Origin of the blood hyperserotonemia of autism
- Skirmantas Janušonis^{1}Email author
Affiliated with
DOI: 10.1186/1742-4682-5-10
© Janušonis; licensee BioMed Central Ltd. 2008
Received: 25 February 2008
Accepted: 22 May 2008
Published: 22 May 2008
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DOI: 10.1186/1742-4682-5-10
© Janušonis; licensee BioMed Central Ltd. 2008
Received: 25 February 2008
Accepted: 22 May 2008
Published: 22 May 2008
Research in the last fifty years has shown that many autistic individuals have elevated serotonin (5-hydroxytryptamine, 5-HT) levels in blood platelets. This phenomenon, known as the platelet hyperserotonemia of autism, is considered to be one of the most well-replicated findings in biological psychiatry. Its replicability suggests that many of the genes involved in autism affect a small number of biological networks. These networks may also play a role in the early development of the autistic brain.
We developed an equation that allows calculation of platelet 5-HT concentration as a function of measurable biological parameters. It also provides information about the sensitivity of platelet 5-HT levels to each of the parameters and their interactions.
The model yields platelet 5-HT concentrations that are consistent with values reported in experimental studies. If the parameters are considered independent, the model predicts that platelet 5-HT levels should be sensitive to changes in the platelet 5-HT uptake rate constant, the proportion of free 5-HT cleared in the liver and lungs, the gut 5-HT production rate and its regulation, and the volume of the gut wall. Linear and non-linear interactions among these and other parameters are specified in the equation, which may facilitate the design and interpretation of experimental studies.
The blood hyperserotonemia of autism is an increase in the serotonin (5-hydroxytryptamine, 5-HT) levels in the blood platelets of a large subset of autistic individuals. It is usually reported as mean platelet 5-HT elevations of 25% to 50% in representative autistic groups [1] that almost invariably contain hyperserotonemic individuals. Since the first report in 1961 [2], this phenomenon has been described in autistic individuals of diverse ethnic backgrounds by many groups of researchers [3–9]. Despite the fact that the hyperserotonemia of autism is considered to be one of the most-well replicated findings in biological psychiatry [1], its biological causes remain poorly understood.
The blood-brain barrier is virtually impermeable to 5-HT and, therefore, free 5-HT in the blood plasma is unlikely to reach cerebrospinal fluid or brain parenchyma. However, biological factors that cause the platelet hyperserotonemia may play a role in the early development of the autistic brain, since the brain and peripheral organs express many of the same neurotransmitter receptors and transporters. The consistency of the platelet hyperserotonemia suggests that many of the genes implicated in autism [15, 16] may control a small number of functional networks. Since blood platelets are short-lived, the altered processes may remain active in the periphery years after the brain has formed. In contrast, most of the brain developmental processes are over by the time an individual is formally diagnosed with autism. SERT is expressed by brain neurons and blood platelets [17] and its altered function may both affect brain development and lead to abnormal 5-HT levels in platelets. To date, most experimental studies have focused on SERT polymorphisms as a likely cause of the platelet hyperserotonemia, but the results have been inconclusive. While SERT polymorphic variants may partially determine platelet 5-HT uptake rates [18] or even platelet 5-HT levels [19], these polymorphisms, alone, are unlikely to cause the platelet hyperserotonemia of autism [18, 20]. Some evidence suggests that the platelet hyperserotonemia may be caused by altered 5-HT synthesis or release in the gut [21–23] or by interactions among several genes [24–26].
To date, most research into the causes of the platelet hyperserotonemia has focused on a specific part of the peripheral 5-HT system. However, this system is cyclic by nature and does not allow easy intuitive interpretation. It is not clear what parameters and their interactions platelet 5-HT levels are likely to be sensitive to, as well as what parameters should be controlled for when others are varied. For instance, an increase in SERT activity may increase platelet 5-HT uptake, but it may also increase 5-HT uptake in the gut and lungs and, consequently, may reduce the amount of free 5-HT in the blood plasma.
Here, we develop an equation that yields platelet 5-HT levels that are consistent with published experimental data. The equation also provides information about the sensitivity of platelet 5-HT levels to a set of biological parameters and their interactions.
where τ = t_{1/2}/ln 2 ≈ 1.44t_{1/2}.
where u_{ i }(t) is the 5-HT uptake rate (mol/min) of platelet i at time t.
