Improved cholesterol phenotype analysis by a model relating lipoprotein life cycle processes to particle size.

Increased plasma cholesterol is a known risk factor for cardiovascular disease. Lipoprotein particles transport both cholesterol and triglycerides through the blood. It is thought that the size distribution of these particles codetermines cardiovascular disease risk. New types of measurements can determine the concentration of many lipoprotein size-classes but exactly how each small class relates to disease risk is difficult to clear up. Because relating physiological process status to disease risk seems promising, we propose investigating how lipoprotein production, lipolysis, and uptake processes depend on particle size. To do this, we introduced a novel model framework (Particle Profiler) and evaluated its feasibility. The framework was tested using existing stable isotope flux data. The model framework implementation we present here reproduced the flux data and derived lipoprotein size pattern changes that corresponded to measured changes. It also sensitively indicated changes in lipoprotein metabolism between patient groups that are biologically plausible. Finally, the model was able to reproduce the cholesterol and triglyceride phenotype of known genetic diseases like familial hypercholesterolemia and familial hyperchylomicronemia. In the future, Particle Profiler can be applied for analyzing detailed lipoprotein size profile data and deriving rates of various lipolysis and uptake processes if an independent production estimate is given.

order kinetics for production and fi rst-order kinetics for all the other processes. The pool size in each subclass can therefore be solved from a linear equation. The model framework contains the new concept that rate constants of processes in different subclasses vary nonlinearly as a function of particle diameter. The parameters of these nonlinear functions can be estimated by comparing experimental data on particle concentrations and fl uxes in lipoprotein size classes to the model prediction for those size classes.
The size-dependent models for production, lipolysis, and uptake are based on biological hypotheses explained below. These hypotheses were translated into mathematical equations to generate a fi rst model instance. The current model, described below, can be considered a fi rst functional implementation to which further biological knowledge can be added in order to arrive at new, more detailed, model instances. Each process's biology, conceptual model, and mathematical model are discussed consecutively. A full motivation of all equations can be found in the supplementary data.

Model input
Production. BIOLOGY. Hepatic production of VLDL particles ( 30 ) is thought to be a two-step process. First, VLDL2 is produced intracellularly through a fi rst lipidation of an ApoB100 molcule. VLDL2 can then be fused to a lipid droplet to form VLDL1 ( 31 ). LDL and intermediate density lipoprotein (IDL) are, for the greatest part, lipolytic products of the VLDL particles. Whether a small fraction or combine both ApoB and triglyceride information ( 21 ). Similar models describing other apolipoprotein kinetics have also been developed (22)(23)(24)(25)(26). The reported models were developed to deal with various density-based lipoprotein separation techniques. Now, new measuring techniques, such as HPLC ( 27 ) and NMR measurements ( 28 ), provide more detailed size-concentration profi les of lipoproteins and their constituents. The analysis of this data is challenging, but because earlier models were not designed for this task, new model approaches are necessary.
Here, we develop the mathematical model framework Particle Profi ler. The central hypothesis of the framework is that the rate of lipoprotein production, remodeling, and uptake processes depends on the size of the lipoprotein particle. First, we show that a model implementation in the Particle Profi ler model framework can reproduce published stable isotope fl ux data ( 29 ) and analyze them using few and physiologically relevant model parameters. The analysis results are then compared with known underlying physiology, specifi cally the LDL peak size according to which patients were categorized in the original paper ( 29 ). The model's ability to sensitively indicate physiological differences between these groups is also demonstrated. Finally, the potential for modeling genetic defects is illustrated by evaluating the result of changes in LPL-related lipolysis rate and ApoB-related uptake rate. We discuss the framework's potential to derive lipoprotein lipolysis and uptake rate information from detailed lipoprotein size measurements and an independent production estimate.

Model framework
The lifecycle of lipoprotein particles consists of three processes: production, remodeling, and uptake. The Particle Profi ler model framework starts from the assumption that the rate of each of these processes depends on the size of the lipoprotein particles. Different hypotheses about how these processes depend on particle size can lead to different model instances within the framework.
The Particle Profi ler model framework is shown schematically in Figs. 1 , 2 , and 3 . The particles in the Particle Profi ler model are subdivided into many very small subclasses according to their size. We refer to "subclasses" for the small size ranges the model uses and to a particle size "class" for the larger size ranges that are measured in experiments, like "LDL", "VLDL1", etc. Figure 1 shows the processes that act on each subclass, Fig. 2 shows the particle size range of the subclasses after subsequent lipolysis steps and fi nally, Fig. 3 shows how the step size due to lipolysis is calculated. Mass balances can be written for each of the subclasses, and also for the sum of all subclasses in any particle size range. These equate what comes in from the lipolysis and direct production with what leaves through lipolysis and direct uptake. The equations in each subclass obey zero- Fig. 1. A schematic overview of the model framework. Particles are contained in very many subclasses, each representing a narrow size range. Processes affecting the particle concentration in a subclass are production, extrahepatic lipolysis by LPL, and liver attachment. Liver attachment can in turn be followed by either hepatic lipolysis through HL, or uptake through ApoB-or ApoE-related mechanisms. If the particle is lipolyzed, it proceeds to the next step in the lipolysis cascade, which is explained in more detail in Figs. 2 and 3 . Subsequent subclasses in the lipolysis cascade are smaller; that is, they generally cover a smaller size range of lipoproteins (see Fig. 2 ). The different thicknesses of the arrows indicate that processes have different importance at each particle size. The particle production fl ux is an input to the model, whereas the model adjusts the lipolysis and uptake processes to fi t the data. The rates of the lipolysis and uptake processes vary continuously with particle size, as shown in the supplementary data. (eq.1) In this equation, p i , j J (d ) is the infl ux due to production into a subclass with average particle diameter d i,j , and subclass resolution r i,j d . The subindices refer to the lipolysis step (i) and the subclass within that lipolysis step range (j) (see Fig. 2 ). These subclasses have a variable resolution, which is always smaller than 0.01nm in the current implementation. J p,LDL is the production rate in the LDL class, which is fi xed based on the production data of each subject ( 29 ) (see Table 1 ).
