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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 12 — Jun. 13, 2005
  • pp: 4465–4475
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Investigation of bi-phasic tumor oxygen dynamics induced by hyperoxic gas intervention: A numerical study

Jae G. Kim and Hanli Liu  »View Author Affiliations


Optics Express, Vol. 13, Issue 12, pp. 4465-4475 (2005)
http://dx.doi.org/10.1364/OPEX.13.004465


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Abstract

This study intends to explore the underlying principle of the bi-phasic behavior of increases in oxygenated hemoglobin concentration that was observed in vivo from rat breast tumors during carbogen/oxygen inhalation. We have utilized the Finite Element Method (FEM) to simulate the effects of different blood flow rates, in several geometries, on the near infrared measurements. The results show clearly that co-existence of two blood flow velocities can result in a bi-phasic change in optical density, regardless of the orientation of vessels. This study supports our previous hypothesis that the bi-phasic tumor hemodynamic feature during carbogen inhalation results from a well-perfused and a poorly perfused region in the tumor vasculature.

© 2005 Optical Society of America

1. Introduction

To explain this biphasic behavior of tumor hemodynamics, we established a mathematical model based on Kety’s approach [13

13. S. S. Kety, “The theory and applications of the exchange of inert gas at the lungs and tissue,” Pharmacol. Rev. , 3, 1–41 (1951). [PubMed]

] in our earlier publication [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

]. We formed a hypothesis in Ref. 10 that tumor vasculature is comprised of a well-perfused and poorly perfused region that can be detected with the two time constants through Δ[HbO2] readings derived from near infrared spectroscopy (NIRS). The mathematical model basically allowed us to associate the bi-phasic Δ[HbO2] amplitudes and time constants to the ratio of vascular coefficients and vascular perfusion rates in the two different regions [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

]. While the mathematical model seemed useful for interpretation of tumor hemodynamics and physiological factors, it was a suggested model without experimental or computational proof or confirmation at the time. To provide solid support and better understanding for this model and to further investigate heterogeneities of tumor vasculature, we have used a computational approach to validate the dynamic NIRS measurements. The computational results given in this study strongly demonstrates that with our bi-phasic mathematical model, tumor vascular dynamics can be determined and monitored non-invasively using NIRS while a perturbation of hyperoxic gas intervention is given.

Fig. 1. Normalized hemodynamic changes of tumor blood oxygenation, Δ[HbO2], obtained with the NIRS measurement from a rat breast tumor while the breathing gas was switched from air to carbogen (Gu et al. Applied Optics, 2003) [12].

2. Review of the our mathematical model of tumor vascular oxygenation

Δ[HbO2]vasculture(t)=γHo[1exp(ftγ)]=A[1exp(tτ)],
(1)

where γ was defined as the vasculature coefficient of the tumor (=Δ[HbO2]vasculature/Δ[HbO2]vein), Ho was the arterial oxygenation input, f represented the blood perfusion rate in cm3/sec, τ is the time constant, A=γHo, and τ=γf.

If a tumor has two distinct perfusion regions and the measured NIRS signals result from the both regions (Fig. 2), it is reasonable to include two different blood perfusion rates, f 1 and f 2, and two different vasculature coefficients, γ 1 and γ 2, in the model. Equation (1) becomes Eq. (2) to count for the double exponential feature observed in the NIRS experiments:

Δ[HbO2]vasculture(t)=γ1Ho[1exp(f1tγ1)]+γ2Ho[1exp(f2tγ2)]
=A1[1exp(tτ1)]+A2[1exp(tτ2)]
(2)

where f 1 and γ 1 are the blood perfusion rate and vasculature coefficient in the well perfused region, respectively; f 2 and γ 2 represent the same respective meanings for the poorly perfused region, and A1=γ 1Ho, A2=γ 2Ho, τ1=γ 1/f 1, τ2=γ 2/f 2. Since A1, A2, τ1, and τ2 can be determined by fitting Eq. (2) with Δ[HbO2] readings taken from the NIRS measurements, we can obtain the ratios of two vasculature coefficients and the two blood perfusion rates as:

γ1γ2=A1A2f1f2=A1A2τ1τ2
(3)

With these two ratios, we are able to understand more about tumor vascular structures and blood perfusion rates. In this paper, we report our computational evidence to support the tumor hemodynamics model by quantifying γ12 and f 1/f 2 from three different locations of the simulated tumor dynamic phantoms.

Fig. 2. A schematic diagram of light transmitting patterns in a tumor when the tumor has two distinct perfusion regions. The right side of tumor with gray color represents the poorly perfused region, whereas the left side of tumor corresponds to a well-perfused region. As shown, different detectors may interrogate different tumor volumes.

3. Computer simulations using the finite element method

The Finite Element Method (FEM) was utilized to simulate the bi-phasic behavior of increases in Δ[HbO2] with FEMLAB software (COMSOL Inc. Burlington, MA). It uses the numerical approach to solve partial differential equations (PDE) in modeling and simulating various engineering problems. The geometry of our FEM simulations is given in Figure 3, representing the simplified tumor vascular model (Fig. 2). E1 represents an overall tumor volume (diameter =4 cm), and R5 shows the location of light source. Blood vessels in two different perfusion regions are represented by several rectangles (0.1 cm×2.4 cm): R1 and R2 represent vessels with a fast flow rate, and R3 and R4 denote vessels with a slow flow rate.

As given in Eq. (4), the diffusion equation was applied to predict the measured NIR light intensities along the boundary of simulated model as the simulated oxygenated blood flows through the blood vessels with two different perfusion rates in tumors:

(1c)(t)ϕ(r,t)D2ϕ(r,t)+μaϕ(r,t)=S(r,t)
(4)

where ϕ(r,t) is the diffuse photon fluence rate at the position r, c is the speed of light in tissue, S(r,t) describes the photon source, D=[3(µa +µs ′)]-1 is the diffusion coefficient, µa is the light absorption coefficient in tissue, and µs is the reduced light scattering coefficient in tissue. For the boundary conditions, the extrapolated boundary condition which mathematically assumes that photon flux vanishes at the extrapolated distance, Ze [14

14. A.H. Hielsher, S. L. Jacquest, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. , 40, 1957–1975 (1995). [CrossRef]

], was applied to the FEM model as given below:

Ze=ϕ(z=0)[(z)ϕ(r,z,t)t=0]=2AD,ϕ(r,z=ze,t)=0
(5)

where A=(1+rd )/(1-rd ), and rd is the internal reflectance caused by the refractive index mismatch between air and tissue. This can be estimated using the following empirically determined equation [15

15. R.A. Groenhuis, A.A. Ferwerda, and J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. , 22, 2456–2462 (1983) [CrossRef] [PubMed]

]:

rd=1.440n2+0.710n1+0.668+0.0636n
(6)

with n=ntissue /nair .

