## Abstract

We develop a novel method for finding sufficient experimental conditions for discriminating and quantifying individual biomolecule production sources in distributed, inhomogeneous multisource systems in vivo, and we apply it experimentally to a complex, unsolved problem in endocrinology. The majority of hormonal triiodothyronine (T_{3}) is produced from prohormone thyroxine (T_{4}) in numerous nonthyroidal organs and, with one exception, the T_{3} production rate has not been fully resolved in any single extrathyroidal organ of any species. Using a readily generalized graphic method called cut-set analysis, we show here that measured steady-state responses in several organs to three independent tracer infusions, two into blood and one directly into the organ(s) of interest, are sufficient to resolve this problem for organs fully accessible to direct infusion in vivo. We evaluated local T_{3} production in rat liver and intestine, which also required T_{3} bile flux measurements, and we found that liver produces ∼31% and whole intestine ∼6% of whole body T_{3} from T_{4}. With thyroidal production included, liver contributes ∼15% and intestine ∼3% of whole rat T_{3} production. This new methodology is broadly applicable, especially to biosystems that include molecular interconversions at multiple sites.

- multisource triiodothyronine production
- cut-set analysis
- local organ hormone production rates
- thyroid hormone interconversion
- intraduodenal-intraileal infusion
- bile hormone flux
- steady-state kinetics
- inhomogeneous multisource biosystems

besides the numerous molecular components common to all cells, many less ubiquitous endogenous substances are produced at multiple and often widely distributed anatomic sites. Many are products of local interconversion processes, thyroid and steroidal hormones, lipoproteins, and D vitamins, for example. For primarily technical reasons, quantification of these distributed sources at individual production sites in vivo remains a very difficult problem, with no general solution yet available. Thyroid hormone (TH), for example, is produced in virtually all mammalian organs, and data-sufficient experimental designs and results on its rate of production in a single organ have been reported only for developing rat brain (46) and for human (38), mammalian (22), fasting trout (44), and adult rat (18) thyroid glands.

It is well established that thyroxine (T_{4}) is the main secretory product of the thyroid gland and that triiodothyronine (T_{3}), the main metabolically active TH, is also secreted by the gland, at least in mammals. However, a substantial fraction of whole body T_{3}, but not T_{4}, is produced extrathyroidally and enzymatically from exchangeable T_{4} in tissues. Notably, the T_{3}-producing (activating) enzymes, the iodothyronine 5′-deiodinases called D1 and D2, are distributed quite differently among different tissues (47). Indeed, from a regulation viewpoint, intracellular T_{3} production appears to be under local control in many organs, e.g., in rat brain and others containing primarily D2 (19, 32), as well as via hypothalamo-pituitary-thyroid feedback “axis” control (23). Quantification of local T_{3} production takes on its primary import in this local regulation context, a problem whose solution has thus far eluded all. For the liver, estimates reported from in vivo kinetic studies vary from substantial extrathyroidal T_{3} produced by liver (3, 51) to nearly none at all (17).

Both T_{3} and T_{4} are exchangeable between blood and cells; thus the T_{3} typically found in any tissue has many sources, indistinguishable without appropriate experimental intervention, problems shared by all distributed-source biomolecules. For practical purposes, the T_{3} typically found in the cell has been classified by thyroidologists as having two sources. The first is that provided indirectly via circulating T_{3} of unknown origin, via capillary uptake, lumped into one and commonly called “plasma-derived T_{3}.” The second is the unique T_{3} produced enzymatically from T_{4} locally, via D1 or D2, called “local T_{3}.” The source identification problems in this field have thus been simplified to that of distinguishing and quantifying the contributions of each of these two sources in a given cell type or organ, most notably in Refs. 45 and 49–51, where local T_{3} derived from local T_{4} was designated “LcT_{3}(T_{4}),” and this pool size parameter was estimated in as many as 34 different tissues by use of a dual-isotope T_{3}/T_{4} kinetic experiment design. Others have used the same approach, e.g., Ref. 43. As we will show, the steady-state pool size estimates provided by the dual-isotope design do not reflect local production rates per se, because this design does not fully distinguish locally derived tracer T_{3} from that produced in any or all other organs and then transported there via blood, at least not without additional independent data. Although we have shown this indirectly in previous work (9, 10, 20–22), this lack of complete information from dual-tracer kinetics was noted explicitly earlier, in Ref. 46, where dual-tracer data were supplemented by additional tracer kinetic study data eliciting the fractional rate of removal of T_{3} from brain tissues. This additional independent data, when combined and analyzed together with the dual-tracer kinetic results, permitted computation of T_{3} production rates in developing rat brain (46).

