British Journal of Radiology 74 (2001),123-126 © 2001 The British Institute of Radiology
Improving the sensitivity of stereotactic core biopsy to diagnose ductal carcinoma in situ of the breast: a mathematical model
N J Coombs, FRCS
1
J R L Laddie, BM
1
G T Royle, FRCS
1
C M Rubin, FRCR
2 and
M S Briley, FRCR
3
1 Departments of Breast Surgery
3 Radiology, Royal South Hants Hospital, Southampton
2 Southampton and Salisbury Breast Screening Unit, Southampton, UK
Correspondence: Mr Gavin Royle, Consultant Surgeon, Royal South Hants Hospital, Southampton SO14 0YG, UK
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Abstract
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Stereotactic core biopsy (SCB) is performed on mammographically suspicious, non-palpable lesions of the breast. Reported sensitivities of SCB for the detection of ductal carcinoma in situ (DCIS) vary from 41% to 93%. We have developed a simple mathematical model to predict the probability of retrieving at least one diagnostic core from a focus of DCIS. We make recommendations of the number of samples needed for different sized areas of microcalcification. The sensitivity of SCB is affected by needle placement accuracy, diameter of the area of microcalcification (d), histological density of DCIS (x) (calculated as 7.5% by previous studies) and number of core samples (n) removed. The probability of achieving at least one representative core sufficient for diagnosis (P(core)) is defined as:
where 
is the probability of a SCB accurately targeting the area of microcalcification. At least seven core samples should be removed in small foci (<5 mm) of DCIS to achieve a 0.75 probability of an accurate diagnosis. The probability of a diagnostic biopsy of larger areas of DCIS (>10 mm) is 0.95 when five cores are removed. This formula serves as an explanation to patients why SCB may fail to diagnose DCIS, and justifies the retrieval of more core samples to increase the probability of an accurate diagnosis and to reduce the chance of a non-representative core. In the absence of sufficient samples, a wire-guided open biopsy is necessary to exclude DCIS.
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Introduction
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Ductal carcinoma in situ (DCIS) of the breast is usually detected on screening mammography as microcalcification. Confirmation of the diagnosis requires guided fine needle aspiration cytology (FNAC), stereotactic core biopsy (SCB) or wire-guided open biopsy. Although FNAC and SCB are performed in an outpatient setting, the reported sensitivities of detection of DCIS vary and are lower than the sensitivity of diagnosing invasive carcinomas [1]. SCB may be more sensitive than FNAC at detecting malignancy [2] and although some authors report SCB sensitivities and specificities in excess of 85%, at least 20% of these patients will require an open biopsy because a definitive diagnosis could not be reached [3].
Previous studies in Southampton have measured the relative proportions of malignant ducts and normal parenchyma in areas of DCIS and have demonstrated that, even in areas of mammographic abnormalities, more than 90% of the tissue is normal breast stroma.
We believe that the failure of SCB to diagnose DCIS is due to sampling the normal breast parenchyma surrounding the DCIS or due to retrieval of insufficient malignant cells to make a diagnosis with confidence. Imaging of the core specimen to confirm the presence of microcalcification may improve the sensitivity of SCB [4, 5], and increasing the number of samples removed will improve the likelihood of achieving an accurate diagnosis [6, 7].
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Mathematical model
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We have developed a mathematical model to predict the probability of retrieving at least one diagnostic core sample by SCB from an area of DCIS. Measurable variables include radiological diameter of the microcalcification (d), number of core samples removed (n), accuracy of stereotactic needle placement and volume density of DCIS ducts within a focus of microcalcification (x).
DCIS is not a homogeneous lesion but consists of islands of scattered malignant ducts separated by areas of benign stroma and breast tissue. Sectioning tissue containing DCIS in any plane allows the malignant ducts to be seen on the slide. The proportion of the slide containing positively involved ducts is equivalent to the histological volume density of DCIS (x) and has been measured as 7.5% in our unit. This figure appears to be independent of foci size, according to studies within the Southampton Pathology Department.
If the focus of DCIS is likened to a series of breast sections, each 1 mm thick, then the probability of the core needle hitting either malignant tissue or benign stroma can be calculated for each section. As more sections are examined deeper into the specimen, so the probability of the core needle hitting an area of malignancy increases and, conversely, the probability of a misrepresentative core decreases as the diameter of the foci (d) increases. The diameter d equates to the number of 1 mm thick sections that can be taken from the foci of DCIS and targeted by a 1 mm diameter core needle.
A diagnostic core sample therefore relies on needle placement within the area of microcalcification and retrieval of sufficient malignant cells from the breast. An insufficient core biopsy is due to either inability of the stereotactic machine to place the needle within the mammographic abnormality or to biopsy of the normal breast stroma that surrounds the malignant ducts.
