysis, a dummy variable of integration. Of major importance, if either of the two terms in the convolution [e.g., [C.sub.E](t) or [IRF.sub.TK] Equations 6 and 7] is a [Delta] function, then that term drops out of the convolution, and the relation reduces to a straight equivalency equivalency
the combining power of an electrolyte. See also equivalent. . That is, if the residence time of a toxicant at the sensitive target is very short, and if there is no subsequent bioavailability bioavailability /bio·avail·a·bil·i·ty/ (bi?o-ah-val?ah-bil´i-te) the degree to which a drug or other substance becomes available to the target tissue after administration.
n. due to release from body stores, then [IRF.sub.TK] is essentially a [Delta] function, and the assumptions implicit in Adj. 1. implicit in - in the nature of something though not readily apparent; "shortcomings inherent in our approach"; "an underlying meaning"
underlying, inherent Equation 4 are valid. On the other hand, if the residence time is significant, then [IRF.sub.TK] is not a [Delta] function, but a curve with some spread in time, and Equation 7 must be used (with Equation 1) instead of Equations 2 and 4 to determine [AUC.sub.T]. In a similar fashion, if there is significant subsequent release from body stores, [IRF.sub.TK] will be multi-peaked or have an initial peak (representing the initial transfer from the environment), followed by non-zero values over time (representing the release from body stores), and Equation 7 must again be used to compute To perform mathematical operations or general computer processing. For an explanation of "The 3 C's," or how the computer processes data, see computer. [AUC.sub.T] via Equation 1.
In essence, one extreme scenario for the IRF and Equations 6 and 7 is that [IRF.sub.TK] is a [Delta] function, implying that the residence time is essentially zero (i.e., the active agent rapidly transits through the sensitive target) and that no release from body stores occurs. The opposite scenario from a toxicokinetic perspective is that the active agent permanently resides in the sensitive target; that is, [IRF.sub.TK] is a constant with time (either because the agent never clears from the sensitive target or because ongoing significant biorelease from body stores constantly replenishes that amount of agent cleared from the target). In general, Equations 1 and 7 can be combined, with [AUC.sub.T] expressed as a double integral, with integration limits from time zero to the current observation time, T:
 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If [IRF.sub.TK] is a [Delta] function, the inner integral drops out, and Equation 8 reduces to that relation implied by Equations 3 and 4:
 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
If [IRF.sub.TK] a constant function, it drops out of the inner integral in Equation 8, but the double integration remains:
 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[Note that [IRF.sub.TK] is causal, which means that it results from an event in real time; thus, [IRF.sub.TK] (t [is less than] 0) = 0. This assumption facilitates the simplification of Equation 8 to either Equation 9 or Equation 10.]
Equations 9 and 10 define opposite scenarios. If the actual [IRF.sub.TK] is closer to a [Delta] function than a constant, the use of Equation 9 in environmental epidemiologic investigations should provide biologically effective dose estimates that are better predictors of health outcomes than would the use of Equation 10. If the actual [IRF.sub.TK] is significantly spread, either because of significant residence time or significant release over time from body stores, then the use of Equation 10 should provide better biologically effective dose estimates, and thus stronger and more consistent associations with the health outcomes under study.
Toxicodynamics: Time-Dependent Measures of Response
In the previous section, we used linear systems theory to characterize the time course of the active form of the agent at the sensitive target, [C.sub.T](t). We now want to characterize the time course of the health outcome or response, R(t). As with [C.sub.T](t), R(t) can be conceptualized as a linear system characterized by an IRF. Since we designated the IRF for the toxicokinetic relation [IRF.sub.TK], a similar IRF, designated [IRF.sub.TD], can be used to characterize the toxicodynamics:
 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From a toxicodynamic perspective, [IRF.sub.TD] is the R(t) curve that would be observed from a single, infinitely short duration [C.sub.T](t) curve [i.e., in the case where both [C.sub.E](t) and [IRF.sub.TK] are (5 functions]. The shape of [IRF.sub.TD] directly indicates reversible reversible,
adj capable of going through a series of changes in either direction, forward or backward (e.g., reversible chemical reaction).
n See hydrocolloid, reversible. versus irreversible (persistent) versus progressive effects from a single exposure, as shown in Figure 1. A reversible effect would yield an [IRF.sub.TD] that goes from zero at the point of exposure to some response value and back to zero. An irreversible effect would yield an [IRF.sub.TD] that goes from zero at the point of exposure to some response value that persists, independent of time. A progressive (increasing) effect (from a previous exposure) would yield an [IRF.sub.TD] that continuously increases with time, starting from zero at the point of exposure.
