Limitations of a TCP model incorporating population heterogeneity

Monte Carlo role in radiobiological modelling of radiotherapy outcomes

Radiobiological models are essential components of modern radiotherapy. They are increasingly applied to optimize and evaluate the quality of different treatment planning modalities. They are frequently used in designing new radiotherapy clinical trials by estimating the expected therapeutic ratio of new protocols. In radiobiology, the therapeutic ratio is estimated from the expected gain in tumour control probability (TCP) to the risk of normal tissue complication probability (NTCP). However, estimates of TCP/NTCP are currently based on the deterministic and simplistic linear-quadratic formalism with limited prediction power when applied prospectively. Given the complex and stochastic nature of the physical, chemical and biological interactions associated with spatial and temporal radiation induced effects in living tissues, it is conjectured that methods based on Monte Carlo (MC) analysis may provide better estimates of TCP/NTCP for radiotherapy treatment planning and trial design. Indeed, over the past few decades, methods based on MC have demonstrated superior performance for accurate simulation of radiation transport, tumour growth and particle track structures; however, successful application of modelling radiobiological response and outcomes in radiotherapy is still hampered with several challenges. In this review, we provide an overview of some of the main techniques used in radiobiological modelling for radiotherapy, with focus on the MC role as a promising computational vehicle. We highlight the current challenges, issues and future potentials of the MC approach towards a comprehensive systems-based framework in radiobiological modelling for radiotherapy.

A model of cellular dosimetry for macroscopic tumors in radiopharmaceutical therapy

PURPOSE

In the radiopharmaceutical therapy approach to the fight against cancer, in particular when it comes to translating laboratory results to the clinical setting, modeling has served as an invaluable tool for guidance and for understanding the processes operating at the cellular level and how these relate to macroscopic observables. Tumor control probability (TCP) is the dosimetric end point quantity of choice which relates to experimental and clinical data: it requires knowledge of individual cellular absorbed doses since it depends on the assessment of the treatment's ability to kill each and every cell. Macroscopic tumors, seen in both clinical and experimental studies, contain too many cells to be modeled individually in Monte Carlo simulation; yet, in particular for low ratios of decays to cells, a cell-based model that does not smooth away statistical considerations associated with low activity is a necessity. The authors present here an adaptation of the simple sphere-based model from which cellular level dosimetry for macroscopic tumors and their end point quantities, such as TCP, may be extrapolated more reliably.

METHODS

Ten homogenous spheres representing tumors of different sizes were constructed in GEANT4. The radionuclide 131I was randomly allowed to decay for each model size and for seven different ratios of number of decays to number of cells, N(r): 1000, 500, 200, 100, 50, 20, and 10 decays per cell. The deposited energy was collected in radial bins and divided by the bin mass to obtain the average bin absorbed dose. To simulate a cellular model, the number of cells present in each bin was calculated and an absorbed dose attributed to each cell equal to the bin average absorbed dose with a randomly determined adjustment based on a Gaussian probability distribution with a width equal to the statistical uncertainty consistent with the ratio of decays to cells, i.e., equal to Nr-1/2. From dose volume histograms the surviving fraction of cells, equivalent uniform dose (EUD), and TCP for the different scenarios were calculated. Comparably sized spherical models containing individual spherical cells (15 microm diameter) in hexagonal lattices were constructed, and Monte Carlo simulations were executed for all the same previous scenarios. The dosimetric quantities were calculated and compared to the adjusted simple sphere model results. The model was then applied to the Bortezomib-induced enzyme-targeted radiotherapy (BETR) strategy of targeting Epstein-Barr virus (EBV)-expressing cancers.

RESULTS

The TCP values were comparable to within 2% between the adjusted simple sphere and full cellular models. Additionally, models were generated for a nonuniform distribution of activity, and results were compared between the adjusted spherical and cellular models with similar comparability. The TCP values from the experimental macroscopic tumor results were consistent with the experimental observations for BETR-treated 1 g EBV-expressing lymphoma tumors in mice.

CONCLUSIONS

The adjusted spherical model presented here provides more accurate TCP values than simple spheres, on par with full cellular Monte Carlo simulations while maintaining the simplicity of the simple sphere model. This model provides a basis for complementing and understanding laboratory and clinical results pertaining to radiopharmaceutical therapy.

Population TCP estimators in case of heterogeneous irradiation: a new discussion of an old problem

PURPOSE

To investigate the capacity of two phenomenological expressions to describe the population tumor response in case of a heterogeneous irradiation of the tumor. The generalization of the individual tumor control probability (TCP) models to include the case of a heterogeneous irradiation is a trivial problem. However, an analytical solution that results in a closed form population TCP formula for the heterogeneous case is, unfortunately, a very complex mathematical problem. Therefore we applied a numerical approach to the problem.

METHOD

Pseudo-experimental data sets are constructed through the generation of dose distributions and population TCP data obtained by a numerical solution of a multi-dimensional integral over an individual TCP model. The capacity of the following two phenomenological - Poisson and equivalent uniform dose (EUD) based - TCP expressions: [Figure: see text] to describe the population tumor response in case of heterogeneous irradiation is investigated through their fitting to the psuedo-experimental data sets. RESULTS AND CONCLUSIONS. While both expressions produce statistically acceptable fits to the pseudo-experimental data within 2% TCP error band, the use of the second expression is preferable since it produces considerably better fits to the data sets.

Choline PET based dose-painting in prostate cancer--modelling of dose effects

BACKGROUND

Several randomized trials have documented the value of radiation dose escalation in patients with prostate cancer, especially in patients with intermediate risk profile. Up to now dose escalation is usually applied to the whole prostate. IMRT and related techniques currently allow for dose escalation in sub-volumes of the organ. However, the sensitivity of the imaging modality and the fact that small islands of cancer are often dispersed within the whole organ may limit these approaches with regard to a clear clinical benefit. In order to assess potential effects of a dose escalation in certain sub-volumes based on choline PET imaging a mathematical dose-response model was developed.

