Multi-Institutional Audit of FLASH and Conventional Dosimetry With a 3D Printed Anatomically Realistic Mouse Phantom.

PURPOSE: We conducted a multi-institutional dosimetric audit between FLASH and conventional dose rate (CONV) electron irradiations by using an anatomically realistic 3-dimensional (3D) printed mouse phantom. METHODS AND MATERIALS: A computed tomography (CT) scan of a live mouse was used to create a 3D model of bony anatomy, lungs, and soft tissue. A dual-nozzle 3D printer was used to print the mouse phantom using acrylonitrile butadiene styrene (∼1.02 g/cm3) and polylactic acid (∼1.24 g/cm3) simultaneously to simulate soft tissue and bone densities, respectively. The lungs were printed separately using lightweight polylactic acid (∼0.64 g/cm3). Hounsfield units (HU), densities, and print-to-print stability of the phantoms were assessed. Three institutions were each provided a phantom and each institution performed 2 replicates of irradiations at selected anatomic regions. The average dose difference between FLASH and CONV dose distributions and deviation from the prescribed dose were measured with radiochromic film. RESULTS: Compared with the reference CT scan, CT scans of the phantom demonstrated mass density differences of 0.10 g/cm3 for bone, 0.12 g/cm3 for lung, and 0.03 g/cm3 for soft tissue regions. Differences in HU between phantoms were <10 HU for soft tissue and bone, with lung showing the most variation (54 HU), but with minimal effect on dose distribution (<0.5%). Mean differences between FLASH and CONV decreased from the first to the second replicate (4.3%-1.2%), and differences from the prescribed dose decreased for both CONV (3.6%-2.5%) and FLASH (6.4%-2.7%). Total dose accuracy suggests consistent pulse dose and pulse number, although these were not specifically assessed. Positioning variability was observed, likely due to the absence of robust positioning aids or image guidance. CONCLUSIONS: This study marks the first dosimetric audit for FLASH using a nonhomogeneous phantom, challenging conventional calibration practices reliant on homogeneous phantoms. The comparison protocol offers a framework for credentialing multi-institutional studies in FLASH preclinical research to enhance reproducibility of biologic findings.

Exploring Deep Learning for Estimating the Isoeffective Dose of FLASH Irradiation From Mouse Intestinal Histological Images.

PURPOSE: Ultrahigh-dose-rate (FLASH) irradiation has been reported to reduce normal tissue damage compared with conventional dose rate (CONV) irradiation without compromising tumor control. This proof-of-concept study aims to develop a deep learning (DL) approach to quantify the FLASH isoeffective dose (dose of CONV that would be required to produce the same effect as the given physical FLASH dose) with postirradiation mouse intestinal histology images. METHODS AND MATERIALS: Eighty-four healthy C57BL/6J female mice underwent 16 MeV electron CONV (0.12 Gy/s; n = 41) or FLASH (200 Gy/s; n = 43) single fraction whole abdominal irradiation. Physical dose ranged from 12 to 16 Gy for FLASH and 11 to 15 Gy for CONV in 1 Gy increments. Four days after irradiation, 9 jejunum cross-sections from each mouse were hematoxylin and eosin stained and digitized for histological analysis. CONV data set was randomly split into training (n = 33) and testing (n = 8) data sets. ResNet101-based DL models were retrained using the CONV training data set to estimate the dose based on histological features. The classical manual crypt counting (CC) approach was implemented for model comparison. Cross-section-wise mean squared error was computed to evaluate the dose estimation accuracy of both approaches. The validated DL model was applied to the FLASH data set to map the physical FLASH dose into the isoeffective dose. RESULTS: The DL model achieved a cross-section-wise mean squared error of 0.20 Gy2 on the CONV testing data set compared with 0.40 Gy2 of the CC approach. Isoeffective doses estimated by the DL model for FLASH doses of 12, 13, 14, 15, and 16 Gy were 12.19 ± 0.46, 12.54 ± 0.37, 12.69 ± 0.26, 12.84 ± 0.26, and 13.03 ± 0.28 Gy, respectively. CONCLUSIONS: Our proposed DL model achieved accurate CONV dose estimation. The DL model results indicate that in the physical dose range of 13 to 16 Gy, the biologic dose response of small intestinal tissue to FLASH irradiation is represented by a lower isoeffective dose compared with the physical dose. Our DL approach can be a tool for studying isoeffective doses of other radiation dose modifying interventions.