In the present studies, we use graph theory and cut-set analysis to show that at least a triple-tracer approach is needed. We found that measured steady-state responses in several organs to three independent tracer infusions of T_{3} and/or T_{4}, two into blood and one directly into the organ of interest, are together sufficient to resolve this local conversion rate problem for organs fully accessible to direct infusion. Enterohepatic system (EHS) organs, although they fulfill this requirement, present an unusual experimental challenge because of their special interconnectivity. We apply these new experimental design models and evaluate local T_{3} production rates in the EHS organs, liver and intestine, separately and together. The separation was achieved by reapplying cut-set analysis and using an additional measurement, the steady-state T_{3} bile flux.

This new graph theory-based approach is generalizable and readily applicable to critical design of experiments for quantifying distributed sources in many other complex, inhomogeneous biosystems in vivo. These include other interconverting biomolecules, such as steroidal androgens and estrogens, vitamins and lipoproteins, and possibly even more ubiquitously distributed molecular pathways.

## METHODS

### New Experimental Design Models Via Cut-Set Analysis

We take advantage of the special physioanatomic interconnections and exchanges among liver, intestine, and blood, and we use the two TH steady-state pool model structures in Figs. 1 and 2 to find solutions for local T_{3} production in both liver and intestine. Both structures and solutions were obtained by extending several new theoretical experimental design results in cut-set analysis of compartmental (and nonhomogeneous) pool models (20–22). We present the models and a summary of new results here. Detailed derivations and an explanation of the cut-set analysis are given in the APPENDIX, and our nomenclature is summarized in Table 1.

In Fig. 1, enterohepatic T_{4}→ T_{3} conversion is separated into liver and intestinal conversions, in separate organ pools. In Fig. 2, it is depicted in combined liver and intestinal EHS pools. Figure 1 has eight and Fig. 2 has six measurable T_{4} and T_{3} pools, with vascular or biliary pathways among them and sinks within them. These pools represent steady-state nonhomogeneous hormone pool sizes (labeled or unlabeled mass units) in blood (BLD), residual carcass (RESTO), and either the liver (LIV) and intestine (INT = tissue + contents) separately, or combined in the EHS, measured in steady state. The several *Q*_{4ORG} and *Q*_{3ORG} values, shown as measurements from each pool (dashed lines), represent the respective T_{4} and T_{3} measurable masses in these same organs, i.e., ORG ≡ EHS, LIV, or INT, or measurements in BLD or in RESTO (in ng or cpm). We remark that ORG can be any other organ that is fully accessible and directly infusible, e.g., brain or kidney, which renders the method and Fig. 2 more generally applicable. The *k*_{ij} parameters (h^{-}^{1}) represent fractional rates of transfer of T_{4} or T_{3} between pools, by arterial, venous, or biliary pathways and within or from pools by degradation or excretion. Numbering of the pools in Fig. 2 is consistent with corresponding separate or combined pools in Fig. 1, and relationships among *k*_{ij} in the two figures have been noted in Fig. 2. Note that these models incorporate all production, metabolism, and other elimination pathways separately and distinctly. Thus the fractional T_{3} production rates, shown by the bolder arrows *k*_{4-3EHS}, *k*_{4-3LIV}, and *k*_{4-3INT}, are the distinct measures only of these T_{4}-to-T_{3} conversions. They are not encumbered by, nor do they include, alternate or subsequent metabolism pathways. We show in the APPENDIX that we can solve for these three *k* values by superimposing three tracer-input infusions in Fig. 1, as shown in Fig. 3.

In Fig. 3, tracer [^{125}I]T_{4} (T* _{4}) in *experiment A* (Expt A), and [^{131}I]T_{3} (T**_{3})in *experiment B* (Expt B), are infused into blood simultaneously, at rates IR_{4BLD} _{A} and IR_{3BLD} _{B} (*group 1*). The additional *experiment C* (Expt C) requires a direct infusion of tracer T_{3} into the intestine, for practical purposes [^{125}I]T_{3} in a separate group of rats (*group 2*). Expt C is done with two separate pumps, each filled with [^{125}I]T_{3} (T*_{3}), one directed intraluminally into the duodenum (IR_{3duodenum} ^{Cint}), the other intraluminally into the distal ileum (IR_{3ileum}^{Cint}), as noted earlier. This is to simulate approximately the natural influx of endogenous T_{3} into intestine from bile and from blood (14).

