Modeling Microelectrode Biosensors: Free-flow Calibration Can Substantially Underestimate Tissue Concentrations

biosensors: free-flow calibration can substantially underestimate tissue concentrations.trode amperometric biosensors are widely used to measure concentrations of analytes in solution and tissue including acetylcholine, adenosine, glucose, and glutamate. A great deal of experimental and modeling effort has been directed at quantifying the response of the biosensors themselves; however, the influence that the macroscopic tissue environment has on biosensor response has not been subjected to the same level of scrutiny. Here we identify an important issue in the way microelectrode biosensors are calibrated that is likely to have led to underestimations of analyte tissue concentrations. Concentration in tissue is typically determined by comparing the biosensor signal to that measured in free-flow calibration conditions. In a free-flow environment the concentration of the analyte at the outer surface of the biosensor can be considered constant. However, in tissue the analyte reaches the biosensor surface by diffusion through the extracellular space. Because the enzymes in the biosensor break down the analyte, a density gradient is set up resulting in a significantly lower concentration of analyte near the biosensor surface. This effect is compounded by the diminished volume fraction (porosity) and reduction in the diffusion coefficient due to obstructions (tortu-osity) in tissue. We demonstrate this effect through modeling and experimentally verify our predictions in diffusive environments. NEW & NOTEWORTHY Microelectrode biosensors are typically calibrated in a free-flow environment where the concentrations at the biosensor surface are constant. However, when in tissue, the analyte reaches the biosensor via diffusion and so analyte breakdown by the biosensor results in a concentration gradient and consequently a lower concentration around the biosensor. This effect means that naive free-flow calibration will underestimate tissue concentration. We develop mathematical models to better quantify the discrepancy between the calibration and tissue environment and experimentally verify our key predictions. MICROELECTRODE BIOSENSORS are valuable experimental tools for accurate measurement of analytes in real time, both in vitro and in vivo (Dale et al. 2005). Biosensors have been used to measure neurotransmitters and neuromodulators including glu-Many mi-croelectrode biosensors developed for brain tissue use oxida-tive enzymes followed by detection via fixed-potential amper-ometry. Such biosensors are typically formed of a platinum or carbon fiber core on which a conductive polymer matrix such as pyrrole or paraphenylene is electrochemically deposited. The thick layer biosensor designs considered in this paper are made when the enzyme is entrapped within the free volume of the polymer matrix (Llaudet et al. 2003) or …

Microelectrode amperometric biosensors are widely used to measure concentrations of analytes in solution and tissue including acetylcholine, adenosine, glucose and glutamate.A great deal of experimental and modelling effort has been directed at quantifying the response of the biosensors themselves; however, the influence that the macroscopic tissue environment has on biosensor response has not been subjected to the same level of scrutiny.Here we identify an important issue in the way microelectrode biosensors are calibrated that is likely to have led to underestimations of analyte tissue concentrations.Concentration in tissue is typically determined by comparing the biosensor signal to that measured in free-flow calibration conditions.In a free-flow environment the concentration of the analyte at the outer surface of the biosensor can be considered constant.However, in tissue the analyte reaches the biosensor surface by diffusion through the extracellular space.Because the enzymes in the biosensor break down the analyte, a density gradient is set up resulting in a significantly lower concentration of analyte near the biosensor surface.This effect is compounded by the diminished volume fraction (porosity) and reduction in the diffusion coefficient due to obstructions (tortuosity) in tissue.We demonstrate this effect through modelling and experimentally verify our predictions in diffusive environments.

NEW & NOTEWORTHY
Microelectrode biosensors are typically calibrated in a free-flow environment where the concentrations at the biosensor surface are constant.However, when in tissue the analyte reaches the biosensor via diffusion and so analyte breakdown by the biosensor results in a concentration gradient and consequently a lower concentration around the biosensor.This effect means that naive free-flow calibration will underestimate tissue concentration.We develop mathematical models to better quantify the discrepancy between the calibration and tissue environment and experimentally verify our key predictions.
Microelectrode biosensors are valuable experimental tools for accurate measurement of analytes in real time, both in vitro and in vivo (Dale et al 2005).Biosensors have been used to measure neurotransmitters and neuromodulators including; glutamate (Hu et al 1994;Oldenziel et al 2006;Tian et al 2009), acetylcholine (Bruno et al 2006;Zhang et al 2010), adenosine triphosphate (Llaudet et al 2005;Frenguelli et al 2007;Gourine et al 2008;Lalo et al 2014;Lopatář et al 2015;Wells et al 2015), glucose (Lowry et al 1998;Dash et al 2013), adenosine, inosine and hypoxanthine (Llaudet et al 2003;Klyuch et al 2012;Dale 2013;Van Gompel et al 2014;Wall and Richardson 2015;Frenguelli and Wall 2016).Many microelectrode biosensors developed for brain tissue use oxidative enzymes followed by detection via fixed-potential amperometry.Such biosensors are typically formed of a platinum or carbon fibre core on which a conductive polymer matrix such as pyrrole or paraphenylene is electrochemically deposited.The thicklayer biosensor designs considered in this paper are made when the enzyme is entrapped within the free volume of the polymer matrix, (Llaudet et al 2003) or bonded with the polymer matrix (Kotanen et al 2014).The relevant feature of enzymatic biosensors for this study is that they break down and remove the quantity that they are measuring.
In order to characterise the response of microelectrode biosensors and assist in their design, extensive mathematical and computational modelling has been used (Cambiaso et al 1996;Lowry et al 1998;Rinken and Tenno 2001), quantifying the influence of substrate and product inhibition (Simelevicius and Baronas 2010; Simelevicius and Baronas 2011), geometry (Stikoniene et al 2010) and enzyme kinetics (Ivanauskas et al 2008;Simelevicius et al 2012).However, interactions of the bulk properties of tissue with the biosensor, and how this scenario is distinct to calibration conditions, does not yet appear to have been fully considered.
Biosensors measure tissue concentrations of analytes by comparing the signal in tissue to that in calibration conditions.The biosensor is calibrated in a standard concentration of analyte, typically in free-flow conditions where the concentration of the analyte at the outer surface of the biosensor can be considered constant because any analyte broken down by the biosensor is rapidly replaced.Details of biosensor calibration are discussed in Frenguelli and Wall (2016).The free-flow calibration conditions differ substantially to those in tissue, where the analyte diffuses to reach the biosensor.Because the biosensor breaks down the analyte, it can be expected that a concentration gradient will be set up with a lower concentration near the biosensor surface than in the bulk tissue.The reduced volume fraction (porosity) and reduced diffusion coefficient due to obstructions (tortuosity) in tissue (Sykova and Nicholson 2008) will compound these effects.
Here we model this effect in free-flow, non-tissue and tissue diffusive environments using an idealised description 1 Articles in PresS.J Neurophysiol (December 7, 2016).doi:10.1152/jn.00788.2016 Copyright © 2016 by the American Physiological Society.
of a single-enzyme electrochemical biosensor.The model does indeed predict that in diffusive environments a density gradient is established with a reduced concentration near the biosensor: this central result is experimentally verified using single-enzyme biosensors (glucose and hypoxanthine) in agar blocks.The mathematical modeling provides a scaling factor that quantifies the discrepancy between freeflow and diffusive conditions.Though the scaling factor is strongly dependent on the properties of the particular analyte and tissue that is being investigated, it is apparent from its functional form that free-flow calibration will lead to significant underestimates of tissue concentrations.

