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Tumor oxygenation status is considered one of the important prognostic markers in cancer since it strongly influences the response of cancer cells to various treatments; in particular, to radiation therapy. Thus, a proper and accurate assessment of tumor oxygen distribution before the treatment may highly affect the outcome of the treatment. The heterogeneous nature of tumor hypoxia, mainly influenced by the complex tumor microenvironment, often makes its quantification very difficult. The usual methods used to measure tumor hypoxia are biomarkers and the polarographic needle electrode. Although these techniques may provide an acceptable assessment of hypoxia, they are invasive and may not always give a spatial distribution of hypoxia, which is very useful for treatment planning. An alternative method to quantify the tumor hypoxia is to use theoretical simulations with the knowledge of tumor vasculature. The purpose of this paper is to model tumor hypoxia using a known spatial distribution of tumor vasculature obtained from image data, to analyze the accuracy of polarographic needle electrode measurements in quantifying hypoxia, to quantify the optimum number of measurements required to satisfactorily evaluate the tumor oxygenation status, and to study the effects of hypoxia on radiation response. Our results indicate that the model successfully generated an accurate oxygenation map for tumor cross-sections with known vascular distribution. The method developed here provides a way to estimate tumor hypoxia and provides guidance in planning accurate and effective therapeutic strategies and invasive estimation techniques. Our results agree with the previous findings that the needle electrode technique gives a good estimate of tumor hypoxia if the sampling is done in a uniform way with 5-6 tracks of 20–30 measurements each. Moreover, the analysis indicates that the accurate measurement of oxygen profile can be very useful in determining right radiation doses to the patients.

Hypoxia is a feature of many solid malignant tumors and influences malignant disease progression, development of metastases, clinical behavior, and response to conventional treatments like radiotherapy [

Several approaches are commonly used to measure hypoxia in patient and experimental tumors, including polarographic electrode techniques and nitroimidazole binding as determined by flow cytometry, immunohistochemistry or PET imaging [_{2} consumption is the most important factor influencing the local _{2} distribution in the tumors. Kohandel et al. [

Here, we introduce a theoretical approach to model tumor hypoxia using known spatial distributions of the perfused tumor vasculature obtained from histological sections. We simulate oxygen distributions and calculate hypoxic fraction in two ways corresponding to sampling with a polarographic electrode and binding of a nitroimidazole agent. We further demonstrate that the simulated results correlate well with hypoxia measured directly in the same tumor sections [

Representative, high-resolution, two-dimensional, histologic images of eight human glioma xenografts were gratefully received from P. F. J. W. Rijken, Department of Radiotherapy, University of Nijmegen, The Netherlands. An immunofluorescence staining technique was used to assess vascularity, perfusion, and hypoxia using 9F1 (mouse-specific endothelial marker), Hoechst 33342, and either pimonidazole or a similar agent 7-[49-(2-nitroimidazol-l-yl)-butyl]-theophylline (NITP), as described previously [

Binary images of one of the eight glioma xenographt cross-sections, illustrating tumor blood vessels, perfused vessels, hypoxic area, and total tumor area, respectively.

Blood vessels

Perfusion

Hypoxia

Tumor cells

Here, the perfused vasculature is considered to be the source of oxygen supply and thus gives the initial spatial distribution of oxygen concentration. This perfused vascular network (at a fixed point in time) is obtained by combining the images of perfused areas (Figure

Following Kohandel et al. [

Similarly, the temporal rate of change of cell concentration is considered as a net result of diffusion and proliferation. If we denote by

Here,

List of parameters.