In normal humans, C_{ s }/C_{ p }has been experimentally estimated to be around 3.58 · 10^{-18} mol/platelet [7]. The half-life of human platelets is approximately 5 days [28, 29], so τ ≈ 1.44t_{1/2} = 1.04 · 10^{4} min. Plugging these values into Eq. (6) yields $\overline{u}$ = 3.44 · 10^{-22} mol/min, or an "average" platelet takes up around 3.5 molecules of 5-HT every second.
where V_{ max }is the maximal 5-HT uptake rate of one platelet, K_{ m }is the Michaelis-Menten constant, and c_{ i }is the local concentration of free 5-HT surrounding platelet i.
In normal humans, V_{ max }≈ 1.26 · 10^{-18} mol/(min · platelet) and K_{ m }≈ 0.60 · 10^{-6} mol/L (these values were obtained by weighting the medians of each of the three groups of [18] by the number of subjects in the study). Plugging these values and the obtained $\overline{u}$ into Eq. (12) yields C_{ f }≈ $\overline{c}$ = 0.16 · 10^{-9} mol/L = 0.16 nM.
Experimental measurement of free 5-HT in the blood plasma poses serious challenges. It is not uncommon to report concentration values of free 5-HT that are a few orders of magnitude higher than those obtained in carefully designed studies (for discussion, see [14, 30, 31]). The theoretically calculated value (0.16 nM) is on the same order as an accurate experimental estimate of free 5-HT in the distal venous plasma (0.77 nM) obtained by Beck et al. [30]. These authors note that new experimental methodologies may further reduce their estimate [30]. Taken together, these theoretical and experimental results suggest that virtually all platelets take up 5-HT at very low free 5-HT concentrations, after most of the 5-HT released by the gut has been cleared from the circulation by the liver and the lungs.
where t is time and λ > 0 is the time constant of the process (the larger is the λ, the slower is the return to R_{00}). We next consider a more general scenario, where the gut 5-HT release rate is controlled by the actual state of the peripheral 5-HT system.
First, we consider a local mechanism that monitors the extracellular 5-HT concentration in the gut wall. The actual sensitivity of the gut 5-HT release rate to extracellular 5-HT levels is not well understood. In the brain raphe nuclei, 5-HT release does not appear to be controlled by 5-HT1A autoreceptors unless extracellular 5-HT levels become excessive [32]. The gut expresses 5-HT1A, 5-HT3, and 5-HT4 receptors [11], but these receptors may not be activated by the normal levels of endogenous extracellular 5-HT in the gut wall [33]. In SERT-deficient mice, 5-HT synthesis appears to be increased by around 50%, but the expression and activity of tryptophan hydroxylases 1 and 2 are not altered [34]. In SERT-deficient rats, the expression and activity of tryptophan hydroxylase 2 are also unaltered in the brain, even though the extracellular 5-HT levels in the hippocampus are elevated 9-fold [35]. From a systems-control perspective, the reported insensitivity of 5-HT synthesis to extracellular 5-HT levels may be due to the inherent ambiguity of the signal. In fact, high extracellular 5-HT levels may signal both overproduction of 5-HT by tryptophan hydroxylase and an excessive loss of presynaptic 5-HT due to its reduced uptake by SERT. If the former is the case, the activity of trypotophan hydroxylase should be decreased; if the latter is the case, it should be increased.
Alternatively, platelet 5-HT levels can be regulated by global peripheral mechanisms. Since platelets take up 5-HT over their life span, their 5-HT levels will change only if an alteration of the peripheral 5-HT system is sustained over a considerable period of time. Since platelets act as systemic integrators, we can assume that, formally, the gut 5-HT release rate is a function of the platelet 5-HT concentration. In essence, we simply assume that the gut 5-HT release is controlled by global, systemic changes in the peripheral serotonin system. In biological reality, this relationship would be mediated by latent variables, because platelet 5-HT is inaccessible to the gut.
where G is the extracellular 5-HT concentration in the gut wall, P is the platelet 5-HT concentration (mol/platelet), and f(., .) is a differentiable function.
where ${\alpha \equiv -\partial f/\partial G|}_{({G}_{0},{P}_{0})}\ge 0$ and ${\beta \equiv -\partial f/\partial P|}_{({G}_{0},{P}_{0})}\ge 0$.
