SENSOR FOR MEASURING pH AND pCQ ₂

Cross-Reference to Related Applications

[0001] This application claims priority to U.S. Provisional Application No. 61/772,201 titled "METHODS FOR USE IN A CALIBRATION-FREE ABSORBANCE SENSOR FOR MEASURING pH AND pC0 ₂ ", filed March 4, 2013, the disclosure of which is incorporated by reference herein in its entirety.

Field of the Invention

[0002] Embodiments described herein relate to methods for measuring analytes, such as with an optical absorbance-based sensor for the detection of pH and pC0 ₂ .

Background

[0003] The precision and robustness of a sensor is closely associated with the quality and robustness of its calibration. Known sensing methods to detect, for example, biologically relevant analytes such as pH require the use of a single- or multi-point calibration by the user before use. This can require a dedicated device for calibration and can take up to 30 minutes. Additionally, once a sensor is calibrated, measurement errors can be introduced by changes in the measured light levels, changes in optical component configurations, and other causes. For example, changes in the light source intensity, bending of optical fibers, and detector temperature changes can be detected incorrectly as pH changes. Furthermore, manufacturing variability between sensors can lead to inconsistent and unreliable measurements between sensors.

[0004] Some known sensors need calibration at many data points and at frequent intervals. Other known sensors may need calibration at only one data point and one time. Known methods of manufacturing a sensor with robust calibration, typically, involve a highly reproducible manufacturing process for both the instrumentation and the sensing material, and implementation of functionalities that can eliminate or reduce the possibility of changes in the sensing material and instrumentation over time. Such known manufacturing methods can also add to the cost and the complexity of the sensor. [0005] Accordingly, a need exits for an analyte sensor that is calibration-free, e.g. that can reliably and accurately measure analytes regardless of manufacturing variability, changes in ambient conditions or operating parameters.

Summary

[0006] Disclosed embodiments relate to relationships between the absorbance of a pH-sensitive sensor molecule and the pH of the solution to which the sensor molecule is exposed. The relationship is shown to be independent of the concentration of the sensor molecule, the sensor molecule path length, and the intensity of the light source. Disclosed embodiments also relate to the absorbance of a pH-sensitive sensor molecule and the pC0 ₂ of the solution to which the sensor molecule is exposed. The relationship between the pH and pC0 ₂ measurements is also shown to be independent of the concentration of the sensor molecule, the sensor molecule path length, and the intensity of the illumination light source.

Brief Description Of The Drawings

[0007] FIG. 1 A shows the molecular structure of the acid form and the base form of Phenol Red.

[0008] FIG. IB shows the change in the absorption spectra of the acid form of Phenol Red and the absorption spectra of the base form of Phenol Red as a function of change of pH of the surrounding solution.

[0009] FIG. 2 is a graph illustrating the independence of pH measurement accuracy from the concentration of the indicator (Phenol Red).

[0010] FIG. 3 is a graph illustrating the pH measurement accuracy over a physiologically relevant range of pH values using Phenol Red indicator.

[0011] FIG. 4 is a graph illustrating the independence of pH measurement accuracy from light intensity over a three-fold change in incident light source intensity.

Detailed Description [0012] The following detailed description is organized into the following three sections: Section 1 addresses the derivation of mathematical expressions for pH measurement;

(ii) Section 2 addresses the derivation of mathematical expressions for pC0 ₂ measurement; and

(iii) Section 3 describes the results of experiments conducted to evaluate the accuracy of the derived mathematical expressions.

SECTION 1 - Derivation of mathematical expressions for nil measurement

[0013] The pH of a solution is a measure of the activity of the (solvated) hydrogen ion [H ⁺ ] in the solution. Hence, the pH of a solution is a measure of the hydrogen ion concentration in the solution. Pure water has a pH of approximately 7 at 25°C. Solutions with a pH less than 7 are said to be acidic and solutions with a pH greater than 7 are basic or alkaline.

[0014] In living organisms, pH affects many essential metabolic and homeostatic functions. For example, pH affects how enzymes function, as they can only work at specific pH depending upon the enzyme. If the pH is not correct then the enzyme's active site can be altered permanently, i.e. the enzyme is denatured. This can prevent key chemical reactions from occurring within the body.