In the simulation, 1.4 and 1.0 were used for ntissue and nair , respectively, to obtain A. A value of D=0.033 cm was chosen for both the background and vasculature of phantom with µs =10 cm-1. The values of 0.03 cm-1 and 1.5 cm-1 were selected as absorption coefficients of the tissue background (E1) and oxygenated blood flowing through the simulated vessels (R1, R2, R3 and R4), respectively. In this simulation model, the absorption coefficients of perfused blood prior to carbogen intervention was assumed to be the same as tissue background since we are measuring only changes of tumor blood oxygenation from the baseline (air) to carbogen intervention. Therefore, the value of 1.5 cm-1 used in R1–R4 can be considered as a difference in absorption between preperfused blood and oxygenated blood after carbogen intervention. The simulation model was generated with FEMLAB having 1147 elements and 609 nodes. Finally, the model was solved using the stationary nonlinear solver type.

Fig. 3. The geometry used in our FEM simulations for a simplified tumor vascular model. R1 and R2 rectangles are located in a fast flow region, while R3 and R4 are in a slow flow region within tumor. The units for both the X-axis and Y-axis are cm. The distances from R1 to R2 and from R2 to R3 are 1 cm and 0.5 cm, respectively.

To model the dynamic NIR signals, multiple FEM runs of the diffusion model with Eqs. (4)(6) were performed repeatedly with different lengths of R1, R2 and R3, R4, where the µa value of 1.5 cm-1 was used to simulate oxygenated blood within R1, R2, R3, and R4 regions. Each of the computed run/frame from the model was associated with the blood perfusion in the two vascular regions at a selected time. By assuming both vascular regions in tumors are having a same vascular density, the perfusion rates of two regions is directly proportional to the blood flow rates. Therefore, the two different flow rates/perfusion rates passing through the two regions in tumors were mimicked by progressing the lengths of R1, R2 from 0 to 2.4 cm with an increment of 0.4 cm per frame to represent the fast flow process, and the lengths of R3, R4 with a smaller increment of 0.02 cm per frame were used to evolve the slow flow.

Figure 4 shows an example of a series of continuous FEM outputs for the fast flow case, where each of the output frames corresponds to a time interval of 2 seconds. The frame rate in the calculation was kept the same for both fast and slow cases; thus, a series of discrete outputs of the FEM model can replicate the time-dependent NIR signals taken from the in vivo tumor model with a flow rate difference as large as 20 times between the two different perfusion regions.

Fig. 4. Light distribution inside of a simplified tumor vascular model simulated by FEM with the increase of R1 and R2 length to mimic the oxygenated blood flow in the well perfused region. (Movie: 267 KB)

To investigate if the bi-phasic hemodynamic feature depends on the orientations of the vessels, we also changed the position of source light to examine the effect of vessel geometry on NIR signals taken from the tumor hemodynamic measurements. We have simulated the light source to be a) perpendicular to the vessels of the phantom and b) at the center of the phantom. In the former case, the light penetrates the slow-flow vessels first and then the fast-flow vessels. Such a simulation allowed us to investigate if in this geometry, the dynamic changes of NIR signals still have the bi-phasic behavior.

ΔO.D.=log(ϕinitialϕtransient)
(7)

where ϕinitial and ϕtransient are the photon fluence rates at the initial and transient states.

4. Results from the FEM simulation

Figure 4 given above shows light distributions in a simplified tumor vascular model from seven simulation outputs to mimic a fast oxygenated blood flow in tumor by increasing the length of R1 and R2 with an increment of 0.4 cm per each frame, or 0.2 cm per second. The 0th frame shows the light distribution in tumor vascular model when there is no blood flow, and all the other frames represent the light distributions with a fast oxygenated blood flow in R1 and R2. In a similar fashion, a slow flow rate of oxygenated blood in the poorly perfused region was simulated by increasing the length of R3 and R4 with a much slower rate of 0.02cm/frame (0.01 cm/sec).

Figure 5(a1) presents the light distribution of the simulated model when a fast oxygenated blood flow passed through R1 and R2 with a rate of 0.2 cm/sec; similarly, Fig. 5(b1) shows the light distribution when an oxygenated blood flow went into R3 and R4 with a slow flow rate of 0.01 cm/sec. Finally, Fig. 5(c1) shows the combined light distribution in the phantom with both fast and slow flows in the two different regions. Figure 5(a1) is the result at the 6th frame, while Fig. 5(b1) results from the 120th frame. Figure 5(c1) is also the simulation output at the 120th frame when the oxygenated blood flows passed through the entire lengths of all the simulated vessels.

In comparison with the results from our animal experiments [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

12

12. Y. Gu, V. A. Bourke, J. G. Kim, A. Constantinescu, R. P. Mason, and H. Liu, “Dynamic response of breast tumor oxygenation to hyperoxic respiratory challenge monitored with three oxygen-sensitive parameters,” Appl. Opt. , 42, 2960–2967 (2003). [CrossRef] [PubMed]

], we extracted the light intensity values (proportional to the photon fluence rates, ϕ) at three positions of (2,0), (-2,0) and (0,2) from each frame of the simulations to calculate ΔO.D. values, which are plotted in the right column of Fig. 5. These three positions are corresponding to D1, D2 and D3 in Fig. 2. The time unit in these plots was obtained by associating each frame to 2 seconds. Thus, ΔO.D. shown in Fig. 5(a1) has 12 seconds to reach the maximum ΔO.D. since it has only 6 frames to simulate a fast flow rate, with a velocity of 0.2cm/sec. Similarly, ΔO.D. values in Fig. 5(b1) and 5(c1) will have 240 seconds to achieve their maximums because they have 120 frames to simulate a slow flow rate, with a velocity of 0.01 cm/sec.

