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(1)A voxel-based multiscale model to simulate the radiation response of hypoxic tumors I. Espinoza, P. Peschke, and C. P. Karger Citation: Medical Physics 42, 90 (2015); doi: 10.1118/1.4903298 View online: http://dx.doi.org/10.1118/1.4903298 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/42/1?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Dose prescription complexity versus tumor control probability in biologically conformal radiotherapy Med. Phys. 36, 4379 (2009); 10.1118/1.3213519 Self-consistent tumor control probability and normal tissue complication probability models based on generalized EUDa) Med. Phys. 34, 2807 (2007); 10.1118/1.2740010 Investigating the effect of cell repopulation on the tumor response to fractionated external radiotherapy Med. Phys. 30, 735 (2003); 10.1118/1.1567735 An analysis of the relationship between radiosensitivity and volume effects in tumor control probability modeling Med. Phys. 27, 1258 (2000); 10.1118/1.599003 Volume and Kinetics in Tumor Control and Normal Tissue Complications Med. Phys. 26, 1407 (1999); 10.1118/1.598815.

(2) A voxel-based multiscale model to simulate the radiation response of hypoxic tumors I. Espinozaa) Institute of Physics, Pontificia Universidad Católica de Chile, Santiago 7820436, Chile and Department of Medical Physics in Radiation Oncology, German Cancer Research Center (DKFZ), Heidelberg 69120, Germany. P. Peschke Clinical Cooperation Unit Molecular Radiooncology, German Cancer Research Center (DKFZ), Heidelberg 69120, Germany. C. P. Karger Department of Medical Physics in Radiation Oncology, German Cancer Research Center (DKFZ), Heidelberg 69120, Germany. (Received 13 May 2014; revised 11 November 2014; accepted for publication 15 November 2014; published 16 December 2014) Purpose: In radiotherapy, it is important to predict the response of tumors to irradiation prior to the treatment. This is especially important for hypoxic tumors, which are known to be highly radioresistant. Mathematical modeling based on the dose distribution, biological parameters, and medical images may help to improve this prediction and to optimize the treatment plan. Methods: A voxel-based multiscale tumor response model for simulating the radiation response of hypoxic tumors was developed. It considers viable and dead tumor cells, capillary and normal cells, as well as the most relevant biological processes such as (i) proliferation of tumor cells, (ii) hypoxiainduced angiogenesis, (iii) spatial exchange of cells leading to tumor growth, (iv) oxygen-dependent cell survival after irradiation, (v) resorption of dead cells, and (vi) spatial exchange of cells leading to tumor shrinkage. Oxygenation is described on a microscopic scale using a previously published tumor oxygenation model, which calculates the oxygen distribution for each voxel using the vascular fraction as the most important input parameter. To demonstrate the capabilities of the model, the dependence of the oxygen distribution on tumor growth and radiation-induced shrinkage is investigated. In addition, the impact of three different reoxygenation processes is compared and tumor control probability (TCP) curves for a squamous cells carcinoma of the head and neck (HNSSC) are simulated under normoxic and hypoxic conditions. Results: The model describes the spatiotemporal behavior of the tumor on three different scales: (i) on the macroscopic scale, it describes tumor growth and shrinkage during radiation treatment, (ii) on a mesoscopic scale, it provides the cell density and vascular fraction for each voxel, and (iii) on the microscopic scale, the oxygen distribution may be obtained in terms of oxygen histograms. With increasing tumor size, the simulated tumors develop a hypoxic core. Within the model, tumor shrinkage was found to be significantly more important for reoxygenation than angiogenesis or decreased oxygen consumption due to an increased fraction of dead cells. In the studied HNSSC-case, the TCD50 values (dose at 50% TCP) decreased from 71.0 Gy under hypoxic to 53.6 Gy under the oxic condition. Conclusions: The results obtained with the developed multiscale model are in accordance with expectations based on radiobiological principles and clinical experience. As the model is voxel-based, radiological imaging methods may help to provide the required 3D-characterization of the tumor prior to irradiation. For clinical application, the model has to be further validated with experimental and clinical data. If this is achieved, the model may be used to optimize fractionation schedules and dose distributions for the treatment of hypoxic tumors. C 2015 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4903298] Key words: radiotherapy, tumor hypoxia, reoxygenation, response simulation 1. INTRODUCTION Cancer is the leading cause of death in many developed countries and will become a major cause of morbidity and mortality in the coming decades in all regions of the world.1 It is estimated that today approximately 50% of all cancer patients receive radiotherapy during their treatment,2 but 90. Med. Phys. 42 (1), January 2015. optimal treatment is still compromised by the limited knowledge of the tumor response to irradiation. With this respect, mathematical models may help to reduce this uncertainty and optimize the radiation treatment. Many analytical mathematical models of tumor growth, radiation response and treatment optimization were developed in the past and have been of great relevance for the understanding. 0094-2405/2015/42(1)/90/13/$30.00. © 2015 Am. Assoc. Phys. Med.. 90.