Concentrations, fluxes and diffusion
Concentrations.The concentrations considered are those in the volume fraction α where the compounds of interest (the analyte A or electro-active breakdown product H) can diffuse freely.Fluxes.In certain regions, such as within tissue, the extracellular space is highly tortuous which has the effect of reducing the diffusion coefficient.For example, for a region denoted by b we use the diffusion permeability θ b such that the effective diffusion coefficient is D Ab = θ b DA where DA is the free diffusion coefficient for a particular compound (Sykova and Nicholson 2008).The flux uses the total concentration per unit volume (αA, for example) so for the analyte would be the radial flux in the region b.Boundary conditions.At the interface between two regions, for example b and g, the concentrations (in the respective fractions of free space) at the boundary are matched as well as the fluxes across the boundary.For example for a boundary at, say, a radius r2 the continuity conditions would be Note that the free diffusion coefficient DA cancels from both sides in the flux condition.Continuity at interfaces.For cylindrical coordinates in which only the radial variable r is considered we have ) . (3) Neglecting the depth variable is a fair approximation for cylindrical boundary conditions and biosensors that are long relative to their radius, as is the case here.

Description of biosensor
The biosensor takes the form of a cylinder comprising an inner electrode core enveloped by an enzyme layer.The core radius is r1, the outer surface of the enzyme layer is at r2 and the length is z b (Fig. 1A) .Within the enzyme layer the analyte A is broken down into an electrically inactive product (which we ignore) and an electrically active product H202 hydrogen peroxide.It is this latter product that is measured at the electrode surface at radius r1.At the electrode we have so that two electrons are liberated for each H202 molecule.
There are many characteristics used to quantify and compare biosensors (Baronas et al 2009); however, here the current measured will be used.The current from the biosensor is equal to the charge on two electrons times the core area times the H202 flux per unit area at the core.
where Faraday's constant F = 96485Cmol −1 is the charge on a mole of electrons, 2πr1z b is the surface area of the biosensor core and JH is the H202 flux (note that contribution to the current coming from the surface area πr 2 1 of the end of the biosensor has been ignored -this is a reasonable approximation for the length-radius ratio of the biosensors considered here).

Model of calibration condition
For this condition the biosensor is placed in a free-flow environment with a constant concentration A * c of the analyte (see Fig. 1A).Within the enzyme layer of the biosensor the analyte diffuses from the surface, with diffusion coefficient D Ab = θ b DA and is broken down at a constant rate v b into H202 which diffuses within the enzyme layer with coefficient D Hb = θ b DH.More complex enzyme kinetics have previously been considered (Baronas et al 2009), but biosensors used in tissue are designed to operate within their linear regime so a linear-rate form is used here.The analyte and H202 concentrations A b (r) and H b (r) in the biosensor enzyme layer therefore obey At the core radius r1 there is zero flux of the analyte and H202 is rapidly removed and so has zero concentration At the surface of the biosensor at r2 there is continuity of analyte concentration and zero concentration of the breakdown product, because it is rapidly washed away in the free-flow environment.Hence, the following boundary conditions hold at the outer surface of the biosensor enzyme layer Finally, the current measured on the biosensor is proportional to minus the radial flux H202 at the core (radius r1) for a biosensor of length z b (end effects are ignored in this model).

Model of diffusive conditions in agar
For this condition we consider the biosensor embedded within a long cylindrical block of agar that is immersed in a bathing medium with concentration A * g at the agar outer surface (see Fig. 1B).The motion of the analyte Ag and the breakdown product Hg within the agar are diffusive with diffusion coefficients DAg and DHg respectively.The free-volume fraction is αg and the diffusion permeability is θg.The equations of motion are therefore Within the biosensor enzyme layer the analyte and H202 obey equations (6, 7) with boundary conditions at the core radius r1 given by equation set (8) and the biosensor current by equation (10).However, at the biosensor surface we require continuity so that with identical continuity conditions for H b and Hg at r2. Similarly at the surface of the agar we have the conditions where it is assumed that there is no breakdown product H in the bathing medium because any that diffuses out is rapidly washed away.2008) corresponding to diffusion in the extracellular space only.If the analyte can enter and leave cells, this generally leads to a more complex model which is beyond the scope of the current paper.The second factor that affects the diffusive motion of the analyte is the tortuosity λ of the tissue resulting from the complex microscopic structure around which the analyte diffuses.This can be modelled by using a diffusion coefficient DAt = θtDA for the analyte, that is reduced from that in a free solution DA by a permeability factor θt = 1/λ 2 .For brain tissue the tortuosity is quoted as λ = 1.6 (Sykova and Nicholson 2008) giving θt ≃ 0.4.We assume that the electro-active breakdown product H is very rapidly removed from tissue (as is the case for hydrogen peroxide).Its concentration is therefore considered to be zero throughout the tissue.

