INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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).

View larger version (20K): [in this window] [in a new window]   Fig. 1.   Path length and hormone secretion. A: calcium molecules arrive at the cell receptor with a frequency that keeps the parathyroid hormone (PTH) secretion at a steady state. B: a decrease in calcium concentration implies a decrease in the frequency of calcium molecules' arrival at the receptor site; subsequently PTH secretion is increased. C: increased presence of PTH increases the calcium molecule path length, thus further reducing the chance of arrival at a receptor and further enhancing PTH secretion.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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.

View larger version (14K): [in this window] [in a new window]   Fig. 2.   A: schematic of our computational gland and its relationship to our cell level environment. The gland boundary consists of line segments 1-6 and the inflow and outflow boundaries. Boundary conditions are described in the text. B: schematic of the cell mode. The source "S" is confined to the cell membrane µ.

Mathematical Model

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 C_cell 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 D_c is defined

where  is a tissue density factor, D_c* is the free calcium diffusion, and  is a model constant. D_p is the PTH diffusion coefficient. In the glandular model, there is in addition an advection term that represents the carrying of blood in the vasculature. In the cell model, the spatial location of the source is confined to the cell membrane [see the APPENDIX for mathematical details (http://www.acm.caltech.edu/~danny/)].

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

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, D_c = D_c* · 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).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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 (C_cell). 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.

View larger version (27K): [in this window] [in a new window]   Fig. 3.   An in vitro experiment in which the calcium concentration is fixed and deposited in a localized area in the domain. Right: a PTH response curve corresponding to the secretion of the cell. Note that it reaches steady state at a PTH level of 5.6 (dimensionless units). The simulation conditions here impose no flux conditions, which means that the calcium concentration remains the same, and the PTH concentration grows from an initial condition of zero to the final maximum at the point when PTH secretion is shut down by calcium inhibition. There is no extracellular loss of PTH in this simulation. (Note that, whereas the calcium concentration is fixed throughout the simulation, its distribution evolves in time until the domain is homogeneous spatially.) The homogeneous or steady-state calcium concentration is 20.0 nondimensional units, well above the inhibitory threshold.

View larger version (27K): [in this window] [in a new window]   Fig. 4.   An experiment analogous to Fig. 3 in which 4 cells are packed closely together. The PTH curve (right) reaches a steady-state value of 33 (dimensionless units). If the PTH curve here were a result from a linear sum of 4 single cells, the total should be no more than 23. The nonlinear increase is due to the effects of geometric spacing and high concentrations of PTH, both of which contribute to the lowering of the effective diffusion. It should be noted that the PTH measures in the cell model experiments shown in Figs. 3 and 4 represent the total PTH in the model domain.

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.

View larger version (42K): [in this window] [in a new window]   Fig. 5.   Simulation of calcium diffusion in the presence of an adenoma. In adenomas (large square cells), the cell population density and overall cell size are increased, whereas the no. of calcium-sensing receptors per cell is reported to be significantly reduced. Left: initially the gland is uniform in calcium concentration, with the exception of a small area of highly concentrated calcium just to the left of the 3 adenomatous cells. Right: as time proceeds, calcium sweeps around the cluster of 3 adenomatous cells. The high [PTH] and narrow spacing combine to make the calcium diffusion path through the adenoma center more tortuous, resulting in a low calcium concentration in the center of the adenoma (implying uninhibited PTH release from the adenomatous cells) and suppression of the PTH release in the normal cells. Adenoma sensitivity to calcium regulation has been shown to be reduced, both in vivo and in vitro (5, 6). Calcium concentration in the whole domain is unchanged throughout, but the distribution is initially localized near the adenomatous cells and diffuses, as seen here.

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 C_cell 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.

View larger version (14K): [in this window] [in a new window]   Fig. 6.   Glandular level simulation. Here, we show the calcium diffusion in a simulated gland. Left: the gland is homogeneous in density, and the diffusion lines proceed from left to right without deformation. Right: note the bending of the calcium diffusion lines around the adenomatous tumor, which corresponds to the tortuous path the calcium molecules must follow in the vicinity of a dense tumor. It is important to note that, although other PTH parameters are the same in normal and in adenomatous glands, a normal gland will show a density of 1.0, whereas an adenomatous gland will show a density of 1.0 in the normal area but that of 0.8 in the area of the adenoma. The effective or apparent diffusion of calcium is slower in the tumor area. In these simulations, the calcium level is set to be initially 1.0 (dimensionless units) and, after time >0, an inflow of constant calcium infusion is kept at 1.18. (See Fig. 7 for the PTH dynamics.) It is important to note that the color legend in Fig. 5 does not apply here. The colored contour lines here are for visualization, and all represent the same concentration of calcium.

View larger version (9K): [in this window] [in a new window]   Fig. 7.   The PTH curve corresponding to Fig. 6, where the solid line represents the normal gland and the dashed line curve represents the adenoma. Note that the adenomatous gland will reach a steady state of ~45% of baseline, whereas the normal gland will reach a steady state of <10% of baseline. This is qualitatively consistent with clinical research reports (see Ref. 3 for a similar depiction).

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).

View larger version (26K): [in this window] [in a new window]   Fig. 8.   Diagram A is taken from Schmitt et al. (4) and represents a calcium infusion clamp experiment on a healthy patient. The calcium is raised at minute 75 (x-axis is time) at a rate of ~0.01 mmol/min for 20 min. PTH (y-axis) drops precipitously. Diagram B is our computational simulation. D: graph depicts the calcium dynamics input into the gland. The calcium is infused at a concentration of 1.20 (dimensionless units), and at minute 75 it is raised at the rate described above to a final concentration of 1.40. The small fluctuations about the mean are randomly generated to mimic the situation described in the clinical studies. C: PTH output for the identical protocol in the absence of small calcium fluctuations. The essential quality is the same. The PTH levels depicted in the glandular level model represent the total PTH arriving at the outflow boundary over time.

View larger version (13K): [in this window] [in a new window]   Fig. 9.   Diagram A is taken from Schmitt et al. (4) and represents a citrate clamp experiment on a healthy patient. The calcium is lowered at minute 75 (the x-axis is time) at a rate of 0.01 mmol/min for 20 min. The PTH (y-axis) rises dramatically. Diagram B is our computational simulation (based on the diffusion principle). C: a representation of the calcium dynamics in the gland. Note that the PTH response peaks at 90 min and descends despite continued calcium lowering, which ends at minute 95. This is likely due to depletion of stored intracellular PTH (see text).

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 C_cell, which is 1.41 concentration units. This was done to accommodate the concentration range indicated by the clinical profile.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

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/