The primary measurements include the tracer masses of T_{3} and T_{4} in all organs depicted in Fig. 3, i.e., the *Q*_{3ORG}^{A,B,Cint} and *Q*_{4ORG}^{A} (in cpm), IR_{3INT}^{Cint} = IR_{3duodenum}^{Cint} + IR_{3ileum}^{Cint} (in cpm/h), and plasma specific activities (SA). Mass fluxes are obtained as products of *k* and *Q* values, with tracer converted to endogenous masses by dividing by SA. As shown in the APPENDIX, the first cut-set analysis of Fig. 3 yields the conversion rate of T_{4} to T_{3} in the EHS 1 where SA_{4plasma}^{A} is the measured specific activity in plasma of [^{125}I]T_{4} in Expt A.

The second cut-set analysis in the APPENDIX yields the result for whole intestine. In this case, the solution also requires direct measurement of the T_{3} bile flux, in Expt C only, as shown in Fig. 3, where . Intestinal mass-flux conversion of T_{4} to T_{3} is 2 The percentage of whole body T_{4} converted to T_{3} (CR_{4-3TOTAL}) attributed to EHS is thus calculated as 3 where TOTAL = liver + intestine + blood + residual carcass, and 4 is the fraction of T_{4} production (secretion) converted to T_{3} (34, 35). Similarly, the percentage of whole body T_{4} converted to T_{3} attributed to intestine is 5 Finally, T_{4} converted to T_{3} in liver is computed as the difference between EHS and intestinal conversion. From *Eqs. 1* and *2*:CR_{4-3LIVER} = CR_{4-3EHS} - CR_{4-3INT} (ng/h). From *Eqs. 3* and *5*: %CR_{4-3LIVER} = %CR_{4-3EHS} - %CR_{4-3INT}. Complete derivations of the equations are given in the APPENDIX.

### Animal Experiments

Animals were prepared and cared for, and outer ring-labeled 3,5,3′-[^{131}I]T_{3} (T**_{3}, ∼3.7 mCi/μg) and [^{125}I]T_{4} (T*_{4}, ∼3 mCi/μg) were synthesized from 3,5-T_{2} and 3,5,3′-T_{3}, respectively, and purified on HPLC, all as described in Ref. 34. Tracer ^{99m}Tc was obtained from the UCLA Department of Nuclear Medicine and was used to label rat donor red blood cells with the Ultratag ^{99m}Tc-RBC Labeling Kit (Malinckrodt, St. Louis, MO). β-Glucuronidase (*Escherichia coli* type VII), aryl-sulfatase (type V), and other reagents were purchased from Sigma; Alzet Osmotic Minipumps (model 2001) were purchased from Alza (Palo Alto, CA). HPLC analyses were done using a gradient system (LKB/Pharmacia), as in Ref. 34, with mobile phase 20:80–40:60 acetonitrile-H_{2}O (0.1% phosphoric acid) for analytic samples and 50:50 MeOH-H_{2}O (0.1% phosphoric acid) isocratic mobile phase for tracer purification. These were run at 1 ml/min, all on a C_{8} reverse-phase 5-μm ChromPack analytic column.

*Pump preparation, study groups, and implantation.* Purified tracer infusates [^{125}I]T_{4} (T*_{4}) and [^{131}I]T_{3} (T**_{3}) were prepared and tested as in Ref. 34. Alzet pumps were filled with the labeled T_{4} or T_{3} and implanted subcutaneously for 7 days, also as in Ref. 34, in two rat groups. In *group 1* (*n* = 8), [^{125}I]T_{4} and [^{131}I]T_{3} were infused subcutaneously and simultaneously from separate pumps. In *group 2* (*n* = 5), [^{125}I]T_{3} in two pumps was infused simultaneously into the duodenum and the distal ileum of the intestine by use of two PE-60 catheters placed in a hole in the duodenum and one in the distal ileum. The holes were sealed with animal glue, and the tubing was sutured to the intestinal wall, as in Ref. 15. To prevent catheter clogging, tip openings were heated and stretched to reduce tubing diameter, thereby increasing efflux velocity, and several pinholes were made near the tip to assure continuous flow (15). The tubing was filled with [^{125}I]T_{3} tracer and attached to two T*_{3} pumps implanted subcutaneously between the shoulders by running it under the skin. On *day 0*, 0.5 ml of blood was drawn from the vena cava for hematocrit and endogenous plasma T_{3} and T_{4} measurements. Rats were housed in individual metabolic cages for the duration of the study.

