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Inspiratory contrast for in vivo optical imaging

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Abstract

We demonstrate the use of inspired oxygen and carbon dioxide as a possible route to increase contrast in optical imaging of cancerous tissue. Differential imaging in human xenograft rodent models of cancer exhibits significant variation in signal between normal and cancerous tissue. This differential cancer-specific contrast is stronger and more consistent than the conventional static contrast. This differential technique exploits the response of abnormal tumor vasculature to inhaled gases and could provide a promising alternative to supplement mainstream cancer imaging modalities such as x-rays and MRI.

©2008 Optical Society of America

1. Introduction

Differential or dynamical optical imaging [1–3] provides a number of advantages over conventional imaging including a significant reduction in boundary effects and improved sensitivity. The advantages have been exemplified previously in two examples of successful optical measurements in vivo that employ differential measurements: functional optical imaging of the brain [4,5], and perhaps the most successful clinical use of light in human tissue, pulse oximetry [6,7]. We are applying differential imaging to cancer detection based on contrast from tumor vasculature.

During the process of angiogenesis [8], tumors develop abnormal vasculature, and as a result, cancerous tissue is often hypoxic [9], a condition that can be observed with hemoglobin oxygenation measurements [10,11]. Hypoxic tumors are also likely to be metastatic [12,13] or invasive, and as a consequence, measurements to monitor blood oxygenation parameters have been carried out in order to guide the treatment of such tumors [14–17]. Since hypoxic cells are more radioresistant than normal cells, increasing tumor oxygenation has been explored as a possible route to improve radiation response during cancer therapy. Of the several methods being explored today [18], inhalation of hyperoxic gases such as carbogen (95% oxygen+5% carbon dioxide) have been tested in the ARCON (accelerated radiotherapy with carbogen and nicotinamide) trial [19,20], in clinical investigations [21–23] and in animal model studies [24–29]. While these various prior studies with hyperoxic gases are directed to improving cancer therapies, we are investigating whether hyperoxic and hypercapnic gas inhalation can provide contrast for cancer detection.

Inhalation of carbogen can produce two types of vascular changes: (1) vasoconstriction or vasodilation and (2) an increase in blood oxygenation. Blood vessels feeding tumor tissue are a combination of host and tumor vasculature and their specific response to vasoactive agents could be a combination of both vessel types [30]. Near infrared optical imaging is a good modality for vascular imaging because the instrumentation is compact and low cost, and there are strong signals for functional imaging. Near infrared light in the tissue transparency window (~650–900 nm) can penetrate tissue to a significant depth (many centimeters), and oxyhemoglobin (HbO2) and deoxyhemoglobin (Hb) molecules are the primary absorbers in this wavelength range. Multi-wavelength measurements allow separate determination of both total hemoglobin concentrations and hemoglobin oxygenation [31–33]. We believe the endogenous contrast afforded by spectroscopic imaging of HbO2 and Hb in combination with differential optical imaging during inhalation of exogenous vasoactive agents (hyperoxic and hypercapnic gases) can strongly accentuate the abnormal vasculature or tumors, providing a method to improve detection of cancerous tumors. We demonstrate the viability of this hypothesis by performing whole animal imaging using optical transillumination imaging.

2. Materials and methods

2.1. Animal models

All experiments using mouse animal models were approved by SRI’s Institutional Animal Care and Use Committee. Differential imaging measurements via transillumination were performed on human xenograft tumor models in athymic nude mice. Athymic female nude mice (20–25g, Harlan Laboratories) bearing human cell lines—U87 brain tumor, A549 lung tumor or DLD colon cancer cells [American Type Culture Collection, Manassas (VA)]—were used for the experiments reported here. An aliquot of 0.1 ml of cell suspension, containing 2–3 million cells was injected subcutaneously into the distal dorsal region of each mouse. Tumor volumes were measured twice weekly once they attained a measurable size. Tumor volumes were measured using the standard ellipsoid rule (pi/6*l*b*h, where l, b, and h are the tumor length, breadth, and height). Prior to imaging, each animal was anesthetized with sodium pentobarbital (30–40 mg/kg, Butler Dublin, Ohio) injected intraperitoneally at the beginning of the experiment.

