optimal design of experiments to estimate ldl transport ......department of biomedical engineering,...

Optimal design of experiments to estimate LDL transport parameters in arterial wall

EVAN D. MORRIS, GERALD M. SAIDEL, AND GUY M. CHISOLM III Department of Biomedical Engineering, Case Western Reserve University, and Department of Vascular Cell Biology and Atherosclerosis Research, Cleveland Clinic Foundation, Cleveland, Ohio 44195

MORRIS, EVAN D., GERALD M. SAIDEL, AND GUY M. CHISOLM III. Optimal design of experiments to estimate LDL transport parameters in arterial wall Am. J. Physiol. 261 (Heart Circ. Physiol. 30): H929-H949, 1991.-To quantify transport processes in atherosclerosis, the arterial wall is often exposed to labeled lipoproteins. In vivo experiments are desirable for estimation of transport parameters, but they are technically difficult. A dynamic mass transfer model has been developed to describe experimental transmural profiles of lipoprotein accumulation as a function of luminal permeability, diffusion, convection, and degradation. To avoid extraneous experiments and to assure successful parameter estimation, an optimal design of experiments is needed. For our purposes a design was considered optimal when it maximized the sensitivity of the model output to changes in parameter values as indicated by the determinant of the Hessian matrix of the objective function. A comparison was made between two designs: dual-time designs prescribing unequal circulation times for two distinguishable injections of labeled low-density lipoprotein (LDL) and dual- species designs requiring simultaneous circulation of LDL and tyramine-cellobiose-modified LDL. Circulation time was optimized for both designs. Although both were heavily dependent on the circulation times, dual-time designs required better preliminary knowledge of parameter values. Because labeled degradation products of the modified tracer become anchored in the arterial tissue, information about the degradation process is retained in the dual-species study. For this reason, dual- species designs were generally superior to dual-time designs.

low-density lipoprotein; tyramine-cellobiose; mathematical model; spatial distribution; parameter estimation; sensitivity analysis; optimal experiment design

ATHEROSCLEROSIS is accompanied by an excessive accumulation of plasma low-density lipoprotein (LDL) in the extracellular spaces of major arteries (20). Arteries are lined on the luminal surface of the intima by the one- cell-thick endothelium and are bounded on the abluminal side of the media by vascularized adventitia (Fig. 1). The accumulation of LDL, which accompanies atherosclerosis, occurs primarily in the intima and inner media of the vessels (34). To examine events related to atherosclerosis, investigators have studied the uptake and distribution of injected tracer macromolecules (e.g., LDL)

by the arteries of experimental animals. Analysis of data from such experiments requires the use of mathematical models of the processes governing transport of macromolecules in arterial tissue.

Mathematical models have been developed and applied to data from tracer studies performed on arteries both in vivo and in vitro (6, 13, 39, 43). To describe and analyze data from in vivo experiments, compartmental models have been used (7) that quantify exchange rates of tracers between plasma and tissue compartments without regard to transport processes within the compartments. An inherent assumption in these models is that the compartments are well mixed (17). To describe transport processes affecting tracer behavior within the tissue, investigators have used spatially distributed models (16). Models have included such mechanisms as filtration of large molecules by the internal elastic lamina (14) or infrequent “holes” in the endothelium (41,43) to explain the pathological accumulation of LDL in the subendo- thelial space. The spatially distributed model used in the present study is an extension of models proposed previously by us (21, 30) and by others (6, 16, 39, 40). An eventual objective of these studies is to estimate the parameters of interest depicted in our model and to evaluate the relative significance of the corresponding processes.

In a mechanistic mass transport model, parameters characterize physiological processes. Parameters are evaluated by estimation algorithms that give the best fit of model-predicted simulations to experimental data. In our system, the data consist of concentration profiles of the tracer across the wall of the rabbit aorta from lumen to adventitia. They are obtained after intravenous injection and circulation of the tracer in the plasma for a specified time. The profile of radioactivity across the artery wall that results from an injection of LDL labeled with radioactive iodine (“I-LDL) consists of the intact, undegraded *I-LDL at each point in the tissue. A com- mon limitation of this approach is that the data are not sufficient to assure precise estimates of all important parameters (13, 40). Factors that reduce precision are variability of biological tissue samples, small signal-to- noise ratio of the lower tracer concentrations, and overly

0363-6135/91 $1.50 Copyright 0 1991 the American Physiological Society H929

H930 OPTIMAL DESIGN OF LDL TRANSPORT EXPERIMENTS

lum

ENOOTHELI

INTIMA

MEDIA ADVENTITIA --

ELASTIC LAYERS

smooth muscle cells

extracellular space

cross-section of

adwentitial blood vessel

cells of adventitia

(fibroblasts, adipocytes...)

collagenous fibrils

I I NORMALIZED

I I

TRACER CONCENTRATION

~

IN TISSUE

t

I

‘1

6 ‘1

0 c- 21

NORMALIZED DISTANCE

FIG. 1. Top: enlargement of section of arterial wall. Accumulation of tracer is via transport from both luminal and adventitial blood supplies. Bottom: curve representing normalized accumulation of tracer (0) in intima-media after fixed circulation times, tl and tB, as a function of dimensionless distance ([). Media is depicted as avascular, although in certain species blood vessels of adventitia (vasa vasorum) penetrate outer media. In that case, profile would only apply to avascular portion [From Saidel et al. (30)] H and < in this figure are defined differently than in text.

complex models, the many parameters of which are not all identifiable.

For a given model and data set, it may not be possible to obtain suitable estimates of the parameters or the estimates may have excessive variance. To obtain estimates with acceptable precision, changes in the model or in the experiments may be n .eeded. Optimal experiment design provides a systematic strategy for discriminating among experimental variations to improve the precision of parameter estimates. In this st,udy, we examine optimal experiment design criteria 1) to determine which parameters of our model are significant and should be estimated, 2) to decide whether insignificant parameters should be fixed or eliminated, and 3) to design appropriate experimental protocols and data-sampling schemes. Such an analysis provides a priori information about the parameter estimates and makes the experimental and computational work more efficient. As a prelude to optimization of the experiment design, we introduce a model of tracer behavior and examine the sensitivity of the model output to its parameters. Before proceeding with difficult experiments, we would like to know if it will be possible to estimate parameter values precisely enough to quantify the physiological processes governing lipoprotein accumulation. The approach outlined here can readily be adapted to other and their mathematical models.

Glossary

W,t)

experimental systems

Concentration of labeled tracer-derived degradation products (“I-TC) in arterial media

CM

C,(t)

D F PO

G H K L P 1

P 4;’ S t h, t2

t01, to2

u ij V W W ij

V Ai X Y

Y Dij Y ij z a9 P

Concentration of tracer (*I-LDL) in arterial me- . da

Plaima concentration of injected tracer (*I-LDL or *I-TC-LDL)

Effective diffusivity, length’/time Coefficient of plasma concentration function Hessian matrix of reduced dimension Hessian matrix Degradation rate constant, time-’ Thickness of media, length Luminal permeability, length/time Adventitial permeability, length/time Half-length of ellipse axis i Sensitivity matrix Time, h Circulation times Exponential time constants for plasma concen-

tration function Weighting matrix element Convective velocity, length/time Hessian weighting matrix Weighting matrix element Eigenvector of Hessian matrix Matrix of eigenvectors Output matrix (includes Cij and Bij + Cij com-

ponents) Experimental data matrix element Predicted model output matrix element Medial position (z = 1 corresponds to 10 pm) Summation indexes referring to two tracer spe-

cies Variance of data error Objective function Parameter vector Optimal parameter vector Difference vector, 8 - 8* Product of XT68 Tolerance of objective function Vector of parameters of interest Vector of nuisance parameters Equilibrium distribution coefficient, dimension-

less Linear dependence index Eigenvalue of Hessian matrix Matrix of eigenvalues Angle between resealed coordinate axis and ma-

jor axis of resealed ellipse

TRACER EXPERIMENTS

The data used in our analysis consisted of simulated concentration profiles of a tracer distributed across the intima and media of a rabbit aorta. Using simulated data, we first examined single-tracer experiments similar to those reported previously (4, 5, 8, 39). In addition, we examined and compared two experimental protocols, each involving two distinguishable tracers, to determine their relative benefit and to quantify their advantage over the single-tracer experiment.

The experimental protocols used to produce concentration profiles typically begin with the intravenous injection of labeled macromolecular tracer(s). At a prede-

OPTIMAL DESIGN OF LDL TRANSPORT EXPERIMENTS H931

0.03 -j

0

c 0.02

‘;;;

Y

:I

0.01

0.00

k

ii AA

AA

2 4 a

z (10yn) FIG. 2. Experimental data from dual-species experiment with 24-h

circulation. Concentrations of iodine-labeled low-density lipoprotein (“I-LDL; q ) and iodine-labeled tyramine-cellobiose adduct linked to LDL (“I-TC-LDL + “I-TC; A) in medial tissue are normalized by initial plasma concentrations of respective injected tracers [Y(z)&(O)]. Hor- izontal axis of medial position (x) is 100 pm full scale. Profile is compiled from 3 sets of concentration measurements corresponding to 3 tissue samples taken from descending thoracic aorta of male New Zealand White rabbit weighing 3.54 kg. These data can be compared qualitatively with 18-h simulation in Fig. 50.

termined circulation time(s) after injection, the experiment is terminated and the thoracic aorta removed. The vessel is opened longitudinally and frozen immediately to prevent further diffusion of tracer. The tissue is sliced into 4- to ZO-pm thick layers in planes parallel to the flattened lumen. Once the tissue layers have been washed to remove undesirable radioactive species, each slice is assayed for intact tracer (9). In single-tracer experiments we are interested in the dynamics of intact tracer only. Without washing the tissue, the desired signal would be masked by other labeled species. These species might include some residual free iodide, which is a by-product of tracer preparation, or labeled degradation products, which are the result of intracellular metabolism of the intact tracer both locally and at remote tissue sites. The data are displayed graphically as tracer concentration vs. discrete position in the intima and media (Fig. 1). The data obtained from such experiments in a normal rabbit aorta do not distinguish intima from media, since the normal intima is thin compared with the thickness of the slices. However, arterial concentration profiles with greater spatial resolution have been achieved in vitro (15, 32), and other improved techniques may be appli- cable to in vivo studies as well (27).

