Estimation of $\mathit{p}{\mathit{O}}_2$ distribution in EPR oximetry

Abstract

Electron paramagnetic resonance (EPR) oximetry, using oxygen-sensing implant such as OxyChip, is capable of measuring oxygen concentration in vivo – a critical tissue information required for successful medical treatment such as cancer, wound healing and diabetes. Typically, EPR oximetry produces one value of the oxygen concentration, expressed as $p{O}_2$ at the site of implant. However, it is well recognized that in vivo one deals with a distribution of oxygen concentration and therefore reporting just one number is not representative_a long-standing critique of EPR oximetry. Indeed, when it comes to the assessment of radiation efficacy one should be guided not by the mean or median but the proportion of oxygenated cancer cells which can be estimated only when the whole oxygen distribution in the tumor is known. Although there is a handful of papers attempting estimation of the oxygen distribution they suffer from the problem of negative frequencies and no theoretical justification and no biomedical interpretation. The goal of this work is to suggest a novel method using the empirical Bayesian approach realized via nonlinear mixed modeling with a priori distribution of oxygen following a two-parameter lognormal distribution with parameters estimated from the multi-implant single component EPR scan. Unlike previous work, the result of our estimation is the distribution with positive values for the frequency and the associated $p{O}_2$ value. Our algorithm based on nonlinear regression is illustrated with EPR measurements on OxyChips equilibrated with gas mixtures containing four values of pO₂ and computation of the proportion of volume with $p{O}_2$ greater than any given threshold. This approach may become crucial for application of the EPR oximetry in clinical setting when the sucsess of the treatment depends of the proportion of tissue oxygenated.

1. Introduction

Electron paramagnetic resonance (EPR) oximetry allows estimation of the oxygen concentration from a single-scan fit. Medical application of EPR oximetry in tumor, wound or diabetes was criticized due to the fact that $p{O}_2$ values are not unique – instead one need to estimate a distribution of oxygen. For example, the efficacy of radiation treatment of cancer crucially depends on the proportion of oxygenated cells in tumor that are responsive to radiation.

A desirable property of the EPR oximetry in vivo is the ability to measure spatial distribution of oxygen. For example, in cancer research, it is critical to evaluate the extent at which tumor is hypoxic and as such the chance that the radiotherapy will be successful or not compared to surgical removal of tumor. Previous techniques of EPR oximetry aimed at reconstruction of a unique value of $p{O}_2$. Only recently, new technological advances in developing microchip-based EPR sensors make it possible to recover the distribution of oxygen using a single composite EPR scan without gradient [1].

2. Estimation of the distribution using Tikhonov regularization and its limitation

Only a few papers have addressed the problem of oxygen distribution and its estimation using EPR measurements. Ahmad et al. [2] suggested estimating the distribution of $p{O}_2$ from a single-scan EPR by solving an ill-posed inverse problems based on the idea of Tikhonov regularization [3]. The basis of that approach was the assumption that the composite signal is the convolution of Lorentzian intensity function with a distribution of linewidth $\gamma$

$$f(B)={\int }_{{\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{{\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}L(B;\gamma )p(\gamma )d\gamma ,$$ (1)

where $B$ is the magnetic field, the interval $\left[{\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}},{\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$ defines the lower and the upper limit of the linewidth, $p(\gamma )$ is the estimated probability density function (pdf) of the linewidth, and $L$ is the Lorentzian intensity function

$$L\left(B_i,\gamma \right)=-\frac{16}{\pi }\frac{\left(B_i-B_0\right)\gamma }{{\left[4{\left(B_i-B_0\right)}^2+{\gamma }^2\right]}^2},i=1,\dots ,N=1024,$$ (2)

where the center field $B_0$ is supposed to be known. The authors discretize function $p$ and reduce its estimation to a linear model with large number of parameters $\mathbf{p}=\left\{p_j,j=1,\dots ,M\right\}$

$$\mathbf{y}=\mathbf{L}\mathbf{p}+\epsilon ,$$

where $\mathbf{y}$ is the $N\times 1$ vector of EPR signal, $\mathbf{L}$ is the $N\times M$ matrix with elements $L_{ij}=L\left(B_i,{\gamma }_j\right)$ where ${\gamma }_j={\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}}+\left({\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\gamma }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)(j-1)/(M-1)$ the grid of the gamma values. Since $M$ is large, say, $M=512$ we must penalize the least squares solution for $\mathbf{p}$ using Tikhonov regularization

$$\underset{\mathbf{p}}{\min }\left[\parallel \mathbf{y}-\mathbf{L}\mathbf{p}{\parallel }^2+\lambda \parallel \mathbf{p}{\parallel }^2\right],$$

where $\lambda >0$ is the penalty coefficient. A positive feature of this approach is given $\lambda$ the closed-form expression for $\mathbf{p}$ exists:

$$\mathbf{p}={\left({\mathbf{L}}^{\prime }\mathbf{L}+\lambda \mathbf{I}\right)}^{-1}{\mathbf{L}}^{\prime }\mathbf{y}.$$

Recall that $p_j$ estimates the probability density function of the linewidth ${\gamma }_j$ and the respective oxygen concentration value derived from ${\gamma }_j$ from previous calibration study.