2.2. Transillumination apparatus

Each anesthetized animal was partially immersed in a temperature controlled (37–38 °C) medium that matched the optical properties of the mice (Ropaque and water, µa=0.18 cm-1, µs’=12 cm-1) [34–36]. For the purpose of this matching, optical properties of a mouse were determined using the immersion technique [37] at a wavelength of 780 nm, and found to be µa=0.21 cm-1, µs’=10 cm-1. Infrared light from the LEDs (780 nm and 840 nm, Epitex, Inc.) passed through the immersion box and was collected with a digital monochrome CCD camera (Dragonfly, Point Grey Research) through a 16 mm f/0.95 lens (JML Optics). Mass flow controllers (MDC) were used to regulate the levels of air, oxygen, and carbon dioxide, which were delivered to the mice via a custom nose cone at a constant total flow rate of 4.2 L/min. The camera, LEDs, gas concentrations, and temperature measurement were interfaced to and controlled by a laptop computer on a mobile platform. A schematic of the instrument is shown in Fig. 1. Continuous wave transillumination images were obtained at each wavelength. The animals inhaled various gases (air, carbogen, oxygen, and air with elevated levels of carbon dioxide) during the course of the experiment. Data images were post-processed using MATLAB (Mathworks) and ImageJ (NIH).

 figure: Fig. 1.

Fig. 1. Experimental apparatus to acquire transillumination images. The LED box consists of two arrays of LEDs at 780 and 840 nm. The mass flow controllers deliver precise concentrations of gases to the animal during the experiment. The temperature of the tissue phantom immersion medium is held constant at around 37 °C.

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2.3. Data analysis

We analyze the images at two wavelengths to derive information on the changes in the concentration of deoxyhemoglobin (Hb) and oxyhemoglobin (HbO2). We assume that the changes in the absorption coefficient Δµa at the two wavelengths are due entirely to changes in the concentrations [Hb] and [HbO2] as follows

Δμa780=ln(10){εHb780Δ[Hb]+εHbO2780Δ[HbO2]}
Δμa840=ln(10){εHb840Δ[Hb]+εHbO2840Δ[HbO2]}

where ε are the molar extinction coefficients for Hb and HbO2 for each wavelength and the factor ln(10) accounts for the fact that µa and ε are by convention expressed in base e and base 10 exponents respectively. The appropriate extinction coefficients at 780 and 840 nm were calculated by integrating the product of the area normalized LED intensity spectrum measured with a spectrometer, I(λ), with the extinction coefficients for hemoglobin (each hemoglobin molecule contains four heme groups) [38] over the wavelength spread in the spectrum as

εHb,HbO2780,840=λε(λ)I(λ)dλ.

Image analysis was performed using a modified Beer-Lambert analysis using differential changes [27,39,40]. Specifically, we calculate the change in the absorption coefficient between transient images when the animal breathes particular hyperoxic or hypercapnic gases, and baseline images where the animal breathes air. Equations (1) and (2) show that a change in absorption corresponds to a change in the concentrations of [Hb] and [HbO2]. The change in absorbance can also be related to the baseline and transient intensities measured during a typical transillumination experiment, IB and IT, with the animal breathing air and various gas mixtures according to

1L780ln(IBIT)780=ln(10){εHb780Δ[Hb]+εHbO2780Δ[HbO2]}
1L840ln(IBIT)840=ln(10){εHb840Δ[Hb]+εHbO2840Δ[HbO2]}

where L is the pathlength factor for the region of change. Inverting equations (3) and (4) yields a matrix equation relating the changes in Hb and HbO2 concentrations with the differential intensity at each wavelength,

(Δ[Hb]Δ[HbO2])=1ln(10)(εHb780εHbO2780εHb840εHbO2840)1(1L780ln(IBIT)7801L840ln(IBIT)840).

It is important to emphasize that the changes in Hb and HbO2 concentrations calculated by this method are only semi-quantitative given that there is no precise measurement to determine the optical pathlength at each wavelength. In the present analysis, L780 is assumed to be equal to L840. The pathlength depends on the different tissue types that the transmitted light traverses as well as the size of the changing volume. Because we use differential measurements, contributions from the tissue phantom and heterogeneity in the unresponsive tissue regions of the animal do not impact the measurements. Although the measurements are only semi-quantitative, these differential measurements may offer a possible route to detect cancerous tissue.

Image analysis was executed via MATLAB code. Raw images acquired by the camera were smoothed via spatial convolution with a 2D Gaussian function of FWHM of 2.5 pixels and a total kernel width of ±5 pixels and then decimated by a factor of 16. Difference images were then generated by subtracting the logarithm of the transient images from the logarithm of the baseline images. Finally, using equations 3 and 4, relative changes in Hb and HbO2 were calculated from the difference images.