Multi-Institutional Audit of FLASH and Conventional Dosimetry with a 3D-Printed Anatomically Realistic Mouse Phantom.

We conducted a multi-institutional audit of dosimetric variability between FLASH and conventional dose rate (CONV) electron irradiations by using an anatomically realistic 3D-printed mouse phantom. A CT scan of a live mouse was used to create a 3D model of bony anatomy, lungs, and soft tissue. A dual-nozzle 3D printer was used to print the mouse phantom using acrylonitrile butadiene styrene ($~1.02 g/cm^3$) and polylactic acid ($~1.24 g/cm^3$) simultaneously to simulate soft tissue and bone densities, respectively. The lungs were printed separately using lightweight polylactic acid ($~0.64 g/cm^3$). Hounsfield units (HU) and densities were compared with the reference CT scan of the live mouse. Print-to-print reproducibility of the phantom was assessed. Three institutions were each provided a phantom, and each institution performed two replicates of irradiations at selected mouse anatomic regions. The average dose difference between FLASH and CONV dose distributions and deviation from the prescribed dose were measured with radiochromic film. Compared to the reference CT scan, CT scans of the phantom demonstrated mass density differences of $0.10 g/cm^3$ for bone, $0.12 g/cm^3$ for lung, and $0.03 g/cm^3$ for soft tissue regions. Between phantoms, the difference in HU for soft tissue and bone was <10 HU from print to print. Lung exhibited the most variation (54 HU) but minimally affected dose distribution (<0.5% dose differences between phantoms). The mean difference between FLASH and CONV from the first replicate to the second decreased from 4.3% to 1.2%, and the mean difference from the prescribed dose decreased from 3.6% to 2.5% for CONV and 6.4% to 2.7% for FLASH. The framework presented here is promising for credentialing of multi-institutional studies of FLASH preclinical research to maximize the reproducibility of biological findings.

Human enteroids as a tool to study conventional and ultra-high dose rate radiation.

Radiation therapy, one of the most effective therapies to treat cancer, is highly toxic to healthy tissue. The delivery of radiation at ultra-high dose rates, FLASH radiation therapy (FLASH), has been shown to maintain therapeutic anti-tumor efficacy while sparing normal tissues compared to conventional dose rate irradiation (CONV). Though promising, these studies have been limited mainly to murine models. Here, we leveraged enteroids, three-dimensional cell clusters that mimic the intestine, to study human-specific tissue response to radiation. We observed enteroids have a greater colony growth potential following FLASH compared with CONV. In addition, the enteroids that reformed following FLASH more frequently exhibited proper intestinal polarity. While we did not observe differences in enteroid damage across groups, we did see distinct transcriptomic changes. Specifically, the FLASH enteroids upregulated the expression of genes associated with the WNT-family, cell-cell adhesion, and hypoxia response. These studies validate human enteroids as a model to investigate FLASH and provide further evidence supporting clinical study of this therapy. Insight Box Promising work has been done to demonstrate the potential of ultra-high dose rate radiation (FLASH) to ablate cancerous tissue, while preserving healthy tissue. While encouraging, these findings have been primarily observed using pre-clinical murine and traditional two-dimensional cell culture. This study validates the use of human enteroids as a tool to investigate human-specific tissue response to FLASH. Specifically, the work described demonstrates the ability of enteroids to recapitulate previous in vivo findings, while also providing a lens through which to probe cellular and molecular-level responses to FLASH. The human enteroids described herein offer a powerful model that can be used to probe the underlying mechanisms of FLASH in future studies.

FLASH-RT does not affect chromosome translocations and junction structures beyond that of CONV-RT dose-rates.

