Symptoms: fatigue, cold-intolerance, hair loss, decreased libido, constipation… sound familiar?
Hypothyroid-like symptoms occur relatively frequently during times of rapid fat loss, and this may be at least partially dependent on diet quality. Another component is the fact of being in an energy deficit, but this is difficult to evade on a weight loss diet.
Tl;dr: large energy deficit = hypothyroid-like symptoms.
Food choices and diet quality is important, too.
The negative impact of low calorie diet-induced weight loss on thyroid biology and energy metabolism is well established, but this figure from Agnihothri and colleagues sums it up rather nicely (2014):
Rapid fat loss causes a decline in TSH and T3.
And in this study on long-term calorie restriction (3-15 years), a similar phenomenon occurred (Fontana et al., 2006). However, TSH was normal (unlike in Agnihothri’s study), possibly because they were not actively losing weight:
Note: calorie restricted diet had adequate protein
But then there’s this study, which shows active weight loss does not necessarily cause reduced TSH levels, if the diet is ketogenic (Yancy et al., 2005):
Also, this study showed reduced T3 after 11 days on a ketogenic diet
And lastly, from the [notorious] Ebbeling study:
Low TSH and low T3, but highest energy expenditure in weight-reduced patients on very low-carbohydrate diet.
Summary
Agnihothri: Low calorie diet-induced active weight loss: low T3 and low TSH; probably low energy expenditure and hypothyroid-like symptoms.
Yancy: Ketogenic diet, active weight loss: normal TSH.
Fontana: Weight-reduced, weight-stable (CR): low T3, normal TSH.
Bisschop: Ketogenic diet, weight-stable (11 days): low T3, normal TSH, and normal energy expenditure (indicative of no hypothyroid-like symptoms)
Ebbeling: Very low-carbohydrate diet, weight-stable (4 weeks): low T3, low TSH, normal energy expenditure (indicative of no hypothyroid-like symptoms).
In all of the above studies, leptin mirrored body fat. And all of the old Rosenbaum & Leibel studies showed the reduction in thyroid function after low calorie diet-induced weight loss could be repaired with leptin treatment. Their 2002 paper had it all (Rosenbaum et al., 2002)… replacing leptin levels to pre-weight loss levels, which required roughly 3 mg/d, completely restored thyroid levels and energy expenditure:
Carbohydrate restriction appears to throw a wrench in the gears.
We also know from rodent studies that excess carbohydrates, particularly fructose, can induce leptin resistance. Perhaps by improving leptin sensitivity, carbohydrate restriction attenuates some of the hypothyroid-like symptoms associated with an energy deficit. This would explain the findings of Bisschop’s and Ebbeling’s studies, and would be in line with the known impact of carbohydrate restriction on lowering plasma triacylglycerols, and the theorized (and somewhat controversial) association between high triacylglycerols and low leptin sensitivity.
If you’re on a weight loss diet and are experiencing cold intolerance, fatigue, low libido, or any other hypothyroid-like symptoms, consider either upping the calories or lowering the carbs (or this?
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is an effective weight loss strategy for many. 
, on the other hand, is a pathological condition caused by insulin deficiency. The common theme is low insulin; however, in ketoacidosis, blood glucose levels are very high. Ketone levels are elevated in both states, although are 10-20x higher in ketoacidosis (~0.5-2 vs. > 20 mM). Nutritional ketosis and ketoacidosis should not be confused with one another, and a ketogenic diet doesn’t cause ketoacidosis.
(GOS) (
can help, too, if they’re administered with either prebiotics or fermented foods (they need something to nourish them in transit). 
was much higher in fat, and as such, increased the prevalence of a microbe involved in fat digestion. “Bilophila.” The article focused on this one and cited a 
increased the levels of Bacteroides and decreased Firmicutes. These are the two that brought the whole gut microbe-obesity connection into the spotlight. The microbiome in obese mice is characterized by low Bacteriodetes and high Firmicutes. Fecal transplants from obese mice to lean mice causes them to gain weight. Little is known about Bilophila relative to Bacteriodetes & Firmicutes, and I suspect the focus was on Bilophila because the authors wanted something negative to say about a meat-based, fiber-free ketogenic diet, and that 
condition. They were then given 4 eggs (~23 g protein) that were labeled with two stable isotopes (15N & 13C, derived from hens fed isotope-enriched diets!). Throughout the entire study duration, the subjects were infused with a third isotope, 2H-glucose. By collecting and analyzing the enrichment of isotopically-labeled metabolites like expired CO2, urea, and glucose, the researchers were able to determine the fate of those 23 grams of protein.
measure urinary acetoacetate (AcAc) and reflect the degree of ketosis in the blood probably about 2-4 hours ago. 
measure beta-hydroxybutyrate (bHB) right now. bHB fluctuates to a greater degree, eg, it plummets after a meal whereas AcAc takes longer to decline. AcAc/bHB is usually around 1, but increases after a meal (
et al., 1997)