3. Results

3.1. U87 and DLD cancer cell lines

We report experiments conducted on four animals bearing subcutaneously implanted U87 brain tumors and one animal bearing a DLD colon cancer. All five animals inspired various combinations of air (100%), hypercapnic (air + 5 or 10% carbon dioxide) and hyperoxic (100% oxygen or oxygen +5–15% carbon dioxide) gases.

Tables Icon

Table 1. Mouse statistics.

Two imaging positions were used to collect data. In the first (ventral) position, the tumor (on the dorsal side of the animal) is on the distal side of the animal as viewed by the camera (i.e., closer to the LED array). Light from the array thus passes through the tumor tissue and traverses through the entire animal body before reaching the camera. In the second (dorsal) position, the tumor faced the camera. In this case, light exiting the tumor reaches the camera directly. Images of three mice as seen from the camera without the immersion medium in the ventral imaging position are shown in Fig. 2. The three animals include one animal (#1) in which an implanted tumor never grew (Fig. 2a). This animal is used as a control animal. The position of the tumor in these ventral-imaging experiments is determined visually when the animal is flipped to collect data in the dorsal imaging position.

 figure: Fig. 2.

Fig. 2. Direct images of mice in the ventral imaging position with the tumor facing away from the camera. (a) Normal, no tumor (#1), (b) U87 brain tumor (#3) and (c) DLD colon cancer (#6).

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In order to visualize signal changes in the transient images, we compute the difference images at each wavelength. These are a measure of the relative change in absorbance (ΔA). A region of interest corresponding to the tumor area is identified and normalized (to the area of the tumor) and difference intensities are obtained as a function of gas intervention. A typical temporal plot following this analysis is shown in Fig. 3a and corresponds to the tumor region in animal #3. The dynamic response due to hyperoxic gas intervention is dramatic. The isobestic point in the Hb and HbO2 absorption spectrum lies close to 800 nm. We note that the change in absorbance observed for both wavelengths is out of phase, indicating the change is primarily a change in oxygenation for this animal, not vasoconstriction. The measured dynamics for Hb and HbO2 concentrations exhibit similar trends to those measured with point (non-imaging) methods [27,41], namely, that the concentration of oxyhemoglobin increases with hyperoxia while the concentration of deoxyhemoglobin decreases, and the total concentration of hemoglobin remains approximately constant. These are shown in Fig. 3b.

 figure: Fig. 3.

Fig. 3. (a) Change in absorbance from the tumor in animal #3 at 780 and 840 nm. The various inspired gases are indicated by colored shading. The Air+CO2 mixture contains 5% CO2, the first carbogen mixture contains 5% CO2 and the second carbogen mixture contains 10% CO2. No color shading is used for inhalation of Air. (b) Changes in relative concentrations of Hb, HbO2 and Hbtotal for the same data.

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The variation in absorbance can be converted to relative changes in Hb and HbO2 concentration using equations 5 and 6. Such an analysis can be performed for all the images obtained in a single experiment and relative concentrations maps for changes in Hb and HbO2 concentration can be used to locate a tumor and its response to gas intervention. To demonstrate the power of this imaging modality, concentration maps from a normal mouse (#1) were compared with those obtained from tumor bearing animals. In Fig. 4, concentration maps for relative changes in HbO2 concentration upon hyperoxic gas intervention are shown for animals imaged in the ventral position, corresponding to Fig. 2. The static images seen in Fig. 4, column (a) do not show consistent signatures in the tumor location. Some increased intensity is seen near the tail region. However, this signal is seen in all animals studied in this imaging position and could be related to the location of the bladder. In contrast to the static images, the tumor position can be localized with relative ease using the HbO2 concentration maps when the animal breathes a hyperoxic gas. All images shown in Fig. 4, columns b-d, were obtained towards the end of the respective gas cycle in order to minimize transient effects. Specifically, for animal #6, significant changes are seen around the boundary of the tumor closer to the center of the animal body and relatively reduced intensity is seen at the center of the tumor (column (c)). Such signal changes localized to tumor boundaries might indicate regions of active tumor growth.

 figure: Fig. 4.

