Microscopic Peritoneal Residual Disease after Complete Macroscopic Cytoreductive Surgery for Advanced High Grade Serous Ovarian Cancer

Background: Epithelial ovarian cancers (EOC) are usually diagnosed at an advanced stage and managed by complete macroscopic cytoreductive surgery (CRS) and systemic chemotherapy. Peritoneal recurrence occurs in 60% of patients and may be due to microscopic peritoneal metastases (mPM) which are neither eradicated by surgery nor controlled by systemic chemotherapy. The aim of this study was to assess and quantify the prevalence of residual mPM after complete macroscopic CRS in patients with advanced high-grade serous ovarian cancer (HGSOC). Methods: A prospective study conducted between 1 June 2018 and 10 July 2019 in a single referent center accredited by the European Society of Gynecological Oncology for advanced EOC management. Consecutive patients presenting with advanced HGSOC and eligible for complete macroscopic CRS were included. Up to 13 peritoneal biopsies were taken from macroscopically healthy peritoneum at the end of CRS and examined for the presence of mPM. A mathematical model was designed to determine the probability of presenting at least one mPM after CRS. Results: 26 patients were included and 26.9% presented mPM. There were no differences in characteristics between patients with or without identified mPM. After mathematical analysis, the probability that mPM remained after complete macroscopic CRS in patients with EOC was 98.14%. Conclusion: Microscopic PM is systematically present after complete macroscopic CRS for EOC and could be a relevant therapeutic target. Adjuvant locoregional strategies to conventional surgery may improve survival by achieving microscopic CRS.


Supplementary Material 1. Mathematical method.
Let be the number of patients included in the study. Let be the number of peritoneal biopsies, which have been carried out in the peritoneum of each patient to histologically determine if microscopic peritoneal metastases (mPM) of epithelial ovarian cancer (EOC) are present (all patients were assumed to have undergone the same number of biopsies). We assumed 100% reliability for biopsy (i.e., no false positives and no false negatives).
Let + be the discrete random variable that gives the number of biopsies with mPM in each patient. Given the above definition of , the possible values for + are 0, 1, 2, …, (note, that uppercase letters are used for random variables, whereas numbers and lowercase letters are related to the possible values of these random variables). Let + = { } be the event corresponding to the occurrence of biopsies with mPM among the performed ones. The probability associated with the event + was denoted by ( + ). All the probabilities ( 0 + ), ( 1 + ), … and ( + ) were assumed to have been estimated from the patient data and therefore to be known. Let (cm 2 ) be the average peritoneal surface area in adults. If we consider that each biopsy samples a surface area of (cm 2 ), a total of = peritoneal biopsies should have been carried out in each patient to assess the entire peritoneal surface. The peritoneum in each patient can therefore be described as the union of possible areas of biopsy. Of course, is far higher than . Let + be the discrete random variable that gives the number of these possible areas of biopsy that actually have mPM in each patient. + can then have the values 0, 1, 2, up to . We denoted by + = { } the event that of the possible areas of biopsy actually have mPM and by ( + ) the probability for this event to occur. The events 0 + , 1 + , …, + , which are mutually exclusive (both events cannot occur at the same time) and together cover all possible values of + , form a partition. As the result, the sum of the probabilities of these ( + 1) events is equal to 1 (Equation (S1)). (S1) The probabilities ( 0 + ), ( 1 + ), … and ( + ) might be very useful for optimizing treatment for patients. The deduction from ( 0 + ), ( 1 + ), … and ( + ), is the probability of the event that at least one of the possible areas of biopsy has mPM. This event, above referred to as ? ?1 + , can be written as the union of the events 1 + , 2 + , …, + , so that its probability can be determined as follows: where represents the union operator and the two last equalities result from the fact that the events 0 + , 1 + , …, + form a partition.
In this paper, we were particularly interested in this probability: the higher it is, the more likely the peritoneum is affected by mPM and the more beneficial the use of an adjuvant treatment to cytoreductive surgery.