[0015] The pH of a solution can be defined as: p¾ ^T = -&s¾ _s C¾ >

Where ¾ ₄ = the Activity of f * = ¾τ> ·

(fa * ) = H ^" * ^" Activity Coefficient (1 for dilute solutions) (¾÷ } = if* Concentration (molality) = { *

≠K = -Z -J *J Eq. 1

[0016] Optical sensor molecules such as, for example, the pH-sensitive dye Phenol Red are typically weak acids. The general dissociation constant of a weak acid (HA) into a hydrogen ion (H ⁺ ) and the acid's conjugate base (A ) can be written as:

HA * 4 A- Eq. 2 [0017] The Dissociation Constant "K" of the acid is defined in terms of the equilibrium molar concentrations (mol/L) of the acid, its conjugate base, and the hydrogen ion:

Solving Equation 3 solved for [H ]

Taking the L& of both sides: liS = Log K + LogiM - Log r Eq. 5

Substituting Equations land K = -Log (K) into Eq. 5:

[0018] This relationship between pH, the dissociation constant, and the concentrations of the acid and its conjugate base is used in the derivations below.

Derivation of an expression for the absorption of both the acid form and the basic form of a pH indicator sensor molecule as a function of pH

[0019] For a sensor molecule, such as a dye, that can exist in either an acid or base form (such as Phenol Red), the total sensor molecule concentration [T] in a solution is equal to the concentration of the acid form of the sensor molecule [HA] plus the concentration of the conjugate base form of the sensor molecule [A ].

= ^" l M Eq. 7

[0020] Combining Equations 6 and 7 produces an expression for pH (of the solution being measured) as a function of either the acid form or base form of the sensor molecule, as well as the sensor molecule's pH and the total sensor molecule concentration.

[0021] Solving for pH using the acid form of the sensor molecule: [0022] Solving for pH using the base form of the sensor molecule: pH = pH -f- LQg Eq. 9

[0023] By combining Beer's Law (absorbance of light by a solution is directly proportional to the concentration of the solution) with Equations 8 and 9, an expression for the absorption (ABS) of both the acid form of the sensor molecule (ABSHA) and the base form of the sensor molecule (ABSA) can be derived.

[0024] Beers Law:

BS£ = the absorption of species x at a wavelength λ

I = the intensity of light transmitted through the sample at wavelength λ ^'

Io =the intensity of light transmitted with no sample present at wavelength

L = the path length of the light through the sample

e = The absorption coefficient of species x at wavelength λ

[X] = The concentration of species x.

[0025] Combining Equations 10 and 8 produces an expression for the absorption of the base form of the sensor molecule:

[0026] Combining Equations 10 and 9 produces expression for the absorption of the acid form of the sensor molecule:

[0027] Equations 1 1 and 12 are useful when under certain conditions (e.g., appropriate temperature, appropriate pH), there is a wavelength, λ, at which the sensor molecule is exclusively in either the acid or the base form. At wavelengths where this is not the case, it is not possible to directly measure the ABS of only the acid or base form of the sensor molecule. Thus Equations 1 1 and 12 are of limited utility by themselves. Hence, there is a need to be able to express the absorption of a pH indicator sensor molecule at a wavelength, λ, at which both the acid and base forms of the sensor molecule are present.

Derivation of an expression for the absorption of a pH indicator sensor molecule at a wavelength, λ, where both the acid and base forms of the sensor molecule are present

[0028] A general expression for the absorption of light passing through the sensor molecule at any wavelength, ABS ¹ , can be written by summing the absorbance of the base and acid forms of a sensor molecule at a wavelength, λ.