It is possible to distinguish two readily available ra- dioisotopes of iodine that can be used to label LDL (1311-LDL and 1251-LDL). By injecting two distinguishable labeled tracers sequentially before terminating the experiment, it is possible to generate distinct concentration profiles representing two different circulation times in a single animal. We refer to such an experiment as a “dual-time” experiment.

An alternative dual-tracer protocol, also using the two iodine labels, involves chemically modified and unmodified LDL. A modification of LDL was introduced in a series of studies by Pittman et al. (28) and Carew et al. (7) in which an iodine-labeled tyramine-cellobiose adduct (*I-TC) was linked to the apoprotein of LDL, yielding

*I-TC-LDL. Whereas the degradation product of *I-LDL tracer, *I-tyrosine, readily escapes cells, intracellular degradation of the *I-TC-LDL yields a labeled product (*I-TC) that is trapped in the lysosomes and does not escape the cell. The labeled degradation product accu- mulates throughout the experiment and increases as long as intact *I-TC-LDL continues to be available for degradation. In contrast, *I-tyrosine, the breakdown product of unmodified *I-LDL, and any radioactive free iodide that remains in the tissue at the end of the experiment are removed during processing. We refer to the experiment in which the two labeled species, TC-modified and unmodified LDL, are injected and circulated simultaneously as a “dual-species” experiment.

Figure 2 is an example of actual experimental data from an animal subjected to the dual-species protocol with a circulation time of 24 h. The data set shown was compiled from three tissue samples of the descending thoracic aorta. Note that even if the thickness of sliced tissue layers was identical from one tissue piece to another (which it was not), there would be at most three independent measurements of concentration for each tracer at any medial point in the concentration profile.

MATHEMATICAL MODEL

The dynamic one-dimensional model we have adapted simulates concentration profiles of LDL tracer in the arterial media at any given time after injection of tracer into the plasma (Fig. 1). The model indicates that, after injection, the tracer permeates the barrier between the plasma and the arterial wall. The endothelium is widely believed to constitute the primary luminal barrier to the entry of macromolecules into the tissue (4, 36, 42); however, the internal elastic lamina, situated between the intima and media, has also been proposed as a significant resistance to transport (14). After traversing the endothelium and the normally thin intimal space, a tracer molecule is depicted as passing through the internal elastic lamina into the media, which we have modeled as uniform and homogeneous (i.e., we make no distinction between intracellular and extracellular). Except within a small “interaction” region (43) adjacent to the lumen, the variation in tracer concentration is considered significant only in the ‘Y-direction, normal to the luminal surface. Details within this region are beyond the resolution of our measurements and are treated in a lumped manner. Because the thickness of the vessel wall is small compared with the characteristic lengths in the axial and circumferential directions, movement is described in terms of a single spatial variable (Fig. 1). Rectangular coordinates are used because the wall thickness is small compared with the radius of the vessel.

Transport of tracer through the extracellular spaces of the tissue is driven by convection and diffusion. The hydrostatic pressure difference across the vessel wall is assumed to produce a constant convection of water and solutes from lumen to adventitia. The concentration gradient is not monotonic, since tracer diffuses into the media from both lumen and adventitia. As the concentration of tracer in the plasma is reduced by metabolism and by distribution to the peripheral tissues, the prevail-


ing concentration gradients may eventually favor diffu- from arteries of animals with thin intima (e.g., a normal sion of tracer from the media to the lumen and adventitia. rabbit), the intima and internal elastic lamina are lumped Within the media, LDL is catabolized by vascular smooth with the endothelium and considered part of the luminal muscle cells. Although metabolism follows a series of steps, we have represented the catabolic loss of *I-LDL as first order.

boundary. Our state equations thus pertain only to the media. The influx from the lumen must equal the convective-diffusive flux on the medial side of the boundary

The general mathematical model describes the concentration distribution in the media of the injected tracer, P

C

C(z,t) (either *I-LDL or *I-TC-LDL), and the trapped - ’ c [ 1 - - C,(t) dC =VC-Dz, z=O (4)

degradation product, B(z,t) (“I-TC). The disposition of the two tracers is depicted schematically in Fig. 3. As In this condition, following Fry and Vaishnav (16), we long as each labeled species is present in no more than introduce a partition coefficient for the media, c, and the trace amounts, we can assume that it follows the biolog- tracer concentration in the plasma, C,(t). ical pathways of native LDL, independent of the other Although there is a less distinct anatomic barrier sep- tracer, at least to the point of metabolism. The governing arating media from the adventitial blood supply than equation for C( z, t) accounts for diffusion, convection, from the luminal blood supply, we have chosen to con- and metabolic degradation ceptualize the media-adventitia boundary (x = L) in an

dC 2 dC analogous manner. Thus another effective membrane

D dC --

x = az2 V --

a2 KC 9 OczcL, t>O (1) with permeability Pz is included that separates the

plasma in the adventitial blood supply from the media

where D is the effective diffusion coefficient, V is the convective velocity of the solute, K is the apparent first- -2- P

C

c - C,(t) = -VC + D$, z=L (5)

order degradation constant, and L is the thickness of the [ 1

media. Because the labeled ligand *I-TC is conserved, its accumulation must be modeled by the same loss term

After injection, the decrease in the tracer concentration

but with the opposite sign in the plasma is depicted as a sum of two exponentials, which is consistent with previous experimental results

dB (4938) -= at

KC 7 t>O (2) C,(t) = F,oexp

( ) -It + [C,(O) - &olexP

-t

Before injection there is no labeled tracer of either type G

i ) G (6)

in the tissue *I-TC-LDL and *I-LDL have been shown to exhibit the

B=C=O, t=O (3) same plasma decay curve in vivo and the same recogni- tion and uptake by a receptor-mediated process in cul-

In the neighborhood of the endothelium (z = 0), we tured human fibroblasts in vitro (7, 28). We have there- regard the molecular flux through the endothelium (and fore assumed that the *I-LDL and *I-TC-LDL tracers intimal region) from the lumen into the media as crossing are transported similarly in the tissue. The only differ- an effective membrane, with permeability P,. The pos- ence in the fate of these tracers is assumed to be the sibility exists that the internal elastic lamina may also retention of labeled metabolites from *I-TC-LDL as modulate luminal permeability (14). For data obtained opposed to the loss of labeled metabolites from *I-LDL.

PLASMA MEDIA

The experimentally observable signals in the tissue

1251 M- diffusion -w

Ok--

convection - tv

LDL degradation

i ’ 2%tyr REMOVED ii==

ADVENTITIA are the following two model outputs: C(z, t), the intact *I-LDL, and B + C, the sum of *I-TC-LDL and *I-TC.

1251

0 tv

LDL

Excluding parameters in the plasma-decay curve, which can be estimated independently with good precision (4, 39), there are six unknown parameters, E, PI, Pp, D, V, and K, associated with transport in the tissue.

-+- diffusion - convection ---4b degradation

t ‘3’l-TC TRAPPED

SIMULATIONS

Model simulations were performed using the values for parameters selected from pertinent literature listed

TC = Tyramine Cellobiose tyr = tyrosine

in Table 1. Distinctions between in vivo and in vitro studies are indicated. Because these transport parameter

FIG. 3. Schematic interpretation of mathematical model of 2 tracer values were chosen based on different models, these

species, 1251-LDL and 1311-TC-LDL (patterned after Ref. 7). Both values should be regarded as a starting point. tracers enter media through conceptual membranes at plasma and To simulate the tracer concentration profiles, the adventitial interfaces and are transported by unidirectional convection model was solved numericallv. Spatial derivatives were and bidirectional diffusion. Whereas degradation products of lz51-LDL (top) are removed and do not contribute to tracer concentration pro-

approximated as finite differences, and the resulting

files, 1311-TC-LDL (bottom) is degraded to labeled product 13’I-TC, initial-value problem was integrated using well-tested which is retained at site of degradation and contributes to observed and extensively commented FORTRAN-callable subrou- profile. tines (19, 31). Profiles of two observable signals were

OPTIMAL DESIGN OF LDL TRANSPORT EXPERIMENTS

TABLE 1. Initial parameter estimates (and sources) used in simulations

H933

Parameter Values Dimensionless Alternative Parameters

Scaled units Conventional units Parameter Values Scaled units

P, = 1.3 x 10-l l(l0 pm/h) =3.6 X 10T8 (cm/s) Pl = 2.6 x 10-l = P1L/D PI = 2.0 x 10-l P2 = 2.0 x 10m2 (10 pm/h) =5.6 X lo-’ (cm/s) P2 = 4.0 x lO-2 = P,L/D Pa = 2.5 x lo-‘1