[ID]TBL1[/ID] is the Gaussian cumulative density function, LDL d stands for the mean diameter of the LDL class, σ is the standard deviation of the distribution curve, and subscripts indicate the class to which a diameter refers and whether it is a minimum or maximum value for that class. In the lower boundary subclass, which lies only partially in the LDL class, ( ) is replaced by the lower border of the LDL class, d LDLmin ; in the upper boundary subclass, ( is replaced by the upper border of the LDL class, d LDLmax . For the IDL and VLDL2 classes, the production is defi ned analogously. For the VLDL1 class, the normal distribution is replaced by the lognormal distribution as follows: Fig. 2. The particle size range (in nm) of the subclasses after subsequent lipolysis cascade steps. Each particle size range of a given lipolysis cascade step, indicated by a black bar in the fi gure, is subdivided into n subclasses, in the current implementation n equals 1149, corresponding to a maximum interval of 0.01 nm per subclass. Particles can be produced at all sizes ranging from approximately 14-60 nm. If a particle is lipolyzed, it will always fl ow into a subclass in the size range (Y-axis) of the next cascade step (X-axis). A particle that is produced in a given subclass with particle size d i,j (i, cascade step number, j, subclass number within cascade step) will always fl ow into a subsequent subclass with particle size d i+1,j . This is because the particle changes its size deterministically and always loses a fi xed percentage of triglycerides at each lipolysis step as illustrated in Fig. 3 . The solid and dotted gray lines in this fi gure indicate two routes that a particle may take as it is lipolyzed, one at the top (d i,1 ) and another at the bottom (d i,n ) of the lipolysis step size ranges. The particle size ranges of subsequent lipolysis steps do not overlap. Fig. 3. Calculation of change in lipoprotein particle diameter due to a percentage change in triglyceride content. The model calculation proceeds in three steps. First, the model calculates the triglyceride content of the particle at its initial size. Second, it calculates the fi nal triglyceride content after lipolysis during which the particle loses 52% of its triglycerides. Third, it calculates the particle size corresponding to the calculated fi nal triglyceride content, which is the particle size after lipolysis. The relation between lipoprotein diameter and composition was based on the model presented by Tuzikov et al. ( 34 ). Solid line, triglyceride mass per particle; striped line, free cholesterol mass per particle; dotted line, cholesteryl ester mass per particle. of these lipoproteins is produced directly is still a matter of debate. Stable isotope fl ux studies generally show a small amount of direct input into these classes.
CONCEPTUAL MODEL. The current production model contains four size classes (VLDL1, VLDL2, IDL, and LDL), and specifi es how the infl ux into these large classes is distributed over many subclasses, which each have a narrow size range. The total production fl ux into each large class is based directly on the lipoprotein fl ux data of each subject ( 29 ).
The expected biological variation in the amount of lipids added to ApoB during the production process makes it very unlikely that production falls in one very narrow size range. The size of VLDL 2 particles is expected to vary around a given mean, because its size is based on the structure of ApoB plus a fi rst lipidation ( 32 ).We therefore assume that the diameter of secreted VLDL2 particles is normally distributed within the VLDL2 range. In the case of VLDL1, the size of the fused lipid droplets in the second production step can vary greatly ( 33 ). Therefore, we used a lognormal size distribution in the VLDL1 range, which allows the incidental production of larger particles in its 'tail'. Inasmuch as the IDL and LDL production processes are under debate, a normal size distribution within these classes was thought to be the best option.
MATHEMATICAL MODEL. The production fl ux into the LDL class can be discretized as follows: bound to the HSPGs can then be transferred to the lipoprotein and mediate the lipolysis of the particle. What exactly determines the rate of this lipolysis is not known, although the available surface area, the biochemical composition of the particle ( 39 ), gene expression changes, activators (e.g., ApoCII, ApoE, ApoAV), inhibitors (e.g., ApoCI, ApoCIII, Angptl4), and modulators of LPL expression (e.g., VLDL receptor) ( 40 ) are all thought to infl uence this rate. CONCEPTUAL MODEL. Because the primary physiological function of lipoproteins is lipid transport, the current model implementation assumes that the rate of remodeling is determined by the lipolysis rate. Therefore, in this model, changes in cholesterol, phospholipid, and protein content are determined by changes in triglyceride content. Therefore, only lipolysis is explicitly modeled and so, in continuation, we will speak about "lipolysis" instead of "remodeling". In the model framework, the lipolysis process is split into two steps: the fi rst step describes whether a particle is bound to a HSPG for lipolysis, the second, how many triglycerides it loses during lipolysis. The fi rst step depends on the particle's binding affi nity to HSPGs and GPIHBP1, which in turn depends on its apolipoprotein composition. The apolipoprotein composition is not modeled explicitly but implicitly by a function relating lipolysis rate to particle size. This means that the total affi nity of the particle for HSPG increases with particle size until a maximum is reached. Once a particle is selected for lipolysis, it proceeds to the next step in the lipolysis cascade. The calculation of lipolysis cascades is described below under "Model calculation -Lipolysis cascades".
