Introduction

The pH value of living tissue is a valuable biomarker for numerous pathologies, e.g. cancer. A non-invasive technique for the determination of pH in vivo is phosphorus (³¹P) magnetic resonance spectroscopy (MRS). As gold standard, the pH-dependent chemical shift of inorganic phosphate (P_i) is used for pH calculation via a modified Henderson-Hasselbalch equation, assuming physiological conditions¹. However, challenges of this method are (I) altered ion homeostasis in pathologies and (II) the low signal of P_i in vivo.
In general, the measured chemical shifts of ³¹P compounds are dependent on pH, but to varying extent. Also the concentration of other ions, e.g. magnesium (Mg), sodium (Na), potassium (K), influences the chemical shifts directly or indirectly (i.e. due to change of pK values)^2-5. Obtaining the information of multliple ³¹P resonances under various conditions allows the implementation of a dictionary from which measured chemical shifts can be mapped to the most probable pH values. Such a dictionary would have the benefit of (I) being more robust to changing chemical conditions across different tissues and (II) enabling pH estimation in cases where the P_i signal is too low.
We propose a dictionary-based approach for the estimation of pH values under various chemical conditions using multiple chemical shifts of ³¹P compounds. Starting from a limited number of samples, the space of dictionary entries is extended using empirically developed approximations of the Hill equation to cover a range of conditions expected in vivo. We demonstrate the feasibility of this approach on a limited subspace using the chemical shifts of adenosine-5’-triphosphate (ATP) and their dependencies on pH and Mg.

Methods

A total of 154 model solutions with a base concentration of 5mM P_i, ATP and phosphocreatine (PCr) were prepared with different pH values and concentrations of Mg, Na and K. ³¹P spectra of all model solutions were acquired at a 9.4T small-animal scanner (Bruker BioSpec) with an unlocalized FID sequence at a temperature of 37°C. Evaluation of data was performed in MATLAB R2020a (The Mathworks). The chemical shifts of PCr, P_i and ATP were quantified using a home-built implementation of the AMARES algorithm^6,7.
For demonstrating the concept, we focused on a subset of 24 model solutions with [Na]$$$\,$$$=$$$\,$$$29mM, [K]$$$\,$$$=$$$\,$$$160mM, and the limited pH range [6.8–7.4], for which the dependency of the chemical shifts δ_i of ATP on the pH value and the ratio R$$$\,$$$=$$$\,$$$[Mg]/[ATP] can be described by empirically developed approximations of the Hill equation:$$\delta_{i}(pH,R)=k-\frac{m\cdot(pH-7.1)+b}{1+\exp(d\cdot{R})}, i=\alpha,\beta,\gamma.\qquad\qquad(1)$$Equation 1 was fitted to the quantified chemical shifts of γ, α and β-ATP, resulting in three defined functions δ_γ(pH, R), δ_α(pH, R) and δ_β(pH, R). These relations define the dictionary, allowing for an assignment of input values δ_γ,input, δ_α,input, δ_β,input to possible pairs of pH and R values.
To incorporate uncertainties in the input chemical shifts$$$\,\sigma_{\delta_{i}}\,$$$(assumed to be 0.01$$$\,$$$ppm), pH was calculated as a normal probability density function$$$\,\mathcal{N}(\mu,\,\sigma^{2})\,$$$using Gaussian error propagation:$$P_{i}(pH,R)=\mathcal{N}\left(pH(\delta_{i})|_{R},\left(\frac{\partial{pH}}{\partial{\delta_{i}}}\sigma_{\delta_{i}}\right)^2\right)\qquad\qquad(2)$$ with $$pH(\delta_{i},R)=-\frac{(\delta-k)\cdot(1+\exp(d\cdot{R}))+b}{m}+7.1.\qquad\qquad(3)$$The estimated pH value is then obtained as the weighted mean pH over the joint probability density function (product of all P_i(pH,R)) with the weighted standard deviation as error estimate: $$pH_{est}=\frac{\sum_{pH,R}pH\cdot\prod_{i}P_{i}}{\sum_{pH,R}\prod_{i}P_{i}}.\qquad\qquad(4)$$

Results

The acquired spectra from the model solutions were of good quality, enabling a robust quantification of the chemical shifts of all contained metabolites and showing clear spectral changes with changing pH (Fig.$$$\,$$$1A) and R (Fig.$$$\,$$$1B). For the investigated subspace of pH$$$\,$$$=$$$\,$$$[6.8–7.4] and R$$$\,$$$=$$$\,$$$[0-2], the dependency δ_i(pH,R) described by Eq.1 is reasonable (Fig.$$$\,$$$2).
The obtained relations δ_i(pH,R) were used to calculate probability density functions P_i(pH,R) (Eq.3) for specific input values δ_i,input. For the shown example (Fig.$$$\,$$$3), the estimated values pH_est$$$\,$$$=$$$\,$$$7.25$$$\,$$$±$$$\,$$$0.05 (with R_est$$$\,$$$=$$$\,$$$0.77$$$\,$$$±$$$\,$$$0.01) are in good agreement with the prepared conditions pH_prep$$$\,$$$=$$$\,$$$7.2 and R_prep$$$\,$$$=$$$\,$$$0.75.
When applying this method to all datasets of the shown subspace, the estimated pH values are in good agreement with the prepared values for datasets with small R (Fig.$$$\,$$$4). For the datasets with higher R, the estimated pH values deviate more strongly from the prepared values (Fig.$$$\,$$$4B)

Discussion

We demonstrated the concept of pH estimation by a dictionary-based approach using multiple chemical shifts of ³¹P metabolites for the example of ATP under different chemical conditions i.e. [Mg]. For the presented subset of model solutions, the estimated pH values are in good agreement with the prepared values. However, the increasing deviations (pH_est$$$\,$$$-$$$\,$$$pH_prep) for increasing R indicate the need for further refinement of the assumed relation δ_i(pH,R) for the range of pH$$$\,$$$=$$$\,$$$[6.8-7.4] and R$$$\,$$$=$$$\,$$$[0-2]. Compared to earlier approaches utilizing complete titration curves^2,3, the herein proposed approach requires only a reduced number of samples within a specific range of chemical conditions, which simplifies its implementation. For the application to in vivo data, the dictionary needs to be further extended by additional influences, e.g. of other ions. This extension can be done in a similar way as demonstrated, resulting in multidimensional relations δ_i(pH,R,[K],[Na]), and is currently under investigation utilizing the entire set of 154 model solutions.

Conclusion

We proposed a dictionary-based approach for pH estimation using ³¹P MRS under various chemical conditions and demonstrated its feasibility for a limited subspace using the chemical shifts of ATP and their dependencies on pH and Mg. Further refinement and extension of the dictionary by other influences is, however, required for potential application to in vivo data.

Acknowledgements

No acknowledgement found.

References

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