= 1.0 X 10-l (dimensionless) b = 5.0 [(lo pm)2/h]

l/E = 1.0 x 1o+l = 2.0 x 10-l =1.4 X lo-’ (cm2/s) a! = 3.0 = VL/D ;, = 2.5

V = 1.5 (10 pm/h) =4.2 X low7 (cm/s) v = 1.5 K = 0.5 (h-l) =1.4 x lo-4 (s-l) P = 1.0 x lO+l = KL2/D K = 1.0

Plasma Concentration Parameters [Truskey (39)]: Fpo/Cpo = 4.72 x 10-l toI = 1.443 (h) to2 = 1.281 X lO+l (h)

Parameter Comment Reference

P, Estimation with 4-parameter model; in vivo Truskey (39) Estimation of initial uptake rate across lumen; in vivo Bratzler et al. (4)

p2 Albumin data extrapolated to obtain initial uptake through Tedgui and Lever (36) adventitia [after Bratzler et al. (4)]; in situ

D Linearization of diffusion equation; in vivo Bratzler et al. (4) Estimation with 4-parameter, l-boundary (i.e., semi-infinite) model; Fry and Vaishnav (16)

in vitro

V Estimation with 3-parameter diffusion model; in vivo In vitro measurement of filtration rate serves as upper bound for

velocity of solute

Holley (2 1) Tedgui and Lever (36)

K

E

Estimation with autoradiographic data from canine aorta; in vitro Value based on data presented as rate of accumulation of

degradation product vs. concentration of *I-LDL; in vivo Direct measurement with [14C]sucrose after in vitro equilibration in

swine aorta (t = 0.45)

Fry (13) Truskey (39)

Harrison and Massaro (18)

Direct measurement with horseradish peroxidase after in vitro equilibration in rat aorta (E = 0.32)

Penn et al. (27)

Direct measurement of equilibrium profile of albumin in canine aorta in vitro (6 = 0.145)

Fry (13)

Direct measurement of equilibrium profiles for albumin in rabbit aorta in vitro; found dependence on convection (E = 0.044-0.064)

Tedgui and Lever (38)

See Glossary for definitions of abbreviations.

simulated via solution of the model for the appropriate circulation times, tl and t2, and discrete spatial points,

21. . l zn, from lumen to adventitia. The spatial distance between discrete simulated data points was chosen to be commensurate with the thickness of an experimental tissue slice (4, 5).

The dashed curves in Fig. 4, A-D, show the progression of the normalized transmural concentration profile of C (“I-LDL) as the circulation time was lengthened from 1 to 36 h. Alternatively, Fig. 4, A-D, can be viewed individually as the predicted outputs from various dual-time experiments in which tracers ( 1311-LDL, lz51-LDL) would be injected sequentially and allowed to circulate in the same animal for two specified times. For shorter circulation (sampling) times, the concentration profiles rise with increasing time. At longer circulation times, >4 h, the profiles become lower with increasing time, reflecting the action of a loss term in the media and the decrease in concentration driving forces at the boundaries.

The model simulations shown in Fig. 5, A-D, represent the predicted outputs of C and B + C of dual-species experiments for circulation times between 0.5 and 18 h after simultaneous injection of 1311-LDL and 1251-TC- LDL. At any point in the tissue, the difference between the two output profiles is the concentration of trapped degradation product or B. For the parameter values chosen, the virtual coincidence of the two curves at 30 min indicates that degradation has an insignificant effect on profiles at short times after injection, which is consistent with a previous study by Truskey (39). In con-

trast, the 18-h simulation predicted that the level of labeled TC would be an order of magnitude greater than 1311-LDL alone. Unlike intact LDL, the TC portion of TC-LDL can continue to accumulate even after the concentration driving force in the plasma has dropped significantly. Despite the decreased level of the 1311-LDL in the tissue at 18 h, the radioactivity has been shown to be measurable (4).

SENSITIVITY ANALYSIS

Sensitivity analysis was performed to assess the influence of small changes in parameter values on the model output. A sensitivity analysis assumes a priori that the model is valid and that the effect of noise on data is negligible (23, 25). If all the parameters of a certain model can be estimated uniquely from specific experiments, then that model is identifiable for the given experiments (22). Even if all the model parameters are theoretically identifiable, noisy data or sampling limitations may compromise the numerical estimation of the parameters, making practical identifiability a more elu- sive quality (17).

A necessary condition of theoretical identifiability is that the sensitivity functions of all the parameters be linearly independent over the set of available data. In other words, the effect on model output of changing a particular parameter cannot be mimicked by changing another parameter or linear combination of parameters. By graphing the sensitivity functions (derivatives of the


0.025 jc

0.005

FIG. 4. Model-derived simulations of dual-time experiment. Profiles of 1311-LDL and 12”1-LDL in media are for circulation times tl and t2, respectively; tl is fixed at 2 h (solid line) and t2 (dashed line) is 1 h (A), 6 h (B), 18 h (C), and 36 h (D). Y(z) designates a generic model output that varies in space for a fixed time ( tl or t2). Distance (z) across media is scaled according to convenient units defined in Table 1. Total thickness of media used in simulations is 100 pm. Tracer concentration values were normalized by initial plasma concentration [C,(O)]. Parameter values used in these simulations are given in Table 1.

0.000 I I I I I 1 10

2 (10 pm)

model output with respect to each parameter) versus respect to each parameter (APPENDIX A). Because these position in the tissue for a fixed circulation time, we were sensitivity functions are expressed in terms of the con- able to assess linear dependence qualitatively. Differen- centrations C and B, they were solved simultaneously tial equations for the sensitivity functions were obtained with the tracer concentration equations (Eqs. l-6). by differentiating the model equations for C and B with Each sensitivity profile was comprised of a sensitivity

0.12

0.00

\ \ /

FIG. 5. Model-derived simulations of dual-species experiment. Profiles of 1311-LDL (solid line) and 1251- TC-LDL (dashed line) are for various circulation times: 0.5 h (A), 6 h (B), 12 h (C), and 18 h (D). Profile for an 12’1-TC-LDL injection is sum of outputs from 12”1-TC and intact tracer. Parameter values are same as those used in Fig. 4, A-D and are given in Table 1.

/ / I’ / / /

2 (lo pm) 2 (10 urn)


function evaluated at points in the media after a given circulation period tl, which was fixed. The selected sensitivity profiles simulated in Fig. 6 were obtained assuming a 6-h dual-species experiment with 1311-LDL and 1251- TC-LDL. These curves indicated the regions of high and low sensitivity of the output concentration profiles C and B + C with respect to the parameters. Each of the sensitivity profiles in Fig. 6 consists of two curves: one, the sensitivity of the 1311-LDL profile to a designated parameter, and the second, the sensitivity of the profile of 12?-TC-LDL + 1251-TC to the same parameter.

Each of the curves in Fig. 6, A-C, is qualitatively similar to its companion curve. Thus the parameters P1, D, and V exhibit a consistent effect on the output, regardless of the tracer species. The sensitivity to P1 was greatest at the luminal boundary (z = 0), whereas its influence diminished further into the media. This result is not surprising since the luminal permeability parameter enters the model through the luminal boundary condition. At the luminal boundary, the sensitivity functions of convection and diffusion would be negative since both pressure and concentration gradients would be negative (i.e., decreasing in the positive z-direction). Near the adventitial boundary (z = L), the pressure gradient would remain negative, while the concentration gradient would become positive. In this case, the sensitivity functions of convection and diffusion were of opposite signs. Thus an increase in either diffusion or convection would be expected to reduce the concentration near the endothelium. Near the adventitia, however, an increase in convection would augment the concentration, whereas an increase in diffusion would lower it. The sensitivity of an *I-LDL profile to degradation is characteristically different from that of an *I-TC-LDL profile (Fig. 6D). The sensitivity function of *I-LDL with respect to deg-

0.40 A 0.003

I B

0.002

radation, dC/dK, is always negative. The sensitivity of *I-TC-LDL to degradation, d(B + C)/dK, is almost always positive. These results comport with our under- standing of the action of the two types of tracers. An increase in degradation will remove additional intact *I- LDL from the tissue while causing a concomitant depo- sition of *I-TC.

In the case of a dual-time experiment, we would anticipate a sensitivity profile for degradation comprised of two wholly negative curves. Figure 7 illustrates such a case. This set of sensitivity profiles corresponds to a dual-time experiment with 1311-LDL and 12”1-LDL in which the circulation times are 2 and 6 h, respectively. The corresponding concentration output is given in Fig. 4B. The sensitivities to degradation reveal the most obvious difference between the output of dual-time and dual-species experiments. Regardless of circulation time or position in the tissue, the sensitivity function for degradation in a dual-time experiment is negative. We interpret the sensitivity profiles to mean that the inclu- sion of an *I-TC-LDL tracer in an experiment endows the resultant concentration data with additional unique sensitivity to degradation. Provided the sensitivity to degradation is distinct from the sensitivities to other parameters, then greater absolute sensitivity of the data to degradation means a more precise estimate for the degradation constant. This result has important implications for experiment design.

We can also use the sensitivity information to identify processes whose contributions to the output are not distinct. Figure 8 shows the sensitivity profiles for E, P1, and P2 from the same 6-h dual-species experiment referred to above (Fig. 5B). One can imagine that a linear combination of the sensitivities to PI and P2 (Fig. 8, A and C) could approximate the sensitivity to E (Fig. 8B).

