1 Applied and Computational Mathematics, California Institute of Technology, Pasadena, California 91125; and 2 Division of Endocrinology, Diabetes and Hypertension, Department of Medicine, UCLA School of Medicine, Los Angeles, California 90095
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ABSTRACT |
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In endocrine glands, vigorous and coordinated responses are often elicited by modest changes in the concentration of the agonist molecule. The mammalian parathyroid gland is a representative case. Small (5%) changes in serum calcium result in 10-fold (1,000%) changes in glandular parathyroid hormone (PTH) release. In vitro, single isolated cells are observed to secrete fewer hormones than cells residing within a connected group, suggesting that a network has emergent regulatory properties. In PTH-secreting tumors, however, the ability to respond quickly to changes in calcium is strongly damped. A unifying hypothesis that accounts for these phenomena is realized by extracellular modulation of calcium diffusivity. A theoretical model and computational experiments demonstrate qualitative agreement with published experimental results. Our results suggest that, in addition to the cellular mechanisms, endocrine glandular networks may have regulatory prowess at the level of interstitial transport.
cell network dynamics; tortuosity; intercellular communication; parathyroid hormone; drug resistance
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INTRODUCTION |
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IN CONTEMPLATING THE DESIGN of endocrine systems, we are confronted by the following question: are there advantages in localizing cells together in a densely packed space over dispersing cells in individual units throughout the body? Mammalian physiology would seem to have answered the question in the form of organizing cells as specialized endocrine organs. The human parathyroid gland provides us with an interesting example. The rapid and coordinated secretion of parathyroid hormone (PTH) by the parathyroid gland is essential for maintaining serum calcium within safe levels. Small decrements in serum calcium concentration are routinely demonstrated to produce five- to tenfold increases in serum parathyroid concentrations within minutes (3, 11, 28, 37). The mechanisms that enable this endocrine cell network to coordinate a collective response are largely unknown. Calcium binding to the extracellular domain of a calcium-sensing receptor is thought to inhibit the exocytosis of stored PTH (40). In vitro studies have shown that denser collections of PTH-producing cells secrete more PTH per unit cell than sparse collections and that this effect was not due to conditioned media (10, 34). Studies in vivo and in vitro have shown that PTH-secreting tumors (adenoma) maintain higher PTH secretion than healthy tissue (3), whereas calcium's regulatory effect on this secretion is significantly dampened (28, 40). However, calcium sensitivity is at once recovered in vitro when the adenoma is dispersed into smaller cellular collections (28). These results suggest that the system (network of cells) has emergent regulatory properties.
We present here a novel physiological mechanism that describes how the parathyroid gland's interstitial environment might influence the diffusional properties of the regulating ligand and endow the healthy cell network with the ability to respond to small changes in calcium concentration. On the basis of this concept, we constructed a mathematical model and explored the consequences of modulating interstitial calcium transport. Computer simulations suggest that regulation of interstitial calcium transport renders a consistent principle that permits the incorporation of the established PTH secretory phenomena described above, raising the possibility that in addition to cellular mechanisms, the parathyroid gland extends its regulation of PTH secretion to the level of interstitial transport.
Calcium's inhibitory control of PTH secretion is presumed to be
dependent on the ability to arrive at a calcium-sensing receptor (37). Hence, the transport of calcium into and within the
intercellular space of the gland should be of fundamental importance.
The transport of calcium ions from arteriole capillaries to the cell
membrane receptors occurs by means of diffusion (7, 37).
