Linear mixed-effects models for predicting sarcoma local recurrence growth rates: Implications for optimal surveillance imaging frequency

Abstract

Purpose:

Sarcomas are rare tumours of mesenchymal origin that are often treated with surgical resection and radiation to prevent local recurrence (LR). Surveillance for LR after surgical resection is often done with contrast-enhanced MRI, however, the optimal frequency of surveillance imaging is unknown. The aims of this study are to calculate LR growth, and to evaluate the factors that affect LR growth and to use this data to predict the optimal imaging surveillance frequency.

Method:

Retrospective cohort study of patients with sarcoma who were treated at a tertiary care academic institution between 01/01/2007 and 01/03/2020 identified 34 patients with 60 histologically confirmed LRs. The maximum LR length was measured on each surveillance MRI. Linear mixed-effects models were used to calculate the maximum LR length growth rate. We evaluated whether age, sex, primary sarcoma tumour size, sarcoma grade, margins, chemotherapy or radiation therapy affected the growth rate of the maximum LR length.

Results:

One patient had 6 LRs (2.9 %), two patients (5.9 %) had 5 LRs, two patients (5.9 %) had 4 LRs, two patients (5.9 %) had 3 LRs, three patients (8.8 %) had 2 LRs, and twenty-four patients (70.6 %) had 1 LR. Most patients had high grade (Grade II/III) disease (88.2 %). 41.2 % of the patients had microscopically positive surgical margins. The median time (range) from surgery to LR was 264 days (17 – 8013) days, and 90 % occurred within 42.8 months (1284 days). Microscopically positive margins were associated with faster growth of the maximum LR length (P = 0.036). Assuming that a 1 cm sarcoma LR is actionable and that the previous sur-veillance MRI was negative, the data predicts that patients with microscopically positive margins should have surveillance MRIs every 6.2 months when LRs are expected to achieve a length of 1.0 cm (95 % CI (0.4–2.3 cm)).

Conclusions:

Patients with microscopically positive resection margins had LRs that grew faster than patients with negative margins. Surveillance imaging with contrast-enhanced MRI could be conservatively performed every 6 months when LRs are expected to be just less than 1.0 cm in length.

1. Introduction

Sarcomas are rare tumours, thought to be of mesenchymal origin [1–3]. These are a heterogeneous group of tumours, with over 60 different subtypes of sarcomas [2,3]. Approximately 15,000 individuals are diagnosed with sarcomas each year in the United States [1]. Sarcomas have a propensity to metastasize to the lungs [4,5]. At our institution, local control for sarcomas is achieved with neoadjuvant chemotherapy and surgical resection for mostly osteosarcoma and Ewing sarcoma, however for other sarcomas, local control is achieved using neoadjuvant radiation therapy and surgical resection [6–8]. Radiation therapy has been shown to result in approximately 90 % local control [6,7]. After surgical resection, patients with sarcomas routinely undergo surveillance imaging for local recurrence (LR).

There is variable frequency of surveillance imaging in clinical practice. While the National Comprehensive Cancer Network (NCCN) has guidelines for imaging surveillance modality and frequency [9], individual physicians/health systems tend to choose their own surveillance protocols. Excessive or too frequent surveillance imaging incurs increased cost to the patient/healthcare system and increased anxiety to the patient. Too infrequent surveillance imaging results in the potential missed opportunity to resect tumour LR before metastases or before the LR becomes inoperable.

We hypothesized that statistical models could be utilized to better predict the optimal imaging surveillance frequency for sarcoma after surgical resection. The aims of this paper are to predict the growth rate of sarcoma recurrences, and to evaluate what factors affect this growth rate. We use the estimates of the sarcoma LR growth rate to determine the optimal imaging surveillance frequency for LR.

2. Material and methods

The study protocol was reviewed and approved by the local Institutional Review Board and the need for signed informed consent from each patient was waived. All data collection was compliant with the Health Insurance Portability and Accountability Act of 1996 (HIPAA). Patients were identified by searching the radiology reports texts for patients with sarcomas evaluated or treated at a tertiary care academic healthcare institution between 01/01/2010 and 03/01/2020 using Nuance mPower (Burlington, MA). The electronic medical record (EMR) was reviewed to confirm that the patient had a histological diagnosis of bone or soft tissue sarcoma.