Note that Eq. (13) is a special case of Eq. (16) when neither G nor P controls the gut 5-HT release rate (i.e., when α = β = 0).
where G(t + s) is the concentration of extracellular 5-HT in the gut wall at time t + s, D is the 5-HT diffusion coefficient across the blood capillary wall, S is the total surface area of the gut blood capillaries, w is the thickness of the capillary wall, Ω_{ g }is the effective extracellular volume of the gut wall, Q_{ tot }is the total cardiac output, z_{ g }is the proportion of the total cardiac output routed to the gut and/or the liver, F(t) is the flow of free 5-HT in the aorta at time t, σ is the proportion of blood volume that is not occupied by cells (approximated well by 1 - Ht, where Ht is the hematocrit), and d_{ g }≡ DS/(w Ω_{ g }). Note that z_{ g }F (t)/(σz_{ g }Q_{ tot }) is the concentration of free 5-HT in the blood plasma that arrives in the gut at time t + s (Fig. 1).
where k_{ g }is the 5-HT uptake rate constant in the gut wall. This constant is likely to be a function of SERT activity (γ), i.e., k_{ g }≡ k_{ g }(γ). Importantly, k_{ g }(0) is not necessarily zero, since 5-HT uptake in the gut may be mediated by low-affinity 5-HT transporters, at least in the absence of SERT [12, 35].
where all parameters and G(t + s) are defined as in Eq. (17), F(t) is the flow of free 5-HT in the aorta, and d_{ b }≡ DS/w (note that d_{ b }/d_{ g }= Ω_{ g }).
After the 5-HT flow leaves the gut, it passes through the liver that removes a large proportion (1 - θ_{ h }) of free 5-HT [13, 14]. After exiting the liver, the 5-HT flow is joined by the 5-HT flow that did not enter the gut and/or the liver and the merged flow passes through the lungs that remove another large proportion (1 - θ_{ p }) of free 5-HT [13, 14]. Experimental results suggest that θ_{ h }≈ 0.25 and θ_{ p }≈ 0.08 [13]. Since the lungs express SERT [36], θ_{ p }may be considered to be a function of SERT activity, i.e., θ_{ p }≡ θ_{ p }(γ). It is likely that θ_{ p }(0)≠ 0, since no obvious toxic 5-HT effects are seen in mice that lack SERT [12].
Platelet 5-HT uptake is a slow process compared with the blood circulation through the gut, liver, and lungs. Therefore, in this circulation, platelet uptake should have a negligible effect on free 5-HT levels in the blood plasma [13, 14]. However, platelets spend a considerable proportion of the circulation cycle in the vascular beds of other organs (the "non-gut" system of Fig. 1), where platelet 5-HT uptake may have an impact on the already low levels of free 5-HT.
where 1 - θ_{ v }is the proportion of free 5-HT cleared by the platelets in the "non-gut" system (Fig. 1) and z_{ ng }= 1 - z_{ g }.
Denote $\stackrel{\wedge}{F}$ the steady-state flow of free 5-HT in the aorta. The system is in its steady state if the following is true: dR/dt = 0, dG/dt = 0, F(t) = F(t - T) = $\stackrel{\wedge}{F}$, and if F(t - s) ≈ F(t - s - x) = $\stackrel{\wedge}{F}$ for all x > 0 for which N_{ tot }exp(-x/τ) ≫ 1, where 0 <s <T (for the last condition, see Eqs. (36) and (47) in Appendix 2).
where k_{ p }≡ k_{ p }(γ) is the 5-HT uptake rate constant of one platelet. In mice lacking SERT, the amount of 5-HT stored in blood platelets in virtually zero [12], suggesting that k_{ p }(0) = 0.
In the derivation, we used the relationship d_{ g }= d_{ b }/Ω_{ g }.