Increasing the path length or the transit time for diffusible calcium
ions would affect the calcium-sensing receptor kinetics and the
subsequent signal transduction pathways, culminating in PTH release
(6, 7, 29). Studies examining ionic transport in the brain
microenvironment show that the presence of large macromolecules
significantly influences intercellular diffusion
(22-24). The formalism commonly used is adapted from
porous media theory (25), where the apparent diffusion coefficient D* is related to the free solute diffusion
coefficient Do by a factor n (the tortuosity)
D* = Do/n. Tortuosity is a
dimensionless parameter that represents a number of implicit factors
that modify the path length of a diffusing species, such as geometry,
viscosity, macromolecular concentration, and the like. The parathyroid
gland geometry is segmented into nests or chords of cells surrounded by
a capillary network (14). Hypothetically, a small
decrement in the calcium concentration near an individual cell
initiates the release of that cell's PTH (and other co-stored
proteins) into an intercellular space shared by neighboring cells. The
presence of large proteins in the extracellular matrix
increases the local tortuosity n and results in an increased
path length for calcium diffusion. The arrival of fewer calcium ions
promotes even more PTH release and ignites a runaway reaction of PTH
secretion within the local nest or chord of cells. The basic idea is
schematically presented in Fig. 1. The
secretory response is perhaps modulated by countering forces such as a
rising calcium concentration, depletion of PTH from stored vesicles
(31, 36), and potentially other counterregulatory
mechanisms, such as autocrine inhibition by degradation products of
co-secreted chromogranin A (30). In this configuration,
the cell network is poised to tune its own secretory response by
locally controlling molecular diffusion. A compelling consequence of
this theory is that altered geometry and deficient or ineffective
calcium-sensing receptors in adenomatous and hyperplastic cells would
establish and maintain a higher intercellular protein concentration and
dampen the regulatory effects of calcium, as is observed in vivo and in
vitro (11, 17, 40).
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METHODS |
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Two separate but related computational models were constructed. The models investigate the calcium-PTH physiology in 1) the context of small cellular networks and 2) the macrophysiological scale of organ level responses, corresponding to published animal and human clinical experiments.
A schematic diagram (Fig. 2) illustrates
the relationship between the two modeling environments.
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Mathematical Model
The general equations are in two dimensions, where
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Here C is the extracellular (interstitial) calcium
concentration, P is the extracellular parathyroid hormone
(PTH) concentration, I is the intracellular PTH, and
S is the source of P that is governed by the
formula given. The calcium threshold Ccell
determines the calcium concentration level above which PTH exocytosis
is inhibited. V (taken to be constant in this model)
represents the synthesis of intracellular PTH, and and k
are scaling constants. The diffusion coefficient
Dc is defined
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To briefly summarize: calcium ions diffuse in the intercellular domain. At each time step, every cell computes the local calcium concentration (mesh points in the glandular model and nodes depicting receptors in the cell level model) and secretes an amount of PTH based on the following rules. 1) PTH is released if the extracellular calcium concentration at the receptor site is below the cell's calcium threshold. 2) The amount of PTH released is modified depending on the availability of intracellular PTH stores.
The computational experiments are performed in two settings (see Fig. 2). 1) They are performed on the scale of cellular level dynamics, where the cells are depicted by squares and the receptors are the nodes on the cell's membrane. In these experiments, we investigate the effects of cell spacing and PTH concentration on calcium transport. We also examine the properties of calcium transport in the vicinity of adenomatous cells. Our cells are created numerically by a "level set" method that is described in the APPENDIX.
2) They are performed on a glandular level, where the cells themselves are the gridpoints and calcium is advected into the gland, diffuses through the gland, and finally is advected out. PTH is being created in the gland and advected out as well.
In these experiments, we examine the clinical research studies that lower and raise serum calcium by means of a "clamp" (11, 28). In addition to examining a "healthy" gland (see Figs. 8 and 9), we also perform simulations on a gland containing an adenoma (see Figs. 6 and 7).
The numerical integration of the model equations is described in the APPENDIX. The equations are in dimensionless form, and the derivation can also be found in the APPENDIX.
Model Assumptions and Justifications
The equations in the cell and glandular models are generally the same, with the exceptions noted above. The advection term is dominant only in the vasculature entering the gland and is negligible in the glandular interstitium.1) We use a 2-d model in all simulations. Each parathyroid gland is on average 7 × 3 × 0.5 mm; therefore, we think that the aspect ratio justifies the 2-d approximation for a first model. In the cell model, the 2-d representation is justified because the cells are pancake-like in geometry.
2) Interstitial transport of calcium and PTH occurs by diffusion.
Our regulatory hypothesis rests upon the factors modifying diffusion
(1, 19). The diffusion of small ions in a polymeric solution has been the subject of several papers in the literature (6, 18, 33). The diffusion coefficient D is
affected by the fraction of the diffusible volume occupied by larger
proteins (PTH, chromogranin A), and the formula we employ is a
modification of the Stokes-Einstein equation that is often seen:
D = D0 · exp(a Mn), where D0 is the free
diffusion coefficient, M is protein concentration, and a and
n are constants (33). This formula motivates
our particular choice, Dc = Dc* · exp(
P).
