Introduction

Most chemical and electrochemical diagnosis is based on detecting a marker compound and determining its concentration in a biological fluid. The marker compound is produced, or its concentration is increased/depleted above/below certain levels when a disease occurs. The diagnosis requires a highly sensitive and selective analytical method to determine the marker compound. Intensive research in this field produced various sensors and biosensors [1,2,3,4]. This kind of diagnostic can be seen as an extreme methodology whose opposite methodological approach is metabolomic. Here, the diagnosis is based on multicomponent analysis defining a metabolic profile to be characterized using chemometric techniques [5,6,7].

Several analytical strategies can be seen as intermediate between these two approaches. Among them are the use of an “external” marker compound such as methylene blue [8, 9] and the determination of non-specific metabolites [10, 11] such as tryptophan, serotonin, and other metabolites whose concentration in biological fluids is sensitive to diseases such as bladder cancer [12,13,14,15,16]. All these approaches have in common the need to record analyte-sensitive signals, which ideally are linearly dependent on the analyte concentration, a common assumption in recent literature [17,18,19,20,21,22,23,24,25,26,27,28].

The application of this analytical approach for diagnostic purposes involves several problems:

  1. 1.

    The concentration of the target analyte(s) in the biological fluid depends on water uptake, circadian cycles, medication, and other factors, thus varying between different emissions from the same individual.

  2. 2.

    The biological fluids constitute complex matrices where various interfering components may exist.

  3. 3.

    The electrochemical sensing processes can involve multistep pathways differing from the single, diffusion-controlled n-electron transfer process.

One paradigmatic example of problem 3 is the determination of neurotransmitter catecholamines (dopamine, noradrenaline) whose oxidation involves a multistep pathway initiated by the two-proton, two-electron oxidation of the parent o-catechol to the corresponding o-quinone, followed by a cyclization reaction [29, 30]. Multistep mechanisms also hold for the oxidation of uric acid [31], serotonin [32], and ascorbic acid [33], among others. The rough description of this process as a reversible-like, diffusion-controlled process catalyzed by the electrode modifier is relatively frequent in recent sensing literature [17,18,19,20,21,22,23,24,25,26,27,28]. Limitations imposed by diffusion [34, 35] and the apparent character of the catalytic effect at porous modified electrodes [36] are additional constraints to be accounted for sensing. These factors can determine deviations from linearity in the current/concentration calibration relationships.

In this context, reported studies have in common [17,18,19,20,21,22,23,24,25,26,27,28] (i) the use of the standard additions method assuming that multistep electrochemical mechanisms can be treated as diffusion-controlled processes and (ii) the use of biological samples where the analyte is absent or below the detection limit of the method, testing it in analyte-spiked samples. Common strategies are also to dilute the biological fluid [17,18,19,20,21,22] or use synthetic model fluids [23,24,25,26,27,28]. The former method requires a significant increase in sensor sensitivity without loss of selectivity, at the same time, while the second omits the presence in natural biological fluids of interfering components, such as serotonin and tryptophan metabolites, as well as the possible presence of foreign components, such as methotrexate, in medicated patients. As a result, despite extensive research, most reported sensors are not applied in clinical practice, as illustrated by the case of bladder cancer, whose golden standard is cystoscopy, an invasive technique [37].

In previous works [10, 11], we reported the suitability of analyzing the entire voltammetric record of urine (rather than detecting an individual marker compound) for the diagnosis of bladder cancer. Here, we study the possibility of expanding this approach to determine individual components in urine. That requires facing the above problems. Two general cases will be treated, the “static” one, when the voltammetric signals of the analyte and the interferent are coincident, but their electrochemical oxidation or reduction is independent, and the “dynamic” case, when there is interaction between these components and/or the intermediate species generated in their oxidation or reduction. This second case refers, to some extent, to the problems of variable optical paths existing in attenuated total reflectance—Fourier transform infrared spectroscopy and X-ray diffraction [38] and can be faced via asymptotic approaches [39]. Theoretical modeling was tested by studying the standard additions of serotonin and noradrenalin to human urine. These were selected as representative examples of compounds in urine and displaying complex electrochemistry potentially involving cross-reactions with other biological fluid components.