Fig. 5. Light distributions inside of a simplified tumor vascular model simulated by the FEM. Left column: (a1) is the output result with an only fast simulated flow rate (R1 and R2), (b1) is the result with an only slow flow rate (R3 and R4) (Movie: 1,763 KB), and (c1) is the result with both fast and slow flow combined (R1, R2, R3 and R4) (Movie: 1,643 KB). Right column: Optical density changes measured at three locations, (2,0), (-2,0), and (0,2), in the FEM simulations during fast flow only (a2), slow flow only (b2), and both fast and slow flow combined (c2).

Figure 5(a2) and 5(b2) show temporal ΔO.D. profiles taken from the three positions during a fast flow only and a slow flow only simulation, respectively. The former one shows that the ΔO.D. is the largest at (-2, 0) position and is the smallest at (2, 0) position when an oxygenated blood flow passes only into the vessels (R1 and R2) near D2 in the simulation. Similar results are observed when the blood flowed only into the vessels in the slow perfusion region (R3 and R4), as seen in Fig. 5(b2). Namely, the ΔO.D. values are much larger at (2, 0) position than at (-2, 0). Moreover, Fig. 5(c2) shows that the temporal profiles of ΔO.D. taken at (2, 0) and (-2, 0) positions do not change significantly in comparison with those given in Figs. 5(a2) and 5(b2). However, in this case, the temporal ΔO.D. profile at (0, 2) position clearly shows a bi-phasic behavior, similar to that shown in Fig. 1, as we often observed in the animal tumor studies [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

12

12. Y. Gu, V. A. Bourke, J. G. Kim, A. Constantinescu, R. P. Mason, and H. Liu, “Dynamic response of breast tumor oxygenation to hyperoxic respiratory challenge monitored with three oxygen-sensitive parameters,” Appl. Opt. , 42, 2960–2967 (2003). [CrossRef] [PubMed]

]. Notice that the portions in the ΔO.D. profile seem to be equally weighted by the fast and slow flows, implying that the fast and slow flows contribute to the measured NIR signals approximately equivalently. This indeed supports that it is necessary to contain two distinct flow or perfusion rates within tumors in order to exhibit the bi-phasic blood oxygenation dynamics during carbogen/oxygen inhalations.

We also changed the position of light source to be perpendicular to the vessels on the simulated phantom, as shown in Fig. 6(a1), or to be at the center of the phantom, shown in Fig. 6(b1). In this way, we can investigate how blood vessel geometry within the tumor/phantom affects the bi-phasic feature of the tumor hemodynamics.

Fig. 6. FEM simulations of light distribution inside of a simplified tumor vascular model. Left column: (a1) shows the results when the light source is located perpendicular to the vessels (Movie: 1,539 KB), (b1) presents the results when the light source is located in the center of the model (Movie: 1,540 KB). Right column: Changes in O.D. measured at three locations, as labeled in Fig. 6(a1), with a fast and slow flow combined. (a2) plots three ΔO.D. profiles measured at the respective locations; (b2) reveals four ΔO.D. temporal profiles during the combined fast and slow flow.

Similar to Fig. 5(c1), Fig. 6(a1) shows light distribution within the simulated model when an oxygenated blood flow passed through R1 and R2 with a fast flow rate of 0.2 cm/sec and through R3 and R4 with a slow rate of 0.01 cm/sec. Temporal profiles of ΔO.D. at the three corresponding positions were quantified and plotted in Figs. 6(a2). It shows that the detector at (0, 2) sees the change in optical density with a single-exponential shape, more dominated by the slow flow, while the detector at (0, -2) detects a small initial rise in ΔO.D. induced by the fast flow, followed by a gradual plateau and then a large increase by the slow flow. Such features can be expected based on their detection positions. Interestingly, the ΔO.D. profile taken at (-2, 0) exhibits an unambiguous bi-phasic exponential curve that results from both the fast and slow flow. This curve resembles very well the previously observed feature in our animal studies (see Fig. 1 as an example).

Figures 6(b1) and 6(b2) simulate the light distribution and ΔO.D. profiles, respectively, obtained from the four positions of the tumor vascular dynamic phantom with the light source located at the center. The latter one displays that the detector at (-2, 0) is most sensitive to the signal from the fast flow only, while the readings of ΔO.D. at (2, 0) is affected by the fast flow at the initial onset and followed by a gradual increase. Furthermore, the ΔO.D. readings at (0, 2) and (0, -2) reveal somewhat bi-phasic behaviors with a fast increase in ΔO.D. initially, followed by an exponential and delayed exponential rise, respectively. These two bi-phasic profiles can be attributed to the fact that the detected NIR signals at (0, 2) and (0, -2) interrogated both fast and slow perfusion regions.

5. Discussion and conclusion

In this study, we employed the FEM method to simulate the bi-phasic behavior that was frequently observed in blood oxygenation from animal tumors during carbogen/oxygen inhalation. We believe that the bi-phasic feature of tumor blood oxygenation during hyperoxic gas inhalation results from two distinct vascular structures of the tumor, namely, a well-perfused and poorly perfused region. Our numerical simulations were performed to explore what can cause tumor hemodynamics to have two time constants, i.e., the bi-phasic feature. From the simulation results, it is confirmed that co-existence of two blood flow velocities can result in a bi-phasic change in optical density, thus leading further to a bi-phasic change in hemodynamics in tumor vasculature.

A comparison between Figs. 5(c2) and 6(a2) reveals that the bi-phasic or bi-exponential feature can be well present if both slow and fast perfusion regions exist within the interrogated area or volume of NIR source and detector, regardless of the orientation of vessels. Single exponential or non-exponential component of ΔO.D. exists if the NIR source and detector interrogates only the fast or slow perfusion area. Moreover, the contribution of each perfusion region to the NIR signal appears to be proportional to the vascular area or volume, i.e., the areas of R1, R2, R3, and R4 in this study. While we learned that different flow velocities gave rise to bi-exponential profiles, such differences in flow velocity could arise from different blood vessel diameters with the same blood flow rate or from different blood flow rates with the same vessel diameter, both of which tumor vasculatures have. Detailed association between the measured NIR signals and vascular density and vessel sizes needs to be explored further in our future studies.

