I've sent an old paper to Louise Roberts regarding the reliability of such measurements. Here is the abstract's findings:
Triiodothyronine and Thyroxine in Urine. II. Renal Handling, and Effect of Urinary Protein
C. W. BURKE AND R. A. SHAKESPEAR*
Department of Endocrinology, Radcliffe Infirmary, Oxford, England
ABSTRACT. Mean urinary clearances of T3 were 164 ml/min in normal subjects, 177 in pregnancy, 221 in thyrotoxicosis, 174 in hypothyroidism, and 194 in 3 persons with undetectable T4 but normal T3 levels. T4 clearances were 38 ml/min in normal subjects, 48 in thyrotoxicosis, and 138 in hypothyroidism. Low creatinine clearance was associated with low clearances of T4 and T3. The data suggest urinary excretion of T3 by glomerular filtration of serum unbound T3, with added tubular excretion; and T4 excretion by glomerular filtration of unbound T4 and tubular reabsorption. However, 3 - 9% of urinary T3 and 5-12% of urinary T4 were bound
to urinary proteins, and increased protein excretion caused markedly increased T4 excretion. In addition, 52% of urinary T3 and 68% of urinary T4 were bound to other substances of approximate mol wt 500-2,000, which may influence tubular handling
of T3 or T4. (J Clin Endocrinol Metab 42: 504, 1976).
The outcome of this paper is first, that T3 clearance in urine does not distinguish euthyroidism from hypothyroidism but Y4 clearance is better. Second however, both T4 and T3 excretion in urine is strongly increased by protein in the urine e.g. albumin. So, on a practical note, if urine is frothy this indicates the presence of protein, and T3/4 clearance rates are no longer diagnostic.