We have assumed that, like many biological and mechanical devices, the accuracy of the stereotactic needle placement follows a normal distribution. Thus, 95% of samples will fall within two standard deviations of the intended site. Although the "phantom check" of stereotactic core biopsy accuracy is within 2 mm, the true accuracy of needle placement in the breast may be worse. For the purpose of this study, we estimate that the error of stereotactic needle placement is 5 mm; 95% of biopsies will therefore be within ±10 mm. If the diameter of mammographic abnormality is 20 mm, then there is a 5% probability that the needle will fail to accurately locate the microcalcification. If the standard error of a stereotactic machine is estimated as 5 mm, the probability of a biopsy needle targeting an abnormality can be calculated for any diameter lesion [8] using a "z-table" to calculate the area under a normal distribution curve within standard deviations of the mean (Table 1
).
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Table 1. "z-table" relating probabilities to multiples of standard deviations in a normal distribution. The probability of localizing an area of mammographic abnormality, knowing the area of the needle placement and the diameter of the target, is calculated (modified from Swinscow [8])
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Mathematical formula
The calculations assume a 1 mm diameter core is removed at core biopsy and cannot correct for loss of tissue or damage during retrieval, embedding or processing.
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Discussion
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Although this formula makes assumptions about needle accuracy and tissue processing, it clearly demonstrates the importance of removing at least five cores to achieve a reasonable probability of a diagnosis of DCIS in microcalcification. Clinical studies by Liberman et al [6] retrieved up to 11 cores samples individually by SCB. Although they included all patients with impalpable lesions, i.e. both invasive and in situ malignancy, their calculations of the probability of a diagnostic core are similar to our mathematical model. Similar results have been shown by retrospective analysis of 500 women undergoing SCB. They demonstrated an increase in the sensitivity of SCB to diagnose DCIS in microcalcification from 70.1% to 94.0% when the number of cores removed increased from two to six [7]. Studies in the Southampton and Salisbury Breast Screening Unit have demonstrated that the median mammographic microcalcification size of screen-detected DCIS is 15 mm. Within the Southampton Pathology Department a study of 80 DCIS specimens has calculated that the proportion of positively involved malignant tissue is 7.5% and is independent of foci size.
Assuming the standard error of the stereotactic machine is ±5 mm and knowing the proportion of malignant tissue, then the probability of a representative core of DCIS increases with the number of samples removed (Figure 1).
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Figure 1. Theoretical probability of retrieving at least one diagnostic core of ductal carcinoma in situ (DCIS) from 10 mm diameter focus of DCIS and 7.5% DCIS duct density.
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These theoretical probabilities are similar to sensitivities of SCB measured by Liberman et al [6] and Rich et al [7]. This formula can provide a useful guide to radiologists performing SCB and demonstrates the importance of retrieving at least five core biopsies to achieve a probability of a diagnostic core in 90% of patients.
Figure 2 demonstrates the theoretical probabilities of obtaining a diagnostic core with varying calcification diameter and number of cores removed. In small areas of microcalcification, at least seven cores should be attempted, but in larger lesions fewer cores may be required. The larger number of cores required to diagnose small lesions is explained by the increased probability of the needle being placed outside of the area of malignant cells. For all mammographically-detected lesions, Dahlstrom et al [9] have demonstrated that the diagnostic yield of three cores is statistically equal to that of five. We suggest that when a diagnosis of DCIS is suspected, three cores could be taken when larger (>20 mm) areas of microcalcification are involved.
Orientation of the needle biopsy relative to the direction of the ducts within the breast will not influence needle accuracy. Delesse's formula [10] demonstrates that the proportion of the volume of breast tissue containing the abnormality is identical to the proportional surface area of the abnormality that is visualized in any plane. A needle inserted into the breast parallel to the ducts therefore has the same probability of retrieving a diagnostic core as a needle placed perpendicular to the duct structure. However, inserting a core needle along the longest axis of the abnormality will increase the volume of abnormal tissue available to be removed by the core along its length.
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Conclusions
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The pre-operative diagnosis of DCIS by SCB can be difficult. The biopsy samples may not be representative in patients where a radiological suspicion of DCIS is not confirmed by the core biopsy. Patients then require wire-guided open biopsy or further SCB to confirm the diagnosis. Patients and lawyers may also question why core biopsy tests yield inaccurate results. An understanding of the probabilities in sampling a non-homogeneous lesion may aid clinicians in justifying why several core biopsies should be performed to achieve a diagnosis or to assist in explaining why a patient may need an open biopsy for a diagnosis.
Achieving a diagnosis of DCIS by SCB relies on accurate placement of the needle within the abnormality and the sampling of sufficient malignant cells. This is prone to errors due to stereotactic machine accuracy, patient movement during the procedure, inability of the patient to tolerate the procedure to allow removal of more than a few cores, damage of the tissue during its retrieval or tissue processing artefacts. The simple mathematical formula used in this paper is unable to account for these variables but presents the theoretical probability of retrieving a diagnostic core. Table 1
gives radiologists a guide as to the number of cores that should be removed and serves as an explanation to patients undergoing SCB. In small (<5 mm) areas of microcalcification, a case could also be made for use of the mammotome to remove a larger volume of tissue or use of wire-guided excision of suspected DCIS. However, these small areas of DCIS can be difficult to diagnose radiologically.
Received for publication March 8, 2000.
Revision received September 20, 2000.
Accepted for publication October 5, 2000.
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References
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Abstract
Introduction
, 1 core; 
, 3 cores; 
, 5 cores; X, 7 cores.