In comparison, such a bi-phasic change in hemodynamics has been observed in MRI (Magnetic Resonance Imaging) studies during hypercapnic intervention or brain functional stimulation [16

16. J. B. Mandeville, J.J.A. Marota, C. Ayata, G. Zaharchuk, M.A. Moskowitz, B. R. Rosen, and R. M. Weisskoff, “Evidence of a cerebrovascular postarteriole windkessel with delayed compliance,” J. Cereb. Blood Flow Metab. , 19, 679–689 (1999). [CrossRef] [PubMed]

,17

17. M. E. Brevard, T. Q. Duong, J. A. King, and C. F. Ferris, “Changes in MRI signal intensity during hypercapnic challenge under conscious and anesthetized conditions,” Magn. Res. Imaging. , 21, 995–1001 (2003). [CrossRef]

]. Mandeville et al. have developed a modified Windkessel model to explain the bi-phasic increase of relative cerebral blood volume (rCBV) during 30 seconds of electrical stimulation on rat forepaw [16

16. J. B. Mandeville, J.J.A. Marota, C. Ayata, G. Zaharchuk, M.A. Moskowitz, B. R. Rosen, and R. M. Weisskoff, “Evidence of a cerebrovascular postarteriole windkessel with delayed compliance,” J. Cereb. Blood Flow Metab. , 19, 679–689 (1999). [CrossRef] [PubMed]

]. They explained the acute increase of rCBV by the fast elastic response from both capillary and vein and the slow increase of rCBV by a delayed venous compliance. Functional MRI can detect signals from large blood vessels such as artery and vein as well as those from small vessels including capillaries, while the NIRS measurement is most sensitive to microvessels [18

18. H. Liu, A. H. Hielscher, F. K. Tittel, S. L. Jacques, and B. Chance, “Influence of Blood Vessels on the Measurement of Hemoglobin Oxygenation as Determined by Time-Resolved Reflectance Spectroscopy,” Medical Physics , 22, 1209–1217 (1995). [CrossRef] [PubMed]

,19

19. Y. Gu, R. Mason, and H. Liu, “Estimated fraction of tumor vascular blood contents sampled by near infrared spectroscopy and 19F magnetic resonance spectroscopy,” Optics Express , 13, 1724–1733 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1724 [CrossRef] [PubMed]

]. Therefore, the bi-phasic changes of Δ[HbO2] observed by NIRS during carbogen intervention may not necessarily follow the same principle as explained by the modified Windkessel model.

In addition, the response of rCBV and relative cerebral blood flow (rCBF) in the brain due to stimulation is much faster than the blood oxygenation changes in tumor during carbogen intervention. The time constants of rapid and slow increases in rCBV were 1.9±0.7 second and 14±13 second [16

16. J. B. Mandeville, J.J.A. Marota, C. Ayata, G. Zaharchuk, M.A. Moskowitz, B. R. Rosen, and R. M. Weisskoff, “Evidence of a cerebrovascular postarteriole windkessel with delayed compliance,” J. Cereb. Blood Flow Metab. , 19, 679–689 (1999). [CrossRef] [PubMed]

], respectively, while the time constants of increases in Δ[HbO2] in rat breast tumors during carbogen inhalation were much slower, varying from 3.9 sec to 150 seconds (mostly 20–60 second) for the rapid increase and 180 to 1500 sec during the gradual increase [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

12

12. Y. Gu, V. A. Bourke, J. G. Kim, A. Constantinescu, R. P. Mason, and H. Liu, “Dynamic response of breast tumor oxygenation to hyperoxic respiratory challenge monitored with three oxygen-sensitive parameters,” Appl. Opt. , 42, 2960–2967 (2003). [CrossRef] [PubMed]

]. This suggests that the bi-phasic feature obtained in Δ[HbO2] during tumor oxygenation may be a physiological and hemodynamic characteristic different from that observed in the brain. We are currently investigating the association between our experimental data and the modified Windkessel model [20

20. M. Xia and H. Liu, “A model of the hemodynamic response of tumor in rats with hyperoxic gas challenge”, Optical Tomography and Spectroscopy of Tissue VII, B. Chance, R. R. Alfano, B. J. Tromberg, M. Tamura, and E. M. Sevick-Muraca, eds., Proc. SPIE5693, in press (2005).

], following the approach that estimates the relative cerebral metabolic rate of oxygen (rCMRO2) developed by Boas et al [21

21. D. A. Boas, G. Strangman, J. P. Culver, R. D. Hoge, G. Jasdzewski1, R. A. Poldrack, B. R. Rosen, and J. B. Mandeville, “Can the cerebral metabolic rate of oxygen be estimated with near-infrared spectroscopy?” Phys. Med. Biol. , 48, 2405–2418 (2003). [CrossRef] [PubMed]

].

Padhani and Dzik-Jurasz have reviewed the heterogeneity in perfusion from extracranial tumors measured by dynamic contrast-enhanced MR imaging (DCE-MRI) [22

22. A. R. Padhani and A. Dzik-Jurasz, “Perfusion MR imaging of extracranial tumor angiogenesis,” Top. Magn. Reson. Imaging , 15, 41–57 (2004). [CrossRef] [PubMed]

]. They have shown that the kinetics of signal intensity changes obtained from T2*- or T1-weighted images are significantly varying within a tumor. Especially, Figs. 4 and 5 in their paper clearly support that the bi-phasic increase of Δ[HbO2] from rat breast tumors during carbogen intervention could be from different perfusion rates in tumors, given that our single channel NIRS obtains global changes in tumor hemodynamics. In addition, other reports presented cerebral oxygenation during a relatively long period of hypercapnic challenge, e.g. 2 minutes [23

23. A. Y. Bluestone, M. Stewart, J. Lasker, G.S. Absoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brainimaging in small animals, part1:hypercapnia,” J. Biomed. Opt. , 9, 1046–1062 (2004). [CrossRef] [PubMed]

] or 10 minutes [24

24. E. Rostrup, I. Law, F. Pott, K. Ide, and G. M. Knudsen, “Cerebral hemodynamics measured with simultaneous PET and near-infrared spectroscopy in humans,” Brain research , 954, 183–193 (2002). [CrossRef] [PubMed]

], and did not show the bi-phasic feature in Δ[HbO2].