3) The cells have calcium-sensing receptors. We assume that the calcium occupancy of the receptors is proportional to the concentration of calcium in a small neighborhood of the receptor's extracellular domain. At the simulation level of a small group of cells, we make use of two additional features: 1) chief cells possess apical lateral secretory polarity, and 2) calcium-sensing receptors are also located in the same apical-lateral region. The justification of the former was cited earlier. In a study by Kifor et al. (17), the calcium sensor was found to be localized in caveolin-rich domains. Caveolin has been associated with exocytotic locations on the cell membrane surfaces (38). The inclusion of these features is not essential for the diffusion-mediated regulation concept but rather represent our current view of the PTH cell physiology. (Data of precise interstitial PTH concentration in proximity to the calveolae-rich spaces are not available.)
4) We empirically assign a rule for PTH release.
The general concept of diffusion-mediated regulation does not depend
strongly on any particular form of the rule, just that there exists a
relationship between ambient calcium concentration and cell PTH
release, a theory for which there is a strong consensus (3, 11, 28, 37). In the model, each cell has a given "calcium threshold" below which PTH is released and above which PTH
exocytosis is inhibited. We justify this assumption by referring to
studies in the literature characterizing calcium-PTH response curves
with set points, thresholds, and saturation levels (10, 34). Furthermore, the quantity of PTH released is
dependent on the available intracellular PTH stores. In the cell level
model, the source is confined to the cell membrane, which is
mathematically achieved by using a characteristic function
(x) (see APPENDIX).
5) Intracellular PTH dynamics. Although precise quantitative data are not available, we make use of recent observations concerning PTH biosynthesis and intracellular storage and decay (36). The synthesis for new PTH is observed to be on a time scale of 30-60 min (20, 39). The natural decay of stored vesicular PTH is on the same time scale (36). Both of these are slow processes compared with diffusion of calcium in the interstitium. PTH release (source "S") is modified by available intracellular PTH and previous history of secretion (30). We use a differential equation to dynamically update the state of each cell in the simulation.
6) In the organ level model, the PTH-secreting cells are arranged
in either a uniform pattern, representing a healthy control, or in a
nonuniform pathological pattern with an area of increased cell density
and low receptor number, representing an adenoma.
At this simulation scale, we do not assign a polarity of hormone
secretion or a receptor distribution. Upon exocytotic release from the
cell, PTH will diffuse to nearby capillaries and eventually be advected
out through an outflowing vein. Transport of calcium and PTH in
capillaries and larger vessels is predominantly by means of passive
advection. The cell density is explicitly given by the mesh size.
Tortuosity is likely to be affected by other secreted macromolecules,
such as chromogranin A. During a prolonged interstitial presence
(30-60 min), chromogranin A may degenerate and break down into
pancreastatin, which is known to inhibit PTH secretion
(30). The simulations presented here focus on events generally shorter than the time intervals generally associated with the
formation of pancreastatin. Additionally, we do not explicitly represent effects such as osmotic pressure, surface tension, and curvature, which to first order can be considered as implied in the
tortuosity factor. To mimic the increased tissue density of an adenoma
(in the glandular level model), we assign a density value
(x,y) to the mesh points that are in the
adenoma region, which represents the increased geometric tortuosity in
the adenoma region (see APPENDIX).
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RESULTS |
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In Figs. 3 and
4, we compare the PTH secretory
response of a single isolated cell with a group of four cells,
respectively. The parameters used in the simulation are as given in
METHODS. A calcium depot is placed in the domain (as an
initial condition) and diffuses into the domain, eventually inhibiting
the PTH seceretion. The boundary conditions are such that there is no
flux of PTH or calcium into the cells or out of the interstitial domain
(see figure in APPENDIX). Therefore, a PTH steady state is
achieved once the calcium concentration near the cell/cells is equal to or greater than the cell threshold level
(Ccell). The depot is placed so that the mean
distance between the depot and the group of four cells is the same as
the distance for the single isolated cell. The total calcium in the
simulation domain remains constant throughout; only the spatial
distribution evolves in time by means of diffusion. The final
calcium concentration is 20.0 nondimensional concentration
units, which is well above the inhibitory threshold. In Fig. 3, the
value of PTH at steady-state value is 5.6 (nondimensional units). In
Fig. 4, it is notable that the value of PTH at steady state is 33 units. If the PTH secretory response of the group of four cells were a
simple linear sum of four single isolated cells, the expected value
should be no more than 23 units. The simulation thus demonstrates the
nonlinear effects of geometric spacing and PTH concentration on calcium
transport. Fitzpatrick and Leong (10) and Sun et al.