Experimental

Urine was collected from healthy volunteer patients randomly selected from the population performing routine analysis in the hospital context. Healthy volunteers were also randomly selected from the hospital population and signed the pertinent informed consent. For each patient, 10 mL of spontaneously produced urine was collected into sterilized tubes and stored at 5 °C for 1–6 h before electrochemical measurements. Serotonin and noradrenalin (Fluka) were used as received and diluted in acetic acid/sodium acetate (HAc/NaAc) aqueous buffer (Panreac) in concentration 0.25 M and pH 4.75 and/or urine diluted to its half with HAc/NaAc 0.50 M.

These were carried out at glassy carbon (BAS, MF 2012, geometrical area 0.072 cm2) (in the following, GCE) and gold (BAS, MF 2014, geometrical area 0.018 cm2) adapting a three-electrode tap to the urine-containing tube using a CH I 660c potentiostatic device (Cambria Scientific, Llwynhendy, Llanelli, Wales, UK). A platinum mesh auxiliary electrode and an Ag/AgCl (5 M NaCl) reference electrode completed the three-electrode arrangement. The working electrode was cleaned and activated using polishing cloths (BAS, West Lafayette, USA) with sequentially applied 1.0, 0.3, and 0.05 mm alumina in nanopure water. That was followed by sonication in a glass vessel for 10 min. Repeatability tests were performed thrice by measuring each urine sample and replicating such experiments with three working electrodes.

Results and discussion

Urine electrochemistry

Figure 1 depicts the square wave voltammograms recorded at the Au electrode on urine samples spontaneously emitted by healthy volunteers and patients diagnosed with bladder cancer at different stages. Upon scanning the potential from − 1.45 V vs. Ag/AgCl in the positive direction, a series of voltammetric peaks appear between 0.0 and 1.2 V, which can be attributed to the oxidation of uric acid, serotonin, tryptophan, creatinine, and/or their metabolites, including melatonin and/or indoleacetic acids [10, 11].

Fig. 1
figure 1

Square wave voltammograms at the Au electrode of urine samples spontaneously emitted by a–c healthy volunteers; d–f patients diagnosed with bladder cancer at pTaG3 (d), pT1G3 (e), and pT2G3 (f) stages. Potential scan initiated at − 1.45 V in the positive direction; potential step increment 4 mV; square wave amplitude 25 mV; frequency 5 Hz. The dotted arrows mark the direction of the potential scan

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Although the absolute values of the peak currents varied from one individual to another, the voltammetric profile was similar for healthy volunteers. Remarkably, the voltammograms for patients diagnosed with bladder cancer showed differing distinguishable profiles, as described in detail [10, 11], from those of healthy individuals.

The complexity of the voltammetric pattern, however, makes quantifying an individual analyte difficult. Figure 2 illustrates this problem by comparing the voltammetric response of serotonin and a urine sample after buffering with 0.25 M HAc/NaAc at pH 4.75. Whereas serotonin displays well-defined signals in square wave and linear potential scan voltammograms, the urine sample exhibits the characteristic overlapping peak profile illustrated in Fig. 1.

Fig. 2
figure 2

Voltammetric response at GCE of a, b a 0.71 mM solution of serotonin in 0.25 M HAc/NaAc and c, d a urine sample buffered with 0.25 M HAc/NaAc at pH 4.75. a, c Square wave voltammograms initiated at − 0.45 V in the positive direction, potential step increment 4 mV, square wave amplitude 25 mV, frequency 5 Hz. c, d Linear potential scan voltammograms initiated in the positive direction, potential scan rate 50 mV s.−1

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As previously commented, the quantification of serotonin or another individual component can be made employing the standard additions methodology. A typical example of standard additions is illustrated in Fig. 3 for square wave voltammograms at the Au electrode of urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 after additions of serotonin and noradrenalin in different concentrations. One can see that the primary urine signal becomes minimally displaced along the potential axis, whereas the peak current changes more or less significantly. However, as shown in Fig. 4, these last variations differ from the linear dependence between peak currents and concentrations expected for non-interacting systems (see below). Accordingly, theoretical modeling is needed, including interactions between biological fluid components.