The utility of Equation 11 is that it identifies issues in the design of an epidemiologic study epidemiologic study A study that compares 2 groups of people who are alike except for one factor, such as exposure to a chemical or the presence of a health effect; the investigators try to determine if any factor is associated with the health effect in which response is measured either cross-sectionally at some time after exposure ceases, or longitudinally. For example, a cross-sectional design uses a surrogate of cumulative exposure and a single measure of response at some later time. Equation 11 tells us that such a design is only applicable if [IRF.sub.TK] is a [Delta] function and if [IRF.sub.TD] is a step function (i.e., the effect is persistent rather than either reversible or progressive). In a longitudinal study longitudinal study
a chronological study in epidemiology which attempts to establish a relationship between an antecedent cause and a subsequent effect. See also cohort study. , it is imperative to consider possible shapes for [IRF.sub.TD], because the observed R(t) implies underlying [IRF.sub.TK] and [IRF.sub.TD] functions that are usually not directly obtainable. For example, the observation of a progressive increase in effect with time could be the result of a) the ongoing presence of active agent at the sensitive target, due to either long residence time or ongoing release from body stores, either of which would cause [AUC.sub.T] to continue to increase with time; b) progressive response from past exposures; or c) a combination of the two. Equation 11 suggests that the observed R(t) could be the result of a significantly non-[Delta] function IRF for either the toxicokinetic or toxicodynamic portion of the toxicologic paradigm. For example, from a purely mathematical point of view, the same R(t) would be observed if either [IRF.sub.TK] or [IRF.sub.TD] were a [Delta] function and the other were a step function.
Application to Longitudinal Data Analysis
Bandeen-Roche et al. (7) have previously described a general model for analysis of data from prospective observational studies observational studies,
n.pl an investigational method involving description of the associations be-tween interventions and outcomes. Outcomes research and practice audits are examples of this investigational method. with multiple outcome measures over time. Their model includes a family of exposure summaries whose mathematical formalism is a convolution integral similar to Equation 7, although they did not approach the overall model's conceptualization con·cep·tu·al·ize
v. con·cep·tu·al·ized, con·cep·tu·al·iz·ing, con·cep·tu·al·iz·es
To form a concept or concepts of, and especially to interpret in a conceptual way: with linear systems. In their data analysis model, outcome or response is a function of exposure history plus a random error. With our notation notation: see arithmetic and musical notation.
How a system of numbers, phrases, words or quantities is written or expressed. Positional notation is the location and value of digits in a numbering system, such as the decimal or binary system. :
 R(t) = F[[C.sub.T](t)] + [Epsilon 1. (language) EPSILON - A macro language with high level features including strings and lists, developed by A.P. Ershov at Novosibirsk in 1967. EPSILON was used to implement ALGOL 68 on the M-220. ](t)
In their approach, F[.] represents a complex function representing the effects of the family of exposure summaries. They use this construct to create a generalized linear model Not to be confused with general linear model.
In statistics, the generalized linear model (GLM) is a useful generalization of ordinary least squares regression. It relates the random distribution of the measured variable of the experiment (the that separates cross-sectional, longitudinal, historic, and regression-to-the-mean effects. Here, "cross-sectional," "longitudinal," and "regression-to-the-mean" have their convention. al epidemiologic definitions, and "historic" addresses the influence of previous exposures on the toxicodynamics resulting from subsequent exposures. If we use separate [Beta] coefficients to designate des·ig·nate
tr.v. des·ig·nat·ed, des·ig·nat·ing, des·ig·nates
1. To indicate or specify; point out.
2. To give a name or title to; characterize.
3. cross-sectional ([[Beta].sub.cs]), longitudinal ([[Beta].sub.l]), historic ([[Beta].sub.h]), and regression-to-the-mean ([[Beta].sub.rm]) terms in the model, then initial (i.e., cross-sectional) and subsequent (i.e., longitudinal) relations can be defined (with our notation):
 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
It is common in data and statistical analyses of longitudinal data to use the difference between successive measurements. In such a case, the model above reduces to:
 [Delta]R = [R.sub.2] - [R.sub.1] = [Beta]* + [[Beta].sub.1] ([AUC.sub.2] - [AUC.sub.1]) + [[Beta].sub.h][AUC.sub.1] + [B.sub.rm][R.sub.1] + [Epsilon]*
This equation is important in generalizing the use of the linear systems approach we propose in longitudinal data analysis. Such analyses go directly to the heart of the toxicokinetic and toxicodynamic implications of the IRF model we have invoked.
Empirical Validation An empirical validation of a hypothesis is required for it to gain acceptance in the scientific community. Normally this validation is achieved by the scientific method of hypothesis commitment, experimental design, peer review, adversarial review, reproduction of results, : Cognitive Effects of Lead
Lead is widely recognized as a significant neurotoxicant, and the development of biomarkers of lead exposure has been vigorously pursued. X-ray fluorescence X-ray fluorescence (XRF) is the emission of characteristic "secondary" (or fluorescent) X-rays from a material that has been excited by bombarding with high-energy X-rays or gamma rays. (XRF XRF X-Ray Fluorescence
XRF X-Ray Flash
XRF Cross Reference
XRF Extended Recovery Facility (IBM)
XRF Extended Reliability Feature
XRF Cross Reference File
XRF External Reference ) measurement of lead in bone has been adopted as the method of choice to assess cumulative exposure (8,9) because lead in blood has a clearance half-time of 30 days, whereas lead in bone has a clearance half-time of 15-30 years. XRF measures of bone lead highly correlate with the integral of the time-course of blood lead concentration [also called the "cumulative blood lead index" (10,11)]. Independent data suggest an association between cumulative blood lead level and cumulative brain uptake uptake /up·take/ (up´tak) absorption and incorporation of a substance by living tissue.
n. (12-13). Because multiple blood-lead level measurements as a function of time, which are necessary to compute the cumulative index, are usually not available in epidemiologic studies and only rarely in occupational studies, single XRF measures of lead in bone are taken to represent "cumulative exposure" (via Equation 2) or "cumulative dose" (via Equation 3).