Another information, the probabilities ( 0 + ), ( 1 + ), … and ( + ) would provide, relates to the spread of EOC within the peritoneum. Let be the integer between 0 and such that the probability of the event that more than possible areas of biopsy have mPM is about (Equation (S3)). (S3) where ? ? + denotes the event that more than possible areas of biopsy have mPM and the equalities result from the fact that the events 0 + , 1 + , …, + form a partition.
The values of we were interested in here were the standard 5%, 25%, 50%, 75% and 95%. The corresponding integers P can be considered as kind of percentiles. Take for example = 25%: 25% is such that the probability to have more than 25% possible areas of biopsy with mPM is equal to 25%; this implies that the probability of having less than 25% possible areas of biopsy with mPM is equal to 75% and therefore 25% , can be viewed as the 75th percentile.
The higher the integers are, the greater the spread of EOC within the peritoneum.
To achieve this, we used the law of total probability. By applying this law with the above-defined partition 0 + , 1 + , …, + , the probability ( + ) and the set of probabilities ( 0 + ), ( 1 + ), … and ( + ) are linked to each other as follows (Equation (S4)). (S4) With ( + | + ) is the conditional probability of + given + , that is, the probability that + occurs given that + occurs (in other words, this is the probability that of the peritoneal biopsies present mPM given that of the possible areas of biopsy have mPM). This conditional probability clearly follows a hypergeometric distribution. A hypergeometric distribution allows to describe the probability of obtaining a certain number of successes (here, biopsies with mPM, ) in a certain number of draws (here, biopsies, ) given a population of a certain size (here, possible areas of biopsy, ) containing a certain number of successes (here, possible areas of biopsy with mPM, ), without replacement. According to the definition of the hypergeometric distribution, ( + | + ) can therefore be expressed as in Equation (S5): where the binomial coefficient ( ) is the total number of subsets of distinct elements in a set of elements, for any integers and such that . Applying Equation (S4) for between 0 and , the following system of linear equations in unknowns were constructed (Equation (S6)): (S6) In this system, only the probabilities ( 0 + ), ( 1 + ), … and ( + ) were unknown. To determine them, this system had to be solved. With many more unknowns than equations, this consistent system has infinite solutions (consistency was ensured by the above definition of ( + | + ) (Equation (S5)).
However, many of these solutions were not relevant from a probabilistic perspective. Bearing in mind that the solution consists of the probabilities of a partition, the solution had to meet two specific constraints: the first one, expressed in Equation (S1), states that the sum of ( However, without additional constraint, particularly on the trend, this solution might contain irregularities. This means that there might be jumps between sequential probabilities: the probability that of the possible areas of biopsy actually have mPM might, in fact, differ significantly from the probability that ( + 1) or ( − 1) of the possible areas of biopsy actually have mPM. Such jumps would be obviously unphysical. This is why we introduced a smoothness constraint designed to discard solutions containing irregularities. The smoothness term ℎ , we used to impose this smoothness constraint on the solution, involves the third-order difference as shown in Equation (S9) (the forward finite difference and the backward finite difference were applied for = 0 and 1 and for = − 1 and , respectively. The central finite difference was used for all the other ): The unspecified elements in matrix are equal to 0.
The closer to 0 this smoothness term is, the smoother the trend of the solution. Note that this smoothness term was designed for constant, linear, and quadratic solutions (the third-order difference and thus the smoothness term for such solutions indeed is zero).
This smoothness term was associated with the above defined fitting error (Equation (S8)) into a total error, which therefore accounts for both the compliance with smoothness and the closeness to the data (Equation (S10)): with the parameter ? controlling the trade-off between the compliance with smoothness and the closeness to the data: for large ? , the solution, resulting from the minimisation of the total error , will be smoother at the expense of being further from the data, whereas for small ? , the solution will be closer to the data but with more irregularities.
The optimal value for ? was determined using the cross-validation method [1]. The total error can be rearranged by using a matrix format so that the least squares method can be applied (Equation (S11)): Applying the least squares method to minimize the total error under constraints of Equations (S1) and (S7) would provide the most suitable solution ( 0 + ), ( 1 + ), … and ( + ) to our problem.
This minimization was performed using the function "lsqlin" of the Matlab software.