Fig. 3
figure 3

Square wave voltammograms at the Au electrode of urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 after additions of a serotonin and b noradrenalin in different concentrations. Potential scan initiated at − 0.45 V in the positive direction, potential step increment 4 mV, square wave amplitude 25 mV, and frequency 5 Hz. u, parent urine buffered solution; concentrations of added serotonin and noradrenalin indicated in the graphs

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Fig. 4
figure 4

Variation of the peak current for the main urine anodic signal on the added concentration of noradrenalin in square wave voltammograms recorded at GCE (solid circles) and Au (circles) electrodes in urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 in conditions such as in Fig. 3. Averaged values from three replicate measurements

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Electrochemical processes

To rationalize voltammetric features, the complexity of the electrochemical pathways involved in the oxidation of analytes and interfering urine components has to be considered. Figure 5 depicts a simplified scheme for the electrochemical oxidation of noradrenaline and serotonin. As dopamine, noradrenaline initially experiences a two-electron, two-proton oxidation yielding o-quinone, followed by a cyclization reaction yielding leucoaminochrome. This species is more easily oxidized than the parent catecholamine and can experience further oxidation to aminochrome, competing with a disproportionation reaction [29, 30]. The oxidation of serotonin in aqueous electrolytes proceeds via an initial one-electron, one-proton oxidation step yielding a phenoxyl radical, which subsequently experiences a second one-electron one-proton oxidation to give a very reactive quinone imine. This species reacts with water to give 4,5-dihydroxytryptamine, which in turn experiences a two-electron, two-proton oxidation to tryptamine-4,5-dione [32]. The simplified schemes in Fig. 5 omit other coupled processes, such as disproportionation reactions of intermediates, dimerizations, and further polymerizations [29,30,31,32].

Fig. 5
figure 5

Simplified scheme for the electrochemical oxidation of serotonin (upper part, R = CH2-CH2-NH2) and noradrenalin (lower part). Two of the possible dimeric forms generated during the oxidation of serotonin are represented but other possible lateral reactions and polymerizations are omitted

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Concerning analytical approaches, it is pertinent to note that even the initial two-proton, two-electron oxidation step of noradrenalin and serotonin consists of a stepped process operating via ECEC or CECE sequences [29, 30, 32]. Semiquinones formed as intermediates in these processes can react with different species of biochemical interest [40], for instance, ascorbic acid [41]. For our purposes, the relevant point is that the usual analytical treatment of the oxidation of dopamine, uric acid, ascorbic acid, etc., as reversible-like, diffusion-controlled processes [17,18,19,20,21,22,23,24,25,26,27,28] is a rough oversimplification. In the following, we discuss the implications for quantifying of analytes in biological matrices of several possible cases adopting a semiempirical approximation.

Modeling

Non-interfering systems

Let us consider an electroactive species X that experiences a reversible, diffusion-controlled electrochemical process under determined electrochemical conditions. Under these conditions, the peak current of the voltammetric signal will be proportional to the concentration of X, cX,

$$I{\text{(X)}} = H_{{\text{X}}} c_{{\text{X}}}$$
(1)

where HJ is the electrochemical coefficient of response under the conditions of operation. Now, let us consider a biological fluid containing different electroactive species, X, Y, Z, … in concentrations cX, cY, …. These concentrations can also vary between different emissions of the same individual. Under fixed electrochemical conditions, the peak current of the different signals assumed to be reversible, diffusion-controlled, and able to be measured separately, will be proportional to the concentration of each one of the electroactive components, i.e.,

$$I{\text{(J)}} = h_{{\text{J}}} c_{{\text{J}}}$$
(2)

where hJ is the electrochemical coefficient of response characterizing each species in the biological fluid under the conditions of operation, and J = X, Y, Z,… are the existing electroactive species. In the biological fluid, complexation, co-diffusion, pH variations, and/or ionic strength effects may modify the electrochemical response of the different electroactive species. Then, the response coefficients in the biological fluid, hJ, in general will differ from those individually determined in J-containing blank solutions, HJ. The HJ coefficients can be determined using external calibration experiments, but the coefficients hJ will be different—and hence, unknown—for each individually emitted biological fluid.

When the X-localized signal is not overlapped with the signals displayed by other electroactive species in the biological fluid, the concentration of component X can be determined using a conventional standard additions method. That approach is systematically used in literature requiring high selectivity [17,18,19,20,21,22,23,24,25,26,27,28]. For this purpose, ideally, we add weighted amounts of X, reaching an added concentration \(c_{{\text{X}}}^{*}\). Then, the intensity of the X-peak should increase as

$$I{\text{(X)}} = h_{{\text{X}}} c_{{\text{X}}} + h_{{\text{X}}} c_{{\text{X}}}^{*} \quad$$
(3)

so that the representation of I(X) vs. \(c_{{\text{X}}}^{*}\) will fit a straight line of slope hX and ordinate at the origin hXcX. Accordingly, the unknown concentration of X in the biological fluid can be calculated as cX = (intercept)/(slope).