(3) 91. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. of the radiation response. However, they become prohibitively complex, when a large number of biological processes and interactions are considered. Thus, computer simulation models became increasingly important as they could handle more complex algorithms and were able to consider the heterogeneous nature of tumors as well as the stochastic properties of the radiation response by Monte Carlo methods. Titz and Jeraj3 classified the existing radiobiological computer models in three categories according to the spatial scale on which they focus. (i) Microscopic models consider individual interacting cells and are usually computationally limited to small tumor sizes (<1 cm in diameter).4–6 (ii) Macroscopic models, in contrast, describe spatiotemporal changes in cell densities modeled by reaction–diffusion equations.7,8 To consider complex relevant biological processes, however, it was necessary to develop hybrid models (iii), which consider different spatial scales.3,9–11 A complete review of in silico radiobiological models is given in Marcu and HarrissPhillips.12 Dionysius et al.9,10 introduced the notion of a “geometrical cell,” an elementary cubic volume containing a quantity of biological cells, which can be transferred to neighboring geometrical cells in order to maintain cell density within the tumor. In their work, metabolic information from medical imaging can be assigned to the spatially distributed geometrical cells. However, they do not directly consider the variation of radiosensitivity due to hypoxia. Titz and Jeraj3 developed another interesting multiscale model that uses medical imaging to assign an oxygenation level to each voxel. They consider, however, only one average value of oxygen partial pressure per voxel and, as Dasu et al.13 stated, it is important to consider the complete distribution of oxygen within a tumor voxel to avoid erroneous prediction of treatment outcome. The underlying idea is that a small fraction of hypoxic and thus radioresistant cells may already cause radiotherapy failure. In addition, it is important to consider the effect of spatially varying vascular density on oxygenation, as shown by Lagerlöf et al.14 In this work, we present a voxel-based multiscale computer model that takes into account patient-specific information of the tumor to simulate the radiation response of hypoxic tumors. The model considers the following relevant biological processes: (i) proliferation of tumor cells, (ii) hypoxia-induced angiogenesis, (iii) spatial exchange of cells leading to tumor growth, (iv) oxygen-dependent cell survival after irradiation, (v) resorption of dead cells, and (vi) spatial exchange of cells leading to tumor shrinkage. Information on oxygenation is incorporated using a previously published tumor oxygenation model (TOM),15 which calculates the oxygen histograms for each voxel, based on its vascular fraction. As this quantity may be obtained from noninvasive medical imaging,16–19 the initial characterization of tumor oxygenation is thus connected to methods applicable in patients. In the future, the developed model may help to further optimize the radiation treatment based on the information obtained from individual patients. Medical Physics, Vol. 42, No. 1, January 2015. 91. 2. MATERIAL AND METHODS 2.A. The tumor response model (TRM). The model was written in C++ and considers a virtual tumor and its surrounding normal tissue as a 3D cubic grid of voxels. Each voxel may contain four different types of cells (Sec. 2.A.1). After the initial characterization of the virtual tumor (Sec. 2.A.1), the radiation response together with several relevant biological processes is simulated (Fig. 1). 2.A.1. Input data to characterize the tumor prior to irradiation. The TRM considers four different types of input information, which are described below. 2.A.1.a. Characterization of the tumor. As a first step of the simulation, the initial condition of the tumor has to be characterized geometrically as well as biologically. The geometric characterization includes the location and shape of the tumor, that of adjacent bony structures as well as the separation of tissue from air. Interfaces to bony structures and air–volumes are important as they may affect tumor growth. This volumetric information can in principle be obtained from the respective structures of the treatment planning CT. In this initial study, however, it was assigned manually to the voxels. In addition to this geometrical characterization, each voxel has to be described biologically by the following additional parameters: All voxels of the TRM are assumed to have a fixed cell density µ (Table I) and each voxel may contain up to four types of cells. The initial biological status of the tumor is characterized by the number of (i) viable tumor cells, (ii) capillary cells, and (iii) normal cells per voxel. Additionally to these cell types, dead tumor cells (iv) may be present after irradiation. The number of capillary cells defines the oxygenation status of the tumor and is specified by a parameter called vascular. F. 1. Flow chart of the tumor response model..