Eq. 13 or

[0029] Equation 14 can be rearranged as shown below to either of two simpler forms:

^ϊϋ- +χ ^{q' 5}

Derivation of an expression for the pH of a pH sensitive sensor molecule as a function of the ratio of the sensor molecules absorbance at two different wavelengths. [0030] Using Equation 15, the ratio of the absorbance of the sensor molecule (ABS ) at two different wavelengths, * smi can be written as:

[0031] Equation 17 can be simplified to:

[0032] Solving Equation 18 for pH gives the following general expression for the relationship between the pH of a solution and the absorbance of a pH indicator sensor molecule, measured at two wavelengths, which is in the solution:

[0033] Since pK = -Log(K), Equation 19 can be expressed as follows:

[0034] Equation 20 is thus a generic expression of the relationship between pH of a solution and the absorption of a pH-sensitive sensor molecule at two different wavelengths.

Derivation of an expression for the pH of a pH sensitive sensor molecule where there is no absorbance of the acid form of the sensor molecule at the base absorption peak, λ ¹ , but both the acid and base forms of the sensor molecule have absorbance at the acid peak, λ .

[0035] For this situation the acid form of the sensor molecule has no absorbance at ¹ thus the absorption coefficient {sf } ⁼ @· Setting to zero in Equation 20 gives the following equation.

[0036] Simplifying and rearranging Equation 21 gives the following equation:

[0037] Since the absorption coefficients of the sensor molecule are constant for a given wavelength, two constants can be defined, C = K ) _? ¾1 €2 = E / ( . Substituting these constants into Equation 22 gives the following:

[0038] A common pH-sensitive sensor molecule is Phenol Red. Equation 23 is the general equation to obtain the pH of a solution in contact with Phenol Red by obtaining absorption values at each of two different wavelengths, λ ¹ and λ ² . Phenol Red is the sensor molecule used to experimentally test the validity of the expression derived below.

[0039] FIG. 1A shows the molecular structure of the acid form and the base form of Phenol Red. Phenol Red is a weak acid with pKa = 8.00 at 20 °C. Phenol Red is used as a pH indicator where its color exhibits a gradual transition from yellow to red over the pH range 6.8 to 8.2. Above pH 8.2, phenol red turns a bright pink color. Phenol Red exits in the acid form at pH below 8.0 and in the base form at pH over 8.0 as can be seen in FIG. IB. FIG. IB also shows that the acid form of Phenol Red has an optically distinct absorption spectra when compared to the optical absorption spectra of the base form on Phenol Red. Additionally, the isosbestic point of Phenol Red is also shown in FIG. IB. The isosbestic point of a sensor molecule such as Phenol Red will be discussed in greater detail herein. Derivation of an expression for the pH of a pH-sensitive sensor molecule where there is no absorbance of the acid form of the sensor molecule at the base absorption peak, A ⁱ ,and the sensor molecule has an isosbestic point at A ² = ^ίφί

[0040] The isosbestic point for a sensor molecule (e.g., a pH-sensitive dye) is the spectroscopic wavelength at which the two species of the sensor molecule, i.e. the acid form of the sensor molecule and the conjugate base form of the sensor molecule, have the same molar absorptivity (ε) or, more generally, are linearly related. When an isosbestic plot is constructed by the superposition of the absorption spectra of the two species (whether by using molar absorptivity for the representation, or by using absorbance and keeping the same molar concentration for both species), the isosbestic point corresponds to a spectroscopic wavelength at which these spectra cross each other. Said another way, the isosbestic point is the spectroscopic location where the absorbance value of the acid form of the sensor molecule and the base form of the sensor molecule are equal.

[0041] If in Equation 20, λ is chosen as the isosbestic point of a sensor molecule, then the absorption coefficients for both the acid and base form of the sensor molecule at A will be equal. That is = (¾f-) = {ff _f ca,}. Additionally, if the acid form of the sensor molecule has no absorbance at A ¹ , then the absorption coefficient ⁼ Substituting these relationships into Equation 20 gives the following expression:

[0042] Rearranging Equation 24 and defining constant K ^' } gives the following equation: Derivation of an expression for the pH of a pH-sensitive sensor molecule where there is no absorbance of the acid form of the sensor molecule at the base absorption peak, A ¹ , and there is no absorbance of the base form of the sensor molecule at the acid absorption peak, 2 ²

[0043] For a sensor molecule with no absorbance of the acid form of the sensor molecule at the base absorption peak, A ¹ , and also no absorbance of the base form of the sensor molecule at the acid absorption peak, λ*, the absorption coefficients (s - ) and (sj^) ^are by definition equal to zero. Thus, Equation 20 can then be