Non-interacting systems

The above treatment assumed that (i) all electrochemical processes behave reversibly under diffusion control without mechanistic complications and (ii) the different species are independently oxidized or reduced with no interaction between them. Although the first condition does not apply in cases such as those condensed in Figs. 2, 3, and 4, it will be assumed that, under several experimental conditions, it can be taken as a reasonable approximation, as is customary in sensing literature [17,18,19,20,21,22,23,24,25,26,27,28].

In the following, several possible cases of interference will be described, assuming that the electroactive analyte X produces a voltammetric signal strongly overlapped with that of an electroactive component Z existing (or formed, vide infra) in the biological fluid.

Now, let us consider the case in which the analyte signal is superimposed with the signal of other components (represented by a unique component Z), which occurs for most analytes in urine. That can be seen in Fig. 2, where the square wave and linear potential scan voltammetric responses of serotonin in acetate buffer at pH 4.75 can be compared with the response of urine buffered at the same pH. Serotonin displays a well-defined anodic peak at 0.47 V vs. Ag/AgCl, which is essentially coincident with the corresponding signals for the oxidation of uric acid and other metabolites dominating the voltammetry of urine, characterized by a prominent anodic peak at 0.55 V under equivalent conditions [10, 11].

Ideally (again assuming reversibility and diffusion control), the peak current measured at 0.55 V will contain the contribution of all species which are oxidized at this potential. If there is no interaction between the X and Z components, the current Io(X) in the native solution will be

$$I_{{\text{o}}} {\text{(X)}} = h_{{\text{X}}} c_{{\text{X}}} + h_{{\text{Z}}} c_{{\text{Z}}}$$
(4)

where the hX and hZ coefficients represent the response of the respective component at the indicated potential. Under the above assumptions, the current measured after a given standard addition of X where the concentration of Z remains unaltered will be

$$I{\text{(X)}} = h_{{\text{X}}} (c_{{\text{X}}} + c_{{\text{X}}}^{*} ) + h_{{\text{Z}}} c_{{\text{Z}}}$$
(5)

so that the I(X) vs. \(c_{{\text{X}}}^{*}\) representation will be a straight line passing by the origin of slope hX and intercept hXcX + hZcZ. Since the hZcZ is unknown, the problem concentration cX cannot be directly determined. Ideally, the hZcZ term can be estimated from the representation of the I(X)/[I(X) − Io(X)] ratio, which tends, when a significant excess of X is added, to

$$\frac{{I{\text{(X)}}}}{{I(X) - I_{o} (X)}} \approx \frac{{h_{{\text{Z}}} c_{Z} }}{{h_{X} c_{X}^{*} }}$$
(6)

When \(h_{{\text{X}}} (c_{{\text{X}}} + c_{{\text{X}}}^{*} )\) >  > \(h_{{\text{Z}}} c_{{\text{Z}}}\), Eq. (4) tends to Eq. (3), and the usual standard additions formalism can be applied. Alternatively, there is a possibility of taking currents at two different potentials and then applying H-point standard addition methods [42,43,44,45].

Chemical reaction and co-diffusion

Let us first consider that the analyte X reacts with an electrochemically silent component of the biological fluid, yielding a second electroactive species, Z,

$$\mathrm{X }+\mathrm{ W }\to \mathrm{ Z}$$
(7)

Assuming that at the beginning of the experiment the reaction is at equilibrium, the equilibrium concentrations of species are dictated by the equilibrium constant K and the initial concentration of X, cX. This situation refers to the well-known scheme of two species in equilibrium simultaneously reduced or oxidized [46,47,48,49,50]. Suppose the reaction rate is slower than the rate of electron transfer processes experienced by both X and Z. In that case, the net peak current equals the sum of the diffusion currents for each one of the components separately. If the chemical reaction is always at equilibrium [49] and the system is under conditions of diffusion control, the “diffusive” terms for the different species can be considered additive. Then, co-diffusion occurs so that the net current will be proportional to the square root of an effective, averaged diffusion coefficient, Deff [46,47,48,49,50].