BACKGROUND AND PURPOSE: The impact of radiotherapy (RT) at ultra high vs conventional dose rate (FLASH vs CONV) on the generation and repair of DNA double strand breaks (DSBs) is an important question that remains to be investigated. Here, we tested the hypothesis as to whether FLASH-RT generates decreased chromosomal translocations compared to CONV-RT. MATERIALS AND METHODS: We used two FLASH validated electron beams and high-throughput rejoin and genome-wide translocation sequencing (HTGTS-JoinT-seq), employing S. aureus and S. pyogenes Cas9 "bait" DNA double strand breaks (DSBs) in HEK239T cells, to measure differences in bait-proximal repair and their genome-wide translocations to "prey" DSBs generated after various irradiation doses, dose rates and oxygen tensions (normoxic, 21% O2; physiological, 4% O2; hypoxic, 2% and 0.5% O2). Electron irradiation was delivered using a FLASH capable Varian Trilogy and the eRT6/Oriatron at CONV (0.08-0.13 Gy/s) and FLASH (1x102-5x106 Gy/s) dose rates. Related experiments using clonogenic survival and γH2AX foci in the 293T and the U87 glioblastoma lines were also performed to discern FLASH-RT vs CONV-RT DSB effects. RESULTS: Normoxic and physioxic irradiation of HEK293T cells increased translocations at the cost of decreasing bait-proximal repair but were indistinguishable between CONV-RT and FLASH-RT. Although no apparent increase in chromosome translocations was observed with hypoxia-induced apoptosis, the combined decrease in oxygen tension with IR dose-rate modulation did not reveal significant differences in the level of translocations nor in their junction structures. Furthermore, RT dose rate modality on U87 cells did not change γH2AX foci numbers at 1- and 24-hours post-irradiation nor did this affect 293T clonogenic survival. CONCLUSION: Irrespective of oxygen tension, FLASH-RT produces translocations and junction structures at levels and proportions that are indistinguishable from CONV-RT.

FLASH-RT does not affect chromosome translocations and junction structures beyond that of CONV-RT dose-rates.

The molecular and cellular mechanisms driving the enhanced therapeutic ratio of ultra-high dose-rate radiotherapy (FLASH-RT) over slower conventional (CONV-RT) radiotherapy dose-rate remain to be elucidated. However, attenuated DNA damage and transient oxygen depletion are among several proposed models. Here, we tested whether FLASH-RT under physioxic (4% O 2 ) and hypoxic conditions (≤2% O 2 ) reduces genome-wide translocations relative to CONV-RT and whether any differences identified revert under normoxic (21% O 2 ) conditions. We employed high-throughput rejoin and genome-wide translocation sequencing ( HTGTS-JoinT-seq ), using S. aureus and S. pyogenes Cas9 "bait" DNA double strand breaks (DSBs), to measure differences in bait-proximal repair and their genome-wide translocations to "prey" DSBs generated by electron beam CONV-RT (0.08-0.13Gy/s) and FLASH-RT (1×10 2 -5×10 6 Gy/s), under varying ionizing radiation (IR) doses and oxygen tensions. Normoxic and physioxic irradiation of HEK293T cells increased translocations at the cost of decreasing bait-proximal repair but were indistinguishable between CONV-RT and FLASH-RT. Although no apparent increase in chromosome translocations was observed with hypoxia-induced apoptosis, the combined decrease in oxygen tension with IR dose-rate modulation did not reveal significant differences in the level of translocations nor in their junction structures. Thus, Irrespective of oxygen tension, FLASH-RT produces translocations and junction structures at levels and proportions that are indistinguishable from CONV-RT.

Clinical Linear Accelerator-Based Electron FLASH: Pathway for Practical Translation to FLASH Clinical Trials.

PURPOSE: Ultrahigh-dose-rate (UHDR) radiation therapy (RT) has produced the FLASH effect in preclinical models: reduced toxicity with comparable tumor control compared with conventional-dose-rate RT. Early clinical trials focused on UHDR RT feasibility using specialized devices. We explore the technical feasibility of practical electron UHDR RT on a standard clinical linear accelerator (LINAC). METHODS AND MATERIALS: We tuned the program board of a decommissioned electron energy for UHDR electron delivery on a clinical LINAC without hardware modification. Pulse delivery was controlled using the respiratory gating interface. A short source-to-surface distance (SSD) electron setup with a standard scattering foil was configured and tested on an anthropomorphic phantom using circular blocks with 3- to 20-cm field sizes. Dosimetry was evaluated using radiochromic film and an ion chamber profiler. RESULTS: UHDR open-field mean dose rates at 100, 80, 70, and 59 cm SSD were 36.82, 59.52, 82.01, and 112.83 Gy/s, respectively. At 80 cm SSD, mean dose rate was ∼60 Gy/s for all collimated field sizes, with an R80 depth of 6.1 cm corresponding to an energy of 17.5 MeV. Heterogeneity was <5.0% with asymmetry of 2.2% to 6.2%. The short SSD setup was feasible under realistic treatment conditions simulating broad clinical indications on an anthropomorphic phantom. CONCLUSIONS: Short SSD and tuning for high electron beam current on a standard clinical LINAC can deliver flat, homogenous UHDR electrons over a broad, clinically relevant range of field sizes and depths with practical working distances in a configuration easily reversible to standard clinical use.