There are several limitations of this approach: (i) the lower and the upper limits of the linewidth must be specified beforehand, (ii) since the penalty coefficient $\lambda$ is unknown it must be estimated using other ad hoc techniques such as cross-validation or L-curve [4,5], (iii) the centerfield must be known, and most importantly, (iv) $p_j$ may take, and in most situations takes, negative values that contradict the assumption that $p$ is the pdf (see an example in the next section).

3. Nonlinear mixed-modeling approach

Mixed modeling is becomeing a popular statistical approach to address clustered and heterogeneous data. For example, Google Scholar returns more that 5 million results for the search ‘‘mixed models” (one of the top few on the list is the book by one of the authors [6]). This approach to handle complex-structure data successfully applies (i) in medicine to account for heterogeneity of treatment responses, (ii) in longitudinal settings to analyze subjects/objects observed in time, (iii) to image analysis when groups of scientific images are analyzed for pattern recognition and detection of statistical significance, etc.

In the nonlinear mixed-modeling approach [6], we assume that the composite EPR signal is the convolution of signals with different linewidth following a distribution with pdf p as above, but we do not discretize the pdf and do not apply penalization. Instead, we propose to model this function as a pdf that belongs to a family of unimodal distributions with unknown parameters. Our previous research [7] on distribution of $p{O}_2$ in vivo studies with Eppendorf polarographic electrode suggests that this distribution, upon taking log transformation, is normal that justifies the lognormal distribution as the distribution of $e^X$ where $X$ is a Gaussian random variable with mean $\mu$ and variance ${\sigma }^2$ [8]. We are not the first who use the lognormal distribution for modeling oxygen distribution in living tissue. The paper [9] was the first where the lognormal distribution was mentioned in the context of studying oxygen distribution in vivo. Recently, this distribution was used in Monte Carlo simulations to model oxygen distribution in tumors [10, 11]. Parameters $\mu$ and $\sigma$ receive a novel interpretation in the framework of the lognormal distribution: $e^{\mu }$ is the median and $\sigma$ is the coefficient of variation, that is, $100\mathrm{\%}\sigma$ is the percent scatter with respect to the median [8].

In general, in the formulation of the nonlinear mixed-modeling approach we assume that the signal intensity, given vector parameter $\theta$, takes the form $L(B;\gamma ,\theta )$, where for Lorentzian function it may be $\theta =B_0$. The key point of our approach is that a priori distribution of oxygen on $(0,\mathrm{\infty })$ is specified by the linewidth pdf $p(\gamma ;\tau )$ where the vector parameter $\tau$ is unknown and subject to estimation. Finally, the mixed model combines the two distributions, similarly to (1), leads to the theoretical signal intensity

$$f(B;\theta ,\tau )={\int }_0^{\mathrm{\infty }}L(B;\gamma ,\theta )p(\gamma ;\tau )d\gamma ,$$ (3)

which is computed using numerical integration. We make a few remarks: (i) the mixed-modeling approach may be interpreted as a hierarchical model in the framework of Bayesian approach [12,13], where function $L$ is the conditional signal intensity, $p$ is a priori pdf, and $f$ as posterior signal intensity, (ii) no limits for the linewidth have to be specified, (iii) no discretization is required, (iv) the problem of negative values of $p$ is solved. The EPR signal intensity $L$ may be complicated and contain several unknown parameters, such as the centerfield $B_0$.

Finally, the problem of the EPR signal deconvolution reduces to a nonlinear regression

$$y_i=f\left(B_i;\theta ,\tau \right)+{\epsilon }_i,$$

where unknown parameters $\theta$ and $\tau$ are estimated by the nonlinear least squares [8, 14]. There are several advantages of this model: (i) one can build a parsimonious model through statistical significance, (ii) this model can be easily generalized to account for heterogeneity across samples/patients by incorporating random effect [6], (iii) the uncertainty of oxygen measurements can be readily obtained through error propagation, known as the standard error of inverse prediction (SEIP) [15].