Model of biosensor in tissue
The tissue is considered to have some steady-state concentration of the analyte A * t far away from the biosensor (see Fig. 1C).This is maintained by an equilibrium between some unspecified release mechanism and a tissue removal mechanism at rate vt.Mathematically, this can be captured in the following description with the behaviour of the analyte and breakdown product in the biosensor enzyme layer obeying the substrate equations (6-7).Note that there is no tissue equation for H because it is considered to be rapidly removed from tissue and so the boundary condition H b (r2) = 0 also applies for the tissue condition.The boundary conditions for the analyte at the interface are now where the last equality ensures that the analyte fluxes across the biosensor interface are matched.

Model of biosensor in tissue with free space
Insertion of the biosensor can sometimes damage surrounding tissue.This aspect was included in the modeling as a free diffusion space in a region from the biosensor surface at a radius r2 to a radius rs beyond which the tissue extends (see Fig. 1D).The dynamics within the biosensor and the tissue follow equations (6-7) and (15) as before.
with Ht(rs) = 0 providing the boundary condition for H.

Analytical and numerical solutions
Analytical solutions.The models considered here have radial symmetry and so the steady-state equations typically take the form with similar equations for H.The first equation is satisfied by a constant plus a logarithm log(r) multiplied by a constant.The second equation is satisfied by a linear combination of the zero-order modified Bessel functions I0(r/ℓ) and K0(r/ℓ).The solutions for each of the cases considered here are provided in the Appendix.Numerical solutions.We sought analytical solutions for the coupled differential equations describing the steady state, and numerical solutions for the partial differential equations describing the time-dependent concentration profiles.
Analytical solutions for the steady-state are derived in the Appendix.For the numerical solution of the partial differential equations the system was discretized in time and space and integrated forward in time using a second-order or fourth-order Runge-Kutta scheme.

Experimental methods
All microelectrode biosensors were obtained from Sarissa Biomedical Ltd (Coventry UK).Microelectrode biosensors consist of an enzymatic biolayer on top of a permselectivity layer around a Pt/Ir wire (diameter 50µm) which has a length of 500µm.A block of agar (0.6-0.9g in 50ml, ∼2mm 3 ) was held submerged in a recording bath and perfused (6ml/min) with recording saline composed of (mM): 127 NaCl, 1.9 KCl, 1 MgCl2, 2 CaCl2, 1.2 KH2PO4, 26 NaHCO3 (pH 7.4 when bubbled with 95% O2 and 5% CO2, 300mOSM) at 32 • C. The block of agar sat on a suspended grid so was perfused from above and below.Biosensors were positioned above the agar block, in the bathing medium, and polarised.The agar and biosensors were perfused with the analyte of interest for 20-30 minutes to allow equilibration.Biosensors were manually inserted (in less than 1s) into the agar block so that the sensing area was completely embedded.Biosensor signals were acquired at 1 KHz with a Micro 1401 interface using Spike 2 (Vs 6.14) software (Cambridge Electronics Design, Cambridge UK).Glucose biosensors have entrapped glucose oxidase in an enzyme layer, it oxidises glucose to D-glucono-1, 5-lactone + H202 (Updike and Hicks 1967;Frayling et al 2011).Hypoxanthine biosensors have entrapped xanthine oxidase in an enzyme layer, which oxidises hypoxanthine to xanthine + H202 and also xanthine + O2 to uric acid + H202.Experiments typically used 50µM glucose or 10µM hypoxanthine.Values are quoted as the mean with standard deviation based on n trials.

Parameter choice
The parameters used are summarised in Table 1.The approach was to use a generic model of a single-enzyme biosensor in which the analyte has relatively simple dynamics in tissue.Reasonable values for the quantities were chosen in respect to specific biosensors, such as hypoxanthine or glucose biosensors.
Biosensor dimensions.Several sizes of biosensors are available, the most common size provided by Sarissa Biomedical Ltd have a core of radius r1 = 25µm.The outer surface of the enzyme layer, though more variable, was typically r2 = 50µm.The length of the core was z b = 500µm.
Volume fractions, tortuosity.The volume fraction α b and tortuosity θ b are rarely if ever specified in publications on biosensor design.The free volume fraction in the polypyrrole matrix is difficult to determine, as it is known to depend on deposition conditions, doping agents, inert additives and thickness (Garcia-Belmonte 2003).A comparison of the free volume of polypyrrole with different dopant-ions measured with nitrogen gas (Hallik et al 2007) has volume fractions between 0.26 and 0.56, but nitrogen is much smaller than either hydrogen peroxide, glucose or hypoxanthine, so may overestimate the relevant free volume.Here the value α b = 0.4 is chosen.The diffusion coefficient in the polymer matrix has been quoted (Sarissa Biomedical Ltd) as being similar to that for free diffusion and so θ b = 1.0 is used.For the agar we assume little excluded volume αg = 1.0 and low tortuosity and so that θg = 1.0 (McCabe 1972;Nicholson et al 1979).The corresponding quantities αt and θt for tissue are described above.
Free diffusion coefficients.The free diffusion coefficients for glucose and hydrogen peroxide at 32 • C are 860µm 2 /s and 1700µm 2 /s (van Stroe-Biezen et al 1993).
Biosensor breakdown rates.The model is meant to be generic and describes diffusion rather than enzyme limited biosensors, so a representative and relatively rapid breakdown rate of analyte in the biosensor enzyme layer of v b = 100/s is chosen.Such a rapid rate is seen for specific example in the hypoxanthine biosensor, which is constructed with 10µl containing the pyrrole monomer and 5U xanthine oxidase (Llaudet et al 2003) of which ∼ 8% is immobilized (Coche-Guerente 1995).Assuming the enzyme is uniformly distributed, the kinetics are not significantly affected by entrapment in the polymer matrix and using average velocity as a maximum velocity provides an estimated removal rate of v b = 178/s (Monda and Mitra 1994).Tissue breakdown rates.Similarly for the tissue breakdown rate a generic value of vt = 0.1/s is chosen.This obviously depends on the analyte in question, but generally it is relatively slow compared to that in the biosensor enzyme layer.This is reasonable for hypoxanthine, for example.In homogenate of the rat cerebrum and cerebellum, xanthine oxidase activity was found to be 19.5mU/g of tissue at 30 • C (Hashimoto 1974).Assuming hypoxanthine clearance is due to xanthine oxidase and this activity is uniformly distributed in the brain suggests a clearance rate vt = 40 × 10 −3 /s (Coche-Guerente 1995).Considering glucose as another example: its metabolism in the brain is a complex process involving multiple metabolic pathways and is coupled with neuronal activity (Bélanger et al 2011).An estimate can be obtained from studies using radio-labelled analogues (Berti et al 2013).Assuming labelled analogue behaviour is the same as glucose, the phosphorylation rate in grey matter is ∼ 1.2×10 −3 /s for human or ∼ 0.9 × 10 −3 /s for rats (Reivich et al 1985).Glucose is transported though the blood-brain-barrier at a similarly slow rate, ∼ 0.2 × 10 −3 /s (Clarke and Sokoloff 1999).