Hu et al. (14) considered two paradigms for the interpretation of skeletal skeletal /skel·e·tal/ (skel´e-t'l) pertaining to the skeleton.
pertaining to the skeleton. See also skeletal muscle. lead, as measured by XRF: bone lead as an indicator of cumulative lead exposure, and bone lead as a source of body lead burden that can be mobilized into the circulation. The first paradigm considers bone lead as a surrogate marker surrogate marker Lab medicine A parameter or measured to detect a pathologic condition when a more specific test doesn't exist, is impractical or not cost-effective; surrogate testing has been used for non-A, non-B hepatitis, measuring ALT and antibodies to HBV for cumulative dose to sensitive targets, whereas the second considers bone lead as an important endogenous endogenous /en·dog·e·nous/ (en-doj´e-nus) produced within or caused by factors within the organism.
1. Originating or produced within an organism, tissue, or cell. source of further exposure. With either paradigm, the dose--response relation could be linear or nonlinear A system in which the output is not a uniform relationship to the input.
nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. and involve a threshold or not (14). In either case, XRF measures of bone lead could be predictive of a given health outcome (such as cognitive decline) as long as a strictly linear relation exists between the XRF measure and [AUC.sub.T]. As discussed above, if either the residence time (assuming the first paradigm) or release from bone stores (assuming the second paradigm) is significant, then the XRF measure, although clearly better than a blood measure, would not correlate as highly with health outcome as a truer [AUC.sub.T] metric. For example, a comparison of the strength of association with health outcomes between [AUC.sub.T] estimated from Equation 9 versus Equation 10 would provide some evidence for which of the two scenarios (short residence time and no release versus long residence time and/or subsequent release) is more likely. This distinction is particularly useful in making hypotheses about whether an effect is likely to be transient or persistent or progressive.
To empirically investigate these scenarios, we reanalyzed data from a longitudinal study of 535 former organolead manufacturing workers, for whom we have already reported results (15-19). Informed consent was obtained before a subject was enrolled in the study. As reported in more detail elsewhere (19), a battery of 19 cognitive tests Cognitive tests are assessments of the cognitive capabilities of humans and animals. Tests administered to humans include various forms of IQ tests; those administered to animals include the mirror test (a test of self-awareness) and the T maze test (which tests learning ability). was obtained annually (Table 1). The results of this battery were compared with blood and bone lead measurements. Current tibial tibial
pertaining to the tibia.
a longitudinal prominence on the cranial border of the proximal tibia. Its proximal end (tibial tubercle) has a growth plate separate from the proximal tibia; hyperflexion injuries to lead was measured via XRF and used to estimate the peak tibial lead value at the time of cessation cessation Vox populi The stopping of a thing. See Smoking cessation. of occupational exposure. To do so, the clearance of lead from bone was modeled with a mono-exponential function, as has previously been demonstrated to fit longitudinal bone lead data (20); the clearance half-time was assumed to be 27 years (21). The [AUC.sub.T] was estimated in three ways: a) by assuming that the current tibial lead was proportional to [AUC.sub.T], b) by assuming that the peak tibial lead was proportional to [AUC.sub.T], and c) by forming and integrating the estimated tibial lead time course (which we designate AUC'). To do the latter, we back-extrapolated from the current tibial lead value to the time at which exposure ended (i.e., the same process used to get the peak tibial lead value), and then assumed a straight line between that peak value and a value of zero at the start of occupational exposure some years earlier, as shown in Figure 2. (We knew the date of start of occupational exposure and cessation of exposure for each subject.)
Table 1. Generalized estimating equation linear modeling results identifying predictors of annual change in neurobehavioral test scores in 535 former organolead manufacturing workers comparing four different dose metrics, 1994-1998. Neurobehavioral measure(a) (used in separate regression models of Current PbB(b) Current TL(c) change) [Beta] (SE [Beta]) [Beta] (SE [Beta]) Block design (Wechsler 0.161 (0.135) -0.058 (0.160) Adult Intelligence Scale) Digit symbol (Wechsler -0.187 (0.126) -0.133 (0.135) Adult Intelligence Scale, revised) Symbol digit -0.012 (0.102) -0.099 (0.099) Serial digit learning -0.160 (0.132) -0.020 (0.153) Rey complex figure, copy -0.001 (0.097) -0.030 (0.091) Rey complex figure, -0.174 (0.088)(**) -0.077 (0.090) delayed recall Rey auditory verbal 0.049 (0.167) -0.255 (0.175) learning test, immediate recall, 5 trials Rey auditory verbal -0.055 (0.055) -0.128 (0.054)(**) learning test, delayed recall Rey auditory verbal -0.009 (0.053) 0.035 (0.062) learning test, recognition Trails A -0.225 (0.252) -0.586 (0.273)(**) Trails B -0.790 (0.629) 0.285 (0.740) Finger tapping, dominant -0.196 (0.124) -0.260 (0.155)(*) hand Finger tapping, -0.137 (0.099) -0.224 (0.128)(*) nondominant hand Pegboard, dominant hand 0.045 (0.087) -0.133 (0.092) Pegboard, nondominant hand -0.093 (0.087) -0.254 (0.100)(**) Pegboard, both hands 0.053 (0.074) -0.048 (0.095) Pegboard assembly 0.461 (0.235)(**) -0.034 (0.329) Stroop (C form - A form) -0.740 (0.389)(*) -0.676 (0.544) Choice reaction time -0.393 (2.541) -0.298 (2.703) average SIGNS of [Beta] 14/19 negative 20/22 negative coefficients Statistical significance -- -- 2 < 0.05 (1+) 3 < 0.05 1 < 0.01 2 < 0.10 Neurobehavioral measure(a) (used in separate regression models of Peak TL(d) AUC-lead(e) change) [Beta] (SE [Beta]) [Beta] (SE [Beta]) Block design (Wechsler -0.223 (0.165) -0.213 (0.173) Adult Intelligence Scale) Digit symbol (Wechsler -0.038 (0.140) 0.058 (0.139) Adult Intelligence Scale, revised) Symbol digit -0.206 (0.107)(**) -0.225 (0.108)(**) Serial digit learning -0.104 (0.161) -0.149 (0.165) Rey complex figure, copy -0.138 (0.094) -0.170 (0.096) Rey complex figure, -0.174 (0.096)(*) -0.196 (0.097)(**) delayed recall Rey auditory verbal -0.571 (0.193)(***) -0.671 (0.204)(***) learning test, immediate recall, 5 trials Rey auditory verbal -0.149 (0.058)(**) -0.134 (0.058)(**) learning test, delayed recall Rey auditory verbal -0.028 (0.072) -0.054 (0.078) learning test, recognition Trails A -0.503 (0.308) -0.451 (0.309) Trails B -0.061 (0.851) -0.212 (0.881) Finger tapping, dominant -0.170 (0.153) -0.056 (0.150) hand Finger tapping, -0.169 (0.135) -0.094 (0.137) nondominant hand Pegboard, dominant hand -0.138 (0.098) -0.122 (0.098) Pegboard, nondominant hand -0.250 (0.101)(**) -0.182 (0.102)(*) Pegboard, both hands -0.094 (0.087) -0.117 (0.086) Pegboard assembly -0.320 (0.320) -0.577 (0.301)(*) Stroop (C form - A form) -1.122 (0.606)(*) -1.366 (0.675)(**) Choice reaction time -2.121 (2.776) -2.968 (2.989) average SIGNS of [Beta] 22/22 negative 21/22 negative coefficients Statistical significance 1 < 0.01 1 < 0.01 3 < 0.05 4 < 0.05 2 < 0.10 2 < 0.10 (a) Adjusted for age, education, visit number, testing technician, and baseline score; tests were standardized for direction so that a negative coefficient indicates worsening performance with increasing blood or tibial lead. Beta coefficients are standardized so that they can be directly compared for each neurobehavioral test. The units of each [Beta] coefficient indicate change in neurobehavioral test score per SD unit increase in the lead biomarker. (b) Current PbB = current blood lead level. (c) Current TL = current tibial lead. (d) Peak TL = peak tibial lead, estimated from current tibial lead and years since last exposure, using an estimated half-time of lead in tibia of 27 years (see "Methods"). (e) AUC-lead = area under the curve of estimated tibial lead levels versus time (see "Methods"). (***) p-value s 0.01; (**) p-value < 0.05; (*) p-value < 0.10.
If the XRF measurement can be conceptualized via Equation 2 (i.e., the XRF value at a given time T represents the integral exposure from time zero to that time T), then the [AUC.sub.T] estimated by the third approach above (AUC') represents
 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Equation 15 is thus identical in form to Equation 10 and implies permanent residence of lead at the sensitive target (central nervous system receptors in the brain, in our case). This is also seen by comparison of Equation 15 with Equation 8; the only way that Equation 8 (which is the general form) could be identical to Equation 15 is if [IRF.sub.TK] in Equation 8 is a constant.
In other words, we have four estimates of cumulative biologically effective dose or [AUC.sub.T]:
1. Current blood-lead level, which reflects both past exposures with a 30-day clearance half-time (i.e., it is a poor index of cumulative exposure) and the reintroduction Noun 1. reintroduction - an act of renewed introduction
intro, introduction, presentation - formally making a person known to another or to the public of lead into the circulation via release from bone stores. This metric assumes a short residence time in brain and is mainly reflective of present release from bone and other stores because our population is no longer occupationally exposed.
2. Current tibial lead level, which reflects cumulative exposure (with a 27-year clearance half-time) and the magnitude of bone stores potentially available for release. This metric assumes a short residence time in brain and allows for the possibility of some release from body stores; it better reflects cumulative exposure than blood lead because of the much slower clearance.
3. Peak tibial lead level, which reflects cumulative exposure (and is corrected for clearance) and the magnitude of bone stores potentially available for release. This metric also assumes a short residence time in brain and allows for the possibility of some release from body stores.
4. AUC, which reflects cumulative biologically effective dose in the case of permanent residence (either because the initial residence time is long and/or there is significant ongoing release from bone stores).
Comparisons of the association of these four estimates of [AUC.sub.T] with each of the 19 cognitive tests were based on linear regression Linear regression
A statistical technique for fitting a straight line to a set of data points. using generalized estimating equations methodology. Beta coefficients for the four estimates of [AUC.sub.T], for each of the 19 cognitive tests, were obtained and assessed for statistical significance. The cognitive test outcomes were z-transformed before modeling so that the [Beta] coefficients could be directly compared. The linear regression models controlled for age, education, visit number, testing technician, and baseline score on each test.