In the case of independent diffusion, introducing the equilibrium constant, K, and assuming that W is in significant excess, the effective concentrations of X and Z ([X], [Z], respectively) will be [49]

$$\quad [{\text{X}}] = \frac{{c_{{\text{X}}} }}{{1 + Kc_{{\text{W}}} }}\;;\quad \quad [{\text{Z}}] = \frac{{Kc_{{\text{W}}} c_{{\text{X}}} }}{{1 + Kc_{{\text{W}}} }}$$
(8)

where cW is the concentration of the biological component in excess. Accordingly, the intensity of the analytical signal in the original solution will be

$$I_{{\text{o}}} {\text{(X)}} = \frac{{h_{{\text{X}}} c_{{\text{X}}} }}{{1 + Kc_{{\text{W}}} }} + \frac{{h_{{\text{Z}}} Kc_{{\text{W}}} c_{{\text{X}}} }}{{1 + Kc_{{\text{W}}} }}\quad \quad$$
(9)

Under the conditions mentioned above of diffusive control, the hJ coefficients are proportional to the square root of the diffusion coefficient of the respective species, DJ; i.e., \(h_{J} = f_{J} D_{J}^{1/2}\). In a standard addition experiment, the biological fluid is enriched with a known additional concentration of X, \(c_{{\text{X}}}^{*}\). The new measured peak current will be

$$I{\text{(X)}} = \frac{{h_{{\text{X}}} + h_{{\text{Z}}} Kc_{{\text{W}}} }}{{1 + Kc_{{\text{W}}} }}c_{{\text{X}}} + \frac{{h_{{\text{X}}} + h_{{\text{Z}}} Kc_{{\text{W}}} }}{{1 + Kc_{{\text{W}}} }}c_{{\text{X}}}^{*}$$
(10)

Equation (10) predicts a linear variation of I(X) on \(c_{{\text{X}}}^{*}\) with slope \((h_{{\text{X}}} + h_{{\text{Z}}} Kc_{{\text{W}}} )/(1 + Kc_{{\text{W}}} )\) and intercept \((h_{{\text{X}}} + h_{{\text{Z}}} Kc_{{\text{W}}} )c_{{\text{X}}} /(1 + Kc_{{\text{W}}} )\), allowing for cX to be determined.

In the case of co-diffusion, the averaged diffusion coefficient can be expressed as

$$D_{{{\text{eff}}}} = \frac{{[X]D_{{\text{X}}} + [Z]D_{{\text{Z}}} }}{{c_{{\text{X}}} + c_{{\text{Z}}} }}$$
(11)

Under the assumed conditions of diffusive control, \({\mathrm{h}}_{\mathrm{J}}={\mathrm{f}}_{\mathrm{J}}{\mathrm{D}}_{\mathrm{J}}^{1/2}\) \(h_{J} = f_{J} D_{J}^{1/2}\), and the effective electrochemical constant heff equals to \({\mathrm{f}}_{\text{eff}}{\mathrm{D}}_{\text{eff}}^{1/2}\) \({\text{f}}_{{{\text{eff}}}} {\text{D}}_{{{\text{eff}}}}^{{1/2}}\). Then, Eq. (11) can be rewritten as

$${\mathrm{D}}_{\mathrm{eff}}={\left(\frac{{\mathrm{h}}_{\mathrm{eff}}}{{\mathrm{f}}_{\mathrm{eff}}}\right)}^{2}=\frac{{\left(\frac{{\mathrm{h}}_{\mathrm{X}}}{{\mathrm{f}}_{\mathrm{X}}}\right)}^{2}+{\mathrm{Kc}}_{\mathrm{W}}{\left(\frac{{\mathrm{h}}_{\mathrm{Z}}}{{\mathrm{f}}_{\mathrm{Z}}}\right)}^{2}}{1+{\mathrm{Kc}}_{\mathrm{W}}}$$
(12)

The peak current for the X-localized process will be I(X) = heff[X] and, assuming that heff = hX = hZ, the peak currents measured in the parent solution and the analyte-enriched one will be

$$I_{{\text{o}}} ({\text{X}}) = \left( {\frac{{h_{{\text{X}}}^{2} + Kc_{{\text{W}}} h_{{\text{Z}}}^{2} }}{{1 + Kc_{{\text{W}}} }}} \right)^{1/2} c_{{\text{X}}}$$
(13)
$$I({\text{X}}) = \left( {\frac{{h_{{\text{X}}}^{2} + Kc_{{\text{W}}} h_{{\text{Z}}}^{2} }}{{1 + Kc_{{\text{W}}} }}} \right)^{1/2} c_{{\text{X}}} + \left( {\frac{{h_{{\text{X}}}^{{2}} + Kc_{{\text{W}}} h_{{\text{Z}}}^{{2}} }}{{{1} + Kc_{{\text{W}}} }}} \right)c_{{\text{X}}} *$$
(14)

respectively. This equation predicts a linear variation of I(X) on \(c_{{\text{X}}}^{*}\) of slope \({(}h_{{\text{X}}}^{{2}} + Kc_{{\text{W}}} h_{Z}^{2} )^{1/2} /(1 + Kc_{{\text{W}}} )^{1/2}\) and intercept \({(}h_{{\text{X}}}^{{2}} + Kc_{{\text{W}}} h_{Z}^{2} )^{1/2} c_{{\text{X}}} /(1 + Kc_{{\text{W}}} )^{1/2}\), again allowing for the determination of cX.

