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

Biological dose optimization incorporating intra-tumoural cellular radiosensitivity heterogeneity in ion-beam therapy treatment planning

Objective.Treatment plans of ion-beam therapy have been made under an assumption that all cancer cells within a tumour equally respond to a given radiation dose. However, an intra-tumoural cellular radiosensitivity heterogeneity clearly exists, and it may lead to an overestimation of therapeutic effects of the radiation. The purpose of this study is to develop a biological model that can incorporate the radiosensitivity heterogeneity into biological optimization for ion-beam therapy treatment planning.Approach.The radiosensitivity heterogeneity was modeled as the variability of a cell-line specific parameter in the microdosimetric kinetic model following the gamma distribution. To validate the developed intra-tumoural-radiosensitivity-heterogeneity-incorporated microdosimetric kinetic (HMK) model, a treatment plan with H-ion beams was made for a chordoma case, assuming a radiosensitivity heterogeneous region within the tumour. To investigate the effects of the radiosensitivity heterogeneity on the biological effectiveness of H-, He-, C-, O-, and Ne-ion beams, the relative biological effectiveness (RBE)-weighted dose distributions were planned for a cuboid target with the stated ion beams without considering the heterogeneity. The planned dose distributions were then recalculated by taking the heterogeneity into account.Main results. The cell survival fraction and corresponding RBE-weighted dose were formulated based on the HMK model. The first derivative of the RBE-weighted dose distribution was also derived, which is needed for fast biological optimization. For the patient plan, the biological optimization increased the dose to the radiosensitivity heterogeneous region to compensate for the heterogeneity-induced reduction in biological effectiveness of the H-ion beams. The reduction in biological effectiveness due to the heterogeneity was pronounced for low linear energy transfer (LET) beams but moderate for high-LET beams. The RBE-weighted dose in the cuboid target decreased by 7.6% for the H-ion beam, while it decreased by just 1.4% for the Ne-ion beam.Significance.Optimal treatment plans that consider intra-tumoural cellular radiosensitivity heterogeneity can be devised using the HMK model.

Rethinking the potential role of dose painting in personalized ultra-fractionated stereotactic adaptive radiotherapy

Fractionated radiotherapy was established in the 1920s based upon two principles: (1) delivering daily treatments of equal quantity, unless the clinical situation requires adjustment, and (2) defining a specific treatment period to deliver a total dosage. Modern fractionated radiotherapy continues to adhere to these century-old principles, despite significant advancements in our understanding of radiobiology. At UT Southwestern, we are exploring a novel treatment approach called PULSAR (Personalized Ultra-Fractionated Stereotactic Adaptive Radiotherapy). This method involves administering tumoricidal doses in a pulse mode with extended intervals, typically spanning weeks or even a month. Extended intervals permit substantial recovery of normal tissues and afford the tumor and tumor microenvironment ample time to undergo significant changes, enabling more meaningful adaptation in response to the evolving characteristics of the tumor. The notion of dose painting in the realm of radiation therapy has long been a subject of contention. The debate primarily revolves around its clinical effectiveness and optimal methods of implementation. In this perspective, we discuss two facets concerning the potential integration of dose painting with PULSAR, along with several practical considerations. If successful, the combination of the two may not only provide another level of personal adaptation ("adaptive dose painting"), but also contribute to the establishment of a timely feedback loop throughout the treatment process. To substantiate our perspective, we conducted a fundamental modeling study focusing on PET-guided dose painting, incorporating tumor heterogeneity and tumor control probability (TCP).

Dosimetric Validation for Prospective Clinical Trial of GRID Collimator-Based Spatially Fractionated Radiation Therapy: Dose Metrics Consistency and Heterogeneous Pattern Reproducibility

PURPOSE

Dose heterogeneity within a tumor target is likely responsible for the biologic effects and local tumor control from spatially fractionated radiation therapy (SFRT). This study used a commercially available GRID-pattern dose mudulated nonuniform radiation therapy (GRID) collimator to assess the interplan variability of heterogeneity dose metrics in patients with various bulky tumor sizes and depths.