0.02

0.08

0.06

4

FIG. 6. Sensitivity as function of distance z across media for 6-h dual-species experiment simulated in Fig. 5B. Sensitivity curves are derived from model outputs Y1 and Y2 [1311-LDL (solid line) and 1251-TC- LDL + 1251-TC (dashed line)], as outlined in APPENDIX

A. A: sensitivity to luminal permeability P,. 23: sensitivity to diffusion coefficient D. C: sensitivity to convective velocity V. D: sensitivity to degradation coefficient K.

2 (10gm) 2 (lOurn)


0.08

z 0.06

> m

0.04

0.008

0.006

-0.006

z (10pm) z (1Opd

Such a linear dependence among sensitivities is incom- patible with all three parameters being identifiable.

A quantitative index of linear dependence (&) has been suggested by Beck and Arnold (2). The larger the value of the index, the upper bound of which is 1, the further the sensitivity functions are from linear dependence. To illustrate the use of this index, we compared the relative merits of a 6-h dual-species experiment to those of either 6-h single-tracer experiment. We found that the value of & was orders of magnitude larger for the six sensitivity functions corresponding to the simulated output of *I-LDL and *I-TC-LDL shown in Fig. 5B (& = 10114) as opposed to the output of either tracer individually K = 10a2’). It may still turn out that the improvement g$ned from the dual-species approach is not adequate to overcome near-linear dependence. In fact, preliminary fits of model output to experimental data (not shown) confirm the sensitivity analysis, namely, that we cannot estimate all the parameters simultaneously from the proposed experiments.

OPTIMAL EXPERIMENT DESIGN

Objective Function

To achieve the best fit of our theoretical model to experimental data, we minimize the least-squares objective function (G) with respect to the model parameters (8) at n data points. The residuals are summed over tissue samples (index i) obtained from an animal and over concentration measurements in each tissue layer (index j) cut from each tissue sample. In the case of the dual-time experiment, the objective function (Eq. 7a)

FIG. 7. Sensitivity as function of distance across media for dual-time experiment [tl (solid line) = 2 h, t2 (dashed line) = 6 h] simulated in Fig. 4B. A: sensitivity to PI. B: sensitivity to D. C: sensitivity to V. D: sensitivity to K.

includes residuals from measurements at times tl and t2

@(@;tl,tZ) = CC $ (Wij[YDij(tI) - yij(e,tl)]” j i

(7 ) a + Uj[ YDij(t2) - Kj(e,t,)]“)

For a dual-species experiment, the objective function (Eq. 7b) contains residuals from tracers a and ,8

@(a;a,P) = xx $ (Kj[Yu,(a) - yij(e,a)]” j i

VW

+ U,j[YDij(P) - yij(e,P)]21

where Ynij and Yij are the data and model outputs, respectively. In either case the weighting factors, Wij and Uij, can be adjusted according to the error structure of the data, typically to emphasize each residual inversely according to the variance of the corresponding datum Yn... If the error in the data has constant variance independent of medial position (z-axis) and of tracer signal, then the weighting matrix reduces to the identity matrix. On the other hand, if the data exhibit constant relative error (i.e., constant coefficient of variation), then weighting inversely according to the variance would be equivalent to normalizing each squared residual by the square of the corresponding YnG. For lack of an a priori measure of the data error structure, we have designed our experiments using the identity as a weighting matrix. To investigate the effect of the constant variance assumption, we also examined the dual-species design assuming constant relative error. The results are presented below.

Indifference Region

The gradient of the objective function about the optimal point in parameter space, together with the error

OPTIMAL DESIGN OF L ,DI J TRANSPORT EXPERIME IN TS H937

0.30

0.25

l-

a 0.20

2 > 0.15

m

0.10

0.05

z (1Qlm) z UOym)

structure of the data, determines the precision of the estimated parameters. This gradient depends on the sensitivity functions. The information contained in the sensitivity functions along with the chosen weighting scheme is encoded in the Hessian matrix (APPENDIX B). The Hessian matrix is approximately inversely proportional to the covariance matrix as long as the error in the data is uncorrelated, Gaussian, and with zero mean and the weighting scheme reflects the error variance structure of the data. If these stipulations on the data are met, then the properties of the Hessian matrix pro- vide quantitative measures that relate to precision and interaction of parameter estimates (26).

There exists an “indifference region,” defined by the tolerance in the objective function (Q), about the optimal point 8* in the parameter space. Within this tolerance, changes in the objective function cannot be distinguished (APPENDIX c). For illustration, we consider only two parameters at a time and fix Q = 1. If it is assumed that the remaining parameters are fixed at their true values, then in a plane of the parameter space sufficiently close to the optimum, the indifference region is approximately an ellipse, the origin of which corresponds to the optimal choice of parameters. The axes of the ellipse are related to the eigenvectors of the 2 x 2 Hessian matrix constructed from the sensitivity coefficients of the two parameters of interest (APPENDIX C).

The coordinates of the longest axis of the ellipse indicate the greatest combined deviation from the optimal parameter values that would result in a value of the objective function within the specified tolerance but would yield essentially the same model output. This is a measure of the potential variability in the two parameter values. It is a convenient single measure in lieu of the standard statistical analysis involving the coefficients of variation and the correlation coefficients and requires only a suitable choice of the tolerance St. For the purpose of evaluating the relative variability of one parameter pair versus another at a fixed tolerance of a given optimal point, it is sufficient to fix the tolerance at 1.

The coordinates of the poorest determined choice of each parameter pair are given in Table 2 in terms of the fractional change of a parameter about its optimal value. The values in Table 2, top, were calculated for a 6-h dual- species experiment. Table 2, bottom, contains values for a 2- and 6-h dual-time experiment. Table 2 indicates a consistent result: the coordinates of the poorest determined point on the indifference region for P1 and c were furthest from their optimal values. Combinations of the

FIG. 8. Sensitivity as function of distance across media for 6-h dual-species experiment simulated in Fig. 5B [1311- LDL (solid line), 12’1-TC-LDL + l=I-TC (dashed line)]. A: sensitivity to P1. B: sensitivity to partition coefficient e. C: sensitivity to adventitial permeability &.

TABLE 2. Coordinates of longest principal axis on elliptical indifference regions for all parameter pairs

Dual-Species Experiment

02: p2 PI D V K-

01 t 23.888 138.150 5.042 1.179 7.575

21.995 73.394 34.293 26.842 16.671

p2 17.144 20.598 57.884 14.315 11.859 38.405 50.737 16.973

P, 1.986 0.922 5.347 33.861 26.891 17.420

D 41.943 41.404 21.241 11.647

v 37.591 15.836

Dual-Time Experiment

02: PP PI D V K

01 E 60.050 312.336 6.390 2.162 93.945

43.333 165.085 83.695 72.243 50.929

p2 36.728 58.547 97.407 74.203 30.738 98.177 104.392 33.974

Pl 2.850 1.542 37.179 83.533 72.287 42.180

D 106.665 93.488 60.487 13.741

V 82.128 14.662

Coordinates are given as an absolute value fraction of the optimal parameter value ] Sej/@j ]. Coordinate pairs in bold indicate indifference regions with largest major axes. Tolerance level Q = 1 (see APPENDIX c). For every @, 8, we have listed the coordinates of the longest axis as a column: Se&3 [ 1 se,/@ *

parameter pair (PI, c) could vary dramatically without altering the value of the objective function. The impli- cation is that P, and E cannot both be estimated precisely from the same data set. Comparing Table 2, top and bottom, we see that that two-dimensional indifference regions for dual-time experiments were generally larger than the regions for dual-species experiments. Any comparison across the table is valid only if the error in the data has the same constant variance for tracers of a single design as well as for pairs of tracers from different designs.

One of the most desirable experimental designs is one that minimizes the volume of the indifference region and thus reduces the range of possible parameter choices that


achieve the same value of the objective function. Various scalar design criteria can be used as comparators of different designs (10, 25, 26). Because the determinant of the Hessian [Det(H)] is inversely proportional to the square root of the volume of the indifference region, the maximum Det( H) is a suitable criterion for an optimal design. The design that maximizes Det(H) is referred to as “D-optimal” (10, 11, 26) under the conditions stipu- lated above when H-l is approximately proportional to the covariance matrix. In examining both of our proposed experiments, we have evaluated the determinant over the range of the design variable t, which is the time of tracer circulation in the plasma.

The dual-time experiment requires the simultaneous choice of two circulation times, tl and t2. To find the combination of times for this experiment, we examined Det(H) over a large matrix of tl, t2 pairs. Time tl was varied between 0.5 and 9 h in increments of 0.5 h, whereas t2 was varied between 1 and 36 h in increments of 1 h. On the basis of our sampling of tl, t2 pairs over a large region of interest, the surface of Det(H) plotted in tl, t2

space appears to be unimodal.

Reduction of the Parameter Space

Not all the parameters were equally well determined, nor are they all of comparable physiological importance. The Hessian matrix can be constructed in different ways to reflect the status of the parameters. The simplest approach was to use the sensitivity functions for all six parameters, E, PI, P2, D, V, K, to construct a 6 x 6 Hessian matrix (H6). This case reflects an equal interest in values for all six parameters. The determinant values

-16

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Tf- -24

r w -25

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0 -27

s -28

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-34

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-36

2 hr

A

1 10

time, t (hr)

FIG. 9. Dependence of log of determinant of Hessian, log[Det(HJ], on time of circulation (t) of single tracer *I-LDL. Hessian examined in this analysis was 4 x 4 matrix constructed from sensitivity matrixes of P1, D, V, and K only. For parameters chosen (see Table l), optimal single-species circulation time was 2 h.

for this Hessian matrix were typically more than four orders of magnitude smaller than those for the 4 x 4 matrices shown below. This means that one or more parameters may have an unacceptable variability.