Parameter values
Parameter | Value | Units | Source | Note |
---|---|---|---|---|
(plt = platelet) | ||||
MW (5-HT) | 176.22 | g mol^{-1} | 1 | |
D | 6.00 · 10^{-8} | m^{2} min^{-1} | [48] | 2 |
d _{ b } | 6.00 | m^{3} min^{-1} | d_{ b }= DS/w | 3 |
d _{ g } | 5.82 · 10^{3} | min^{-1} | d_{ g }= d_{ b }/Ω_{ g } | 4 |
G _{0} | 1.00 · 10^{-6} | mol m^{-3} | Table 1 of [32] | 5 |
k _{ g } | 4.00 | min^{-1} | Fig. 4A of [35] | 6 |
k _{ p } | 2.12 · 10^{-15} | m^{3} min^{-1} plt^{-1} | [18] | 7 |
R _{0} | 1.65 · 10^{-5} | mol m^{-3} min^{-1} | [14] | 8 |
P _{0} | 3.58 · 10^{-18} | mol plt^{-1} | [7] | 9 |
S | 1.00 · 10^{2} | m^{2} | Table 8.3 of [48] | 10 |
Q _{ tot } | 5.60 · 10^{-3} | m^{3} min^{-1} | [14] | 11 |
t _{1/2} | 7.20 · 10^{3} | min | [28, 29] | 12 |
w | 1.00 · 10^{-6} | m | Table 8.2 of [48] | 13 |
z _{ g } | 0.27 | Fig. 1 of [14] | 14 | |
z _{ ng } | 0.73 | z_{ ng }= 1 - z_{ g } | 15 | |
α | ≥ 0 | min^{-1} | See note | 16 |
β | ≥ 0 | plt m^{-3} min^{-1} | See note | 17 |
θ _{ h } | 0.25 | [13] | 18 | |
θ _{ p } | 0.08 | [13] | 19 | |
θ _{ v } | 0.50 | [13] | 20 | |
ρ | 9.70 · 10^{4} | m^{-1} | ρ = S/Ω_{ g } | 21 |
σ | 0.56 | See note | 22 | |
τ | 1.04 · 10^{4} | min | τ = 1.44t_{1/2} | 23 |
Ω_{ b } | 5.40 · 10^{-3} | m^{3} | Table 8.3 of [48] | 24 |
Ω_{ g } | 1.03 · 10^{-3} | m^{3} | [49] | 25 |
Equation (22) represents the minimal set of relationships that have to be taken into account in experimental studies. It provides information about the sensitivity of platelet 5-HT levels to biological parameters and their interactions, some of which have not been considered or controlled for in experimental approaches. Here, we limit sensitivity analysis to the simplest case when parameters in Eq. (22) can be considered independent.
Sensitivity of platelet 5-HT concentration to changes in parameters
Parameter, Δ = +10% | Platelet 5-HT, Δ% | Platelet 5-HT, Δ% |
---|---|---|
α = 0 min^{-1} | α = 20 min^{-1} | |
β = 0 | β = 0 | |
d _{ b } | 6.7 · 10^{-3} | 3.5 · 10^{-2} |
G _{0} | 0 | 5.5 |
k _{ g } | -0.27 | -0.24 |
k _{ p } | 10.0 | 10.0 |
R _{0} | 10.0 | 4.5 |
P _{0} | 0 | 0 |
Q _{ tot } | -9.7 | -8.6 |
α | 0 | 4.3 |
θ _{ h } | 9.8 | 8.6 |
θ _{ v } | 0.29 | 0.26 |
θ _{ p } | 10.1 | 8.9 |
σ | -9.7 | -8.6 |
τ | 10.0 | 10.0 |
Ω_{ g }(S constant) | 9.7 | 8.6 |
Ω_{ g }(ρ constant) | 9.7 | 8.6 |
Parameter changes causing 25% and 50% increases in platelet 5-HT concentration
$\stackrel{\wedge}{P}$ + 25% | $\stackrel{\wedge}{P}$ + 25% | $\stackrel{\wedge}{P}$ + 50% | $\stackrel{\wedge}{P}$ + 50% | |
---|---|---|---|---|
Parameter | Value | Δ, % | Value | Δ, % |
d _{ b } | DNE | DNE | DNE | DNE |
G _{0} | DNE | DNE | DNE | DNE |
k _{ g } | DNE | DNE | DNE | DNE |
k _{ p } | 2.65 · 10^{-15} | 25 | 3.18 · 10^{-15} | 50 |
R _{0} | 2.06 · 10^{-5} | 25 | 2.48 · 10^{-5} | 50 |
P _{0} | DNE | DNE | DNE | DNE |
Q _{ tot } | 4.45 · 10^{-3} | -21 | 3.68 · 10^{-3} | -34 |
α | 4.80 | - | 9.92 | - |
θ _{ h } | 0.31 | 26 | 0.38 | 52 |
θ _{ v } | DNE | DNE | DNE | DNE |
θ _{ p } | 0.10 | 25 | 0.12 | 49 |
σ | 0.44 | -21 | 0.37 | -34 |
τ | 1.30 · 10^{4} | 25 | 1.56 · 10^{4} | 50 |
Ω_{ g }(S constant) | 1.30 · 10^{-3} | 26 | 1.57 · 10^{-3} | 52 |
Ω_{ g }(ρ constant) | 1.30 · 10^{-3} | 26 | 1.57 · 10^{-3} | 52 |