Various mathematical models have been proposed to understand the cerebral hemodynamic parameters, including BOLD MRI signal, rCMRO2, rCBV, rCBF during stimulation or hypercapnic intervention [16

16. J. B. Mandeville, J.J.A. Marota, C. Ayata, G. Zaharchuk, M.A. Moskowitz, B. R. Rosen, and R. M. Weisskoff, “Evidence of a cerebrovascular postarteriole windkessel with delayed compliance,” J. Cereb. Blood Flow Metab. , 19, 679–689 (1999). [CrossRef] [PubMed]

,25

25. R.B. Buxton, E.C. Wong, and L.R. Frank, “Dynamics of blood flow and oxygenation changes during brain activation: the balloon model,” Magn. Reson. Med. , 39, 855–864 (1998). [CrossRef] [PubMed]

30

30. K. Lu, J. W. Clark Jr., F. H. Ghorbel, C. S. Robertson, D. L. Ware, J. B. Zwischenberger, and A. Bidani, “Cerebral autoregulation and gas exchange studied using a human cardiopulmonary model,” Am. J. Physiol. Heart. Circ. Physiol. , 286, H584–H601 (2004). [CrossRef]

]. Since Δ[HbO2] was showing large changes during carbogen inhalation in our NIRS measurements, we have adopted Δ[HbO2] as a sensitive parameter to obtain tumor hemodynamic features, just like changes in deoxyhemoglobin concentration used in BOLD MRI. Solid tumors are known to have both temporal and spatial heterogeneity in blood flow [31

31. R. K. Jain, “Determinants of tumor blood flow: a review,” Cancer Res. , 48, 2641–2658 (1988). [PubMed]

], and tumor blood vessels are much leakier and more porous than normal blood vessels [32

32. R. K. Jain, “Barriers to drug delivery in solid tumors,” Sci. Am. , 271, 58–65 (1994). [CrossRef] [PubMed]

]. Therefore, tumor hemodynamics may not follow the currently established mathematical models that estimate cerebral hemodynamics by considering autoregulation and vessel reactivity. Based on the fact that solid tumors develop hypoxic regions which are poorly perfused in the center as they grow [33

33. R. Mazurchuk, R. Zhou, R. M. Straubinger, R. I. Chau, and Z. Grossman, “Functional magnetic resonance (fMR) imaging of a rat brain tumor model: implications for evaluation of tumor microvasculature and therapeutic response,” Magn. Reson. Imaging , 17, 537–548 (1999). [CrossRef] [PubMed]

,34

34. Y. Song, A. Constantinescu, and R.P. Mason, “Dynamic breast tumor oximetery: the development of prosgnostic radiology,” Technology in Cancer Research & Treatment , 1, 1–8 (2002).

], we hypothesized that the bi-phasic feature of Δ[HbO2] stems from two different perfusion rates in tumors. Our approach is a simpler mathematical model in comparison with those presented for cerebral hemodynamic models. In our current study, we do not measure blood flow changes in tumor, and thus we could only obtain the ratio of vascular coefficients and perfusion rates by fitting the increase of Δ[HbO2] [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

]. To overcome this limitation, recently, we have followed an approach that estimates the rCMRO2 developed by Boas et al. [21

21. D. A. Boas, G. Strangman, J. P. Culver, R. D. Hoge, G. Jasdzewski1, R. A. Poldrack, B. R. Rosen, and J. B. Mandeville, “Can the cerebral metabolic rate of oxygen be estimated with near-infrared spectroscopy?” Phys. Med. Biol. , 48, 2405–2418 (2003). [CrossRef] [PubMed]

] to evaluate changes in tumor blood flow and metabolic rate of oxygen in tumor [20

20. M. Xia and H. Liu, “A model of the hemodynamic response of tumor in rats with hyperoxic gas challenge”, Optical Tomography and Spectroscopy of Tissue VII, B. Chance, R. R. Alfano, B. J. Tromberg, M. Tamura, and E. M. Sevick-Muraca, eds., Proc. SPIE5693, in press (2005).

].

In this numerical study, we have simplified the physiological complex of tumors in their hemodynamic structures by assuming the same absorption coefficients for the perfused blood prior to carbogen intervention and tissue background. While this simplified assumption is not realistic in actual tumors, the overall trend of mathematical simulations would remain the same. This is because the numerical simulations will always provide us with changes in light intensity during tumor blood oxygenation, if an absorption difference exists in blood vasculature between the baseline (air) and carbogen intervention. With our current assumption and modeling setup, we are able to enhance the bi-phasic feature for easy observation.

While the tumor vasculature and hemodynamics is very complex and chaotic, a simple numerical model, as we demonstrated in this study, can support an initial mathematical hypothesis for tumor modeling, help us understand and interpret our experimental data, and lead to further development of more complex and realistic models for tumor investigations. The goal of this numerical study is not to develop a comprehensive computational model for tumors, but rather focusing on numerical support to better understand our experimental observation during tumor oxygenation measured with NIRS.

In summary, we have previously used a single-channel NIRS system for global measurements of Δ[HbO2] in tumors during respiratory challenges [10

10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

12

12. Y. Gu, V. A. Bourke, J. G. Kim, A. Constantinescu, R. P. Mason, and H. Liu, “Dynamic response of breast tumor oxygenation to hyperoxic respiratory challenge monitored with three oxygen-sensitive parameters,” Appl. Opt. , 42, 2960–2967 (2003). [CrossRef] [PubMed]

], demonstrating that NIRS is a portable, low cost, and real time measurement system that can monitor changes of vascular oxygen levels in tumor tissues non-invasively. Now, the current study confirms that an NIRS multi-channel approach has great potential to detect and monitor tumor heterogeneity under therapeutic or adjuvant interventions. With an appropriate mathematical model, tumor vascular dynamics can be determined and monitored non-invasively while a perturbation of hyperoxic gas intervention is given. Our future work includes 1) to further investigate and understand the meaning of vasculature coefficient, γ and 2) to develop an NIR imaging system to be used as a monitoring tool for the efficacy of cancer therapy.