If we were not concerned with estimating values for all of the parameters, then we could base the optimization procedure on a matrix composed only of the sensitivity functions of the parameters of interest. Accordingly, we examined Det(H) values for a Hessian matrix corresponding to the sensitivity functions for PI, D, V, and K. Use of this 4 x 4 matrix (H4) assumed that the remaining parameters had been fixed at their true values. For our purposes the parameters P2 and E, although unknown, are of relatively little interest. We therefore treated them as “nuisance” parameters (3) and constructed a transformed 4 x 4 Hessian matrix (G) from all six sensitivity coefficients (APPENDIX D). We also examined determinant values for another 4 x 4 Hessian matrix (G’) in which P2 is regarded as a nuisance parameter, while a value for c was fixed at a representative value (Table 1).

Some of the results of examining these four optimal design criteria, Det(HG), Det(H4), Det(G), and Det(G’), are presented. Values of the determinants were calculated assuming the values in Table 1 to be the true values. Figure 9 is a plot of Det(H*) versus circulation time for a single-tracer experiment. The D-optimal circulation time was -2 h. Figure 10 shows the dependence of Det(H& Det(H4), Det(G), and Det(G’) on circulation time tl for dual-species experiments according to the four design criteria; each curve has a peak determinant at

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z -15

1 -16

+ a> -17

n -18

ol -19

0 - -20

-21

-22

-23

-24

18 hr

/?A-

I I I11l11~ i I rIIll1~ I 111111-q

10 -’ 1 10

time, t (hr) 10 2

FIG. 10. Dependence of log[Det(H)] on circulation time for dual- species experiment. Hessian was constructed in 4 different ways depending on interpretation of parameters P2 and 6, as described in text. For 0, P, and t are nuisance parameters; q , no nuisance parameters are designated; A, P2 and E are fixed constants; X, P2 is nuisance parameter and e is fixed. For dual-species (*I-LDL and **I-TC-LDL) experiment, there is 1 circulation time, tl, which varies along x-axis. In each curve, optimal time is between 12 and 18 h.


either 12 or 18 h. The contours in Fig. 11 show the dependence of Det(H4) (i.e., the 4dimensional indifference region) on the simultaneous choice of circulation times tl and t2 for the dual-time case; the plot has a maximum at tl = 1 h and tz = 5 h. Contour plots of the other three design criteria were comparable in shape and location of the optimum. As in Fig. 10, the magnitudes of the determinants of the other Hessian matrixes were smaller. Comparison of the range of Det(HJ in Fig. 9 with the values for Det(H4) in Figs. 10 and 11 shows that either dual-tracer design was superior to the single-tracer experiment. The valley in the contour map near the imaginary tl = t2 line in Fig. 11 reflects our finding that there would be relatively little advantage in using two unmodified LDL tracers for the exact same circulation period. Such an experiment (with tl = t2) would be only marginally better than the optimal single-tracer design.

Effect of Error Structure

Figures 9-11 were derived from Hessian matrixes constructed using the identity matrix as the weighting matrix. Figure 12 is analogous to the dual-species optimization of determinants shown in Fig. 10 except that the weighting matrix was changed to reflect an assumption of constant relative error in the data (see APPENDIX B). By assuming that the error in the data is proportional to the signal and reweighting accordingly, we find two max- ima. Although Det(H) tends to be larger as circulation time approached zero, its practical significance is limited by the signal-to-noise ratio of the data. Enough time must be given for tracer to enter the tissue in statistically significant quantity.

The fact that the results for dual-species designs were qualitatively consistent regardless of which of the four Hessian criteria was examined suggests that the same optimal experiments would be prescribed whether or not

some subset of the parameters e, PI, and P2 were to be treated as nuisance parameters. That is, even if it is not possible to estimate all parameters simultaneously and some must be held constant, the design results presented here remain valid.

Secondary Design Criterion

Depending on the particular Hessian criterion that was examined, there was generally a range of circulation times over which Det(H) was nearly optimal for the dual- species case. Besides some practical considerations that may favor a 6- to 12-h experiment over an 18-h experiment, there exists a formal criterion that can be used to distinguish D-optimal equivalent designs. The condition number of the Hessian matrix, which is the ratio of the largest to smallest eigenvalues of the matrix, is a measure of the eccentricity (the ratio of lengths of the major to minor axes of an ellipse) of the indifference region. For a fixed volume of the indifference region [a constant value of Det(H)], a small condition number is desirable. The more eccentric the indifference region for a given volume, the more nonuniform the variability among parameters and the less likelihood that an optimal point can be calculated in the presence of noise. Figure 13 displays the condition number of the Hessian matrix for dual-species experiments for all circulation times examined in Fig. 10. Figure 13 indicates that although the eccentricity of the region deteriorated slightly from 6 to 18 h, there was very little distinction among values over this range of D-optimal circulation times.

Effect of Parameter Values

To assess the dependence of our results on the particular choice of parameter values, we repeatedthe analysis with an alternative parameter set (shown in Table 1),

6.0

F

FIG. 11. Contour plot of log[Det(H)] as function of circulation times tl and t2 for dual-time experiment. Hessian used for these calculations was 4 x 4 matrix H4, where within contour A, for which determinant is at a maximum

E and P2 are fixed. Optimal circulation time pair is found

-15.50


4

-1

-2

-3

6 hr

.l 1 10

time, t r (h )

100

FIG. 12. Dependence of log[Det(H)] on circulation time where weighting matrix (W), assuming constant relative error in data, was used to construct Hessian for dual-species experiment. Diagonal elements of W are inversely proportional to square of corresponding model outputs. This figure is analogous to Fig. 10, except for change in W. Hessian was constructed in 4 ways depending on interpretation of parameters P, and E. Symbols are as in Fig. 10.

IO -’ 1 10 10 2

time, t (hr) FIG. 13. Dependence of log of condition number of Hessian,

log[ cond( H)], on circulation time for dual-species experiment. Hessian was constructed in 4 ways depending on interpretation of parameters P, and E. Symbols are as in Fig. 10.

which intentionally deviated from the previous set but which still satisfied the requirement that the corresponding concentration profiles should reflect experimental data. This parameter set differed from the other by twofold increases in K, Peclet number (the dimensionless ratio of convection to diffusion), and E. The dimensionless permeabilities were altered to increase P1 and to

I I I I I I I I I I I lllllrl I I I111111

decrease P2 by three- to fourfold. Figures 14 and 15 are analogous to Figs. 10 and 11 for the alternative parameter choices.

Figure 14 displays Det(H) versus circulation time for dual-species experiments, with the assumption that the alternate parameter choices are the true values. The results for a large range of simultaneous tracer circulation times were nearly identical to those presented above. Although not shown, the single-tracer D-optimal time for the alternate parameter set was -2 h, as before. Figure 15 indicates that for this parameter set the D- optimal times are tl = 0.5 h and t2 = 3 h. For the contours of the other determinant criteria (not shown) the optimum was found at (0.5, 2). The dual-species design was again the preferable design. This does not represent an exhaustive approach; however, Figs. 10, 12, and 14 suggest a robust optimal design result for the dual-species experiment. Ideally, the optimal design analysis would be repeated after obtaining preliminary parameter estimates. A shift in the optimal combination of times tl and t2 for the dual-time experiment (Figs. 11 and 15) reflects a stronger dependence of the choice of these times on knowledge of the true parameter values. This shift is apparently dependent on the value of K.

DISCUSSION

Key Results

We have focused primarily on two possible experimental protocols as means of estimating macromolecular transport parameters in the arterial wall. A dual-species experiment was defined as an intravenous injection of both *I-LDL and **I-TC-LDL (where *I and **I are different isotopes of iodine), circulation of both tracers

-10

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-24

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-26

-27

18 hr a

1 11II111~ I r111IIl~ I 11I1111~

1 10

time, t (hr)

FIG. 14. Dependence of log[Det(H)] on circulation time alternative parameter set was used to evaluate Hessian (see for dual-species experiment. Hessian was constructed in 4 pending on interpretation of parameters PI and E. Symbols Fig. 10. Optimal circulation time tl of dual-species experimen at 18 h.

10 2

where an Table 1) ways de- are as in

t remains

25-


for a predetermined time tl, followed by measurement of tracer concentration profiles in the aorta. A dual-time experiment was defined by two tracers *I-LDL and **I- LDL, concentration profiles of which are obtained at circulation times tl and t2. To investigate the relative potential of dual-species and dual-time experiments for estimating parameter values, we used model simulations, sensitivity analysis, and optimal design methods.

Starting with model simulations, we were able to gain some insight about TC-label experiments and expected results. These simulations, performed with parameter values compiled from several sources, were necessary because distributed profiles of the modified tracer have not been obtained previously. The predicted profiles agreed with the finding (39) that normal degradation in the media is negligible for short times (e.g., labeled LDL circulation of 30 min) (Fig. 5A). Simulated profiles of tracer concentration and sensitivity coefficients suggest that a lengthy (i.e., 18 h) dual-species experiment with the labeled tracers *I-TC-LDL and **I-LDL is preferable to a single-tracer experiment or even a dual-time experiment with two distinguishable tracers (*I-LDL and ““I-LDL).

