Suppose blood platelets are produced at a constant rate, their half-life is t_1/2, and we are interested in the steady state when the total number of platelets (N_tot) remains constant. Then the number of the platelets whose age ranges from x ≥ 0 to x + dx is given by (Appendix 1)

d N ( x ) = N t o t τ e − x / τ d x , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaqMaemOta4KaeiikaGIaemiEaGNaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGobGtdaWgaaqaaiabdsha0jabd+gaVjabdsha0bqabaaabaGaeqiXdqhaaOGaemyzau2aaWbaaSqabeaacqGHsislcqWG4baEcqGGVaWlcqaHepaDaaGccqWGKbazcqWG4baEcqGGSaalaaa@450B@

where τ = t_1/2/ln 2 ≈ 1.44t_1/2.

The 5-HT uptake rate of an "average" platelet at time t can be defined as follows:

u ¯ ( t ) ≡ 1 N t o t ∑ i = 1 N t o t u i ( t ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGGOaakcqWG0baDcqGGPaqkcqGHHjIUjuaGdaWcaaqaaiabigdaXaqaaiabd6eaonaaBaaabaGaemiDaqNaem4Ba8MaemiDaqhabeaaaaGcdaaeWbqaaiabdwha1naaBaaaleaacqWGPbqAaeqaaOGaeiikaGIaemiDaqNaeiykaKcaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGobGtdaWgaaadbaGaemiDaqNaem4Ba8MaemiDaqhabeaaa0GaeyyeIuoakiabcYcaSaaa@4C00@

where u_i(t) is the 5-HT uptake rate (mol/min) of platelet i at time t.

At any two times t₁ and t₂ (t₁ ≠ t₂), at least some of the individual platelets in the circulation will be physically different, because platelets are constantly removed from the circulation and replaced by new platelets. Also, at least some individual platelets will be routed by the circulation to different blood vessels, which may have different concentrations of free 5-HT in the blood plasma. Since the platelet uptake rate depends on the 5-HT concentration in the surrounding plasma, generally, u_i(t₁) ≠ u_i(t₂). However, the 5-HT uptake and distribution of platelets appear to be little affected by their age or by how much 5-HT they have already accumulated 1427. Also, the numbers of platelets in blood vessels are very large and can be considered constant. Therefore, u¯(t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGGOaakcqWG0baDcqGGPaqkaaa@3083@ should be immune to these replacements and permutations, and the time-dependence of u¯(t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGGOaakcqWG0baDcqGGPaqkaaa@3083@ can be dropped:

u ¯ ≡ u ¯ ( t ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGHHjIUcuWG1bqDgaqeaiabcIcaOiabdsha0jabcMcaPiabc6caUaaa@3509@

The total amount of 5-HT that has been taken up by the subpopulation of platelets whose age ranges from x to x + dx is given by (Appendix 1)

d U ( x ) = N t o t τ e − x / τ ( u ¯ x ) d x . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaqMaemyvauLaeiikaGIaemiEaGNaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGobGtdaWgaaqaaiabdsha0jabd+gaVjabdsha0bqabaaabaGaeqiXdqhaaiabdwgaLnaaCaaabeqaaiabgkHiTiabdIha4jabc+caViabes8a0baacqGGOaakcuWG1bqDgaqeaiabdIha4jabcMcaPiabdsgaKjabdIha4jabc6caUaaa@49B4@

If the total volume of the circulating blood is Ω_b and the numerical concentration of platelets is C_p = N_tot/Ω_b, the concentration of platelet 5-HT is

C s = 1 Ω b ∫ 0 ∞ d U ( x ) = 1 Ω b ∫ 0 ∞ N t o t τ e − x / τ ( u ¯ x ) d x = τ C p u ¯ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@66D3@

It follows that

u ¯ = C s τ C p . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGH9aqpjuaGdaWcaaqaaiabdoeadnaaBaaabaGaem4CamhabeaaaeaacqaHepaDcqWGdbWqdaWgaaqaaiabdchaWbqabaaaaOGaeiOla4caaa@373D@

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 2829, so τ ≈ 1.44t_1/2 = 1.04 · 10⁴ min. Plugging these values into Eq. (6) yields u¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebaaaa@2D60@ = 3.44 · 10^-22 mol/min, or an "average" platelet takes up around 3.5 molecules of 5-HT every second.

What concentration of free 5-HT in the blood plasma corresponds to this uptake rate? Since platelet 5-HT uptake obeys Michaelis-Menten kinetics 1418,

u i = V m a x c i K m + c i , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyDau3aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpjuaGdaWcaaqaaiabdAfawnaaBaaabaGaemyBa0MaemyyaeMaemiEaGhabeaacqWGJbWydaWgaaqaaiabdMgaPbqabaaabaGaem4saS0aaSbaaeaacqWGTbqBaeqaaiabgUcaRiabdogaJnaaBaaabaGaemyAaKgabeaaaaGccqGGSaalaaa@404D@

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.