2.1. Patients

Patients had to undergo either computed tomography (CT)-guided or ultrasound-guided biopsies of lesions for suspected sarcoma LR. Each LR had to be histologically confirmed by a pathologist. Patient age at diagnosis in years, sex, sarcoma tumour type, tumour grade (Grade I, II or III), primary tumour size in millimeters measured on the pretreatment magnetic resonance imaging (MRI), and surgical margins were obtained from the medical record. LRs were verified to be within the original primary tumour bed by a radiologist. Regional metastases (e.g. lymph nodes) were excluded and were not considered LRs.

We retrospectively reviewed the original pathology report from the primary tumour surgical resection to evaluate whether the resection had negative margins, microscopically positive margins or unknown margins (AJCC 8th Edition) [10]. Patients with macroscopically positive resections were excluded from the analysis because these patients often undergo re-resection. Patients could have one or more synchronous or asynchronous sarcoma LR.

2.2. MRI protocol and measurements

At our institution, all patients with sarcoma undergo routine contrast-enhanced magnetic resonance imaging (MRI) for surveillance imaging after surgical resection. Surveillance imaging is usually performed every 3–6 months depending on the referring physician. All MRI examinations between the resection of the primary tumour and the resection of the LR were reviewed. The time in days from the resection of the primary tumour to the date of each MRI was recorded to evaluate the LR growth rate.

MRIs were performed using either 1.5 T (Siemens [Siemens Healthineers, Erlangen, Germany] Espree, Siemens Aera, Siemens Avanto, Siemens Symphony, General Electric (GE) [GE Medical Systems, Chicago, IL, USA] Genesis Signa, GE Signa Excite, GE Optima MR450, Philips [Philips, Amsterdam, Netherlands] Intera, Philips Ingenia, Philips Achieva, Toshiba [Canon Medical Electronics, Tochigi, Japan] Titan) or 3.0 T (Siemens Verio, Siemens Skyra, Siemens TrioTim) MRI systems. Measurements were obtained on T1-weighted post-contrast sequences (Repetition Time (TR) = 200–800 ms, Echo Time (TE) = 8–24 ms, matrix 192–768 × 256–768, number of excitations (NEX) 1–4, slice thickness 1–8 mm). Between 8 and 15 mL of Multi-Hance (gadobenate dimeglumine 529 mg/mL (Bracco, NJ, USA)), or Dotarem (gadoterate meglumine (Guerbet LLC, Princeton, NJ, USA)) was administered intravenously for each contrast-enhanced study.

The maximum LR length was assumed to be 0 mm immediately after surgical resection. The maximum LR length was measured along the same axes on subsequent MRI studies was recorded (Fig. 1). Measurements were obtained to the nearest 0.1 mm using SECTRA PACS (Linkoping, Sweden) measurement tool.

Fig. 1. 45-year-old man with UPS of the right thigh.

(A) Baseline post-surgical T1-weighted post-contrast axial MRI image with fat saturation demonstrating no LR.

(B) Surveillance post-surgical T1-weighted post-contrast axial MRI image with fat saturation demonstrating a 2 cm LR involving the right thigh. White line demonstrates how maximum LR lengths were obtained.

White arrow points to a 2 cm LR which developed on post-operative day 236.

The LR volume was assumed to be 0 mm³ immediately after surgical resection. The volume of each of LR on each imaging study was measured using manual segmentation using ITK-SNAP software [11]. The radiologist manually segmented the tumor on each MRI slice where the LR was visible, and the software automatically calculates the LR volume (Fig. 2). Measurements were obtained by a fellowship-trained musculoskeletawtat bootsql radiologist and reconfirmed by a second musculoskeletal radiologist with over 7 years of experience.</ce: sectck,mn Rasajhabaar]31065177777mmm48/432sa’eeeo tion>

Fig. 2. LR volumetric measurements in a 74-year-old man with myxofibrosarcoma of the thigh.

(A) Manual volumetric segmentation of LR on post-operative day 135.

(B) Manual volumetric segmentation of LR on post-operative day 269.

(C) Manual volumetric segmentation of LR on post-operative day 419.

2.3. Statistics

Summary statistics for clinical and demographic variables were calculated. The maximum LR length and LR volume were evaluated for normality using the Shapiro-Wilks test. The maximum LR length (W = 0.636, P < 2.2 × 10⁻¹⁶ ) and the LR volume (W = 0.178, P < 2.2 × 10⁻¹⁶) were right skewed, so these variables were log transformed.