Two types of linear regression models were used. The first ([R.sub.1] in Equation 13) focused on cross-sectional data Cross-sectional data in statistics and econometrics is a type of one-dimensional data set. Cross-sectional data refers to data collected by observing many subjects (such as individuals, firms or countries/regions) at the same point of time, or without regard to differences in time. from the baseline measurements. Here, the baseline score on each of the 19 cognitive tests were the dependent variables in the linear regression models. The second focused on longitudinal data, starting with the baseline data, and adding three subsequent measurements, each 1 year apart, for each of the 19 cognitive tests. Here, the dependent variables were the annual change scores (via Equation 14), in practice defined as ([R.sub.i] - [R.sub.0])/[Delta] time rather than [R.sub.i] - [R.sub.i-1], and generalized estimating equations methods were used to examine associations of the lead measures with change in test scores over time.
Cross-sectional analyses. We previously reported the associations of current and peak tibial lead levels with 19 neurobehavioral test scores (16); we now report associations for blood-lead level and AUC'. Taking a p-value [is less than] 0.65 as significant, current blood lead level was significantly associated with 4/19 tests, current tibial lead level with 9/19, peak tibial lead level with 11/19, and AUC' with 14/19. All significant [Beta] coefficients indicated that increasing lead levels were associated with lower neurobehavioral test scores.
Longitudinal analyses. Table 1 shows the results of the linear regression analyses using the baseline values for blood lead, current tibial lead, peak tibial lead, and AUC'. Taking a p-value [is less than] 0.05 as significant, current blood lead level was associated with 2/19 neurobehavioral test change scores (although one of the two significant [Beta] coefficients was in the opposite direction than expected), current tibial lead level with 3/19, peak tibial lead level with 4/19, and AUC' with 5/19. Of interest, current blood lead level had the highest [Beta] for 3/19 associations, current tibial lead level for 4/19 associations, peak tibial lead level for 3/19 associations, and AUC' for 9/19 associations. There was only one instance where AUC' did not have a significant [Beta] coefficient when one of the other estimates did; in this case, current tibial lead level produced the only [Beta] coefficient that achieved statistical significance.
Because the cognitive test outcomes were standardized standardized
pertaining to data that have been submitted to standardization procedures.
standardized morbidity rate
see morbidity rate.
standardized mortality rate
see mortality rate. , both the direction and magnitude of the associations ([Beta] coefficients) of each [AUC.sub.T] lead measure with each neurobehavioral test change score could be directly compared. Current blood-lead level had only two significant [Beta] coefficients, one of which was positive (i.e., in the opposite direction than expected). Use of current blood level would thus lead to the conclusion of no association between lead and cognitive decline. In contrast, progressing from current tibial lead level to peak tibial level to AUC' increased the number of significant [Beta] coefficients, and AUC' produced the largest [Beta] coefficient in more of the 19 tests than any of the other 3 measures.
Even though the [Beta] coefficients changed depending on which of the four estimates of AUC was used, some association between the four estimates was present. Pearson's correlations were significant (p [is less than] 0.01) between a) blood and current tibial lead (r = 0.44), peak tibial lead (r = 0.26), and AUC' (r = 0.18); b) current tibial lead and peak tibial lead (r = 0.86), and AUC' (r = 0.70); and c) peak tibial lead and AUC' (r = 0.94).
A critical issue in the interpretation of the longitudinal test outcomes is whether the change in cognitive performance over time could be completely (and thus solely) explained by the increase in AUC' with time (since the limit of integration for any AUC measure progressively increases with time). In other words, is any progressive cognitive decline simply the result of progressive cumulative dose? Accordingly, we also evaluated a generalized model in which [Delta]-AUC' ([Delta]-AUC' = AUC' end-of-interval -- AUC' baseline) was used instead of baseline AUC'. With this model, only 2/19 tests had significant beta coefficients, and the change in cognitive test outcomes could not be explained by the change in AUC alone. This suggests that a progressive model for [IRF.sub.TD] needs to be considered.
In epidemiology, assessing the association between exposure to a putative toxicant and subsequent health outcome implies the existence of an underlying biologically based dose--response relation. The goal of exposure or internal dose assessment is thus to find an index that best represents the cumulative biologically effective dose of the active form of the toxicant at the sensitive target. In practice, this index should be proportional to the integral of the time course of the concentration of active agent at the sensitive target (i.e., Equation 1), designated [AUC.sub.T]. The use of cumulative exposure or internal dose as a surrogate for this time--concentration integral will only prove useful if the toxicokinetics are approximately linear over the concentration range expected, and if the effects are approximately cumulative (4,6). This is simply a restatement Restatement
A revision in a company's earlier financial statements.
The need for restating financial figures can result from fraud, misrepresentation, or a simple clerical error. of Haber's rule Haber's rule is a mathematical statement of the relationship between the concentration of a poisonous gas and how long the gas must be breathed to produce death, or other toxic effect. The rule was formulated by German chemist Fritz Haber in the early 1900s. : tissue damage should be related to the product of the mean exposure intensity and time (6). Thus, a major issue in the development and use of biomarkers is the degree to which these assumptions hold. In other words, it is important to understand whether the toxicokinetics and toxicodynamics are linear and time-invariant mathematically, and whether the toxicodynamics represent a reversible, persistent, or progressive process.