4. Testing the nonlinear mixed-modelling approach by phantom experiments

To test our novel approach outlined in the previous section we used the EPR spectrum data obtained from a phantom experiment described in [1]. Four randomly placed oxygen-sensing OxyChips were separately calibrated with a gas mixture containing $p{O}_2$ values 0, 7.6, 15.2, and 38.0 mmHg. A single L-band EPR spectrum was measured without of a magnetic field gradient with the purpose of reconstruction of the $p{O}_2$ distribution. The composite EPR fit based on the penalized approach described in [2] yielded mean and median $p{O}_2$ 15.3 and 19.3 mmHg, respectively. Although the mean and median of oxygen distribution are important characteristics of the level of oxygenation, from clinical perspective, we may want to know what percent of tumor is well oxygenated to make cancer therapy effective.

Fig. 1 depicts the density values, $\left\{p_j,j=1,2,\dots ,M=512\right)$ computed by linear least squares with Tikhonov regularization as a function of the penalty coefficient $\lambda$. As mentioned above, this method does not guarantee positiveness of $p_j$ for any $\lambda$.

Fig. 1.

512 density values $p_j$ estimated using Tikhonov regaularization technique as a function of $\lambda$. The density values may take unwanted negative values since no restriction on ${\mathrm{p}}_{\mathrm{j}}$ values is applied.

Now we turn our attention to a parametric technique where a unimodal distribution of oxygen (and the linewidth) is modeled by a few unknown parameters, such as lognormal distribution justified above:

$$p\left(\gamma ;\mu ,{\sigma }^2\right)=\frac{1}{\gamma \sqrt{2\pi {\sigma }^2}}{e}^{-\frac{1}{2{\sigma }^2}(\mathrm{l}\mathrm{n}\gamma -\mu {)}^2},$$ (4)

where $\mu$ and ${\sigma }^2$ are the mean and variance of the log scale or in the previous notation $\theta =\left(\mu ,{\sigma }^2\right)$. This density serves as a priori distribution for the linewidth parameter $\gamma$ in the Lorentzian curve.

According to the nonlinear mixed-modeling approach the theoretical composite EPR signal takes the form

$$f\left(B;A,\mu ,{\sigma }^2\right)=A{\int }_0^{\mathrm{\infty }}\frac{B\gamma }{{\left(4B^2+{\gamma }^2\right)}^2}p\left(\gamma ;\mu ,{\sigma }^2\right)d\gamma ,$$ (5)

where $B$ is the centeraized magnetic field and $A$ is the unknown amplitude coefficient. Alternatively, one may facilitate integration by change of variable as

$$f\left(B;A,\mu ,{\sigma }^2\right)=A{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{x{e}^u}{{\left(4x^2+e^{2u}\right)}^2}\varphi \left(u;\mu ,{\sigma }^2\right)du,$$

where $\varphi$ is the Gaussian density with the mean $\mu$ and standard deviation $\sigma$. These integrals do not admit a closed-form solution and therefore some numerical integration is required (we used a built-in function integrate in statistical package $\mathbf{R}$).

The three-parameter estimates were further obtained by application of the nonlinear least squares ($\mathbf{n}\mathbf{l}\mathbf{s}$) by fitting $f$ to the EPR spectrum. The output of the R package nls function is shown in Fig. 2. All three parameters are statistically significant with residual standard error of the fit 0.031. Parameters mu and sigma have special meanings by referring to the lognormal density (4) as described in [8]. The mean, median, and mode of the lognormal distribution is $e^{\mu +{\sigma }^2/2}$, $e^{\mu }$, and $e^{\mu -{\sigma }^2}$, respectively. Since the lognormal distribution has long right tail, the mean is difficult to interpret, but the median and mode have clear interpretation (see below).

Fig. 2.

The output of the nls function in R from fitting the EPR spectrum by function (5). Parameter estimate mu defines the center of $p{O}_2$ distribution (mean, mode, or median) and sigma defines the scatter of the distribution (coefficient of varaition).

The lognormal density in Fig. 3 is derived from (4) where the linewidth $\gamma$ is rescaled to $p{O}_2$ using the linear calibration rule

$$p{O}_2=\frac{\left(\frac{\gamma }{2.6}-0.066\right)}{0.0105}.$$

Fig. 3.

The nonlinear mixed- modeling approach for estimation of the distribution of $p{O}_2$ from phantom experirment with four micro chips using the fiting function (5).