Design and validation of a dosimetric comparison scheme tailored for ultra-high dose-rate electron beams to support multicenter FLASH preclinical studies.

BACKGROUND AND PURPOSE: We describe a multicenter cross validation of ultra-high dose rate (UHDR) (>= 40 Gy/s) irradiation in order to bring a dosimetric consensus in absorbed dose to water. UHDR refers to dose rates over 100-1000 times those of conventional clinical beams. UHDR irradiations have been a topic of intense investigation as they have been reported to induce the FLASH effect in which normal tissues exhibit reduced toxicity relative to conventional dose rates. The need to establish optimal beam parameters capable of achieving the in vivo FLASH effect has become paramount. It is therefore necessary to validate and replicate dosimetry across multiple sites conducting UHDR studies with distinct beam configurations and experimental set-ups. MATERIALS AND METHODS: Using a custom cuboid phantom with a cylindrical cavity (5 mm diameter by 10.4 mm length) designed to contain three type of dosimeters (thermoluminescent dosimeters (TLDs), alanine pellets, and Gafchromic films), irradiations were conducted at expected doses of 7.5 to 16 Gy delivered at UHDR or conventional dose rates using various electron beams at the Radiation Oncology Departments of the CHUV in Lausanne, Switzerland and Stanford University, CA. RESULTS: Data obtained between replicate experiments for all dosimeters were in excellent agreement (±3%). In general, films and TLDs were in closer agreement with each other, while alanine provided the closest match between the expected and measured dose, with certain caveats related to absolute reference dose. CONCLUSION: In conclusion, successful cross-validation of different electron beams operating under different energies and configurations lays the foundation for establishing dosimetric consensus for UHDR irradiation studies, and, if widely implemented, decrease uncertainty between different sites investigating the mechanistic basis of the FLASH effect.

Real-time optical oximetry during FLASH radiotherapy using a phosphorescent nanoprobe.

The rapid depletion of oxygen during irradiation at ultra-high dose rate calls for tissue oximeters capable of high temporal resolution. This study demonstrates a water-soluble phosphorescent nanoprobe and fiber-coupled instrument, which together are used to measure the kinetics of oxygen depletion at 200 Hz during irradiation of in vitro solutions.

Abdominopelvic FLASH Irradiation Improves PD-1 Immune Checkpoint Inhibition in Preclinical Models of Ovarian Cancer.

Treatment of advanced ovarian cancer using PD-1/PD-L1 immune checkpoint blockade shows promise; however, current clinical trials are limited by modest response rates. Radiotherapy has been shown to synergize with PD-1/PD-L1 blockade in some cancers but has not been utilized in advanced ovarian cancer due to toxicity associated with conventional abdominopelvic irradiation. Ultrahigh-dose rate (FLASH) irradiation has emerged as a strategy to reduce radiation-induced toxicity, however, the immunomodulatory properties of FLASH irradiation remain unknown. Here, we demonstrate that single high-dose abdominopelvic FLASH irradiation promoted intestinal regeneration and maintained tumor control in a preclinical mouse model of ovarian cancer. Reduced tumor burden in conventional and FLASH-treated mice was associated with an early decrease in intratumoral regulatory T cells and a late increase in cytolytic CD8+ T cells. Compared with conventional irradiation, FLASH irradiation increased intratumoral T-cell infiltration at early timepoints. Moreover, FLASH irradiation maintained the ability to increase intratumoral CD8+ T-cell infiltration and enhance the efficacy of αPD-1 therapy in preclinical models of ovarian cancer. These data highlight the potential for FLASH irradiation to improve the therapeutic efficacy of checkpoint inhibition in the treatment of ovarian cancer.

Multicellular Spheroids as In Vitro Models of Oxygen Depletion During FLASH Irradiation.