In this report, we present a conceptual framework based on linear systems theory as an aid to identifying and considering these issues. With respect to toxicokinetics, we have attempted to relate different surrogates of cumulative biologically effective dose and to identify the conditions under which certain assumptions are implicitly invoked. We have introduced the common linear systems concept of an impulse response function, [IRF.sub.TK], to describe the toxicokinetics following an infinitely short duration exposure. Use of linear systems theory and this concept allows us to define a general relation between the exposure time-course and the time-concentration curve of the active form of the agent at the sensitive target, whose integral, [AUC.sub.T], is likely best correlated with response. This general relation (Equation 8) could be transformed into two more specific relations, one representing the case where the residence time in the sensitive target is infinitely short and no biorelease occurs (Equation 9), and another representing the case where the effective residence time is infinitely long (either because the agent never dears from the sensitive target or because ongoing significant biorelease from body stores constantly replenishes that amount of agent cleared from the target; Equation 10).
We also sought to use linear systems theory to conceptualize con·cep·tu·al·ize
v. con·cep·tu·al·ized, con·cep·tu·al·iz·ing, con·cep·tu·al·iz·es
To form a concept or concepts of, and especially to interpret in a conceptual way: the time-course of response, which was particularly important for the longitudinal data presented here, in which we studied the association of four estimates of [AUC.sub.T] with cognitive decline in an occupationally exposed cohort of 535 workers. We found that progressively more [Beta] coefficients were statistically significant as we moved from current blood lead level to current tibial lead level to peak tibial lead level to AUC' in both cross-sectional and longitudinal analyses.
Conceptualizing [AUC.sub.T] in this way and generating multiple estimates of [AUC.sub.T] is helpful in several ways: it clarifies the distinctions between exposure, internal dose, and biologically effective dose; it guides the development of different estimates of [AUC.sub.T]: and the results of a comparison of the association of these different estimates of [AUC.sub.T] with health outcomes provide indirect evidence of the underlying biological phenomena. For example, for those health outcomes in which a measure of recent or current dose has the highest association, the health outcome is likely an acute, reversible process Noun 1. reversible process - any process in which a system can be made to pass through the same states in the reverse order when the process is reversed
physical process, process - a sustained phenomenon or one marked by gradual changes through a series of states; , whereas for those health outcomes in which a measure of cumulative dose has the highest association, the health outcome is likely a persistent or progressive accumulative LEGACY, ACCUMULATIVE. An accumulative legacy is a second bequest given by the same testator to the same legatee, whether it be of the same kind of thing, as money, or whether it be of different things, as, one hundred dollars, in one legacy, and a thousand dollars in another, or whether process. For those health outcomes in which a measure that includes consideration of residence time has the highest association, the agent's toxicokinetics likely include significant residence in the sensitive target and/or significant release from body stores.
Significant successes and utility have been reported with the use of pharmacokinetic modeling in predicting the toxicokinetics of environmental agents such as lead (4,22). Our approach differs from these previous mathematical efforts in that it is "model free" (i.e., it does not assume a certain "topology topology, branch of mathematics, formerly known as analysis situs, that studies patterns of geometric figures involving position and relative position without regard to size. " or relationship among a series of anatomic or physiologic compartments In developmental biology, compartments are fields of cells of distinct cell lineage, cell affinity, and genetic identity. In a developing organ, all cells within a compartment possess similar affinities, and so intermingle with each other. , as do classical pharmacokinetic modeling approaches). Rather, our approach makes use of linear systems theory to describe the IRF of the system nonparametrically. This nonparametric IRF exists independent of compartment-based descriptions. In other words, there are no predetermined pre·de·ter·mine
v. pre·de·ter·mined, pre·de·ter·min·ing, pre·de·ter·mines
1. To determine, decide, or establish in advance: parameters whose presence, number, and character are fixed by an a priori a priori
In epistemology, knowledge that is independent of all particular experiences, as opposed to a posteriori (or empirical) knowledge, which derives from experience. hypothetical model.
Having stated this, we emphasize that we are not asserting that our approach is intrinsically better than multicompartment modeling; rather, we view the two approaches as highly complementary. For example, both may lead to a useful prediction of [AUC.sub.T], but they require different assumptions and independent data. The choice of approach depends on prior knowledge and on the types of information and relations desired. If compartments and relations can be identified and appropriate quantitative rate constants determined, then the multi-compartment toxicokinetic approach yields accurate predictions (4,22). On the other hand, when less is known about the potential compartments, and particularly when little is known about rate constant values, the conceptualization of the toxicokinetics via a linear system with an IRF may prove useful. Such is the case when an empirical time course in an organ or structure of interest is already known (23). In this regard, the empirical time course need not come from a [Delta] input function because deconvolution In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data. The concept of deconvolution is widely used in the techniques of signal processing and image processing. analysis can be used to obtain [IRF.sub.TK] from the combination of any arbitrary but known input function and the empirical time course (5). Once [IRF.sub.TK] is obtained in this way, the time-course for any other arbitrary but known input function can be predicted via Equation 6.