The above schemes could be operative for processes such as the reaction with water of the quinone imine generated in the initial oxidation of serotonin (see Fig. 5).

Adduct formation

Let us consider that there is some interaction between analyte X and any species Z of the biological fluid. In the simplest case, they form a non-electroactive adduct W,

$$\mathrm{X }+\mathrm{ Z}\leftrightarrows\mathrm{ W}$$
(15)

Assuming that the equilibrium of adduct formation is established in the biological fluid and that Z is in significant excess, the effective concentration of W and X (respectively, [W], [X]) will be

$$[{\text{W}}] = \frac{{Kc_{{\text{X}}} c_{{\text{Z}}} }}{{1 + Kc_{{\text{Z}}} }}\;{; }[{\text{X}}] = \frac{{c_{{\text{X}}} }}{{1 + Kc_{{\text{Z}}} }}\quad \quad$$
(16)

Accordingly, the intensity of the analytical signal in the original solution will be

$$I_{{\text{o}}} {\text{(X)}} = \frac{{h_{{\text{X}}} c_{{\text{X}}} }}{{1 + Kc_{{\text{Z}}} }} + h_{{\text{Z}}} c_{{\text{Z}}} \quad \quad \quad$$
(17)

The additions of X-standard in concentration \(c_{{\text{X}}}^{*}\) will determine a change in the peak current to

$$I{\text{(X)}} = h_{{\text{Z}}} c_{{\text{Z}}} + \frac{{h_{{\text{X}}} c_{{\text{X}}} }}{{1 + Kc_{{\text{Z}}} }} + \frac{{h_{{\text{X}}} c_{{\text{X}}}^{*} }}{{{1} + Kc_{{\text{Z}}} }}$$
(18)

According to this equation, the I(X) vs. \(c_{{\text{X}}}^{*}\) representation will be a straight line of slope \(h_{{\text{X}}} {/(1} + Kc_{{\text{Z}}} {)}\) and ordinate at the origin \(h_{{\text{Z}}} c_{{\text{Z}}} + h_{{\text{X}}} c_{{\text{X}}} {/(1} + Kc_{{\text{Z}}} {)}\). Now, the determination of cX requires the knowledge of the gZcZ term.

The linearity is lost when the stoichiometry of the process of adduct formation is not 1:1. For a process such as

$$qX+\mathrm Z\leftrightarrows\mathrm W$$
(19)

in the presence of a significant excess of Z, the concentration of W satisfies the relationship

$$\frac{{[{\text{W}}]^{\frac{1}{q}} }}{{c_{{\text{X}}} - q[{\text{W}}]}} = (c_{{\text{Z}}} K)^{\frac{1}{q}}$$
(20)

ultimately tending to \(c_{{\text{X}}} /q\). In these conditions, the concentration of X tends to \(\left( {c_{{\text{X}}} /qKc_{{\text{Z}}} } \right)^{\frac{1}{q}}\) and the peak current recorded in the biological fluid will be

$$I_{{\text{o}}} (X) = h_{{\text{Z}}} c_{{\text{Z}}} + h_{{\text{W}}} \left( {\frac{{c_{{\text{X}}} }}{q}} \right) + h_{{\text{X}}} \left( {\frac{{c_{{\text{X}}} }}{{qKc_{{\text{Z}}} }}} \right)^{\frac{1}{q}}$$
(21)

In a standard addition experiment, the recorded current will be

$$I({\text{X}}) = h_{{\text{Z}}} c_{{\text{Z}}} + h_{{\text{W}}} \left( {\frac{{c_{{\text{X}}} + c_{{\text{X}}}^{*} }}{q}} \right) + h_{{\text{X}}} \left( {\frac{{c_{{\text{X}}} + c_{{\text{X}}}^{*} }}{{qKc_{{\text{Z}}} }}} \right)^{\frac{1}{q}}$$
(22)

That results in a non-linear variation of I(X) on \(c_{{\text{X}}}^{*}\). This scheme could be applied to various reactions between electrochemically generated intermediates and the original components of urine.