RESULTS
First the steady-state concentration profile for the idealised biosensor is modelled in the free-flow calibration condition.
We then examine the response when the biosensor is placed in an environment where the analyte reaches the sensor by diffusion.These model results are then tested experimentally in a condition where the biosensor is inserted into a block of agar in which the analyte reaches the biosensor through diffusion.This verifies the key finding that in diffusive environments the biosensor measures a smaller concentration than in free-flow conditions due to the density gradient being set up.The implication for this effect in tissue is then modelled using a general model for an analyte being generated and cleared in tissue.

Model of free-flow calibration
A biosensor is typically calibrated by placing it in a free-flow environment in which the analyte A has a fixed concentration (A * c ) at its outer surface (Frenguelli and Wall 2016).As the analyte is absorbed into the biosensor enzyme layer it diffuses until broken down by the enzyme at a rate v b into a certain number of molecules of the electrically active product H202, the concentration of which we denote by H. Mathematically this can be described by equations ( 6) and ( 7) with boundary conditions given by equations ( 8) and ( 9).Biophysically, these conditions assume that the analyte A can freely cross the outer boundary of the biosensor enzyme layer but cannot diffuse into the solid electrode core; whereas for the breakdown product H we assume that the concentration is zero outside the biosensor due to the free-flow condition and that H is broken down when coming into contact with the biosensor core.It is therefore the flux of H into the biosensor core that is proportional to the measured signal.An example, for steadystate concentrations, is given in Figure 2A using parameters from Table 1.For the parameter values used the biosensor rapidly metabolises the analyte such that very little reaches the biosensor core before being broken down.The resulting breakdown product either diffuses out of the biosensor and is lost, or diffuses to the core and is electrolysed and measured as a signal.Mathematically, the signal measured takes the form where A b (r) is the analyte concentration in the biosensor enzyme layer at a radius r (see the Appendix).Interestingly, this quantity does not depend on the diffusion coefficient of the break-down product H in the biosensor.A characteristic diffusion length ℓ b can be derived for the analyte in the biosensor which is given by ℓ 2 b = D Ab /v b .This is the typical distance that a molecule of analyte will diffuse into the biosensor enzyme layer before being broken down and will be relatively small for a diffusion limited biosensor e.g.ℓ b = 1.9µm for parameters used here.Note that if ℓ b ≪ (r2 − r1) then there is little chance that the analyte reaches the core before break down, so that A b (r1) ≪ A b (r2); then the current simplifies to where the boundary condition A b (r2) = A * c has been used.This is a particularly simple form in that it does not depend on the diffusion coefficient of H or the breakdown rate v b .These free-flow calibration results (Fig. 2) will now be compared with diffusive environments (agar and tissue).

Model of a biosensor in agar
Agar provides a diffusive environment similar to tissue (though there is no excluded volume) but without the added complexities of endogenous analyte dynamics.We now consider the case of a long biosensor embedded in a cylindrical block of agar with concentration A * g at the surface of the agar (Fig. 3 inset).The steady-state current at the biosensor for this configuration is straightforward to derive and is provided in the Appendix.Because the biosensor acts as a sink and a diffusion gradient is set up, the concentration of the analyte at the biosensor surface is reduced by a factor cg so that A(r2) = cgA * g where cg is defined (57).Provided the length scale is small relative to the size of the biosensor core, so that ℓ b ≪ r1, we have For the parameters used in Figure 3 this reduction is substantial (88%).However, the current itself is not so significantly attenuated because some of the electro-active breakdown product H202 generated by the biosensor and initially lost into the agar diffuses back to the biosensor matrix.The form of the current can be written for a biosensor of length z b , where the constant is The inferred concentration of analyte in agar, obtained by comparison of the biosensor current in agar with that in calibration condition, underestimates the true concentration For diffusion-limited biosensors the quantity X(r1) ≪ 1 and so a good approximation is a reduction by a factor κ b Hence, even if the tortuosities and the free volume fractions are the same in the enzyme layer and surrounding agar block, and r2/r1 = r3/r2 the current-equivalent concentration predicted would be 50% lower.This clearly indicates a mismatch between the free-flow calibration conditions and the diffusive experimental environment (in agar or tissue) that is likely to result in underestimation of analyte concentrations.