*Day 7 operations.* Both groups were treated as in Refs. 13, 34, and 35, with one exception. Labeled ^{99m}Tc-RBC was injected intravenously 5 min before termination to correct for hormone trapped in blood in tissues. Blood was collected by cardiac puncture to provide ^{99m}Tc concentrations, final hematocrits, sera for radioimmunoassays, steady-state concentrations of total ^{125}I or ^{131}I in plasma, and T*_{4} or T*_{3} or T**_{3} plasma concentrations and their metabolites after chromatography. Intestine and liver were collected and rinsed in ice water. Residual carcass and liver, and rinsed intestinal tissue after separation from contents (13, 35), were quickly frozen in liquid N_{2} and processed as described below. Small and large intestinal contents were premixed in a single vessel and left for 2 h at room temperature. This assured nearly complete hydrolysis of small intestine conjugates by large intestinal bacteria, with minimal deiodination of tracer, as measured in chromatography and demonstrated earlier in Refs. 13 and 35. In *group 2* rats, the common bile duct was also cannulated with PE-10 tubing, and bile was collected for 1 h before termination, weighed, and counted for total ^{125}I.

*Homogenization, extraction, and chromatography.* Individual organ and residual carcass samples were pulverized and homogenized in a threefold dilution of ice-cold extractant, as in Ref. 34. Premixed intestinal content samples were combined with frozen, crushed intestinal tissue samples and homogenized together. Up to six ∼1-g aliquots of each tissue homogenate were weighed and counted, and total organ ^{99m}Tc, ^{125}I, and ^{131}I radioactivity (*Q*_{mORG}^{* or **}) was calculated as the measured ^{99m}Tc, ^{125}I, and ^{131}I concentrations (C_{mORG}^{* or **}) times the organ weight (*M*_{ORG}), as in Refs. 34 and 35. Extracts, including untreated and hydrolyzed bile (see *Bile hydrolysis*), were chromatographed on Sephadex G25 and by HPLC (34, 35).

*Blood and tissue T*_{3} *and T*_{4}. Total blood and tissue T**_{3} and T*_{4} were measured as in Refs. 34 and 35. Briefly, labeled fractions of T_{3} and T_{4} in tissues on *day 7* (f_{3} or f_{4}), multiplied by the concentration of ^{125}I and ^{131}I activity in the tissue sample, provided the concentrations of T*_{4}, T*_{3}, or T**_{3} in that sample. The latter multiplied by organ weight (*M*_{ORG}) provided the total T*_{4}, T*_{3}, or T**_{3} in that organ, including activity in residual blood trapped in the organ. Total tissue T*_{4} and T**_{3} pool sizes (*Q*_{4or3ORG}^{* or **}) were corrected for residual trapped blood radioactivity in dissected tissues and the residual carcass (35). Plasma samples were radioimmunoassayed in triplicate for unlabeled T_{3} and T_{4} concentrations, as in Ref. 34.

*Bile hydrolysis.* We used a minor modification of the method described in Ref. 13. Six hundred microliters of bile were added to 500 μlof β-glucuronidase (10 units/μl bile) and 500 μl of aryl-sulfatase (11 units/μl bile) in 0.2 M sodium acetate, pH 5.0 buffer. Samples were incubated for 4 h at 37°C and stopped by freezing on dry ice for 2 h. Completeness of hydrolysis was verified by comparing total T**_{3} recovered in treated and untreated bile samples.

## RESULTS

On *day 7*, neither hematocrits nor plasma hormone concentrations were significantly different in the two rat groups. Hematocrits were 0.48 ± 0.012 (SD) for *group 1* (*n* = 8) rats and 0.47 ± 0.061 for *group 2* (*n* = 5) rats [not significant (NS)]. Endogenous plasma hormone concentrations for T_{4} were 32.9 ± 3.47 ng/ml in *group 1* and 39.4 ± 7.09 ng/ml in *group 2* (NS), and for T_{3}, 0.69 ± 0.019 ng/ml in *group 1* and 0.68 ± 0.016 ng/ml in *group 2* (NS).

### Steady-State T_{3} Production Rates from T_{4} in Liver, Intestine, and Whole Body

Table 2 provides mass fluxes, fractional rates of conversion, and relative values based on our measurement of the percentage of whole body T_{4} converted to T_{3}, 23.7 ± 1.7% (SE).