Fig. 4. Comparison between static transillumination images and relative HbO2 concentration maps for a normal mouse (#1) and two tumor bearing mice (#3, 6). Images from a particular animal are shown in a single row. Column (a) includes static transillumination images obtained at 780 nm. Column (b) shows relative HbO2 concentration maps when the animals breathe air. Column (c) shows the relative change in HbO2 concentration upon hyperoxic gas intervention (oxygen, carbogen and oxygen for #1, 3, and 6, respectively). Column (d) shows relative HbO2 concentrations fall back to those noted in column (b) when the hyperoxic gas is turned off. Tumor and animal outlines are shown in black. The units on the intensity scale bar for images in columns (b)–(d) are cm*mM/L. Note that U87 #3 is a relatively shallow and wide tumor while DLD #6 is a relatively deep and narrow tumor.

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When the tumor is on the distal side of the animal (away from the camera) as in the ventral imaging position, the signal from the tumor diffuses out prior to reaching the camera due to its passage through the remaining animal body. Even so, differential imaging shows significant contrast from the tumor tissue. An approximate estimate of the increase in HbO2 concentration in the tumor tissue under hyperoxic gas intervention can be calculated by determining the value of the pathlength L for the tumor. The pathlength L equals the product of the differential pathlength factor (DPF) and d, the thickness of the changing volume. We estimate [42] the DPF value for the mouse tissue to be approximately 6 and is calculated from the µa and µs’ values obtained from measurements in our lab (Materials and Methods). Using this number, an approximate thickness of 0.5 cm for the mouse tumor tissue (this was measured during tumor volume measurements), and subtracting the absorbance change for the normal mouse, we find the magnitude of the maximum signal obtained from the tumor tissue alone is 8 µM. Although this value is approximate because we have not measured the optical properties of the tumor itself, it is very close to values in the literature for the increase in HbO2 concentration upon carbogen intervention in xenograft studies [43]. The images of Fig. 4 exhibit heterogeneity of oxygenation improvement. Spatial heterogeneity also has been observed by inserting oxygen-measuring probes into the tumor mass at different sites [44].

To further increase contrast from the tumor, some animals were imaged in the dorsal position, corresponding to the tumor facing the camera. Animal images as seen by the camera without the immersion medium are shown in Fig. 5. Part of the tumor tissue appears to be touching the wall of the immersion box with some immersion medium trapped between the animal body and the box.

 figure: Fig. 5.

Fig. 5. Direct images of mice in the dorsal imaging position with the tumor facing the camera. (a) U87 brain tumor (#3), (b) U87 brain tumor (#4) and (c) DLD colon cancer (#6).

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Again, relative changes in HbO2 concentration were calculated. These are shown in Fig. 6 where static transillumination images are again shown to compare contrast from the tumor between the static and differential images. There appear to be some bright zones in the tumor region in the static images. However, these may be artifacts due to contact of the tumor with the wall of the immersion box or air pockets. If these are indeed artifacts, the absence of a signal from these zones in the difference images would demonstrate one advantage of differential measurements–the removal of features not responding to the gas protocol. In any case, we again observe significant contrast from the tumor tissue in the dynamic HbO2 concentration maps. Specifically, tumor signal from animal #3 during hyperoxic gas intervention is considerably larger that obtained during air inhalation. Animals 4 and 6 show similar response but with decreased comparative signal. As might be expected when the tumor faces the camera, the dorsal imaging data shown in Fig. 6 appears to provide better contrast from the tumor and allow for more accurate tumor localization.

Another interesting observation from column (d) of Fig. 6 is that a residual response persists in the tumor region even when the inhaled gas has switched back to air, specifically for animals 3 and 6. This persistent response may be related to the abnormal tumor vasculature, such as leaks causing the retention of oxygenated blood in a manner similar to dynamic imaging with MRI for cancer imaging, such as the wash in/wash out of gadolinium contrast agent [45,46]. This second type of signature (delayed response) could prove useful for clinical applications of differential optical imaging.

Data from animals 2 and 5 are not shown. While animal #2 showed expected contrasts from tumor tissue, animal #5 did not. For the U87 brain tumor cell line, our results demonstrate enhanced differential contrast following hyperoxic or hypercapnic gas inhalation, in three out of four animals. Dissections were performed on animals 2 and 3. Significant vasculature was observed to be feeding the tumor. The most successful data set also corresponds to the smallest measured tumor size, i.e. animal #3. This observation may relate to the degree of tumor necrosis with increasing tumor size.

 figure: Fig. 6.

