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MBW:Modeling of Oxygen Transport Across Tumor Multicellular Layers

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Article review by Kumar Thurimella & Nikolas Kauffman

Article

Our paper by Rod Braun and Alexis Beatty, discusses oxygen and its role in tumor cells [1].

Executive Summary

In this article the authors were trying to understand how different levels of oxygen within tumors affect the response of those tumors with different treatments. They furthered their question by modeling the oxygen diffusion across a lab grown melanoma cell using Fick's diffusion equation. The authors had initially sought out to figure out any correlation between a cancer patients outcome with oxygen transport in tumor parenchyma. Each tumor was examined on a multicellular level and thus the authors tested different multicellular levels (MCL) of a certain tumor. At each MCL there was a measured oxygen tension and the authors fit those profiles to a on-dimensional diffusion model. The model was justified with experimental data that was taken from the experiment to describe the oxygen tension at each of those multicellular levels. This model is able to analyze certain tumor cell lines in MCL that would not normally be analyzed in the more common, spheroid tumor. The MCL can be studied in this manner since, unlike spheroids, they grow on an underlying substrate. The authors discuss the importance of oxygen transport, specifically its impact on MCL and how it can affect the treatment of cancer.

Context/Biological Events

Figure 1: This shows how MCL is grown (like a sheet compared to a sphere)


The oxygen content of a tumor is a very important factor in determining whether or not some tumors respond to various treatment. Additionally a patients outcome can be directly linked to a tumor's oxygen level [2]. Specific research within MCL is crucial towards any oncological study due to very limited information on in vitro studies. It has been very hard to study certain cell lines within tumors however the research within MCL (in vitro) helps extend that research. This research would allow for drug transport and oxygen content to be studied quantitatively, which is very useful in studying drugs whose uptake is oxygen dependent.

History

The notion behind studying oxygen levels and the correlation to tumor cells has been studied previously by researchers, W. Mueller-Klieser, J. P. Freyer, and R. M. Sutherland. [3] This history extends to the notion to believe that tumor cells are dependent on oxygen and there is a correlation between a cell line survival with oxygen access. This topic has examined the idea that tumor spheroids (like those in human bodies) adapt their metabolism based on oxygen and glucose intake. At that time the results had shown to be inconclusive and the authors consensus was that: although the metabolism can vary with glucose and oxygen intake, there are too many other possibilities to simply state that growth is solely dependent on glucose and oxygen. When the researchers modeled their oxygen tension profiles in the tumor spheroids they were fitting experimental data based on oxygen diffusion and plotting against the diameter of the spheroid. There was no model presented formally and the ideas and notions of Fick's diffusion equation was an underlying part of the experiment. Fick's equation in our setting would relate the diffusive flux between the outside and the inside of the spheroid with the concentration of the oxygen in the environment. These concepts lay a foreground into the insight of this interesting research and is able to help start a connection between oxygen levels and tumor growth.

Mathematical Model

Parameters of the Model

Parameter Description
D the oxygen diffusion coefficient (cm^2/sec)
k the oxygen solubility (ml O2/ml tissue/mm Hg)
Dk the oxygen permeability or Krogh’s diffusion coefficient (ml O2/cm/mm Hg/sec)
q oxygen consumption (ml O2/ml tissue/sec)



Our Mathematical Model

To model the oxygen transport across OCM-1 cultured MCL's we must use Fick's one-dimensional oxygen diffusion equation:

                                                   {\frac  {dP}{dt}}=Dk*{\frac  {d^{2}P}{dx^{2}}}-q

From Fick's diffusion equation and assumptions made for the method of experimentation that the change in oxygen tension per unit time (dP/dt) is equal to zero, i.e. the steady-state, we obtain a new equation that models oxygen tension with respect to distance x as follows:

                                                           {\frac  {d^{2}P}{dx^{2}}}={\frac  {q}{Dk}}
Figure 2: Oxygen profiles w.r.t to distance x across MCL

where q/Dk is the oxygen consumption across the MCL. In order to adequately determine a best-fitted model for the oxygen diffusion across the MCL, the authors decided to fit four different diffusion models to the experimental data. These models only differed with respect to the number of boundary layers between the media layer and membrane layer of the MCL. By solving the second-order inhomogenous ODE using the method of undetermined coefficients the following equation was obtained:

                 P_{i}={\frac  {q_{i}}{Dk}}+A_{i}x+B_{i}~|~for~i=1~to~m-1

The law of mass conservation tells us that the PO_{2} and oxygen consumption levels at the boundary of adjacent layers must be equal, yielding:

                                                                 P_{i}=P_{{i+1}}
                                                              {\frac  {dP_{i}}{dx}}={\frac  {dP_{{i+1}}}{dx}}

The constants A_{i} and B_{i} are determined by the boundary conditions of the MCL which are defined as follows:

                                                             P_{i}(L_{0})=P_{{med}}
                                                             P_{m}(L)=P_{{mem}}