We utilized linear mixed-effects models for estimation of growth curves [12] to allow for multiple measures on the same patient (2 or more LRs) and repeated serial MRI measurements. Let Y_ijk be the maximum LR length k at time j in patient i where k = (1, 2, …, 5), j is the number of days from diagnosis, and i = (1, …, n), where n is the number of study patients. Let the vector of all outcomes for all patients be Y; let X_i be a matrix of the independent variable(s) for subject i (such as age at diagnosis, maximum size of the primary sarcoma at diagnosis, sarcoma subtype, and tumour grade); let Z_i be a matrix associated with subject-specific random effects (LR and patient); let ε_i be a vector of random errors, which are distributed as a multivariate normal vector with mean vector 0 and covariance matrix Ri. β is an unknown vector of fixed effects, and γ_i is an unknown vector of random effects with mean E[γ_i] = 0 and covariance matrix Var[γ_i] = Σ [12].

Then, Y = X_iβ + Z_iγ_i + ε,

where

$${\mathit{\gamma }}_{\mathit{i}}\sim \mathit{\text{N}}\left(0,\mathrm{\Sigma }\right)$$

and

$${\mathit{\epsilon }}_{\mathit{i}}\sim \mathit{\text{N}}\left(0,{\mathit{\text{R}}}_{\mathit{\text{i}}}\right)$$ (1)

This model accounts for within-patient variation from MRI study to MRI study and within-patient variability associated with LR.

The growth rate of the maximum LR length was modelled, since this is of clinical interest and determines if a LR is large enough to biopsy or undergo surgical resection. First, we estimated the fixed effects for time, using three models – the first with time only, the second with time and time² and the third with time, time ² and time³, so that model 1 is nested within model 2 and model 3, and so that model 2 is nested within model 3. For each model, we assumed an uncorrelated variance-covariance matrix for the random effects (random slopes and intercepts for each person). Models were compared using likelihood ratio tests (LRT). The random effects were evaluated using various combinations of random intercepts and slopes for each LR for each person. LRT/the Akaike Information Criterion (AIC) were used to identify the best random effects model. Next, several different variance-covariance matrices for the random effects were evaluated. LRT/AIC were used to identify the best variance-covariance matrix. We tested the significance of the random effects by comparing the model with the fixed effects only, to the optimal baseline model with fixed and random effects using the LRT. Analyses were performed adding each variable (age at diagnosis, tumor type, tumor grade (high versus low), primary tumour size measured by MRI, and surgical margins) and its interaction term with time, to each model. Each model was compared to the optimal baseline model using LRT/AIC. The final best model was created by including the variables that were significant at the Type 1 error rate of 0.05 in the previous analysis. Full details of model selection and construction are available in the Appendix.

We repeated the analysis to evaluate the LR volume growth rate. Again, we first estimated the fixed effects for time with three models - the first with time only, the second with time and time² and the third with time, time² and time³, so that model 1 is nested within model 2 and model 3, and so that model 2 is nested within model 3. We assumed random intercepts and slopes for each LR within each person and an unstructured variance-covariance matrix for the random effects. LRT were used to select the best model. The random effects were evaluated next. Various combinations of random intercepts and slopes for each LR for each person were evaluated. LRT/AIC were used to evaluate the best random effects model. Next, various variance-covariance matrices for the random effects were evaluated. LRT/AIC were used to identify the best variance-covariance matrix for the model. The significance of the random effects was tested by comparing the optimal model with both random and fixed effects to the model with only fixed effects using the LRT.

Next, analyses were performed adding each variable (age at diagnosis, tumor type, tumor grade (high versus low), primary tumour size measured by MRI, and surgical margins) and its interaction term with time, to each model. LRT were used to evaluate the significance of each variable and the variable’s interaction with time. Clinically, the maximum LR length is used to determine whether a LR is large enough to biopsy/surgically resect, so we did not build a multivariable model to predict the LR volume growth rate. Likelihood ratio tests were used to evaluate the significance of each variable. Full details of model selection and construction are available in the Appendix. Pearson’s correlation coefficient was used to evaluate the association between the maximum LR length and LR volume.

Analyses were performed using R v3.4 statistical software (Vienna, Austria). Linear mixed effects models were built using the lme function from the nlme package [13]. All test statistics were two-sided and P-values<0.05 were considered statistically significant.