Acknowledgments

This work was supported in part by the Department of Defense Breast Cancer Research grants DAMD17-03-1-0353 (JGK) and DAMD17-00-1-0459 (HL) as well as by the National Institutes of Health 1R21CA101098-01 (HL).

References and links

1.

P. Vaupel, O. Thews, D. K. Kelleher, and M. Höckel, “Current status of knowledge and critical issues in tumor oxygenation,” In: Hudetz and Bruley (eds), Oxygen Transport to Tissue XX, 591–602 (Plenum Press, New York, 1998).

2.

P. Vaupel, “Vascularization, blood flow, oxygenation, tissue pH, and bioenergetic status of human breast cancer,” In: Nemoto and LaManna (eds), Oxygen Transport to Tissue XVIII, 243–253 (Plenum Press, New York, 1997).

3.

P. Vaupel, “Oxygen transport in tumors: Characteristics and clinical implications,” Adv. Exp. Med. Biol. , 388, 341–351 (1996). [CrossRef] [PubMed]

4.

R. H. Thomlinson and L. H. Gray, “The histological structure of some human lung cancers and the possible implications for radiotherapy,” Br. J. Cancer , 9, 539–549 (1955). [CrossRef] [PubMed]

5.

E. E. Schwartz, The biological basis of radiation therapy (Lippincott, Philadelphia, 1966).

6.

B. Teicher, J. Lazo, and A. Sartorelli, “Classification of antineoplastic agents by their selective toxicities toward oxygenated and hypoxic tumor cells,” Cancer Res. , 41, 73–81 (1981). [PubMed]

7.

J. D. Chapman, C. C. Stobbe, M. R. Arnfield, R. Santus, J. Lee, and M. S. McPhee, “Oxygen dependency of tumor cell killing in vitro by light activated photofrin II,” Radiat. Res. , 126, 73–79 (1991). [CrossRef] [PubMed]

8.

P. Bergsjo and P. Kolstad, “Clinical trial with atmospheric oxygen breathing during radiotherapy of cancer of the cervix,” Scand. J. Clin. Lab. Invest. Suppl. , 106, 167–171 (1968). [PubMed]

9.

H. D. Suit, N. Marshall, and D. Woerner, “Oxygen, oxygen plus carbon dioxide, and radiation therapy of a mouse mammary carcinoma. Cancer,” Cancer , 30, 1154–1158 (1972). [CrossRef] [PubMed]

10.

H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, and R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt. , 39, 5231–5243 (2000). [CrossRef]

11.

J. G. Kim, D. Zhao, Y. Song, A. Constantinescu, R. P. Mason, and H. Liu, “Interplay of Tumor Vascular Oxygenation and Tumor pO2 Observed Using NIRS, pO2 Needle Electrode and 19F MR pO2 Mapping,” J. of Biomed. Opt. , 8, 53–62 (2003). [CrossRef]

12.

Y. Gu, V. A. Bourke, J. G. Kim, A. Constantinescu, R. P. Mason, and H. Liu, “Dynamic response of breast tumor oxygenation to hyperoxic respiratory challenge monitored with three oxygen-sensitive parameters,” Appl. Opt. , 42, 2960–2967 (2003). [CrossRef] [PubMed]

13.

S. S. Kety, “The theory and applications of the exchange of inert gas at the lungs and tissue,” Pharmacol. Rev. , 3, 1–41 (1951). [PubMed]

14.

A.H. Hielsher, S. L. Jacquest, L. Wang, and F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. , 40, 1957–1975 (1995). [CrossRef]

15.

R.A. Groenhuis, A.A. Ferwerda, and J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. , 22, 2456–2462 (1983) [CrossRef] [PubMed]

16.

J. B. Mandeville, J.J.A. Marota, C. Ayata, G. Zaharchuk, M.A. Moskowitz, B. R. Rosen, and R. M. Weisskoff, “Evidence of a cerebrovascular postarteriole windkessel with delayed compliance,” J. Cereb. Blood Flow Metab. , 19, 679–689 (1999). [CrossRef] [PubMed]

17.

M. E. Brevard, T. Q. Duong, J. A. King, and C. F. Ferris, “Changes in MRI signal intensity during hypercapnic challenge under conscious and anesthetized conditions,” Magn. Res. Imaging. , 21, 995–1001 (2003). [CrossRef]

18.

H. Liu, A. H. Hielscher, F. K. Tittel, S. L. Jacques, and B. Chance, “Influence of Blood Vessels on the Measurement of Hemoglobin Oxygenation as Determined by Time-Resolved Reflectance Spectroscopy,” Medical Physics , 22, 1209–1217 (1995). [CrossRef] [PubMed]

19.

Y. Gu, R. Mason, and H. Liu, “Estimated fraction of tumor vascular blood contents sampled by near infrared spectroscopy and 19F magnetic resonance spectroscopy,” Optics Express , 13, 1724–1733 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1724 [CrossRef] [PubMed]

20.

M. Xia and H. Liu, “A model of the hemodynamic response of tumor in rats with hyperoxic gas challenge”, Optical Tomography and Spectroscopy of Tissue VII, B. Chance, R. R. Alfano, B. J. Tromberg, M. Tamura, and E. M. Sevick-Muraca, eds., Proc. SPIE5693, in press (2005).

21.

D. A. Boas, G. Strangman, J. P. Culver, R. D. Hoge, G. Jasdzewski1, R. A. Poldrack, B. R. Rosen, and J. B. Mandeville, “Can the cerebral metabolic rate of oxygen be estimated with near-infrared spectroscopy?” Phys. Med. Biol. , 48, 2405–2418 (2003). [CrossRef] [PubMed]

22.

A. R. Padhani and A. Dzik-Jurasz, “Perfusion MR imaging of extracranial tumor angiogenesis,” Top. Magn. Reson. Imaging , 15, 41–57 (2004). [CrossRef] [PubMed]

23.