PURPOSE: The differential response of normal and tumor tissues to ultrahigh-dose-rate radiation (FLASH) has raised new hope for treating solid tumors but, to date, the mechanism remains elusive. One leading hypothesis is that FLASH radiochemically depletes oxygen from irradiated tissues faster than it is replenished through diffusion. The purpose of this study was to investigate these effects within hypoxic multicellular tumor spheroids through simulations and experiments. METHODS AND MATERIALS: Physicobiological equations were derived to model (1) the diffusion and metabolism of oxygen within spheroids; (2) its depletion through reactions involving radiation-induced radicals; and (3) the increase in radioresistance of spheroids, modeled according to the classical oxygen enhancement ratio and linear-quadratic response. These predictions were then tested experimentally in A549 spheroids exposed to electron irradiation at conventional (0.075 Gy/s) or FLASH (90 Gy/s) dose rates. Clonogenic survival, cell viability, and spheroid growth were scored postradiation. Clonogenic survival of 2 other cell lines was also investigated. RESULTS: The existence of a hypoxic core in unirradiated tumor spheroids is predicted by simulations and visualized by fluorescence microscopy. Upon FLASH irradiation, this hypoxic core transiently expands, engulfing a large number of well-oxygenated cells. In contrast, oxygen is steadily replenished during slower conventional irradiation. Experimentally, clonogenic survival was around 3-fold higher in FLASH-irradiated spheroids compared with conventional irradiation, but no significant difference was observed for well-oxygenated 2-dimensional cultured cells. This differential survival is consistent with the predictions of the computational model. FLASH irradiation of spheroids resulted in a dose-modifying factor of around 1.3 for doses above 10 Gy. CONCLUSIONS: Tumor spheroids can be used as a model to study FLASH irradiation in vitro. The improved survival of tumor spheroids receiving FLASH radiation confirms that ultrafast radiochemical oxygen depletion and its slow replenishment are critical components of the FLASH effect.

Abdominal FLASH irradiation reduces radiation-induced gastrointestinal toxicity for the treatment of ovarian cancer in mice.

Radiation therapy is the most effective cytotoxic therapy for localized tumors. However, normal tissue toxicity limits the radiation dose and the curative potential of radiation therapy when treating larger target volumes. In particular, the highly radiosensitive intestine limits the use of radiation for patients with intra-abdominal tumors. In metastatic ovarian cancer, total abdominal irradiation (TAI) was used as an effective postsurgical adjuvant therapy in the management of abdominal metastases. However, TAI fell out of favor due to high toxicity of the intestine. Here we utilized an innovative preclinical irradiation platform to compare the safety and efficacy of TAI ultra-high dose rate FLASH irradiation to conventional dose rate (CONV) irradiation in mice. We demonstrate that single high dose TAI-FLASH produced less mortality from gastrointestinal syndrome, spared gut function and epithelial integrity, and spared cell death in crypt base columnar cells compared to TAI-CONV irradiation. Importantly, TAI-FLASH and TAI-CONV irradiation had similar efficacy in reducing tumor burden while improving intestinal function in a preclinical model of ovarian cancer metastasis. These findings suggest that FLASH irradiation may be an effective strategy to enhance the therapeutic index of abdominal radiotherapy, with potential application to metastatic ovarian cancer.

FLASH Irradiation Results in Reduced Severe Skin Toxicity Compared to Conventional-Dose-Rate Irradiation.

Radiation therapy, along with surgery and chemotherapy, is one of the main treatments for cancer. While radiotherapy is highly effective in the treatment of localized tumors, its main limitation is its toxicity to normal tissue. Previous preclinical studies have reported that ultra-high dose-rate (FLASH) irradiation results in reduced toxicity to normal tissues while controlling tumor growth to a similar extent relative to conventional-dose-rate (CONV) irradiation. To our knowledge this is the first report of a dose-response study in mice comparing the effect of FLASH irradiation vs. CONV irradiation on skin toxicity. We found that FLASH irradiation results in both a lower incidence and lower severity of skin ulceration than CONV irradiation 8 weeks after single-fraction hemithoracic irradiation at high doses (30 and 40 Gy). Survival was also higher after FLASH hemithoracic irradiation (median survival >180 days at doses of 30 and 40 Gy) compared to CONV irradiation (median survival 100 and 52 days at 30 and 40 Gy, respectively). No ulceration was observed at doses 20 Gy or below in either FLASH or CONV. These results suggest a shifting of the dose-response curve for radiation-induced skin ulceration to the right for FLASH, compared to CONV irradiation, suggesting the potential for an enhanced therapeutic index for radiation therapy of cancer.