3. Results

We reviewed the records of 1137 patients with known or suspected sarcoma and identified 347 patients with histologically confirmed sarcomas. Of these 347 patients, there were a total of 34 patients (9.8 %) with 60 sarcoma LRs (range 1–6 LRs each). One patient had 6 LRs (2.9 %), two patients (5.9 %) had 5 LRs, two patients (5.9 %) had 4 LRs, two patients (5.9 %) had 3 LRs, three patients (8.8 %) had 2 LRs, and twenty-four patients (70.6 %) had 1 LR. Most patients (61.8 %) were female. The median age (range) at diagnosis was 55 (16–86) years. Most patients had high grade (Grade II/III) disease (88.2 %). 41.2 % of patients had microscopically positive surgical margins. The median time (range) from surgical resection of the primary sarcoma to LR was 264 days (17–8013) days. Biopsy or surgical resection was performed for LRs greater than 1.0 cm (range 1.0–36.7 cm). In our study, we found that 50 % of LR occurred within 8.5 months (256 days), 90 % occurred within 42.8 months (1284 days) and 100 % occurred within 267 months (8013 days) (range 17–8013) days. Patient clinical and demographic characteristics are shown in Table 1.

Table 1.

Demographic and clinical characteristics of patients.

Variable
Mean age at diagnosis in years (SD)	53.8 (17.0)
Male gender (n, %)	13 (38.2 %)
Tumor type
Chondrosarcoma	1 (2.9 %)
Chordoma	1 (2.9 %)
Liposarcoma	3 (8.8 %)
Epithelioid sarcoma	2 (5.9 %)
Ewing	3 (8.8 %)
Fibromyxoid sarcoma	1 (2.9 %)
Leiomyosarcoma	3 (8.8 %)
Myxoinflammatory sarcoma	1 (2.9 %)
Malignant myoepithelioma	1 (2.9 %)
Myxofibrosarcoma	5 (14.7 %)
UPS	13 (38.2 %)
Location
Ankle	2 (5.9 %)
Arm/Forearm	3 (8.8 %)
Thigh/femur	15 (44.1 %)
Calf/leg	3 (8.8 %)
Chest wall	5 (14.7 %)
Other	6 (17.6 %)
Mean maximal tumour size in cm (SD)	9.3 (5.4)
Grade (n, %)
I	3 (8.8 %)
II	5 (14.7 %)
III	25 (73.5 %)
Unknown	1 (2.9 %)
Margins (n, %)
Negative margin	12 (35.3 %)
Microscopic positive margin	14 (41.2 %)
Unknown	8 (23.5 %)
Median number of MRI examinations per patient (range)	4 (2–17)

SD – standard deviation.

UPS – undifferentiated pleomorphic sarcoma.

The maximum LR length showed exponential cubic growth in time. Microscopically positive margins were associated with faster growth of the maximum LR length (P = 0.036). The data showed that patient age at diagnosis, patient sex, tumour grade, tumour size, radiation therapy and chemotherapy did not significantly affect the growth of the maximum LR length (P > 0.05 for all) (Table 2).

Table 2.

Association between variables and LR growth rate.

Outcome of interest	Variable	LRT	df	P-value
Maximum LR Length	Age at diagnosis in years	5.89	2	0.053
	Gender	1.70	2	0.426
	Grade	1.40	2	0.497
	Maximal primary tumour size	3.63	2	0.163
	Margins	6.67	2	0.036
	Radiation	1.96	2	0.375
	Chemotherapy	0.28	2	0.871
LR Volume	Age at diagnosis in years	4.70	2	0.095
	Gender	0.02	2	0.989
	Grade	2.82	2	0.245
	Maximal primary tumour size	2.57	2	0.277
	Margins	0.45	2	0.798
	Radiation	2.39	2	0.303
	Chemotherapy	0.18	2	0.913

The growth rate of sarcoma LRs can greatly inform the frequency of surveillance imaging. Sarcoma LRs were histologically confirmed and required an image-guided biopsy. These lesions were biopsied/surgically resected when the sarcoma LRs were at least 1 cm in size. Based on this data, assuming the prior MRI was negative, surveillance imaging with contrast-enhanced MRI could be performed every 7.2 months, when the maximum LR length is expected to be 1.0 cm (95 % CI (0.4–2.6) cm) in a patient with negative margins. The interval between surveillance imaging should decrease in patients with microscopically positive margins at surgery. For example, assuming that when the maximum LR length is 1.0 cm, then it is actionable, and assuming that the prior surveillance MRI was negative, the data predicts that patients with microscopically positive margins should have surveillance MRI every 6.2 months when the maximum LR length is expected to be 1.0 cm (95 % CI (0.4–2.3 cm)).

At 103 days, the maximum LR length is expected to be 0.5 cm (95 % CI (0.2, 1.0) cm) in patients with microscopically positive margins. The model also predicts that in 103 days, the maximum LR length is expected to be 0.4 cm (95 % CI (0.1, 0.9) cm) in patients with negative margins.