To compare oxygen consumption rates across the MCL it was necessary to define these rates according to their changes across different boundary regimes. The rates were taken at the media interface, membrane interface, and across the MCL. These were calculated using the following relations:

      {\frac  {q_{{av}}}{Dk}}=\sum _{{i=1}}^{{m}}{\frac  {q_{{i}}}{Dk}}*({\frac  {L_{{i}}-L_{{i-1}}}{L-L_{0}}})~,i=1~to~m,~L_{m}=L
      {\frac  {q_{{s}}}{Dk}}=\sum _{{i=1}}^{{a}}{\frac  {q_{{i}}}{Dk}}*({\frac  {L_{{i}}-L_{{i-1}}}{x_{{min1}}-L_{0}}})~,where~a=layer~containing~x_{{min1}},~L_{a}=x_{{min1}}
      {\frac  {q_{{m}}}{Dk}}=\sum _{{i=b}}^{{m}}{\frac  {q_{{i}}}{Dk}}*({\frac  {L_{{i+1}}-L_{{i}}}{L-x_{{min2}}}})~,where~b=layer~containing~x_{{min2}},~L_{b}=x_{{min2}}

The latter two of the above three equations involve x-values which are defined as the minimum values from the media and membrane interfaces where dP/dx=0. The oxygen consumption rates for these two equations are thus calculated as the average rate from the boundary layer to these minimum x-values.

As stated earlier, four different diffusion models were used to determine the best-fit model across the MCL. The accuracy of the fit was determined using MATLAB written programs that evaluated the sum-of-squares error, root-mean-squared error, and the coefficient of determination (r^{2}).

Understanding the Mathematical Model

From Fick's diffusion model in the previous section the assumption that dP/dt=0 was made, thus, the model with respect to time is in steady-state. This is understood through the realization that upon experimentation the tumor MCL's have already assumed a uniform oxygen profile and therefore, oxygen levels across the cells only differ with respect to the x location within the MCL. Solving the steady-state then tells us that the second-derivative of the oxygen levels with respect to position equal the oxygen consumption rate across the entire MCL. The distance across the MCL was then broken into different layers as to obtain different profiles. This allowed for multiple profiles to be examined against the experimental data, which, allowed for the model to better represent the behavior of the test results.

The second-order ODE produced by finding the steady-state of the Fick's one-dimensional diffusion model made it possible to then solve for the PO_{2} levels across each layer using the characteristic equation and then using the principle of superpostion to additively combine the homogeneous and particular solutions of the ODE. For a quick refresh, lets solve for the roots of the characteristic equation ay''+by'+cy=0. This models C.E. parameters are a=1, b=0, c=0; yielding a repeated root of 0 by solving the quadratic equation. Since we have a repeated root solutions to the homogeneous ODE P=A*x*e^{{rt}}+B*e^{{rt}} are not linearly independent and are made so by multiplying one of the terms in the linear combination by x. By setting r=0 and combining the particular solution (i.e. the oxygen consumption rate), we arrive at the solution:

                                                  P_{{i}}={\frac  {q_{i}}{Dk}}+A_{i}*x+B_{i}

This defines the PO_{2} across each layer within the MCL.

Results

A. Measuring Profiles Across OCM-1 MCL

PO_{2} profiles were measured using polarographic microelectrodes in MCL's ranging from 400 to 1400 micrometers. These measurements demonstrated that thicker MCL's gave bigger regions of anoxia across the MCL.

B. Diffusion Modeling of the Profiles

Depending on the extent of anoxia across the MCL it was found that two of the four models used to describe the PO_{2} profiles most accurately displayed the behavior of the experimental data. For MCL's with insignificant anoxic regions the two-region model was found to be the best representation for the data, whereas; thicker MCL's with longer anoxic regions were found to be best represented by the model consisting of one extra layer (the central layer) where the PO_{2} levels were set to be zero, i.e. the three-region model. Upon reexamination of the mathematical model, the two-region representation requires five unknown parameter fits for L,~L_{0},~L_{1},~q_{1}/Dk,~q_{2}/DK and the three-region model requires seven unknown parameter fits for L,~L_{0},~L_{1},~L_{2}~q_{1}/Dk,~q_{2}/DK,~q_{3}/Dk. Because the one-region model does not accurately represent the data it can be said that the boundaries of the MCL at the membrane and media are metabolically different, i.e. the cells near these boundaries are consuming oxygen at different rates.

C. Oxygen Consumption of OCM-1 MCL

It was found that the oxygen consumption rates of the cancerous cells at the media and membrane boundaries were different, indicating different metabolic rates at the two interfaces. The metabolic rate at the membrane surface was found to be higher than that of the free surface (lower values of PO_{2} at the MCL membrane boundary). Unfortunately, it is not known why this phenomena occurs, but the authors speculate that the metabolism levels of these cells close to the boundary could be affected by the collagen matrix interacting with the membrane interface. Another important characteristic of the oxygen consumption is that the average consumption decreased with increasing MCL thickness. Based upon correlations from earlier, it was seen that increasing MCL thickness indicated longer anoxic regions in the profiles. Trivial examination of this demonstrates that the inclusion of zero oxygen consumption for these anoxic regions decreased the average oxygen consumption, q_{{avg}}/Dk. There were also differences in the oxygen consumption parameters when the MCL was exposed to air saturated RPMI as compared to 5% O^{2} saturated RPMI. In both instances, it was evident that more oxygen was being supplied at the free surface, but in the 5% O^{2} saturated RPMI test procedure it was shown that all of the oxygen consumption parameters decreased in value.