If the concentration of free 5-HT were the same in all blood vessels (c_i ≡ C_f), we would obtain

u ¯ = V m a x C f K m + C f MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGH9aqpjuaGdaWcaaqaaiabdAfawnaaBaaabaGaemyBa0MaemyyaeMaemiEaGhabeaacqWGdbWqdaWgaaqaaiabdAgaMbqabaaabaGaem4saS0aaSbaaeaacqWGTbqBaeqaaiabgUcaRiabdoeadnaaBaaabaGaemOzaygabeaaaaaaaa@3D5E@

and

C f = u ¯ K m V m a x − u ¯ . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qam0aaSbaaSqaaiabdAgaMbqabaGccqGH9aqpjuaGdaWcaaqaaiqbdwha1zaaraGaem4saS0aaSbaaeaacqWGTbqBaeqaaaqaaiabdAfawnaaBaaabaGaemyBa0MaemyyaeMaemiEaGhabeaacqGHsislcuWG1bqDgaqeaaaakiabc6caUaaa@3D72@

However, in some blood vessels (such as the ones leaving the gut) the concentration of free 5-HT may be considerably higher than in others. We can define

c ¯ ≡ 1 N t o t ∑ i = 1 N t o t c i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4yamMbaebacqGHHjIUjuaGdaWcaaqaaiabigdaXaqaaiabd6eaonaaBaaabaGaemiDaqNaem4Ba8MaemiDaqhabeaaaaGcdaaeWbqaaiabdogaJnaaBaaaleaacqWGPbqAaeqaaaqaaiabdMgaPjabg2da9iabigdaXaqaaiabd6eaonaaBaaameaacqWG0baDcqWGVbWBcqWG0baDaeqaaaqdcqGHris5aaaa@4473@

and rely on the evidence that c_i ≪ K_m 141830. Then

u ¯ ≈ V m a x K m c ¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebacqGHijYUjuaGdaWcaaqaaiabdAfawnaaBaaabaGaemyBa0MaemyyaeMaemiEaGhabeaaaeaacqWGlbWsdaWgaaqaaiabd2gaTbqabaaaaOGafm4yamMbaebaaaa@398E@

and it follows that

c ¯ ≈ K m V m a x u ¯ = K m C s τ V m a x C p . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4yamMbaebacqGHijYUjuaGdaWcaaqaaiabdUealnaaBaaabaGaemyBa0gabeaaaeaacqWGwbGvdaWgaaqaaiabd2gaTjabdggaHjabdIha4bqabaaaaOGafmyDauNbaebacqGH9aqpjuaGdaWcaaqaaiabdUealnaaBaaabaGaemyBa0gabeaacqWGdbWqdaWgaaqaaiabdohaZbqabaaabaGaeqiXdqNaemOvay1aaSbaaeaacqWGTbqBcqWGHbqycqWG4baEaeqaaiabdoeadnaaBaaabaGaemiCaahabeaaaaGccqGGUaGlaaa@4B3D@

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 u¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyDauNbaebaaaa@2D60@ into Eq. (12) yields C_f ≈ c¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafm4yamMbaebaaaa@2D3C@ = 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 143031). 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.

We denote the gut 5-HT release rate R, where R is expressed per unit volume of the gut wall and includes all 5-HT released by the gut. Specifically, R includes the 5-HT that (i) is taken back up into gut cells, (ii) remains in the extracellular space of the gut wall, and (iii) diffuses into the blood circulation. If the gut 5-HT release rate fluctuates but homeostatic mechanisms keep it near some constant value R₀₀ > 0, then we can write

λ d R d t = R 00 − R , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdWwcfa4aaSaaaeaacqWGKbazcqWGsbGuaeaacqWGKbazcqWG0baDaaGccqGH9aqpcqWGsbGudaWgaaWcbaGaeGimaaJaeGimaadabeaakiabgkHiTiabdkfasjabcYcaSaaa@3AFE@

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 R₀₀). 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.

If the gut release rate is controlled by any of the discussed mechanisms,

λ d R d t = R 00 − R + f ( G , P ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdWwcfa4aaSaaaeaacqWGKbazcqWGsbGuaeaacqWGKbazcqWG0baDaaGccqGH9aqpcqWGsbGudaWgaaWcbaGaeGimaaJaeGimaadabeaakiabgkHiTiabdkfasjabgUcaRiabdAgaMjabcIcaOiabdEeahjabcYcaSiabdcfaqjabcMcaPiabcYcaSaaa@4207@

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.