We also emphasize that our conceptual framework covers both toxicokinetics and toxicodynamics, whereas pharmacokinetic modeling only addresses predictions of toxicokinetics. In this regard, we highlight the complementary nature of pharmacokinetic modeling and linear systems analysis by suggesting that pharmacokinetic modeling can be used, when available, to predict [IRF.sub.TK] for subsequent use in linear systems analysis of the toxicodynamics. In essence, either pharmacokinetic modeling or our approach could be used to predict the needed kinetic time-course if the right data are in hand; it is important that the types of data are strikingly different. In practice, we used the toxicokinetic portion of our conceptual framework to create four different estimates of [AUC.sub.T]; we do not claim that any of these estimates is more accurate than one obtained from pharmacokinetic modeling. Rather, the conceptual framework provides a different understanding of the meaning of each of these estimates than one derived from pharmacokinetic modeling, and this understanding helps in the interpretation of the actual data.
The conceptual framework as presented, embodied em·bod·y
tr.v. em·bod·ied, em·bod·y·ing, em·bod·ies
1. To give a bodily form to; incarnate.
2. To represent in bodily or material form: in specific equations, and used so far, requires that the toxicokinetics and toxicodynamics be linear and time invariant. With respect to toxicokinetics, when the active form of the agent is produced by metabolism, nonlinear effects, especially at high doses, are expected as the processes saturate sat·u·rate
v. Abbr. sat.
1. To imbue or impregnate thoroughly.
2. To soak, fill, or load to capacity.
3. To cause a substance to unite with the greatest possible amount of another substance. . Nonlinear kinetics kinetics: see dynamics.
Kinetics (classical mechanics)
That part of classical mechanics which deals with the relation between the motions of material bodies and the forces acting upon them. could also arise from changes in individual uptake or susceptibility susceptibility
the state of being susceptible. Refers usually to infectious disease but may be to physical factors such as wetting or to psychological factors such as harassment. with time, from synergistic synergistic /syn·er·gis·tic/ (sin?er-jis´tik)
1. acting together.
2. enhancing the effect of another force or agent.
1. or antagonistic effects antagonistic effect The negative effect that one chemical or family of chemicals has on other chemicals related to concurrent mixed exposures to other toxicants, from allergic al·ler·gic
1. Of, caused, or characterized by an allergy.
2. Having an allergy or exhibiting an allergic reaction to a substance.
pertaining to or caused by allergy. responses, from changes produced by the initial exposure to that agent (e.g., upregulation of cytochrome cytochrome (sī`təkrōm'), protein containing heme (see coenzyme) that participates in the phase of biochemical respiration called oxidative phosphorylation. P450), or from dose-rate effects (6). With respect to toxicodynamics, for stochastic processes stochastic process
In probability theory, a family of random variables indexed to some other set and having the property that for each finite subset of the index set, the collection of random variables indexed to it has a joint probability distribution. like carcinogenesis car·ci·no·gen·e·sis
The production of cancer.
production of cancer.
viruses and some parasites are capable of initiating neoplasia. , dose-response relations may be linear or nonlinear. In the case of a direct genotoxic genotoxic /ge·no·tox·ic/ (je´no-tok?sik) damaging to DNA: pertaining to agents known to damage DNA, thereby causing mutations, which can result in cancer.
adj. carcinogen carcinogen: see cancer.
Agent that can cause cancer. Exposure to one or more carcinogens, including certain chemicals, radiation, and certain viruses, can initiate cancer under conditions not completely understood. , a linear or linear-quadratic relation between [AUC.sub.T] and response is expected (4). For nonstochastic processes, a linear relation with a threshold is commonly observed. As with the toxicokinetics, up- or downregulation of receptors or tolerance effects can introduce nonlinearities. To the extent to which strict linearity is not present, the power of the conceptual framework and actual approach decreases; the degree of linearity may be different for the toxicokinetic and toxicodynamic portions of the analysis in a given application.
In contrast to the requirement for linearity, time invariance is not strictly required in our approach. For the sake of simplicity of presentation and implementation, we have invoked the assumption of time invariance; this assumption is what leads to the specific convolution integral given in Equation 7, and repeated below:
 [C.sub.T](t) = [k.sub.E] [integral of] [C.sub.E]([Tau])[IRF.sub.TK](t-[Tau])d[Tau]
If [IRF.sub.TK] is time varying, the convolution integral given in Equation 7 must be modified, as follows:
[7'] [C.sub.T](t) = [k.sub.E] [integral of] [C.sub.E]([Tau])[IRF.sub.TK](t; [Tau])d[Tau]
Equation 7' differs in a subtle way from Equation 7: [IRF.sub.TK] is now a function of both t and [Tau], not just t-[Tau]. An analogous situation holds for [IRF.sub.TD]. This implies the need to obtain a family of IRFs (as a function of t) rather than just a single IRF. In practice, if the IRF changes only slowly, then it becomes possible to treat segments of time as being time invariant and to use a single IRF during that time period.