MATHEMATICAL MODEL. To describe the size-dependency of extrahepatic lipolysis, a cumulative density function of the Rayleigh distribution was chosen. The formula for the lipolysis binding rate in extrahepatic tissue k l (d) (unit: day Ϫ 1 ) then becomes: where d is the particle diameter, d lmin is the minimum size at which lipolysis occurs, k lmax is the maximum lipolysis binding rate, and l is a shape constant for describing how the lipolysis rate depends on particle size (see Table 1 ).
Liver -lipolysis and uptake. BIOLOGY. In the liver, VLDL and IDL particles fi rst bind to liver HSPG via ApoE, whereas LDL binds directly to LDL receptors via ApoB100. LDL particles are directly taken up, but ApoE-mediated binding of larger particles need not result in uptake and can lead to lipolysis instead. The lipolysis in the liver is primarily mediated by HL, an enzyme that functions primarily on smaller ApoB-and ApoE-containing lipoproteins such as IDL, and to a lesser extent on VLDL2 ( 41 ). Uptake of VLDL lipoproteins mainly takes place via the LDL receptor but can also take place via low density lipoprotein receptor-related protein (LRP) ( 42 ). Roles for scavenger In this equation, F is the lognormal cumulative density function starting at d = d VLDL1min with mean μ VLDL1 . In the lower boundary subclass, which lies only partially in the VLDL1 range, ( ) is replaced by d VLDL1min ; in the upper boundary subclass ( The formula for translating the expectation VLDL1 d and standard deviation σ VLDL1 for the particle diameter in the VLDL 1 class to the mean ( μ ) and standard deviation ( σ ln ) of a lognormal distribution is given by: Values for d and σ in the various production classes can be found in Table 1 . The mean size of the VLDL2 and VLDL1 classes were derived by comparing the triglyceride (TG) to ApoB ratio of the production in these classes presented by Adiels et al. ( 21 ) to the TG-particle size relation given by Tuzikov et al. ( 34 ). Because no data for IDL and LDL are available, the class middle was taken as distribution mean. The standard deviation of the curves was taken as half the distance from the distribution mean to the lower class border.

Model optimization
Extrahepatic tissue -lipolysis and remodeling. BIOLOGY. In extrahepatic tissues, VLDL particles are only remodeled; uptake of whole particles is negligible ( 35 ). During remodeling, changes occur in triglyceride content as well as changes in phospholipid, cholesteryl ester, and protein composition. Changes in triglyceride content occur through lipolysis. Lipolysis of lipoproteins in extrahepatic tissues is carried out mainly by LPL. This enzyme mainly lipolyzes larger lipoproteins such as VLDL1 whereas VLDL2 and IDL are lipolyzed to a subsequently lesser extent ( 36 ). The particle binds to cellsurface heparan sulfate proteoglycans (HSPGs) and GPI-anchored HDL-binding protein 1 (GPIHBP1) ( 37 ) mainly through LPL itself whereas ApoE modulates the binding affi nity ( 38 ). Multiple LPLs that are already HL ( 46 ). The resulting fl ux of particles from size classes with larger particles to size classes with smaller particles has often been measured using stable isotope and radioactive tracer techniques. In vitro free fatty acid release from VLDL through LPL has been found to be related to tri glyceride-ApoB ratios ( 47 ) and triglyceride-cholesterol ratios ( 48 ). However, exactly how large a step the lipoprotein particles make per lipolysis event cannot be measured in vivo. CONCEPTUAL MODEL. A particle is produced with a certain diameter and can go through a variable number of lipolysis steps before being taken up. Figure 2 shows how the size of a particle decreases as the particle goes through subsequent lipolysis steps. Each lipolysis step has a corresponding size range, which generally becomes smaller as the particles are smaller. A lipolysis step size range is always divided into the same number of subclasses, 1149 in the current implementation (0.01 nm resolution at the crudest). This arrangement makes it possible for all particles that are produced in a particular subclass to fl ow through to the same subsequent subclass in the lipolysis cascade. In this way, the concentration in each particle size range can be calculated effi ciently.
For calculating a particle's size reduction due to lipolysis, we tested two alternative models. Both models fi rst calculate the particle's triglyceride content but differ in the subsequent calculations. The fi rst model takes out a fi xed quantity of triglyceride molecules, the second takes out a fi xed percentage of triglycerides from the particle as is shown in Fig. 3 . Finally, the model calculates the fi nal particle size from the amount of triglyceride molecules that remain.