METHODS AND MATERIALS

The 3-dimensional heterogeneity metrics of 14 bulky tumors, ranging from 155 to 2161 cm3 in volume, 6 to 23 cm in equivalent diameter, and 3 to 13 cm in depth, and treated with GRID collimator-based SFRT were studied. A prescription dose of 15 Gy was given at the tumor center with 6 MV photons. The dose-volume histogram indices, dose heterogeneity parameters, and peak/valley dose ratios were derived; the equivalent uniform doses of cancer cells with various radiosensitivities in each plan were estimated. To account for the spatial fractionation, high dose core number density of the tumor target was defined and calculated.

RESULTS

Among 14 plans, the dose-volume histogram indices D5, D10, D50, D90, and D95 (doses covering 5%, 10%, 50%, 90%, and 95% of the target volume) were found within 10% variation. The dose ratio of D10/D90 also showed a moderate consistency (range, 3.9-5.0; mean, 4.4). The equivalent uniform doses were consistent, ranging from 4.3 to 5.5 Gy, mean 4.6 Gy, for radiosensitive cancer cells and from 5.8 to 6.9 Gy, mean 6.2 Gy, for radioresistant cancer cells. The high dose core number density was within 20% among all plans.

CONCLUSIONS

GRID collimator-based SFRT delivers a consistent heterogeneity dose distribution and high dose core density across bulky tumor plans. The interplan reproducibility and simplicity of GRID therapy may be useful for certain clinical indications and interinstitutional clinical trial design, and its heterogeneity metrics may help guide multileaf-collimator-based SFRT planning to achieve similar or further optimized dose distributions.

Tumor control probability in hypofractionated radiotherapy as a function of total and hypoxic tumor volumes

Clinical studies in the hypofractionated stereotactic body radiotherapy (SBRT) have shown a reduction in the probability of local tumor control with increasing initial tumor volume. In our earlier work, we obtained and tested an analytical dependence of the tumor control probability (TCP) on the total and hypoxic tumor volumes using conventional radiotherapy model with the linear-quadratic (LQ) cell survival. In this work, this approach is further refined and tested against clinical observations for hypofractionated radiotherapy treatment schedules. Compared to radiotherapy with conventional fractionation schedules, simulations of hypofractionated radiotherapy may require different models for cell survival and the oxygen enhancement ratio (OER). Our TCP simulations in hypofractionated radiotherapy are based on the LQ model and the universal survival curve (USC) developed for the high doses used in SBRT. The predicted trends in local control as a function of the initial tumor volume were evaluated in SBRT for non-small cell lung cancer (NSCLC). Our results show that both LQ and USC based models cannot describe the TCP reduction for larger tumor volumes observed in the clinical studies if the tumor is considered completely oxygenated. The TCP calculations are in agreement with the clinical data if the subpopulation of radio-resistant hypoxic cells is considered with the volume that increases as initial tumor volume increases. There are two conclusions which follow from our simulations. First, the extent of hypoxia is likely a primary reason of the TCP reduction with increasing the initial tumor volume in SBRT for NSCLC. Second, the LQ model can be an acceptable approximation for the TCP calculations in hypofractionated radiotherapy if the tumor response is defined primarily by the hypoxic fraction. The larger value of OER in the hypofractionated radiotherapy compared to the conventional radiotherapy effectively extends the applicability of the LQ model to larger doses.

Local Disease-Free Survival Rate (LSR) Application to Personalize Radiation Therapy Treatments in Breast Cancer Models

Cancer heterogeneity represents the main issue for defining an effective treatment in clinical practice, and the scientific community is progressively moving towards the development of more personalized therapeutic regimens. Radiotherapy (RT) remains a fundamental therapeutic treatment used for many neoplastic diseases, including breast cancer (BC), where high variability at the clinical and molecular level is known. The aim of this work is to apply the generalized linear quadratic (LQ) model to customize the radiant treatment plan for BC, by extracting some characteristic parameters of intrinsic radiosensitivity that are not generic, but may be exclusive for each cell type. We tested the validity of the generalized LQ model and analyzed the local disease-free survival rate (LSR) for breast RT treatment by using four BC cell cultures (both primary and immortalized), irradiated with clinical X-ray beams. BC cells were chosen on the basis of their receptor profiles, in order to simulate a differential response to RT between triple negative breast and luminal adenocarcinomas. The MCF10A breast epithelial cell line was utilized as a healthy control. We show that an RT plan setup based only on α and β values could be limiting and misleading. Indeed, two other parameters, the doubling time and the clonogens number, are important to finely predict the tumor response to treatment. Our findings could be tested at a preclinical level to confirm their application as a variant of the classical LQ model, to create a more personalized approach for RT planning.