Tables 2, 3 indicate that platelet 5-HT concentration is highly sensitive to the platelet 5-HT uptake rate constant (k_{ p }), the baseline gut 5-HT release rate (R_{0}), the proportion of 5-HT cleared in the liver and lungs (θ_{ h }, θ_{ p }), and the volume of the gut wall (Ω_{ g }). Some experimental evidence suggests that k_{ p }is altered in autistic individuals [37, 38]. The analysis also suggests that the hyperserotonemia of autism may be caused by altered extracellular 5-HT-dependent regulation of the gut release rate (α). We have recently shown that mice lacking the 5-HT1A receptor, expressed in the gut [39], develop an autistic-like blood hyperserotonemia [23], which may be caused by altered regulation of the gut 5-HT release rate. Another potentially important 5-HT receptor is the 5-HT4 receptor that is expressed throughout the gastrointestinal tract in humans [40]. The analysis also shows that the 5-HT uptake rate constant in the gut wall (k_{ g }) and the rate constant of 5-HT diffusion into the blood (d_{ b }) should have little effect on platelet 5-HT levels. A recent study has found no link between platelet hyperserotonemia and increased intestinal permeability in children with pervasive developmental disorders [41].
After this correction, the sensitivity of platelet 5-HT concentration to the gut volume remains virtually unchanged (Tables 2, 3).
Care should be exercised in manipulating the parameters k_{ p }, k_{ g }, θ_{ p }, and θ_{ v }, which may not be independent. All of them may be determined, at least in part, by SERT activity (γ). Given the lack of experimental data regarding their actual relationships, two extreme scenarios can be considered. As assumed in the sensitivity analysis, these parameters can be considered to be virtually independent, since each of them is likely to be determined (in addition to SERT) by other factors in the platelet, gut, and lungs. Alternatively, all four parameters may be functions of only one variable, γ. In this case, platelet 5-HT levels may increase or decrease with different γ values, even if each of the functions were linear. This behavior of $\stackrel{\wedge}{P}$ as a function of γ is important to consider in SERT polymorphism studies. The ambiguity could be resolved if an experimentally-obtained covariance matrix for k_{ p }, k_{ g }, θ_{ p }, and θ_{ v }were available. Equation (22) also suggests that platelet 5-HT levels may be highly sensitive to interactions among the platelet uptake rate, the proportion of 5-HT cleared in the liver and lungs, the gut 5-HT release rate, and the volume of the gut wall. The length of the human gut is known to be remarkably variable [42], which may underlie some variability in platelet 5-HT levels. This possibility has not been investigated experimentally or theoretically. It is worth noting that 5-HT itself plays important roles in gastrulation [43] and morphogenesis [44], and that changes in gut length may have had a major impact on the evolution of the human brain [45].
It should be noted that Eq. (22) remains valid if some or all of the parameters are expressed as functions of new, independent parameters. In this case, the original parameters may no longer be independent and changing one of the new parameters may alter more than one of the original parameters. For instance, serotonin uptake in blood platelets has been recently shown to be dependent on interaction between SERT and integrin α IIbβ 3 [46]. Denoting the activity of the integrin y, we can write k_{ p }= k_{ p }(γ, y). It is possible that some other parameters in Eq. (22) can also be expressed as functions of integrin α IIbβ 3 activity. All of these functions can be plugged into Eq. (22), which remains to be correct and now allows calculation of platelet concentration as a function of integrin α IIbβ 3 activity, i.e., $\stackrel{\wedge}{P}$ = $\stackrel{\wedge}{P}$(y). Generally, further theoretical progress will largely depend on understanding the relationships among the current set parameters. Whether they can be expressed as functions of a smaller set of parameters is not known.