The median of the linewidth is $e^{-0.41436}$ and the median of $p{O}_2$ is computed as $\left(e^{-0.41436}/2.6-0.066\right)/0.0105=17.9\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$. The mode, as the typical concentration of oxygen $\left({pO}_2\right)$ is computed as $(e^{-0.41436-{0.67746}^2}/2.6-0.066)/0.0105=9\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$.

Fig. 3 depicts the reconstructed distribution of oxygen by the lognormal density and cumulative distribution function (cdf). As follows from the left plot, 50% of oxygen values are below 17.9 mmHg with the most typical value (if chosen at random) is at 9 mmHg. An attractive property of the lognormal distribution, especially in vivo experiments, is a clear interpretation of the scatter of the oxygen distribution. The coefficient of variation (CV), as the proportion of standard deviation to the mean, can be approximated by sigma. This means that the oxygen variation is around 68%, as follows from the fitting result shown in Fig. 2.

The plot at right helps to understand the quality of the oxygen distribution reconstruction by comparing the theoretical lognormal cdf with the empirical cdf as the probability to get the value of $p{O}_2$ less than those given at the x-axis. For example, to obtain the $p{O}_2$ value less than 10 mmHg is

$$\mathrm{P}\mathrm{r}\left({pO}_2<10\right)=\mathrm{P}\mathrm{r}(\gamma <0.445)=\mathrm{\Phi }\left(\frac{\mathrm{l}\mathrm{n}0.445+0.41436}{0.67746}\right)=0.28.$$

The possibility of computing the probability of hypoxia is an invaluable feature of our approach for measurements of the oxygen distribution in vivo.

5. Extension to multimodal distributions

The approach with a single lognormal component outlined above is possible to extend to two or more components by employing the concept of finite mixture distribution [16]. For example, the two-compartment lognormal distribution of oxygen is expressed as

$$p_{12}\left(\gamma ;\lambda ,{\mu }_1,{\sigma }_1^2,{\mu }_2,{\sigma }_2^2\right)=\lambda p\left(\gamma ;{\mu }_1,{\sigma }_1^2\right)+(1-\lambda )p\left(\gamma ;{\mu }_2,{\sigma }_2^2\right)$$

as a linear combination of lognormal densities defined by equation (4). Again, parameters of this two-compartment distribution can be determined by the nonlinear least squares with the regression function defined by equation (5). The two-compartment log-normal density is illustrated in Fig. 4 where two distribution compartments combined with $\lambda =0.4.$ The first density has the mean around 15 and the second density has the mean around 37 mmHg. The two-mixture density is bymodal.

Fig. 4.

Illustration of the two-mixture density for modeling the bymodal oxygen distribution density.

An attractive feature of this approach is ability to test this distribution against the one-compartment distribution (4) by testing the null statistical hypothesis $H_0:\lambda =1$ using the Wald or likelihood-ratio test [8].

6. Discussion and future work

Heterogeneity of oxygen concentration in tissue is well documented and yet the techniques based on the solution of the ill-posed linear system suffer from several limitations. This work suggests a novel parametric approach by assuming that oxygen scatters through the tissue (or tumor) following a lognormal distribution. By measuring a single convoluted EPR signal from several OxyChips the problem is reduced to a nonlinear regression with a function expressed as a convolution integral. An important advantage of our approach is that knowing the number of the EPR oxygen probes is not required because it does not attempt estimating $p{O}_2$ at each probe but the whole oxygen distribution.

Another advantage of the parametric approach is that the parameters of the distribution can be rigorously tested using classic statistical hypothesis tests that enables developing robust models of the oxygen distribution and its probabilistic assessment and uncertainty ivolved such as the error of inverse prediction that transforms the EPR signal linewidth to $p{O}_2$ [15]. These probabilities may become crucial for application of the EPR oximetry in clinical setting when the sucsess of the treatment depends of the proportion of tissue oxygenated. Moreover, the distribution of oxygen may answer a paramount question on what proportion of oxygenated tumor is required to make chemorerapy or radiation sucsesful in clinic.

Since our parametric approach enables asessing the uncertainty of oxygen estimation, such as standard error of inverse prediction (SEIP) of oxygen concentration [15], we can apply this quantity to answer an important question: how many OxyChips are required to reach the desired statistical power of testing that an additional implantation of the OxyChip will not improve the oxygen density estimation or proportion of oxygenated cancer cells? To answer this question, we will employ the theory of the sample size determination well developed in statistics [17].

This study is limited and should be viewed as a the proof of priciple. Much work has to be done to verify its potentials in vivo. We hypothesize that our approach could be especially beneficial for multiple probes dispersed throuthout the tissue.