Prediction of Tumor Control in 90Y Radioembolization by Logit Models with PET/CT-Based Dose Metrics

The aim of this work was to develop models for tumor control probability (TCP) in radioembolization with ⁹⁰Y PET/CT-derived radiobiologic dose metrics. Methods: Patients with primary liver cancer or liver metastases who underwent radioembolization with glass microspheres were imaged with ⁹⁰Y PET/CT for voxel-level dosimetry to determine lesion absorbed dose (AD) metrics, biological effective dose (BED) metrics, equivalent uniform dose, and equivalent uniform BED for 28 treatments (89 lesions). The lesion dose-shrinkage correlation was assessed on the basis of RECIST and, when available, modified RECIST (mRECIST) at first follow-up. For a subset with mRECIST, logit regression TCP models were fit via maximum likelihood to relate lesion-level binary response to the dose metrics. As an exploratory analysis, the nontumoral liver dose-toxicity relationship was also evaluated. Results: Lesion dose-shrinkage analysis showed that there were no significant differences between model parameters for primary and metastatic subgroups and that correlation coefficients were superior with mRECIST. Therefore, subsequent TCP analysis was performed for the combined group using mRECIST only. The overall lesion-level mRECIST response rate was 57%. The AD and BED metrics yielding 50% TCP were 292 and 441 Gy, respectively. All dose metrics considered for TCP modeling, including mean AD, were significantly associated with the probability of response, with high areas under the curve (0.87-0.90, P < 0.0001) and high sensitivity (>0.75) and specificity (>0.83) calculated using a threshold corresponding to 50% TCP. Because nonuniform AD deposition by microspheres cannot be determined by PET at a microscopic scale, radiosensitivity values extracted here by fitting models to clinical response data were substantially lower than reported for in vitro cell cultures or for external-beam radiotherapy clinical studies. There was no correlation between nontumoral liver AD and toxicity measures. Conclusion: Despite the heterogeneous patient cohort, logistic regression TCP models showed a strong association between various dose metrics and the probability of response. The performance of mean AD was comparable to that of radiobiologic dose metrics that involve more complex calculations. These results demonstrate the importance of considering TCP in treatment planning for radioembolization.

The alfa and beta of tumours: a review of parameters of the linear-quadratic model, derived from clinical radiotherapy studies

BACKGROUND

Prediction of radiobiological response is a major challenge in radiotherapy. Of several radiobiological models, the linear-quadratic (LQ) model has been best validated by experimental and clinical data. Clinically, the LQ model is mainly used to estimate equivalent radiotherapy schedules (e.g. calculate the equivalent dose in 2 Gy fractions, EQD₂), but increasingly also to predict tumour control probability (TCP) and normal tissue complication probability (NTCP) using logistic models. The selection of accurate LQ parameters α, β and α/β is pivotal for a reliable estimate of radiation response. The aim of this review is to provide an overview of published values for the LQ parameters of human tumours as a guideline for radiation oncologists and radiation researchers to select appropriate radiobiological parameter values for LQ modelling in clinical radiotherapy.

METHODS AND MATERIALS

We performed a systematic literature search and found sixty-four clinical studies reporting α, β and α/β for tumours. Tumour site, histology, stage, number of patients, type of LQ model, radiation type, TCP model, clinical endpoint and radiobiological parameter estimates were extracted. Next, we stratified by tumour site and by tumour histology. Study heterogeneity was expressed by the I² statistic, i.e. the percentage of variance in reported values not explained by chance.

RESULTS

A large heterogeneity in LQ parameters was found within and between studies (I² > 75%). For the same tumour site, differences in histology partially explain differences in the LQ parameters: epithelial tumours have higher α/β values than adenocarcinomas. For tumour sites with different histologies, such as in oesophageal cancer, the α/β estimates correlate well with histology. However, many other factors contribute to the study heterogeneity of LQ parameters, e.g. tumour stage, type of LQ model, TCP model and clinical endpoint (i.e. survival, tumour control and biochemical control).