Linearization of f(G, P) in the neighborhood of "normal" values of G and P (denoted G₀ and P₀, respectively) yields

f(G, P) = f(G₀, P₀) + α(G₀ - G) + β(P₀ - P),

where α≡−∂f/∂G|(G0,P0)≥0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaqGaaeaacqaHXoqycqGHHjIUcqGHsislcqGHciITcqWGMbGzcqGGVaWlcqGHciITcqWGhbWraiaawIa7amaaBaaaleaacqGGOaakcqWGhbWrdaWgaaadbaGaeGimaadabeaaliabcYcaSiabdcfaqnaaBaaameaacqaIWaamaeqaaSGaeiykaKcabeaakiabgwMiZkabicdaWaaa@41E6@ and β≡−∂f/∂P|(G0,P0)≥0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaqGaaeaacqaHYoGycqGHHjIUcqGHsislcqGHciITcqWGMbGzcqGGVaWlcqGHciITcqWGqbauaiaawIa7amaaBaaaleaacqGGOaakcqWGhbWrdaWgaaadbaGaeGimaadabeaaliabcYcaSiabdcfaqnaaBaaameaacqaIWaamaeqaaSGaeiykaKcabeaakiabgwMiZkabicdaWaaa@41FA@.

By denoting R₀ = R₀₀ + f (G₀, P₀) we obtain

λ d R d t = R 0 − R + α ( G 0 − G ) + β ( P 0 − P ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdWwcfa4aaSaaaeaacqWGKbazcqWGsbGuaeaacqWGKbazcqWG0baDaaGccqGH9aqpcqWGsbGudaWgaaWcbaGaeGimaadabeaakiabgkHiTiabdkfasjabgUcaRiabeg7aHjabcIcaOiabdEeahnaaBaaaleaacqaIWaamaeqaaOGaeyOeI0Iaem4raCKaeiykaKIaey4kaSIaeqOSdiMaeiikaGIaemiuaa1aaSbaaSqaaiabicdaWaqabaGccqGHsislcqWGqbaucqGGPaqkcqGGUaGlaaa@4B1E@

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).

The concentration of extracellular 5-HT in the gut wall increases due to synthesis and release of 5-HT by EC cells and neurons of the gut. It decreases due to two processes: (i) local 5-HT uptake by SERT (and perhaps by other, low-affinity transporters 1235) and (ii) 5-HT diffusion into gut blood capillaries. Suppose that the blood that has exited the heart through the aorta at time t reaches the gut at time t + s (s > 0). The decrease rate of extracellular 5-HT concentration in the gut wall due to the diffusion into blood capillaries is given, according to Fick's First Law, by

D S w Ω g [ G ( t + s ) − z g F ( t ) σ z g Q t o t ] = d g [ G ( t + s ) − F ( t ) σ Q t o t ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@682B@

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/(wΩ_g). Note that z_gF (t)/(σz_gQ_tot) is the concentration of free 5-HT in the blood plasma that arrives in the gut at time t + s (Fig. 1).

If all three discussed processes are taken into consideration,

d G d t = R ( t ) − k g G ( t ) − d g [ G ( t ) − F ( t − s ) σ Q t o t ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqWGKbazcqWGhbWraeaacqWGKbazcqWG0baDaaGccqGH9aqpcqWGsbGucqGGOaakcqWG0baDcqGGPaqkcqGHsislcqWGRbWAdaWgaaWcbaGaem4zaCgabeaakiabdEeahjabcIcaOiabdsha0jabcMcaPiabgkHiTiabdsgaKnaaBaaaleaacqWGNbWzaeqaaOWaamWaaeaacqWGhbWrcqGGOaakcqWG0baDcqGGPaqkcqGHsisljuaGdaWcaaqaaiabdAeagjabcIcaOiabdsha0jabgkHiTiabdohaZjabcMcaPaqaaiabeo8aZjabdgfarnaaBaaabaGaemiDaqNaem4Ba8MaemiDaqhabeaaaaaakiaawUfacaGLDbaacqGGSaalaaa@59B6@

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 1235.

We next consider the flow (mol/min) of free 5-HT in the blood circulation from the time blood exits the heart through the aorta (at time t) to the time it returns to the aorta after one circulation cycle (at time t + T; Fig. 1). Since blood transit times from organ to organ are relatively short (seconds), we will ignore 5-HT diffusion parallel to the flow. After the blood leaves the heart, a proportion (z_g) of the total cardiac output is routed to the gut and/or the liver. On arrival in the gut at time t + s (0 <s <T), the blood is replenished with new 5-HT synthesized in the gut wall. According to Fick's First Law, this flow of 5-HT into the blood is

D S w [ G ( t + s ) − z g F ( t ) σ z g Q t o t ] = d b [ G ( t + s ) − F ( t ) σ Q t o t ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@666A@

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 1314. 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 1314. 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 1314. 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.