Experimental verification of calibration mismatch
Because it is technically difficult to cut an agar block into a near perfect cylinder, we performed an experiment with a slightly altered geometry.A large rectangular block of agar was immersed in a free-flowing bath with a constant concentration of analyte.Compared to the free-flow conditions, there was a substantial and rapid drop in the signal when a biosensor was inserted into the agar for both hypoxanthine 52.4% (22.6%, n= 5) and glucose 43.2% (8.0%, n= 5) biosensors.A typical experiment is illustrated in figure 4.
Here two glucose biosensors, with similar sensitivity, were initially held within the flow (equivalent to calibration conditions) with 50µM glucose and both registered a current of ∼ 2.5nA (average of first 100s).The first biosensor was then fully inserted into the agar.The signal then decreased with a slow decay rate (∼ 600s) and reached a new value near 1.75nA.To test that the decrease is due to a density gradient being set up, the second biosensor was then introduced into the agar near the first biosensor.This resulted in a further decrease in the current on the first biosensor, with both biosensors reaching a new, lower steady-state current of ∼ 1.5nA (a similar result was obtained with hypoxanthine biosensors, with the decrease depending on both the distance between them and difference in the angle of insertion).On removal of the second biosensor the signal on the first biosensor recovered to its previous value of 1.75nA.Note that when the first biosensor was removed and placed back into calibration conditions the recovery to the calibration current was very rapid (< 5s).This makes it clear that the slow dynamics seen during the insertion into the agar are not inherent to the biosensor but rather to the setting up of the diffusion gradient, as predicted by the model in the previous section.When a single biosensor is inserted together with a null sensor (one lacking an enzyme layer) there was little or no effect on the biosensor current; however, a small increase in the null-sensor current (∼ 50 − 100pA) was seen as would be expected from H202 diffusive over-spill from analyte breakdown in the active biosensor.

Model of a biosensor in tissue
Though the analyte diffuses to reach the biosensor in both agar and tissue there are key differences between the conditions.In tissue there is a significantly reduced volume fraction αt and the tortuosity is greater resulting in a smaller diffusion constant DAt.Additionally the analyte, rather than being externally applied, is generated within the tissue itself.We model the latter property by considering a balance between the release rate and breakdown rate vt, resulting in a steady-state concentration of A * t that is homogeneous throughout the tissue (far away from the biosensor).On the insertion of the biosensor the analyte will begin to diffuse into the enzyme layer and be broken down, resulting in a density gradient being set up in the tissue around the biosensor.An analysis of the system gives a length ℓt, defined through ℓ 2 t = DAt/vt, that gives a scale for the range of influence of the biosensor in tissue.For the parameters used this length was 59µm.The tissue case can be modelled by using the biosensor equations for A and H within the enzyme layer coupled to equation (15) for A in the tissue.The boundary condition in this case is that far (r ≫ ℓt) from the biosensor that the tissue concentration approaches A * t .There are a number of subsidiary conditions we could use for H202 in tissue.In this section we firstly consider that the breakdown product H is instantly removed from tissue.Under these circumstances the profile of the analyte A b and breakdown product H b in the biosensor is functionally the same as the calibration case (though with different amplitudes): all that remains is to find the reduction in analyte concentration at the biosensor surface.Writing this in the form A(r2) = ctA * t the constant, as shown in the Appendix, can be written where X(r) and Y (r) and their derivatives are also defined in the Appendix.Note that this quantity is strictly positive as the gradient of Y (r) is negative, cancelling out the apparent minus sign.Because of the boundary conditions the biosensor current takes the same form as in equation ( 21) but with (A2 − A1) reduced by the factor ct so that Because any H202 leaving the biosensor is rapidly broken down once it enters the tissue, the mismatch between calibration and tissue measurements is substantial in this model: for the example in Fig. 5 the tissue concentration would be measured at only 1.5% of its value in tissue far away from the biosensor.

Effect of space around biosensor in tissue
We now consider the effect of a thin region around the biosensor that allows free diffusion.As well as being of biophysical relevance -it can be caused by the insertion of the biosensor into the tissue -the analysis also serves to demonstrate how sensitive calibration-correction factors are to details of the biosensor-tissue interactions.The free space considered extends from the biosensor surface at r2 to a radius rs and then for radii greater than rs the tissue conditions are the same as in the previous case.The concentration mismatch now takes the form where κ b and γ b have a fairly complex dependency on the parameters given by equations ( 75) and ( 79) of the Appendix.The effect of adding a free-diffusion space of rs = r2 + 5µm is shown in figure 6.As can be seen, even a small region of free diffusion can have a significant effect on the concentration.For the parameters used here the percentage of the true concentration measured rises to 2.5% from the 1.5% seen for the previous case of no free-diffusion region (rs = r2).This effect underlies that the calibration mismatch for tissue conditions is a complex quantity that has a strong dependence on the diffusion of H202 into tissue and back.To examine the functional dependency of the calibration mismatch we varied a number of key parameters: Free-diffusion space.The effect of increasing rs led to a broadly linear (Fig. 6Da) improvement in the biosensor measure, over a range of rs − r2 up to 20µm.
Biosensor free volume and permeability.The accuracy of the biosensor measurement increases markedly with decreasing free-volume fraction α b or diffusion permeability θ b of the biosensor enzyme layer (Fig. 6Db and 6De).Breakdown rates.The calibration mismatch becomes worse with decreasing biosensor reaction rate v b (Fig 6Dc).This can be expected as the biosensor is destroying less of the analyte and so the density gradient is reduced.However, in constructing a biosensor it is not desirable to have too low a reaction rate as this leads to a poor signal-to-noise ratio, similar to the reduced size of the biosensor.For a faster reaction rate in tissue vt the calibration mismatch is less severe (Fig. 6Df) which can be understood as fixing A * t meaning that the relative replenishment rate of analyte is higher too.Biosensor size.Biosensors are available in various sizes, here we consider the core radius to be equal to the thickness of the enzyme layer and consider a range of sizes from 10µm (with 5µm core and enzyme layer), to 100µm (with 50µm core and enzyme layer).The calibration mismatch is less for smaller biosensors (Fig. 6Dd) due primarily to the smaller enzyme layer.However the typical distance the analyte diffuses through the enzyme layer before being oxidised is ℓ b = 1.9µm, so even for the smallest size considered, most of the electro-active product is lost to the tissue.Diffusion constants.For the model assumptions made in this paper the steady-state biosensor current is unaffected by the diffusion coefficient of H202 and the calibration mismatch is not substantially altered by the diffusion coefficient of the analyte.
In summary, the sensitivity analysis demonstrates that conventional calibration will substantially underestimate the concentration of analyte in tissue for a range of parameters relevant to current microelectrode-biosensor usage.