Figures 4 and 5 contrast the progression of the tissue- tracer profiles for either dual-tracer experiment as circulation time increases. The predicted outputs for *I- LDL at various times show a net positive accumulation in the tissue for approximately the first 2-4 h after which point the net flux of tracer is out of the tissue. Presum- ably, this evolution in the profiles is caused by the cumulative degradation in the tissue and the reduction or even reversal of the concentration driving forces across the boundaries.

Information value of the data. Because the goal of our optimal design procedure was to design experiments that would yield data that promote precise estimates of the parameters, we were interested in identifying the char- acteristics of a concentration profile that dictate its information value. With regard to dual-time experiments, we have observed that the *I-LDL profiles at

-14.50

-15.00

-15.50

-16.00

-16.50

FIG. 15. Contour plot of log[Det(H)] as function of circulation times tl and t:! for dual-time experiment with alternative parameter set. Hessian used for these calculations was 4 x 4 matrix H4, where E and P2 are fixed. Optimal circulation time pair is found within co~~tour A, for which determinant is at a maximum.

times >6 h assumed a pseudo-steady-state appearance. That is, their shape changed little, but their scale continued to change (decreasin .g) with time. As the circulation time was extended, the concentration gradi .ent in the luminal region disappeared. Finally, the average level (i.e., average tracer concentration value with respect to medial position) of the near-flat profile decreased to a level that would be difficult to distinguish from noise. Conversely, the TC-label profile continued to accumulate with increasing time. This profile retained curvature, gradients, and average level; thus it serves as a fingerprint of the tracer dynamics in the tissue. A long time after injection, this fingerprint of deposited degradation products will continue to demonstrate the occurrence of certain processes in the tissue even when intact labeled tracer is no longer detectable.

In our analysis the dual-species experiments, which utilize the TC label, were repeatedly found to be superior to the dual-time case in terms of precision of parameter estimates. How can we relate these findings to our observations of the time progression of simulated profiles of each pair of labeled tracers? We begin by examining the derivation of sensitivity functions, as in Eq. A2 of APPENDIX A. Only one of the last three terms on the right-hand side of Eq. A2 will be nonzero for any sensitivity coefficient. For example, if ej = D, then the sensitivity coefficient for diffusion is dependent on the second spatial derivative in the concentration profile while the first-order and zero-order terms in C are zero. In other words, the sensitivity of the output to diffusion is dependent on the second-order spatial character (i.e., curvature) in the profile. Without sufficient curvature in the profile, the effect of diffusion on the model output will be too small to yiel .d an estimate of the parameter. Similarly, sensitivity to convection is dependent on concentration gradient, and sensitivity to degradation requires sufficiently high tracer levels. The same reasoning can be applied to the boundary conditions to understand the sensitivities to P1, P2, and E. Because the TC label retains information with time, a fairly lengthy dual-


species experiment was prescribed by our analysis, perhaps counter to one’s intuition.

We note that the potential for obtaining an estimate of a parameter does not necessarily correlate with its relative effect on the output. A system in which diffusion is much greater than convection when compared on a dimensionless basis (Peclet number ~1) would attain a pseudo-steady state very quickly. If the concentration profile were sampled after the spatial curvature had disappeared, then there would be little sensitivity to diffusion. Thus the chances for estimating the diffusion coefficient would be reduced even though the system might be described as diffusion dominant.

Applicability of the optimal design. Our results for D- optimal circulation times were consistent whether or not we regarded some of the model parameters as nuisance parameters. After choosing representative parameter values from the literature and assuming that data errors would have constant variance regardless of medial position or tracer component, we optimized the experimental tracer circulation time(s) by minimizing the volume of six-, five-, and four-dimensional indifference regions in parameter space.

Examination of pairwise indifference regions identified those parameter pairs that will be the most difficult to estimate simultaneously. The two-dimensional indifference regions, which were constructed for an arbitrary fixed tolerance in the objective function, were used to compare the greatest combined uncertainty in pairs of parameters for a given design and parameter vector. The pair p1, 6 was repeatedly the worst determined, but the indifference regions in Table 2 also indicated that the length of the major axis of the V, Pg regions was very large. In fact, in preliminary attempts at fitting the model outputs to experimental data (not shown), we were not able to estimate both P, and V simultaneously.

Because the adventitial “permeability” Pp represents the combined effects of many processes (e.g., delivery of blood to the vasa vasorum, efflux of tracer to lymphatic capillaries, etc.), it is physically uninterpretable. There- fore it seems reasonable to regard it as a nuisance parameter. It is needed simply to construct a boundary condition that will represent data near the adventitia. We included P, in the model when, for numerical reasons, simulations using a simpler (Dirichlet) form of the adventitial boundary condition caused artifacts in the simulated profiles (30). Recently, Tedgui and Lever (37) obtained profiles of *I-albumin in vitro by exposing the outer medial boundary (after removing the tissue that surrounds the adventitia) directly to a bath of *I-albumin. From their profiles, we were able to calculate an initial uptake rate (5) for *I-albumin through the adventitia. This uptake rate led us to a rough estimate of adventitial permeability, which then was subject to the additional requirement that the resulting simulated profiles resemble actual in vivo profiles of *I-LDL obtained with the slicing method in our laboratory. By varying the value of P2 we were able to obtain the desired slope of the profiles near the media-adventitia border. Both the primary and alternative parameter sets examined above incorporated values of Pa that were in the neighborhood of the values obtained from Tedgui and Lever (37). These

generated model outputs were representative of the available in vivo data (4, 21, 39; G. M. Chisolm and P. Ganz, unpublished data).

As with any model that is nonlinear in parameters, the sensitivity coefficients and thus Det(H) values are functions of the parameter values. Therefore the optimal experiment design will be dependent on the choice of parameter values (24), which in the present work is based on a survey of related studies. By verifying our analyses with an alternative set of parameter values, we have attempted to demonstrate that the apparent optimal circulation time for the dual-species experiment is rea- sonably robust. This, however, is not an exhaustive study. In the case of the dual-time experiment, the D- optimal times tl and t2 found for the alternative parameter set differed from those found for the original parameter set. The contour lines for a given value of Det(H) shifted uniformly to earlier circulation times when simulations were rerun with the alternative parameter set, the degradation rate of which was chosen to be twice as high. This result indicates a dependence of the optimal circulation times on parameter values. This dependence can, however, be understood by studying the simulated model output for the alternative parameter values (not shown). A larger degradation constant caused faster re- moval of tracer from the tissue, reduced the information value of the data more quickly, and placed a premium on shorter circulation times. From the perspective of an investigator designing a preliminary study, the dependence of optimal circulation times on knowledge of the true parameter values is more critical for the dual-time experiments than for the dual-species experiments.

In any case, our methodology for optimization of a data measurement scheme for a multi-input, multi-output distributed system is useful for suggesting a preliminary experiment design. If assumptions inherent in our optimization (e.g., choice of parameter values) turn out to be in error, then subsequent reoptimizations may be required. We have tried to anticipate the need for further redesign of the experiments by examining the sensitivity of the results to our choices of parameter values and error structure of the data. Generally though, the process of experimental design is iterative. On the basis of available information, one first constructs a model and determines a preliminary optimal experiment design. Next, one modifies the model as needed to accommodate preliminary data. One then reoptimizes the experiment design.

Related Studies

Other experiment designs. Many of the attempts to estimate transport parameters for macromolecules in the arterial wall have been based on data generated in vitro (12, 13, 16) or in situ (36-38). An estimate of luminal permeability is compromised in such cases because it is difficult to maintain the integrity of the endothelium outside the animal. In vivo studies using either autoradiographic or slicing methods yield *I-LDL profiles that are generally noisy, and estimation of proposed parameters is difficult. Fry (13) listed only estimates of diffusion coefficient and convective velocity for 7”51-albumin pro-


files in deendothelialized canine aortas in vitro as having acceptable coefficients of variation. Fry’s convective- diffusive model included terms for partition coefficient, degradation, and activity coefficient. In an attempt to improve the precision of certain parameter estimates, other parameters were fixed. All profiles used in data fitting were obtained for 3.5-h exposures to constant concentration baths of either albumin or serum. Al- though autoradiograms of iodinated tracer profiles pro- vide improved spatial resolution over the slicing technique, there is no way to distinguish one iodinated tracer from another and thus no way to use them in conjunction with a dual-tracer experiment.

Other tracers. Although the implicit intent of many studies is to investigate macromolecular transport as it pertains to atherosclerosis, labeled albumin, and not LDL, has often been used as a tracer (5,13,38). Albumin is convenient because it is commercially available in stable form, it can be labeled more efficiently than LDL, and it is not degraded to the same degree in the arterial media. It is difficult to generalize from findings on processes affecting albumin to processes that might affect LDL. For example, a defect in the normal degradation rate of LDL, a reduction in the convective flux of LDL across the arterial wall, or an increase in the available interstitial space could conceivably lead to abnormal accumulation of LDL in tissue and perhaps contribute to an atherosclerotic plaque. Only the alteration in convection could be correctly detected by a study using albumin as the tracer, since the fluid space accessible to it may be quite different from that available to LDL and since smooth muscle cells do not degrade it to the same degree. In arteries with gross atherosclerotic plaques it has been observed that certain cells migrate into the intima and degrade LDL at higher rates than do the smooth muscle cells of the media (45). Although no quantitative estimate was made of a degradation rate constant, it is reasonable to suspect a link between disease and alteration of degradation rate. Labeled LDL would be required to investigate such a link.