(4) 92. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. T I. Biological parameters of the simulated tumor model. Parameter. Symbol. Cell density Vascular fraction of normal tissues Tumor cells proliferation (doubling time) Capillary cells proliferation (doubling time) Half-life of dead cells resorption. µ v fn. Radiosensitivity (LQ-model) Intertumoral variation of radiosensitivity Maximum value of the OER Oxygen partial tension at OER = (m + 1)/2. Value 106 cells/mm3 (Ref. 21) 3.6% (see Sec. 2.B.2). tp. 122.4 h (Ref. 22). ta. α β σα. 612 h [5 times slower than tp. (Ref. 23)] 168 h (1 week) [within the range given by Harting et al. (Ref. 6)] 0.273 Gy−1 (Ref. 24) 0.045 Gy−2 (Ref. 24) 0 or 0.1α. m k. 3 (Ref. 25) 3 mm Hg (Ref. 25). tr. fraction (relative volume of capillary cells within a voxel). The initial number of normal cells per voxel is given by the difference of the total number of cells per voxel (according to the cell density µ) and the quantity of both viable tumor cells and capillary cells. Including normal cells allows assigning arbitrary tumor cell densities to each voxel without changing the overall cell density within the voxel. Although some information about the density distribution of tumor cells might be obtained by medical imaging, a quantitative estimation is currently not feasible and therefore a fixed number of viable tumor cells was manually assigned to each voxel. The number of capillary cells per tumor voxel is obtained from a file containing the vascular fractions for each voxel. Although this information may be in principle obtained from positron-emission-tomography (PET)16,17 or magnetic resonance imaging (MRI),18,19 it was assigned manually for the purpose of this study. Both tumor as well as capillary cells may proliferate. For tumor cells, proliferation rates were taken from the literature and assigned to all voxels, while proliferation of capillary cells was assumed to be connected to the presence of hypoxia (see angiogenesis in Sec. 2.A.2). Both proliferation rates are described by corresponding doubling times (tp and ta) specified in Table I. 2.A.1.b. Oxygen histograms. To calculate the radiation response in hypoxic regions of the tumor, the amount of oxygen available to the irradiated cells is required, since hypoxic cells are known to be more resistant to irradiation than well-oxygenated cells. As the microscopic oxygen distribution is highly heterogeneous, this information is included by voxel-specific “oxygen-histograms” specifying the fraction of tumor cells that are exposed to a certain oxygen partial pressure (PO2). The shape of these oxygen histograms depends strongly on the value of the vascular fraction of the voxel as well as on the assumed vascular architecture. Medical Physics, Vol. 42, No. 1, January 2015. 92. In a previous study,15 we developed a TOM, which simulates the microscopic oxygen distribution in a reference volume for a given vascular fraction based on three different vascular architectures. By sampling the oxygen distribution, the oxygen histograms can be calculated. In this study,15 the so-called “parallel linear vessel architecture (PLVA),” where linear vessels are randomly distributed in 2D (rPLVA) over the reference volume, resulted in realistic oxygen histograms, if a fixed intravascular oxygen concentration of 40 mm Hg was assumed. Additional input parameters of the TOM are the oxygen diffusion coefficient and a PO2-dependent oxygen consumption rate of the tumor cells. Based on these parameters, the diffusion reaction equation for oxygen was solved for a given vascular fraction of the reference volume and the respective oxygen histogram was calculated.15 For the present study, the TOM was used as described in our previous publication for the rPLVA using the same parameters. For consistency of the oxygen distribution in the border region between tumor and normal tissue, a fixed vascular fraction, v f n , was assigned to voxels containing only normal cells (Table I). For an efficient calculation of the tumor response, TOM was used to precalculate the oxygen histograms for vascular fractions in the range of 0.1%–10.0%, in steps of 0.1%. As dead cells were not supposed to consume oxygen, the oxygen histograms were additionally calculated for reduced average consumption rates corresponding to fractions of dead cells in the range of 0%–100%, in steps of 10%. Within the TRM, these precalculated oxygen histograms are stored in a data base, which allows the TRM to access voxel-specific oxygen histograms depending on their individual vascular fraction and fraction of dead cells. In this way, the heterogeneity of the oxygen distribution on a microscopic scale is incorporated into the TRM on a voxel-scale. The availability of the oxygen histograms allows also to calculate the clinically important hypoxic fraction (HF), which is here defined as the fractional volume with PO2 < 5 mm Hg. 2.A.1.c. Tumor response parameters. Tumor response includes the radiosensitivity parameters α and β of the linearquadratic (LQ) model20 as well as the parameter σα describing the intertumoral variation of α. Although different parameter values of α, β, and σα may be assigned to each voxel, fixed values were used for all simulations in this study. In addition, the two parameters m and k, describing the functional dependence of the oxygen enhancement ratio (OER) on PO2 have to be specified (see “cell survival after irradiation” in Sec. 2.A.2). Finally, the rate at which dead cells are resorbed is described by the resorption half-life tr (see “resorption of dead cells” in Sec. 2.A.2). Table I provides the parameter values used in the present study. 2.A.1.d. Treatment parameters. Irradiations were simulated according to the fractionation scheme, described by the start of radiotherapy: the number of fractions, the temporal spacing between fractions as well as the delivered dose distributions. In this study, target volumes were assumed to be covered homogeneously by a fixed dose, so the prescribed dose per fraction was the only required parameter. The dose at each fraction was assumed to be delivered instantaneously.