(34) have observed that a cell in a connected group of
PTH-secreting cells secretes more on average than an isolated single
cell. If we assume that the release of PTH depends on the transport of
calcium, we should therefore expect a delay in the release of PTH
corresponding to the speed of diffusion (in the linear case): distance
traveled proportional to the square root of time. To the contrary, the
model predicts a rise in PTH slope at least as fast as or faster for
the group than the single cell. This can be seen by noting that the
average group PTH level (total divided by 4) at time 5,000 is ~7 vs. 5.4 for a single cell. Thus one could argue that the rapid
nonlinear group response is not a simple sum of the rapid response of
individual cells.
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Our computational model does not include the possible inhibitory effects of chromogranin A. Drees and Hamilton (9) showed that, in vitro, chromogranin A degenerates and breaks down into pancreastatin, which is known to inhibit PTH secretion (30). These effects are reported to occur on a time scale of 30-60 min. Chromgranin A is a relatively large protein, ~50 kDa, and in high enough concentrations could potentially increase the tortuosity of calcium transport. One can argue that the in vivo probability of the molecule remaining in the interstitial area long enough to degrade into pancreastatin is not high and thus not dominant in autocrine regulation of PTH secretion. Designing experiments to clarify this issue would be valuable.
To illustrate how interstitial calcium diffusion might play a role in
hyperplasia or adenoma, we designed a simulation of an in vitro
experiment that contains a small number of adenomatous (larger and
denser) cells within a cluster of normal cells. We place a small depot
of calcium near the adenomatous cells and let it diffuse. In Fig.
5, the smaller square cells represent healthy PTH-secreting chief cells, with the apical portions (secreting ends) oriented toward one another. The larger, more closely spaced cells represent adenomatous cells. In this simulation, we track the
diffusion of calcium, which initially is 0.60 nondimensional units
(blue), everywhere except for a small patch of increased calcium
concentration [3.00 (white)] just to the left of the adenomatous cells. Figure 5 (left) shows the calcium diffusion early in
the simulation, and Fig. 5 (right) shows a later view. As is
evident from the figure, the interstitial calcium diffusion spreads
among the normal cells but sweeps around a cluster of three more
densely packed adenomatous cells (Fig. 2, A and
B). In the configuration of Fig. 5, the low calcium
concentration in the heart of the adenoma implies higher PTH secretion,
whereas the higher calcium levels in the surrounding normal
cells suggest relatively suppressed PTH levels, a physiology consistent
with that seen in clinical experience (8, 11, 16, 40). In
vitro experimental evidence of this effect is suggested in Yu et al.
(40), who showed that adenomatous cells, although residing
in a connected group, persisted in PTH secretion at a given calcium
concentration that was subsequently inhibitory to the same group after
their dispersal into smaller groups. Additional experimental support is
found in Sun et al. (34), who showed that denser cell
collections on average secreted more PTH. The possibility that this was
an autocrine or paracrine effect was ruled out when the conditioned
medium was shown to be inert on fresh cells. A potential analysis is as
follows. The smaller intercellular distance creates a greater obstacle
for calcium diffusion and increases the calcium ion path length. An increased path length is in effect slower calcium diffusion.
The more closely spaced cells have a lower probability of being
inhibited by incoming calcium ions and thus continue to secrete more
PTH until the stores are exhausted or the cell network reaches a steady state.
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The principle extends easily into the macroscopic or organ level case.
In Fig. 6 (left and
right), simulations of calcium diffusion in a healthy gland
and in a gland containing an adenoma, respectively, are depicted. In
both cases, the initial calcium concentration in the domain is 1.0 unit, and a fixed infusion of calcium (1.18 units) is turned on at the
left inflow vessel. As in the case of the cell-level simulation, the
adenoma creates an environment of increased tortuosity, exhibited by
the slowing of calcium diffusion in its vicinity; this is seen as a
bending of the calcium diffusion lines in Fig. 6 (right).