### Bile Flux of T_{3} and Steady-State Kinetic Parameters and Distribution Pools of T_{3} and T_{4}

The steady-state biliary influx of total T_{3} into the intestine was 2.37 ± 0.201 (SE) ng·h^{-}^{1}·100 g body wt^{-}^{1} (fractional influx rate = 0.265 ± 0.0098 h^{-}^{1}). T_{3} and T_{4} kinetic parameters are given in Table 3. Organ tracer pool size data for Expts A, B, and C, normalized by the different T_{4} or T_{3} tracer infusion rates in each rat studied, are given with their variabilities in Table 4. Finally, absolute and relative organ pool sizes and concentrations of endogenous T_{3} and T_{4} are given in Table 5, based on organ and plasma specific activity measurements. In steady state, blood contained very little T_{3} (∼3.5%), but 27% of the T_{4}; liver had only 6–9% of total body T_{3} or T_{4}; and intestines contained about as much T_{3} as residual carcass and somewhat less T_{4}.

## DISCUSSION

### General Considerations

Discriminating among distributed sources in biosystems is difficult, and the literature on distributed source localization is sparse. Quantification is even more difficult, because measurements are typically, of necessity, indirect. The problems are invariably methodological and depend on the measured signal types and experimental design models used. In biomagnetic and bioelectric systems, including imaging modalities like MRI and computed tomography, pattern recognition, signal processing, and localization algorithms have been applied successfully to reconstruct equivalent source patterns (52). Problems are exacerbated when only tracer signals are available. For example, with tracer imaging processes, like functional positron emission tomography, the imaging mechanism does not distinguish between precursor and product of the tracer. Even blood-borne data, when collected from biosystems with more than one endogenous source, typically can provide only indirect, minimum source rate estimates, unless all sources enter blood directly. Otherwise, a more complex experiment involving multiple tracers is required (6, 37).

The multisource quantification problem for thyroid hormone shares some of the structural complexities of other biosystems with unidirectional precursor-product mechanisms and has been addressed using various methods and models, for example, in pharmacokinetics Refs. 4 and 41 and in metabolism studies Refs. 23 and 28. Several studies by our own group (12, 20–22) and others (26, 31, 38, 39) have also addressed this problem for thyroid hormone, utilizing unidirectionally linked submodels of iodothyronine distribution and metabolism to establish T_{3} or reverse-T_{3} (rT_{3}) production from T_{4} in lumped slow- and lumped fast-exchanging tissue groups. The different approaches to quantifying or distinguishing the multiple sources share the same difficulty: information is limited by the number and kinds of probes available for obtaining independent data about individual or collective sources, with more such probes needed for more complex biosystems. On the basis of our results, we conjecture that the number of such probes needed is roughly proportional to the number of sources to be quantified.

### Local T_{3} Production

Consider quantifying the rate at which T_{3} is produced in an organ of interest, or the local T_{3} contribution relative to that in other organs in vivo. In this open and exchanging system, the problem is that T_{3} (or tracer T*_{3}) derived from T_{4} (or tracer T*_{4}) and found in that organ in steady state could have been produced either in that organ or transferred from other organs. As noted above, the key to resolving the ambiguities is multiple tracer probes and measurements that provide sufficient independent information. We have shown earlier that two tracers, when the conventional dual-isotopic T**_{3}/T*_{4} injection or infusion design is used, do not provide enough data for local source quantification, not even for collective tissue groupings (9, 10, 20–22). Two tracers do, however, provide ranges for “slow” and “fast” tissue groupings (8, 12) and for the thyroidal T_{3} secretion rate in mammals when multicompartmental analysis of sufficient tracer kinetic data is used (20–22, 38, 44). In the current work, we have established that direct infusion of tracer T_{3} into a T_{3}-producing organ, a third probe in addition to the conventional dual T_{3}/T_{4} tracer infusion, plus appropriate measurements of the three different tracer responses in blood and tissues (and bile, when liver or intestinal T_{3} production is of interest), is a sufficient condition for quantifying that organ T_{3} source.