Parameters | Symbol | Value | Reference |
---|---|---|---|

Diffusion constant for oxygen | 2.5 × 10^{−5} (cm^{2} s^{−1}) | [ | |

Rate of oxygen supply | 8.2 × 10^{−3} (O_{2} s^{−1}) | [ | |

Cellular oxygen consumption* | 3.8 × 10^{−13} (cm^{2} O_{2} s^{−1} (cells)^{−1}) | [ | |

Diffusion constant for cells | 4.05 × 10^{−9} (cm^{2} s^{−1}) | [ | |

Proliferation rate | 1.85 × 10^{−6} (s^{−1}) | [ | |

Carrying capacity | 2.1 × 10^{11} (cells s^{−1}) | [ | |

Cellular growth rate (effect of vasculature) | 2.96 × 10^{−6} (s^{−1}) | [ |

*Assuming mass of 1 cell = 10^{−9} Kg.

The linear quadratic (LQ) model is the most commonly used approach for studying the survival response of tumor cells to radiotherapy and the concomitant clinical results [^{−1} and ^{−2}) gives a survival fraction of 48% at a dose _{2}_{m}_{2} at which half the maximum ratio is achieved) [

In general, the OER can be also a function of radiation dose, and some studies have suggested that the maximal oxygen enhancement varies in the range of 2.5 to 3 with differences in radiation dosage [

Here, we use this revised LQ model to study effects of heterogeneous oxygen distribution on the predicted survival rates after radiation therapy. To this end, we calculate the cell survival fraction while varying the dosage _{2} at each grid point). For cases (d)–(f), where the oxygen distribution is not uniform, we calculate the final survival fraction by taking the weighted average (with

The spatial distribution of hypoxia in each tumor section was simulated using the mathematical model (6 mm square computational domain with _{2} threshold less than values 2.5%, 5%, and 10% were calculated, simulating image analysis of a hypoxic marker. Second, _{2} was sampled along linear measurement tracks and the percentage of values less than these thresholds again calculated to simulate polarographic needle electrode measurements. The spatial distribution of hypoxia and the summary measures derived from each of these approaches were compared to the known distribution of hypoxia and hypoxic fraction from the nitroimidazole analysis (Figure

The focus of this work is to simulate the spatial distribution of hypoxia at a snapshot in time that will result from a particular distribution of perfused vessels and intravascular O_{2} concentrations, rather than tracking the time evolution of hypoxia. Yet it takes some computational “time” to arrive at this snapshot from our initial domain (the computation begins on a domain in which only the vasculature has nonzero oxygen concentration). The absolute oxygen distribution evolves as computational time proceeds. Therefore, to avoid dependence of our hypoxia quantification on computational time, we require a definition of hypoxia that considers relative, rather than absolute, quantities. Since the blood vessels act as constant source of oxygen, we assume for computational convenience that at any time

Polarographic needle electrode measurements of hypoxia were simulated as linear tracks through the tumor. Four sampling patterns were used as illustrated in Figure

Different needle electrode reading methods illustrated over one representative sample (for a random approach, only one realization is shown).

Random

Uniform

Radial

Radial (full circle)

The total variance in sampling the oxygenation status is the combined effect of within-tumor variance and between-tumor variance. Measurement of tumor _{2} is considered to be a predictive outcome assay only when the within-tumor _{2} variability is smaller than the variability among different tumors [_{2} values from the electrode simulations, the variability of oxygenation status within and between tumor samples is estimated through variance components analysis by computing the ratio of within-tumor variance over the total variance for each reading method (uniform, random, and radial) [_{2} estimates and thus to obtain an optimum number of needle probes. To study the percentage of variation in evaluating the hypoxic proportions, the variance analysis is also performed using the two different approaches to quantify hypoxia (simulated percentage hypoxic area and needle electrode measurements):

The oxygenation status of a heterogeneous tumor is often quantified using polarographic electrode measurements or through nitroimidazole binding and biopsies. These invasive techniques have varying accuracy due to the restricted sampling space as well as limited accessibility. In this paper, we present an alternative theoretical approach that permits an exploration of the spectrum of hypoxic distributions that could possibly be associated with a particular vascular configuration. We used two-dimensional binary images of tumor cross-sections, with perfused tumor vasculature as a computational domain, on which a simple model for the oxygen distribution and tumor cell density was solved. The resulting hypoxic area was quantified through two different approaches and compared against the hypoxic proportions determined from the original biopsy images (Figure

Comparison of hypoxic estimations at mild (HP10), moderate (HP5) and severe (HP2.5) hypoxic levels for a representative sample case.