The corresponding PTH secretory responses are seen in Fig.
7, where the gland with an adenoma and
the healthy gland are distinguished. The calcium infusion is able to
suppress the normal gland from an initial (normalized) value of 100%
to a final value of less than 10%, whereas the same infusion
suppresses the adenomatous gland from an initial value of 100% to a
final value of 45%. In both cases, the Ccell is
the same, as are the other parameters (see METHODS and
Table 1). Clinical research studies have consistently observed that
adenomatous and hyperplastic glands are far more resistant to being
inhibited by calcium infusion than are normal controls (11). Typically, patients with adenoma are able to
suppress their serum PTH concentrations to a level of 30-40% of
baseline, whereas normal control subjects are able to suppress them to
~5-10% (11). The model appears to produce
qualitatively similar results. The denser geometries and higher PTH
levels in the interstitial spaces of these pathologies support the
thesis that calcium's ability to inhibit PTH in such conditions is
limited by the effect on intercellular transport.
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Finally, using the glandular level model, we make comparisons
with established clinical calcium and sodium citrate clamp experiments (28). Figure 8A
depicts the result of a calcium clamp experiment on a healthy patient.
In the clinical study protocol, the subject's serum calcium and PTH
are monitored every 2 min. At a designated time (75th min), a calcium
infusion is introduced (0.01 mm/min for 20 min) to monotonically raise
the subject's serum calcium to a designated target concentration. The
new calcium concentration (1.40 mmol) is then maintained for the
remainder of the study (until minute 180). We follow this
protocol computationally by inputing a steady calcium infusion of 1.20 units into the left inflow for the first 75 time units. At minute
75, we raise the calcium infusion linearly over 20 min
(dimensionless time units) to reach the target concentration of
1.40 by minute 95, and we maintain this concentration for
the remainder of the experiment. To mimic the random calcium
fluctuations, we add a small random fluctuation about the mean of our
calcium input (see graph of calcium input in Fig. 8B). To
demonstrate the effect of the calcium fluctuation input, we display the
PTH outputs for both nonfluctuating and fluctuating calcium inputs
(Fig. 8C). The essential behavior is the same and appears to
qualitatively agree with the clinical study. Analogously, in Fig.
9, we follow the clinical study protocol for the lowering of calcium. In the study, the subject's serum calcium
and PTH are monitored every 2 min. At a designated time (the 75th min),
a sodium citrate infusion is introduced to monotonically lower the
subject's serum calcium to a designated target concentration (0.01 mm/min for 20 min). The new calcium concentration (1.00 mmol) is then
maintained for the remainder of the study (minute 180).
Computationally, we input a steady calcium infusion of 1.20 units into
the left inflow for the first 75 time units. At minute 75,
we lower the concentration linearly to reach a target of 1.00 unit by
minute 95, and we maintain this concentration for the remainder of the simulation. As was done in the calcium clamp simulation, we add a small random fluctuation about the calcium mean
(see Fig. 9C).
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In the actual clinical study, a vigorous rise in PTH secretion is seen within 5 min of initiation of the experimental protocol at minute 75 (see Fig. 9A). This is followed by a slower fall to a new steady state for the remainder of the experiment (28). An important feature in the PTH response is that the peak concentration is reached by minute 90 and then begins to more slowly descend, despite the fact that the calcium concentration continues to drop until minute 95 (see the accompanying calcium dynamics graph in Fig. 8D). The computational analog seems to capture this quality (Fig. 9B). On the basis of our model, we put forward the following analysis: having initiated the runaway rise in PTH release, the cell stores begin to deplete their PTH stores and cannot maintain the maximum secretory capacity. The literature seems to support this, as is shown in Schwarz et al. (30). The fall to the new steady state is governed by a dynamic balance between the evolving state of intracellular PTH stores and continued exocytotic release. The parameters used for the clamp studies are identical to the ones listed in METHODS, with the exception of Ccell, which is 1.41 concentration units. This was done to accommodate the concentration range indicated by the clinical profile.