The LR volume showed exponential growth cubic in time, similar to the maximum LR length. The data showed that patient age at diagnosis, patient sex, tumour grade, tumour size, tumor margins, radiation therapy and chemotherapy did not significantly affect the LR volume growth (P > 0.05 for all) (Table 2). The LR volume was strongly correlated with maximum LR size (r = 0.95, P < 2.2 × 10⁻¹⁶).

Fig. 3 shows the expected growth curves and 95 % CI for LRs with negative margins and microscopically positive margins respectively.

Fig. 3. Plots of the expected LR growth rate for patients with negative margins and microscopically positive margins respectively.

Black lines show the average maximum LR length.

Dashed lines show the 95 % confidence intervals.

4. Discussion

The results of this study show that LR following surgical resection is not common, and occurs in less than 10 % of cases. Sarcoma LRs have an exponential growth pattern. Radiologists are able to detect subcentimeter LRs on MRIs. These subcentimeter LRs are generally too small to be biopsied because they may not be detected by ultrasound or CT (the two most common methods for image-directed biopsies) and could result in a false negative biopsy if the biopsy is attempted at this size. None of the LRs was biopsied or surgically resected before it had a maximum length of 1.0 cm in our study. Generally, treatment does not begin until there is histological confirmation of a local recurrence. Therefore, we define actionable as the minimum size at which biopsy or surgical resection is attempted, and the data suggest that 1 cm is the smallest actionable size for sarcoma LR. Assuming 1 cm is an actionable size, LRs are expected to have a maximum length of 1 cm within 6.2 months for patients with microscopically positive margins and within 7.2 months for patients with negative margins.

LRs with microscopically positive margins were likely to grow faster than LRs with negative margins. We hypothesize that this is because a larger number of tumour cells are present in patients with microscopically positive margins than in patients with negative margins. However, we did note that two patients with LR had initial negative margins determined by an outside pathologist, which were subsequently determined to be microscopically positive margins by another pathologist with sarcoma expertise and more experience reviewing sarcoma cases at a tertiary care center. This suggests that there is variability in pathologists’ expertise, and suggests that there may be sampling error in determining tumour margins. We also noted variability in the surveillance frequency for patients evidenced by the number of MRIs per patient, however, we believe this may be due to referring physician preference and patient insurance restrictions on the frequency of surveillance imaging.

The current NCCN guidelines are based on the opinions of an expert panel. The NCCN recommends surveillance imaging every 3–6 months, which is conservative relative to the results of our analysis. Decreasing the surveillance rate to every 6 months has the potential to decrease imaging healthcare costs and decrease patient anxiety related to uncertainty about MRI findings. For example, the Medicare reimbursement for a contrast-enhanced MRI of the thigh is approximately €330 ($390 USD). Therefore, surveillance imaging every 6 months instead of every three months could save €2310 ($2730 USD) over three and a half years of surveillance.

The analysis also showed that all LRs did not grow at the same rate, and that there were significant random effects for intercept and slope within patient for the maximum LR length. This means that each patient may deviate from the population mean values calculated, and personalized estimates may be more appropriate in select cases. We also suspect that tumor volume growth is slightly different from the growth of the maximum LR length, which may explain why surgical margins was not significant in the volumetric analysis. The optimal duration of surveillance is also variable among referring physicians [14,15], however, our data suggests that 90 % of the LRs occurred within 3.5 years. Radiation therapy and surgical resection with wide margins have a 90 % LR control rate, so approximately 1% of patients with sarcoma will be expected to have LR beyond 3.5 years if treated with radiation therapy and surgical resection. Further research is required to determine the total duration of surveillance, and the cost-effectiveness of using a cheaper imaging modality such as ultrasound or physical examination instead of MRI for surveillance in these patients.

The proposed surveillance imaging interval should not be consistent with the surveillance imaging interval for other cancers (e.g. lymphoma, breast, colon). Sarcoma tumor biology is very different from other tumors, and naturally, the growth rate is expected to be different. Dynamic contrast-enhanced (DCE) MRI, diffusion-weighted imaging (DWI) MRI and positron emission tomography (PET) are likely more sensitive for detection of sub-centimeter LRs, however this increased sensitivity is not actionable. The primary problem is not the sensitivity of imaging for the detection of the recurrence. The primary problem is that the histological confirmation of the recurrence may be difficult/impossible with subcentimeter LRs using current biopsy methods. Furthermore, the surgeon is unlikely to find a subcentimeter LR for resection. Instead, the patient typically gets a follow-up imaging study when the referring physician believes the LR would have grown to an actionable length (1 cm). For example, in current practice, a patient with a 2-mm LR detected by DCE MRI would have a follow-up imaging study, however, the timing of that follow-up study would be variable between physicians. Based on our data, the 2-mm LR would be clinically expected to be 1 cm in maximum length and now actionable in approximately 5.2 months assuming the patient had microscopically positive surgical resection margins.