Herein, we have presented and utilized definitions of hypoxia corresponding to three different commonly considered threshold levels, that is, mild (HP10), moderate (HP5), and severe (HP2.5) hypoxic conditions. The percentage of total area that is hypoxic and the percentage of hypoxic readings (as determined by simulated needle electrode measurement) are then calculated with respect to these hypoxic thresholds. The results are compared against the known proportions estimated from the biopsy images (Figure

The hypoxic proportion, as estimated from the original image, is shown in yellow. The red and green boxes represent simulated hypoxia, which is quantified by estimating the percentage of total area that is hypoxic and through theoretical needle electrode measurements, respectively. In the case of HP2.5 (Figure

The binary images of hypoxic area obtained through biomarker staining reflect a number of factors related to tissue preparation, staining absorption, staining threshold, image acquisition, and image brightness. In several experimental studies [_{2} needle electrodes correlate with pimonidazole binding surface area with a systematic offset of 36%, and this offset is smallest for HP2.5 (18%). Our analysis of simulated hypoxia using eight tumor samples (Figure

Spatial correlation (%) of hypoxic area.

Tumor sample | Percentage of correlations | ||

HP2.5 | HP5 | HP10 | |

1 | 77.6738 | 73.8076 | 68.3374 |

2 | 81.1242 | 79.7579 | 74.8348 |

3 | 80.701 | 83.1489 | 81.847 |

4 | 87.6884 | 85.6093 | 78.3471 |

5 | 73.3323 | 71.2458 | 68.0627 |

6 | 80.498 | 74.8546 | 66.3771 |

7 | 75.8521 | 71.6368 | 65.637 |

8 | 76.2209 | 73.0056 | 66.1667 |

Change in hypoxia as a function of relative changes in production and consumption rates of oxygen. The figure indicates that a relative decrease in consumption might be an effective way to decrease the hypoxia.

Our definition of hypoxia also plays an important role in dictating the reliability of the estimates of hypoxic proportions obtained through computation. To test sensitivity of hypoxia estimates found using this definition with respect to changes in computational diffusion time, we analyzed HP10 values at different (nondimensional) time values and found that, for both theoretical measurement approaches, the hypoxic proportions estimated are similar for each time (result not shown). This supports the validity of our hypoxic definition, since a given tumor microenvironment with a fixed vascular network (fixed in the sense that we consider timescales too small to permit changes in perfused vascular geometry) should yield an approximately fixed hypoxic proportion over these small time intervals.

To study the relative sensitivity of various parameters involved in the present mathematical model, we have performed a comparative analysis of the variations in tissue hypoxia with respect to the relative changes in production/supply (increasing

Figure

HP5 estimations for eight tumor cross-sections using three different needle measurement approaches. (Yellow: percentage of hypoxic area from the original image, red: percentage of hypoxic area from model, and green: HP5 estimation using needle electrode.)

Uniform

Random

Radial

Here, we use statistical analysis with two purposes in mind: to consider the fraction of within-tumor variance (relative to total variance) associated with each pattern of needle insertion in an effort to predict an optimum number of tracks required for satisfactory measurement, and secondly, to determine the best tracking pattern by considering the fraction of variability between two different estimation methods among the tumor samples (relative to total variance). We note that the differences between these three different approaches to needle tracking are not clearly evident in Figure

(a) The percentage of total variance due to within tumor variance (for the entire eight samples), as a function of number of tracks. The plot shows that 5-6 tracks of 20–30 measurements give an optimal reading of hypoxia. (b) The percentage of total variance due to the variance between two methods of hypoxia estimation for three different electrode measurements approaches (analysis of eight samples). This shows that radial sampling is less accurate than uniform and random sampling.