CONCLUSIONS

The value of LQ parameters for tumours as published in clinical radiotherapy studies depends on many clinical and methodological factors. Therefore, for clinical use of the LQ model, LQ parameters for tumour should be selected carefully, based on tumour site, histology and the applied LQ model. To account for uncertainties in LQ parameter estimates, exploring a range of values is recommended.

The alfa and beta of tumours: a review of parameters of the linear-quadratic model, derived from clinical radiotherapy studies

Background

Prediction of radiobiological response is a major challenge in radiotherapy. Of several radiobiological models, the linear-quadratic (LQ) model has been best validated by experimental and clinical data. Clinically, the LQ model is mainly used to estimate equivalent radiotherapy schedules (e.g. calculate the equivalent dose in 2 Gy fractions, EQD₂), but increasingly also to predict tumour control probability (TCP) and normal tissue complication probability (NTCP) using logistic models. The selection of accurate LQ parameters α, β and α/β is pivotal for a reliable estimate of radiation response. The aim of this review is to provide an overview of published values for the LQ parameters of human tumours as a guideline for radiation oncologists and radiation researchers to select appropriate radiobiological parameter values for LQ modelling in clinical radiotherapy.

Methods and materials

We performed a systematic literature search and found sixty-four clinical studies reporting α, β and α/β for tumours. Tumour site, histology, stage, number of patients, type of LQ model, radiation type, TCP model, clinical endpoint and radiobiological parameter estimates were extracted. Next, we stratified by tumour site and by tumour histology. Study heterogeneity was expressed by the I² statistic, i.e. the percentage of variance in reported values not explained by chance.

Results

A large heterogeneity in LQ parameters was found within and between studies (I² > 75%). For the same tumour site, differences in histology partially explain differences in the LQ parameters: epithelial tumours have higher α/β values than adenocarcinomas. For tumour sites with different histologies, such as in oesophageal cancer, the α/β estimates correlate well with histology. However, many other factors contribute to the study heterogeneity of LQ parameters, e.g. tumour stage, type of LQ model, TCP model and clinical endpoint (i.e. survival, tumour control and biochemical control).

Conclusions

The value of LQ parameters for tumours as published in clinical radiotherapy studies depends on many clinical and methodological factors. Therefore, for clinical use of the LQ model, LQ parameters for tumour should be selected carefully, based on tumour site, histology and the applied LQ model. To account for uncertainties in LQ parameter estimates, exploring a range of values is recommended.

Electronic supplementary material

The online version of this article (10.1186/s13014-018-1040-z) contains supplementary material, which is available to authorized users.

Tumor control probability modeling for stereotactic body radiation therapy of early-stage lung cancer using multiple bio-physical models

This work is to analyze pooled clinical data using different radiobiological models and to understand the relationship between biologically effective dose (BED) and tumor control probability (TCP) for stereotactic body radiotherapy (SBRT) of early-stage non-small cell lung cancer (NSCLC). The clinical data of 1-, 2-, 3-, and 5-year actuarial or Kaplan-Meier TCP from 46 selected studies were collected for SBRT of NSCLC in the literature. The TCP data were separated for Stage T1 and T2 tumors if possible, otherwise collected for combined stages. BED was calculated at isocenters using six radiobiological models. For each model, the independent model parameters were determined from a fit to the TCP data using the least chi-square (χ²) method with either one set of parameters regardless of tumor stages or two sets for T1 and T2 tumors separately. The fits to the clinic data yield consistent results of large α/β ratios of about 20Gy for all models investigated. The regrowth model that accounts for the tumor repopulation and heterogeneity leads to a better fit to the data, compared to other 5 models where the fits were indistinguishable between the models. The models based on the fitting parameters predict that the T2 tumors require about additional 1Gy physical dose at isocenters per fraction (⩽5 fractions) to achieve the optimal TCP when compared to the T1 tumors. In conclusion, this systematic analysis of a large set of published clinical data using different radiobiological models shows that local TCP for SBRT of early-stage NSCLC has strong dependence on BED with large α/β ratios of about 20Gy. The six models predict that a BED (calculated with α/β of 20) of 90Gy is sufficient to achieve TCP⩾95%. Among the models considered, the regrowth model leads to a better fit to the clinical data.