The model version that assumes the loss of a fi xed quantity of triglyceride molecules per lipolysis step proved to be unable to fi t the fl ux data (model fi ts not shown). Therefore, we opted for the model version that assumes the particle loses a fi xed percentage of triglycerides per lipolysis step. This option fi ts well with the importance of the tri glyceride-ApoB and triglyceride-cholesterol ratio for enzyme activity observed in vitro ( 47,48 ) because larger particles that contain more triglycerides also lose more triglycerides. Alternatives are possible such as the stochastic step-size model Adiels ( 39 ) presented. However, with the current model and the dataset we consider here, we cannot discriminate between the simpler mechanism with a fi xed step size and a more complicated mechanism with a stochastic step size. We have therefore chosen the simplest model that can reproduce the data, in which a particle loses a fi xed percentage of triglycerides per lipolysis step.
MATHEMATICAL MODEL. The relation between particle diameter and both particle cholesterol and triglyceride content was based on an empirical model presented by Tuzikov et al. ( 34 ) shown in Fig. 3 . This model is a combination of two exponential functions fi tted to literature and newly measured data on the percentage composition of lipoproteins at different particle sizes. Although it is known that particles of a given size do not always have the same biochemical composition, the current approach gives a fi rst receptor class B type I (SR-BI) and direct incorporation via HSPGs have also been suggested ( 43,44 ).
CONCEPTUAL MODEL. In the model, liver binding and further processing are described as a two-step process. First, binding takes place, mainly mediated by ApoE, but with a small contribution from ApoB. Although small, this contribution is important, especially in the LDL size range where a small uptake affi nity combined with large amounts of particles can result in a considerable uptake fl ux. Subsequently, the part of the lipoproteins bound via ApoB is taken up. The part bound through ApoE can result in either lipolysis or uptake. Here, the lipolyzed part is mainly processed by HL and because HL mainly lipolyzes smaller particles, the lipolysis/uptake ratio decreases with increasing particle size.
MATHEMATICAL MODEL. Because binding fi rst increases and can subsequently decrease with particle size, a Rayleigh probability density function is used to describe this pattern. In order to have its maximum at one, it is scaled using the maximum of this same function, which lies at σ a,apoE d= .
The resulting liver uptake function is given by: Because binding to the liver can result in either uptake or lipolysis (see conceptual model above), lipolysis in the liver is given by: l,liver a,liver u,liver The model for these liver processes does not include a separate pool of attached particles. The model, rather, considers that attachment is directly followed by either uptake or lipolysis.

Model calculation
Lipolysis cascades. BIOLOGY. ApoB-containing lipoproteins go through a sequence of delipidation steps during which they become successively smaller. The lipoprotein particle that has been bound to the vascular wall is lipolyzed extrahepatically mainly by LPL ( 45 ) and, in the liver, mainly by

Model output
Particle profi le. CONCEPTUAL MODEL. The model calculates a detailed size-concentration lipoprotein profi le with the resolution that was used for calculation; in this study, a 0.01 nm resolution was used. This profi le can be shown as such or as size classes that correspond to experimental measurements. Examples of other possible output classes include the classical VLDL1, VLDL2, IDL, and LDL classes and size classes measured by HPLC ( 27 ) or NMR ( 28 ) techniques. The model output in the fi rst instance includes steady-state particle concentrations but can also show steady-state fl uxes of production, lipolysis, and uptake at each particle size.
MATHEMATICAL MODEL. The equation for the total steadystate output pool Q out in a given diameter range [d a d b ] is given by: Triglyceride and cholesterol concentrations. The particle profi les are translated to triglyceride and cholesterol profi les by the same size-composition relation used above to calculate the lipolysis cascades. It uses an empirical model presented by Tuzikov et al. ( 34 ) shown in Fig. 3 .

Model implementation
The model was implemented in MATLAB version 7.5.0(R2007b).

Model testing
Data used. In order to test the model's capability to reproduce measured lipoprotein fl ux data, the model was fi tted to fl ux data preanalyzed by a multi-compartment model from a stable isotope labeling study by Packard et al. ( 29 ). Packard and coworkers divided their subjects into three groups based on the 'LDL peak size'. Phenotype A had an LDL peak size greater than 26 nm; phenotype I, an LDL peak size between 25 and 26 nm; and phenotype B, an LDL peak size smaller than 25 nm.
Packard et al. ( 29 ) analyzed the ApoB fl ux data of each subject by a multi-compartment model. This yielded the pool size of each lipoprotein density fraction, the infl ux approximation, which may still be improved in future versions of the model.
The change in particle diameter through lipolysis is calculated as follows. Particle diameters are denoted by d i,j where the particle's cascade step (see Fig. 2 ) is indicated with index i and the subclass within that cascade step is indicated with index j. The size change due to the cascade step from d i,j to d i+1,j is determined by a model parameter specifying the triglyceride fraction a particle loses during lipolysis (f tg ). The equation for the loss of triglycerides is given by: where tg i,j n d is the initial and tg i+1,j n d the fi nal number of triglyceride molecules in a lipoprotein particle, as a function of particle diameter.