Fully 3D refraction correction dosimetry system.

The irradiation of selective regions in a polymer gel dosimeter results in an increase in optical density and refractive index (RI) at those regions. An optical tomography-based dosimeter depends on rayline path through the dosimeter to estimate and reconstruct the dose distribution. The refraction of light passing through a dose region results in artefacts in the reconstructed images. These refraction errors are dependant on the scanning geometry and collection optics. We developed a fully 3D image reconstruction algorithm, algebraic reconstruction technique-refraction correction (ART-rc) that corrects for the refractive index mismatches present in a gel dosimeter scanner not only at the boundary, but also for any rayline refraction due to multiple dose regions inside the dosimeter. In this study, simulation and experimental studies have been carried out to reconstruct a 3D dose volume using 2D CCD measurements taken for various views. The study also focuses on the effectiveness of using different refractive-index matching media surrounding the gel dosimeter. Since the optical density is assumed to be low for a dosimeter, the filtered backprojection is routinely used for reconstruction. We carry out the reconstructions using conventional algebraic reconstruction (ART) and refractive index corrected ART (ART-rc) algorithms. The reconstructions based on FDK algorithm for cone-beam tomography has also been carried out for comparison. Line scanners and point detectors, are used to obtain reconstructions plane by plane. The rays passing through dose region with a RI mismatch does not reach the detector in the same plane depending on the angle of incidence and RI. In the fully 3D scanning setup using 2D array detectors, light rays that undergo refraction are still collected and hence can still be accounted for in the reconstruction algorithm. It is found that, for the central region of the dosimeter, the usable radius using ART-rc algorithm with water as RI matched medium is 71.8%, an increase of 6.4% compared to that achieved using conventional ART algorithm. Smaller diameter dosimeters are scanned with dry air scanning by using a wide-angle lens that collects refracted light. The images reconstructed using cone beam geometry is seen to deteriorate in some planes as those regions are not scanned. Refraction correction is important and needs to be taken in to consideration to achieve quantitatively accurate dose reconstructions. Refraction modeling is crucial in array based scanners as it is not possible to identify refracted rays in the sinogram space.

Effects of refractive index mismatch in optical CT imaging of polymer gel dosimeters.

PURPOSE: Proposing an image reconstruction technique, algebraic reconstruction technique-refraction correction (ART-rc). The proposed method takes care of refractive index mismatches present in gel dosimeter scanner at the boundary, and also corrects for the interior ray refraction. Polymer gel dosimeters with high dose regions have higher refractive index and optical density compared to the background medium, these changes in refractive index at high dose results in interior ray bending. METHODS: The inclusion of the effects of refraction is an important step in reconstruction of optical density in gel dosimeters. The proposed ray tracing algorithm models the interior multiple refraction at the inhomogeneities. Jacob's ray tracing algorithm has been modified to calculate the pathlengths of the ray that traverses through the higher dose regions. The algorithm computes the length of the ray in each pixel along its path and is used as the weight matrix. Algebraic reconstruction technique and pixel based reconstruction algorithms are used for solving the reconstruction problem. The proposed method is tested with numerical phantoms for various noise levels. The experimental dosimetric results are also presented. RESULTS: The results show that the proposed scheme ART-rc is able to reconstruct optical density inside the dosimeter better than the results obtained using filtered backprojection and conventional algebraic reconstruction approaches. The quantitative improvement using ART-rc is evaluated using gamma-index. The refraction errors due to regions of different refractive indices are discussed. The effects of modeling of interior refraction in the dose region are presented. CONCLUSIONS: The errors propagated due to multiple refraction effects have been modeled and the improvements in reconstruction using proposed model is presented. The refractive index of the dosimeter has a mismatch with the surrounding medium (for dry air or water scanning). The algorithm reconstructs the dose profiles by estimating refractive indices of multiple inhomogeneities having different refractive indices and optical densities embedded in the dosimeter. This is achieved by tracking the path of the ray that traverses through the dosimeter. Extensive simulation studies have been carried out and results are found to be matching that of experimental results.