This is not the only additional probe capable (in principle) of providing the needed additional information, along with the dual T_{3}/T_{4} blood infusion study; indeed, there are undoubtedly other possibilities, e.g., T_{4} instead of T_{3} infusion in the organ of interest, or independent measurement of T_{3} degradation rates in that organ (46). However, as a corollary, we have shown again that the dual-tracer infusion study data alone do not provide enough information for source rate quantification, in particular the independent information needed to isolate and quantify the local source. The key is *Eq. A3* in the APPENDIX, which has one unknown on the right-hand side, *k*_{65}+*k*_{75}, the fractional transfer rate of T_{3} from blood to EHS (via hepatic and mesenteric arterial pathways). This is in addition to the unknown of interest, the fractional T_{4}-to-T_{3} conversion rate in the EHS, *k*_{4-3EHS}, on the left-hand side of *Eq. A3*. Thus there are two unknowns, with only one equation to resolve them, and *k*_{65}+*k*_{75} clearly cannot be established from Expt A and Expt B, the dual-tracer study, on the basis of analysis of Fig. 3 (with notation defined in Fig. 1) developed in the first several lines of the APPENDIX and *Eqs. A1* and *A2*.

Early estimates of the origins of the T_{3} found in liver were based on T_{3} receptor kinetic studies. Theoretical analyses of tracer T_{3} kinetics across plasma, cytoplasm, and nuclear compartments indicated that maximum nuclear T_{3} occupancy occurs when specific activities of nuclear T_{3} and plasma T_{3} are equal, suggesting that nuclear T_{3} in liver depends primarily on the plasma T_{3} concentration (36, 45). The contribution of intracellular (local) T_{4} to T_{3} conversion to nuclear (not whole cell) T_{3} was reported to be ∼30% relative to plasma-derived T_{3} also found in the liver nucleus (45). However, nuclear T_{3} represents <10% of intracellular T_{3} in rat liver (36), so it is difficult to glean the relative contributions for the whole cell, an important factor given that D1 mono-deiodination occurs in the plasma membrane (40, 48) and D2 in the endoplasmic reticulum (33). In any case, these values represent relative steady-state pool sizes, not production rates.

In situ liver perfusion studies, in contrast, address the production rate question more directly and are capable in principle of providing data on actual rates of local T_{3} production from T_{4} in liver, as well as effects of experimental interventions on this T_{3} source. Also, this approach obviates many of the problems associated with the far more numerous in vitro liver homogenate studies, which do not retain the metabolic control systems associated with intact cellular and organ structure. T_{3} production was shown to be a direct function of liver size, T_{4} uptake, and 5′D activity in one liver perfusate preparation, indicating that this organ has a large capacity for T_{4} uptake and T_{3} production from T_{4} via 5′D (30). Unfortunately, the nonphysiological perfusates used in these studies did not provide realistic estimates of the in vivo contribution of liver to T_{3} production. In two other perfused rat liver studies, local T_{4} conversion to T_{3} was estimated as 10% (29) and 4% (25), disparate values that, again, probably can be attributed to (intentional) nonphysiological perfusates.

With use of the dual-isotope steady-state approach, local T_{3} produced from T_{4}, denoted LcT_{3}(T_{4}), was estimated in 34 rat tissues (49). To establish relative measures, this group defined *%*LcT_{3}(T_{4}) as the percent contribution of the locally derived T_{3} from local T_{4} relative to the total T_{3} resident in a given tissue, the remainder being the plasma-derived T_{3} pool. At face value, these data suggest that both liver and intestine contribute significantly to the total circulating T_{3} pool: *%*LcT_{3}(T_{4}) was ∼40% for liver and ∼29% for all intestinal tissues combined (43, 49–51). It is important to note, however, that these important data represent pool size measures, not production rates, i.e., these estimates are not the same as, nor are they proportional to, local conversion rates, or even local conversion rates relative to those in other tissues.

Normal human liver appeared to produce no T_{3} of its own, on the basis of interpretations of kinetic studies using simultaneously intravenously injected [^{125}I]T_{4}, -T_{3}, -rT_{3}, and ^{131}I in normal human volunteers, and Eng et al. (17) remarked that slowly equilibrating tissue sites are thus primarily responsible for peripheral T_{4}-to-T_{3} conversion and regulation of circulating T_{3} levels in humans. The literature supports the significance of slow pools, like muscle, in overall T_{3} production (7, 12, 42), but there is also reasonable evidence that liver is an important site of T_{3} production in humans (27, 38) and in the rat, as we have shown here by specific quantification.