Particle concentrations in cascade. CONCEPTUAL MODEL. Particles are produced in many small subclasses. In the subclasses containing very large particles, the only particle infl ux is due to production but, in subclasses containing smaller particles, there is an additional infl ux due to the lipolysis process. This is shown schematically in Fig. 1 . A steady-state particle pool is calculated in each subclass by dividing the total particle infl ux (particles/s) due to either production or lipolysis by the total effl ux rate (1/s) due to lipolysis and uptake. This calculation proceeds from the mass balance in each subclass at steady state. In this way, the particle concentration in each small subclass can be calculated effi ciently using the steady-state assumption.
The resolution of the subclasses becomes smaller as particle size becomes smaller. This is because particles that are produced in the same subclass all proceed through the same sequence of subclasses as they are lipolyzed (see Figs. 1, 2 ). The initial size difference of the large particles within one subclass is slowly mitigated through the lipolysis process.
MATHEMATICAL MODEL. The steady-state pool ss i,j Q (d ) at each mean subclass particle diameter d i,j is given by: where p i , j J (d ) is the particle infl ux resulting from production, l i , j J (d ) is the particle infl ux resulting from extrahepatic lipolysis, l,liver i,j J (d ) is the particle infl ux resulting from hepatic lipolysis, k l is the extrahepatic lipolysis rate, k l,liver is the hepatic lipolysis rate and k u,liver is the particle uptake rate. These infl uxes due to lipolysis are calculated iteratively as follows: The infl ux due to production p i . j J (d ) is given in equations 1 and 2.
were selected as input to a new fi tting round; of these outcomes, then, the three best were selected for a fi nal fi tting round. The three fi nal parameter sets were then compared. If the fi nal parameters were found to differ, the whole procedure was repeated using the minima and maxima of each parameter in the set of fi nal parameters as starting points for a new experimental design. If the difference was negligible (parameter difference <1%), the best fi tting parameter set was chosen.
Statistics. Differences in fi tted parameters between the groups defi ned by Packard et al. ( 29 ) were inspected using the nonparametric Kruskal-Wallis test as implemented in MATLAB.
Output parameters. To improve interpretation, we derived various process-indicating parameters from the fi tted model parameters. A fi rst group contains process indicators, such as the maximum HL activity, the particle size at which it HL affi nity is at a maximum, and the average ApoE-related uptake affi nity over the VLDL1 range. A second group contains size-class specifi c indicator parameters of process, age, or size averages per particle in that class. For example, the average lipolysis binding rate per particle in the VLDL1 size class was calculated. This is an improvement over the 'transfer from VLDL1 to VLDL2' variable presented by Packard and coworkers, because it takes into account all lipolysis steps of VLDL1 particles, including those that do not cause the particle to change class.
Simulation of defects. Finally, the model's potential for modeling biological defects was investigated. Two defects were simulated. The fi rst is a polymorphism in the ApoBrelated uptake, which leads to hypercholesterolemia, the second a defect in LPL lipolysis, which leads to hypertriglyceridemia. Data from Patient 20 in Packard et al. ( 29 ) was chosen for the in-silico experiment because this patient has both a substantial amount of LPL lipolysis and ApoB-related uptake. The patient is in the B category with low LDL peak size and higher cardiovascular disease risk. The profi le of the patient was fi rst compared with a situation in which the ApoB-related uptake activity was halved, corresponding to a heterozygous LDL-receptor defi ciency (familial hypercholesterolemia). This was simulated by setting the k a,apoB parameter to half its original value. Second, the LPL-mediated lipolysis activity was reduced, corresponding to an LPL defect. This was simulated by setting the k lmax value to 50% of its original value. The output of the model is reproduced for the LDL, IDL, VLDL1, and VLDL2 size classes.

Feasibility of model approach
The pool and fl ux data were well fi tted with the Particle Profi ler model. In all patients, the model fi t converged to a difference of less than 1% between parameters in the from the previous class into the reported class, here interpreted as the lipolysis fl ux, and the direct catabolism from the fraction, here presented as uptake. Also, the direct production into each class was quantifi ed. Production fl uxes were used as input to our model whereas the pool sizes (4 data points), lipolysis fl uxes (3 data points), and uptake fl uxes (4 data points) were fi tted using six parameters (see Table 1 ).
The reported dataset of a subject was considered to be suitable as input to our model if it passed a "steady-state" test. The total infl ux should equal the total effl ux in each class of the dataset. Datasets with a large imbalance (infl ux-effl ux difference >10%) in one class were disregarded, leading to the exclusion of four patients (numbers 10, 13, 15, and 16). This selection is necessary because the current model assumes steady state, which therefore needs to be present in the data.
Because the original paper separated the VLDL1, VLDL2, IDL, and LDL categories, the model was adapted to reproduce these size classes as specifi ed under "size classes" in the supplementary data.
Fitted parameters. The parameters as arising from the equations above (see Table 1 ) were found to be correlated with respect to the used dataset. A parameter set was designed in which correlations were kept to a minimum, which is shown under reparametrization in Table 1 . The defi nitions can be found in the supplementary data.
The datasets used here to fi t our model parameters do not contain enough information to estimate the values of all unknowns. We therefore estimated two constants related to the particle-size dependence of extrahepatic lipolysis, d lmin and σ l , as shown in Table 1 . We also fi xed the value for the triglyceride loss per lipolysis step ( tg f ) at 0.52, a 52% triglyceride loss per lipolysis step. We fi tted all patients with a range of values for f tg and 0.52 fi tted best overall (analysis not shown). Other model parameters showed a limited covariation with this choice as shown for one patient in the supplementary data.
