The principle of the tracer dilution model is relatively simple as it determines the extent of dilution of tracer relative to tracee at isotopic and metabolic steady state. Fig. 5 shows a schematic principle of how the dilution model works. Here we provide an example of in vivo quantification of R_a FFA in the fasted steady state. If the pool size is constant, we know that R_a FFA is equal to R_d FFA. To determine R_a FFA, a representative tracer (i.e., palmitate tracer, denoted as red circle) is constantly infused at a fixed rate (F) intravenously into body to estimate the R_a FFA (i.e., tracee, denoted as black circle). As a function of time, relative concentration of tracer to tracee rises until it reaches plateau, which is called plateau enrichment or isotopic equilibrium (E_p). At E_p, R_a tracee palmitate is equal to R_d tracee palmitate, which is the same for R_a (i.e., F) and R_d palmitate tracer. And the ratio of these 2 (i.e., TTR or t/T) is the direct reflection of the ratio of R_a palmitate and F. In the dilution model, therefore, R_a tracee is inversely proportional to TTR for a given F. With regard to the calculation of R_a FFA, there are 2 steps: 1) R_a palmitate needs to be calculated by dividing F by E_p and then 2) R_a palmitate is then divided by the palmitate fractional contribution (FC) to total FFA to give R_a FFA. Here we provide a numerical example (Fig. 5). If F is 2.5 μmol/min delivered into system, and the measured E_p is 0.25, then R_a palmitate would be 10 μmol/min (i.e., F/E_p=2.5 μmol/min/0.25). If we assume the FC of palmitate to total FFA in the basal fasted state (e.g., FC=0.65),26 then R_a FFA can be calculated to be 15.4 μmol/min (i.e., R_a palmitate/its FC=10 μmol/min/0.65=15.4 μmol/min), which is also R_d FFA, as the pool size stays same. This principle applies to the assessment of R_a of any compound (e.g., glycerol, amino acids, lactate, glucose etc.) that appears in a compartment and leaves the same compartment (e.g., plasma), to which a tracer of the same compound is introduced.

The principle of the tracer incorporation model is predicated on determining the rate of precursor (e.g., fatty acid, acetate unit of fatty acids, glycerol, and so on) incorporation to product (e.g., TAG). Fig. 6 shows a schematic principle of the tracer incorporation model to determine the synthesis rate of a product. For example, TAG is comprised of fatty acids and glycerol in 3:1 stoichiometric ratio. Fatty acids comprising TAG are combinations of various fatty acid species such as palmitate (saturated fatty acid containing 16 carbons with 8 repeating acetate monomers), stearate (saturated fatty acid containing 18 carbons with 9 repeating acetate monomers), oleate (monounsaturated fatty acid containing 16 carbons with 8 repeating acetate monomers) and so on. For simplicity, here we provide an example of determining rate of synthesis of a theoretical polymer that is comprised of 2 repeating monomers (2 same units). Quantification of more complex species such as TAG (3 fatty acids+1 glycerol) and palmitate (8 repeating acetate moieties) is identical in principle. To assess synthesis rate of TAG bound to VLDL, various tracers can be used such as fatty acid, acetate, and glycerol (backbone of TAG) (e.g., [1-¹³C]palmitate, [U-¹³C₁₆]palmitate, and [²H₃₂]palmitate). After achieving an isotopic equilibrium of a precursor enrichment (i.e., E_P=1.0), which is far greater than what would normally be achieved (for a conceptual understanding) with a representative precursor tracer, labelled and unlabeled precursors will be incorporated into product polymers in proportional to their relative abundance. Thus, increases in product enrichment is dependent on 1) the precursor tracer enrichment achieved at isotopic equilibrium, 2) the pool size, and 3) the rate of polymer synthesis. With an isotopic steady state of 1.0, the possible permutations of labeled compound (i.e., tracer precursor) and unlabeled compound (i.e., tracee precursor) are 4, i.e., unlabeled-unlabeled (25%), unlabeled-labeled (25%), labeled-unlabeled (25%), and labeled-labeled (25%) with the exact same probability of each case. And regardless of the turnover rate and size of the pool of the product, the product enrichment will continue to increase until it becomes equal to the precursor enrichment. Measurement of the synthesis rate of a polymer is determined as a change in product enrichment over a given time period divided by steady state precursor enrichment, multiplied by 100 (to express the rate as %/time). For example, if product enrichment was changed from 0.01 at time=0 hour (t₁) to 0.05 at time=4 hour (t₁) (thus, a change in product enrichment=0.05−0.01=0.04) with a steady state precursor enrichment of 0.10 from t₁ and t₂. Then fractional synthesis rate (FSR) of the polymer can be calculated as [0.04/(0.10×4)]×100=10%/h, which means that 10% of the pool size was newly synthesized over 1 hour. This value actually does not provide absolute synthesis rate. To obtain the absolute rate of polymer synthesis, the FSR must be multiplied by the pool size. For an example of plasma TAG with assumptions of effective volume of distribution of TAG (i.e., plasma volume) being 3 L and TAG concentration being 100 mg/dL (or 1 g/L), the pool size is 3 g of plasma TAG (i.e., 100 mg/dL×3 L=3 g). If we use the same FSR (i.e., 10%/h, shown above) and the pool size of 3 g, then absolute rate of TAG synthesis will be 300 mg TAG/h (i.e., 3 g×10%/h). Fig. 6A shows the principles of how the tracer incorporation model works: precursor enrichment is at isotopic equilibrium (E_p) and product is being labeled at a constant rate. As time goes, product enrichment will rise and then plateaued where enrichments of precursor and products become equal, the fractional synthetic rate can be calculated before this happens (Fig. 6B). Fig. 6C represents the time course changes in enrichments of precursor and product between 2 time points (t₁ and t₂) where tissues (e.g., plasma) are typically harvested to measure changes in product enrichment and precursor enrichments for quantification of TAG FSR. The FSR term is also used in determining various polymers such as protein (from amino acids),1418192027 glucose (from 2 trioses),2829 DNA (from bases),2830 RNA (from bases),3132 and so on. In the following section, we will show flux calculation of selected variables of lipid kinetics using stable isotope tracer methodology and mass spectrometry.