### Liver and Intestinal T_{3} Production vs. Thyroidal Secretion

We estimate T_{3} secretion (SR_{3}) from our data as 50% of total body T_{3} production (PR_{3}), as follows. PR_{3} = SR_{3} + CR_{3-4} > PR_{3}^{min} = PAR_{3} (9), where CR_{3-4} is the measured total rate of production of T_{3} from T_{4}, 6.2 ng·h^{-}^{1}·100 g body wt^{-}^{1}, and PAR_{3}, 12.4 ng/h, is the measured plasma appearance rate (Table 2). This yields a lower bound on SR_{3}, with SR_{3}^{min} = 12.4 - 6.2 = 6.2 ng/h. Then %SR_{3} is estimated as 100 SR_{3}^{min}/PR_{3}^{min} = 100 × 6.2/12.4 = 50%. No upper limit on SR_{3} is available from our data, but this ratio of minima is likely to be a reasonable best estimate. It is about the same as two reported estimates, ∼50% in Ref. 5, and 50% calculated from the data in Ref. 18, the latter obtained by achieving complete tissue euthyroidism in T_{3}+T_{4}-replaced female adult rats, the T_{3} component being 6.25 ng/h, i.e., 100 × 6.25/(6.25+6.2) = 50%. It is also comparable to the 43% computable from whole body study data for PAR_{3} (=SR_{3}^{min} + CR_{3-4}) reported in Table 1 of Ref. 43. Thus, from our data, liver T_{3} production is estimated as 15% and intestinal T_{3} production as 3% of total body T_{3} production. This leaves 32% to be accounted for in other organs.

### Plasma-Derived T_{3} and the Elusive Liver

It is of interest to compare the liver T_{3} production rate from local T_{4}, 2.18 ng·h^{-}^{1}·100 g body wt^{-}^{1}, with other computed or computable T_{3} mass influxes and effluxes in liver. The influx of T_{3} from peripheral plasma, T_{3} uptake via the hepatic artery, is ≥119 ng/h, with use of the values of liver fractional uptake rate (*k*_{uptake} = 40/h) and plasma T_{3} pool size (3 ng) we measured (14, 35). We more recently obtained a more precise estimate of *k*_{uptake} for liver, 46.2/h, using the same database, plus additional early kinetic data: another 6 rats at *t* = 1.5 min, thereby yielding a T_{3} arterial influx of 139 ng/h. Thus plasma-derived T_{3} is ≥64 (=139/2.18) times greater than locally derived T_{3} from T_{4}. Inclusion of the unmeasured influx of T_{3} from portal vein would render plasma-derived T_{3} about two orders of magnitude greater than locally derived T_{3} from T_{4}.

Liver T_{3} efflux/elimination occurs along the hepatic vein and via the biliary and local hepatic degradative pathways. We measured the T_{3} biliary efflux directly, as 2.37 ng/h. The venous efflux also has been roughly estimated from our same transient kinetic database (14) and is of the same order of magnitude as the arterial influx, >100 ng/h. The degradative pathway is more difficult to quantify, but it cannot amount to more than total body T_{3} production, ∼12 ng/h, minus fecal excretion, ∼3.6 ng/h (13), and therefore it too is of the order of magnitude of local T_{3} production.

These comparisons explain why liver kinetic processes are so difficult to quantify in the intact biosystem. Arteriovenous T_{3} fluxes exceed local production and degradation rates by two orders of magnitude, swamping local metabolic processes overall, thereby rendering the liver kinetically indistinct from the plasma pool, relatively speaking. This, of course, does not diminish the major importance of liver as a T_{3} producer and metabolic processor, but it provides an explanation of why these processes are so elusive to experimental probes.

### Graph Theory/Cut-Set Analysis

Our novel approach can be readily applied to design critical experiments for quantifying distributed, inhomogeneous biomolecular sources in many other biosystems with accessible organs or organelles, e.g., distributed sources of steroid hormones, vitamins, and lipoproteins (e.g., 1, 2, 16). It also can be used to exclude infeasible designs, on the basis of informationally insufficient probe site availability.

## APPENDIX

Cut-set analysis of Fig. 3 provides the required individual pool T_{4}-to-T_{3} conversion rates, from graphic and algebraic considerations, as follows. These are independent of the statistical properties of the data, treated separately. Two factors (18) govern the solution. First, in steady state, the sum of the mass influx arrows into a closed cut-set curve, in particular each of the two curves labeled A, B, C in Fig. 3, the first, A_{1},B_{1},C_{1}, cutting the EHS organs (liver and intestine) together and the second, A_{2},B_{2},C_{2}, cutting the intestine only, must be equal to the sum of the mass efflux arrows leaving the closed curve. Second, the curves must actually represent different experiments, three different input infusions, denoted A, B, and C, each capable of providing additional analytically independent information about parameters of the model graph or structure, and the inputs and measurements associated with each experiment must be clearly distinguished in the analysis. The only assumption made is that the model rate constants remain the same in each experiment, justified because they are small-perturbation tracer experiments.