Prior studies support some of our findings. First, several papers have shown the efficacy of adjuvant and neoadjuvant radiation therapy for local control of sarcoma, and have estimated efficacy between 80–95 % [6–8]. Our study suggests that surgery and radiation therapy have an efficacy of about 90 % for local control. Prior studies show that most local recurrences occurred within 2 years of resection, similar to our own findings [6]. Approximately 90 % of LR occur within 3.5 years of the surgical resection (range 17–8013 days). The patient with the recurrence at 8013 days transferred care from another institution and had a low-grade fibromyxoid sarcoma, which likely had a slow growth rate and low metastatic potential.

The study has a few limitations. The retrospective nature of this study made it susceptible to ascertainment bias. The study was conducted at a tertiary care academic medical center, and the referral patterns and treatment/management patterns may have influenced the distribution of sarcoma types in the study. The small study sample size is expected given the prevalence of sarcomas and the efficacy of radiation therapy for local control. We believe that the confidence intervals are wide because the sarcoma local recurrence growth rate varies between patients and because of the sample size. The number of patients did not allow for internal validation of the model on new data, and external validation of this study is required for confirmation.

5. Conclusions

Sarcoma LRs grow faster in patients with microscopically positive resection margins compared to patients with negative margins. Assuming the previous surveillance MRI was negative and that a LR is actionable if it has a maximum length of 1.0 cm, then surveillance imaging with MRI can therefore be conservatively performed every 6 months, when the maximum LR length is expected to be less than 1.0 cm in size. Linear mixed effects modeling is a statistical technique can be utilized for other diseases to allow for better imaging utilization.

Abbreviations:

AJCC

American Joint Committee on Cancer

CI

Confidence Interval

LR

Local recurrence

LRT

Likelihood Ratio Test

MRI

magnetic resonance imaging

NCCN

National Comprehensive Cancer Network

NEX

Number of excitations

SD

Standard deviation

UPS

undifferentiated pleomorphic sarcoma

Appendix A

Maximum LR length growth rate

First, we estimated the fixed effects to predict the rate of the log (maximum LR length + 1 mm) growth. Linear mixed-effects models with random intercepts and slopes for each person, were fitted with an uncorrelated variance-covariance matrix for the random effects and restricted maximum likelihood (REML).

The first model included time only, the second model included time and time squared, and the third model included time, time squared and time cubed. The models were then refitted using maximum likelihood and compared using likelihood ratio test (LRT) statistics to identify the best model. The model with time squared (LRT Chi-squared statistic = 39.32, df = 1, P<0.0001), and time cubed (LRT Chi-squared statistic = 68.30, df=2, P<0.0001) were significantly better than the model with time only. The model with time cubed was significantly better than the model with time squared (LRT Chi-squared statistic =29.00, df =1, P<0.0001).

Next, the random effects were evaluated. Seven different random effects models were created: random intercepts only for each LR (Model 1), random intercepts for each person (Model 2), random slopes only for each LR (Model 3), random slopes only for each person (Model 4), random slopes and intercepts for each LR (Model 5), random slopes and intercepts for each person (Model 6) and random slopes and intercepts for each LR within each person (Model 7). The model with random intercepts and slopes for each person (Model 6) (AIC=690.89) was better than Model 1 (LRT Chi-squared statistic=21.36, df=2, P<2.2 × 10⁻¹⁶), better than Model 2 (LRT Chi-squared statistic=11.54, df=2, P=0.003), better than Model 3 (LRT Chi-squared statistic=18.76, df=2, P=1 × 10⁻⁴), better than Model 4 (LRT Chi-squared statistic=10.18, df=2, P=0.006) and better than Model 5 (AIC=709.51). There was no difference between Model 6 and Model 7 (LRT Chi-squared statistic=7.08, df=3, P=0.069). Model 6 was chosen as the best model.