Fig. 6. Comparison between static transillumination images and relative HbO2 concentration maps for three tumor-bearing mice (#3, 4 and 6). Images from a particular animal are shown in a single row. Column (a) includes static transillumination images obtained at 780 nm. Column (b) shows relative HbO2 concentration maps when the animals breathe air. Column (c) shows the relative change in HbO2 concentration upon hyperoxic (oxygen) gas intervention. Column (d) shows relative HbO2 concentrations fall back to those noted in column (b) when the hyperoxic gas is turned off. Tumor and animal outlines are shown in black. The units on the intensity scale bar for images in columns (b)–(d) are cm*mM/L.

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To better visualize these results, videos corresponding to the entire image sequence for animals 3 and 4 are shown in Figs. 7 and 8 respectively. Relative changes in HbO2 concentration are shown as a function of various inspired gases. These videos demonstrate the power of this imaging modality to detect cancerous tissue via differential imaging.

 figure: Fig. 7.

Fig. 7. (2.1 MB) Movie showing the dynamic change in relative HbO2 concentration as a function of inspired gases for animal #3. The units on the intensity scale bar are cm*mM/L. [Media 1]

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 figure: Fig. 8.

Fig. 8. (2.1 MB) Movie showing the dynamic change in relative HbO2 concentration as a function of inspired gases for animal #4. The units on the intensity scale bar are cm*mM/L. [Media 2]

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3.2. A549 Lung cancer cell line

We found rather different results for the A549 tumor from a single animal bearing the tumor on its dorsum, closer to the right flank. Images were taken with the animal in two positions, one with the tumor facing the camera and the other with the tumor facing the light source. Differential image analysis did not result in dramatic contrast at the location of the tumor when the animal inspired carbogen (or another hyperoxic/hypercapnic gas mixture). However, the change in absorbance and correspondingly, the relative change in the Hb and HbO2 concentration as a function of time showed some interesting variation with the inspired gases. These data are shown in Fig. 9 and correspond to normalized integrated signal from the tumor tissue. From the figure, it appears that inspired gases containing carbon dioxide result in a vasoconstriction effect. This is concluded by observing the net decrease in the total hemoglobin concentration when the animal breathes carbogen or a mixture of air and carbon dioxide. In this experiment, elevated levels of carbon dioxide were used, i.e. 15% and this may contribute to the observed vasoconstriction. Histology or dissection studies were not performed on this animal and the data from this run remain inconclusive. It is unclear whether the tumor had significant vasculature associated with it or whether the tumor was largely necrotic.

 figure: Fig. 9.

Fig. 9. Dynamic change in the differential intensity observed for the tumor zone during the inhalation of various gases. The gas mixtures cycled over the entire experiment were in the following order: Air, carbogen (85% oxygen + 15% carbon dioxide in this case), oxygen (85% oxygen + 15% air) and carbon dioxide (85% air + 5% carbon dioxide).

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4. Discussion and conclusions

The present technique demonstrates the viability of using a differential imaging technique in the near infrared to image cancerous tissue. Vasoactive exogenous agents are used to increase contrast by altering the concentration of HbO2 in the tumor vasculature as compared to normal tissue. Highly vascular tumors such as the U87 brain tumors appear to provide the best contrast in this new imaging modality. Three of four U87 tumors and one DLD colon cancer tumor showed substantial contrast from tumor tissue when analyzed using differential imaging algorithms in the present study. In this case we are comparing the tumors to the mouse’s internal organs, which are themselves quite vascularized. We can expect an easier situation for breast imaging, which is the ultimate goal. Sodium pentobarbital can impact blood flow and oxygenation. We plan to use other methods of anesthesia such as inhalation of isoflurane, which is known to have a lesser effect on vascular response [47]. Future studies are aimed at tracking tumor growth with this imaging modality for a syngeneic rat tumor (R3230 AC) and implementation of this technique to study breast tumors in preliminary clinical studies. These studies will be performed on women with suspected breast cancer using breathing protocols adapted for human studies with pre-prepared gas mixtures.

Acknowledgements

This research was supported by funds from the DOD Breast Cancer Research Program, Grant Number DAMD17-02-1-0570 and the California Breast Cancer Research Program of the University of California, Grant Number 10EB-0168. A.D. Gibbs was supported by the National Science Foundation through the Research Experiences for Undergraduates Program. We acknowledge helpful conversations with Dr. Mark Dewhirst of Duke University Medical Center and the assistance of William N. Boenig for the measurements of the mouse optical properties.