Truskey (39) has quantified degradation, convection, and diffusion of LDL, using both cell culture and in vivo experiments in normal rabbits. Furthermore, they used a [‘4C]sucrose ligand attached to LDL (analogous to *I- TC-LDL) in vivo, but because the [‘4C]sucrose-LDL had such a low specific activity, their estimates of degradation were derived from cell culture work and from lumped averages of [14C]sucrose accumulation in tissue rather than from distributed profiles in vivo.

Other specific chemical modifications of LDL can be used, along with appropriate changes in the general mathematical model, to resolve parameters into receptor- mediated and receptor-independent components. Meth- ylation of the lysyl residues on LDL is known to block the receptor-mediated component of degradation of the lipoprotein by smooth muscle cells of the media (44). Simultaneous injection of methylated *I-LDL and methylated *I-TC-LDL could lead to a precise estimate of the non-receptor-mediated component of LDL degradation. The concentration C would apply to both the TC-modified methyl (Me)-LDL as well as the “native” Me-LDL as required by the model. The disappearance from the

plasma of these methylated tracers ought to be equivalent and could be incorporated into the model’s boundary equations by an appropriate time-varying function C,( t ). We would expect the non-receptor-mediated degradation rate to be less than the degradation rate proposed in the present study, but values of other parameters should be invariant. Therefore a valuable sequence of events would be 1) estimate parameter values from optimal dual- species experiments, 2) modify the model, and 3) reop- timize experiments with two methylated tracers using the results from 1) and an adjusted value of K as initial parameter estimates.

Carew et al. (7) have used the TC-ligand in conjunction with Me-LDL in a related experiment. They measured the rate of degradation of LDL in rabbit aortic media or intima alone by comparing the presence of total TC-Me- LDL with intact LDL. The difference between the tracer levels in the tissue was taken as a measure of non- receptor-mediated degradation. However, they conceded some difficulty with this assessment since the unmeth- ylated tracer was removed from the blood more rapidly than the methylated, resulting in unequal amounts of TC-Me-LDL and LDL being presented to the tissue. No quantitative estimate of a rate constant was made in this study.

Practical Considerations

Data auailability. To perform the analysis presented in this paper, we used idealized data. For each study except one, we assumed that the error variance of all data points would be equal for each tracer component and each tissue slice. In Fig. 12, we presented a similar design analysis assuming constant relative error of the data measurements as an alternative. In most cases we believe that one or the other of the examined error structures would be adequate to model our data. The general agreement of optimization results for the two different error models implies that sensitivity of analysis to the error structure is minimal.

The disparity between the results of different error assumptions (see Figs. 10 and 12) arises at circulation times ~1 h. The constant relative error assumption appears to favor very short circulation times approaching zero; as the signal approaches zero, the error approaches zero. At very low signal levels, though, other factors concerning the measurability of the data, which have not been included in the model, become significant. If it is considered that tracer radioactivity at internal points in the media for early times approaches l/1,000 of the initial plasma radioactivity (typically lo7 counts l min-‘. ml plasma-‘), then the number of counts per minute in a l- cm2 tissue slice IO-pm thick is on the order of background in a gamma counter. In other words, the internal regions of the artery wall, which at early times have not yet been exposed to much incoming tracer, might actually exhibit greater relative error than regions near the medial boundaries. Thus the design recommendations between the two plausible error models proposed are in general agreement, and neither may apply when low signal-to-noise limitations are most likely (e.g., very short or very long circulation times). We must consider these limitations


when comparing optimal designs based on different objective function weighting matrixes, which are chosen to reflect different possible error structures in the data.

Although the data at 18 h (assuming constant variance) may lead to greater parameter sensitivity than at 12 h, here too the effect of a decreased signal-to-noise ratio in LDL data might mitigate this advantage. In other words, there may be a trade-off between less noise relative to the signal of the data and a slightly larger indifference region predicted for a 12-h rather than an 18-h circulation of *I-LDL and **I-TC-LDL. A priori knowledge of the data error supporting the constant relative error assumption would be an additional argu- ment in favor of the 12-h dual-species circulation. The condition number, a secondary design criterion, suggests a reduction in the eccentricity of the indifference region at 12 vs. 18 h. Both times yield nearly equivalent D- optimal designs in the dual-species case. If we choose to repeat our optimal design analysis following a preliminary estimation, recalculation of the Hessian matrix could incorporate an empirically determined weighting matrix W to account for the error associated with the data points (APPENDIX B) in addition to the new parameter estimates.

Experimental limitations. A clear advantage of in vivo studies is that an estimate of permeability will reflect the actual physiological barrier between the lumen and the media. Once the vessel has been removed from the animal as in an in vitro system, endothelial and smooth muscle cells may not survive. If endothelial cells comprise the primary barrier for entry of plasma macromolecules, then even a small experimental disruption in the con- fluent cell lining of the vessel would lead to an artifactual estimate of permeability (43). In some experiments the endothelium is denuded intentionally to examine transport in the absence of endothelium (37, 42). In such experiments, the internal elastic lamina may be damaged or medial smooth muscle cells may be injured (29).

Because the internal elastic lamina may play a major role in trapping LDL in the intima (14), harsh experimental methods could bias estimates of permeability in the intentionally denuded system as well. Damage to smooth muscle cells would almost surely affect the partition coefficient, diffusion coefficient, and degradation constant. Recently, Tedgui and Lever (38) demonstrated a dependence of the equilibrium partition coefficient on convective flux’ of macromolecules into the interstitial spaces of the artery wall. Thus any deviation from the in vivo transmural pressure distribution that might accompany cannulation of the vessel in vitro could induce an apparent change in E.

Having highlighted some shortcomings of in vitro experiments, we must also acknowledge the experimental and theoretical caveats that accompany the interpretation of parameter estimates from in vivo experiments. The data that we are modeling are generated by the slicing technique described above, which 1) obliterates any two-dimensional information in the tracer profiles and 2) has a spatial resolution of 4-20 pm depending on the thickness of the slice. Because of these limitations, the model was devised to explain one-dimensional profiles in the media, while all structural components of the

arterial wall situated within 10 pm of the lumen were necessarily lumped into the boundary condition. Thus any action of the endothelium, the intima, or the internal elastic lamina to restrict the entry of plasma macromolecules into the media will affect our estimate of luminal permeability P1. Nevertheless, reliable estimates of this “lumped” permeability in normal animals may be a quantitative measure of arterial wall health.

A change in this parameter in preatherosclerotic animals would have mechanistic implications for the early development of the disease. In addition to autoradiography, there are higher resolution digital imaging techniques being developed to measure profiles using peroxidase activity as a tracer (27). These techniques may yield better conditioned profiles with which to estimate parameters. Optimal design analysis based on simulated profiles with the appropriate number of spatial points would apply equally well to experiments using such techniques.

Conclusions

Sensitivity analysis and optimal experiment design are commonly used with compartmental models. We have applied these techniques to a spatially distributed system that is nonlinear in parameters. Although the sensitivity equations are more difficult to manipulate than in compartmental modeling, we believe there are advantages in performing such an analysis before extensive experimental work. In the particular case of modeling LDL concentration profiles in the arterial media, our analysis suggests optimal circulation times that likely would have been overlooked had we not first performed the optimization. Our simulations of the sensitivity functions im- plied that we would not be able to resolve the contributions of each proposed transport process. In light of this, we examined pairwise indifference regions to identify poorly determined combinations of parameters in the hope of focusing our efforts on the parameters that are both useful and accessible.

Although our specific results are dependent on the validity of our assumptions, we also considered the effects of these assumptions via an alternative parameter set and another plausible data error structure. Neverthe- less, it is clear from our analysis of two dual-tracer protocols that either the dual-species or dual-time tracer experiments are definite improvements over single labeled-tracer techniques for measuring concentration profiles. Furthermore, the TC-labeling technique developed and applied by Pittman et al. (28) and Carew et al. (7) offers a means of improving the information value of tracer profiles in the artery wall. The benefit of the TC label in enhancing the precision of parameter estimates is that it generates unique concentration profiles without requiring additional parameters or complexity in a model that describes the data. We have identified the optimal dual-species experiment to estimate parameters of interest with a precision adequate to quantify transport processes in normal tissue and possibly processes in diseased tissue.


APPENDIX A run-time if it is the sole job on a VAX/ll-780.