Mostly stored in adipose tissue, TAG is constantly broken down into FFA and glycerol in 3:1 stoichiometric ratio in a process of lipolysis via a series of enzymes including adipose triglyceride lipase (ATGL). Glycerol appears in the circulation at a rate in parallel with rate of lipolysis. Glycerol derived as the result of lipolysis does not participate in an esterification process in the adipose tissue. Glycerol kinase, an enzyme required to convert glycerol to glycerol phosphate, is absent from adipose tissue. Glycerol phosphate is the activated form of glycerol that forms the backbone of TAG and is thus essential for re-esterification. Thus, R_a glycerol reflects lipolysis and is the best proxy for lipolysis.253334 To determine R_a glycerol, a glycerol tracer (e.g., [1,1,2,3,3-²H₅]glycerol) is primed constantly infused intravenously (e.g., prime, 1 μmol/kg and F, 0.08 μmol/kg/min). Blood samples are collected before the tracer infusion for determining background enrichment and after achieving E_p. Enrichment is typically determined with use of GC-MS or LC-MS. If we assumed E_p to be 0.05, R_a glycerol is calculated to be 1.6 μmol/kg/min as the result of F divided by E_p (0.08 μmol/kg/min÷0.05=1.6 μmol/kg/min). In a steady state, R_a glycerol is equal to R_d glycerol (i.e., 1.6 μmol/kg/min).

FFA that are released as a result of lipolysis can either appear in the circulation at a varying rate, called R_a FFA, or may be re-esterified back to TAG by combining with glycerol phosphate. Thus, while R_a FFA is related to the rate of lipolysis, it is not a quantitative reflection of lipolysis. To assess R_a FFA, a representative tracer (e.g., [1-¹³C]palmitate, labeled with carbon 13 at 1 carbon position) can be infused intravenously into the circulation at a constant rate, F (e.g., 0.04 μmol/kg/min). Blood samples are collected before the tracer infusion and after achieving E_p. A priming is not required to reach an equilibrium with a palmitate tracer due to its fast turnover rate.35 If we assume E_p to be 0.04, R_a palmitate is then calculated by dividing F by E_p: 0.04 μmol/kg/min÷0.04=1 μmol/kg/min. Palmitate is not the only source of fatty acid but one species among many fatty acids. Thus, to account for contribution from other fatty acid species, R_a palmitate can be divided by its FC (assumed to be 0.65 or 65%) to estimate R_a FFA. Thus, R_a FFA can be calculated to be 1.54 μmol/kg/min as the result of R_a palmitate of 1 μmol/kg/min divided by its FC of 0.65.