The variance-covariance matrix for the random effects was then evaluated. Five different variance-covariance matrices were used: first order continuous autoregressive (CAR1) (Variance 1), compound symmetry (Variance 2), linear spatial (Variance 3), normal/Gaussian spatial (Variance 4), and uncorrelated random errors (Variance 5). The AIC or LRT were used to select the best variance-covariance matrix. The model with uncorrelated random errors (Variance 5) was not significantly worse than the model with the CAR1 variance-covariance matrix (Variance 1) (LRT Chi-squared statistic = 1.30, df=1, P=0.255), not significantly worse than the model with the compound symmetry variance-covariance matrix (Variance 2) (LRT Chi-squared statistic = 0.04, df=1, P=0.834), not significantly worse than the model with the linear spatial variance-covariance matrix (Variance 3) (LRT Chi-squared statistic = 1.65, df=1, P=0.199), and not significantly worse than the model with the normal/Gaussian spatial variance-covariance matrix (Variance 4) (LRT Chi-squared statistic = 1.64, df=1, P=0.200). The uncorrelated random errors variance-covariance matrix (Variance 5) was chosen as the best variance-covariance matrix.

To investigate the significance of the random effects, the fit of the best model was compared to a model with only fixed effects. The model with random slopes for each person was significantly better than the model with only fixed effects (LRT Chi-squared statistic=21.35, df=3, P = 1.0 × 10⁻⁴).

To evaluate the significance of patient age at diagnosis in years, sex, tumour type, tumour grade (high=grade II/III vs low=grade I), tumour size in mm measured by MRI, and surgical resection margins, we included each variable and its interaction with time individually. Likelihood ratio tests were used to evaluate the significance of each variable.

The final multivariable model for predicting the log (maximum LR length +1 mm) was created by utilizing variables that were significant at the Type I error rate of 0.05 in the previous analysis (margins, patient age at diagnosis), and their interactions with time. The final model was:

$$Y+1mm={\beta }_0+{\beta }_{1}Margins+{\beta }_{2}time+{\beta }_{3}tim{e}^2+{\beta }_{4}tim{e}^3+{\beta }_{5}Margins\ast time$$

Coefficient	Estimate	Standard Error	T-statistic	P-value
β 0	0.62289	0.24761	2.52	0.014
β 1	0.17863	0.32942	0.54	0.593
β 2	0.00924	0.00104	8.84	<0.001
β 3	−4.6 × 10−6	7.0 × 10−7	−6.13	<0.001
β 4	−4.76 × 10−10	9.9 × 10−11	4.81	<0.001
β 5	0.00013	0.00107	0.13	0.901

Correlations between fixed effects

	β 0	β 1	β 2	β 3	β 4
β 1	−0.723
β 2	−0.262	0.068
β 3	0.148	0.065	−0.690
β 4	−0.138	−0.020	0.647	−0.958
β 5	0.143	−0.190	−0.460	−0.049	−0.049

Therefore, the size of an LR, assuming negative resection margins can be estimated using the formula,

$$\text{exp}\left(Y+ 1mm\right) = \text{exp}({\beta }_0+{\beta }_{1}Margins+{\beta }_{2}time+{\beta }_{3}tim{e}^2+{\beta }_{4}tim{e}^3+{\beta }_{5}Margins\ast time)$$

Using this equation, the expected maximum LR length, assuming negative resection margins at 215 days is 1.0 cm (95 % CI (0.4, 2.6) cm). Similarly, the expected maximum LR length, assuming microscopically positive resection margins at 187 days is 1.0 cm (95 % CI (0.4, 2.3) cm).

The data can be used to determine the timing of the next surveillance imaging study. For example, if a 2-mm LR identified on the surveillance MRI for a patient with microscopically positive margins, then the next MRI should be performed in 155 days (5.2 months) when the LR would be expected to have a maximum length of 1 cm, and be actionable.

The variance of the log (maximum LR length +1 mm), assuming negative resection margins is:

$$\begin{gathered} Var\left(Maximum LR length+1mm\right) =Var\left({\beta }_0\right) +Var\left({\beta }_1\right) +tim{e}^2\ast Var\left({\beta }_2\right) +tim{e}^4\ast Var\left({\beta }_3\right) +tim{e}^6\ast Var\left({\beta }_4\right) +tim{e}^2\ast Margin{s}^2\ast Var\left({\beta }_5\right) +Margins \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ast Cov\left({\beta }_0, {\beta }_1\right) +time\ast Cov\left({\beta }_0, {\beta }_2\right) +tim{e}^2\ast Cov\left({\beta }_0, {\beta }_3\right) +tim{e}^3\ast Cov\left({\beta }_0, {\beta }_4\right) +time\ast Margins\ast Cov\left({\beta }_0, {\beta }_5\right) +time \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ast Margins\ast Cov\left({\beta }_1, {\beta }_2\right) +tim{e}^2\ast Margins\ast Cov\left({\beta }_1, {\beta }_3\right) +tim{e}^3\ast Margins\ast Cov\left({\beta }_1, {\beta }_4\right) +time\ast Margin{s}^2\ast Cov\left({\beta }_1, {\beta }_5\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+tim{e}^3\ast Cov\left({\beta }_2,{\beta }_3\right) +tim{e}^4\ast Cov\left({\beta }_2,{\beta }_4\right) +tim{e}^2\ast Margins\ast Cov\left({\beta }_2,{\beta }_5\right) +tim{e}^5\ast Cov\left({\beta }_3,{\beta }_4\right) +tim{e}^3\ast Margins \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ast Cov\left({\beta }_3, {\beta }_5\right) +tim{e}^4\ast Margins\ast Cov\left({\beta }_4, {\beta }_5\right) \end{gathered}$$