DISCUSSION
Microelectrode biosensors provide good spatio-temporal resolution for measurements of a range of physiologically relevant substances in vitro and in vivo (Dale et al 2005) and are widepread in their use.Here we presented models of a generic single-enzyme biosensor under calibration conditions, in agar and in tissue.It clearly demonstrates a discrepancy between the biosensor's response in tissue and during calibration.However, when the biosensor response is linear our modelling suggests that, in principle, the calibration can be corrected by scaling factors: Eq. ( 26) for agar; Eq (29) for tissue; and Eq (30) for tissue when the effects of the free diffusion space around the biosensor are significant.If the parameters of the scaling factors can be estimated or constrained, then this provides an approach for improved estimates of analyte concentrations, or bounds on concentrations, in tissue.It must be noted that the models do not imply that the concentration recorded in tissue is incorrect.The biosensors are indeed measuring the local concentration of the analyte; however, they have themselves reduced the local concentration through their measurement mechanism and this concentration differs from that in tissue far from the biosensor.
Biosensors could be designed to have less of an impact on the analyte in tissue, so that calibration and tissue conditions are better matched.This would require a lower total reaction rate for the enzyme in the biosensor, which would in turn result in a lower signal-to-noise ratio and a biosensor response potentially more dependent on the enzyme kinetics than the analyte concentration (Baronas et al 2009).
Agar was used to provide a diffusive environment and denser concentration of agar could be considered as a means of altering the porosity or permeability.However the diffusive parameters of agar are not greatly influenced by the density (McCabe 1972) and at higher densities the agar damaged the enzyme layer of the biosensor.Silica microbeads could also be mixed with the agar to create a controlled experimental model of excluded volume if desired: this would produce a stronger effect than that see in Figure 4.

Bath application
When experiments are carried out with biosensors placed within tissue the current produced by bath applying analyte is much smaller than that produced in free-flow conditions.For example, when adenosine is bath applied to the rat neocortex there is reduction of 90% in the current measured in tissue compared to that observed in free-flow conditions (Wall and Richardson 2015) and when ATP is bath applied to the rat hippocampal slices only around 5% could be detected, compared to free-flow calibration (Frenguelli et al 2007).Such a difference between free-flow and tissue measurements supports the calibration mismatch proposed by our modelling.The effects of the density gradient produced in tissue by diffusion and tortuosity will be compounded by the presence of active removal mechanisms such as uptake into neurons and glia and metabolism.However during adenosine application blocking nucleoside transport had only minor effects on the current measured in tissue (unpublished observations) suggesting that the effect maybe principally due to the density gradient set up by the biosensor.

Existing biosensor models
Models of biosensors are typically applied to the case where the biosensor is placed in a well-stirred medium (Schulmeister 1990;Rinken and Tenno 2001;Baronas et al 2009).Even in a well-stirred medium a narrow layer around the biosensor will persist where transport is primarily due to diffusion and not convection.The substrate concentration will asymptotically increase towards the bulk concentration with distance from the biosensor.The Nernst diffusion layer approximation assumes transport in a region around the biosensor is solely due to diffusion, the thickness of this layer depends on the flow and viscosity of the medium.Consequently it would not substantially effect the calibration signal as the flow in the bath is relatively rapid.Other modelling has examined the effects of substances that oxidise at the holding potential of the biosensor (+500mV) and could be potential sources of positive or negative (antioxidants like ascorbic acid) interference (Lowry and O'Neill 1994;Lowry et al 1994;Lowry et al 1998).To reduce the interference biosensors are coated in a perme-able membrane and this can be modelled by an additional diffusion layer (Baronas et al 2014).Modelling this additional layer around the biosensor is beyond the scope of the current paper, but its inclusion would not alter the paper's broad conclusions.

Biosensor geometry
Here we have considered a cylindrical design of biosensor often used in in experiments (Chen et al 2002;Mikeladze et al 2002;Shigetomi et al 2013;Kotanen et al 2014).The size of the biosensor has little impact on the calibration mismatch provided the thickness of the enzyme layer is greater than the typical distance the analyte diffuses before being oxidised (ℓ b ), i.e. the biosensor is diffusion controlled.This work could be extended to consider other biosensor geometries such as: discs (Chen et al 1998; Kobayashi and Hoshi 2001;Razola et al 2003;Patel et al 2011), microelectrode arrays (Burmeister andPalmer 2003;Walker et al 2007;Hascup et al 2013) or twisted pairs of electrodes (Santos et al 2015).While considering geometries is useful in optimising design, such as for a plate-gap biosensor (Ivanauskas and Baronas 2008), and could help better quantify the discrepancy between free-flow and diffusive environments, it would not refute our key finding.Irrespective of the geometry, the consumption of the analyte by the biosensor will set-up a concentration gradient resulting in lower signals in diffusive (tissue or agar) than in a free-flow calibration environment at equivalent concentrations.