Fitting routine. Parameters were fi tted to the data using MATLAB's nlinfi t method of version 7.5.0 (R2007b), which is an implementation of the Levenberg-Marquardt algorithm. The supplementary data shows the error function used.
Because we are fi tting six parameters to 11 data points, there is a real possibility of fi nding multiple optima in the parameter landscape. We maximized the security of fi nding the global optimum by fi rst scanning the parameter space for good initial conditions and subsequently starting the algorithm from multiple starting points. The parameter space was scanned by applying a full 'experimental design' on estimates of the upper and lower bound of each parameter. The model was evaluated at each of these 64 points. We then evaluated the middle points in parameter space between six points with the lowest error values. In total, this results in 79 model evaluations. Of these 79, we used 12 with the lowest error value as starting points for the fi tting routine. After one fi tting round, the six best fi ts tion of the particle size ranges of each density class, the model still qualitatively reproduced an LDL particle size shift.
The particle size shift was also visible in the model parameters. The parameter most directly associated to this shift is the lipolysis minimum size, because, if small particles are lipolyzed more, they will become even smaller. The value of this parameter seems realistic for all patients except possibly patient 19, where the lipolysis minimum size hits the minimum boundary of 11 nm. The Kruskal-Wallis test showed that the median of the lipolysis minimum size signifi cantly decreased from group A and I to group B, with P = 0.014. This shows that both the modeled size-concentration profi les and the model parameters qualitatively reproduced the LDL size shift between the groups. Table 3 shows the derived parameters that indicate the status of the various physiological processes. Next to the lipolysis minimum size, the HL peak binding rate and the ApoB-related uptake rate also have signifi cantly different medians between the groups. Table 4 shows the size-class specifi c indicator parameters with a signifi cantly changed median between the groups. These include the VLDL1, VLDL2, and LDL aver-three best fi t parameter sets. The minimum with the smallest error value was chosen. Table 2 shows the parameters that have been estimated for all subjects from the study by Packard et al. and the corresponding deviations, the definition of which can be found in the supplementary data.

Process identifi cation
[ID]TBL2[/ID] The deviation ranged from 1.6% to 16.6% with an average of 7.2%. Only patients 4, 8, and 18 have a deviation above 10%. It is striking that these patients have high particle uptake from both the LDL and VLDL1 classes but low to very low uptake from the intermediate IDL and VLDL2 classes. The current model was not able to reproduce this pattern. Therefore, our model could reproduce the fl ux data of 13 out of 16 patients accurately.

Prediction of LDL size shift
With Particle Profi ler, we simulated detailed particle size profi les, although the model was fi tted to pools and fl uxes of only four density categories (VLDL 1, VLDL 2, IDL, and LDL). These detailed profi les were averaged for all patients in each phenotype class defi ned by Packard et al. In Fi g. 4 , these averaged profi les are shown. Although the A and I category profi les overlap, a shift toward lower LDL sizes was observed as the phenotype changes from A and I to B, corresponding to the size shift measured by Packard. This result points to the physiological realism of the model because, with no size data other than an estima-Only subjects with a data set corresponding to steady-state were selected. The patients were grouped by Packard et al. into three phenotype classes, according to their 'LDL peak size'. Class A had a peak size >26 nm, class I between 25 and 26 nm and class B <25 nm. Lower LDL peak size is thought to correspond to a higher risk for cardiovascular disease. The fi tted model parameter average for each of these classes is given and the signifi cance of inter-group difference according to the nonparametric Kruskal-Wallis test. An asterisk indicates the group that differs signifi cantly from the other two groups with P < 0.05.

DISCUSSION
This study presents Particle Profi ler, a model framework capable of analyzing cholesterol and triglyceride data by describing how lipoprotein production, remodeling, and uptake processes depend on lipoprotein particle size. The model was applied to existing preanalyzed stable isotope tracer data presented by Packard et al. ( 29 ). Our model implementation was able to reproduce the original model fi ts by Packard et al., requiring only six parameters to describe all modeled lipolysis and uptake processes. The Particle Profi ler results were able to predict the LDL size shift that was measured in the original study, only using reported fl ux data measured in four density classes by Packard et al. Furthermore, Particle Profi ler was able to sensitively indicate relevant differences in physiology between the groups. Finally, the potential for modeling the effects of genetic variants was demonstrated by simulating reductions in ApoB-related uptake affi nity and lipolysis affi nity.
The biological realism of the Particle Profi ler model is largely determined by the correctness of the hypotheses describing how different processes depend on particle size. Although the current set of assumptions reproduces the fl ux data well, uncertainties still exist. The three datasets that were not fi tted well indicate that improvements are needed in the uptake function because the modeled uptake is too high in the IDL and VLDL2 classes. Also, the model for particle triglyceride loss per lipolysis event age particle age; the IDL and LDL average particle size; the VLDL1, VLDL2, and LDL average lipolysis binding rate in general and specifi cally for HL in VLDL1 and LDL; and the LDL uptake rate.
A similar group-comparison analysis was done based on the fl ux parameters Packard et al. ( 29 ) report in their paper and using the same patients we did. In that case, next to the pool sizes, only the transfer rate of VLDL1 to VLDL2 differs signifi cantly between the groups.