LR volume growth rate

We estimated the fixed effects to predict the rate of the log (volume + 1 mm³) of the LR. Linear mixed-effects models with random intercepts and slopes for each LR within each person, were fitted with an uncorrelated variance-covariance matrix for the random effects and restricted maximum likelihood (REML).

The first model included time only, the second model included time and time squared, and the third model included time, time squared and time cubed. The models were then refitted using maximum likelihood and compared using likelihood ratio test (LRT) statistics to identify the best model. The model with time squared (LRT Chi-squared statistic = 25.26, df=1, P<0.0001), and time cubed (LRT Chi-squared statistic = 61.93, df=2, P<0.0001) were significantly better than the model with time only. The model with time cubed was significantly better than the model with time squared (LRT Chi-squared statistic =36.67, df=1, P<0.0001).

Next, the random effects were evaluated. Seven different random effects models were created: random intercepts only for each LR (Model 1), random intercepts for each person (Model 2), random slopes only for each LR (Model 3), random slopes only for each person (Model 4), random slopes and intercepts for each LR (Model 5), random slopes and intercepts for each person (Model 6) and random slopes and intercepts for each LR within each person (Model 7). The model with random slopes and intercepts for each LR within each person (Model 7) was better than Model 1 (LRT Chi-squared statistic=27.08, df=5, P=1.0 × 10⁻⁴), better than Model 2 (LRT Chi-squared statistic=23.76, df=5, P=2.0 × 10⁻⁴), better than Model 3 (LRT Chi-squared statistic=25.11, df=5, P=1.0 × 10⁻⁴), better than Model 4 (LRT Chi-squared statistic=21.99, df=5, P=5.0 × 10⁻⁴), better than Model 5 (LRT Chi-squared statistic=24.84, df=3, P=<0.0001) and better than Model 6 (LRT Chi-squared statistic=15.07, df=3, P=0.0018). Model 7 (random slopes and intercepts for each LR within each person) was chosen as the best model.

Next, the variance-covariance matrix for the random effects was evaluated. Five different variance-covariance matrices were used: uncorrelated random errors (Variance 1), first order continuous autoregressive (CAR1) (Variance 2), linear spatial (Variance 3), normal/Gaussian spatial (Variance 4), and compound symmetry (Variance 5). The AIC and/or LRT were used to select the best variance-covariance matrix.

The model with the uncorrelated random errors variance-covariance matrix (Variance 1) (AIC=1014.34) was not worse than the model with the CAR1 variance-covariance matrix (Variance 2) (LRT Chi-squared statistic=0.13, df=1, P=0.714), not worse than the model with linear spatial variance-covariance matrix (Variance 3) (LRT Chi-squared statistic=0.21, df=1, P=0.647), not worse than the model with a normal spatial variance-covariance matrix (Variance 4) (LRT Chi-squared statistic=1.02, df=1, P=0.998), and not worse than the model with the compound symmetry variance-covariance matrix (Variance 5) (LRT Chi-squared statistic=2.21, df=1, P=0.137). The uncorrelated variance-covariance matrix for the random errors (Variance 1) was chosen as the best model.

To investigate the significance of the random effects, the fit of the best model was compared to a model with only fixed effects. The model with random intercepts and slopes for each LR within each person and an unstructured variance-covariance matrix for the random errors was significantly better than the model with only fixed effects (LRT Chi-squared statistic=27.07, df=6, P = 1.0 ×10⁻⁴).

To evaluate the significance of patient age at diagnosis, sex, tumour type, tumour grade, tumour size measured by MRI, and surgical resection margins, we included each variable and its interaction with time individually in the best model. Likelihood ratio tests were used to evaluate the significance of each variable.