(5) 93. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. and complete repair of sublethal damage between fractions was assumed. 2.A.2. Components of the TRM. The TRM includes the following components (Fig. 1): 2.A.2.a. Proliferation of tumor cells. To simulate proliferation of tumor cells, the number of tumor cells in each voxel is multiplied by a proliferation factor PF given by ) ( ln(2) ∆t , (1) PF = exp tp where ∆t is the simulation time step and tp is the doubling time (Table I). 2.A.2.b. Angiogenesis. Hypoxia is assumed to induce angiogenesis.26 In the TRM, capillary cells proliferate, if the HF in each voxel differs from zero. In this case, the fraction of capillary cells that corresponds to the hypoxic fraction is multiplied by a proliferation factor PFangio, similar to PF in Eq. (1) ) ( ln(2) ∆t , (2) PFangio = exp ta where ta is the doubling time for capillary cells (Table I). 2.A.2.c. Cell survival after irradiation. In the TRM, only the radiation response of viable tumor cells is considered, while capillary and normal cells are considered as completely radioresistant. The survival fraction (SF) of irradiated tumor cells is calculated by the LQ-model, including the OER as a dose modifying factor to consider the dependence of the radiation response on oxygenation status given by the PO2:6 ( )  β 2 2 α (3a) SF = exp − d · OER PO2 − 2 d OER PO2 , m m where the OER is given by  mPO2 + k . OER PO2 = PO2 + k. (3b). In Eq. (3a), the values of α and β refer to oxic conditions. The parameters k and m describe the functional dependence of the OER on PO2 (Table I). According to its vascular fraction, a specific oxygen histogram is connected to each voxel. For the fraction of tumor cells corresponding to each oxygen level of the histogram, SF is calculated using the respective PO2—value. This procedure is applied for each oxygen level of the histogram and the total number of surviving cells in the voxel is then obtained by adding up the surviving cells over all oxygenation levels. If the number of surviving tumor cells at each oxygenation level is ≥10, the number of surviving cells is taken simply as the SF [Eq. (3a)]. Only if the number of surviving tumor cells is <10, the fate of each tumor cell was determined randomly according to the probability SF, using a random number generator. In this way, stochastic effects occurring for small cell numbers are considered. 2.A.2.d. Resorption of dead cells. Cells killed by irradiation are resorbed only after some time. For this, the resorption fraction is modeled by an exponential probability distribution Medical Physics, Vol. 42, No. 1, January 2015. 93. indicating a fixed probability of resorption per time interval, which is characterized by the resorption half-life tr ( ) ln(2) RF = 1 − exp − ∆t . (4) tr For ≥10 remaining dead cells per voxel, RF directly indicates the fraction that is resorbed. If less than 10 dead cells remain per voxel, the same stochastic approach as described in Sec. 2.A.2.c is used. 2.A.2.e. Exchange of cells between voxels. If the number of cells in voxels decreases due to radiation-induced cell kill and subsequent resorption, or if it increases due to proliferation, it is necessary to redistribute cells between neighboring voxels to maintain the cell density. This may lead to shrinkage or growth of the tumor. In the TRM, this redistribution of cells takes place only if the number of cells per voxel changes more than ±10% from the initial cell density µ. If the cell number in some voxels exceeds the upper limit, these voxels transfer cells to their 26 neighboring voxels [“voxel cell exchange (I)” in Fig. 1] according to the following rules. (i) Only neighboring voxels with lower cell densities (and that do not correspond to bone) receive cells. (ii) The number of transferred cells is proportional to the differences of cell numbers between exchanging voxels and inversely proportional to the Euclidean distances between their centers. (iii) The number of transferred cells of each cell type is handled independently and hence reflects the distributions of the different cell types in the “donor” voxel (i.e., if 10% of the cells in the donor voxel are capillary cells, 10% of the transferred cells are capillary cells too, and the same holds for the other cell types). (iv) This exchange process is repeated iteratively until the cell number of all voxels has arrived in the accepted range of µ ± 10%. As a consequence, the number of voxels containing tumor cells increases, which leads to growth of the virtual tumor. On the other hand, if the density in some voxels decreases below the lower limit of the accepted range, these voxels receive cells from the neighboring voxels [“voxel cell exchange (II)” in Fig. 1] using the same rules as described above. To assure that this process leads to shrinkage of the tumor, the voxels receive cells only from neighboring voxels that are equally close or closer to the surface of the tumor. This distance is measured as the Euclidean distance between the center of the respective voxel and the center of the closest voxel that does not contain any viable tumor cells. The process starts with the voxel that has the largest distance to the border of the tumor. As a result, the virtual tumor shrinks. 2.B. Simulations 2.B.1. Normal tissue geometry and biological tumor characteristics. To demonstrate the main features of the model and to visualize the impact of the different biological effects, artificially created virtual tumors with simple geometries were used in this study. In addition, all simulations consider the same spatial configuration of bone, soft tissue, and air regions,.

(6) 94. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. as shown in Fig. 2. To characterize the biological properties of the tumor as well as the radiation response, the parameters given in Table I, corresponding to head and neck squamous cells carcinomas (HNSSC), were used. 2.B.2. Tumor growth. Tumor growth is demonstrated by starting with a single voxel containing only tumor and capillary cells (Fig. 2). For this initial tumor voxel, the initial vascular fraction was set to that of the surrounding normal tissue since hypoxia is expected to develop only after continued tumor growth. Starting with this initial condition, tumor growth is simulated until the tumor reaches a diameter of approximately 3 cm. To obtain a median hypoxic fraction of 38%, as measured with Eppendorf electrodes in HNSCCs by Nordsmark et al.,27 v f n was set to the realistic value of 3.6%. As no dead and normal cells are considered in this single initial voxel, the remaining cells (96.4%) were assumed to be tumor cells. At each time point during the simulations, the spatial distributions of tumor cells and vascular fraction on the central slice of the tumor were saved for later analysis. 