[0044] Since the two absorption coefficients are constants for a given sensor molecule and wavelength, Equation 26 can be rewritten as follows:

[0045] Equation 27 could also be rewritten as:

Where:

[0046] The derivation of the mathematical expressions discussed in detail above dealt with detecting the pH of a solution with the use of pH-sensitive sensor molecules with different spectral and functional characteristics. The goal of these derivations was to generate an expression that can relate the pH of a solution that is in operable contact with the pH sensitive sensor molecule to the ratio of the optical absorbance of the sensor molecule at two different wavelengths. This can enable a pH sensor that: a) has the potential to eliminate the need for a user to perform a calibration; b) can be highly tolerant of changes in manufacturing due to chemistry and mechanical variability; c) can eliminate or reduce issues caused by fiber bending or source light changes; d) can be stable over time; and e) can have a low cost of manufacture. SECTION 2 - Derivation of mathematical expressions for uCOi measurement

[0047] Carbon dioxide (C0 ₂ ) plays an important role in the human body. It is a waste product of cellular metabolism, exhaled by the lungs. This waste product is involved in the transportation of oxygen from the blood to the cells of the body. C0 ₂ helps to dilate the smooth muscle tissues and helps regulate the cardiovascular system.

[0048] When C0 ₂ is dissolved in water it exists in equilibrium with carbonic acid, and is a primary regulator of the alkaline/acid balance of the body. Moreover, C0 ₂ plays a role in the proper functioning of the digestive system. Thus, carbon dioxide plays a very important role in the body. The level of carbon dioxide in the blood must be approximately 40 mm of Hg for the proper functioning of the body.

[0049] Carbon dioxide is present in three forms in the human body; dissolved C0 ₂ , carbonic acid and bicarbonates. Most of the C0 ₂ content in the body is in the form of bicarbonate. Thus, when laboratory tests are conducted to check the C0 ₂ level in the blood, it is actually measuring the blood bicarbonate level. The appropriate blood bicarbonate level for adults is approximately 23- 29 milli-moles per liter (mmol/L). Deviations in the amount of C0 ₂ in blood can lead to dizziness, respiratory or cardiac arrest, and even death.

[0050] Presented below is a derivation of a relationship between the absorbance of a pH- sensitive sensor molecule and the C0 ₂ concentration (hereinafter referred to as pC0 ₂ ) of the solution with which the sensor molecule is in contact. This relationship is intended to be independent of the concentration of the sensor molecule, the sensor molecule path length, and the intensity of the illumination light source.

Derivation of an expression for the pCC>2 of a sensor molecule in a bicarbonate buffer as a function of the ratio of the sensor molecule's absorbance at two different wavelengths.

[0051] When carbon dioxide is dissolved in water the following equilibrium is established:

£¾ 4 ¾€¾ ** B* mC&g Eq. 29

[0052] The Henderson - Hasselbalch equation for then becomes as follows: mm

Eq. 30

[0053] When [ ££¾] is substituted for [.¾£¾¾ ] in Equation 30, the ¾e _% becomes pK' to denote two related but different values. Also, since the solubility of 0¾ is 0.0301 mmol/L/mmHg of £¾ Equation 30 can be rewritten in terms of pH and PC0 ₂ as follows:

= Log ..,.„ f f ¾ .. Eq. 31

[0054] Combining Equations 31 and 20 allows elimination of pH from the relationship and the following expression can be obtained:

[0055] Solving Equation 32 for pC0 ₂ gives the following general equation describing the relationship between the pC0 ₂ of a bicarbonate buffer and the absorbance of a pH-sensitive sensor molecule (e.g., Phenol Red) in the buffer or sample, measured at two different wavelengths:

[0056] Note that the proposed mechanism to measure pC0 ₂ uses a pH-sensitive sensor molecule such as Phenol Red to measure pC0 ₂ indirectly. It is proposed that C0 ₂ from the sample being tested comes into operative contact with the bicarbonate ions [Μ£ _§ ] and raises the pH in the vicinity of the pH sensitive sensor molecule (e.g., Phenol Red). This changes the absorbance properties of Phenol Red which can then be correlated with pC0 ₂ in the sample being tested. Derivation of an expression for the pCC>2 of a bicarbonate buffer with a sensor molecule, where there is no absorbance of the acid form of the sensor molecule at the base absorption peak, 2 ^a , but both forms of the sensor molecule have absorbance at the acid peak, λ ²

[0057] For this situation, the acid form of the sensor molecule has no absorbance at A ¹ , thus the absorption coefficient (sj^ f = . Setting to zero in Equation 33 gives the following equation:

[0058] Note that the concentration of the bicarbonate ions, fifCOjJ, in Equation 34 and the subsequent equations refers to the concentration of bicarbonate ions present in the sensor environment surrounding a sensor molecule, such as embedded within a polymer complex, that can be inserted into a test site. It is not the concentration of bicarbonate ions present in a biological sample such as, for example, blood, interstitial fluid, etc.

[0059] Simplifying and rearranging Equation 34 gives the following equation:

[0060] For a given sensor molecule and a fixed bicarbonate concentration, f CUg ], and since the absorption coefficients of the sensor molecule are constant for a given wavelength, two additional constants can be defined as:

[0061] Substituting Kl and K2 into Equation 35 gives the following expression:

Derivation of an expression for the pC0 ₂ of a bicarbonate buffer with sensor molecule, where there is no absorbance of the acid form of the sensor molecule at the base absorption peak, λ , and the sensor molecule has an isosbestic point at A ² = A ^is *

[0062] If in Equation 32, A ^z is chosen as the isosbestic point of a sensor molecule, then the absorption coefficients for both the acid and base form of the sensor molecule at Λ ^,' will be equal. That is, = = Additionally, if the acid form of the sensor molecule has no absorbance at A ¹ , the absorption coefficient = 0. Including these two substitutions into

Equation 33 gives the following expression:

[0063] Rearranging Equation 39 gives the following relationship:

[0064] Two new constants can be defined,

[0065] Substituting K3 and K4 into Equation 40 gives the following equation:

Derive an expression for the pCC>2 of a bicarbonate buffer with a sensor molecule, where there is no absorbance of the acid form of the sensor molecule at the base absorption peak, A ¹ , and there is no absorbance of the base form of the sensor molecule at the acid absorption peak, A ^z

[0066] For a sensor molecule with no absorbance of the acid form of the sensor molecule at the base absorption peak, A ¹ , and likewise no absorbance of the base form of the sensor molecule at the acid absorption peak, J*, the absorption coefficients and (« ^ are by definition equal to zero. Equation 33 can then be simplified as follows:

[0067] For a given sensor molecule and a constant bicarbonate concentration (fif ! j) the absorption coefficients will be constant. Lumping all the constants together, the following constant can be obtained, substituting Kl into

Equation 44, the following equation can be obtained:

SECTION 3 - Experimental results validatins the derived mathematical expressions

[0068] Experimental studies were performed to test the validity of the different mathematical expressions derived above. The experiments performed investigated three phenomena: (i) the ability of the derived theory to correctly measure pH over a broad range of dye concentrations at a fixed pH; (ii) the robustness of the theory to predict the pH of a solution; and (iii) test the ability of the theory to accurately predict pH over a broad range of light source intensities. The experiments performed and the results obtained are discussed below. Experiment 1- test the ability of the theory to correctly measure pH over a broad range of sensor molecule concentrations at a fixed pH

[0069] A series of ten sensor solutions were prepared. First, nine Phenol Red solutions were prepared that ranged in concentrations from 0.03 to 3.19 mg/dl, all with a pH of 7.337. Subsequently, a tenth solution of Phenol Red was prepared by taking one of the previous solutions (concentration = 1.07 mg/dl), splitting it into two parts (A and B), and changing the pH of B to 6.850 while maintaining the same dye concentration. Part A was left unchanged.