Conclusions

Measurement of PO_{2} profiles

The authors had measured various profiles between a thickness of 400 to 1400 um and their finding in this area led them to look at anoxia at those various profiles. The authors generalized that the more thick the MCL the less oxygen (or the higher anoxia). This finding warrants itself to the idea that with very a thick cell layer there is not very much oxygen that can interact with the cells. However, that being said there is no quantitative element in describing at what specific thickness oxygen is completely cutoff or is very prevalent in. The differences in the PO_{2} profile within the same MCL region is thought to be cause by the various metabolic needs in several regions of the MCL. Figure one describes the general correlation between the thickness level of the MCL and the anoxia (lack of oxygen) there is around that region.

Figure 3

Comparison of Oxygen Measurements in Spheroids vs. MCL

The authors stand behind their claim that in order to properly study oxygen uptake in tumor cells using an MCL model is better than a spheroid model. For one spheroids are difficult to measure PO_{2} under. When measuring oxygen tension in a spheroid it becomes very difficult because everything has to be measured radially which is technically challenging and requires quite a bit of skilled manipulation. However, an MCL is grown perpendicularly with relation to a flat surface and there is much less positional error. There are some disadvantages to using an MCL model, however. Even within an MCL model not all cell lines can be grown which limits the ability of tumor cells to be studied. Since not all cell lines can be grown, oxygen consumption levels cannot even be determined which also limits sides of this experiments. The MCL model still opened up more possibility for study in the oxygen levels within tumor cells.

Diffusion Modeling of PO_{2} Profiles

Throughout this paper several region models were introduced and were tried in order to fit the experimental data. When there was no anoxia (presence of oxygen) the authors found that a 2 region model best described the data. The 1 region model was not able to fit data at different boundaries of the MCL but the 2 region model could account for the metabolic differences among the various boundaries. The authors found that cells growing near the membrane of the MCL had a higher metabolism even though the oxygen tension was lower at the free surface of the MCL which helps back the oddity found when using the 2 region model. When the authors studied the three region model for the MCL they found that it matched all of the oxygen tension profiles quite well. It matched the profiles so well that they didn't need to include particular parameters to model all of the data since there were 3 layers in the MCL. When previous research was done on spheroids there tended to be a zone where there was no oxygen diffusion. In the MCL there was no evidence of such a zone which may be because of the large step size of the electrode in the MCL. Everything was dependent on specific oxygen permeability and that was all calculated through thickness of the MCL. In fact there is a possibility that every diffusion difference between certain profiles can come from cell lines and other cell lines need to be tested to further resolve any issue presented.

Oxygen Consumption of MCL

Figure 4

One of the most interesting results of this paper was the idea that the metabolism of the cells on the membrane surface was larger than the rate of cells exposed to a free surface. This result could possibly be explained because the cells on the membrane surface was exposed to a collagen matrix, which was not the case for the free surface cells. On the whole the average oxygen consumption decreased with the increasing MCL thickness. This helped illustrate the correlation between anoxia and MCL thickness. The fact there is a thick layer of MCL, there is not enough oxygen just due to the sheer density around the region. This experiment cannot be fully evaluated in terms of oxygen consumption because only three MCL were exposed to the oxygen levels. The goal was not to seek the effects of oxygen level on consumption but to simply fit oxygen tension profiles under several different conditions. However, the overall consensus was that the consumption parameters decreased in the MCL in the 5% environment versus the 65% environment. Figure 4 shows the differences between oxygen consumption along the membrane surface and oxygen consumption along the free surface.

Future Applications

This experiment can be expanded out towards certain aspects of cancer research. Issues such as growth conditions, oxygen levels, glucose concentration and other factors can model tumor metabolism. Hopefully in the future there can be an impact towards understanding pharmaceutical drugs and their effect with respect to oxygenation. The MCL model brings a new aspect to the study of tumor oxygenation and can be very useful in the future.

Our Presentation

Our presentation (Thurimella, Kauffman) in .pdf format

References

  1. R. D. Braun and A. L. Beatty. Modeling of oxygen transport across tumor multicellular layers. Microvascular Research, 73(2):113-123, 2007
  2. Brizel DM, et al. Tumor oxygenation predicts for the likelihood of distant metastases in human soft tissue sarcoma. Cancer Res. 1996;56:941–943
  3. Mueller-Klieser W, et al. Influence of glucose and oxygen supply conditions on the oxygenation of multicellular spheroids. Br J Cancer. 1986;53:345–353.