Multi-enzyme biosensors
The idealised model of a biosensor considered here with a single-enzyme layer is likely to display the same characteristics as the more complex biosensors such as those that use a cascade of enzymes for adenosine or inosine (Llaudet et al 2003;Wall and Richardson 2015;Frenguelli and Wall 2016), glutamate (Mikeladze et al 2002) or acetylcholine (Chen et al 1998).The main barrier for such analysis is the paucity of published information regarding the properties of the biosensors, which in many cases is proprietary.Additionally, there is substantial variation between biosensors in key properties such as enzyme layer thickness and reaction rate, which also change with repeated use.Accurate estimation of tissue concentrations would also necessitate more sophisticated models of the dynamics of the analyte and its breakdown quantities in tissue.such that the analyte flux at the core, and therefore gradient of A b , is zero at r1 and, additionally, has been normalised so that X(r2) = 1.Provided the length that the analyte diffuses into the enzyme layer is much smaller than the size of biosensor core ℓ b ≪ r1 the arguments of the modified Bessel functions will be large, so an asymptotic approximation (Eq.34) can be used Breakdown-product concentration and flux.The solution for H b can be separated into two components Applying the boundary condition H b (r1) = 0 at the core and using the solution given in equation ( 31) allows h b to be written in the form where A1 = A b (r1) can be found from equation ( 37) and γ is a constant.In the calibration case, using the condition at the biosensor surface H b (r2) = 0, gives γ = 1 so that The flux (and therefore gradient) of the analyte at r1 is zero so the flux of the breakdown product at r1 is simply so the current for a length z b of biosensor takes the form Note this does not depend on the diffusion coefficient of H202 in the biosensor enzyme layer.When r2 − r1 ≫ ℓ b , meaning the analyte is almost certainly broken down before reaching the core, the biosensor is said to be diffusion limited (Baronas et al 2009) as the current no longer depends on v b .In this case A2 ≫ A1 and the current may be approximated as where the boundary condition A b (r2) = A * c has been used.

B. Biosensor in agar
In cylindrical geometry the agar region extends from r2 → r3 within which the steady-state equations DAg∇ 2 Ag = 0 and DHg∇ 2 Hg = 0 are obeyed by the concentrations.At the agar surface r3 the analyte has a fixed concentration A * g and the breakdown product has zero concentration, corresponding to free-flow bath conditions.Within the biosensor the concentrations obey formulae (35,36) with continuity of concentration and flux across the biosensor-agar interface at r2.Note that these conditions allow for loss and diffusion of H202 from the biosensor into the agar and hence require a different boundary condition at r2 to the zero H b (r2) used for the calibration case.Analyte concentration.Within the biosensor enzyme layer A b has the same functional form as equation ( 37) except that here the concentration A2 at r2 is a fraction cg of A * g so that A2 = cgA * g .Within agar the solution takes the form (31) which, on matching concentrations at the biosensor surface r2, can be written Matching the analyte fluxes provides an equation for cg This can be re-arranged to find the constant which solves the problem for the analyte A. Provided the length scale ℓ b is smaller than the biosensor core radius (ℓ b ≪ r1) the same expansion used (Eq.39) shows X ′ (r2) ≃ 1 ℓ b , which gives the result (Eq.23).Breakdown product concentration and flux.The concentration H b within the biosensor takes the form given by equation ( 41).However, the concentration at the biosensor surface is non-zero because H is not washed away but diffuses into the agar.The analyte concentration appears in equation ( 40) for the biosensor and so, to simplify the boundary matching at r2, it proves convenient to add the analyte concentration within agar also Boundary conditions related to the analyte component are automatically matched so just the components h b and hg remain to be matched.At r1 and r3 the following boundary conditions hold constraining the equation ( 31) solutions to give The constants κ b and κg are fixed by matching conditions at the biosensor-agar interface at r2.The concentration match requires h b (r2) = hg(r2) and the flux match Solving for the constants gives and κg = 1 The H flux at the biosensor core is This gives the following form for the current We can now compare this form with the current for the calibration condition (Eq.44).In terms of the function X(r) we have for the agar case for the calibration case we have (A2 − A1) = A * c (1 − X1).Dividing out the current for the agar condition Iagar and calibration I cal gives the ratio which can be re-arranged to provide a formula for the calibration mismatch and hence the concentration measured is a factor of the concentration A * g at the outer surface of the agar block.Note that for diffusion-limited biosensors, where A2 ≫ A1 we have X1 ≪ 1 and so A inferred ≃ A * g κ b is a good approximation.

C. Biosensor in tissue
In tissue the analyte is modelled as having a steady-state extracellular concentration A * t due to the balance between production and removal rates.In the steady-state the analyte dynamics (Eq.15) reduce to The breakdown product H202 is considered to be removed very rapidly from tissue and so has zero concentration outside the biosensor.The boundary conditions at the biosensor-tissue interface at r2 are given by equation set ( 16) with the conditions at the biosensor core given by equation (8).Analyte concentration.Within the biosensor the functional form is given by equation ( 37) with the concentration at the biosensor surface a fraction ct of A * t so A2 = ctA * t .Within the tissue the limit r → ∞ should remain finite so the solution is constructed from the zero-order modified Bessel function K0(r/ℓt) where ℓt is a length scale defined through ℓ 2 t = DAt/vt.The function Y (r) has been normalised such that Y (r2) = 1 so concentrations are already matched at r2 and it remains only to render the flux continuous This can be re-arranged to give for the fractional reduction of the bulk concentration A * t at the surface of the biosensor due to the combined effects of diffusive transport and analyte removal by the biosensor.Note this fraction is strictly positive because the gradient of Y (r) is negative.The gradients required take the form for X and for the function Y we have Breakdown product concentration and flux.Because the H202 concentration in tissue is zero, the boundary conditions on H are the same as for the calibration case, except that instead of A * c we have ctA * t for the concentration at the biosensor surface.The concentration of the breakdown product and the biosensor current take the same forms as equations ( 42) and (44), respectively, for the calibration case but now calculated by replacing * c with ctA * t throughout.Hence, the ratio of the currents in calibration and tissue conditions used to infer the concentration would give an erroneous result underestimating the true extracellular concentration A * t by a factor ct given by equation (64).