2.B.3. Tumor shrinkage during radiotherapy. Using the resulting virtual tumor of Sec. 2.B.2, the response of the tumor to radiation is simulated using a standard fractionation regime: 2 Gy per fraction (uniformly distributed over the tumor), 5 fractions per week excluding the weekends until tumor control is achieved. For simplicity, all simulations started on Mondays. At each time point during the simulations, the distributions of tumor cells and vascular fraction of a central tumor slice were saved. 2.B.4. Reoxygenation. During treatment simulation, hypoxic tumors may reoxygenate. Within the TRM, there are three different mechanisms that may contribute to reoxygenation of the tumor:. F. 2. Bone, air, and normal soft tissue regions. For the geometry of these regions, axial symmetry was assumed. The position of the voxel containing the initial tumor and capillary cells is displayed in the center of the figure. Medical Physics, Vol. 42, No. 1, January 2015. 94. (i) Hypoxia-induced angiogenesis. (ii) Decrease of the oxygen consumption rate of cells due to the presence of dead cells. (iii) Increase of the vascular fraction due to transport of capillary cells from the tumor periphery toward tumor core as a result of tumor shrinkage. To study the impact of each of these three mechanisms on reoxygenation separately, the simulation of tumor response to irradiation was repeated three times, each time by inactivating two of the three mechanisms [for example, to consider only angiogenesis, the mechanisms (ii) and (iii) were inactivated]. Angiogenesis (i) was inactivated directly in the code. The decrease of the oxygen consumption rate due to dead cells (ii) was inactivated directly in the model by not considering reduced oxygen consumption rates. The transport of cells due to tumor shrinkage (iii) was inactivated by switching off the resorption of dead cells. In this case, no tumor shrinkage and accordingly no transfer of capillary cells occurred. 2.B.5. Sensitivity analysis. As the simulations results in Secs. 2.B.3 and 2.B.4 may depend on the selected parameters, we performed a sensitivity analysis for the most important parameters. For this, we varied separately the values of the doubling times tp and ta for tumor and capillary cells, respectively, as well as the value of the resorption half-life tr (see Sec. 2.A.2). The alternative doubling times of the tumor cells were set to one fourth and four times of the value in Table I (30.6 and 489.6 h, respectively). Impact of angiogenesis was simulated by inactivating proliferation of capillary cells and by setting the doubling time to the value of the tumor cells (122.4 h). Finally, the resorption half-life was changed to one half and two times of the value in Table I (84 and 336 h, respectively). For comparison, the tumor simulated with the parameters in Table I was used as reference. 2.B.6. Tumor control probability (TCP). A TCP curve was generated by simulating the tumor response to radiotherapy for many tumors generated as described in Sec. 2.B.3 (using the parameters values of Table I, which results in an initial median HF of 38%). A tumor was considered as locally controlled, if all tumor cells were killed by the irradiation. Intertumor variation of radiosensitivity was introduced by varying the value of α. In the simulation of each tumor, α takes the value obtained by sampling a normally distributed random variable centered at the value of α and having a standard deviation of σα = 0.1 α. For the selected values, negative values of α were very unlikely and were disregarded, if they occurred. Ten tumors were simulated for each dose level, which results in incidence rates between 0% and 100%. Using these incident rates, dose response curves were adjusted and the tumor control dose TCD50 (dose at 50% TCP) was determined. To study the impact of oxygenation on TCD50, the simulations were repeated with a second parameter set characterizing.

(7) 95. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. 95. F. 3. Central slices of the simulated tumor growth showing tumor cell density (left, in a logarithmic scale from 100 to 107) and vascular fraction (right). The anatomy is underlayed in gray (bone: white, soft tissue: gray, and air: black, see Fig. 2 for details).. a better oxygenated tumor. For this tumor, a median hypoxic fraction of only 2.4% was used. The initial state of this second tumor (also having a diameter of 3 cm) was also obtained by simulating the tumor growth starting with a single tumor voxel. In this case, the vascular fraction of this initial voxel (and the surrounding normal tissue) was set to 7.2%, which is twice as large as the previously used value of 3.6%. For this tumor, the TCP-curve was simulated as well. 2.C. Conditions of simulation and evaluation. During the simulations, an iteration time step ∆t equal to 6 h was used. The image processing software Gnuplot (version 4.2) was used to convert the spatial distributions of tumor cells and vascular fraction in the central tumor slice for each of these time steps into PNG images. In all these images, bone, air, and normal soft tissue regions are indicated as an underlying gray image. Simulations were carried out on a PC equipped with a 3 GHz Intel® Core™2 Quad processor and 3 GB RAM and took up to a few hours.. a diameter of approximately 3 cm is reached. In soft tissue, the tumor grows spherically without infiltrating into the bone. As the virtual tumor grows, the density of tumor cells remains higher in the center as compared to the peripheral region of the tumor. With increasing tumor size, the vascular fraction decreases relative to that of the surrounding normal tissue and a hypoxic tumor core develops. 3.B. Tumor shrinkage during radiotherapy. Figure 4 shows tumor shrinkage during radiotherapy over time. The accumulated dose is indicated in the left column. As a starting point for radiotherapy, a reference tumor was used, generated as described in Sec. 2.B.2. After start of treatment, the virtual tumor of approximately 3 cm diameter starts shrinking and the tumor cell density decreases from approximately 106 tumor cells per voxel in the tumor center to less than 100 after 68 Gy. At the same time, an increase in the vascular fraction in the center from about 2% to almost 4% is observed. Figure 5 displays profiles of the tumor cell density and the vascular fraction at different time points during treatment.. 3. RESULTS 3.A. Tumor growth. 3.C. Reoxygenation. Figure 3 shows the distribution of tumor cell density and vascular fraction for an untreated growing virtual tumor until. For the virtual tumor simulated in Sec. 3.B, Fig. 6(a) displays the oxygen histograms of a central voxel at different. Medical Physics, Vol. 42, No. 1, January 2015.