Taking all these considerations together, the 5-HT flow that leaves the heart after one full circulation cycle is

F ( t + T ) = [ ( z g F ( t ) + d b [ G ( t + s ) − F ( t ) σ Q t o t ] ) θ h + θ v z n g F ( t ) ] θ p , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6BED@

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 F∧ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOrayKbaKaaaaa@2CFA@ 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) = F∧ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOrayKbaKaaaaa@2CFA@, and if F(t - s) ≈ F(t - s - x) = F∧ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOrayKbaKaaaaa@2CFA@ 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).

At the steady state, the platelet 5-HT concentration is (Appendix 2)

P ∧ = τ k p F ∧ σ Q t o t , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaacqGH9aqpjuaGdaWcaaqaaiabes8a0jabdUgaRnaaBaaabaGaemiCaahabeaacuWGgbGrgaqcaaqaaiabeo8aZjabdgfarnaaBaaabaGaemiDaqNaem4Ba8MaemiDaqhabeaaaaGccqGGSaalaaa@3D15@

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.

Solving Eqs. (16), (18), (20), and (21) at the steady state yields

P ∧ = S 1 + β P 0 S 2 + β , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaacqGH9aqpjuaGdaWcaaqaaiabdofatnaaBaaabaGaeGymaedabeaacqGHRaWkcqaHYoGycqWGqbaudaWgaaqaaiabicdaWaqabaaabaGaem4uam1aaSbaaeaacqaIYaGmaeqaaiabgUcaRiabek7aIbaacqGGSaalaaa@3BA0@

where

S₁ = R₀ + αG₀

and

S 2 = 1 τ k p [ σ Q t o t Θ ( K g d b + 1 Ω g ) + K g ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4uam1aaSbaaSqaaiabikdaYaqabaGccqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabes8a0jabdUgaRnaaBaaabaGaemiCaahabeaaaaGcdaWadaqaaiabeo8aZjabdgfarnaaBaaaleaacqWG0baDcqWGVbWBcqWG0baDaeqaaOGaeuiMde1aaeWaaeaajuaGdaWcaaqaaiabdUealnaaBaaabaGaem4zaCgabeaaaeaacqWGKbazdaWgaaqaaiabdkgaIbqabaaaaOGaey4kaSscfa4aaSaaaeaacqaIXaqmaeaacqqHPoWvdaWgaaqaaiabdEgaNbqabaaaaaGccaGLOaGaayzkaaGaey4kaSIaem4saS0aaSbaaSqaaiabdEgaNbqabaaakiaawUfacaGLDbaacqGGSaalaaa@5211@

where for brevity we defined

Θ ≡ 1 − z g θ h θ p − z n g θ v θ p θ h θ p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiMdeLaeyyyIOBcfa4aaSaaaeaacqaIXaqmcqGHsislcqWG6bGEdaWgaaqaaiabdEgaNbqabaGaeqiUde3aaSbaaeaacqWGObaAaeqaaiabeI7aXnaaBaaabaGaemiCaahabeaacqGHsislcqWG6bGEdaWgaaqaaiabd6gaUjabdEgaNbqabaGaeqiUde3aaSbaaeaacqWG2bGDaeqaaiabeI7aXnaaBaaabaGaemiCaahabeaaaeaacqaH4oqCdaWgaaqaaiabdIgaObqabaGaeqiUde3aaSbaaeaacqWGWbaCaeqaaaaaaaa@4D86@

and

K_g ≡ k_g + α.

In the derivation, we used the relationship d_g = d_b/Ω_g.

The values of the parameters can be approximated based on published experimental results (Table 1). Since little is known about the regulation of 5-HT release from the gut, we can initially assume that α = β = 0 (in this case, platelet 5-HT concentration is independent of G₀ and P₀). Plugging the parameter values into Eq. (22) yields P∧ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmiuaaLbaKaaaaa@2D0E@ = 2.40 · 10^-18 mol/platelet, or 4.23 · 10^-16g/platelet. Since the platelet concentration in the blood has been estimated to be 4.28 · 10⁸ platelets/mL 7, the obtained value is equivalent to the whole-blood 5-HT concentration of 1.02 μM or 0.18 μg/mL. These values are well within the range of normal 5-HT concentrations obtained in experimental studies (Fig. 2). Platelet 5-HT concentrations when α > 0 are plotted in Fig. 2.