D. Biosensor in tissue with free space
To model the effect of damage due to the insertion of the biosensor, a scenario in which there is an additional free diffusion space between the biosensor surface at r2 out to a radius rs is considered.The tissue continues as before at radii greater than rs.Within the free space there is unimpeded diffusion so that in the steady state DA∇As = 0 and DH∇Hs = 0 (68) where here the free diffusion coefficients are used and there is no tortuosity or porosity.Analyte concentration.We can adapt the solutions for A used previously so that   The steady-state analyte concentration sharply decreases from the surface for this diffusion-limited biosensor, so peak breakdown H 2 0 2 production is at the biosensor surface.(B) Corresponding concentration profile for H 2 0 2 .(C) The dynamics of the biosensor current demonstrating the rapid (< 1s) responsiveness.The bath concentration of analyte was A * c = 1µM with other parameters given in Table 1.Fig. 3. Model: biosensor in cylindrical agar block (see schematic inset) with the biosensor enzyme layer extending from radii 25µm to 50µm and the agar block from radii 50 to 150µm.(A) The steady-state analyte concentration exhibits a density gradient from the agar surface that is induced by breakdown within the biosensor enzyme layer.The inset shows the detail within the enzyme layer itself.(B) Corresponding concentration of H 2 0 2 .Note that H 2 0 2 is lost through diffusion into the agar and then washed away.(C) Dynamics of the current response for the biosensor inserted into the bath (calibration condition) from 0-3s, into the agar from 3s-18s during which the density gradient builds up, and finally back into the bath from 18s onwards.The increase at the point of insertion into the agar is due to the transient increase in local H 2 0 2 concentration, which was often seen in experiment (see Fig. 4) Fig. 4. Experiment: diffusive transport in agar markedly reduces the biosensor current.Two glucose biosensors, with almost identical sensitivities, were moved in and out of an agar block.The top panels show configurations of the two biosensors (black and grey) either above the agar block (free-flow conditions) or inserted into the agar, with the respective, superimposed current traces illustrated in the bottom panel.Initially both biosensors were held in free-flow calibration conditions in the presence of 50µM glucose (current of ∼ 2.5nA).The first biosensor (black) was then fully inserted into the agar block.The current recorded dropped to ∼ 1.75/2.5 = 70% of its calibration value due to the establishment of the diffusion gradient.To verify the presence of the concentration gradient the second biosensor was then inserted close to the first.An initial rise due to a transient and localised increase in H 2 0 2 can be seen, as predicted by the model (Fig. 3).The second biosensor steady-state signal was lower than that of the first biosensor previously; however, the first biosensor signal also dropped to the same lower value.These results are what would be expected if each biosensor established a density gradient of analyte and that these gradients superpose.On removal of the second biosensor from the agar the first biosensor recovered to the earlier steady current of ∼ 1.75nA.Both biosensor signals returned to their calibration values when removed from the agar.The current is initially high, but then decreases as the analyte around the biosensor is broken down and the density gradient is set up.In this particular example the biosensor measures a steady-state current equivalent to only ∼ 1.5% of the bulk tissue concentration.This mismatch is much greater than the case for agar, largely due to the instantaneous removal of H 2 0 2 in this model of tissue.

Fig. 1 .
Fig.1.Schematic of the different model configurations.The biosensor electrode core has an outer radius of r 1 and the biosensor enzyme layer extends from radii r 1 to r 2 .The length of the biosensor is z b .(A) Calibration conditions.(B) Biosensor in agar, with the agar block extending out to a radius r 3 .(C) Biosensor in tissue, where the tissue is considered to be infinite in extent.Practically, this means extending out for a distance that is much greater than the tissue length constant ℓt (i.e. a few 100µm) described in the related section in Results.(D) Biosensor in tissue with a free-diffusion region caused by insertion damage extending from r 2 to rs beyond which the tissue begins.

Fig. 5 .
Fig. 5. Model: biosensor in tissue.(A) Concentration of analyte and H 2 0 2 in the biosensor enzyme layer.Note the same forms as Fig. 2. (B) Distribution of the analyte in tissue.A sharp decrease in density around the biosensor from the bulk value of 1µM is apparent.(C) Time course of the biosensor current.The current is initially high, but then decreases as the analyte around the biosensor is broken down and the density gradient is set up.In this particular example the biosensor measures a steady-state current equivalent to only ∼ 1.5% of the bulk tissue concentration.This mismatch is much greater than the case for agar, largely due to the instantaneous removal of H 2 0 2 in this model of tissue.

Fig. 6 .
Fig. 6.Model: biosensor surrounded by a free-diffusion layer in tissue.(A) Concentrations of analyte and H 2 0 2 in the biosensor enzyme layer.Here the functional forms of the concentration profiles differ from Fig. 2 due to the boundary conditions for H on the biosensor surface.(B) The distributions of the analyte in tissue.(C) Time-course of the biosensor current.In this example the biosensor measures a steady-state signal equivalent to only ∼ 2.5% of the concentration in tissue far from the biosensor.This is nevertheless less of a mismatch than for the model without a free-space around the biosensor, in Fig 5. (D) Examination of the calibration mismatch as key parameters are varied; (a) size of the free space rs − r 2 , (b) porosity α b of the biosensor enzyme layer, (c) breakdown rate v b in biosensor, (d) the radius r 2 of the biosensor (with equal core radius and enzyme layer thickness), (e) diffusion permeability θ b in the biosensor enzyme layer and (f) breakdown rate vt in tissue.The parameters used in (A-C) are shown in grey.

Table 1 .
Parameters used in the paper, unless otherwise stated.