(8) 96. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. 96. F. 4. Central slices showing tumor cell density (left, in a logarithmic scale from 100 to 107) and vascular fraction (right) of the simulated tumor shrinkage during radiotherapy.. time points during radiotherapy. In addition, the oxygen histograms are shown for a case where only one of the three reoxygenation mechanisms of the TRM is activated [Figs. 6(b)–6(d)]. From Fig. 6(a), it can be seen that reoxygenation is largely completed before t = 300 h, with only minor changes thereafter. A very similar effect is observed in case of tumor. shrinkage only [Fig. 6(d)], however with slightly higher hypoxic fraction values. The other two effects [Figs. 6(b) and 6(c)] show only marginal changes of the hypoxic fraction, which also take place before t = 300 h. Beyond t = 300 h, no significant changes are seen in Figs. 6(b) and 6(c). Only a very slight decrease of the HF is observed between t = 600 h and t = 900 h in Fig. 6(b).. F. 5. Profiles of tumor cell density (left) and vascular fraction (right) during radiotherapy on a line through the center of the virtual tumor. Medical Physics, Vol. 42, No. 1, January 2015.

(9) 97. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. 97. F. 6. Oxygen histograms of a central voxel during the radiotherapy course and impact of the three different mechanisms that contribute to tumor reoxygenation within the TRM: (a) sum of the three effects, (b) angiogenesis only, (c) decreased oxygen consumption due to dead cells only, and (d) tumor shrinkage only. The light gray bins in the histograms represent the HF.. 3.C.1. Sensitivity analysis. Figure 7 displays the results of the sensitivity analysis. Increasing the doubling time of tumor cells has a larger impact on tumor cell density as well as on vascular fraction than a decrease. Changing the doubling time of capillary cells has almost no influence on tumor cell density but may still be of significant importance for the vascular fraction, if angiogenesis is increased. Changing the resorption half-life has no influence on tumor cell density, however, it can significantly reduce the vascular fraction for a decreased resorption time. 3.D. Tumor control probability. Figure 8 displays the TCP-curve for two simulated tumors using two different initial vascular fractions for the tumor and the normal tissue. The curve on the right represents the Medical Physics, Vol. 42, No. 1, January 2015. simulated TCP curve of the hypoxic virtual tumor used as reference in Sec. 3.C (vascular fraction 3.6%). The tolerance dose TCD50 is 71.0 Gy. The simulated TCP curve for a well-oxygenated tumor (vascular fraction 7.2%) is shown as well (left curve). As compared to the hypoxic case, the curve is shifted to the left by 17.4 Gy, showing a TCD50 of 53.6 Gy. 4. DISCUSSION It is widely assumed that therapy outcome could be significantly improved if reliable models for the prediction of the tumor and normal tissue response were available. Prediction of the tumor response is usually based on the Poisson model, which assumes that local control of the tumor is achieved if all clonogenic cells are sterilized by the radiation. Apart from the fact that this assumption might not be strictly fulfilled in.

(10) 98. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. 98. F. 7. Impact of decreased and increased proliferation of tumor cells (proliferation) and capillary cells (angiogenesis) as well as resorption of dead tumor cells (resorption), respectively, on tumor cell density and vascular fraction.. real tumors, a major problem with tumor response modeling is the intrinsic heterogeneity of tumors. This heterogeneity is observed on three different levels: (i) the radiosensitivity of tumors may vary from patient to patient, (ii) within one patient, the tumor characteristic may exhibit regional differences, and (iii) the response of tumor cells may vary on a microscopic scale. The interpatient heterogeneity (i) explains the fact that clinical dose response curves are less steep as compared to predictions of the Poisson model and in the latter, this is considered by averaging TCP-values over a population of Medical Physics, Vol. 42, No. 1, January 2015. patients.28 Examples for regional differences (ii) are variations of tumor cell density or the presence of hypoxic regions. While a higher number of tumor cells require a higher dose to arrive at the same survival level, hypoxia modulates the radiosensitivity of the cells. As a consequence, an analytical modeling of the whole tumor is not possible and a voxel-based modeling approach is more appropriate. Considering all tumor cells within one voxel as a homogeneous sample, however, is also a too simplistic approach, e.g., in case of hypoxia, it is known that the oxygen levels vary strongly on a microscopic scale29–32 (iii) and a relatively small hypoxic fraction of tumor.