A. Y. Bluestone, M. Stewart, J. Lasker, G.S. Absoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brainimaging in small animals, part1:hypercapnia,” J. Biomed. Opt. , 9, 1046–1062 (2004). [CrossRef] [PubMed]

24.

E. Rostrup, I. Law, F. Pott, K. Ide, and G. M. Knudsen, “Cerebral hemodynamics measured with simultaneous PET and near-infrared spectroscopy in humans,” Brain research , 954, 183–193 (2002). [CrossRef] [PubMed]

25.

R.B. Buxton, E.C. Wong, and L.R. Frank, “Dynamics of blood flow and oxygenation changes during brain activation: the balloon model,” Magn. Reson. Med. , 39, 855–864 (1998). [CrossRef] [PubMed]

26.

K. J. Friston, A. Mechelli, R. Turner, and C. J. Price, “Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics,” NeuroImage , 12, 466–477 (2000). [CrossRef] [PubMed]

27.

A. Mechelli, C. J. Price, and K. J. Friston, “Nonlinear Coupling between Evoked rCBF and BOLD Signals: A Simulation Study of Hemodynamic Responses,” NeuroImage , 14, 862–872 (2001). [CrossRef] [PubMed]

28.

Y. Zheng, J. Martindale, D. Johnston, M. Jones, J. Berwick, and J. Mayhew, “A Model of the hemodynamic Response and Oxygen Delivery to Brain,” NeuroImage , 16, 617–637 (2002). [CrossRef] [PubMed]

29.

R. B. Buxton, K. Uludağ, D. J. Dubowitz, and T. T. Liu, “Modeling the hemodynamic response to brain activation,” NeuroImage , 23, S220–S233 (2004). [CrossRef] [PubMed]

30.

K. Lu, J. W. Clark Jr., F. H. Ghorbel, C. S. Robertson, D. L. Ware, J. B. Zwischenberger, and A. Bidani, “Cerebral autoregulation and gas exchange studied using a human cardiopulmonary model,” Am. J. Physiol. Heart. Circ. Physiol. , 286, H584–H601 (2004). [CrossRef]

31.

R. K. Jain, “Determinants of tumor blood flow: a review,” Cancer Res. , 48, 2641–2658 (1988). [PubMed]

32.

R. K. Jain, “Barriers to drug delivery in solid tumors,” Sci. Am. , 271, 58–65 (1994). [CrossRef] [PubMed]

33.

R. Mazurchuk, R. Zhou, R. M. Straubinger, R. I. Chau, and Z. Grossman, “Functional magnetic resonance (fMR) imaging of a rat brain tumor model: implications for evaluation of tumor microvasculature and therapeutic response,” Magn. Reson. Imaging , 17, 537–548 (1999). [CrossRef] [PubMed]

34.

Y. Song, A. Constantinescu, and R.P. Mason, “Dynamic breast tumor oximetery: the development of prosgnostic radiology,” Technology in Cancer Research & Treatment , 1, 1–8 (2002).

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.5280) Medical optics and biotechnology : Photon migration
(170.6510) Medical optics and biotechnology : Spectroscopy, tissue diagnostics
(290.1990) Scattering : Diffusion

ToC Category:
Research Papers

History
Original Manuscript: April 13, 2005
Revised Manuscript: May 23, 2005
Manuscript Accepted: May 30, 2005
Published: June 13, 2005

Citation
Jae G. Kim and Hanli Liu, "Investigation of bi-phasic tumor oxygen dynamics induced by hyperoxic gas intervention: A numerical study," Opt. Express 13, 4465-4475 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4465