Pseudo-kinetically driven interaction

Let us consider that X and Z are not independently reduced/oxidized. In the cases of serotonin, dopamine, noradrenalin, etc., intermediate species generated during the electrochemical oxidation processes can experience cross-reactions between them and/or between the parent compounds.

The following treatment assumes that the measured signal reflects the original concentrations of X and Z accompanied by an interaction term. By analogy to formal kinetics, that can be represented by an intensity term depending on both cX and cZ, as

$$I_{{\text{o}}} {\text{(X)}} = h_{{\text{X}}} c_{{\text{X}}} + h_{Z} c_{Z} + g_{{{\text{XZ}}}} c_{{\text{X}}}^{a} c_{{\text{Z}}}^{b}$$
(23)

where the exponents a and b represent the kinetics of the process and gXZ represents the corresponding coefficient of electrochemical response (notice that the units of gXZ differ from those of the coefficients hX, hZ). The peak current measured after the addition of the X-standard of concentration \(c_{{\text{X}}}^{*}\) will be

$$I{\text{(X)}} = h_{{\text{X}}} (c_{{\text{X}}} + c_{X}^{*} ) + h_{Z} c_{Z} + g_{{{\text{XZ}}}} (c_{{\text{X}}} + c_{X}^{*} - y)^{a} (c_{{\text{Z}}} - y)^{b}$$
(24)

where y represents the interaction advance under the conditions of operation. At the beginning of the standard addition experiment, i.e., at low \(c_{{\text{X}}}^{*}\) values, one can expect that y will be low, and the I(X) − Io(X) difference can be approximated by the expression

$$I{\text{(X) - }}I_{{\text{o}}} {\text{(X)}} = h_{{\text{X}}} c_{X}^{*} + g_{{{\text{XZ}}}} \left[ {(c_{{\text{X}}} + c_{X}^{*} )^{a} - c_{{\text{X}}}^{a} } \right]c_{{\text{Z}}}^{b} \quad \quad \quad$$
(25)

That is a non-linear function of \(c_{{\text{X}}}^{*}\). In the case of a large excess of X, the peak current difference tends to

$$I{\text{(X) - }}I_{{\text{o}}} {\text{(X)}} = h_{{\text{X}}} c_{X} * + g_{{{\text{XZ}}}} (c_{X}^{*} )^{a} c_{{\text{Z}}}^{b} \quad \quad \quad$$
(26)

or, equivalently,

$$\frac{{I{\text{(X) - }}I_{{\text{o}}} {\text{(X)}}}}{{c_{X}^{*} }} = g_{{\text{X}}} + g_{{{\text{XZ}}}} (c_{X}^{*} )^{a - 1} c_{{\text{Z}}}^{b} \quad$$
(27)

This equation predicts a potential variation of the \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\) which can experimentally be tested. In the case of a = 1, Eq. (27) predicts a linear variation of the \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\), which can also be tested experimentally.

Application to quantification of serotonin and noradrenalin in urine

The above modeling was applied for quantifying noradrenalin and serotonin in urine. As expected, our experimental data for noradrenalin and serotonin do not fit the expected parameters for single, uncomplicated diffusion-controlled reversible oxidation processes. Since no adsorption-characteristic features (symmetric peaks in linear potential scan voltammetry, increasing peak currents in repetitive voltammetry) were detected, one can conclude that the observed voltammetry is consistent with the multistep pathways in Fig. 5. However, peak current vs. concentration plots in blank solutions were satisfactorily fitted to straight lines suggesting that, under several conditions (and as unanimously made in literature [17,18,19,20,21,22,23,24,25,26,27,28]), the processes can reasonably be approximated to diffusion-controlled ones. That can be seen in Fig. 6, where the calibration graph recorded for serotonin in urine-free HAc/NaAc buffer is depicted. The same figure illustrates the variation of I(X) on \(c_{{\text{X}}}^{*}\) obtained for serotonin-enriched urine samples in the corresponding standard addition experiment. A curved path, indicative of existing interactions, is obtained. The external calibration data have been shifted along the peak current axis to facilitate comparing the above graphs. Apparently, the initial slope of the I(X) vs. \(c_{{\text{X}}}^{*}\) curve in urine is close to the slope of the linear graph in urine-free solutions.

