A Mathematical Model for Predicting Patient Responses to Combined Radiotherapy with CTLA-4 Immune Checkpoint Inhibitors

Abstract

The purpose of this study was to develop a cell-cell interaction model that could predict a tumor's response to radiotherapy (RT) combined with CTLA-4 immune checkpoint inhibition (ICI) in patients with hepatocellular carcinoma (HCC). The previously developed model was extended by adding a new term representing tremelimumab, an inhibitor of CTLA-4. The distribution of the new immune activation term was derived from the results of a clinical trial for tremelimumab monotherapy (NCT01008358). The proposed model successfully reproduced longitudinal tumor diameter changes in HCC patients treated with tremelimumab (complete response = 0%, partial response = 17.6%, stable disease = 58.8%, and progressive disease = 23.6%). For the non-irradiated tumor control group, adding ICI to RT increased the clinical benefit rate from 8% to 32%. The simulation predicts that it is beneficial to start CTLA-4 blockade before RT in terms of treatment sequences. We developed a mathematical model that can predict the response of patients to the combined CTLA-4 blockade with radiation therapy. We anticipate that the developed model will be helpful for designing clinical trials with the ultimate aim of maximizing the efficacy of ICI-RT combination therapy.

Keywords: immune checkpoint inhibitor; mathematical modeling; radiation therapy; tremelimumab.

Figure 1

Schematic diagram of model fitting procedure. The yellow highlighted part is a procedure different from the previous study. Abbreviations: RT = radiation therapy; ICI = immune checkpoint inhibitor; RECIST = response evaluation criteria in solid tumors.

Figure 2

The effect of the 𝛿_trem (tremelimumab term) on RECIST1.1 responses for 10,000 virtual patients. The green, blue, orange, and red lines indicate complete response (CR), partial response (PR), stable disease (SD), and progressive disease (PD), respectively. Abbreviations: RECIST = response evaluation criteria in solid tumors.

Figure 3

Based on the RECIST 1.1 response to the constant 𝛿_trem, a figure was used to find the distribution of tremelimumab suitable for the values for PR (17.6%), SD (58.8%), and PD (23.6%) [28]. (a) A heatmap showing the difference in relation to the reference according to the mean and standard deviation of tremelimumab. (b) Distribution of suitable-for-reference values and RECIST 1.1 responses. Abbreviations: PR = partial response; SD = stable disease; PD = progressive disease; RECIST = response evaluation criteria in solid tumors.

Figure 4

Tumor diameter change over time for 100 virtual patients treated with tremelimumab. Each line represents the tumor dynamics of a single patient. The green, blue, orange, and red lines indicate complete response (CR), partial response (PR), stable disease (SD), and progressive disease (PD), respectively.

Figure 5

Tumor size changes for 100 virtual patients after one year of treatment (upper: irradiated tumor, lower: non-irradiated tumor). Each bar represents a single patient. The RECIST response 1.1 is expressed as CR: green, PR: blue, SD: orange, and PD: dark red. Abbreviations: CR = complete response; PR = partial response; SD = stable disease; PD = progressive disease; RECIST = response evaluation criteria in solid tumors; RT = radiation therapy; ICI = immune checkpoint inhibitor.

Figure 6

Diameter change in the irradiated tumor and non-irradiated tumor after 1 year according to the irradiated tumor fraction ((a): irradiated tumor, (b): non-irradiated tumor). The green line indicates CTLA-4 blockade ICI + RT; the blue line indicates RT only; the gold line indicates CTLA-4 blockade ICI only; and the red line indicates none. Abbreviations: RT = radiation therapy; ICI = immune checkpoint inhibitor.

Figure 7

Predicted results of tumor diameter change with different modality combination sequences. (a) Schematic to explain (b,c). Blue and green arrows indicate the start of treatment. On the time axis, negative values (− sign with red arrow in (a)) indicate that RT started before the ICI, whereas positive values (+ sign with red arrow in (a)) indicate that RT started after the ICI. (b) Response to treatment sequence when the irradiated tumor fraction was 99%. (c) Response to treatment sequence when the irradiated tumor fraction was 1%. Blue error bars indicate irradiated tumors and light blue error bars indicate non-irradiated tumors. Abbreviations: RT = radiation therapy; ICI = immune checkpoint inhibitor.