(11) 99. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. 99. It is the aim of this publication to present the main features of the TRM. Although only the most important biological processes are included, we are aware that the model is already highly complex as it includes a number of biological mechanisms and input parameters. The related problems are manifold and the most significant issues are discussed in the following. 4.A. Tissue oxygenation. F. 8. TCP curves simulated for a human squamous cell carcinoma with two different oxygenation levels. An increase of the vascular fraction from 3.6% to 7.2% decreases TCD50 by 17.4 Gy.. cells may determine whether the tumor is controlled or not. In addition, the intrinsic radiosensitivity is known to vary during cell cycle2 and resting cells are known to be more resistant to irradiation.9 To fully consider this type of heterogeneity, a modeling approach on a microscopic level is in principle required. Although some work in this direction has been done, this faces severe limitations. First, for tumors of realistic sizes, the required computing resources are beyond current capabilities and second, even if this technical issue could be solved, the problem of obtaining suitable input data on a microscopic scale remains. With the presented TRM, we avoid these problems by following a multiscale approach. This approach models the tumor response on three different scales: (i) On a microscopic scale, the oxygen distribution is simulated only once, based on the TOM (Ref. 15) using a reference volume together with different vascular fractions and numbers of oxygen-consuming tumor cells. The resulting dose distributions are then condensed into oxygen histograms, which are stored in a database. (ii) On a mesoscale, each voxel is connected to its individual vascular fraction and its number of viable tumor cells and thus with one of the precalculated oxygen histograms. By sampling the response of the tumor cells within one voxel according to the respective oxygen histogram, the microscopic features of the oxygen distribution are incorporated into the voxel-scale. Moreover, as the oxygen histograms are precalculated, this procedure is computationally efficient. As a consequence of this approach, the spatial information on cells and oxygen levels within one voxel is lost. This, however, is not critical as the regional properties of the tumor within one voxel may be considered to be constant, while the heterogeneity on the microscopic scale is still maintained. (iii) Finally, on a macroscopic scale, proliferation and radiation-induced cell kill lead to growth or shrinkage of the tumor. These are morphometric parameters, which can be easily accessed by imaging of the patient. As the underlying cell-exchange process includes capillary cells, the influence of proliferation and cell kill on oxygenation is incorporated from the voxel into the macroscopic scale. Medical Physics, Vol. 42, No. 1, January 2015. On a microscopic level, the TRM makes use of the TOM to calculate the oxygen histograms and the underlying problems have been discussed in detail in our previous publication.15 While the mechanism of oxygen transport in tissue is well described by the oxygen reaction–diffusion equation and the diffusion coefficient in tissue can be considered as being relatively accurate, the assumed vascular architecture may be more critical. Although we compared linear capillaries in 2D and 3D and found no significant differences in the oxygen histograms, the difference relative to real vascular architectures has not been investigated. In addition, the detailed shape of the PO2-dependent oxygen consumption rate as well as the impact of systemic factors like perfusion may be a critical issue. Here, future research comparing modeling approaches with thorough experimental work is necessary. Nevertheless, the developed TOM provides initial estimations for the oxygen histograms, which are comparable to those obtained from measurements.33–37 4.B. Tumor growth. Within the TRM, proliferation of tumor cells is a critical parameter. As neoangiogenesis was assumed only within the hypoxic fraction, proliferation of tumor cells is the driving factor for the development of hypoxia in our model. This was shown by simulating growth of an untreated tumor starting from a single voxel. The result is a hypoxic core, which is in agreement with the experimental finding that hypoxia typically appears in central tumor regions due to the fast proliferation of tumor cells and the inappropriate supply of oxygen. This is a consequence of chaotic vascular architectures and insufficient angiogenesis.21,34 In spite of this simplistic approach, realistic median hypoxic fractions were obtained.27 The impact of changing proliferation rates of tumor and capillary cells was studied as well. While the impact of a significantly enhanced angiogenic process is very small, substantial changes in tumor size and vascular fraction were seen in the results when tumor cell proliferation rate was changed. Great impact on the oxygenation level was seen also when modified vascular fraction values for the initial tumor voxel and the surrounding normal tissue were used, e.g., the median hypoxic fraction was reduced from 38% to 2.4% when a vascular fraction of 7.2% was used instead of 3.6%. Therefore, an adequate choice of tp and v f n is of major importance. The simulation of tumor growth was performed to demonstrate the mechanism of proliferation and its impact on hypoxia, however, it is not expected that the complete.

(12) 100. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. development of real tumors can be described by the underlying growth model. Instead, the TRM focuses on the development of the tumor under irradiation. In this case, tumor shrinkage due to radiation-induced cell kill with subsequent resorption of dead cells counteracts with tumor growth, and for a short-term prediction, the relative impact of both is relevant. To be independent of the pretreatment growth phase of the tumor and to adapt the model to the individual patient, the initial status of the tumor is assumed to be characterized by external input data, preferably from imaging, rather than by a simulation of the tumor ab initio. 4.C. Radiation response. Within the TRM, the radiation response is simulated using the linear-quadratic model including the OER as a dose modifying factor. These model components are well established and were also used by others.6,38 To our knowledge, however, sampling the radiation response by means of the voxel-specific oxygen histograms represents a new approach and allows considering the microscopic heterogeneity of the oxygen distribution on a voxel-scale. While the development of hypoxic regions is a consequence of enhanced tumor growth, the TRM includes three reoxygenation mechanisms during treatment, which may be important at least for some tumors.2 With this respect, our simulations illustrate the predominant influence of tumor shrinkage, while angiogenesis and reduced oxygen consumption due to dead tumor cells are of minor importance. This result, however, may depend on the specific parameter values selected for the simulation. As shown by the results of the sensitivity analysis, the temporal development of the vascular fraction (and hence oxygenation) depends critically on the balance of proliferation of tumor and capillary cells as well as on the resorption rate of inactivated tumor cells. For example, a drastically reduced resorption rate would maintain