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DISCUSSION |
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In health, endocrine organs have an inherent capacity to exhibit strong and coherent secretory responses to specific stimuli. The diffusion-mediated regulation suggested in the present work is perhaps an intermediate control mechanism that serves as a bridge between the cellular and the system level dynamics. The cell network can exploit diffusional transport for the purposes of coordinating a range of glandular responses downstream from the cellular mechanisms and upstream from the systemic physiology. Cell network geometry and biochemistry provide the parathyroid gland with a dynamic capacity to control intercellular diffusivity. Conversely, perturbations to the cellular and extracellular geometric structure will influence the glandular dynamics, as seen in pathology. The current model is a simple representation of the biophysics and physiology, and it does not include effects such as receptor adaptation and signal transduction pathways that are known to be operant in these cells (5). Quantitative data on calcium-sensing receptors, kinetics, and signal transduction are largely unavailable. Regulatory details concerning vitamin D (32), phosphate (4), and synthetic calcium mimetics (21) are now emerging. However, despite the omission of those cellular processes, the model is able to capture essential qualitative behavior seen in the published studies. Once the cell biology and biochemistry details are understood better, the model can be intelligently refined. Because we are examining time scales that are shorter than those required for de novo synthesis of PTH and receptors, we believe those effects will not contribute to the essential mechanisms discussed here (20, 29). An important secretory phenomenon, which has been the subject of significant research, is the episodic or pulsatile pattern of serum PTH (12, 13, 27). The consensus is that PTH pulses occur with a frequency of 6-7/h. The amplitude and frequency of the pulses have also been observed to vary in pathology (12, 13, 27). The origin of PTH pulsatility is unclear. To date, no "pacemaker" cells have been identified. A potential cause might be through a feedback process via calcium, although the evidence from studies is not established. As is clearly seen in the clinical clamp studies (11, 28), a change in the calcium concentration of 20-50 µmol already induces PTH responses significantly beyond the baseline fluctuations. However, most clinical ion-sensitive electrodes in general are able to resolve serum calcium concentrations only to the 10 µmol range (26), perhaps frustrating the ability to correlate the dynamic calcium-PTH relationship at nonstimulated conditions. We consider the present work as a preliminary study in addressing the complexity of the episodic secretory patterns. Identifying the dominant regulatory principles in the setting of in vitro and clinical clamp studies would help to clarify a new set of questions necessary to approach the full physiological feedback cycle.
The extracellular diffusional mechanism proposed provides a consistent argument for 1) higher secretion of single cells in a connected network compared with isolated cells, 2) the rapid nonlinear response seen in healthy glands, as well as 3) the pathological responses seen in hyperplasia and adenoma. Because the proposed diffusional regulation strongly depends on the existence of a connected cell network (gland), it also suggests a rationale for the advantages of cell networks as organs vs. a dispersed system of isolated cells.
The diffusion-mediated concept is experimentally accessible. The effects of diffusion modulation in parathyroid gland interstitium might be visualized, for example, by means of confocal microscopy with calcium fluorophobes or perhaps by diffusion-weighted magnetic resonance imaging (DWI) (4). In DWI, the anisotropy of molecular diffusion is often accounted for by underlying structural features. A large and active adenoma may possess sufficient tortuosity to enable a visual confirmation of our hypothesis. Consistent with this thesis is the recent work of Pluen et al. (24), demonstrating the strong effect of the extracellular matrix on the diffusion of macromolecules in the setting of tumors. Experimental validation of the diffusion principle may potentially create opportunities for new diagnostic and therapeutic tools and perhaps initiate further exploration of regulatory themes in cell-network communication.
A mathematical APPENDIX of this work can be viewed at http://www.acm.caltech.edu/~danny/
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ACKNOWLEDGEMENTS |
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We express our gratitude to Drs. W. Goodman, I. Salusky, and A. Van Herle of UCLA and Dr. S. Fraser of Caltech for many enlightening and helpful discussions.
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FOOTNOTES |
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This work was supported by the W. M. Keck Foundation Fund for Discovery in Basic Medical Research at the California Institute of Technology. D. Petrasek was also supported by the Burroughs Wellcome Fund's Computational Molecular Biology program at Caltech and by the UCLA STAR program.
Address for reprint requests and other correspondence: D. Petrasek, 217-50 Firestone Bldg., Caltech, Pasadena, CA 91125 (E-mail: petrasek{at}caltech.edu).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
10.1152/ajpendo.00306.2001
Received 11 July 2001; accepted in final form 25 February 2002.
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