Many of the assumptions in the model are "natural" in the sense that they are commonly used to explain experimental results (even though they may not be explicitly stated). In essence, the model simply formalizes the idea that peripheral 5-HT is produced in the gut, from which it can diffuse into the systemic blood circulation, where it can be transported into blood platelets. The strength of the model is in its "bird's-eye" view of the entire system. In particular, the model does not allow focusing on one parameter without explicitly stating what assumptions are made regarding the other parameters (some of which may be equally important in determining platelet 5-HT levels). For example, studies on SERT polymorphisms often focus on 5-HT uptake in platelets but do not explain how the same polymorphisms may affect 5-HT release from the gut (which also expresses SERT). The model also indicates which parameters and their interactions platelet 5-HT concentration is likely to be sensitive to, thus limiting one's freedom in choosing which factors can fall "outside the scope" of a study. By its very nature, the platelet hyperserotonemia of autism is a systems problem.
Some of the model assumptions are not critical, such as the assumption that the gut 5-HT release rate can be controlled by extracellular 5-HT in the gut wall or by platelet 5-HT levels. In the model, the absence of control is simply a special case of this more general scenario, since we can always set α = β = 0. If control is present, the assumption of its linearity (Eq. (16)) is necessary to obtain Eq. (22). While the Taylor series, used in Eq. (15), guarantees near-linear behavior of the control mechanisms in the neighborhood of G_{0} and P_{0}, nothing is said about how far one can move away from G_{0} and P_{0} before non-linearities can no longer be ignored.
The assumption of the independence of the parameters in Eq. (22) is not necessary and is used here only to simplify the numerical sensitivity analysis. Some or all of the parameters may be tightly linked, which does not change Eq. (22) (but it may change the results obtained in the sensitivity analysis). Interdependent parameters can be expressed as functions of other, independent parameters (or "parameterized" in the mathematical sense), and these functions can be substituted for the parameters in Eq. (22). In this case, $\stackrel{\wedge}{P}$ becomes a function of the new parameters, as already discussed with regard to integrin α IIbβ 3.
The model assumes that the gut 5-HT release rate is constant at the steady-state. Strictly speaking, this assumption is incorrect, since gut activity exhibits circadian and other rhythmic behavior. Likewise, platelet counts exhibit normal fluctuations due to a number of factors, such as exercise, digestion, exposure to ultraviolet light, and others [47]. However, platelets accumulate 5-HT over days; therefore, R and N_{ tot }can be thought of as "baseline" values.
A potentially important assumption is made regarding the nature of the 5-HT diffusion from the gut into the blood circulation. Passive diffusion is assumed, and the value of the diffusion coefficient (D) is considered to be comparable to typical values observed in liquids. Virtually no experimental data are available on the exact nature of the 5-HT diffusion (which may be facilitated), and its D value remains to be determined.
where F_{0} and β' are constants and β' ≠ 0, the solution in Eq. (22) would no longer be correct.
These critical assumptions define the limits within which the model should perform reasonably well. New experimental data will be needed to further refine it.
We developed an equation that allows calculation of platelet 5-HT levels as a function of biological parameters. While the main goal is to understand the origin of the hyperserotonemia of autism, the equation can also be used to calculate platelet 5-HT levels in normal individuals and in individuals whose peripheral 5-HT system may be altered due to conditions unrelated to autism. In the simplest case when each parameter is manipulated independently, theoretical analysis predicts that platelet 5-HT concentration should be sensitive to changes in the platelet 5-HT uptake rate constant, the proportion of free 5-HT cleared in the liver and lungs, the gut 5-HT production rate and its regulation, and the volume of the gut wall. The equation also specifies linear and non-linear interactions among these and other parameters, some of which may also play a role in the developing autistic brain.