[0070] The absorbance of all 10 solutions was measured in a spectrometer (Perkin-Elmer Lambda 4B) at 486 and 582 nm, respectively. These wavelengths correspond to Phenol Red's isosbestic point and the base absorbance peak, respectively. With the absorbance data from parts A and B of the 10 ^th sample (same concentration, different pH), constants CI and C2 (used in

Equation 23) were calculated from the expressions C = K ) i id€2 = M \ . The t«# {*» ^■ #.! values were: CI = 8.79 x 10 ^~8 and C2 = 3.25x 10 ^~8 . Using Equation 23 and the calculated CI and C2 constants above, the pH for each of the 10 solutions was calculated.

[0071] FIG. 2 shows the plot of the Phenol Red concentration versus the difference between the calculated and measured pH values, or measurement bias (pH _ca i _cu i _at ed - pH _mea sured). FIG. 2 clearly shows that the change in dye concentration had no discernible effect on the accuracy of the pH calibration. Hence the measured pH values correlated with the calculated pH values at all the dye concentrations.

Experiment 2- test the robustness of the theory to predict the pH of a solution

[0072] The second experiment was performed to examine how robustly the theory based on the mathematical derivations presented above predicts the pH of a solution. Nine new Phenol Red solutions were prepared covering a pH range of 6.955 to 7.759 in approximately 0.1 pH unit steps. The sensor concentration of each solution was held constant at 2.0 mg/dl.

[0073] In this experiment, the absorbance was measured for each solution at the Phenol Red's acid peak (λ ² = 434 nm) and base peak (λ ¹ = 557 nm). Using the absorbance readings for two of the solutions, two new CI and C2 constants were calculated for Equation 23 (not shown). The solutions used for calculating CI and C2 had a pH of 7.053 pH units and 7.659 pH units respectively.

[0074] FIG. 3 shows the plot of the pH of the Phenol Red solution versus the measurement bias (pH _C aicuiated - pH _mea sured). It is clear from FIG. 3 that the mathematical expressions presented above accurately predict pH over nearly the entire physiological range. The data in FIG. 3 had a mean bias of -0.001 pH units, and a precision (i.e., the standard deviation of the bias) of 0.007 pH units.

Experiment 3- test the ability of the theory to accurately predict pH over a broad range of light source intensities

[0075] The third experiment was performed to examine how accurately the theory based on the mathematical derivations presented above can predict the pH of a solution over a broad range of incident light source intensities. In this experiment, the light source intensity was varied by approximately three-fold and the pH of the sample solution was calculated from the measured absorbance ratio. The actual pH of the solution was held constant at 7.246 units and the temperature maintained at 37 C during the entire experiment. The results of this experiment are presented in FIG. 4. FIG. 4 clearly shows that the method of predicting the pH of the sample solution according to the derivations presented above is insensitive to changes in light intensity.

[0076] The equations derived above provide a theoretical means for making a sensor or a sensing technique that can measure pH and/or pC0 ₂ and meet two of the criteria set forth in the problem that is being addressed, namely, measurements independent of manufacturing variability, for example, dye concentration and path length, and measurements independent of the light source intensity changes caused by any sources, including fiber bending. Although the derivations are done using a pH-sensitive sensor molecule such as Phenol Red, the equations can apply generally to any sensor molecule that exists in two states, in one of which the sensor molecule is reversibly bound to a species or an analyte to be measured and in the other of which it is not bound to the species, and the two sensor molecule states are optically distinct and measurable. These equations can enable the development of a sensing technique (or a sensor) that can potentially be applied broadly to measure many different analytes (e.g., sodium, potassium, lactate, etc.) provided that a sensor molecule with the required two forms can be found.

[0077] The various embodiments described herein should not to be construed as limiting this disclosure in scope or spirit. It is to be understood that no limitation to the scope of the disclosure is intended thereby. It is to be further understood that resort may be had to various other embodiments, modifications, and equivalents thereof which may suggest themselves to those skilled in the art without departing from the spirit of the present disclosure and/or scope of the appended claims.

[0078] Those skilled in the art will recognize, or be able to ascertain, using no more than routine experimentation, numerous equivalents to the specific embodiments described specifically herein. Such equivalents are intended to be encompassed in the scope of the following claims.