Fig. 6
figure 6

Variation of the peak current for the main urine anodic signal on the added concentration of serotonin in square wave voltammograms recorded at Au electrode in urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 in conditions such as in Fig. 3 (solid circles) and the same variation for the peak current for serotonin oxidation in blank experiments in 0.25 M HAc/NaAc at pH 4.75 (circles). These have been vertically displaced to show the hypothetical calibration graph in urine in the ideal case of no interactions. Averaged values form three replicate measurements and tendency lines corresponding to linear and polynomial fit, respectively, of experimental data

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Figure 7 shows the variation of the \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\) for square wave voltammograms recorded in noradrenalin-enriched urine samples buffered with 0.25 M HAc/NaAc at pH 4.75. Remarkably, the data points for glassy carbon and gold electrodes are identical and can satisfactorily be fitted (correlation coefficient of 0.9998) to a potential function such as Eq. (27) taking a = 0.14 ± 0.02.

Fig. 7
figure 7

Variation of the \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\) for square wave voltammograms recorded at GC (solid circles) and Au (circles) electrodes in noradrenalin-enriched urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 in conditions such as in Fig. 3. The continuous line corresponds to the potential fit of both (indistinguishable) data sets. Error bars omitted for clarity

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The variation of the \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\) for serotonin data in Fig. 6 is depicted in Fig. 8. Again, experimental data can satisfactorily be fitted to a potential function (r = 0.992) reproduced by Eq. (27) taking a = 0.24 ± 0.03, a value close to that estimated for noradrenaline.

Fig. 8
figure 8

Variation of the \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\) for square wave voltammograms recorded at Au electrode in serotonin-enriched urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 in conditions such as in Fig. 3. The continuous line corresponds to the potential fit of experimental data

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Interestingly, the interaction term introduced in Eqs. (23–27) can be eventually taken as negative. That refers to a situation in which the intermediate species were electrochemically silent so that the interacting system decreases the measured currents. This situation is illustrated in Fig. 9 for the standard additions of serotonin to a urine sample buffered with 0.25 M HAc/NaAc at pH 4.75 recorded at the glassy carbon electrode (GCE). In contrast with the voltammograms of the same samples recorded at the Au electrode (Fig. 3a), the peak current of the main anodic peak ca. 0.60 V decreases on increasing the concentration of added serotonin. Pertinent peak current data are shown in Fig. 10, where the I(X) vs. \(c_{{\text{X}}}^{*}\) (a) and \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) vs. \(c_{{\text{X}}}^{*}\) (b) plots are depicted. As in the case of the gold electrode (Fig. 7), a satisfactory fit of experimental data to a potential function such as predicted by Eq. (27) is obtained. The value of a calculated from GCE data (0.20 ± 0.04) is consistent with that estimated from data at Au electrode (0.24 ± 0.03).

Fig. 9
figure 9

Square wave voltammograms at GCE of urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 after additions of serotonin in different concentrations. Potential scan initiated at − 0.45 V in the positive direction, potential step increment 4 mV, square wave amplitude 25 mV, and frequency 5 Hz. u, parent urine buffered solution; concentrations of added serotonin indicated in the graph

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Fig. 10
figure 10

Variation of a I(X) and b \([I({\text{X}}) - I_{{\text{o}}} ({\text{X}})]/c_{{\text{X}}}^{*}\) ratio on \(c_{{\text{X}}}^{*}\) for square wave voltammograms recorded at GCE in serotonin-enriched urine samples buffered with 0.25 M HAc/NaAc at pH 4.75 in conditions such as in Fig. 3. The continuous lines correspond to the fit of experimental data to potential functions

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However, it is pertinent to note that the semiempirical, operational treatment described here depicts a simplified scenario for the involved electrochemistry. A complete, rigorous description of such phenomena should include the consideration of diffusion mass transport, electrode kinetics, kinetics, etc.

Conclusions

The appearance of non-linear effects in the voltammetric determination of analytes in biological fluids is discussed. Apart from the deviations from reversibility and diffusion control, ideal conditions for voltammetric sensing, these can be attributed, at least partially, to interactions between the analyte and the fluid components. Various interaction models are described based on the equilibrium-like formation of non-electroactive adducts and the reaction under kinetic-like control, optionally including co-diffusion effects. These processes can involve the parent compounds and the intermediate species generated during electrochemical sensing. Application to the determination of serotonin and noradrenalin in buffered urine using the standard additions strategy provides non-linear results in satisfactory agreement with the theoretical modeling. These features evidence the need to consider electrochemical mechanisms for electrochemical sensing accurately.