Similar analyses comparing the variability of different oxygen measurements have been carried out in several experimental studies [

The variance analysis results of Figure

The differences among the three different needle tracking approaches are further studied with variance analysis by calculating the percentage of total variance (between-estimation methods and between-tumor sections) due to the variations between two different estimation methods (i.e., by finding the hypoxic area and through the needle electrode method). This is repeated for all three sample electrode tracking approaches and the results are shown in Figure

Plots showing the variance comparison between two different types of radial approaches (considering eight sample cases) to investigate the decreased accuracy of radial sampling.

(a) Simulated oxygen distribution as histograms of width 5 mm Hg (for a representative case), (b) oxygen modification factor (OMR) as a function of the oxygen concentration, and (c) survival fraction for different cases of oxygen profiles considering the oxygen distribution of a representative case.

The oxygenation status of a tumor is generally considered to be an important intrinsic factor in determining radiation response, where viable hypoxic cells are more resistant to radiation than well-oxygenated cancer cells [

To study the effects of oxygenation status on tumor cell survival fraction, we considered six different cases of oxygen sensitivity profiles (listed in Section _{2} concentration. Hence, considering the sensitivity of the heterogeneous distribution of oxygen at each grid point, a much higher dosage is required to get the same survival fraction of cells compared to the other four cases (Figure

Tumor hypoxia is a common feature of advanced solid tumors wherein the metabolic demand for oxygen exceeds its supply or availability [

The oxygenation status of a heterogeneous tumor is often quantified using polarographic electrode measurements or through nitroimidazole binding and biopsies. These invasive techniques often have varying accuracy due to the restricted sampling space as well as limited accessibility. In this paper, we present an alternative theoretical approach that might allow us to explore the spectrum of hypoxic distributions that could possibly be associated with a particular vascular configuration. Herein, we have used two-dimensional images of eight tumor cross-sections with perfused tumor vasculature as a computational domain on which a system of partial differential equations describing the distribution of hypoxia has been solved. As discussed in the previous sections, the resulting hypoxic area has then been quantified by two different approaches. To validate the findings from our mathematical model, these hypoxic estimates have been compared against the hypoxic proportions determined from the original images showing the hypoxic area according to pimonidazole binding.

In most tumors, the hypoxia that occurs is of mixed type [

In conclusion, we have presented a simple diffusion model, which can satisfactorily estimate the oxygenation maps of a heterogeneous tumor with a given vascular network. We have shown that an estimate can be made of average tumor hypoxia that appears to be less sensitive to the characteristics of the vascular network as compared to the variations in O_{2} consumption. Thus, this approach can be used to quantify average tumor hypoxia knowing only the distribution of tumor vessels. Using this model, we have found that the polarographic electrode measurements accurately quantify the oxygenation status of the tumor microenvironment. Our studies show that five to six uniformly distributed equidistant measurement tracks with 20–30 measurements per track give the optimum balance between accuracy and invasiveness. The radiation response under various oxygenation conditions has also been analyzed using a simple model for the radiation effect and we have found that consideration of the heterogeneous distribution of oxygen plays an important role in the accurate prescription of radiation dosage. This type of theoretical study may be used to provide an alternative method of estimating hypoxia distribution in solid tumors, which may possibly help in the design of optimal, patient-specific, and accurate invasive estimation methods.

The authors would like to thank Paul F. J. W. Rijken (Department of Radiotherapy, University of Nijmegen, The Netherlands) who provided them with original images of glioma xenographt cross-sections. They are also thankful to Colin Turner for carefully reviewing the manuscript. The support of NSERC (to M. Kohandel and S. Sivaloganathan) and NSERC/CIHR (to M. Kohandel, M. Milosevic, and S. Sivaloganathan) is gratefully acknowledged.

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_{2}histograph, [

^{3}H]misonidazole binding and paired survival assay