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Supplementary Material (2)

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Figures (9)

Fig. 1.
Fig. 1. Experimental apparatus to acquire transillumination images. The LED box consists of two arrays of LEDs at 780 and 840 nm. The mass flow controllers deliver precise concentrations of gases to the animal during the experiment. The temperature of the tissue phantom immersion medium is held constant at around 37 °C.
Fig. 2.
Fig. 2. Direct images of mice in the ventral imaging position with the tumor facing away from the camera. (a) Normal, no tumor (#1), (b) U87 brain tumor (#3) and (c) DLD colon cancer (#6).
Fig. 3.
Fig. 3. (a) Change in absorbance from the tumor in animal #3 at 780 and 840 nm. The various inspired gases are indicated by colored shading. The Air+CO2 mixture contains 5% CO2, the first carbogen mixture contains 5% CO2 and the second carbogen mixture contains 10% CO2. No color shading is used for inhalation of Air. (b) Changes in relative concentrations of Hb, HbO2 and Hbtotal for the same data.
Fig. 4.
Fig. 4. Comparison between static transillumination images and relative HbO2 concentration maps for a normal mouse (#1) and two tumor bearing mice (#3, 6). Images from a particular animal are shown in a single row. Column (a) includes static transillumination images obtained at 780 nm. Column (b) shows relative HbO2 concentration maps when the animals breathe air. Column (c) shows the relative change in HbO2 concentration upon hyperoxic gas intervention (oxygen, carbogen and oxygen for #1, 3, and 6, respectively). Column (d) shows relative HbO2 concentrations fall back to those noted in column (b) when the hyperoxic gas is turned off. Tumor and animal outlines are shown in black. The units on the intensity scale bar for images in columns (b)–(d) are cm*mM/L. Note that U87 #3 is a relatively shallow and wide tumor while DLD #6 is a relatively deep and narrow tumor.
Fig. 5.
Fig. 5. Direct images of mice in the dorsal imaging position with the tumor facing the camera. (a) U87 brain tumor (#3), (b) U87 brain tumor (#4) and (c) DLD colon cancer (#6).
Fig. 6.
Fig. 6. Comparison between static transillumination images and relative HbO2 concentration maps for three tumor-bearing mice (#3, 4 and 6). Images from a particular animal are shown in a single row. Column (a) includes static transillumination images obtained at 780 nm. Column (b) shows relative HbO2 concentration maps when the animals breathe air. Column (c) shows the relative change in HbO2 concentration upon hyperoxic (oxygen) gas intervention. Column (d) shows relative HbO2 concentrations fall back to those noted in column (b) when the hyperoxic gas is turned off. Tumor and animal outlines are shown in black. The units on the intensity scale bar for images in columns (b)–(d) are cm*mM/L.
Fig. 7.
Fig. 7. (2.1 MB) Movie showing the dynamic change in relative HbO2 concentration as a function of inspired gases for animal #3. The units on the intensity scale bar are cm*mM/L. [Media 1]
Fig. 8.
Fig. 8. (2.1 MB) Movie showing the dynamic change in relative HbO2 concentration as a function of inspired gases for animal #4. The units on the intensity scale bar are cm*mM/L. [Media 2]
Fig. 9.
Fig. 9. Dynamic change in the differential intensity observed for the tumor zone during the inhalation of various gases. The gas mixtures cycled over the entire experiment were in the following order: Air, carbogen (85% oxygen + 15% carbon dioxide in this case), oxygen (85% oxygen + 15% air) and carbon dioxide (85% air + 5% carbon dioxide).

Tables (1)

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Table 1. Mouse statistics.

Equations (6)

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Δ μ a 780 = ln ( 10 ) { ε Hb 780 Δ [ Hb ] + ε HbO 2 780 Δ [ Hb O 2 ] }
Δ μ a 840 = ln ( 10 ) { ε Hb 840 Δ [ Hb ] + ε HbO 2 840 Δ [ Hb O 2 ] }
ε Hb , Hb O 2 780 , 840 = λ ε ( λ ) I ( λ ) d λ .
1 L 780 ln ( I B I T ) 780 = ln ( 10 ) { ε Hb 780 Δ [ Hb ] + ε HbO 2 780 Δ [ HbO 2 ] }
1 L 840 ln ( I B I T ) 840 = ln ( 10 ) { ε Hb 840 Δ [ Hb ] + ε HbO 2 840 Δ [ HbO 2 ] }
( Δ [ Hb ] Δ [ HbO 2 ] ) = 1 ln ( 10 ) ( ε Hb 780 ε HbO 2 780 ε Hb 840 ε HbO 2 840 ) 1 ( 1 L 780 ln ( I B I T ) 780 1 L 840 ln ( I B I T ) 840 ) .
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