FFA that has appeared in the circulation can either be taken up by tissues such as muscle, where it is completely oxidized into CO₂ in the process of mitochondrial oxidation, or re-esterified into TAG in tissues other than adipose tissue mostly in liver, called extracellular cycling. Determination of the FFA oxidation rate is based on the incorporation of carbon from FFA to CO₂. To trace CO₂ formation derived from oxidation of FFA and thus quantify the rate of FFA oxidation, carbon-13 labeled fatty acid tracer (e.g., [1-¹³C]palmitate) can be used with which R_a palmitate (and ultimately R_a FFA) can be simultaneously determined as shown above. For example, [1-¹³C]palmitate can be intravenously infused at a constant rate (e.g., F, 0.04 μmol/kg/min). In order to estimate the FFA oxidation rate, 4 distinct values must be determined: 1) R_d FFA (=R_a FFA in a steady state), 2) carbon dioxide production rate (VCO₂/kg/min), determined via indirect calorimetry, 3) breath CO₂ enrichment (i.e., ratio of ¹³CO₂ of ¹²CO₂), determined by isotope ratio mass spectrometry, 4) % of FFA uptake that enters mitochondrial oxidation, determined as ratio of ¹³CO₂ excretion rate (VCO₂×breath CO₂ enrichment) to infusion rate of ¹³C unit of palmitate tracer. In addition, account must be taken of the fraction of the carbon label that is lost via isotopic exchange in the TCA cycle. For simplicity, however, we assumed that ¹³CO₂ produced from the labeled palmitate as the result of mitochondrial oxidation is completely recovered in the breath, and thus the recovery correction factor (CF) is 1. The equation for estimating rate of FFA oxidation is as follows: (% of FFA uptake oxidized×R_d FFA)÷CF. Using the following data, we will show how to calculate FFA oxidation rate: 1) R_d FFA of 1.54 μmol/kg/min, 2) ¹³CO₂ enrichment of 0.0002, 3) VCO₂ of 100 μmol/kg/min, 4) F of ¹³C unit from [1-¹³C]palmitate of 0.04 μmol/kg/min, and 5) CF of 1 (full recovery). Then, rate of FFA oxidation =[(0.0002×100)÷(0.04÷1)]×1.54 μmol/kg/min=0.77 μmol/kg/min.

TAG-fatty acid substrate cycling has 2 components: 1) intracellular cycling and 2) extracellular cycling. As stated above, FFA resulting from lipolysis in adipose tissue can either be released into the circulation (R_a FFA) or re-esterified back to TAG, the process being called “intracellular cycling”.253436 FFA that appears in the circulation (i.e., R_a FFA), on the other hand, can be either completely oxidized into CO₂ in the process of mitochondrial oxidation in tissues such as muscle or re-esterified into TAG in tissues other than adipose tissue, mostly in liver. The later process is called extracellular cycling.2534 These cycling rates can then be determined based on the information provided above (i.e., R_a FFA, R_a glycerol, FFA oxidation rate). Intracellular cycling rate is calculated as (3×R_a glycerol)−R_a FFA, later of which is predicated on the fact that 3: 1 stoichiometric relation exists between FFA and glycerol derived as the result of lipolysis. The extracellular cycling rate is calculated as R_a FFA – FFA oxidation rate, and the total cycling rate is sum of the these 2 (i.e., intracellular cycling and extracellular cycling) which is calculated as [(3×R_a glycerol)−R_a FFA]+[R_a FFA−FFA oxidation rate]=(3×R_a glycerol)−FFA oxidation rate. Using values shown above for each component, we can calculate intracellular cycling rate [(3×R_a glycerol)−R_a FFA = 3.26 μmol/kg/min], extracellular cycling rate (R_a FFA−FFA oxidation rate=0.77 μmol/kg/min), and total cycling rate [(3×R_a glycerol)−FFA oxidation rate=4.03 μmol/kg/min].

VLDL-TAG synthesis is a large fraction of the extracellular recycling, regulation of which is implicated in clinical conditions such as cardiovascular events.3738 The first step in determining the VLDL-TAG synthesis rate (and subsequent VLDL-TAG secretion rate) is to determine VLDL-TAG FSR. Several tracers can be used for assessing FSR of VLDL-TAG, including palmitate,39 acetate,40 glycerol34 and even deuterium oxide.41 Here we will discuss the calculation of VLDL-TG FSR again using [1-¹³C]palmitate. As stated above, FSR is calculated as changes in product enrichment divided by precursor enrichment and time, multiplied by 100 (to express it as %/time).24 For example, if plasma palmitate enrichment achieved at isotopic steady state (i.e., E_p) is 0.04 above background, and changes in enrichment of palmitate derived from plasma VLDL-TAG (i.e., product enrichment) elapsed over an experimental time period (e.g., 3 hours) is 0.03, then VLDL-TAG FSR is calculated to be 0.03÷(0.04×3 hours)=0.25/h (or 25%/h). This means that 25% of plasma VLDL-TAG pool is newly synthesized per hour. To obtain an absolute synthesis rate of VLDL-TAG, FSR (i.e., 0.25) needs to be multiplied by the pool size (VLDL-TAG concentrations x volume of distribution). If the TAG concentration is assumed to be 100 mg/dL and the volume of distribution of VLDL-TAG is 3 L, then calculation of absolute VLDL-TAG synthesis rate is as follows: 0.25/h×100 mg/dL×3 L=750 mg/h.