Symbols used in the text
Symbol | Definition |
---|---|
C _{ p } | Numerical concentration of platelets in the blood |
C _{ s } | Amount of 5-HT per platelet (platelet 5-HT concentration) |
D | Diffusion coefficient of 5-HT diffusion from the gut wall into gut blood capillaries |
d _{ b } | Rate constant of 5-HT influx into the blood due to 5-HT diffusion from the gut |
d _{ g } | Rate constant of 5-HT loss in the gut due to 5-HT diffusion into the blood |
F = F(t) | Flow of free 5-HT in the aorta as a function of time |
$\stackrel{\wedge}{F}$ | Steady-state flow of free 5-HT in the aorta |
G _{0} | Theoretical concentration of extracellular 5-HT in the gut wall around which the control of gut 5-HT release is near-linear |
G = G(t) | Extracellular 5-HT concentration in the gut wall as a function of time |
$\stackrel{\wedge}{G}$ | Steady-state extracellular 5-HT concentration in the gut wall |
k _{ g } | 5-HT uptake rate constant in the gut wall |
k _{ p } | 5-HT uptake rate constant in blood platelets |
N _{ tot } | Total number of platelets |
R _{0} | Theoretical, steady-state gut 5-HT release rate achieved when α = β = 0 |
R = R(t) | Gut 5-HT release rate as a function of time |
$\stackrel{\wedge}{R}$ | Steady-state gut 5-HT release rate |
P _{0} | Theoretical 5-HT concentration in blood platelets around which the control of gut 5-HT release is near-linear |
$\stackrel{\wedge}{P}$ | Steady-state platelet 5-HT concentration |
S | Total surface of the blood capillaries in the gut wall |
Q _{ tot } | Total cardiac output |
t | Time |
t _{1/2} | Half-life of blood platelets |
$\overline{u}$ | 5-HT uptake rate of an "average" blood platelet |
T | Period of blood circulation |
w | Wall thickness of gut capillaries |
z _{ g } | Proportion of cardiac output routed to the gut and/or liver |
z _{ ng } | Proportion of cardiac output not routed to the gut and/or liver |
α | Gain of the gut 5-HT release control that monitors the extracellular 5-HT concentration in the gut wall |
β | Gain of the gut 5-HT release control that monitors the 5-HT concentration in platelets |
γ | SERT activity |
θ _{ h } | 1- proportion of free 5-HT cleared by the liver in one blood circulation cycle (Fig. 1) |
θ _{ p } | 1- proportion of free 5-HT cleared by the lungs in one blood circulation cycle (Fig. 1) |
θ _{ v } | 1- proportion of free 5-HT cleared by the vascular beds of the "non-gut" system (Fig. 1) |
ρ | Surface area of blood capillaries per unit volume of the gut wall |
σ | Proportion of blood volume not occupied by cells |
τ | Time constant of the decay of platelet numbers due to their aging |
Ω_{ b } | Total volume of circulating blood |
Ω_{ g } | Total volume of the gut wall |
where $\overline{u}$ is the 5-HT uptake rate of an "average" platelet, defined in Eqs. (2) and (3). If Δx is allowed to tend to zero, Eqs. (33) and (34) become Eqs. (1) and (4).
Consider the circulation of peripheral 5-HT (Fig. 1). We start by dividing the peripheral 5-HT system into the "gut" system (G-system) and the "non-gut" system (NG-system). In the G-system, arterial blood exits the heart through the aorta, perfuses the gut and/or the liver, joins the venous blood flow to the heart, passes through the lungs, and returns to the heart with the oxygenated blood. In the NG-system, arterial blood exits the heart through the aorta, perfuses various peripheral organs, and joins the venous blood flow. In further considerations, the blood flow rate (measured in m^{3}/min) is clearly distinguished from the 5-HT flow rate (measured in mol/min). In fact, if a blood vessel carrying 5-HT-enriched blood is joined by another blood vessel with virtually no 5-HT in its blood, the blood flow rate of the merged vessel increases, but its 5-HT flow rate remains the same. We intentionally avoid the term "flux", which often denotes flow rate per unit area.
where σ is the proportion of blood volume not occupied by cells.
where U_{ ng }(t) ≡ U_{ ng }(-∞, t).
For the convenience of notation, we will further consider Eq. (55) to be exact.
We investigate the sensitivity of $\stackrel{\wedge}{P}$ to changes in the parameters, which for the purpose of this analysis are considered to be independent. For the convenience of notation, we denote the set of parameters in Eq. (22) X = (X_{1}, X_{2}, X_{3}, X_{4}, ...) ≡ (α, β, k_{ g }, k_{ p }, ...), $\stackrel{\wedge}{P}$ (X) ≡ $\stackrel{\wedge}{P}$. Two approaches are used.
for each X_{ i }, where q = 1.25 or q = 1.5. The results are given in Table 3.
This study was supported, in part, by the Santa Barbara Cottage Hospital – UCSB Special Research Award. I thank the anonymous reviewers for their constructive comments and Vaiva for her support.
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