METHODS

Based on different assumptions for alpha/beta, gamma 50, sensitivity and specificity of choline PET, the influence of the whole prostate and simultaneous integrated boost (SIB) dose on tumor control probability (TCP) was calculated. Based on the given heterogeneity of all potential variables certain representative permutations of the parameters were chosen and, subsequently, the influence on TCP was assessed.

RESULTS

Using schedules with 74 Gy within the whole prostate and a SIB dose of 90 Gy the TCP increase ranged from 23.1% (high detection rate of choline PET, low whole prostate dose, high gamma 50/ASTRO definition for tumor control) to 1.4% TCP gain (low sensitivity of PET, high whole prostate dose, CN + 2 definition for tumor control) or even 0% in selected cases. The corresponding initial TCP values without integrated boost ranged from 67.3% to 100%. According to a large data set of intermediate-risk prostate cancer patients the resulting TCP gains ranged from 22.2% to 10.1% (ASTRO definition) or from 13.2% to 6.0% (CN + 2 definition).

DISCUSSION

Although a simplified mathematical model was employed, the presented model allows for an estimation in how far given schedules are relevant for clinical practice. However, the benefit of a SIB based on choline PET seems less than intuitively expected. Only under the assumption of high detection rates and low initial TCP values the TCP gain has been shown to be relevant.

CONCLUSIONS

Based on the employed assumptions, specific dose escalation to choline PET positive areas within the prostate may increase the local control rates. Due to the lack of exact PET sensitivity and prostate alpha/beta parameter, no firm conclusions can be made. Small variations may completely abrogate the clinical benefit of a SIB based on choline PET imaging.

BGRT: biologically guided radiation therapy-the future is fast approaching!

Rapid advances in functional and biological imaging, predictive assays, and our understanding of the molecular and cellular responses underpinning treatment outcomes herald the coming of the long-sought goal of implementing patient-specific biologically guided radiation therapy (BGRT) in the clinic. Biological imaging and predictive assays have the potential to provide patient-specific, three-dimensional information to characterize the radiation response characteristics of tumor and normal structures. Within the next decade, it will be possible to combine such information with advanced delivery technologies to design and deliver biologically conformed, individualized therapies in the clinic. The full implementation of BGRT in the clinic will require new technologies and additional research. However, even the partial implementation of BGRT treatment planning may have the potential to substantially impact clinical outcomes.

Analytic investigation into effect of population heterogeneity on parameter ratio estimates

PURPOSE

A homogeneous tumor control probability (TCP) model has previously been used to estimate the alpha/beta ratio for prostate cancer from clinical dose-response data. For the ratio to be meaningful, it must be assumed that parameter ratios are not sensitive to the type of tumor control model used. We investigated the validity of this assumption by deriving analytic relationships between the alpha/beta estimates from a homogeneous TCP model, ignoring interpatient heterogeneity, and those of the corresponding heterogeneous (population-averaged) model that incorporated heterogeneity.

METHODS AND MATERIALS

The homogeneous and heterogeneous TCP models can both be written in terms of the geometric parameters D(50) and gamma(50). We show that the functional forms of these models are similar. This similarity was used to develop an expression relating the homogeneous and heterogeneous estimates for the alpha/beta ratio. The expression was verified numerically by generating pseudo-data from a TCP curve with known parameters and then using the homogeneous and heterogeneous TCP models to estimate the alpha/beta ratio for the pseudo-data.

RESULTS

When the dominant form of interpatient heterogeneity is that of radiosensitivity, the homogeneous and heterogeneous alpha/beta estimates differ. This indicates that the presence of this heterogeneity affects the value of the alpha/beta ratio derived from analysis of TCP curves.

CONCLUSIONS

The alpha/beta ratio estimated from clinical dose-response data is model dependent--a heterogeneous TCP model that accounts for heterogeneity in radiosensitivity will produce a greater alpha/beta estimate than that resulting from a homogeneous TCP model.

Fundamental form of a population TCP model in the limit of large heterogeneity

A population tumor control probability (TCP) model for fractionated external beam radiotherapy, based on Poisson statistics and in the limit of large parameter heterogeneity, is studied. A reduction of a general eight-parameter TCP equation, which incorporates heterogeneity in parameters characterizing linear-quadratic radiosensitivity, repopulation, and clonogen number, to an equation with four parameters is obtained. The four parameters represent the mean and standard deviation for both clonogen number and a generalized radiosensitivity that includes linear-quadratic and repopulation descriptors. Further, owing to parameter inter-relationship, it is possible to express these four parameters as three ratios of parameters in the large heterogeneity limit. These ratios can be directly linked to two defining features of the TCP dose response: D50 and gamma50. In the general case, the TCP model can be written in terms of D50, gamma50 and a third parameter indicating the ratio of the levels of heterogeneity in clonogen number and generalized radiosensitivity; however, the third parameter is unnecessary when either of these two sources of heterogeneity is dominant. It is shown that heterogeneity in clonogen number will have little impact on the TCP formula for clinical scenarios, and thus it will generally be the case that the fundamental form of the Poisson-based population TCP model can be specified completely in terms of D50 and gamma50: TCP= 1/2 erfc[square root of pi(gamma50)(D50/D-1)]. This implies that limited radiobiological information can be determined by the analysis of dose response data: information about parameter ratios can be ascertained, but knowledge of absolute values for the fundamental radiobiological parameters will require independent auxiliary measurements.