Sensitivity Equations and the Sensitivity Matrix The set of all sensitivity functions make up the sensitivity

matrix S

For all experiments, we define an output vector Ylk = C(t,,zJ, where k = 1, . l . n for circulation time tl. For the dual- time experiment, we define a second output vector Y2k = C( t2,zk). For the dual-species experiment, YZk = C( tl,zk) + B( tl,zJ. The output sensitivity functions are the derivatives of the outputs with respect to the parameters

S’ rkj (Al)

(A7)

where @j is the jth parameter of et PB, P1, D, V, K. To obtain s&j, we must differentiate the state variables C and B individ-

where ST is the transpose of S.

ually and sum the results where appropriate. The general An index for assessing near-linear dependence has been

approach is as follows: differentiate the state equations with recommended by Beck and Arnold (2)

respect to any parameter, interchange the order of differentia- Det(STS) tion, and solve for the newly constructed dependent variables, c

’ = [p-lTr(STS)lp (A8)

namely, the sensitivity functions. The sensitivity equations fo the state variables are where p is the number of parameters and Tr( ) indicates the

trace of a matrix. I f lp is near zero, then two or more of the sensitivity functions are close to being linearly dependent.

and

OCZCL, t>o

d at

The

-p1 t

boundary conditions are at z = 0

(g-)-[+cp(t)][gJ+P,c[$]

=v(g)+$+:(g)

and at z = L

APPENDIX B

Objective Function and the Hessian Matrix

To find the optimal parameter set we seek to minimize the weighted least-squares objective function + as follows

@(8) = $[YD - Y(W’W[YD - W@l (Bl) L, t > 0 (A3)

[ I W)

dC dD -- - dZ &3j

where Yn is the vector of experimental observations and Y is the vector of model outputs. Thus YD - Y is the residual vector. The weighting matrix W is chosen to reflect the variance of the data when possible. If the data error structure is not known a priori, then the weighting matrix must be chosen based on assumptions about the error. The ith component of the gradient vector of first derivatives of the objective function with respect to the parameters is

- yk@)] (B2)

The summation over h corresponds to the n data points. For uncorrelated errors, W is a diagonal matrix, the elements of

I \ I . _ WV which are the inverse variances (0~~) of Yn, [i.e., W = diag(ai2)]. When the variance is constant, W reduces to a constant matrix. For a constant relative error (i.e., constant coefficient of variation), each element ak2 of W is chosen

The initial conditions are proportional to [ Yke]-2. dC dB P--x de, - 00, O9 t = O (A6

Note that the derivatives of the parameters (in square brackets 83. are evaluated as follows: L [ I i@j

= 1 for i = j and = 0 for i # j.

The Hessian is the matrix of second derivatives of the objective function with respect to the parameters. For the case where YD - Y is small, the ijth element of the Hessian follows

The sensitivity functions are coupled to the state variables. and the matrix can be expressed as the product Thus the whole system of partial differential equations must be solved simultaneously. In general, the number of partial H ES g’s (B4) differential equations to be solved in such a system is n(p i- 1), where n is the number of state equations and p is the

where the components of s are

number of parameters. We solve our problem for concentration values and for derivative values with respect to parameters E, $ikj ’ aYik =-

ck 6@j

VW P1, P2, D, V, and K at 21 evenly spaced points across the z- axis. For example, for two tracers (n = 2) and six parameters, When the model is linear in parameters, the higher-derivative we solve 2(6 + 1)21 = 294 ordinary differential equations terms with respect to the parameters 8; and @j are zero and simultaneously (19, 31). This takes approximately 1 min of Eq. B4 is an exact equality (1).


APPENDIX C

Indifference Regions

A Taylor series expansion of the objective function in the neighborhood of the optimal point a(@*), truncated after the second derivative term, yields

where H is a p x p Hessian matrix and 60 = 8 - 8*. When H is positive definite, all choices of the parameter vector that give values of the objective function at a given tolerance, G!, from the optimal point lie approximately on the surface of a p- dimensional ellipsoid in parameter space referred to as the “indifference region” (1). The tolerance can be fixed to specify a particular level of confidence about 8* if the variances of the data are known (2, 24, 26).

One way to examine the nature of the ellipsoid is to consider the two-dimensional ellipses formed by the intersection of the ellipsoid and each of the coordinate planes in parameter space (1, 23, 24). Each ellipse corresponds to a 2 X 2 Hessian matrix with Q equal to 1. By examining these matrixes, we can determine the range of linear combinations of parameters @I and aZ that yield values of @ within the given tolerance of the optimum. Of course, this approach requires that for each 2 x 2 matrix we

space as Se,, 663, and is oriented along the directions xii, xi2 of the eigenvectors of H (Fig. 16).

All points on the perimeter of the ellipse (i.e., combinations of the parameters 8, and O2 that satisfy Eq. Cl ) yield identical values for a(e), which is an approximation of the objective function. Therefore, insofar as Eq. Cl is a good approximation to the surface of the indifference region, the greatest combined uncertainty in the estimates is indicated by the greatest Eu- clidean distance from the optimal point. Such estimates lie at the ends of the major axis of the ellipse. If the ellipse axes do not line up with the coordinate axes, then the combined uncertainty, as measured by distance from the center, is greater than the uncertainty in either parameter individually.

Because parameters are not all defined on the same scale, we normalize the coordinates of each axis by the corresponding parameter value before comparing the relative variabilities of parameter pairs (Fig. 16). In terms of the resealed axes (Se;/ 8i), which is a different eigenvalue/eigenvector problem, the coordinates of the poorest determined points can be calculated as follows. First, letting tildes denote quantities computed in the resealed domain, we define

Km

fix the remaining parameters. Doing so results in an optimistic characterization of the overall uncertainty on 8, and 8,.

as the angle formed by the resealed major axis and the Se&

We can transform any real symmetric matrix axis. Then the length of the projections (proj; indifference intervals) are, respectively, in the Se,/@, direction

H = XAX’ wa

where X is the orthogonal matrix, the columns of which, [X,, x2, l l ‘1 are the eigenvectors of H, and the matrix

Al 0 A = x2

[ I .

. .

0 L

sOy”j 1 - = - l&l cos e

8 1 8 1

= (&)cos[tan+J]

Jii- x11 =

is the diagonal matrix of the eigenvalues of H, where 0 < X1 5 x,5 . . . X,. Forp = 2 and in the S@/O, direction

6eTH6e = 6eTxfume = QTAQ

or

segroj 1 -=I- l&l sin*

e2 = [&)sin[t an

(Cl4

x12 (‘-)1 (Cl5) x11

f-2 = xlq: + x2q; (c6) &i 52

This is the equation for an ellipse with axes aligned along the = 8, Jx,(x:, + 32) Km

ql, q2 (“canonical variables”) coordinate axes. The half-lengths along the axis 4; can be found by setting qi = 0 (i # j). We see that the lengths of the axes are inversely proportional to the

APPENDIX D

square roots of the Hessian eigenvalues Nuisance Parameters

Consider a p-dimensional indifference region described by

I4 I = (c7) Eq. CL If the p-dimensional parameter vector 8 contains q i parameters of interest and p - q nuisance parameters, we

partition the parameter vector as To recover the coordinates of the axes in terms of the eigen- vet tars of H, we substitute back for Q, above

Q = XT68 = (;;;;;)(;g:)

x11@ + x12s@2 =

x,,ae, + x2&

(W

(W

Therefore ql, q2 is an orthogonal basis that spans the same

(Dl)

where 4 is the vector of parameters of interest and $ is the vector of nuisance parameters. Next, we partition the Hessian as

W)


FIG. 16. A: elliptical indifference region in @, 8, parameter space. Lengths of projections, xl1 and x12, of point furthest from origin on perimeter of indifference region determine joint confidence region limits on parameters. B: same ellipse is shown after axes have been normalized by respective parameter values. Axes represent percent changes in each parameter about optimal point. Major axis of resealed ellipse makes angle \k with coordinate axis. Projections of point furthest from optimum on resealed axes are indicated. See Glossary for definitions of abbreviations.

where the off-diagonal matrixes are defined as

VW

These submatrixes reflect the degree of interaction between parameters of interest and nuisance parameters. As shown in Eqs. B3 and B4, for the least-squares objective function and small residuals, the Hessian matrix is given by STS.

The expression for the indifference region (Eq. C4) becomes

(04)

in terms of a transformed nuisance parameter set, 0, the terms of interaction with the parameters of interest have been eliminated. A maximum value of the criterion, Det( G,i,,,J, corresponds to the optimal design for obtaining precise estimates of the parameter subset of interest, 4 (3). Similarly, to minimize the eccentricity of the region described by G++, we minimize the condition number of the matrix.

Consider the case of a three-parameter model with two parameters of interest and one nuisance parameter. The 3 x 3 Hessian is

where h++ is a scalar, and

is the vector of interaction terms. This expression can be rewritten in terms of a block-diagonal If there were no nuisance parameter, the limits of the indif- matrix ference regions, as in APPENDIX C, would be inversely related

to the eigenvalues of the 2 x 2 Hessian

(012)

where The eigenvalues are

hll + h,, k d(hll - h,,)” + 4(h12)2 x1, A2 =

2 (013)

(W

and

uw with h12 = h21. With the single nuisance parameter the eigenvalues of the transformed 2 x 2 Hessian Gdj4 are shown in Eq. 014, and each term in the eigenvalue expressions is corrected

W) by an interaction term. As the interactions of the two parameters with the nuisance parameter become negligible, the indif-

I-I948 OPTIMAL DESIGN OF LDL TRANSPORT EXPERIMENTS

ference region limits of this three-parameter those of the simpler 2 x 2 system.

system approach

This work was supported by National Heart, Lung, and Blood Institute Grants HL-29582 and HL-07242.

Present address of E. D. Morris: Div. of Neuropsychiatric Imaging, Dept. of Psychiatry, Case Western Reserve University, Hanna Pavilion, Cleveland, OH 44106.

Address for reprint requests: G. M. Chisolm, Dept. of Vascular Cell Biology and Atherosclerosis Research, Cleveland Clinic Foundation, 9500 Euclid Ave., Cleveland, OH 44195.

Received 7 July 1988; accepted in final form 8 March 1991.

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optimal design of experiments to estimate ldl transport ......department of biomedical engineering,...

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