The Particle Profi ler model, therefore, seems to be able to sensitively indicate relevant differences in physiology between groups with a differing LDL peak size. Figure 5 shows the model fi t of patient 20 and simulated defects affecting ApoB-mediated uptake and LPL lipolysis affi nity. The cholesterol and triglyceride concentrations in different size classes for the simulated ApoB-mediated uptake reduction show the expected hypercholesterolemia ( 49 ). The halved ApoB-related uptake affi nity results in a 1.7-fold increase of the LDL-cholesterol concentration in plasma.

Simulation of genetic defects
The modeled lipolysis affi nity reduction also reproduces the expected hypertriglyceridemia ( 27 ), although less severely than the hypercholesterolemia induced above. Reducing the LPL lipolysis affi nity (by 50%) results in a 1.5-fold increase of VLDL1-triglyceride concentration in plasma. The modeled genetic variants, therefore, qualitatively resemble the observed phenotype. Fig. 4. Particle Profi ler model results: average particle, total cholesterol (TC) and triglyceride (TG) concentrations of model fi ts based on fl ux data of VLDL1, VLDL2, IDL, and LDL only. The three curves represent averages of the subjects in the three phenotype groups as determined by Packard et al. ( 29 ). The striped line indicates phenotype A (LDL peak size >26 nm), the solid line phenotype I (LDL peak size between 25 and 26 nm) and the dotted line phenotype B (LDL peak size <25 nm). Although the fl ux data did not contain any particle size information further than the four mentioned classes, the model results do show more small LDL particles in the class of patients with phenotype B. This corresponds to the LDL size shift measured by Packard et al. and, therefore, gives confi dence in the physiological realism of the model. Our analysis shows a clear difference in several parameters between groups A and B with a measured LDL peak size above 26 nm and under 25 nm, respectively, as defi ned in ( 29 ). The position of the three fi tted patients in the intermediate I group with measured LDL peak size between 25 and 26 nm, as defi ned in ( 29 ), is ambiguous in our analysis. The LDL peak size reproduced by our model resembles the A group, but some parameters derived from the model, such as ApoB-related uptake and the LDL and VLDL 1 particle lipolysis rates, resemble the B group. Because of this ambiguity, we will disregard the intermediate would benefi t from more biological underpinning. The data we have analyzed in this paper do not contain enough information to identify this process exactly. We could see that the hypothesis that a particle loses a fi xed quantity of triglycerides per lipolysis step is highly unlikely as the corresponding model cannot fi t the data. We could not distinguish the possibility that a particle that uses a fi xed percentage of triglycerides per lipolysis step from the possibility that a particle loses a stochastic percentage of triglycerides per lipolysis step. Therefore, we have chosen the simplest possible model. Future investigation with more detailed datasets on both particle concentration and particle composition at different particle sizes may allow more insight into this issue. Also, the values for the constants d lmin and σ l describing how extrahepatic lipolysis depends on particle size require further investigation. The current model framework does provide the structure to investigate these issues.
In order to check the correspondence between our analysis and the original analysis by Packard et al., we can compare the patients' parameter values. A direct comparison is impossible, because the different models have different parameters. However, there should be a qualitative correspondence between the "fractional catabolic rate of LDL" and the "direct catabolism of VLDL1" in Packard's model and the "ApoB-related uptake rate" and the "average ApoE-related uptake binding of VLDL1" in the Particle Profi ler model. We can see that in general patients with high or low values for these parameters in Packard's model have high or low values for the corresponding parameters in our model. The qualitative correspondence between similar parameter values in the model by Packard et al. and Particle Profi ler is therefore correct.  Data as in T able 2 . When we tested the patients, we selected using variables from the original publication. This showed a difference between groups in one process: transfer from VLDL1 to VLDL2. The current analysis showed fi ve signifi cantly different processes. It indicated lipolysis changes in the LDL, VLDL2, and VLDL1 region, as well as indicating a changed HL activity in the LDL and VLDL2 range. These changes were found to be biologically plausible (see Discussion). of particle size. It then uses these specifi cations to calculate the particle concentration in size classes that can be freely chosen. Particle Profi ler allows the analysis of various hypotheses concerning how the modeled physiological processes vary as a function of lipoprotein size whereas multi-compartment models explicitly model fl uxes between measured classes.
The most similar model to Particle Profi ler is that of Hübner et al. ( 57 ), which also uses a single-particle perspective. Their approach attempts to include all relevant biochemical reactions and then simplify to obtain numerical traceability. In doing so, the emphasis differs from ours. The model of lipoprotein composition is more detailed with cholesterol and triglyceride content as well as a simplifi ed apolipoprotein content explicitly modeled. The Hübner et al. model also includes HDL and associated processes such as cholesteryl ester transfer protein activity. This model aims to include as much biochemical detail as possible. In contrast, our model focuses on integrated physiological process rates and how these rates depend on the size of the particle. For example, whereas in the Hübner et al. model ApoB-containing particles are produced at a single size, our model includes the full range of production sizes measured in the study we use for validation ( 29 ). Starting from integrated physiological rates, Particle Profi ler also allows zooming in on biochemical processes that appear to be important. In this way, we introduced the distinction between LPL and HL lipolysis of ApoBcontaining lipoproteins, a distinction that Hübner et al. do not make. In the present study, the distinction was important because these two enzymes showed a different response to changing LDL peak size.