In essence, our approach has permitted initial comparisons of different assumptions about the residence time of lead in brain, release from bone stores, and the persistence or progression of lead-associated neurobehavioral effects. Using both cross-sectional and longitudinal data analysis, we have been able to show, in preliminary form, that a) a measure of cumulative lead dose (AUC') that implies either long residence time of lead in brain or significant ongoing release of lead from body stores is the best predictor of both test scores at cross-section and test score declines over time; b) the change in this AUC metric over time is a poor predictor of longitudinal test score change; and c) the observed longitudinal change in test scores is consistent with a model of progressive neurobehavioral effect. Given our knowledge about the clearance half-time of lead in brain and the current blood and tibial lead levels in the former workers, we believe that the observed annual test score declines are likely due to a combination of newly induced effects from lead released from bony stores and, more significantly, progressive effects from past exposures to lead. In any event, the model requires that, at a minimum, effects persist for many years past the exposure that triggered the effect. These data thus support the hypothesis that this is not a transient neurochemical neu·ro·chem·is·try
The study of the chemical composition and processes of the nervous system and the effects of chemicals on it.
neu effect, that, by necessity, would depend on the continued presence of lead in brain to sustain the effect, but rather a persistent structural change (which may have been initially triggered by neurochemical events).
In general, elimination of the toxicant from the sensitive target site probably represents the rate-limiting step between exposure and response (6). For example, Rappaport (6) has alluded to the physiologic damping damping
In physics, the restraint of vibratory motion, such as mechanical oscillations, noise, and alternating electric currents, by dissipating energy. Unless a child keeps pumping a swing, the back-and-forth motion decreases; damping by the air's friction opposes the that "resulted from accumulation of lead over several months owing to owing to
Because of; on account of: I couldn't attend, owing to illness.
owing to prep → debido a, por causa de the slow rate of elimination and distribution of this metal from the blood." If this elimination is slow, the accumulated burden is large relative to the amount of toxicant received (6). In such a case, knowledge of [AUC.sub.T] itself (compared with either integrated exposure or internal dose) will better predict response because even large short-term fluctuations in exposure will not directly provide important information on biologically effective dose.
Ultimately, we are interested in the development, validation, and application of biomarkers whose behavior we understand vis-a-vis the toxicologic paradigm. In this regard, the use of the IRF has already been described in noninvasive non·in·va·sive
1. Not penetrating the body, as by incision. Used especially of a diagnostic procedure.
2. Not invading healthy tissue. imaging (24) and may provide a means to obtain impulse response functions for the toxicokinetics and toxicodynamics of environmental agents of interest, such as lead, which has been radio-labeled (25).
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1. Mechanically determined.
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Within a living organism.
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The process by which a molecule encircles and binds to a metal and removes it from tissue.
Mentioned in: Heavy Metal Poisoning
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n. polymorphism polymorphism, of minerals, property of crystallizing in two or more distinct forms. Calcium carbonate is dimorphous (two forms), crystallizing as calcite or aragonite. Titanium dioxide is trimorphous; its three forms are brookite, anatase (or octahedrite), and rutile. on the accumulation of lead in bone and blood in lead smelter workers. Environ Res 77:49-61 (1998).
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Any of a group of fat-soluble alcohols important in calcium metabolism in animals to form strong bones and teeth and prevent rickets and osteoporosis. It is formed by ultraviolet radiation (sunlight) of sterols (see steroid) present in the skin. receptor in former organolead manufacturing workers. Environ Health Perspect 108:199-211 (2000).
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fix - something craved, especially an intravenous injection of a narcotic drug; "she needed a fix of chocolate" . Neurotoxicology 19:197-207 (1998).
Jonathan M. Links,(1) Brian S The name Brian (sometimes spelled Bryan) comes from an Irish backround. It is of Celtic origin and its meaning may be "hill" or "strong, noble, and high". . Schwartz,(1,2) David Simon David Simon can refer to:
Departments of (1) Environmental Health Sciences, (2) Epidemiology, and (3) Biostatistics biostatistics /bio·sta·tis·tics/ (-stah-tis´tiks) biometry.
The science of statistics applied to the analysis of biological or medical data. , The Johns Hopkins Noun 1. Johns Hopkins - United States financier and philanthropist who left money to found the university and hospital that bear his name in Baltimore (1795-1873)
2. School of Hygiene and Public Health, Baltimore, Maryland "Baltimore" redirects here. For the surrounding county, see Baltimore County, Maryland. For other uses, see Baltimore (disambiguation).
Baltimore is an independent city located in the state of Maryland in the United States. , USA
Address correspondence to J.M. Links, Department of Environmental Health Sciences, Johns Hopkins School of Hygiene and Public Health, 615 North Wolfe Street, Baltimore, MD 21205-2179 USA. Telephone: (410) 955-9622. Fax: (410) 955-6222. E-mail: firstname.lastname@example.org
We thank J. Prince and A. Todd for helpful discussions and the reviewers for important clarifying comments.
This research was supported in part by NIH grants R01 AG10785 from the National Institute on Aging The National Institute on Aging is a division of the U.S. National Institutes of Health, located in Bethesda, Maryland.
Formed in 1974, NIA's mission is to improve the health and well-being of older Americans through research. It is the primary U.S. and P30 ES03819 from the National Institute of Environmental Health Sciences The National Institute of Environmental Health Sciences (NIEHS) is one of 27 Institutes and Centers of the National Institutes of Health (NIH),which is a component of the Department of Health and Human Services (DHHS). The Director of the NIEHS is Dr. David A. Schwartz. .
Received 22 September 2000; accepted 25 October 2000.