Direct evaluation of radiobiological parameters from clinical data in the case of ion beam therapy: an alternative approach to the relative biological effectiveness

The relative biological effectiveness (RBE) concept is commonly used in treatment planning for ion beam therapy. Whether models based on in vitro/in vivo RBE data can be used to predict human response to treatments is an open issue. In this work an alternative method, based on an effective radiobiological parameterization directly derived from clinical data, is presented. The method has been applied to the analysis of prostate cancer trials with protons and carbon ions.Prostate cancer trials with proton and carbon ion beams reporting 5 year-local control (LC5) and grade 2 (G2) or higher genitourinary toxicity rates (TOX) were selected from literature to test the method. Treatment simulations were performed on a representative subset of patients to produce dose and linear energy transfer distribution, which were used as explicative physical variables for the radiobiological modelling. Two models were taken into consideration: the microdosimetric kinetic model (MKM) and a linear model (LM). The radiobiological parameters of the LM and MKM were obtained by coupling them with the tumor control probability and normal tissue complication probability models to fit the LC5 and TOX data through likelihood maximization. The model ranking was based on the Akaike information criterion.Results showed large confidence intervals due to the limited variety of available treatment schedules. RBE values, such as RBE = 1.1 for protons in the treated volume, were derived as a by-product of the method, showing a consistency with current approaches. Carbon ion RBE values were also derived, showing lower values than those assumed for the original treatment planning in the target region, whereas higher values were found in the bladder. Most importantly, this work shows the possibility to infer the radiobiological parametrization for proton and carbon ion treatment directly from clinical data.

A method to visualize the uncertainty of the prediction of radiobiological models

A method for quantitative visualization of the uncertainty in the predicted tumor control probability (TCP) and normal tissue complication probability (NTCP) in radiotherapy has been developed. Uncertainties of TCP and NTCP due to inter-individual variation of the underlying radiosensitivity parameters was simulated by sampling the prescribed dose from a uniform distribution and the radiosensitivity-parameters from a Gaussian distribution. The result is visualized as a scatter-plot superimposed to the population-based dose response curves using the prescribed dose as the common dosimetric variable. In addition, probability histograms are derived quantifying the probability of specific TCP- or NTCP-values for individual patients from the underlying population. The method is exemplified with a pleural mesothelioma case with the lung as organ at risk. A prescribed dose of 54 Gy together with radiosensitivity variations of 6% (tumor) and 10% (normal tissue) results in a TCP of 85% (range 68-94%, 90% confidence interval, CI) and an NTCP of 4% (range 3-6%, 90% CI), respectively. Increasing the radiosensitivity variation of the tumor to 15% and reducing the lung tolerance dose by 25% results in values of 84% (range 51-97%, 90% CI) for TCP and 9% (range 6-12%, 90% CI) for NTCP. Increasing the dose to 60 Gy leads to TCP- and NTCP-values of 93% (range 69-100%, 90% CI) and 12% (range 8-17%, 90% CI), respectively. The new method visualizes the uncertainty of TCP- and NTCP-values and hence of the therapeutic window. This can help the clinician to assess the treatment plan for the individual patient.

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.

Biologically effective dose-response relationship for breast cancer treated by conservative surgery and postoperative radiotherapy

PURPOSE

To find a biologically effective dose (BED) response for adjuvant breast radiotherapy (RT) for initial-stage breast cancer.

METHODS AND MATERIALS

Results of randomized trials of RT vs. non-RT were reviewed and the tumor control probability (TCP) after RT was calculated for each of them. Using the linear-quadratic formula and Poisson statistics of cell-kill, the average initial number of clonogens per tumor before RT and the average tumor cell radiosensitivity (alpha-value) were calculated. An alpha/beta ratio of 4 Gy was assumed for these calculations.

RESULTS

A linear regression equation linking BED to TCP was derived: -ln[-ln(TCP)] = -ln(No) + alpha(*) BED = -4.08 + 0.07 (*) BED, suggesting a rather low radiosensitivity of breast cancer cells (alpha = 0.07 Gy(-1)), which probably reflects population heterogeneity. From the linear relationship a sigmoid BED-response curve was constructed.

CONCLUSION

For BED values higher than about 90 Gy(4) the radiation-induced TCP is essentially maximizing at 90-100%. The relationship presented here could be an approximate guide in the design and reporting of clinical trials of adjuvant breast RT.

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.