The advantage of Particle Profi ler over earlier multicompartment models is two-fold. First, it can analyze and reproduce more detailed lipoprotein size profi les. The detailed calculation is important, because the output can then be given in any set of size classes required, such as those corresponding to the classical VLDL, IDL, and LDL fractionation but equally well for HPLC ( 27 ) and NMR ( 28 ) measurements. These different types of data can all be analyzed using our model framework. To actually start using these types of data for model optimization will require an independent estimate of the production fl uxes as input to the model. This is due to the fact that data from an HPLC or NMR measurement does not contain the production fl ux information the study by Packard et al. ( 29 ) provides. Lipoprotein profi le data alone are therefore insuffi cient to fi t the production fl uxes. Methods to determine production fl uxes include stable isotope methods [see ( 58 ) for a review], Intralipid ( 59 ), and estimation based on other plasma biomarkers [e.g., see ( 60 )]. Together with independent production estimates, data from detailed lipoprotein size profi les suffi ce to calculate lipolysis and uptake through different mechanisms.
The second advantage of Particle Profi ler above multicompartment models is its more direct link with the physiology of individual lipoprotein particles. The model analysis leads to 'integrated physiological process rates' of processes affecting lipoprotein particles. It is important to group and focus on the differences between the A and B groups in continuation.
Comparing our results to those in the original paper by Packard et al. ( 29 ), we see that Particle Profi ler backs up the statement by Packard, who considers the effi ciency of VLDL1 clearance a major controlling factor in small dense LDL formation. Reanalyzing their dataset using Particle Profi ler indeed showed a decreased lipolysis affi nity in the VLDL1 size range, but also in the VLDL2 size range in patients with smaller LDL. Additionally, in our study, we see that these patients have increased LDL particle lipolysis, a variable that Packard's analysis ( 29 ) cannot identify. Therefore, we back up their claim on the relationship between smaller LDL particle size and VLDL1 lipolysis and extend it to VLDL2 and LDL through a more subtle within-class lipolysis analysis. Another additional observation we could make is a decrease in the (ApoB-related) uptake in the LDL range with decreasing particle size.
Therefore, our study identifi es several changes in lipoprotein metabolism that are associated with decreased LDL peak size. In agreement with Packard's analysis ( 29 ), we fi nd that subjects with decreased LDL peak size have lower LPL lipolysis activity. In addition, we fi nd that HL activity is increased. These fi ndings are in agreement with a study by Campos et al ( 50 ). Another study by Tan et al. ( 51 ) also fi nds an increased HL activity when LDL peak size is decreased, but no association with LPL activity. Indeed, HL activity is well known to cause smaller LDL particles ( 46 ). What causes the activity of LPL to drop is not clear. Packard et al. ( 29 ) indicated that postheparin LPL activity and plasma triglycerides are only weakly correlated and suggested that other factors such as the ApoC-II content or the ApoC-II/C-III ratio in VLDL might determine the lipolysis rate. More recent research has identifi ed additional candidates ( 40 ). Finally, our observation that ApoB-related particle uptake decreases with decreasing particle size fi ts in with earlier studies (52)(53)(54). This decrease can be explained by a conformational change in the ApoB molecule on the smaller particles, which results in a less effi cient interaction with the LDL receptor ( 52 ). We conclude that the processes identifi ed by our analysis as changing with decreasing LDL peak size are biologically plausible.
There are several important ApoB lipoprotein-associated processes that are not yet included in the Particle Profi ler model. One is the exchange of triglycerides and cholesteryl esters between VLDL or LDL and HDL particles through the cholesteryl ester transfer protein ( 55 ). Incorporation of this process may be an interesting future development. Another candidate is endothelial lipase, although it seems that this enzyme mainly infl uences HDL metabolism ( 56 ). Its infl uence on ApoB-containing particles needs to be clarifi ed further. These two mechanisms can be incorporated as soon as the relation between lipoprotein size and enzyme activity is clear.
In contrast with earlier modeling approaches ( 6 -26 ), Particle Profi ler is not an explicit multi-compartment model; instead, it specifi es the lipoprotein production, remodeling, and uptake processes as continuous functions note that enzymes other than those identifi ed and incorporated into the model may contribute to determining these rates. Still, the integrated process rates are a step closer to actual enzyme activities than the fl uxes between classes reported by earlier models. For example, the affi nity of a VLDL1 particle for the lipolysis process gives more information about LPL and HL activity than the ApoB transfer rate from VLDL1 to VLDL2. At the same time, the integrated process rates are one organization level higher than the enzyme activities used by Hübner et al. ( 57 ). This means the description can be simpler. A simpler description in turn leads to fewer parameters to be estimated and, therefore, less data is needed for parameter estimation. In this way, Particle Profi ler fi nds an effi cient balance between the need for biological insight and the practical identifi ability of model parameters based on available data.
Particle Profi ler can calculate rates of various lipoprotein lipolysis and uptake processes from detailed lipoprotein size measurements and an independent production estimate. This information will be useful for diagnostic purposes.