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References

  1. P. Vaupel, O. Thews, D. K. Kelleher, M. Höckel, “Current status of knowledge and critical issues in tumor oxygenation,” In: Hudetz, Bruley (eds), Oxygen Transport to Tissue XX, 591–602 (Plenum Press, New York, 1998).
  2. P. Vaupel, “Vascularization, blood flow, oxygenation, tissue pH, and bioenergetic status of human breast cancer,” In: Nemoto, LaManna (eds), Oxygen Transport to Tissue XVIII, 243–253 (Plenum Press, New York, 1997).
  3. P. Vaupel, “Oxygen transport in tumors: Characteristics and clinical implications,” Adv. Exp. Med. Biol., 388, 341–351 (1996). [CrossRef] [PubMed]
  4. R. H. Thomlinson, L. H. Gray, “The histological structure of some human lung cancers and the possible implications for radiotherapy,” Br. J. Cancer, 9, 539–549 (1955). [CrossRef] [PubMed]
  5. E. E. Schwartz, The biological basis of radiation therapy (Lippincott, Philadelphia, 1966).
  6. B. Teicher, J. Lazo, A. Sartorelli, “Classification of antineoplastic agents by their selective toxicities toward oxygenated and hypoxic tumor cells,” Cancer Res., 41, 73–81 (1981). [PubMed]
  7. J. D. Chapman, C. C. Stobbe, M. R. Arnfield, R. Santus, J. Lee, M. S. McPhee, “Oxygen dependency of tumor cell killing in vitro by light activated photofrin II,” Radiat. Res., 126, 73–79 (1991). [CrossRef] [PubMed]
  8. P. Bergsjo, P. Kolstad, “Clinical trial with atmospheric oxygen breathing during radiotherapy of cancer of the cervix,” Scand. J. Clin. Lab. Invest. Suppl., 106, 167–171 (1968). [PubMed]
  9. H. D. Suit, N. Marshall, D. Woerner, “Oxygen, oxygen plus carbon dioxide, and radiation therapy of a mouse mammary carcinoma. Cancer,” Cancer, 30, 1154–1158 (1972). [CrossRef] [PubMed]
  10. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, R. P. Mason, “Noninvasive Investigation of Blood Oxygenation Dynamics of Tumors by Near-Infrared Spectroscopy,” Appl. Opt., 39, 5231–5243 (2000). [CrossRef]
  11. J. G. Kim, D. Zhao, Y. Song, A. Constantinescu, R. P. Mason, H. Liu, “Interplay of Tumor Vascular Oxygenation and Tumor pO2 Observed Using NIRS, pO2 Needle Electrode and 19F MR pO2 Mapping,” J. of Biomed. Opt., 8, 53–62 (2003). [CrossRef]
  12. Y. Gu, V. A. Bourke, J. G. Kim, A. Constantinescu, R. P. Mason, H. Liu, “Dynamic response of breast tumor oxygenation to hyperoxic respiratory challenge monitored with three oxygen-sensitive parameters,” Appl. Opt., 42, 2960–2967 (2003). [CrossRef] [PubMed]
  13. S. S. Kety, “The theory and applications of the exchange of inert gas at the lungs and tissue,” Pharmacol. Rev., 3, 1–41 (1951). [PubMed]
  14. A.H. Hielsher, S. L. Jacquest, L. Wang, F. K. Tittel, “The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol., 40, 1957–1975 (1995). [CrossRef]
  15. R.A. Groenhuis, A.A. Ferwerda, J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt., 22, 2456–2462 (1983) [CrossRef] [PubMed]
  16. J. B. Mandeville, J.J.A. Marota, C. Ayata, G. Zaharchuk, M.A. Moskowitz, B. R. Rosen, R. M. Weisskoff, “Evidence of a cerebrovascular postarteriole windkessel with delayed compliance,” J. Cereb. Blood Flow Metab., 19, 679–689 (1999). [CrossRef] [PubMed]
  17. M. E. Brevard, T. Q. Duong, J. A. King, C. F. Ferris, “Changes in MRI signal intensity during hypercapnic challenge under conscious and anesthetized conditions,” Magn. Res. Imaging., 21, 995–1001 (2003). [CrossRef]
  18. H. Liu, A. H. Hielscher, F. K. Tittel, S. L. Jacques, B. Chance, “Influence of Blood Vessels on the Measurement of Hemoglobin Oxygenation as Determined by Time-Resolved Reflectance Spectroscopy,” Medical Physics, 22, 1209–1217 (1995). [CrossRef] [PubMed]
  19. Y. Gu, R. Mason, H. Liu, “Estimated fraction of tumor vascular blood contents sampled by near infrared spectroscopy and 19F magnetic resonance spectroscopy,” Optics Express, 13, 1724–1733 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1724 [CrossRef] [PubMed]
  20. M. Xia, H. Liu, “A model of the hemodynamic response of tumor in rats with hyperoxic gas challenge”, Optical Tomography and Spectroscopy of Tissue VII, B. Chance, R. R. Alfano, B. J. Tromberg, M. Tamura, E. M. Sevick-Muraca, eds., Proc. SPIE5693, in press (2005).
  21. D. A. Boas, G. Strangman, J. P. Culver, R. D. Hoge, G. Jasdzewski1, R. A. Poldrack, B. R. Rosen, J. B. Mandeville, “Can the cerebral metabolic rate of oxygen be estimated with near-infrared spectroscopy?” Phys. Med. Biol., 48, 2405–2418 (2003). [CrossRef] [PubMed]
  22. A. R. Padhani, A. Dzik-Jurasz, “Perfusion MR imaging of extracranial tumor angiogenesis,” Top. Magn. Reson. Imaging, 15, 41–57 (2004). [CrossRef] [PubMed]
  23. A. Y. Bluestone, M. Stewart, J. Lasker, G.S. Absoulaev, A. H. Hielscher, “Three-dimensional optical tomographic brainimaging in small animals, part1:hypercapnia,” J. Biomed. Opt., 9, 1046–1062 (2004). [CrossRef] [PubMed]
  24. E. Rostrup, I. Law, F. Pott, K. Ide, G. M. Knudsen, “Cerebral hemodynamics measured with simultaneous PET and near-infrared spectroscopy in humans,” Brain research, 954, 183–193 (2002). [CrossRef] [PubMed]
  25. R.B. Buxton, E.C. Wong, L.R. Frank, “Dynamics of blood flow and oxygenation changes during brain activation: the balloon model,” Magn. Reson. Med., 39, 855–864 (1998). [CrossRef] [PubMed]
  26. K. J. Friston, A. Mechelli, R. Turner, C. J. Price, “Nonlinear Responses in fMRI: The Balloon Model, Volterra Kernels, and Other Hemodynamics,” NeuroImage, 12, 466–477 (2000). [CrossRef] [PubMed]
  27. A. Mechelli, C. J. Price, K. J. Friston, “Nonlinear Coupling between Evoked rCBF and BOLD Signals: A Simulation Study of Hemodynamic Responses,” NeuroImage, 14, 862–872 (2001). [CrossRef] [PubMed]
  28. Y. Zheng, J. Martindale, D. Johnston, M. Jones, J. Berwick, J. Mayhew, “A Model of the hemodynamic Response and Oxygen Delivery to Brain,” NeuroImage, 16, 617–637 (2002). [CrossRef] [PubMed]
  29. R. B. Buxton, K. Uludağ, D. J. Dubowitz, T. T. Liu, “Modeling the hemodynamic response to brain activation,” NeuroImage, 23, S220–S233 (2004). [CrossRef] [PubMed]
  30. K. Lu, J. W. Clark, F. H. Ghorbel, C. S. Robertson, D. L. Ware, J. B. Zwischenberger, A. Bidani, “Cerebral autoregulation and gas exchange studied using a human cardiopulmonary model,” Am. J. Physiol. Heart. Circ. Physiol., 286, H584–H601 (2004). [CrossRef]
  31. R. K. Jain, “Determinants of tumor blood flow: a review,” Cancer Res., 48, 2641–2658 (1988). [PubMed]
  32. R. K. Jain, “Barriers to drug delivery in solid tumors,” Sci. Am., 271, 58–65 (1994). [CrossRef] [PubMed]
  33. R. Mazurchuk, R. Zhou, R. M. Straubinger, R. I. Chau, Z. Grossman, “Functional magnetic resonance (fMR) imaging of a rat brain tumor model: implications for evaluation of tumor microvasculature and therapeutic response,” Magn. Reson. Imaging, 17, 537–548 (1999). [CrossRef] [PubMed]
  34. Y. Song, A. Constantinescu, R.P. Mason, “Dynamic breast tumor oximetery: the development of prosgnostic radiology,” Technology in Cancer Research & Treatment, 1, 1–8 (2002).

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