the structural reasons of a reduced vascular fraction and could only be compensated by hypoxia-induced angiogenesis. In this case, increased angiogenesis may be of higher relative impact for reoxygenation than in our simulation. As tumor shrinkage can be assessed by imaging of the tumor, this problem might be solved by adapting the resorption rate such that tumor volumes from imaging series are reflected in the simulation. In our model, we have assumed capillary cells as being completely resistant against radiation. This assumption represents a simplification in our model and contributes to the uncertainty of its predictions as capillary response may lead to a decreased nutrition supply as well as to the development of hypoxia.39 The impact of capillary cell response has been clearly shown for single high-dose and hypofractionated irradiations,40,41 while the importance for conventionally fractionated treatment schedules is more controversial. In the latter, most of the in vivo studies indicate a minor role of radiation-induced vascular damage in tumors as a determinant of response for the endpoint of permanent local control.42–44 Moreover, the underlying mechanisms are highly complex and most likely different at different dose levels.45 Although Medical Physics, Vol. 42, No. 1, January 2015. 100. the radiosensitivity of capillary cells could in principle be implemented in a similar way as for tumor cells, the underlying biological mechanisms remain still rather unclear. Thus, for the investigated 2 Gy fractionation schedules, we neglected the radiosensitivity of capillary cells and considered this as a first approximation in our model. 4.D. Input parameter. The problem of finding adequate input parameters is common to all biological models and it increases with the complexity of the model. This is of special importance, if predictions on a macro- or mesoscopic scale involve processes on a microscopic scale. The presented TRM combines two important design features: first, it is a voxel-based model, which in principle allows the characterization of individual patient tumors prior to irradiation based on medical imaging techniques. Second, although hypoxia is modeled on a microscopic scale using the TOM, the respective input parameter, vascular fraction, is required only on a voxel-scale. In principle, all parameters of Table I may be provided as 3D-parameter maps, although only the vascular fraction and the tumor cell density were specified in this way in the present study. This approach relies on the optimistic assumption that future imaging techniques will allow measuring these parameters spatially resolved and as long this is not realized, reasonable global values have to be used. In a review paper, Bentzen and Gregoire46 discuss the most important PET techniques used to assess tumor burden (FDG and choline), proliferation maps (FLT) and hypoxia maps [[18F] fluoromisonidazole, EF3, EF5, and 64Cu-labeled copper(II) diacetyl-di(N4-methylthiosemicarbazone)] in the context of dose painting. Blood-oxygen-level dependent (BOLD) dynamic contrast-enhanced MR imaging has been also investigated as a method for mapping for hypoxia regions in tumors.47 However, despite these very promising preliminary studies, more research in this area is needed to achieve reliable quantitative information. This work focuses on hypoxia since this is the most prominent example where the microscopic heterogeneity influences tumor response and where significant efforts are made to derive information from imaging. Although there are still open issues with the underlying TOM (see Sec. 4.A), there already exist approaches to determine specifically the vascular fraction as the most important input parameter by spatially resolved imaging using dynamic PET and high-resolution contrast-enhanced MR.16–19 Regarding the radiation response, which depends on the number of tumor cells as well as on the response parameters α and β, it appears unlikely that a reliable description of a clinical tumor response can solely be derived from in vitro tumor cell measurements.48 Moreover, tumor cell density and distribution of radiosensitivity are currently difficult to estimate.49 Nevertheless, the influence of these parameters is relevant for assessing tumor response, as studied by previous relevant theoretical works regarding intratumor heterogeneities in the context of dose painting.50–53 In our work, tumor cell density and intrinsic radiosensitivities may be adjusted.

(13) 101. Espinoza, Peschke, and Karger: Simulation of the response of hypoxic tumors. such that the simulated dose response curves reflect the clinically observed tumor control doses. As shown by Fig. 8, realistic values may be obtained. Finally, also other parameters such as the proliferation and resorption rates might be adjusted from longitudinal imaging series in patients. Most likely, the determination of the different input parameters has to be investigated in dedicated preclinical and clinical studies. 4.E. Potential clinical applications. Although the TRM can in principal simulate TCP-curves, it is not a typical TCP-model. As tumor response for the individual patient is intrinsically probabilistic and the decision whether the tumor is controlled or not can be made only after months or even years, the uncertainties involved in the predictions increase with time. Therefore, a prediction of the complete tumor development after irradiation appears difficult. Rather, the potential value of the TRM lies in the short-term prediction of the tumor response during the radiotherapy course. If the model could be adjusted to clinical data on this time scale, short-term predictions of cell kill and associated parameters like tumor shrinkage and reoxygenation might offer the opportunity to compare the tumor response for different treatment plans and to optimize dose distributions and fractionation schedules. This may result in boost-treatments of hypoxic regions or even “dose painting” approaches,54,55 however, these approaches require reliable information of the temporal development of the tumor. This information may be generated by imaging supported by predictions from tumor response models. Finally, the TRM offers the possibility to study the relative biological importance of different tumor or treatment parameters on tumor development in general as well as on the TCP-curves. As an example for analysis of tumor-specific factors, Fig. 8 demonstrates the impact of hypoxia on TCP. In addition to this, the comparative analysis of standard against hyperfractionated, hypofractionated, or accelerated treatment schedules is of high clinical relevance. 5. CONCLUSION In this work, a multiscale TRM has been developed. After characterizing the initial status of the tumor prior to treatment, the TRM simulates the response of hypoxic tumors to radiotherapy. As a special feature, the initial tumor characteristics have to be specified only on a voxel-scale, which allows using morphological or functional images of the patient. The model predicts the spatiotemporal development of the tumor cell density, the vascular fraction as well as the resulting oxygenation of the tumor. Yet, the model still has to be validated by preclinical and clinical studies. 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Figure

Figure 3 shows the distribution of tumor cell density and vascular fraction for an untreated growing virtual tumor until
Figure 7 displays the results of the sensitivity analysis.

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