To obtain the conversion rate in the EHS, we begin with Expt A data only, and thus cut-set A_{1} surrounding liver and intestine in Fig. 3. Expt B and Expt C do not play a role yet. By inspection, the steady-state mass-flux balance equation for cut-set curve A_{1} is A1a Now, with reference to Fig. 2, we note that CR_{4-3LIV A} ^{A} + CR_{4-3INT}^{A} ≡ *k*_{4-3EHS}*Q*_{4EHS}^{A}. Similarly, (*k*_{56} + *k*_{06})*Q*_{3LIV}^{A} + *k*_{07}*Q*_{3INT}^{A} ≡ *k*_{3EHS}*Q*_{3EHS}^{A}, where we have defined *k*_{3EHS} as the total EHS T_{3} turnover rate constant, the sum of the two arrows *k*_{x}+*k*_{y} in Fig. 2. Thus *Eq. A1a* becomes A1b Then, solving for the key parameter of interest, *k*_{4-3EHS}, we get A2 Note that there is no term in *Eqs. A1a* or *A1b* for Expt B and Expt C inputs shown in Fig. 3, because only Expt A data are appropriate here.

Two rate constants still need to be resolved. To eliminate (solve for) *k*_{3EHS}, we use the independent steady-state Expt B (cut-set B_{1}), without influence of Expt A and Expt C. Summing influxes and effluxes yields (*k*_{65} + *k*_{75})*Q*_{3BLD}^{B} = *k*_{3EHS}*Q*_{3EHS}^{B}. Therefore *Equation A2* then becomes A3 To eliminate the factor (*k*_{65}+*k*_{75}), we apply cut-set analysis a third time to Fig. 3, in this case using Expt C and not A or B. The solution for *k*_{65}+*k*_{75} is obtained, as above, from the mass-flux balance equation for the EHS (cut-set C_{1}). After some algebra, this yields A4 Substitution of *Eq. A4* into *Eq. A3* then gives A5 Finally, CR_{4-3EHS} = *k*_{4-3EHS}*Q*_{4EHS} (ng/h), as given in *Eq. 1.*

We remark here on the need for the additional information provided by the independent Expt C in solving the individual organ T_{3} production rate problem, information over and above that provided by the conventional dual-tracer infusion studies A and B. *Equation A3*, in particular, requires independent knowledge of *k*_{65}*k*_{75} for solution, and this is not provided unless Expt C augments A and B. However, we hasten to add that our specific Expt C is not the only such third independent infusion study that can be used to solve the problem. For example, it can be shown that infusion of T*_{4}, instead of T*_{3}, directly into the organ (EHS in this case) would suffice. However, our derivation here clarifies why studies using data only from A/B dual-tracer experiments, like those reported in Refs. 43, 44, 47–49, cannot by themselves provide local hormone production rates for this multisource system.

Evaluation of intestinal T_{3} production from T_{4} is a bit more involved algebraically. The steady-state mass-flux balance equation for *pool 7* in the 8-pool model of Fig. 3, with Expt A data only (cut-set A_{2}), is A6 where *k*_{77} is the turnover rate of *pool 7* (intestine): *k*_{77} = *k*_{07}+*k*_{67} (see also Fig. 1). The key parameter of interest here is *k*_{4-3INT} A7 As above, Expt B data then yield a simpler *pool 7* equation, from cut-set B_{2} and thus A8 Then, from *Eq. A7* A9 To eliminate *k*_{75} and *k*_{76} in *Eq. A9*, we first apply cut-set analysis a third time to Fig. 3, using Expt-C. Mass-flux balance for cut-set C_{2} gives A10 We need an independent equation in *k*_{76} to complete the task, with *k*_{77} obtained from *Eq. A8* and *k*_{75} from *Eq. A10*. From Figs. 1 and 3, let . Recall that is additionally measured. This provides the final unknown from measurements: Then *Eq. A10* becomes Rearranging and gathering terms and solving for *k*_{75} yields A11 Then, substitution of *k*_{76} ≡ *k*_{3bile} and *k*_{75} from *Eq. A11* into *Eq. A9* yields the desired result for intestine: A12 Finally, CR_{4-3INT} = *k*_{4-3INT}*Q*_{4INT} (ng/h), as given in *Eq. 2*.

## Acknowledgments

This research was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-34839 and a TRAC grant from Knoll Pharmaceutical (now Abbott Laboratories).

## Footnotes

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- Copyright © 2003 by American Physiological Society